aA CORNELL UNIVERSITY LIBRARY 924 059 413 249 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059413249 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. 1991. CHAPTEES ON THE MODEEN GEOMETEY OF THE POINT, LINE, AND CIECLE. VOL, IL CAMBKIDGE : PKINTED BY WILUiM METCALFE, CEEEN STHEET. CHAPTERS MODEEN GEOMETEY POINT, LINE, AND CIRCLE; BEING THE SUBSTANCE OF LECTURES DELIVERED IN THE UMVEESITY Of DDBLIN TO THE CANDIDATES FOE HONORS OF THE FIRST YEAR IN ARTS. BT THE REV. RICHARD TOWNSEND, M.A., FELLOW AKD TT7T0E OF TKINITY COLLEOE. VOL. II. DUBLIN: HODGES, SMITH, AND CO., PUBLISHEKS TO THE UNIYERSITY. 1865. CONTENTS OF VOL. II. ART. CHAPTER XIII. THEORY OP HARMONIC SECTION. 213 Harmonic Section of a Line or Angle. Modulus of. Harmonic Conjugates. Harmonic Systems . .1 214 Equation of Harmonicism of a System of four Points or Rays . 2 215 In Harmonic Section either pair of Conjugates may be imaginary 2 216 Particular positions of Conjugates for particular values of the Hatio of Harmonic Section . . , . .3 217 Conjugate Lines with respect to a Segment. Conjugate Points with respect to an Angle. Pole of a Line with respect to a Segment. Polar of a Point with respect to an Angle. . 4 218 Erery Segment or Angle cutting another Harmonically is cut Harmonically by the other . . . .5 219 Relations connecting the three pairs of opposite Segments or Angles determined by the four Points or Rays of an Harmonic System 6 220 Relations connecting the three pairs of Segments or Angles deter- mined by two Points or Lines with three others forming with one of them an Harmonic System ... 6 221 Every Harmonic Row determines an Harmonic Pencil at every Vertex, and every Harmonic Pencil determines an Harmonic Row on every Axis. Great Importance of this Property . 8 222 Consequences from the General Property of the preceding Article 8 223 The four Polars of an Harmonic Row, and the four Poles of an Harmonic Pencil, with respect to any Circle are Harmonic. Important General Consequence resulting from this Property . 1 1 a VI CONTENTS. ART. PAOE 224 When two Angles cut each other Harmonically, every Chord of either parallel to a Side of the other is bisected internally by the second Side of the other ... 12 225 When two Segments cut each other Harmonically, Square of half either = Kectangle under distances of its Middle Point from Extremities of other , . . . .13 226 Properties of a variable Segment of a fixed Axis cutting a. fixed Segment of the Axis Harmonically , , . 16 227 When two Segments cut each other Harmonically, Square of distance between their Middle Points = Sum of Squares of their Semi- Lengths . . . , .17 228 Every Circle cutting Harmonically any Diameter of another is or- thogonal to the other ; and, conversely, every Circle orthogonal to another cuts Harmonically every Diameter of the other 1 8 229 Properties of any two Segments of a Line and of the Segment intercepted between the two Harmonic Conjugates of the Extremities of either with respect to the other . . 20 230 Solutions of the Problem " to determine the Segment or Angle which cuts two given Segments or Angles Harmonically " 21 231 Statement of the Harmonic Relation of four CoUinear Points in terms of the three Distances of any three of them from the fourth. General Formulae resulting from the Relation so stated 22 232 Harmonic Progression. Harmonic Mean and Extremes . 24 233 General Properties of Magnitudes in Harmonic Progression resulting from the General Formulje of Art. 231 . . .24 234 Statement of the Harmonic Relation of four CoUinear Points in terms of their four Distances from any fifth Point on their Common Axis. Consequences ... 26 23a General Relations connecting three Segments of a Line one of which cuts Harmonically and is cut Harmonically by the other two. Consequences . . . , .27 CHAPTER XIV. HARMONIC FUOFEKTIES OF TUB POINT AND LINE. 236 Fundamental Harmonic Properties of the Tetragram and Tctrastigm 32 237 Those Properties, as well as the Demonstrations given of them, are Reciprocal . . ... 33 238 They follow also each from the other without the aid of the Reci- procating Process , . . . .3d CONTENTS. Vll ART. PAGE 239 Particular Cases, when one of the four Lines of the Tetragram, and of the four Points of the Tetrastigm, is at Infinity . 36 240 General Consequences resulting from the Fundamental Reciprocal Properties of Art. 236 . . . . .37 241 Reciprocal Problems solved by aid of the same Properties . 38 242 Reciprocal Harmonic Properties of Triangles, expressed by a different Statement of the same . . . .39 243 Other Harmonic Properties of Triangles, in pairs Reciprocals of each other. Consequences . . . . 41 214 Reciprocal Properties of the Tetragram and Tetrastigm resulting from the Fundamental Properties of Art. 236 . . 44 245 Other Harmonic Properties of the Tetragram and Tetrastigm, in pairs Reciprocals of each other. Consequences . 45 246 Polar of a Point with respect to an Angle, and Pole of a Line with respect to a Segment, Harmonic Properties of . .50 247 Polar of a Point with respect to any two Lines, and Pole of a Line with respect to any two Points, for any two Multiples, General Properties of . . . . .51 248 Polar of a Point with respect to any System of Lines, and Pole of a Line with respect to any System of Points, for any System of Multiples. General Properties of . . .54 249 The two General Properties of the preceding Article, and also those to which they have been reduced, are Reciprocal . 58 250 Polar Properties of Triangles, in pairs Reciprocals of each other. Consequences . ... 60 25 1 Polar Properties of the Centres and Axes of Perspective of Triangles in Perspective. Consequences ... 62 CHAPTER XV. HARMONIC PUOPERTIES OF THE CIRCLE. 252 Fundamental Harmonic Properties of the Circle. Property of four Points. Property of four Tangents . . .67 253 Those Properties are Reciprocal ... 67 254 Harmonic Systems of Points on and Tangents to a Circle. Conju- gate Lines and Points with respect to a Circle , . 68 255 Reciprocal Inferences from the Fundamental Properties of Art. 252 69 256 Metric Relations connecting the three pairs of opposite Chords determined by an Harmonic System of Points on a Circle 70 a 2 VI 11 CONTENTS. AHT. FADE 257 Conjugate Points lie each on the Polar of the other, and Conjugate Lines pass each through the Pole of the other, with respect to the Circle. Consequences . . . .71 258 Reciprocal Harmonic Properties of the Connectors of the Terminal Points and of the Intersections of the Terminal Tangents of two Arcs of a Circle which cut each other Harmonically ■ 76 259 Reciprocal Harmonic Properties of Conjugate Points and Lines with respect to a Circle. Consequences . . .78 260 Remarkable Conclusions, from the Properties of the preceding Article, respecting " The Two Circular Points at Infinity" 81 261 Reciprocal Polar Properties of the Tetrastigm determined by any four Points on, and of the Tetragram determined by any four Tangents to, a Circle. Consequences ... 83 262 Reciprocal Properties of a Triangle with respect to an arbitrary Circle . . . . . .Si 263 Reciprocal Solutions of the Reciprocal Problems " To construct a Triangle at once exseribed or inscribed to a given Triangle and inscribed or exseribed to a given Circle." Peculiarity of the case when the Triangle is Self- Reciprocal with respect to the Circle ...... 9o 264 Metric Relation connecting the three Triangles determined by any arbitrary Point on a Circle with the three Sides of any Triangle Self-Reciprocal with respect to the Circle . . .98 265 Reciprocal Harmonic Properties of the Tetrastigm determined by four Points on and of the Tetragram determined by four Tan- gents to a Circle. Consequences . . . 100 26S Reciprocal Harmonic Properties of a Point and Line Pole and Polar to each other with respect to a Circle . . 103 267 Reciprocal Properties of a Point and its Polar, and of a Line and its Pole, with respect to any System of Circles, for any System of Multiples ..... 104 CHAPTER XVI. THEOHY OP ANHAKHONIC SECTION. 268 Anharmonic Ratios of the Section of a Line or Angle by two Points or Lines of section. Always Reciprocals of each other. Reason why so called . . . . .108 269 Have always the same Sign, but any absolute Values from to » . Common Peculiarity of the Section for the particular values 0, 00 , and + 1 . . . .108 CONTENTS. IX ART, FAOB 270 Simplification of Anhannonic Ratios in the case of a Line when one Point of Section is at Infinity .... 109 271 Equianharmonic Section, Nature of. Mode of Expression often used foT Shortness with lespect to . . . 109 272 When a Segment or Angle AB is cut Equianharmonically by two pairs of seetors C and D, C and D', it is also cut Equianhar- monically by the two pairs C and C", D and D' ; and conversely 110 273 Every two Segments having a Common Axis or Angles having a Commcm Vertex, AB and CD, cut each other Equianhar- monically ...... Ill 274 Every four CoUinear Points or Concurrent Lines A, B, C, D, deter- mine six different Anharmonic Ratios, in pairs Reciprocals of each other . . . . .112 273 Simplification of all Six in the former case when one of the four Points is at Infinity . . . . .113 276 Of the three pairs of Ratios determined in all cases by the four constituent Points or Rays, two are always Positive, and the third always Negative . . . . 113 277 General Relations connecting the six Anharmonic Ratios determined by any System of four constituent Points or Rays . .114 278 Every two Systems of four Constituents,which have one Anharmonic Ratio common, have all six common . . 115 279 Dr. Salmon's Notation for the Relation of Equianharmonicism between two or more Systems of four Constituents. Sense in which the expression " Anharmonic Ratio " is to be regarded when applied to such in the singular number . .116 280 Any two Constituents of an Anhannonic System may be interchanged without affecting any Anharmonic Ratio of the System, provided the other two be interchanged also . . . 117 281 One case, and one only, in which the double interchange is un- necessary ; viz. when the System is Harmonic, and when the Constituents of a single interchange are either pair of Conjugates 119 282 General Properties of two variable Constituents which determine in every position Equianharmonic Systems with two Triads of fixed Constituents . . ■ ■ .119 283 General Property of six Constituents, corresponding two and two in three Conjugate pairs, any four of which are Equianharmonic with their four Conjugates . . . .124 284 General Property of eight Constituents, corresponding two and two in four Conjugate pairs, any two Systems of four of which are Equianharmonic with their four Conjugates . .126 285 Every Anharmonic Pencil determines an Equianharmonic Row on every Axis, and every Anharmonic Row determines an Equian- harmonic Pencil at every Vertex. Great Importance of this Property in Modern Geometry . . . 128 X CONTENTS. ABT. fAOX 286 Consequences from the General Property of the preceding Article . 130 287 Heciprocal Problems solved by aid of the same . . 131 288 Every two Equianharmonic Systems having two Triads of Corre- sponding Constituents in Perspective, are themselves in Perspective ..... 133 289 Every two Equianharmonic Systems having a coincident pair of Corresponding Constituents are in Perspective . . 1 34 290 Reciprocal Conditions that two Equianharmonic Rows should determine Pencils in Perspective at two Points, and that two Equianharmonic Pencils should determine Rows in Perspective on two Lines . . . . . 1 36 291 Solutions by Linear Constructions of the two Reciprocal Problems, Given three pairs of Corresponding Constituents of two Equian- harmonic Systems of Points or Rays, and the fourth Constituent of either System, to determine the fourth Constituent of the other System . . . . .137 292 Every four Collinear Points, or Concurrent Lines, and their four Concurrent Polars, or Collinear Poles, with respect to any Circle are Equianharmonic. Important General Consequence resulting from this Property , . . 1 38 CHAPTER XVIL ANUABHOXIC FROFEKTIES OF THE FOINT AND LIKE, 293 General Property of two Triads of Collinear Points or Concurrent Lines. Anharmonic Relations existing between them and their derived Triad. Cyclic Connexion of all three . .140 294 Reciprocal Inferences from the General Property of the preceding Article. Anharmonic Properties of a Cycle of three Triangles each inscribed to one and exscribed to the other of the remain- ing two ...... 142 295 General Property of two Triads of Points or Lines in Perspective. Anharmonic Relations existing between them and their Centre and Axis of Perspective. Particular case of Harmonic Section. 148 296 Reciprocal Inferences from the General Property of the preceding Article. Anharmonic Properties of two Triangles in Perspective 1 o3 297 General Anharmonic Relation existing between any two Figures iu Perspective. Anharmonic Ratio of Perspective, Particular case of Harmonic Perspective. Examples . .156 298 Important Properties of Pigures in Perspective as regards Recipro- cation to an arbitrary Circle. £.\amples . . 158 300 301 CONTENTS. XI *"■<■■ PAUE 299 Reciprocal A nharmonic Properties of the Tetraatigm and Tetragram. Involve their Harmonic Properties as particular cases. Conse- quences . . . . . .160 Reciprocal Problems solved by aid of the Properties of the preceding Article ...... 164 General Property of a System of six Points, or Lines, any four of which connect, or intersect, EquiEinharmonically with the remaining two. Consequences .... 165 302 Reciprocal Theorems of Pascal and Brianchon. Nature of the Hexagons involved in the general cases of both. General Property of Triangles in Perspective with regard to such Hexagons . . . .168 303 General Properties resulting from the Theorems of Pascal and Brianchon, combined with the Fundamental Property of Triangles in Perspective . . . .170 304 General Criteria, that three pairs of Points on the three Sides of a Triangle should determine a Pascal Hexastigm, and that three pairs of Lines through the three Vertices of a Triangle should determine a Brianchon Hexagram. Consequences resulting from their application. Examples . . . 173 CHAPTER XVIII. ANHARMONIC PUOPEKTIES OF THE CIRCLE. 305 Fundamental Anhannonic Properties of the Circle. Property of four Points, Property of four Tangents. Both true of all Figures into which the Circle can be transformed by Recipro- cation . . . . . .177 306 Different mode of stating the Reciprocal Properties of the preceding Article. Meaning of the expression " Constant Anharmonic Ratio " as applied to a system of four fixed Points on, or Tangents to, a Circle . . . .178 307 Every System of four Points on a Circle is Equianharmonic with the corresponding System of four Tangents to the same Circle ; and conversely . . . . .179 308 Values of the six Anharmonic Ratios of any System of four Points on a Circle in terms of the six Chords they determine in pairs 180 309 Equianharmonic Systems of Concyclic Points or Tangents. Nature of. Notation representing . . . 181 310 Pencils in Perspective determined by Equianharmonic Systems of Points on, and Rows in Perspective determined by Equianhar- monic Systems of Tangents to, the same Circle . . 182 Xll CONTENTS. AST. FAOB 311 Reciprocal Anhannonic Properties of the Tetrastigm determined by any four Points on, and of the Tetragram determined by any four Tangents to, a Circle. Consequences . . 183 312 Bemarkable Conclusion, from the Properties of the preceding Article, respecting the two Circular Points at Infinity • 189 313 Reciprocal Anharmonic Properties of two Coney clic Triads of Points and Tangents in Perspective. Consequences . . 190 314 Reciprocal Anharmonic Properties of two Coney clic Quartets of Points and Tangents in Perspectiye, Consequences . 192 315 Bifferent Statement and Proof of the two Reciprocal Properties of the preceding Article .... 195 316 Anharmonic Properties of a System of two Circles with respect to their Centres and Axes of Perspective. Consequences . 198 317 General Property of any two Triads of Concyclic Points or Tan- gents. Anharmonic Relations thence resulting . 200 318 Reciprocal Inferences from the General Property of the preceding Article. Anharmonic Properties of the Pascal Line, or Brian- chon Point, of any Hexagon inscribed, or exscribed, to a Circle 203 319 Important Reciprocal Problems solved by aid of the same. Simpli- fication in the particular case of Perspective . . 206 320 Reciprocal Properties of the two Triangles determined by two Concyclic Triads of Points or Tangents in Perspective. Con- sequences ..... 207 CHAPTER XIX. THEOBT OF HOMOGKAFHIC DIVISION. 321 Homographic Systems of CoUinear Points and Concurrent Lines. Homographic Division of Lines and Angles . . 210 322 Homographic Systems of Points on and Tangents to Circles . 210 323 Systems Homographic with a Common System, or with different Homographic Systems, are Homographic with each other . 211 324 Notation for representing Symbolically the Relation of Homography between two or more Systems . . . 212 325 Fundamental Examples of Cases of Homographic Division ; grouped in Reciprocal pairs ..... 213 326 Additional Examples of Cases of Homographic Division ; grouped ill Inverse pairs ..... 216 CONTENTS. XUl AHT. PAOK 327 General Helation connecting two yariable Constituents of any species generating by their movement two Homographic Systems. Three pairs of Corresponding Constituents in all cases determine completely the Systems to which they belong . . 220 328 General Relation connecting two variable Points or Rays generating by their movement two Homographic Rows or Pencils. Equally general with the Preceding as regards such Systems . 22 1 329 Two variable Sectors dividing a fixed Segment or Angle in any constant Anharmonic Ratio generate two Homographic Systems. The Extremities of the Segment or Angle how related to such Systems. Converse Property true of all Homographic Systems of the same species having a Common Axis or Vertex . 222 330 Criterion of Similarity between two Homographic Rows of Points. Points at Infinity always Corresponding Constituents of such Rows . . . . . .223 331 Peculiarity of the Points corresponding to those at Infinity in any two Homographic Rows. Rectangle under their Distances from all pairs of Corresponding Points always constant both in magnitude and sign . , . ■ 224 332 Application of the General Property of the preceding Article to the particular case of a fixed Segment cut in any constant An- harmonic Ratio by a variable pair of Sectors . • 225 333 Examples, grouped in Reciprocal pairs, of cases of Homographic Division, reducible to the case referred to in the preceding Article . . . . . .226 331 'When, of two Homographic Rows of Points on difiierent Axes or Pencils of Rays through different Vertices, or Concyclic Systems of either species, any two Triads are in Perspective, the Systems are in Perspective ..... 229 335 Criteria of Perspective oftwo Homographic Rows and Pencils. Coin- cidence of a pair of Corresponding Constituents with the single Point or Ray common to their Axes or Vertices, Consequences 230 S36 Examples, grouped in Reciprocal pairs, of the application of the preceding Criteria ..... 231 337 Directive Axis of any two Homographic Systems of Points on two different Axes or on a Common Circle. Directive Centre of any two Homographic Systems of Tangents to two different Points or to a Common Circle. Reason why so denominated. Properties, Uses, and Determination of . . 234 338 General Properties of any two Homographic Systems of Points on two different Axes, or on a Common Circle, as regards the Connectors of their several pairs of Corresponding Points. Reciprocal Properties of any two Homographic Systems of Tangents to two different Points, or to a Common Circle, as regards the Intersections of their several pairs of Corresponding Lines. Consequences .... 236 XIV CONTENTS. ART. PAGE 339 General Property of any two Homographic Pencils of Raya, having a pair of Corresponding Constituents whose Directions are Parallel. Consequences . . . ■ 240 340 Keciprocal Properties of any two Homographic Rows of Points on different Axes and Pencils of Rays through different Vertices. Different Consequences thence resulting. Long familiar in the Theory of Conic Sections . . . .241 CHAPTER XX. OV THE DOUBLE POINTS AND LINES OF HOMOaRAPBIC BTSTEMS. 341 Double Points and Rays of Coaxal Rows and Concentric Pencils. Reason why so named. Fundamental Properties of . 246 342 Segment or Angle determined by cut in the same constant Anhar- monic Ratio by every pair of Corresponding Constituents of the Systems ...... 247 343 Same Segment or Angle cut in the square of the same constant Anharmonic Ratio by the Correspondents in the two Systems of every Point on the common Axis or Ray through the common Vertex ...... 247 344 Character of Coaxal Rows having one or both of their Double Points at Infinity ..... 249 345 General Properties of Coaxal Rows whose Double Points are Real 250 346 General Properties of Coaxal Rows whose Double Points are Imaginary . . . . .251 347 Double Points and Tangents of Concyclic Systems. Properties identical with those of Coaxal Rows and Concentric Pencils . 252 348 General Solutions of the four cases of the Problem " Given three pairs of CorrespondingConstituents of two Homographic Systems of Points on a Common Line or Circle, or of Tangents to a Common Point or Circle, to construct the Double Points, or Lines, of the Systems " .... 253 349 Chasles' Particular Construction for the case of Collinear Systems on a Common Axis ..... 254 350 Another Particular Construction for the same case. Simplification of all constructions when one Double Point or Line is known 255 351 Remarkable Results from the General Constructions for Concyclic Systems of Points and Tangents, when applied to the particular cases of Similar and Similarly Ranged Systems separated by an interval of any finite magnitude . . . 255 CONTENTS. ART. 3.52 General Property of every two Homographie Pencils of Rays deter- mined by the Sides of a variable Angle of invariable form revolving round a fixed Vertex, resulting from that of the preceding Article . . . . '256 353 Examples of Problems in nomographic Division reducible to the determination of the Double Points or Lines of Homographie Systems ...... 257 354 Preliminary Process necessary in all cases when the three pairs of Corresponding Constituents which determine the Systems are not given ...... 266 356 Method of Trial. Analogy of to the " Method of Palse Position " in Arithmetic ..... 267 356 Miscellaneous Examples of Problems solved by the Method of Trial 267 CHAPTER XXI. ON THE BELATION OF IITVOLUTION BETWEEN HOMOQRAPUIC SYSTEMS. 357 Fundamental Property of Coaxal Rows of Points and Concentric Pencils of Rays having an Interchangeable pair of Correspond- ing Constituents ..... 276 35S Relation of Involution between two nomographic Rows of Points or Pencils oi Rays having a common Axis or Vertex . 277 359 Two pairs of Corresponding Constituents sufficient to determine two Homographie Rows or Pencils in Involution with each other , 277 360 Homographie Systems of Points on or Tangents to a Common Circle in Involution with each other . , . 278 361 Fundamental Examples of the Relation of Involution between two nomographic Systems generated by a variable pair of Conju- gates connected by a known law .... 278 362 Additional Examples left as Exercises to the Reader . 279 363 Any two Homographie Systems of Points on a Common Line or Circle, or of Tangents to a Common Point or Circle, may be placed in Involution, by the movement of either, or both, on the Common Line or Circle, or round the Common Point or Circle . . . . . .280 361 Every three pairs of Conjugates of two Homographie Systems in Involution determine a System of six Constituents every four of which are Equianharmonic with their four Conjugates , 231 XVI CONTENTS. iRT. J-AOF. 365 Fundamental Examples, gronped in Reciprocal pairs, of three pairs of Corresponding Constituents satisfying the Criterion of Invo- lution supplied by the Property of the preceding Article . 282 366 Additional Examples left as Exercises to the Reader . 283 367 Metric Relations connecting the three pairs of Corresponding Seg- ments or Angles determined by the six pairs of non-correspond- ing Constituents of any two Collinear or Concurrent Triads of Points or Rays in Involution . . . ■ 285 368 Examples of two Collinear and Concurrent Triads of Points and Rays satisfying the Criteria of Involution supplied by the Metric Relations of the preceding Article . . ■ 286 369 Additional Examples left as Exercises to the Reader . . 288 370 Eyery two Conjugate Points or Lines of two Horaographic Systems in Involution are Harmonic Conjugates with respect to the two Double Points or Lines of the Systems . . 289 371 Consequences resulting from the General Property of the preceding Article . . . . . .289 372 Property of " The Centre " of two nomographic Rows of Points in Involution on a Common Axis. A particular case of the General Property of article 331 . . . 292 373 The General Property of article 363 an easy Consequence from the Property of the Centre in the cnse of any two Homographic Rows of Points on a Common Axis . . . 292 374 Examples of Involution between three or more pairs of Points on a Common Axis resulting from the Property of the Centre 293 375 Properties of Involution with respect to the Double Points or Lines of any two Homographic Systems of Points on a Common Line or Circle, or of Tangents to a Common Point or Circle . 294 376 Reciprocal Problems in Involution solved by virtue of the General Proi)erty of article 370 .... 296 377 Reciprocal Problems in Involution solved by virtue of the Equian- harmonic Relations of article 364, combined with the General Property of article 327 .... 296 378 Reciprocal Properties of Involution with respect to the Directive Centre of any two HomographicPencils of Rays through different Vertices, and the Directive Axis of any two Homographic Rows of Points on different Axes . . , 298 379 Reciprocal Problems in Involution solved by aid of the Directive Centre of two Homographic Pencils of Rays through different Vertices, and of the Directive Axis of two Homographic Rows of Points on different Axes .... 300 CONTENTS. CHAPTER XXII. METHODS OF GEOHETSICAL T&ANSFORUATION. THEORY OF HOMOOKAFHIC FIQUSES. ART. PAOP. 3S0 Definition of Homographic Figures. Every two Figures in Per- spective with each other are Homographic. Every two Figures Homographic with a Common Figure are Homographic with each other . . . . .301 381 Fundamental Properties of Figures satisfying the four Preliminary Conditions, whether Homographic or not . . . 301 382 General Property of any two Homographic Figures, with respect to any two pairs of Corresponding Points or Lines of the Figures. Property of the two Lines whose two Correspondents coincide at Infinity. Consequences. Property of two Figures having two Corresponding Lines coinciding at Infinity. Consequences 307 383 General Property of any two Homographic Figures, with respect to any three pairs of Corresponding Points or Lines of the Figures. Consequences . . . . .312 381 Chasles' General Construction for the Double Generation of two Homographic Figures by the simultaneous variation of a pair of connected Points or Lines . . . .314 38d Consequences respecting the Homographic Transformation of Figures, resulting from the General Constructions of the preceding Article . , . . .317 386 Properties of Figures Homographic to the Circle, deduced by Homographic Transformation from the Corresponding Proper- ties of the Circle . . . . -321 387 General Properties of any two Homographic Figures. Conditions of Perspective of two Homographic Figures. Chasles' Con- struction for placing any two Homographic Figures in Per- spective with each other. Becomes Indeterminate for Figures having a Double Line at Infinity. Construction for that case. Observation ..... 333 CHAPTER XXIII. UETHODS OF QEOUETBICAL TBANSFORUATION. THEORY OF CORRELATIVE FIGURES. 388 Definition of Correlative Figures. Every two Figures Reciprocal Polars to each other with respect to a Circle are Correlative. Every two Figures Correlative with a Common Figure are Homographic with each other , . .33 XVIII CONTENTS. AET. PAGE 389 Fundamental Properties of Figures satisfying the Four Preliminary Conditions, whether Correlative or not . • 339 390 General Property of any two Correlative Figures, with respect to any two pairs of Corresponding Points or Lines of the Figures 345 391 General Property of any two Correlative Figures, with respect to any three pairs of Corresponding Points or Lines of the Figures. Consequences . . . . ,346 392 Chasles' General Construction for the Double Generation of two Correlative Figures, by the simultaneous variation of a con- nected Point and Line, or Line and Point . . 349 393 Consequences respecting the Correlative Transformation of Figures, resulting from the General Constructions of the preceding Article . . . . . .352 394 Properties of Figures Correlative to a Circle, deduced by Correlative Transformation from the Corresponding Properties of the Circle. Identity of Figures nomographic and of Figures Correlative to the Circle ..... 355 395 General Properties of any two Correlative Figures. Conditions of Interchangeability between their several pairs of Corresponding Elements. Chasles' General Construction for placing any two Correlative Figures so as to fulfil those conditions. The Figures when so placed are Reciprocal Polars to each other with respect to a Figure Homographic to a Circle . . . 358 CHAPTER XXIV. METHODS OF OEOMETRIOAL TKAirsFORMATION. ThEOBT OF INVERSE FIGTJBES. 396 Definition of Inverse Figures. Every two Concentric Circles Inverse to each other with respect to the Concentric Circle the Square of whose Radius is equal to the Rectangle under their Radii 363 397 Two parts of the Same Figure may be Inverse to each other with respect to the Dividing Circle. Every Line or Circle is thus divided by every Circle intersecting it at right angles . 363 398 The Same two Figures may be Inverse to each other with respect to more than one Circle. Every two Circles are thus related to each other with respect to each of their two Circles of Antisimilitude 363 399 Transformation of any Figure into an Inverse Figure. Process of Inversion. Circle, Centre, and Radius of Inversion . 363 400 Observations respecting the Process of Inversion . . 364 CONTENTS. XIX AST. PAGE 401 General Properties of any two Figures Inverse to each other with respect to any Circle .... 364 402 General Properties of any two pairs of Figures Inverse to each other with respect to a Common Circle . . .367 403 Figures Inverse to the Line and Circle with respect to any Circle of Inversion. Different cases of under different circumstances of Magnitude and relative Position . . . 368 404 Position of the Centre and Length of the lladius of the Circle Inverse to a given Line or Circle .... 372 405 Construction for the Centre of the Circle Inverse to a given Line or Circle. Consequences .... S73 406 Anharmonic Equivalence of all pairs of CoUinear and Concyclic Quartets of Points Inverse to each other with respect to any Circle ...... 374 407 Angle of Intersection of any two Circles Equal or Supplemental to that of their two Inverses with respect to any Circle . 374 408 Metric Relation between the Squares of the Common Tangents and the Rectangles under the Radii of any two Intersecting Circles and of their two Inverses with respect to any Circle . , 375 409 Other Properties of Intersecting Circles with respect to Inversion 376 410 Important Properties of Inverse Points with respect to one Circle as regards Inversion with respect to another Circle. Consequences 378 411 General Property of any two Inverse Figures with respect to one Circle as regards Inversion with respect to another Circle. Example . . . • . 380 412 Important Properties of Coaxal Circles with respect to Inversion . 381 413 Converse Properties of Concurrent Lines and Concentric Circles with respect to Inversion .... 382 414 Utility of the Preceding Properties in the application of the Process of Inversion to the Investigation of the Properties of Coaxal Circles. Examples .... 333 415 Utility of the same in the application of the Process of Inversion to the Solution of Problems connected with Coaxal Circles. Ex- amples ...... 384 416 Other useful Property of Coaxal Circles with respect to Inversion. Consequences ..... 3S4 417 Relations connecting the Distances, absolute and relative, between pairs of Points and their Inverses with respect to any Circle. Consequences ..... 386 418 Facilities supplied by the Process of Inversion in the Investigation of Certain Classes of Properties in the Geometry of the Line and Circle. Examples , . .388 XX CONTENTS. ART. PAGE 419 Facilities supplied by the same in the Solution of Certain Classes of Problems in the Geometry of the Line and Circle. Examples 389 420 Miscellaneous Examples illustrative of the Fertility of the Process of Inversion as an Instrument for the Evolution of Certain Classes of Properties in the Geometry of the Line and Circle 390 THE MODERN GEOMETRY OF THE POINT, LINE, AND CIRCLE. CHAPTER XIII. THEORY OF HARMONIC SECTION. 213. A line AB cut at two points X and Y (fig. 1), or an angle AB cut bj- two lines Xand Y (fig. 2), is said to he cut harmonically when the ratios {AX:BX &n^AY:BY) of the two pairs of segments into which it is divided in the former case, or (sin AX: sin BX and sin A Y: sin B Y) of the sines of the two pairs of segments into which it is divided in the latter case, are equal in magnitude and opposite in sign. The absolute mag- nitude common to the two ratios (or to their reciprocals according to the extremities of the line or angle from which VOL. II. « 2 THEORY OF HARMONIC SECTION. the antecedents and consequents are respectively measured) is called the ratio^ and sometimes the modulus^ of the harmonic section ; the two points or lines of section X and Y are termed harmonic conjugates to each other with respect to the extremities of the line or angle A and 5; and the four points or lines A and B, X and F, taken together, are said to form an har- monic system. As the ratio of the harmonic section of a line or angle may have any value, real or imaginary, a line or angle may be cut harmonically in an infinite number of ways ; but the ratio of the harmonic section, or the position of one of the two conjugates, is of course sufficient to determine the particular harmonic section in the case of a given line or angle. 214. The relation characteristic of the harmonic section of a line or angle AB by a pair of conjugates X and Y, viz. AX : BX =- AY -.BY, or f^iTL AX: siaBX=- s\nAY: sinBY, may obviously be stated in the more symmetrical form AX: BX+AY: JSr=0, or sin^X: sin^X+sin^ F: sin J5r=0, which is that most generally employed, and which is called the equation of harmonicism of the row or pencil of four points or rays A, B, X, Y. When three points or rays of an harmonic row or pencil are given, the fourth evidently is implicitly given with them ; provided, of course, it be known to which one of the given three it is to be conjugate. 215. In the theory of harmonic section either pair of con- jugates A and 5, or X and Y, may be imaginary; and as cases of each, of the section of a real line or angle by an imaginary pair of conjugates, and of an imaginary line or angle by a real pair of conjugates, are of familiar and necessary occurrence in every application of the theory to the geometry of the circle, the reader must be prepared from the outset to encounter and not be embarrassed by them. When a line or angle and its ratio of harmonic section are both real, the two points or lines of harmonic section are of course real also, and necessarily one external and the other internal to the line or angle ; the former corresponding to the positive, and the latter to the negative sign of the ratio. THEOBT OF HARMONIC SECTION. 3 216. Conceiving the ratio of harmonic section of a real line or angle AB to take successively all real values from to oo , the following particulars respecting the simultaneous positions and changes of position of the two conjugates X and Y are evident from the mere definition of harmonic section (213), viz. 1°. In the extreme case when the ratio = 0, the two antecedents AX and A Y, or sin AX and sin ^ F, in the two ratios of section simultaneously vanish; and, therefore, the two conjugates X and I^ coincide at the extremity A of the line or angle from which the antecedents are measured, 2°. In the extreme case when the ratio = eo , the two consequents BX and BY^ or smBX and sin^F, in the two ratios of section simultaneously vanish ; and, therefore, the two conjugates X and Y coincide at the extremity B of the line or angle from which the consequents are measured. Hence, for the two extreme values and cc of the ratio of harmonic section of a real line or angle, the two points or lines of harmonic section coincide with each other and with an extremity of the line or angle. 3°. In the particular case when the ratio = 1, the two con- jugates X and Y are the two points or lines of bisection, external and interna], of the line or angle ; and are, therefore, at their greatest distance asunder ; being infinitely distant from each other in the case of the line, and at right angles to each other in the case of the angle. Hence, for the mean value, 1, of the ratio of harmonic section of a real line or angle, the two points or lines of harmonic section are in their position of greatest separation from each other ; being infinitely distant from each other in the former case, and at right angles to each other in the latter case. 4°. For all values of the ratio < 1, the two antecedents AX and A Y, or sin AX and sin A Y, in the two ratios of section, are less than the two consequents AST and BY, or sin 5X and fsinBY, and diminish or increase simultaneously with the diminution or increase of the ratio ; therefore the two conjugates X and Y lie in the same segment or angle, intercepted between the two points or lines of bisection of the line or angle, with the extremity A from which the antecedents are measured, and B2 4 THEORY OF HARMONIC SECTION. simultaneously approach to or recede from that extremity and each other as the ratio approaches to or recedes from 0. 5°. For all values of the ratio > 1, the two consequents BX and ^F or sin^JiT and sin BY in the two ratios of section are less than the two antecedents AX and AY, or sin AX and sin AY, and diminish or increase simultaneously with the in- crease or diminution of the ratio ; therefore the two conjugates X and Y lie in the same segment or angle, intercepted between the two points or lines of bisection of the line or angle, with the extremity B from which the consequents are measured, and simultaneously approach to or recede from that extremity and each other as the ratio approaches to or recedes from oo . Hence, _^r all values of the ratio of harmonic section of a real line or angle different from 1, the two points or lines of harmonic section lie in the same segment or angle intercepted between the two points or lines of bisection of the line or angle ; and move or revolve in opposite directions with the change of the ratio; ap- proaching to or receding from each other and the extremity of the line or angle at the side of which they lie as the ratio recedes from or approaches to 1 . These several particulars undergo, as will appear in the sequel, considerable modifications when the extremities A and B of the line or angle are, as they often are, imaginary. 217. Every two lines whose intersections with the axis of a segment cut the segment harmonically are termed conjugate lines with respect to the segment; and every two points whose connectors with the vertex of an angle cut the angle harmoni- cally are termed conjugate points with reject to the angle. It is evident, from the definition of harmonic section, that every line has an infinite number of conjugates with respect to every segment, all passing through the point on the axis of the segment which with the intersection of the line and axis cuts the segment harmonically, and which is termed the pole of the. line with respect to the segment ; and that every point has an infinite number of conjugates with respect to every angle, all lying on the line through the vertex of the angle which with the con- nector of the point and vertex cuts the angle harmonically, and which is termed the polar of the point with respect to the angle ; ?rHEOEY OF HARMONIC SECTION. 5 the origin and appropriateness of these several names, based as they have been on the analogy of the circle (165), will appear in a subsequent chapter. For every two lines or points M and iV, conjugates to each other with respect to a segment or angle AB, it is evident, from the equation of harmonic section (213), that in either case AM: BM=- AN: £N (Euc. vi. 4, and Art. 61), or more symme- trically jiM . BM+ AN : BN= (2U) ; a relation which, conversely, may be regarded as a criterion of two lines or points M and N being conjugates to each other with respect to a segment or angle AB. 218. When a line or angle AB is cut harmonically hy two points or lines X and F, then, reciprocally, the line or angle XY is cut harmonically by the two points or lines A and B. For, the relation, AX: BX+AY: BY= 0, or sin^X: sinBX+s\n AY: smBY=0, which (214) expresses the harmonic section of AB by X and Y, gives at once, by simple alternation, the relation XA: YA + XB: YB=0, or amXA:amYA + 3mXB:smYB=Q, which expresses the harmonic section of XYhy A and B (214). Hence, When four points on a common axis, or rays through a cotnmon vertex A, B, X, Y form an harmonic system (213)/ tlie two segments or angles AB and XY, intercepted between the two pairs of conjugate points or rays, cut each other harmonically ; and the equation of harmonicism of the system (214) lis the ex- pression of the fact of their mutual harmonic section. In exactly the same manner it may be shewn from the closing relation of the preceding article (217), that When two lines or points M and N are conjugates to each other with respect to a segment or angle AB, then, reciprocally, the two points or lines A and B are conjugates to each other with respect to the angle or segment MN; a very important property of harmonic section which will be presently considered under another form. It follows, of course, from the above, that every property of harmonic section which is true of X and Y in relation to A and B, is true reciprocally of A and B in relation to X and Y, and conversely. 6 THEORY OP HARMONIC SECTION. 219. When four collinear points or concurrent lines, in con- jugate pairs A and B, X and F, form an Jiarmonic system ; the three pairs of opposite segments or angles they determine (82) are connected two and two. a. In the former case by the three following relations AX.BY+ AY.BX=0 (1), AB.XY+2AY.BX=0 (2), AB. rX+ 2AX.BY=0 (3). a. In the latter case by the three corresponding relations smAX.myBY+ %mAY.BmBX=Q (1'), BmAB.^mXY+2 smAY.smBX=(i (2'), Bln.4J5.3m ZZ+2 siii^X.smJ?r=0 (3'), the signs as well as the magnitudes of the several segments or angles being regarded in all. For, the first relation of each group is manifestly equivalent to the equation of harmonicism of the system (214), which it expresses in perhaps its most convenient form ; and the second and third of each follow immediately from the first, in virtue of the general relation (82) connecting the six segments or angles determined by any four points on a common axis (82) or rays through a common vertex (82, Cor. 3°). Since, in virtue of the general relation in question, any one of the three relations in each group involves the other two; each, therefore, by itself singly, may be regarded as characteristic of an harmonic system, and sufficient to determine it. 220. When four collinear points or concurrent lines, in con- jugate pairs A and B, X and F, form an harmonic system ; the three "pairs of segments or angles determinedby any one of them A, and by any arbitrary fifth collinear point or concurrent line K, with the remaining three X, F, and B, are connected. a. In the former case by the following relation KX:AX+KY:A F= 2. KB: AB. a. In the latter case by the corresponding relation BinKX: sin^X+sinZ'F: sin-4F=2.8in^5 : BtaAB; THEORY OP HARMONIC SECTION. 7 the signs as well as the magnitudes of the several segments or angles involved being regarded in each. For, whatever be the position or direction of K^ since, in the former case, by the general relation of Art. 82, BY.KX- BX.KY= XY.KB; and, in the latter case, by the corresponding relation. Cor. 3°, of the same article, sin^r.sin^Z- smBX.&aiKY= smXY.mxKB ; and, again, from the harmonicism of the system -4, J5, X, Y, since, in the former case, by relations (a) of the preceding article, BY.AX=- BX.AY=:^.XY.AB; and, in. the latter case, by the corresponding relations (a') of the same article, BmBY.8inAX=-amBX.amAY=^.sinXY.smAB; therefore, in the former case, KX:AX+KY:AY, or its equivalent KX.BY: AX.BY+KY.BX: AY.BX, = XY.KB: ^.XY.AB=^2.KB: AB', and, in the latter case, ainKX : sinAX-^ sin^F : sin^ F, or its equivalent sinZX.sin^F: sin^X.sin5F+ sinKY.amBX: aiuA Y.ainBX, = amXY.amKB:^.a\nXY.amAB=2.a\nKB:amAB; and therefore &c. By taking the arbitrary fifth point or ray K to coincide successively with the three B, X, and F of the four A, B, X, Y constituting the harmonic system, the above useful relations become obviously those of the preceding article in the order of their enumeration ; which accordingly they include as particular cases, and equally with which they may be regarded as charac- teristic of the relation of harmonicism between four points or rays, and sufficient to determine it. 8 THEORY OF HARMONIC SECTION. 221. Every harmonic pencil of rays determines an harmonic row of points on every axis ; and^ conversely, every harmonic row of points determines an harmonic pencil of rays at every vertex. For, if, in either case, (tig. 2, Art. 213) be the vertex of the pencil, and A, B, X, Y the four points of the row ; then since, by (65), AX_AO smAOX fiZ_fi^ ^aAOY BX'BO'smBOX' ^""^ BY ~ BO' sinBOY' therefore at once, by division of ratios, AX AY sin AOX smAOY ^ BX' BY~ &mBOX' sin 50 F' consequently, if either equivalent = — 1 so is the other also, that is, if either the row or the pencil be harmonic (213) so is the other also ; and therefore &c. There is one case, and one only, in which the above demon- stration fails, that, viz. when the vertex of the pencil is at an infinite distance ; but in that case, the four rays of the pencil being parallel (16), the property is evident without any demon- stration (Euc. VI. 10). Of all properties of harmonic section the above, from which it appears that the relation of harmonicism of a row of points or pencil of rays is preserved in perspective (130), is by far the most important. As an abstract proposition it was known to the Ancients, but it was only in modem and comparatively recent times that its importance was perceived. It is to it indeed mainly that the theory of harmonic section owes its utility and power as an instrument of investigation in modern geo- metry. 222. Among the many consequences deducible from the general property of the preceding article, the following are of repeated occurrence in the applications of the theory of har- monic section. 1°. When a pencil of four rays determines an harmonic row of points on any axis, it does so on every axis ; and, reciprocally, when a row of four points determines an harmonic pencil of rays at any vertex, it does so at every vertex. T^EOEY OF HARMONIC SECTION. 9 For, the pencil, in the former case, as determining an har- monic row of points on an axis, is itself harmonic ; and the row, in the latter case, as determining an harmonic pencil of rays at a vertex, is itself harmonic ; and therefore &c. 2°. When a row of four points or pencil of four rays is har- monic, the perspective of either to any centre and axis is also harmonic. For, the row and its perspective, in the former case, connect with the centre of perspective by the same pencil of rays ; and the pencil and its perspective, in the latter case, intersect with the axis of perspective at the same row of points; and there- fore &c. 3°. When two harmonic rows of points on different axes A,Bj X, Y and A', B', X\ Y' are such that any pair of their points A and A', the conjugate pair B and B\ and either of the re- maining pairs X and X' connect by lines AA', BB\ XX' passing through a common point P, the fourth pair Y and Y' connect also by a line YY' passing through the same point P. For, the two rows of four points A, jB, X, Fand A', B\ X\ Y' being, by hypothesis, harmonic, so therefore, ^ by the preceding, are the two pencils of four rays P{A,B,X,Y)a.niPiA\ B\X\ F); butthree pairs of corresponding rays of those two harmonic pen- cils PA and PA', PB and PB', PX and PX', by hypothesis, coincide; there- fore (214) the fourth pair PFand PY' coincide also ; and there- fore &c. 4°. When two harmonic pencils of rays through different vertices A, B, X, Y and A', B', X', Y' are such that any pair of their rays A and A', the conjugate pair B and B', and either of the remaining pairs X and X' intersect at points AA', BB', XX' lying on a common line L, the fourth pair Y and Y' intersect also at a point YY' lying on the same line L. 10 THEORY OP HARMONIC SECTION. For, the two pencils of four rays A, B, X, Y and A', B', X', Y' being, by hypothesis, harmonic, so therefore, by the preceding, are the two rows of four points L{A, B,X, Y), and L{A',B\X', Y') ; but three pairs of corre- sponding points of those two harmonic rows LA and LA', LB and LB', LX and LX', by hypothesis, coincide, therefore (214) the fourth pair LY and LY' coincide also; and there- fore &c. 5°. When of two harmonic rows of points on different axes A,B, X, Y and A', B', X', Y', any pair of points A and A' coincide at the intersection of the axes, the conjugate pair B and B' are collinear with the two centres of perspective P and Q of the two segments XY and X' Y' determined hy the remaining two pairs XandX', YandY. For, as in 3°, of which this is evidently a particular case. the two rows of four points A, B, X, Y and A!, B', X', Y' being, by hypothesis, harmonic, so therefore, by the preceding, are the four pencils of four rays Pi^A, B, X, Y) and P{A', B', X, T), Q{A, B, X, Y) and Q{A', B', X', T); but, for both pairs of harmonic pencils, three pairs of corresponding rays, viz. PX and PX, PF and PY, PA and PA' for the first pair, and, QX and QY', QY and QX', QA and QA' for the second pair, by tSeoet op harmonic section. 11 hypothesis, coincide, therefore the fourth pairs for both, viz. PB and PB' for the first, and, QB and QB' for the second, coincide also ; and therefore &c. 6°. When^ of two harmonic pencils of rays through different vertices A, B, X, Y and A', B', X\ F', any pair of rays A and A' coincide along the connector of the vertices ^ the conjugate pair B and B' are concurrent with the two axes of perspective L and M of the two angles XY and X' Y' determined hy the remaining two pairs X and X'^ Y and Y'. For, as in 4°, of which this is evidently a particular case, the two pencils of four rays A, B, X, Y and A\ B', X', Y' being, by hypothesis, harmonic, so therefore, by the preceding, are the four rows of four points L{A, B, X, Y) and L{A', B', X\ F'), M{A, B, X, F) and M{A', B\ X\ T) ; but, for both pairs of harmonic rows, three pairs of corresponding points, viz. LX and LX', LY and LY\ LA and LA' for the first pair, and, MX and MY\ MY and MX\ MA and MA' for the second pair, by hypothesis, coincide, therefore the fourth pairs for both, viz. LB and LB' for the first, and, MB and MB' for the second, coincide also ; and therefore &c. 223. When four points form an harmonic row^ their four polars with respect to any circle form an harmonic pencil ; and, conversely, when four lines form an harmonic pencil, their four poles with reject to any circle form an harmonic row (166, Cor. 1°). For, in either case, the pencil determined by the four rays being similar to that subtended by the four points at the centre 12 THEORY OP HARMONIC SECTION. of the circle (171, 2°), the harmonicism of either, consequently, involves and is involved in that of the other (213) ; but, hy virtue of the general property of Art. 221, the harmonicism of the latter pencil involves and is involved in that of the row determined by the four points, and therefore &c. In the applications of the theory of harmonic section, the above property, from which it appears that the relation of har- monicism of a row of points or pencil of rays is preserved in reciprocation (172), ranks next in importance to that of Art. 221, from which, as above demonstrated, it is indeed an inference. By virtue of it all harmonic properties of geometrical figures are in fact double^ every harmonic property of any figure being accompanied by a corresponding harmonic property of its reciprocal figure to any circle (172), the establishment of either of which involves that of the other without the necessity of any further demonstration (173). The principal harmonic properties of figures consisting only of points and lines, which will form the subject of the next chapter, will be found arranged throughout in reciprocal pairs, placed in immediate connection with each other, and marked by corresponding letters, accented and unaccented, so as to keep the circumstance of this remarkable duality con- tinually present before the reader, and supply him at the same time with numerous examples by which to keep up the valuable exercise of inferring one from the other by the reciprocating process described in Art. 172. The principal harmonic proper- ties of figures involving circles, which will form the subject of the following chapter, will also, when their reciprocals are properties involving no higher figures (173), be arranged as far as possible on a similar plan. 224. When two angles having a common vertex cut each other harmonically, every chord of either parallel to a side of the other is bisected internally by the second side of the other. For, by the general property of Art. 221, every chord of either, whatever be its direction, is cut harmonically by the sides of the other ; but for the particular direction in question, one point of harmonic section is at an infinite distance (16), and therefore the other is the middle point of the chord (216, 3°). Conversely, When two angles having a cmnmon vertex are THEORY OF HARMONIC SECTION. 13 such that a side of one bisects while its second side is parallel to any chord of the other ^ they cut each other harmonically. For, the extremities of the chord with its points of internal and external bisection form an harmonic row (216, 3°); and therefore, by the same general property (221), subtend an harmonic pencil at every vertex. CoR. The preceding furnishes a rapid method of constructing the fourth ray of an harmonic pencil conjugate to any assigned one of three given rays ; for, drawing any transversal parallel to the assigned conjugate, and bisecting its segment intercepted between the other two rays, the line connecting the point of bisection with the vertex of the pencil is the fourth ray required. 225. When two segments having a common axis cut each other harmonically^ the rectangle under the distances of the extremities of either from the middle point of the other ^ is equal in magnitude and sign to the square of half the other. Let AB and XY (fig. 1, Art. 213) be the segments, C and Z their middle points ; then since, by hypothesis, AX:BX+AY:BY=% therefore [AX+BX) : [AX-BX) :: {AY- BY) : {AY+BY), but (76, (1), and 75) AX+BX=2CX, AY+BY=2CY, AX-BX=AY-BY=AB, therefore 2CX : AB :: AB : 2CY, and therefore ACX.CY=AB', or CX.GY={^ABy; and in the same manner exactly it may be proved, that 4ZA.ZB=XY', or ZA . ZB = {^XY)% and therefore &c. Conversely, When two segments having a common axis are such, that the rectangle under the distances of the extremities of one from the middle point of the other is equal in magnitude and sign to the square of half the other, they cut each other har- monically. For, since, by hypothesis, CX.CY={^AB)\ or 4.CX.CY=AB% 14 THEORY OF HARMONK! SECTION. therefore 2CX : AB :: AB : 2CY, but (76, (1), and 75) 2CX=AX+BX, 2CY=AY+BY, AB=AX- BX=AY-BY, therefore AX+BX:AX-BX::AY-BY:AY+BY, and therefore AX : BX+ AY:BY=Q- and similarly, if it had been given that ZA.ZB='{^XY% or 4ZA.ZB = XY', and therefore &c. Of all properties of the harmonic section of lines, the above leads to the greatest variety of consequences, and, as a criterion of the relation between two segments having a common axis, is generally found the most readily applicable, especially in questions relating to the circle. An analogous criterion of harmonic section between two angles, having a common vertex might be established, in precisely the same manner, with or without the aid of Trigonometry, but the general property (221) renders this unnecessary, and reduces at once all questions re- specting the harmonic section of angles to the corresponding questions respecting the harmonic section of lines. Cor. 1°. Since CA.CB=-[ ^sAB)\ and ZX. ZY=- {^XYy, the preceding relations may obviously be stated in the forms CX.CY+CA.CB=0, and, ZA.ZB+ZX.ZY=0; which, therefore, equally with their equivalents, express each the mutual harmonic section of the two coaxal segments XY and AB. Cor. 2°. A convenient and rapid construction, for determin- ing in any number segments of a given axis cutting a given segment AB harmonically, is supplied immediately by the above ; drawing a line, in any direction diflferent from that of the given axis, through the middle point G of the given segment AB, and taking upon it any segment PQ for which the rectangle CP.CQ is equal in magnitude and sign to the square of half the given segment AB; every circle passing through Pand Q will intercept on the given axis a segment XY cutting harmonically the given segment AB. For, Euclid in. 35, 36, GX.CY= CP.CQ, which by con- struction = {^ABf, and therefore &c. fbEORY OF HARMONIC SECTION. TT" 15 From this construction it appears at once, as observed in (215), that when A and B are real, and therefore [\AB)'' posi- tive (fig, a), X and Y may be, as they often are, imaginary ; and that when A and B are, as they may be and often are, imaginary, and therefore {^ABy negative (fig. ^), X aaA. Fare always real. COE. 3°. Since for a fixed segment AB, real or imaginary, cut harmonically by a variable pair of conjugates X and F, the rectangle CX.GY, as appears from the above, is constant, and equal to the square, positive or negative, of half the fixed segment; the following particulars respecting the simultaneous positions and fluctuations of X and Y may be immediately inferred : 1°. When A and B are real, and {^ABf therefore positive ; they lie at the same side of the point C, move in opposite direc- tions on the axis AB, and coincide with each other at each of the points A and B (fig. a). 2°. When A and B are imaginary, and {^AB)' therefore negative ; they lie at opposite sides of the point C, move in the same direction on the axis AB, and are at their least distance asunder when equidistant from C (fig. yS). 3°. Whether A and B be real or imaginary ; when either of them is at or passes through C, the other is at or passes through infinity; and, conversely, when either of them is at or passes through infinity, the other is at or passes through C (figs, a and j3). Cob. 4°. Again, for a fixed angle AB, real or imaginaiy, cut harmonically by a variable pair of conjugates X and F, if 16 THEORY OF HARMONIC SECTION. be its vertex and C and D its two lines of bisection (fig. 2, Art. 213) ; the following analogous particulars, respecting the simultaneous positions and movements of X and Y, follow im- mediately from the preceding by virtue of the general property of Art. 221, viz. : 1°. When A and B are real ; they lie in the same region of the angle CD, revolve in opposite directions round the vertex 0, and coincide with each other at each of the lines A and B. 2°. When A and B are imaginary; they lie in different regions of the angle CD^ revolve in the same direction round the vertex 0, and are at their least separation asunder when equally inclined to C or D. 3°. Whether A and B be real or imaginary ; when either of them is upon or passes over either bisector G or 2), the other is upon or passes over the other bisector D or C. 226. If a varicAle segment XY of a fixed axis cut a fixed segment AB of the axis harmonically — 1°. The circle on the variable segment XY as diameter deter- mines a coaxal system (184), whose limiting points (184) are the extremities J real or imaginary^ of the fixed segment AB. 2°. The circle on the variable segment XY as chord which passes through any fixed point P, not on the axis, passes also through a second fixed point Q, on the line connecting the first, real or imaginary, with the middle point C of the fixed segment AB. Both these properties follow at once from the preceding. The first from the consideration that for the variable circle of which XYis diameter, and therefore Z centre, CZ^- (JXF)", which (Euc. II. 5, 6) = CX.CY, is constant and = {^ABf (184) ; and the second from the consideration that for the variable circle PXY, if Q be the second point in which it intersects the line FC (figs, a and j8, Cor. 2°, Art. 225), CP.CQ, which (Euc. III. 35, 36) = CX.GY, is constant and = {^ABf. Conversely, Ikery circle of a coaxal system cuts harmonically the segment, real or imaginary — 1°. Of the line of centres intercepted between ike two limiting points cfthe system. fllEORY OF HARMOKIC SECTION. 17 2 . Of any line intercepted between its two points of contact with circles of the system. For, AB as before being the segment of the line, XY the diameter or chord of the circle, and C and Z the middle points of AB and XY; then since in the case of 1°, by (184), CZ' - iiXT)' = CA' = CB^ = {^ABy, therefore CX.CY={iAB)% and therefore &c. (225), and since in the case of 2°, by (182, Cor. 9°), C is on the radical axis of the system, therefore CX.CY= CA''= CB'' = {^ABy, and therefore «Scc. (225). 227. When two segments having a common axis cut each other harmonically, the square of the distance between their middle points is equal to the sum of the squares of their semi-lengths. Let, as before, AB and XY be the segments, C and Z their middle points ; then, since (Euc. ii. 5, 6) CZ' = CX. CY-{- {^XY)" ov=ZA.ZB+{^ABY, and since (225) CX.CY={iABf and ZA.ZB= (^ZI7, therefore CZ^= {iABy+i^XYY, and there- fore &c. Conversely, when two segments having a common axis are such that the square of the distance between their middle points is equal to the sum of the squares of their semi-lengths, they cut each other harmonically. For, since, by hypothesis, GZ'={^ABY+{iXYy, therefore CZ'-i^XY)', or CX.GY, =[\ABy\ and CZ'-i^ABy, or ZA.ZB, = i^XY)", and therefore &c. (225). Cor. 1°. Since, in a right-angled triangle, the square of the side subtending the right angle is equal to the sum of the squares of the sides containing the right angle, and, conversely, (Euc. I. 47, 48), it appears immediately, from the above, that- — If two coaxal segments which cut each other harmonically be turned round their middle points and made conterminous in position, they will form a right angh ; and, conversely, If two conterminous segments which form a right angle be turned round their middle points and made coincident in direction, they will cut each other harmonically. Cor. 2°. Since, when two circles intersect at right angles, the square of the distance between their centres is equal to the VOL. II. c 18 THEORY OF HARMONIC SECTION. sum of the squares of their radii, and conversely (23), it appears again, from the above, that — When two coaxal seginents cut tacli oilier harmonically^ the two circles of which they are diameters intersect at right angles ; andj conversely, when two circles intersect at right angles, their two diameters which coincide in direction cut each otlier har- monically. Cor. 3°. The above, ako, supplies obvious solutions of the three following problems : 1°. Given one segment AB of a line and the length XY of another cutting it harmonically^ to determine the middle point Z of the other. 2°. Given one segment AB of a line and the middle point Z of another cutting it harmonically, to determine the length XY of the other. S°. Given two segments AB and A'B' of a line, to determine the middle point Z and the length XY of the segment which cuts both harmonically. 228. TT'7^e« two segments having a common axis cut each other harmonically, every circle passing through the extremities of either cuts ortlwgonally the circle of which the other is a diameter. Let, as before, AB and XY be the segments, C and Z their middle points; then since (225) GX.CY=[^ABY, therefore (Euc. III. 35, 36) square of tangent from C to any circle passing through X and F= square of radius of circle of which AB is diameter; and since ZA.ZB=[\XYY, therefore square of tangent from Z to any circle passing through A and B = square of radius of circle of which XY is diameter ; and therefore &c. (23). Conversely, when two circles of any radii cut each other orthogonally, every diameter of either is cut harmonically by the other. Let AB be any diameter of either, C its middle point, and X and Fthe two points, real or Imaginary, at which it intersects the other; then since (Euc. iii. 35, 36) CZ.C7r= square of tangent from C to the latter, that is, as the circles cut ortho- gonally, = square of radius of foi-mer, = [^ABf, therefore &c. (225). THEORY OF HARMONIC SECTION. 19 COE. 1°. Since a variable circle passing through a fixed point, and cutting a fixed circle orthogonally, passes through a second fixed point, the inverse of, the first with respect to the fixed circle (149), it appears at once from the above, as already noticed in (226, 2°), that — A variable circle passing through a fixed point, and cutting afijced segment of a fisced axis harmonically, passes also through a second fixed point, on the line connecting the first with the centre of the fixed segment. CoR. 2°. Again, since a variable circle cutting two fixed cii'cles orthogonally determines a coaxal system, whose radical axis is the line of centres, whose line of centres is the radical axis, and whose limiting points are the intersections, real or imaginary, of the fixed circles (185) ; it appears also, from the above, that — A variable circle cutting two fixed segments of two fixed axes harmonically determines a coaxal system, whose radical axis is the line of centres, whose line of centres is the radical axis, and whose limiting points are the intersections, real or imaginary, of the circles of which the fixed segments are diameters, COR. 3°. Since (156, Cor. 4°) a circle may be described, 1° passing through two given points and cutting a given circle orthogonally ; 2° passing through a given point and cutting two given circles orthogonally ; 3° cutting three given circles ortho- gonally; the radical centre of the given group and its tan- gential distance from each circle of the group, evanescent or finite, being the centre and radius of the cutting circle in each case ; the above furnishes solutions at once simple and obvious of the three following problems, viz. To describe a circle, 1° passing through two given points and cutting a given segment of a given axis harmonically ; 2° parsing through a given point and cutting two given segments of two given axes harmonically ; 3° cutting three given segments of three given axes harmonically. Cor. 4°. As three segments of three axes may be the three sides of the triangle determined by the axes, the problems of the preceding corollary (3°) consequently include as particular cases the three following, respectively, viz. — c 2 20 THEORY OF HAIiMONIC SECTIOX. To describe a circle, 1° ijassing through two givin points and cutting a side of a given triangle harmonically y 2° passing through a given point and cutting two sides of a given triangle liarmonically ; 3° cutting the three sides of a given triangle har- mouically. CoE. 5°. Since (168) the three circles of which the sides of any triangle are diameters are cut orthogonally by the polar circle, real or imaginary, of the triangle ; that is, by the circle round the intersection of its three perpendiculars as centre, the square of whose radius is equal, in magnitude and sign, to the common value of the three equal rectangles under the segments into which they mutually divide each other ; hence again, from the above, it appears that — In every triangle the polar circle, real or imaginary, cuts the three sides harmonically. Cor. 6°. Since (189, 1°, Cor. 1°) the three circles of which the three chords of intei-section of any tetragram are diameters are coaxal, and since consequently (185) every circle cutting two of them orthogonally cuts the third also orthogonally; hence also, from the above, it appears that — Every circle cutting two of the three chords of intersection of any tetragram liarmonically cuts the third also harmonically. 229. If a line AB be cut harmonically by two pairs of con- jugates X and Y, X' and Y', both pairs being arbitrary. a. The three circles on XX', YY', and AB (and also the three on XY', YX', and AB) as diameters are coaxal. b. The three circles on XX', YY', and AB [and also the three on XY', YX', and AB) as chords, which pass through any common point P not on the line, pass also through a second common point Q not on the line. To prove (a). Since by hypothesis AX:AY=- BX: BY and AX' : AY' = -BX' : BY', therefore, by composition of ratios, AX. AX' lAY.AY':: BX.BX' : BY. BY', and therefore &c. (192, Cor. 1°). To prove (6), Since, by (a), there exists a point on AB for which OX.OX' = OY.OY' =^ OA.OB, therefore, if Q be THEORY OF HARMONIC SECTION. 21 the point on OP for which each = OP. 0^, the three circles XPX\ YPT, and APB all pass through $, and therefore &c. The point on AB for which OX.OX' = OY.OY\ and each therefore = OA . OB, is evidently that determined by the relation OZ.X'Y' + OZ'.XY=(i, Zand Z' being the middle points of XFand XT'; for, since when OX. OX'=OY.OY' then OX:0Y=OT : 0X\ therefore 0X+ OY: OX- 0Y= OT + OX' : OT - OX', or 2. OZ: YX = 2. OZ" : X' Y, and therefore &c. 230. In the applications of the theory of harmonic section to the geometry of the circle, the solutions of a variety of problems are reduced to those of the following : Given two segments or angles AB and A'B' having a common axis or vertex, to determine the segment or angle XY which cuts both harmonically. By virtue of the general relation of Art. 221, the case of the angle is of course reduced at once to that of the segment, which is given immediately by any of the three following construc- tions, all based on the property of Art. 225, viz. : 1°. Describing the two circles of which AB and A'B', bisected at O and C respectively, are diameters ; any circle cutting them both orthogonally will intercept on the given axis the required segment XY. For [228) CX.CY={\ABY, &ni. C'X.C'Y={^A'B'Y, and therefore &c. (225). 2°. Taking arbitrarily any point P not on the given axis, and describing the two circles PAB and PA'B' ; their chord of intersection PQ will intersect the given axis at the middle point Z of the required segment XY; and the circle round Zas centre, the square of whose radius is equal to the rectangle ZP.ZQ, will intercept on the given axis the required segment itself. For, (Euc. in. 35, 36) ZA.ZB = ZP.ZQ= {\XYf, and ZA.ZB' = ZP.ZQ = {^XY)% and therefore &c. (225). 3°. Taking arbitrarily any point P not on the given axis, connecting it with the middle points G and C of AB and A'B', and taking on the connecting lines PC and PC the two points Q and Q, for which CP. CQ = {^ABf, and C'P. C Q' = [^A'B'Y ; 22 THEORY OF HAHMGNIC SECTION. the circle QPQ' will intercept on the given line the required segment XY. For, (Euc. in. 35, 36) CX.CY= CP.GQ^ii^AB)-, and C'X.C'Y= G'P.C'Q'=[\AB')% and therefore &c. (225). If either or both of the given segments AB and A'B' be imaginary, the last alone of the preceding constructions is appli- cable ; and the problem, as solved by it, is obviously in its most general form equivalent to the following, viz. : On a given line to determine the two points X and Y the rectangles under whose distances from each of tioo given points on the line C and C are given in magnitude and sign. When the two given segments or angles AB and A'B' are such that A and B alternate with A' and B' in order of suc- cession, the segment or angle XY which cuts them both har- monically is of course necessarily imaginary ; its two points or lines of bisection are however in all cases real (225, Cor. 2°). 231. The harmonic relation of a system of four points on a common axis A, B, X, Y may be expressed in terms of the three distances of any three of them from the fourth as follows : If A be the point from which the distances of the remaining three are measured ; substituting for BX and 5 F their equiva- lents AX - AB ami AY- AB, the fundamental proportion of harmonic section (213) becomes AX: AY:: AX-AB:AB-AY (1). If B be the point ; substituting for AX and A Y their equiva- lents BX- BA and BY-BA, it becomes BX: BY:: BX-BA : BA - BY (2). If X be the point ; substituting for YA and YB their equi- valents XA - XY and XB- XY, it becomes XA : XB:: XA-XY: XY-XB (3). And if Y be the point ; substituting for XA and XB their equivalents YA - YX and YB- YX, it becomes YA: YB:: YA-YX: YX-YB (4), in each of which the relation is expressed in terms of the dis- tances of three of the points from the fourth, in a form which is precisely the same from whichever of the four the three dis- tances are measured. rHEORY OF HARMONIC SECTION, 23 CoR. r equalities The four preceding relations give at once the ( 2.AX.AY={AX+AY).AB' 2.BX.BY={BX + BY).BA 2.XA. XB = {XA + XB) .XY 2. YA. YB={YA + YB).YX J from which it follows immediately, that •2. AX. AY AX. AY (I), AB = BA = XY= YX= AX+AY ' 2.BX.BY BX+ BY '' 2.XA.XB XA + XB '' 2.YA.YB ^ {AX+AY) BX.BY i{BX+BY) XA.XB ' \ {XA+XB) YA.YB (11), YA+YB ^{YA+YB) j relations which express the distance of any point of an harmonic system from its conjugate, in terms of its distances from the remaining two points of the system. CoR. 2°. The reciprocals of the four latter relations (II), give again immediately 1 AX'^ AY~ BX^ BY' XA ■*" XB '' YA 1 YB 2 \ 1 AB 2 BA 2 YY or 2 Yx] f— =-f AB 2 V AX^ A = 1^ + AY) BA~ 2 \BX " BY J_ _ 1 /_!_ J. XY~ 2\XA^ XB (HI), YX~ 2\YA^ Yb] ^ which express, in a remarkably simple manner, the harmonic relation of four points, in terms of the reciprocals of the distances of any three of them from the fourth. CoR. 3°. If G and Z (fig. 1, Art. 213) be the middle points of the two conjugate segments AB and XY respectively ; then since (76, 1) AX+AY=2.AZ, BX+BY=2.BZ, XA + XB^2.XC, YA+YB=2.YC, 24 THEORY OF HARMONIC SECTION. the four general relations (I) of Cor. 1°, are obviously equivalent to the following : (ax.ay=az.ab \bx.by=bz.ba ^ XA.XB = XG.XY\ YA.YB=YC. YX\ (IV), relations of considerable utility, each of which, like any of the preceding, is characteristic of an harmonic system, and sufficient to determine it. 232. When four points on a common axis A and B, X and Y form an harmonic system, the three distances from any one of them to the remaining three, regard being had to their signs as well as to their magnitudes, are said to be in harmonic progression, and the distance from each to its conjugate is termed the harmonic mean of the distances from it to the other two. Thus, A and B, X and Y being the two pairs of conjugates, the four sets of their magnitudes AX, AB, and A Y; BX, BA, and BY; XA, XY, and X5; YA, YX, and YB, taken all with the proper signs due to their several directions, are each in harmonic progression, AB,. BA, XY, and YX being the har- monic means in the four cases respectively. From the invariable order of the four points of an harmonic system when all real, it is evident, from the above definition, that the harmonic mean of two magnitudes has the sign common to both when their signs are similar, and that of the nume- rically lesser of the two when their signs are opposite. According to the analogy of arithmetic and geometric progression, any number of magnitudes are said to be in har- monic progression when every consecutive three of them are in such progression. 233. The several groups of relations of Art. 231 and its corollaries. Interpreted in accordance with the above definitions, express all the ordinary properties of three or more magnitudes in harmonic progression, regard being had to their signs as well as to their absolute values in every case. THEOKY OF HARMONIC SECTION. 25 The group of proportions (1), (2), (3), (4) express im- mediately that — 1°. When three magnitudes are in harmonic progression^ the first : the third :: the first — the second : the second— the third. The group of equalities (II) that — 2°. When three magnitudes are in harmonic progression, the mean = twice the product of the extremes divided hy their sum, or = the product of the extremes divided hy half their sum. The group of equalities (III) that — 3°. When three magnitudes are in harmonic progression, the sum of the reciprocals of the extremes = twice the reciprocal of the mean ; or, the reciprocal of the mean — half ike sum of the reciprocals of the extremes. As half the sum of two magnitudes = their arithmetic mean, and the product of two magnitudes = the square of their geometric mean ; the group of equalities (I) or (IV) shew that — 4°. The product of the arithmetic and harmonic means of tico magnitudes = the square of their geometric mean. As three magnitudes are in geometric progression when the product of the first and third = the square of the second ; it appears, from 4°, that — 5°. The arithmetic, geometric, and harmonic means of two magnitudes are in geometric progression. As three magnitudes are in arithmetic progression when the sum of the first and third = twice the second, or, the second = half the sum of the first and third; it appears, from 3°, that— 6°. When three or any number of magnitudes are in har-monic progression, their reciprocals are in arithmetic progression. Between the three kinds of progression, arithmetic, geo- metric, and harmonic, the following relation appears from 1° — 7°. For every three, consecutive terms a, h, c, the difference [a-h) : the difference [h — c), in arithmetic progression •.•.a:a,in geometric progression : : a : Z>, and in harmonic progression : : a : c. An extension of the term harmonic mean from two to any number of magnitudes, by the same kind of analogy by which the terms arithmetic mean and geometric mean have been similarly extended, has been suggested by 3°. 26 THEORY OF HARMONIC SECTION. 8°. As, for two magnitudes a, b, we say that — Arithmetic mean = half of (a + 7)), Geometric mean = square root of (a x b), Harmonic mean = reciprocal of ^ (- + r] • So, by analogy, for n magnitudes a, J, c, J, &c., we say that — Arithmetic mean = «'" part oi{a + b + c + d+ &c.) , Geometric mean = n" root of [axbxcxdx &c.). Harmonic mean = reciprocal of-(- + TH h -,4 &c. ) , ^ n\a c d J The harmonic mean of any number of magnitudes thus signify- ing the magnitude tohose reciprocal = the arithmetic mean of the reciprocals of the magnitudes. 234. More generally (231) the harmonic relation of a system of four points on a common axis A^ B, X, Y, may be expressed in terms of their four distances from any arbitrary point P on the axis of the system, as follows : — In the fundamental proportion of harmonic section (213) sub- stituting for AX, A Y, BX, and BY their equivalents PX- PA, PY-PA, PX-PB, and PY- PB, the result {PX-PA):{PX-PB) + {PY-PA):{PY-PB) = 0...{1), or, which is the same thing, iPX-PA).{PY- PB) + {PY- PA).{PX- PB) = 0. . . (1'), expresses the relation in terms of the four distances in question, and may, like any of the preceding, be regarded as characteristic of an harmonic system, and sufficient to determine it. Dividing both terms of the proportion (1) by the ratio PA : PB, or of the equality (1') by the product PA.PB.PX.PY, the resulting proportion \PX~PAJ • [pX~Pb)'^\PY~Pa) '■ \PY~ PSr^-^-^'l' or the resulting equality \px~pa) • [py~pb)'^\py~pa) ■ \px~PBr^-^^'^' THEORY OF HARMONIC SECTION. 27 expresses again the relation in terms of the reciprocals of the four distances, in precisely the same form as in terms of the distances themselves. Cor. 1°. The first of the preceding proportions (1), or its equivalent {!'), gives at once the equality ^.PX.PY+2.PA.PB={PA + PB).{PX+PY)...{Z), and the second (2), or its equivalent (2'), the corresponding equality ■+ -f ' \PA ^ Pb) • \PX ^ Py) • • ■ ^^^' PX.PY^ PA.PB in which the forms again, as they ought to he, are identical. Cor. 2°. If C and Z be the middle points of AB and XY respectively, then, as PA + PB=2.PC and PX+PY=2.PZ, the first of these latter equalities (3) becomes PX.PY+PA.PB=2.PC.PZ (5), a relation of considerable utility in the applications of the theory of harmonic section. Cor. 3°. If Q and Q' be the harmonic conjugates of P with respect to AB and JCl' respectively, then, as 1 1_2 ,1 1_2 PA'^ PB~ PQ' '''''^ PX'^ PY- PQ'' ^-^^ '"•^' the second (4) becomes 1 1 (6); PX.PY^ PA.PB ~ PQ.PQ a relation again identical in form, and, as may be easily seen from (233, 6°), in meaning too, with that for the direct distances (5) to which it corresponds. 235. If a line AB cut harmonically at tioo points X and Y he again cut harmonically at two other points P and Q, both pairs of conjugates being arbitrary, then PX.PY=2PC.BZ (a), QX.QY=2QC.BZ (/S), C, B, and Z being the middle points of the three segments AB, PQ, and XY respectively. (See fig. 1, Art. 213). 28 THEORY OK HARMONIC SECTION. For, by (234, Cor. 2°), PX.FY+ PA.PB= 2PC.PZ, and, by (231, Cor. 3°), PA.PB = PG.PQ = 2.PC.PR, therefore PX.PY=2. PC. {PZ- PR) = 2. PC.RZ; and, similarly, QX.QY^I.QC.^QZ- QB) = 2.QC.BZ, and therefore &c. Otherwise thus, by (Euc. ii. 5, 6), PX.PY- CX.CY=PZ'- CZ' = [PZ- CZ).[PZ+ CZ) = PC.{PZ+ QZ- QC) = 2.PC.BZ-PC.QC, and, by (225), CX.CY=CP.CQ, therefore PX.PY=2.PC.BZ; and, similarly, QX.QY=2.QC.BZ; and therefore &c. The latter proof, depending only on the single consideration that the two rectangles CX.CY and CP.CQ are equal in magnitude and sign, shews that the relations themselves, (a) and (/S), depend on that circumstance alone, and are therefore independent of the accident as to whether the two points A and B are real or imaginary. Cor. 1". Taking successively the sum, difference, product, and quotient of the above equalities (a) and (/S), we get at once the four following relations : — 1°. Adding, remembering that P(74 QC=2.BC, we get PX.PY+ QX.QY=4:.BC.BZ (1). 2°. Subtracting, remembering that PC— QC= PQ, we get PX.PY- QX.QY=2.PQ.BZ (2). 3°. Multiplying, remembering that i.CP.CQ = AB' (225), we get PX.PYx QX.QY=AB\BZ' (3). 4°. Dividing, we get at once, without any reduction, PX.PY: QX.QY-.-.PC: QC (4), relations which, like those from which they are derived, are perfectly general, and independent alike of the position of either pair of conjugates X and F, or P and Q, and of the accident of A and B being real or imaginary. THEORY OF HARMONIC SECTION. 29 Cor. 2'. From (a) and (/8), and from (3°, Cor. 1°), we get at once the equalities PX.PY QX.QY ^^-^Trz-^ ^^—^jrz- ^^)' , PX.FY.Q X.QY ^^ RZ" W, which are the simplest formulae by which to calculate in numbers the position and length of AB when those of PQ and XY are given; a problem for which, it will be remembered, various constructions were given in Art. 230. CoE. 3°. If, while P and Q, and therefore B,^ are supposed to remain fixed, X and Y", and therefore Z, be conceived to vary, and in the course of their variation to coincide all three first at A and then at^; we see, from (4°, Cor. 1°), that PX.PY: QX.QY = a constant ratio, and also that PX.PY _ PA^ _PB^ _PC QX.QY' QA^~ QB'~ QG ^^^' relations which, for the particular positions of A and B, may be easily verified from the fundamental conception of harmonic section. See Arts. 150, and 161, Cor. 1°. Cor. 4°. If XY, X'T, X"Y'\ &c. be any number of seg- ments cutting the same segment AB harmonically, Z, Z', Z", &c. their several middle points, P and Q as before any arbitrary pair of conjugates, and B their middle point ; then since, from (a) and {(3), PX.PY iPX'.PY' : PX".PY", &c. = QX. Q Y: QX'. Q T : QX". Q Y'\ &c. = RZ: RZ : RZ\ &c., if the several distances RZ, RZ\ RZ'\ &c. form an arithmetic, geometric, or harmonic series, so do the two sets of rectangles PX.PY, PX'.PY, PX'.PY", &c., and QX.QY, QX'.QY, QX". Q Y", &c., whatever be the positions of P and Q. Cor. 5°. If XY, X' Y, X" Y" be any three segments cutting the same segment AB harmonically, Z, Z, Z" their three middle points, andP any arbitrary point on the axis of the segments ; then PX.PY. ZZ" + PX'.PY'. Z"Z + PZ".Pr".ZZ' = 0...(8), 30 THEORY OF HARMONIC SECTION. a theorem due to Chasles, and made much use of by him in the theory of involution. For, Q being the harmonic conjugate of P with respect to AB, and B the middle point of PQ, therefore by (a), FX.PY=2PC.BZ, PX'.PT=2PaBZ', PX".PY"=2PC.BZ", and E, Z, Z\ Z" being four points on a common axis, therefore, ty (82), BZ. Z'Z" + BZ'. Z"Z+ BZ". ZZ' = 0] and therefore &c. This proof, it will be observed, is independent of the cir- cumstance as to whether A and B are real or imaginary. Cor. 6°. If POQ and XOY be two angles cutting harmoni- cally the same angle A OB, then, all three being otherwise entirely arbitrary, amPOX.smPOY _ sin'PQ^ _ sin^PO -B . . smQOX.smQOY~ em^'QOA ~ sin'QOB ^^' which are the formulae by which to calculate In numbers the positions of the sides of the angle AOB when those of the angles POQ and XOFare given. For, if PQ, XY, and AB be the three segments intercepted by the three angles on any arbitrary line not passing through their common vertex 0, then since, by (65), BinPOXPX PO^ smPOY PY PO ~BmQOX~QX' QO' smQOY~ QY' QO' therefore, at once, by composition of ratios, BinPOXsinPQF _ P X.PY PO" BmQOX.smQOY~ QX.QY' QG'' and since, by the same again directly, sln'PQ^ _ ^ -^2! A sin'POP _ PB^ PO^ siii'QOA' QA'' QO"^ im'QOB~ QB"' QG'' therefore &c. ; the rest being evident from relation (7), Cor. 3°. COR. 7°. In the particular case when the angle POQ is right, that is, when the two conjugates OP and OQ are the two bisectors, internal and external, of the angle AOB (216, 3°), the sines of the several angles measured from OQ may be re- TlffeORY OF HARMONIC SECTION. 31 placed by the cosines of the corresponding angles measured from OP^ or conversely, and the above relation (9) becomes for the harmonic section of an angle what that of Art. 225 is for that of a line, viz. tanC0XtanC0r=tan''(7C^ = tan''C05 (10), OC being either bisector, internal or external, of the angle ^05. This latter relation, however, appears more immediately from that of the article referred to, by drawing the arbitrary line in the general proof of (9) perpendicular to the direction of 0(7, and then dividing the relation of that article, CX. GY= CA^ = CB'', by the square of OG (60). ( ^2 ) CHAPTER XIV. HARMONIC PROPERTIES OF THE POINT AND LINE. 236. Of the various harmonic properties of figures of points and lines, the two following, reciprocals of each other (173), lead to the greatest number of consequences, and may be re- garded as fundamental. a. In every tetragram the three pairs of opposite intersections (106) divide harmonically the three sides of the triangle deter- mined hy their three lines of connection, a. In every tetrastigm the three pairs of opposite connectors (106) divide harmonically the three angles of the triangle deter- mined by their three points of intersection. To prove a. If Zand X\ Fand T, Za.ni Z' be the three pairs of opposite intersections of the tetragram determined by the four lines in the figure on which they lie, three and three, and ABC the triangle determined by their three lines of con- nection ; the three segments XST' , YY\ ZZ' cut harmonically and are cut harmonically by the three 5C, C7^, AB. HARMONIC PROPERTIES OF THE POIN'T AND LINE. 33 For, in the triangle X.YX'^ having any one of the three former XX' for a side, and either extremity Y of either of the remaining two YY' for the opposite vertex ; the axis of the third ZZ' intersecting with the three sides at three collinear points Z, Z\ B, and the other extremity Y' of the second con- necting with the three vertices by three concurrent lines XZ, X'Z\ YG, therefore, by relations a and b', Art. 134, XB _ XZ^ YZ j:£^__ ^ X^ X'B ~^ YZ" XZ' X'C " YZ' • XZ' which evidently (213) prove the property for the pair of segments XX' and BC; and, as it may be proved exactly similarly for the remaining two pairs YY' and C4, ZZ' and AB, there- fore &c. To prove a. If QR and PS, RP and QS, PQ and RS be the three pairs of opposite connectors of the tetrastigm determined by the four points P, Q, R, S in the same figure, and ABC the triangle determined by their three points of intersection ; the three angles PAQ, QBR, RGP cut harmonically and are cut harmonically by the three BA C, CBA, A CB. For, in the triangle PAQ, having any one of the three former PA Q for an angle, and either side PQ of either of the re- maining two RCP for the opposite side ; the vertex of the third QBR connecting with the three vertices by three concurrent lines PR, QS, AB, and the other side RS of the second inter- secting with the three sides at three collinear points R, S, C, therefore, by relations a and b, Art. 1 34, sinP^^ _ _ si" PQS sm APR sin QAB ~ sin AQS ' sm QPR ' and slnPAC _ sin PQS sm AP R smQAC~'^ sinAQS' sin QpR ' which evidently (213) prove the property for the pair of angles PAQ ani BAG; and, as it may be proved exactly similarly for the remaining two pairs QBR and CBA, RCP and ACS, therefore &c. 237, That the two properties just established are reciprocals of each other, in the sense explained in Art. 173, may readily be shewn, in general terms, as follows : — VOL. u. r> 34 HAlt.MOXIC PROPERTIES OF THE POINT A\U LINE. If i, M^ N, be any four lines, and P, Q, B, 8 their four poles with respect to any circle, either system being arbitrary ; U and U', V and V\ W and W the three pairs of opposite intersections MN and L 0, NL and MO, LM and NO of the four lines ; X r.nd X\ Y and Y\ Z and Z' the three pairs of opposite connectors QR and PS, RP and QS, PQ and RS of the four points ; A, £, C the three vertices of the triangle de- termined by the three connectors UU', VV, WW ; and B, E, F the three sides of the triangle determined by the three inter- sections XX\ YY', ZZ' \ then, since, by the fundamental property of poles and polars (167), the several pairs of points and lines U and X, V and F, W and Z; U' and X, V and Y\ W and Z' ; A and Z), B and E, G and F are pole and polar to each other with respect to the circle, therefore, by the general property of Art. 223, the harmonicism of the three rows of four coUinear points B, C, U, U'; (7, A, V, V; A, B, W, W' involves and is involved in that of the three pencils of four concurrent lines E, F, X, X ; F, D, F, Y' ; D, E, Z, Z ; and therefore &c. The reader understanding the spirit of the above mode of reasoning is recommended to apply it for himself to the several other examples of pairs of reciprocal properties which will be given him in abundance in the course of the sequel. He will in general find the transformation of a property into its re- ciprocal to be a process almost purely mechanical, consisting ordinarily of little more than merely changing in its statement all points into lines and lines into points, all connectors of points into intersections of lines and intersections of lines into con- nectors of points, all points on a circle into tangents to the circle and tangents to a circle into points on the circle, &c. In cases presenting any exceptional peculiarity, or involving the necessity of any intermediate considerations, the reciprocality of the properties will occasionally be proved for him, but in all ordinary cases, like the above, the process of tracing it will be left as an exercise to himself; especially when, as in the preceding article, the demonstrations actually given of the re- ciprocal properties are themselves also reciprocal ; a circumstance which in that article would have been rendered more apparent by the employment, as above, of corresponding notation applied to the reciprocal parts of separate figures for both properties, HARMONIC PROPERTIES OF THE POINT AND LINE. 35 had not, for other reasons which will appear in the sequel, the figures for the two heen combined in their case instead. 238. It is easy to see, from the general property of Art. 221 , that the harmonicism of any one of the three rows of four points X, X', B^O; Y, Y\ C, A; Z, Z\ A, B in property a, or of any one of the three pencils of four rays QR^ PS, AB, ACj RP, QS, BO, BA ; PQ, BS, CA, CB in property a of Art. 236, (see figure of that article), involves that of the other two ; for, in the former case, the two rows for every two of the three connectors XX\ YY', ZZ' being in perspective at both ex- tremities of the third, viz. F, F', C, A and Z, Z',A,B at X and X' ; Z, Z\ A, B and X, X\ B, C at Fand F' ; X, X\ B, C and F, F', G, A at Z and Z' (see fig.) ; and, in the latter case, the two pencils for every two of the three intersections A, B, C being in perspective on both lines determining the third, viz. BP, QS, BC, BA and PQ, BS, CA, CB on QR and PS; PQ, RS, CA, CB and QR, PS, AB, AC ou RP and QS; QR, PS, AB, AC and RP, QS, BC, BA on PQ and RS see fig.) ; therefore, by property 2°, Art. 222, if any one of the three rows in the former case, or of the three pencils in the latter case, be harmonic, so are the other two ; and there- fore &c. It is again easy to see, from the same, that of the two re- ciprocal properties themselves, a and a, either involves the other directly without the aid of the reciprocating process explained in Art. 173, and applied in Art. 237. For, in the triangle ABC (see fig.), if the three sides BC, CA, AB are cut harmonically by the three pairs of conjugates X and X', Y and F', Z and Z', then, by the general relation of Art. 221, the three opposite angles BAG, CBA, ACB are cut harmonically by the three pairs of conjugates QR and PS, RP and QS, PQ and RS; and, conversely, if the three angles BA C, CBA, A CB are cut har- monically by the three pairs of conjugates QR and PS, RP and QS, PQ and RS, then, by the same, the three opposite sides BG, CA, AB are cut harmonically by the three pairs of con- jugates X and X, Y and F, Z and Z' ; and therefore &c. A tetragram and tetrastigm, related as in the figure to the same central triangle ABC, possess many interesting har- D2 36 HARMONIC PROPERTIES OF THE POINT AND LINE. monic properties In connexion with each other and the triangle, some of which will be noticed in the course of the sequel. 239. In the particular cases, when, in property a of Art. 236, one of the four lines X' Y'Z' constituting the tetragram in the general case is the line at infinity (131), and when, in pro- perty a of the same article, one of the four points S constituting the tetrastigm in the general case is the polar centre of the triangle PQR determined by the remaining three (168) ; since. In the former case, the three pairs of harmonic conjugates X and X', Y and y, Z and Z' connect by infinite intervals, they bisect, internally and externally, the three sides 5(7, CA, AB of the triangle ABC determined by their three lines of connec- tion (216, 3°); and since, in the latter case, the three pairs of harmonic conjugates QR and PS', RP and QS^ PQ and R8 intersect at right angles, they bisect, extei'nally and internally, the three angles BAC, CBA, ACB of the triangle ABC deter- mined by their three points of intersection (216, 3°) ; hence, the two reciprocal properties themselves, a and «', shew for these particular cases, as is otherwise evident, that — a. In every triangle the three vertices bisect the three sides of the triangle determined by the directions of ike three parallels through them to the opjwsite sides. a. In every triangle the three sides bisect the three angles of the triangle determined by the intersections of the three perpen- diculars to them through the opposite vertices. These latter properties are not reciprocals in the same sense as those from which they have been inferred ; each, to an arbitrary circle, reciprocating, not into the other, but into the more general property of which the other is a par- ticular case. In reciprocating the first, the line at infinity (136), on which the three parallels through the vertices in- tersect with the opposite sides of the triangle, must be taken into account, with the latter, in order to complete the tetragram of the general property, under which, as above shewn, it comes as a particular case. In the particular case when the tetragram in property a of Art. 236 is a parallelogram ; since then one chord of inter- section, XX' suppose (see figure of that article), of the figure. HARMONIC PEOPERTtES OF THE POINT AND LINE. 37 and with it, of course, the side BC of the triangle ABC, is at infinity ; therefore, by virtue of that property, the other two chords of intersection Y¥' and ZZ' mutually bisect each other at the opposite vertex A of the triangle ABC. Hence the familiar property that in every parallelogram the two diagonals mutually bisect each other, comes as another particular case under the same general property a ■ and, to an arbitrary circle, reciprocates (like the above o) into the general property a reciprocal to a. 240. From the two fundamental properties of Art. 236, the following general consequences, in pairs reciprocals of each other, may be immediately inferred, viz. — a. The two centres of perspective of any two segments (131) divide harmonically the segment intercepted on their line of con- nexion by the axes of the segments. a. The two axes of perspective of any two angles (131) divide harmonically the angle subtended at their point of inter- section by the vertices of the angles. For, if, in the figure of that article, any two of the three pairs of opposite intersections X and X', Y and Y' of the tetragram be regarded as the extremities of the two segments in a ; then are the remaining pair Z and Z' the two centres of perspective (131) of those segments, and, by property a of the article in question, they divide harmonically the segment AB intercepted on their line of connection by the axes of the segments ; and therefore &c. And, if, in the same figure, any two of the three pairs of opposite connectors QR and PS, RP and QS of the tetrastigm be regarded as the sides of the two angles in a ; then are the remaining pair PQ and RS the two axes of perspective (131) of those angles, and, by property a' of the same article, they divide harmonically the angle A CB subtended at their point of intersection by the vertices of the angles ; and therefore &c. b. The two centres of perspective of any two segments connect harmonically with the vertex of the angle determined by the axes of the segments. v. The two axes of perspective of any two angles intersect harmonically with the axis of the segment determined by the vertices of the angles. 38 HARMONIC PKOPERTIES OF THE POINT AND LINE. For, if, in the figure of the same article, any pair of opposite connectors QB and PS of the tetrastigm be regjirded as the two segments in b; then are the two intersections B and C of the other two pairs BP and QS, PQ and BS the two centres of perspective of those segments, and, by property a' of the article in question, they connect harmonically with the vertex A of the angle determined by the axes of the segments; and therefore &c. And, if, in the same figure, the two pairs of lines determining any pair of opposite intersections X and X' of the tetragram be regarded as the two angles in b' ; then are the two connectors YY' and ZZ' of the other two pairs Y and Y', Z and Z' the two axes of perspective of those angles, and, by property a of the same article, they intersect harmonically with the axis BG of the segment XX' determined by the vertices of the angles ; and therefore *S:c. c. The two centres of 'perspective of any two segments are conjuffate points (217) with respect to the angle determined by the axes of the segments, c. The two axes of pempecttve of any tieo angles are conjugate lines (217) with respect to the segment determined by the vertices of the angles. These, by Art, 217, are obviously but another mode of stating the two general properties b and &'; which, though proved independently above by reciprocal demonstrations, follow at once, it may be observed, from the two a and a', by virtue of the general property of Art. 221. For, since, by a, the three rows of four points B, C, X, X' ; (7, A, Y, Y' ; A, B, Z, Z' (see fig.) are harmonic, therefore, by the general property in question, the three pencils of four rays ZZ\ YY', AX, AX' ; XX', ZZ, BY, BY'; YY', XX, CZ, CZ' are harmonic, and therefore &c. And, since, by a', the three pencils of four rays AB, AG, QB, PS; BG, BA, BP, QS; AB, AG, PQ, BS (see fig.) are harmonic, therefore, by the same general property, the three rows of four points B, G, X, X; G, A, F, Y; A, B, Z', Z are harmonic, and therefore &c. 241. The two fundamental properties of Art. 236 supply also obvious solutions, by linear constructions only without the aid of the circle, of the two following reciprocal problems, viz. — IIARM0X1C*PR0PKRT1ES OF THE POINT AND LINE. 39 a. Given three points of an harmonic row, to determine the fourth conjugate to any assigned one of the given three. a . Given three rays of an harmonic pencil, to determine the Jourth conjugate to any assigned one of the given three. Thus, in the figure of that article, of the harmonic row B, C, X, X' given the three points X, X', C to determine the fourth B conjugate to C; and, in the same figure, of the harmonic pencil AB, AG, AX, AX' given the three rays AX, AX', AB to determine the fourth AG conjugate to AB. To solve the first ; on any line GA, drawn arbitrarily through the point G whose conjugate is to be determined, taking arbitrarily any two points Y and Y' ; their connectors with the other two points X and X' determine the two centres of perspective Z and Z' of the two segments XX' and YY', whose line of connection ZZ', by a. Art. 236, intersects with the axis of the given points at the required conjugate B. And, to solve the second, through any point B, taken arbitrarily on the ray AB whose conjugate is to be determined, drawing arbitrarily any two lines BY an^ BY' ; their intersections with the other two rays AX and AX' determine the two axes of perspective PQ and BS of the two angles XAX' and YBY', whose point of intersection C, by a. Art. 236, connects with the vertex of the given rays by the required conjugate A G. CoE. Every point of an harmonic row being the pole of every Hue through its conjugate with respect to the segment determined by the remaining two points, and every ray of an harmonic pencil being the polar of every point on its conjugate with respect to the angle determined by the remaining two rays (217) ; the above reciprocal constructions give, consequently, solutions of the two following reciprocal problems, as well as of those for which they have been given, viz.— To determine by linear constructions only without the aid of the circle : a. the pole of a given line with respect to a given seg- ment ; a', the polar of a given point with respect to a given angle. 242. The two fundamental properties themselves, of Art. 236, may obviously be stated in the following equivalent forms, in which they express two reciprocal harmonic properties of triangles, viz. — 40 HARMONIC PROPERTIES OF THE POINT AND LINE. a. Every three coUinear points on the sides of a triangle determine with the opposite vertices three segments dividing har- monically the sides of the triangle determined by their axes. a'. Every three concurrent lines through the vertices of a triangle determine with the opposite sides three angles dividing harmonically the angles of the triangle determined by their vertices. For, if (fig. of Art. 236) XYZ be any triangle, and X', F', Z' any three coUinear points on its three sides ; then, since, in the tetragram determined by the line of collinearity with the three sides of the triangle, the three segments XX', YY\ ZZ\ by property a of that article, are intersected harmonically each by the axes of the other two, therefore &c. And if (same fig.) PQR be any triangle, and PS, QS, R8 any three concurrent lines through its three vertices ; then, since, in the tetrastigm determined by the point of concurrence with the three vertices of the triangle, the three angles XAX\ YBY\ ZCZ\ by pro- perty a' of the same article, are subtended harmonically each by the vertices of the other two, therefore &c. COE. 1°. As every three lines through the vertices of a triangle which intersect coUinearly with the opposite sides determine an exscribed triangle in perspective with it, and as every three points on the sides of a triangle which connect concurrently with the opposite vertices, determine an inscribed triangle in perspective with it (141) ; it appears consequently, from the above reciprocal properties, or from those of Cors. 1° and 2°, Art. 139, with which they are evidently identical, that — a. When a triangle exscribed to another is in perspective with it, its sides are cut harmonically by the corresponding vertices and sides of the other. a. When a triangle inscribed to another is in perspective with it, its angles are cut harmonically by the corresponding sides and vertices cf the other. Cor. 2°. Since, for every two triangles in perspective, the three pairs of corresponding vertices connect through the centre of perspective, and the three pairs of corresponding sides intersect on the axis of perspective (140) ; it follows consequently, from the two reciprocal properties of the preceding corollary, that — a. When a triangle exscribed to another is in perspective with HARMONIC PROPERTIES OF THE POINT AND LINE. 41 itj its sides are the polars of the centre of persiiective with respect to the corresponding angles of the other. a. When a triangle inscribed to another is in perspective with it^ its vertices are the poles of the axis of perspective with respect to the corresponding sides of the other. 243. Of the various other harmonic properties of triangles, the following, in pairs reciprocals of each other, result imme- diately from the four general relations of Art. 134. a. Whe7i three points on the three sides of a triangle are colli- near^ their three harmonic conjugates loith respect to the sides connect concurrently with the opposite vertices ; and conversely. a'. When three lines through the three vertices of a triangle are concurrent^ their three harmonic conjugates icith respect to the angles intersect collinearly loifh the opposite sides ; and conversely. For, in the case of a, if A, B, G be the three vertices of the triangle ; X^ Y, Z any three points on its three opposite sides ; and X\ Y\ Z' their three harmonic conjugates with respect to the three segments BC, CA, AB respectively ; then, since, by the definition of harmonic section (213), BX BX CY Cr_ AZ AZ'_ CX'^ aX'~' AY'^ AY'~' BZ^ BZ'~^^' therefore at once, by composition of ratios, BX CY AZ BX CY^ ^Z' _ CX ' AY' BZ^ CX' ' AY" BZ'~ ' consequently, when either compound = ± 1, the other then = + J , and therefore &c. (134). And, in the case of «', if ^, B, Che the three sides of the triangle ; X, Y, Z any three lines through its three opposite vertices ; and X\ Y', Z their three harmonic conjugates with respect to the three angles jBC, C4, AB re- spectively ; then, since, by the definition of harmonic section (213), sin^X sin^Z ' _ sinCF sinCF _ ^i^^'^sinOA"" ' sin^ir"''sin^r ' sin^Z sin^Z'_ %m.BZ smBZ' ' 42 HARMONIC PROPERTIES OF THE POINT AND LINE. therefore at once, by compoaltlon of ratios, sin^Z BinCr sm AZ sinBX' s lnCF sin AZ' _ + -• /iv • „:„ 4 v ■ „:„ WS" ' SID GX' aiaAY' smBZ sin CX' ' sm AY' ' amUZ' consequently, when either compound = + 1, the other then = +1, and therefore &c. (134). b. When three points on the three sides of a triangle are coUinear, their three harmonic conjugates with respect to the sides determine with them the three pairs of opposite intersections of a tetragram (106). b'. When three lines through the three vertices of a triangle are concurrent, their three harmonic conjugates with respect to the angles determine with them the three pairs of opposite connectors of a tetrastigm (106). For, in the case of 5, employing the same notation as in the proof of a, the four compounds BX CY AZ CY' AZ' BX CX- AY' BZ' AY'- BZ" CX' AZ BX' CY BX' CY' AZ BZ'- CX' •AY' CX' ' AY" BZ' being always equal in magnitude and sign, when any one of the four =+1, the remaining three each =+1, that is (134, a) when any one of the four groups of three points X, Y, Z; Y',Z',X; Z',X', Y; X', T, Z is coUinear, the remaining three are also coUinear, and, the four lines of collinearity consequently determining a tetragram of which X and X', Y and F', Z and Z' are the three pairs of opposite intersec- tions (106), therefore &c. And, in the case of 6', employing the same notation as in the proof of a', the four compounds ixaBX sinCF sin^Z sin CY' ain AZ ' ainBX ainCX ■ sin^ F" ainBZ' sin J Y' " sin BZ' ' sinCX ' ai n AZ' amBX' am CY sinBX' sin CY' ain AZ sin BZ' ' sin GX' ' sin A Y ' sin GX' " sin^F' * sin^Z ' being always equal in magnitude and sign, when any one of the four = — 1, the remaining three each = - 1, that is (134, a) when any one of the four groups of three lines X, F, Z; F', Z', X ; Z', X', Y ; X', Y', Z is concurrent, the remaining HARMONIC PROPERTIES OF THE POINT AND LINE. 43 three are also concurrent, and the four points of concurrence consequently determining a tetrastigm of which X and X\ Y and Y'^ Z and Z' are the three pairs of opposite connectors (106), therefore &c. c. When three points on the three sides of a triangle are collinear, their three polars (217) with respect to the three oppo- site angles are concurrent. c . When three lines through the three vertices of a triangle are concurrent^ their three poles (217) with respect to the three opposite sides are collinear. These two reciprocal properties follow at once from the two c and a\ by virtue of the general property of Art. 221 ; the harmonic conjugate of each point in c with respect to its own side connecting with the opposite angle by the polar of the point with respect to that angle ; and, the harmonic con- jugate of each line in c with respect to its own angle inter- secting with the opposite side at the pole of the line with respect to that side. d. The three poles of any line with respect to the three sides of a triangle connect concurrently with the opposite vertices. d'. The three polars of any point with respect to tlie three angles of a triangle intersect collinearly with the opposite sides. These two reciprocal properties are obviously identical with the two a and a! ; the three poles of the line with respect to the three sides in d being the three harmonic conjugates of its three points of intersection with the sides ; and the three polars of the point with respect to the three angles in d' being the three harmonic conjugates of its three lines of connection with the vertices (217). In the particular cases when the line in d is the line at infinity, and the point in d' any point at infinity ; since, in the former case, the three poles of the line at infinity with respect to the three sides are the three middle points of the sides (216, 3°) ; and since, in the latter case, the three polars of the point at infinity with respect to the three angles bisect inter- nally the three segments Intercepted by the angles on any line passing through the direction of the point (224) ; the two re- ciprocal properties d and d' become, consequently, those already 44 HAEMONIlJ PROPERTIES OF THE POINT AND LINE. established on other principles in examples (l°and 13°, Art. 137), viz. — In every triangle^ a., the three middle points of the sides con- nect concur nntly with the oiiposite vertices; ol. the three lines cminecting the vertices loith the middle iwints of the segments intercepted bi/ the corresponding angles on any line, intersect collinearly with the opposite sides. Of the several pairs of reciprocal properties established in this article, it may be observed that either reciprocal would follow directly from the other by virtue of the general property of Art. 121 ; from which it follows, evidently, for a triangle, that every two points harmonic conjugates with respect to any side connect harmonically with the opposite angle, and, that every two lines harmonic conjugates with respect to any angle in- tersect harmonically with the opposite side. 244. From the fundamental properties of Art. 236, com- bined with the two a and a of the preceding article, the two following reciprocal properties of the tetragram and tetrastigm may be readily infeired, viz. — a. In every tetragram, tlie three pairs of opposite intersections connect with the opposite vertices of the triangle determined by their three lines of cminection hy six lities parsing three and three through four points, and thus determining the three pairs of op- posite connectors of a tetrastigm. a'. In every tetrastigm, the three pairs of opiposite connectors intersect with the opposite sides of the triangle determined hy their thres points of intersection at six points lying three and three on four lines, and thus determining the three pairs of opposite inter- sections of a tetragram. For, in the case of the tetragram, the three pairs of opposite intersections X and A", Y and T, Zand Z' (fig. Art. 236), dividing harmonically, b)' (236, a), the three sides of the triangle ABC determined by their three lines of connection, and lying, by hypothesis, three and three on four lines YZX', ZXY', XYZ', X' Y'Z' ; therefore, by (243, a), they connect with the opposite vertices A, B, C by three pairs of lines AX and AX', BY and BY', CZ and CZ' passing three and three through four points P, Q,R,S; and_ therefore &c. And, in the case of HARMONIC 'Properties op the point and line. 45 the tetrastigm, the three pah-s of opposite connectors QR and PS, BP and QS, PQ and RS (same fig.) dividing har- monically, by (236, a), the three angles of the triangle ABC determined by their three points of intersection, and passing, by hypothesis, three and three through four points P, Q, B, S; therefore, by (243, a), they intersect with the opposite sides BC, CA, AB at three pairs of points X and X', Y a.ni Y', -2' and Z' lying three and three on four lines YZX\ ZXY\ XYZ', X' Y'Z' ; and therefore &c. It will be seen in the sequel that the four lines YZX\ ZXY', XYZ\ X' Y'Z' and the four points P, Q, R, S, related as above to eacli other, possess also several other reciprocal harmonic relations in connection with the triangle ABC. 245. From the same fundamental relations, combined with the two (7) and (9) of Art. 235, two other important reciprocal properties of the tetragram and tetrastigm may again be readily inferred, viz. — a. In the triangle determined in a tetragram hy the axes of the three chords of intersection of the figure (107), when three points on the sides are either collinear or concurrently connectant with the opposite vertices, their three harmonic conjugates with respect to the three chords of intersection are also either collinear or concurrently connectant with the opposite vertices. a. In the triangle determined in a tetrastigm hy the vertices of the three angles of connection of the figure (107), iihen three lines through the vertices are either concurrent or collinearly intersectant with the opposite sides, their three harmonic conjugates with respect to the three angles of connection are also either concurrent or collinearly intersectant' with the opposite sides. To prove a. If, as in the figure of Art. 236, X and X', Fand Y' Zand Z' be the three pairs of opposite intersections of the tetragram ; ABC the triangle determined by their three lines of connection ; U, V, W any three points on its three sides BC CA, AB; and U', V, W their three harmonic conjugates with respect to the three chords of intersection XX', YY', ZZ' of the figure; then since, by hypothesis and (236, a), the three latter segments are cut harmonically at once by the three UU', 46 HARMONIC PROPERTIES OF THE POIXT AND LINE. W, WW", and also by the three BC, CA, AB, therefore by (7) Art. 235, BU.BU' _ BX^ _ BX2 CU.GU' " CX' CX'" CV.CV _ CY^ ^ GY'-' AV.AV ~ AY'" AY'^' AW. AW _ AZ^ ^ AZ^ BW.BW~ BZ' BZ'"^ and since, by a and b', Art. 134, the two compounds BX' CY' AZ' , BJT CY" AZ' CX^ ' AY^'BZ' CX"' 'AY'-'' BZ" both = + 1, therefore the compound BU.BU' CV.CV AW.AW _ . . CU.CU'' AV.AV" BW.BW'~'^ ^°'^' consequently, when either of the two compounds BU GV AW BU Cr AW GU'AVBW^^ CU" AV BW = ± 1, the other also = ± 1, and therefore &c. (134, a and b'). To prove a. If, as in the same figure, QR and PS, RP and QS, PQ and RS be the three pairs of opposite connectors of the tetrastigm ; ABC the triangle determined by their three points of intersection ; AU, BV, GW any three lines through Its three vertices A, B, C; and AU', BV, GW their three harmonic conjugates with respect to the three angles of connection XAX', YBY', ZGZ' of the figure ; then since, by hypothesis and (236, a), the three latter angles are cut harmonically at once by the three UAU', VBV, WCW, and also by the three BAG, CBA, A GB, therefore by (9) Art. 235, fimBAU^^aBAU^ _ s,m'BAX _ sm^'BAX' sin CA U. sin GA U' ~ sin' GAX ~ sin" GAX' ' s lnC^F.sinC^F ' _ sin' G BY _ sin' GBY' sln^^F.sin^^r ~ slnM^T ~ aWABY' ' 8ln,4(7Tr.sin.^l(7TF' _ am" ACZ _ sm'AGZ ' amBGW.smBCW' ~ am'BGZ ~ am^BGZ' ' HARMONIC 1>R0PERT1ES OF THE POINT AND LINE. 47 and since, by b and a', Art. 134, the two compounds sin'^^X sin''CJ5F sinMCZ and sWBAX' sWGBY' sla' ACZ' airi'OAX' ' sixi'ABY' " ^m'BCZ' both =+1, therefore the compound smBAU.smBAV &mCBV.sm.CBV' smAGW.smACW am CAU. am CAU' ' sin^-BF.sin^^F' ' amBCW.amBCW = +1 («'), consequently, when either of the two compounds amBAU sin CBV sin^CTT sinC^CT" '^SKABV amBCW or sin^^C^' sin CBV &mACW sin CA U' • sin^5F' * sin^CTT' = + 1, the other also = + 1, and therefore &c. (134, h and a). CoK. 1°. Since, by (217), every two lines which intersect with the axes of any number of segments harmonically are conjugate lines with respect to all the segments ; and, every two points which connect with the vertices of any number of angles harmonically are conjugate points with respect to all the angles ; it follows, consequently, from the first (and more im- portant) parts of the above properties a and a', that — h. Every two lines conjugates to each other with respect to two of the three chords of intersection of a tetragram are conjugates to each other with respect to the third also. v. Every two points conjugates to each other with respect to two of the three angles of connection of a tetrastigm are conjugates to each other with respect to the third also. Every two lines / and /', thus conjugates to each other with respect to the three chords of intersection of a tetragram, are said to be conjugate lines with respect to the tetragram; and, every two points and 0', thus conjugates to each other with respect to the three angles of connection of a tetrastigm, are said to be conjugate points with respect to the tetrastigm. Every two conjugates in both cases are evidently interchangeable. CoE. 2°. Since, by the general property of Art. 221, every 48 HARMOMC PROPERTIES OF THE POINT AND LIXE. two lines 1 and T conjugates to each other with respect to a tetragram (Cor. 1°, h) divide harmonically the three angles subtended at their point of intersection IT by the three chords of intersection XX', YT, ZZ' of the figure ; and, every two points and 0' conjugates to each other with respect to a tetrastigm (Cor. 1°, V) divide harraonically the three segments intercepted on their line of connection 00' by the three angles of connection XAX\ YBY', ZGZ of the figure ; hence, from the same, the reciprocal properties that — c. Every two conjugate lines with resjject to a tetragram divide harmonically the three angles subtended at their point of intersection hy the three chords of intersection of the figure. c. Every two conjugate points icith respect to a tetrastigm divide harmoniccdly the three segments tntei'cepted on their line of CA)miection hy the three angles of connection of the figure. Cor. 3°. Since again, by the first parts of a and a', com- bined with the genei'al property of Art. 221, the two lines, real or imaginary (230), which divide harmonically two of the three angles subtended at any arbitrary point by the three chords of intersection XX\ YY\ ZZ' of a tetragram (107) divide harmonically the third also ; and, the two points, real or imagi- nary (230), which divide harmonically two of the three segments intercepted on any arbitrary line / by the three angles of con- nection XAX', YBY\ ZGZ of a tetrastigm (107) divide har- monically the third also ; hence, from the same again, the reciprocal properties that — d. The three angles subtended at any ptoint hy tlie three chords of intersection of a tetragram have a common angle of harmonic section, real or imaginary. d'. The three segments intercepted on any line by the three angles of connection of a tetrastigm have a common segment of harmonic section, real or imaginary. COK. 4°. Every harmonic pencil of rays, whatever be its vertex, detennining an harmonic row of points on every axis, and every harmonic row of points, whatever be its axis, deter- mining an harmonic pencil of rays at every vertex (221) ; it appears consequently, from the two reciprocal properties of the preceding coi-oUary (3°) applied to the particular cases when the point in d and the line in d' are at infinity, that — HARMON IC*PROPERTIES OS THE POINT AND LINE. 49 e. The three, segments determined on any axis^ ly the three pairs of perpendiculars, or any other isoclinals, through the three pairs of opposite intersections of any tetragram^ have a common segment of harmonic section^ real or imaginary. e. The three angles determined at any vertex, by the three pairs of parallels J or any other isoclinals, to the three pairs of opposite connectors of any tstrastigm, have a common angh of harmonic section, real or imaginary. COE. 5°. Every two points harmonic conjugates to each other with respect to any segment being each the pole of every line through the other with respect to the segment (217), and, every two lines harmonic conjugates to each other with respect to any angle being each the polar of every point on the other with respect to the angle (217) ; the first parts of the original properties a and a may consequently be stated otherwise thus as follows — f In every tetragram, the three poles of any line with respect to the three chords of intersection are collinear. f. In every tetrastigm, the three polars of any point with respect to the three angles of connection are concurrent. Ct"»R. 6°. In the particular cases where the line in /(Cor. 5°) is the line at infinity, and the point in /' (same Cor.) any point at infinity ; since, in the former case, the three poles of the line at infinity with respect to the three chords are the three middle points of the chords (216, 3°), and since, in the latter case, the three polars of the point at infinity with respect to the three angles bisect internally the three segments intercepted by the angles on any line passing through the direction of the point (224) ; from the properties themselves (/ and _/", Cor. 5°) ap- plied to those cases, it appears, consequently, that — a. In every tetragram, the three middle points of the three, chords of intersection are collinear. q. In every tetrastigm, the three lines connecting the vertices of the three angles of connection with the middle points of the three segments they intercept on any arbitrary line are concurrent. Of these properties the first {g), it will be observed, is identical with that already established on other principles in (189, Cor. 2°). Cor. 7°. In the particular cases, when, in the original pro- E 50 HARMONIC PROPERTIES OF THE POINT AND LINE. perty a, one of the four lines X' Y'Z' constituting the tetragram is the line at infinity (136), and, in the original property o', one of the four points 8 constituting the tetrastigm is the polar centre of the triangle determined by the remaining three P, Q, B (168) ; since, in the former case, the three pairs of harmonic conjugates 2: and X\ Zand F, Zand Z\ connecting by infinite intervals, bisect, internally and externally, at once the three segments UU', VV, WW and the three BC, CA,AB (216,3°); and since, in the latter case, the three pairs of harmonic con- jugates QR and PS, RP and QS, PQ and BS, intersecting at right angles, bisect, internally and externally, at once the three angles f7j:C7', VBV, WCW a.ad the three BAG, CBA, ACB (216, 3°); hence the two reciprocal properties themselves (a and a) shew, for these particular cases, that — h. When three points on the sides of a triangle are either colUnear or concurrently connectant with the opposite vertices, the conjugate three equally distant from the biseriions of the sides are also either collinear or concurrently connectant with the opposite vertices. Ji. When three lines through the vertices of a triangle are either concurrent or collinearly intersectant with the opposite sides, the conjugate three egyually inclined to the bisectors of the angles are also either concurrent or collinearly intersectant with the opposite sides. Properties which, it will be remembered, have been already established, on other principles, in Examples 11° and 12°, Art. 137. 246. The two following reciprocal properties are evident from the fundamental relation of harmonic section (214), com- bined with the general property of Art. 221, viz. — a. If on a variable line L, turning round a fixed point 0, and intersecting with two fixed lines A and B at two variable points X and Y, a variable point P be taken so as to satisfy in every position the relation PX PY „ OX^ OY ' the point P moves on a fixed line I, passing through the intersec- tion of A and B; the polar, viz., of the point with respect to the angle AB (217). HAEMONK? PROPERTIES OF THE POINT AND LINE. 51 a. If through a variable point P^ moving on a fixed line I^ and connecting with two fijxed points A and B by two varf'rhle lines U and F, a variable line L be drawn so as to satisfy in every position the relation smLU sinZF_ the line L turns round a fixed point 0, lying on the connector of A and B / the pole, viz. of the line I with respect to the segment AB (217). For, in the case of a, the two points and F, being harmonic conjugates (214) with respect to the two X and Y, connect harmonically (221) with the vertex of the angle AB; and in the case of a, the two lines / and X, being harmonic conjugates (214) with respect to the two U and V, intersect harmonically (221) with the axis of the segment AB; and therefore &c. 247. The two reciprocal properties of the preceding article are evidently particular cases of the two following, viz. — a. If on a variable line L, turning round a fixed point 0, and intersecting loith two fixed lines A and B at two variable points X and Y, a variable point P be taken so as to satisfy in every position the relation PX . PY „ a and b being any two finite multiples, positive or negative ; the point P moves on a fixed line I, passing through the intersection of A and B, and termed the polar of the point with respect to the two lines A and Bfor the two multiples a and b. a'. If through a variable point P, moving on a fixed line /, and connecting with two fixed points A and B by two variable lines U and V, a variable line L be drawn so as to satisfy in every position the relation sin LU , sinLV a. —. ==+0.-: — f^ =0, sin/?7 sin/F a and b being any two finite multiples, positive or negative ; the line L turns round a fixed point 0, lying on the connector of A and B, and termed the pole of the line I with respect to the two points A and Bfor the two multiples a and b. n2 52 harmonk; properties of the point and like. To prove which. In the case of «, if OA and OB, PA and PB be the four perpendiculars from and Pupon A and B; then since, by (Euc. VI. 4), PX _PA PY PB 0X~ OA' OY OB' therefore, by the relation determining the position of P on L, PA , PB ,^ "■OA^^-OB^"' from which, the ratio of PA to PB being constant, it follows, consequently, that P lies on the line 1 which divides the angle ^^into segments whose sines are in the constant ratio (61) ; and therefore &c. And, in the case of a\ itAI and BI, AL and BL be the four perpendiculars from A and B upon / and L ; then since, by (61), imLU _AL , ^mLV _BL ImlU ~ AI ' sin/F ~ BI ' therefore, by the relation determining the direction of L through P, AL , BL ^ ''■AI+^-BI = '^' from which, the ratio of AL to BL being constant, it follows, consequently, that L passes through the point which divides the interval AB into segments in the constant ratio (Euc. VI. 4) ; and therefore &c. Cor. r. From the relations in properties a and a above, it is evident, by mere inversion of ratios, that — b. When two points P and Q are such that one of them P lies OH the 2>oIar of the other Q with respect to two lilies A and B for two muhijjles a and b; then the latter Q lies on the polar of the former P with respect to tlie two lines for the reciprocals of the two multiples. b'. When two lines L and M are such that one of them L passes through the pole of the other M with respect to two points A and B for two multiples a and b ; then the latter M passes through the pole of the former L with respect to the two points for the reciprocals of the tioo multiples. For, the relations of condition that the first parts be true, viz. — PA , PB ^ , AL ^ BL ^ HAliMOXIC PUOPEKTIES OF THE POINT AND LINE. 53 give immediately, by inversion of the two ratios in each, the relations \ QA I QB ^ - 1 AM^l BM ^ which are the relations of condition that the second parts le true ; and therefore &c. CoK. 2°. From the same relations again, it is evident, by mere alternation of proportions, that, for two points P and Q, two lines L and M, and two multiples a and b — c. When P lies on the polar of Q with respect to L and M for a and b ; then L passes through the p)ole of M icith respect to P and Q for a and b. c. When L passes through the pole of M with respect to P and Q for a and b ; then P lies on the polar of Q with respect to L and Mfor a and b. For, the relations of condition that both parts of each be true, viz. — PL ^ PM ^ , PL . QL ^ «-^+^-eir^' *°"^ ''■pri+^-QM^''' are evidently identical, by mere alternation of either; and therefore &c. COE. 3°. In the same case, if X and Y be the two points of intersection with L and J/ of the line PQ, and if U and V be the two lines of connection with P and Q of the point LM; then again, by mere alternation of proportions, it is evident that — d. When P lies on the polar of Q with respect to L and M for a and b ; then X lies on the polar of Y with respect to U and V for a and b. d'. When L passes through the pole of 21 with respect to P and Q for a and b ; then U passes through the pole of V with respect to X and Y for a and b. For, the relations of condition that the first parts of each be true, viz. PX , PY ^ , sinip- . siniF „ «-^:Y+^-^=^' ""^ ''•iiKM/+^-s-bJ7F=^' give immediately, by alternation of the proportions in each, XP , XQ . , sin&X , , %mUM ^ YP YQ sin VL sm VM ' 54 HARMONIC PROPERTIES OF THE POINT AND LINE. which are the relations of condition that the second parts of each be true ; and therefore &c. CoE. 4°. In connection with the subject of the present article, it may be readily shewn, that generally — When four collinear points P, Q, X, Y lie on four concurrent lines i, M, U, V, or, when four concurrent lines L, M, ?7, V pass through four collinear points P, Q, X, Y ; then,, for every two finite multiples a and &, the two relations PX , FY ^ - &mLU , siniF „ a.-— p. + 6.-^^=0, and a.-. — jj7=.+ o. . ■... ,= 0, QX QY ' svaMU sin if F ' loith the two equivalent relations derived from them hy alternation^ XP ^ XQ ^ J sini7i ^ sm UM YP YQ ' sin VL sin VM ' mutually involve and are involved in each other. For, evidently, of the two additional relations PU . PV ^ , XL ,XM ^ the first is equivalent to each of the first two, and the second to each of the second two, of the above ; and therefore &o. 248. The two reciprocal properties of the preceding article are again evidently particular cases of the two following ; which follow readily, the first from the general property of Art. 120, respecting the central axis of any system of lines for any system of multiples, and the second from the general property of Art. 86, respecting the mean centre of any system of points for any system of multiples ; viz. — a. If on a variable line L, turning round a fixed pohit and intersecting with any system of fixed lines A, B, C, dx. at a system of variable points X, Y, Z, &c., a variable point P he taken so as to satisfy in every position the relation PX ^ PY PZ , ''-OX'^^-OY-^'-OZ + '^'- = ^' a, b, c, c&c. being any system of finite multiples, positive or nega- tive; the point P moves on a fiaxd line 7, termed the polar of the HARMONI? PROPERTIES OF THE POINT AND LINE. 55 point with respect to the system of lines A, B, C, &c. for the system of multiples a, 6, c, &c. a'. If through a variable point P, moving on a fixed line /, and connecting ivith any system of fixed points A, B, C, &c. hy a system of variable lines CT, F, W, &c., a variable line L be drawn so as to satisfy in every position tlie relation sinLU J sinZF smLW . smlU amlV smlW ' a, 6, e, (&c. being any system of finite multiples^ positive or nega- tive ; the line L turns round a fixed point 0, termed the pole of the line I with respect to the system of points A, B^ C, d;c. for the system of multiples o, J, c, &c. To prove a. From the two points and P conceiving the two systems of perpendiculars OA, OB, OC, OB, &c. and PA, PB, PC, PB, &c. let fall upon the system of lines A, B, C, B, &c. ; then since, by (Euc. vi. 4), PX_PA PY^PB PZPC 0X~ OA' 0Y~ OB' 0Z~ 00' therefore, by the relation determining the position of P on L, PA ^ PB PG , PB ^ ''■OA^^-OB + '-OG + '^-OB-^^''-^^' from which, as it folkws, by (120), that the point P lies on the central axis I of th" 'system of lines A, B, C, B, &c. for the system of multiples u,-v- OA, b-i-OB, c-i-OC, d -i-OB, &c., there- fore &c. To prove a'. From the system of points A, B, C, B, &c. conceiving the two systems of perpendiculars AI, BI, CI, BI, &c. and AL, BL, CL, BL, &c. let fall upon the two lines /and L; then since, by (61), BmLU_AL smLV _BL sinLW _ CL inU~ZIl' smIV~ BI' BinlW CI' ' therefore, by the relation determining the direction of i throughP, AL , BL CL , BL ^ g a.j;j + h.^j+c.-^j + d.^ + &c.=0, from which, as it follows, by (86), that the line L passes through 56 HARMONIC PROPERTIES OF THE POIXT AND LINE. the mean centre of the system of points A, B, C, X>, &c. for the system of multiples a-i- AI, h^BI, c-i-CI, d-i- BI, &c., therefore &c. Cor. 1°. It having been shewn in the demonstrations just given, that — b. The polar of a point with respect to any system of lines A,B, C, D, &c. for any system of multiples a, 6, c, c?, dsc. is the central axis (120) of the system of lines for the system of multiples a^OA,b-rOB,c-rOG,d^ OD, &c. v. The pole of a line I with respect to any system of points A., B, G, B, &c. for any system of multiples a, 6, c, rf, &;c. is ike mean centre (86) of the system of points for the system of multiples a^AI,b-i-BI,c-^ CI, d-irBI, &c. It follows, consequently, that the two general problems: " To determine" a. " the polar of a given point with respect to a given system of lines for a given system of multiples ;" a. " the pole of a given line with respect to a given system of points for a given system of multiples ;" are reduced at once to the two: "To determine" h. "the centi-al axis of a given system of lines for a given system of multiples ;" b'. " the mean centre of a given system of points for a given system of mul- tiples ;" constructions for which in their most general forms have been already given in articles (120) and (92). Cor. 2°. In the particular case when the fixed point 0, in property a, is at infinity in any direction ; since then, whatever be the position of the variable line L passing through it, the several ratios OX : OY: OZ: &c. = 1 ; therefore, for the variable point P, by the relation determining its position on L, a.PX+h.PY+c.PZ-{-&c. = 0. And, in the particular case, when the fixed line i, in property a', is the line at infinity; since then, whatever be the posi- tion of the variable point F lying upon it, the several ratios AI : BI : CI : &c. all = ] ; therefore, for the variable line i, by the relation determining its direction through P, a.AL + b.BL + c.CL + &c. = 0. Hence by (126) and (86) it appears that — r. The polar with respect to any system of lines, for any HARMONIC PROPERTIES OF THE POINT AND LINE. 57 system of multiples^ of a point at infinity^ is the diameter of the system of lines^ for the system of multiples^ corresponding to the direction of the point. c. The pole with respect to any system of points, for any system of multiples, of the line at infinity, is the mean centre of the system of points, for the system of multiples. Cor. 3°. In the particular case when the several fixed lines A, B, C, D, &c., in property a, pass through a common point P; since then, for the particular line L passing through the two points and P, the several segments PA', FY, PZ, &c. all = ; thei'efore the point P, on that line L passing through 0, satisfies, for every system of finite multiples a, h, c, d, &c., the relation PX PY PZ ''•m+^-oY^'-oz-^^^-=^' ■ and consequently lies on the polar of the point with respect to the system of lines A, B, G, D, &c. for the system of multiples a, h, c, d, &c. And, in the particular case, when the several fixed points A, B, C, D, &c., in property «', lie on a common line L ; since then, for the particular point P lying on the two lines / and L, the sines of the several angles L U, L V, L W, &c. all = ; therefore the line L, through that point P lying on /, satisfies, for every system of finite multiples a, b, c, d, &c., the relation sinLU , siniF siniTF „ a.-. — fjf+b.-. — fj^+c. -. — i^=+&c. = 0, smlll am IV smlW ' and consequently passes through the pole of the line / with re- spect to the system of points A, B, C, D, &c. for the system of multiples a, h, c, d, &c. Hence it appears that — d. For a concurrent system of lines, the polar of every point, for every system of finite multiples, passes through the point of concurrence. d'. For a collinear system of points, the pole of every line, for every system of finite multiples, lies on the line of col- linearity. Cor. 4°. In the particular case when the several multiples a, h, c, d, &c. each = 1 ; it may be easily shewn that in every posi- tion of the variable line L, in property a, the distance OP is the harmonic mean of the several distances OX, OY, OZ, &c. (233, 8°.). 58 HARMONIC PKOPEETIES OF THE POINT AND LINE. For, in the relation determining the position of P on Zr, substituting for the several distances PX, PY, PZ^ &c. their equivalents OX- OP, OY-OP, OZ-OP, &c, [To), and divid- ing by the interval OP, there results immediately the relation a b c „ a + b + c + &c. OX^ OY^ OZ^ OP ' which, when a = b = c = d, &c. = 1, becomes OX'^ OY'^ OZ^ OP' 11 being the number of multiples; and therefore &c. (233, 8°.). N.B. It is this latter case (that, viz. in which the several multiples a, &, c, les, any arbitrary point, I the polar of with respect to the three lines for the three multiples, and F, Q, B the three intersections of 1 with A, B, C respectively ; then always ^FB ^ FG ^ QC ^ QA ^ RA , RB ^ ^•OB^''-OG^''^ ''■OC+''-OA = ^' ''■OA + ^-OB = ^' a. If A, B, G be any three points, a, b, c any three corre- sponding muUip)les, I any arbitrary line, the pole of I with respect to the three points for the three multiples, and L, M, N the three connectors of with A, B, C respectively ; then always . BL ^ GL ^ GM , AM „ AN ^ BN ^-m+'-'c-r^^ '■-ci-^''-Ai=^^ ''■Al^^-BI=^- HARMONIcf PROPERTIES OF THE POINT AND LINE. 61 For, in the case of a, since for every three points P, Q, R on the line /, by property a of the preceding article (248), PA , PB PC QA , QB QC a RA RB RC _ ■ n'A'^"- nn'^^- TTn-^t OA^ OB'^^- 00 therefore for the three particular points P, Q, R on it for which respectively PA = 0, QB=0, RC=0 the above relations are true ; and therefore &c. And, in the case of a, since for every three lines L, 31, N through the point 0, by property a of the preceding article (248), AL , BL CL ^ ''■-AT^^-B-I+'-^=''^ AM ^ BM CM AN ^ BN ON "-AI + ^-BT-^'-CI^"^ therefore for the three particular lines L, M, N through it for which respectively AL = 0, BM= 0, CN= the above relations are true ; and therefore &c. Cor. 1°. It is evident, from the above relations, that, in a, the three points P, Q, R connect with the three BO, OA, AB hy the three polars of the point with respect to the three pairs of lines B and 0, and A, A and B for the three p>a,irs of multiples b and c, c and a, a and b respectively (247, a) ; and, that. In a', the three lirtes L, 71/, N intersect tcith the three BO, CA, AB at the three poles of the line I with respect to the three pairs of points B and C, and A, A and B for the th-ee pairs of multiples b and c, c and a, a and b respectively (247, a!). Properties which, for any given system of three lines or points A, B, C, supply obvious and rapid constructions for determining, for any given system of corresponding multiples «, i, r, the polar / of any given point with respect to the former, or the pole of any given line /with respect to the latter. 62 HARMONIC PROPEKTIES OF THE POINT AND LINE. Cor. 2°. In the particular case when a = i = c = 1 , it is evident, from Cor. 1°, that, in a, the three points P, Q, It on the three lines A, £, C are conjugates to the point with respect to the three opiJOsite angles BC, CA, AB of the triangle determined by the lines (217)/ and that, in a', the three lines L, Jf, N through the three points A, B, C are conjugates to the line I with respect to the three opposite sides BC, CA, AB of the triangle determined hy the points (217). It is evident again, from the same, that, the line J, in the former case, is the line of collinearity of the three points P, Q, R at which the three polars of the point with respect to the three angles intersect with the opposite sides of the triangle determined hy the lines (243, d) ; and that, the point (?, in the latter case, is the point of concurrence of the three lines i, J/, N hy which the three poles of the line I with respect to the three sides connect with the opposite angles of the triangle determined by the points (243, d). And it is evident also, from the same, that when, for any triangle ABC, a line I is the polar of a point with respect to the three sides, then, reciprocally, the point is the pole of the line 1 with respect to the three vertices, and conversely (139, Cor. 3°) ; a property which we shall presently see is true generally, not only for the particular system of multiples each = 1, but for any system of finite multiples as well. In the figure of Art. 236, the four points P, Q, B, and 8 are connected with the four lines YZX', ZXT, XYZ', and X'TZ' respectively by the above relation of being pole and polar to each other with respect to the vertices and sides of the central triangle ABC; and, in the figure of Art. 139, the point is connected with the line XYZ by the same relation of being pole and polar to each other with respect to the vertices and sides, not only of the original triangle ABC, but of the several derivatives of both species A'B'C and A,B,C^, A"B"C" and ^„5„C,„ A"'B"'C"' and ^,„5,„C„„ &c. obtained from it, through their directing agency, by the continued application of the two inverse processes of construction described in Cors, 4° and 5° of that article. 251. If ^, B, C be the three vertices, and B, E, Fxhe three opposite sides of any triangle ; and I any arbitrary point and HARMONIC*t>ROPERTIES OF THE POINT AND LINE. 63 line ; L, M, N the three connectors of with the vertices ; P, Q, R the three intersections of / with the sides; Z7, F, TFthe three connectors of P, Q, R with the vertices ; and Z, F, Z the three intersections of i, M, iVwith the sides; then — 64 HAEMONIC PROPERTIES OP THE POINT AND LINE. a. The point is the pole of the line I with respect to the three points A, B^ C for any three multiples a, b, c such that ^ BX CX ^ CY , AY ^ AZ^.BZ a. The line I is the polar of the point with respect to the three lines i), E^ F for any three multiples a, h, c such that sin DW , sm EW ^ ^•~"= — tTt^ + 0.—. — Tprr = 0. sm DN smEN For, the three relations, in the case of a, being evidently equivalent to the three ^ BL CL ^ CM AM „ AN , BN ^ ^'BI+'--CI='^^ ''•W'-*-^=^' ''■AI + ^-Bl = ^' and the three, in the case of a', to the three PE PF ^ QF QD ^ RD , BE ^ ^■OE-^'-OF^^' ''•0F^"'-6D = ^' ''•OD-^^-OE'^^' ■which being identical with those in a and a of the preceding article (250), therefore &c. Cor. 1°. Since, for the same triangle, by Cor. 4°, Art. 247, the two groups of three relations in a and a' of the above mutually involve each other for the same system of multiples a, J, c ; hence, generally, as noticed in Cor. 2° of the preceding article for the particular system of multiples each = 1. — TFAisw, for any triangle ABC, a line I is the polar of a point with respect to the three sides for any system of multiples a, i, c; then, reciprocally, the pioint is the piole of the line I with respect to the three vertices for the same system of multiples a, b, c. This property, it may be observed, would have followed also from those of Cor. 1° of the preceding article (250), combined with those of Cor. 3° of Art. 247, Cor. 2°. Since, for two triangles in perspective (140), the three lines of connection L, M, N of the three pairs of corre- sponding vertices A and A', B and B', C and C pass through a common point 0, their centre of perspective, and the three points of intersection P, Q, B of the three pairs of corresponding sides D and D', E and E', F and F lie on a common line /, HARMON IC*PEOPEETIES OF THE POINT AND LINE. 65 their axis of perspective (see figure) ; hence again, from the above relations a and a combined with the general property of Cor. 4°, Art. 247, the two following polar properties of two triangles in perspective — h. For every two triangles in perspective, the centre is the poJe of the axis of perspective with respect to the vertices of both triangles for the same system of multiples. h'. For every two triangles in perspective, the axis is the polar of the centre of perspective with respect to the sides of both tri- angles Jor the same system of multiples. For, the four lines /, L, M, N and the four points 0, P, Q, B, in the relations a and a, being the same for both triangles (see figure), if U\ V\ W and X', T, Z be for the triangle AB'C what U, V, W and X, Y, Z as above stated are for the triangle ABC; then since the three pairs of corresponding systems of four coUinear points B, C, X, F and B', C, X', P; C, A, Y, Q and C", A', Y\Q; A, B, Z, B and A', B\ Z, B are in per- spective at the point (130), therefore, by the general property, Cor. 4°, Art. 247, the three pairs of corresponding relations ^ BX CX ^ , ^ BX' CX' „ o.^+c. -^ = 0, and * • ^g^p + c . -^rp- = 0, CY AY „ , C'Y' A'Y' „ AZ ^ BZ ^ , A'Z' ^ B'Z „ "•AB + ^-BB'^'^^ ^""^"-mi^^-WB =^' (see property a) mutually involve each other for the same system of multiples a, b, c, and therefore &c. as regards b ; and since the three pairs of corresponding systems of four concurrent lines U, F, U, L and E', F, U\L; F, D, F, M and F, I)', V, M; D, F, W, JV and D', F', W, N are in perspective on the line I (130), therefore, by the same general property, Cor. 4°, Art. 247, the three pairs of corresponding relations ^ sin^fT" smFU ^ , , filnE'U' , sinF'C7' ^ 0--:— £^ +c.-. — ^TF =0, and 6.-^ — ™^+c.-. — jjvy =0, sinJFT sinI>F „ , , sini?"r , sinZ)'F' ^ c. -. — i=jT>+a--= — 77Tv = 0, and 0.—. — j^piv +c.—. — httf =0, sm FM sm DM ' &mFM smDM ' am DW , sin^TF „ , smB'W . sin^'TF' ^ sm DN sm EN ' smDiV smEM ' VOL. II. F 66 HAEMONIU PROPERTIES OF THE POINT AND LINE. (see property a') mutually involve each other for the same system of multiples a, b, c, and therefore &c. as regards b'. N.B. These two reciprocal properties, though thus estah- lished independently, evidently involve each other by virtue of Cor. 1°. CoR. 3°. It follows, of course, from the two reciprocal pro- perties of the preceding corollary, that when any number of triangles ABC, A'B'C, A"B"C", NlC PEOPEETTES OF THE CIECLE. 77 Conversely, since, in the former case, the harmonicism of any one of the four pencils of four rays A.JBXY^ B.BAXY^ X.XYAB^ Y. YXAB involves, by (252, a), that of the system of four points A^ B, X, Y on the circle ; and, since, in the latter case, the harmonicism of any one of the four rows of four points ACFQ, BOBS, XZFB, YZQ8 involves, by (252, a'), that of the system of four tangents AC, BC, XZ, YZ to the circle; therefore, the above reciprocal projjerties are criteria, the former of the harmonicism of four points on a circle, and the latter of the harmonicism of four tangents to a circle. The above demonstrations, as establishing directly the col- linearity of the two triads of points X, Y, C and A, B, Z for an harmonic system of points A, B, X, Y oa a circle, and the concurrence of the two triads of lines PR, QS, AB and PQ, MS, XYfor an harmonic system of tangents AG, BC, XZ, YZ to a circle, and conversely, establish therefore, in a manner applicable to higher figures as well, the two reciprocal proper- ties established in the preceding article by a method applicable to the circle alone. Cor. 1°. By virtue of the above, the two points C and Z (see figure) being the poles of the two lines AB and X Y with respect to the two segments XY and AB (217) ; and the two lines AB and XY being the polars of the two points C and Z with respect to the two angles XZYa.niACB (217) ; it appears, consequently, from it, that — The intersection of the two terminal tangents, and the connector of the two terminal points, of any arc of a circle, are pole and polar to each other xcith respect, at once to the segment determined by every two points on the circle which connect through the former, and to the angle determined by every two tangents to the circle which intersect on the latter. Cor. 2°. Again, the point and the line 1 (see figure) being, by the above, pole and polar to each other with respect at once to the two segments XY and AB, and to the two angles XZY and ACB; and, the three points C, Z, and 0, and the three lines AB, XY, and /, being, by (167), pole and polar to each other with respect to the circle itself; hence, again, from the above, for every point and line pole and polar to each other with respect to a circle, it appears that — 78 HARMONIC PROPERTIES OF THE CIRCLE. Every point and line, pole and polar to each other with respect to a circle, are also pole and polar to each other with respect, at once to the segment determined by every two points on the circle which connect through the former, and to the angle determined hy every two tangents to the circle which intersect on tfie latter, COK. 3°. The triangle determined by the three points C, Z, and 0, or by the three lines AB, XY, and 7, (see figure) being self-reciprocal with respect to the circle (168), each vertex and its opposite side being pole and polar to each other with respect to the circle ; hence, also, from the above, see Cor. 2° of the preceding article. In every triangle self-reciprocal with respect to a circle, the circle divides harmonically the three sides, and subtends harmoni- cally the three vertices ; and, conversely, the circle which divides harmonically the three sides or subtends harmonically the three vertices of a triangle is the polar circle of the triangle (168). 259. Of the various reciprocal properties of points and lines, pole and polar to each other with respect to a circle, the two following, termed their harmonic properties, and obviously tantamount to those just stated in Cor. 2° of the preceding article, are second only in importance to those of Art. 166, and lead, next to them, to the greatest number and variety of re- markable consequences in the modem geometry of the circle : — a. Every two conjugate points with respect to a circle are harmonic conjugates with respect to the two collinear points on the circle y and, conversely, every two points harmonic conjugates with respect to tJie two collinear points on a circle are conjugate points with respect to the circle (174). a'. Every two conjugate lines with respect to a circle are har- monic conjugates with respect to the two concurrent tangents to the circle ; and, conversely, every two lines harmonic conjugates with respect to the two concurrent tangents to a circle are conjugate lines with respect to the circle (174). These properties follow immediately, indirectly, from those of the preceding article ; the two points and C (see figure of that article) being at once conjugate points with respect to the circle, and harmonic conjugates with respect to the two collinear points X and Y on the circle, and the two lines ZO and ZC HARM^IC PKOPERTIES OF THE CIRCLE. 79 being at once conjugate lines with respect to the circle and har- monic conjugates with respect to the two concurrent tangents ZX and ZY to the circle ; the two points and Z being at once conjugate points with respect to the circle, and harmonic conjugates with respect to the two collinear points A and B on the circle, and the two lines CO and CZ being at once conjugate lines with respect to the circle and harmonic conjugates with respect to the two concurrent tangents CA and GB to the circle ; the two points C and Z being at once conjugate points with respect to the circle and harmonic conjugates with respect to the two imaginary collinear points on the circle, and the two lines OC and OZ being at once conjugate lines with respect to the circle, and harmonic conjugates with respect to the two imaginary concurrent tangents to the circle ; and therefore &c. From their importance, however, we subjoin the ordinary direct demonstrations of them, based on the fundamental de- finition of poles and polars with respect to the circle given in Art. 165. If, as regards (a), Pand Q be any two points, X and F the two collinear points on any circle C, and B the inverse of either of them P with respect to the circle; then since, by (221), the harmonicism of the row of four points PQXY involves and is involved in that of the pencil of four rays B.PQXY, and since, by (216, 3°), for the harmonicism of the pencil, the ray ^P being always equally inclined to the two rays BXxaA. BY (163), it is necessary and sufficient that the conjugate ray BQ be at right angles to the ray BP, that is, that the point Q be 80 HARMONIC PEOPEKTIES OF THE CIRCLE. on the polar of tlie point P with respect to the circle (165), therefore &c. And, if, as regards (a'), ZP and ZQ be any two lines, ZX and ZY the two concurrent tangents to the circle, and P and Q the two points at which their chord of contact XY intersects with ZP and ZQ; then since, by (221), the harmonicism of the pencil of four lines Z.PQXY involves and is involved in that of the row of four points PQXY, and since, by (175, 5°), P and Q are conjugate points when ZP and ZQ are conjugate lines with respect to the circle, and conversely, therefore &c., the rest being evident from (a). From these properties, thus, or in any other manner, inde- pendently established, those of the preceding article, with all the consequences to which they lead, follow of course indirectly ; both pairs of reciprocal properties, as above shewn, being, in fact, virtually identical. Cor. r. From the first parts of the above, by virtue of the properties (225) and (235, Cor. 7°), it is evident that (see Art. 176)— a. Every two conjugate points with respect to a circle de- termine with the polar centre of their line of connection two seg- ments^ whose product is constant and equal in magnitude and sign to the square of the semi-chord intercepted by the circle on the line. a. Every two conjugate lines with respect to a circle determine with the polar axis of their point of intersection two angles, the product of whose tangents is constant and equal in magnitude and sign to the square of that of the semi-angle subtended by the circle at the point. CoE. 2°. By vii-tue of the general property of Art. 218, it is evident, also, from the same, that (see Art. 178) — a. When a line intersects one of two circles at a pair of con- jugate points with respect to the other, then, reciprocally, it intersects the latter at a pair of conjugate points with respect to the former. a'. When a point subtends one of two circles by a pair of conjugate lines with respect to the other, then, reciprocally, it subtends the latter by a pair of conjugate lines with respect to the former. haemOnic properties of the circle. 81 Cor. 3°. For the particular case when the circles, in the preceding corollary, intersect at right angles, from the same again, by virtue of properties/ and/. Art.. 208, it appears that — a. Every line intersecting two orthogonal circles in an har- monic system of points passes through one or other common pole of one axis of perspective with respect to one circle and of the other axis of perspective with respect to the other circle (208, e.) a. Every point subtending two orthogonal circles in an har- monic system of tangents lies on one or other common polar of one centre of perspective with respect to one circle and of the other centre of perspective with respect to the other circle (208, e.) Cor. 4°. By aid of the solutions (227, Cor. 3°) and (257, Cor. 5°), the second parts of the above supply obvious solutions of the four following problems, viz. — a. On a given line to determine two points conjugates to a given circle and either separated by a given interval or having a given middle point. a'. At a given point to determine two lines conjugates to a given circle and either separated by a given interval or having a given middle line. And by aid of the solutions (230) and (257, Cor, 4°), the same, again, supply obvious solutions of the two following problems — b. On a given line to determine the pair of points conjugates at once to two given circles. b'. At a given point to determine the pair of lines conjugates at once to two given circles. 260. The line at infinity being the polar of any point with respect to any circle having its centre at the point (165), and the points of intersection of any circle with any line being the points of contact of the tangents to the circle from the pole of the line (165) ; the following remarkable consequences result from the reciprocal properties of the preceding ai-ticle, applied to the particular cases of conjugate points at infinity, and of con- jugate lines through the centres of circles : 1°. Every two points at infinity in directions at right angles to each other being conjugate points with respect to every circle (174), and every two lines through any point at VOL. IT. G 82 HARMONIC PROPEKTIES OF THE CIRCLE. right angles to each other being conjugate lines with respect to every circle having its centre at the point; hence, from pro- perties a and a! of the preceding article, respectively — a. Every two points at infinity in directions at right angles to each other are harmonic conjugates with respect to the two imaginary points at which any circle^ however situated^ intersects with the line at infinity. a'. Every two lines through any point at right angles to each other are harmonic conjugates with respect to the two imaginary tangents from the point to any circle having its centre at the point. 2°. If 0, 0', 0", &c. be the several centres of any number of circles situated in any manner; Xand Y, X' and F', X" and F", &c., the several pairs of imaginary points at which they in- tersect with the line at infinity ; and, P and Q, R and 8 a,nj two pairs of points at infinity in directions at right angles to each other; then since, by the same, the several segments XY, X'Y', X"F", &c. divide harmonically the same two segments PQ and ES, therefore (230) they coincide with each other ; and since, by the same again, the several angles XOY, X'0'Y\ X"0"Y", &c. divide harmonically the several pairs of parallel angles POQ and BOS, PO'Q and BO'S,PO"Q and BO"S, &c. therefore (230) they are parallel to each other ; consequently — h. All circles J however situated, intersect with the line at infinity at the same pair of imaginary points, termed the two circular points at infinity. b'. All circles, however situated, subtend at their several centres pairs of imaginary tangents parallel to the directions of the two circular points at infinity. 3°. In the particular case where the several points 0, 0', 0", &c. coincide, that is, when the several circles are concentric; since then not only the several segments XY, X'T, X"Y", &c. but also the several angles XOY, X'OT, X"OY", &c. coincide, and since, consequently, the several circles have not only a common pair of imaginary points X and Y at infinity, but also a common pair of imaginary tangents OX and OF at those points, therefore — All concentric circles not only intersect but also touch at the two circular points at infinity. HARMONIC PROPERTIES OF THE CIRCLE. 83 4°. The following property of rectangular lines, which is one of considerable importance in the higher branches of geometry, is evident from the preceding properties 1° and 2° combined, viz. — Every two lines intersecting at right angles are conjugate lines with respect to the two circular points at infinity ; and^ conversely^ the two circular points at infinity are conjugate points with respect to every two lines intersecting at right angles (217). Paradoxical as the above conclusions 2° and 3°, like those of Art. 136, always appear when first stated, all doubt of their legitimacy soon vanishes on consideration of their meaning; every system of figures, in perspective two and two, which, like circles however situated, have a common axis of perspective, intersecting (141) at the same system of points, real or imaginary, on their axis of perspective, and touching (20) at that system of points, if, like concentric circles, they have also a second common axis of perspective coinciding with the first (181, 4°. and 207). 261. The two following reciprocal properties, one of every tetrastigm determined by four points on a circle, and the other of every tetragram determined by four tangents to a circle, result also immediately from the two reciprocal properties of Art. 259, viz. — a. In every tetrastigm determined hy four points on a circle, the intersections of the three pairs of opposite connectors determine a self reciprocal triangle with respect to the circle (170). a. In every tetragram determined hy four tangents to a circle, the connectors of the three pairs of opposite intersections determine a self reciprocal triangle with respect to the circle (170). To prove (a). If P, Q, B,, S be the four points on the circle ; A,£,C the three intersections of their three pairs of opposite g2 84 HARMONIC PROPERTIES OP THE OIRCLE. connectors QB and P5, RP and Q8, PQ and ^5; and, £7" and ZT, V and V, W and W the three pairs of intersections of the same pairs of connectors with the three opposite sides BC, CA, ABoi the triangle ABC; then since, by the fundamental property [a) of Art. 236, the three pencils of four rays A.BCVU', B.CAVV, C.ABWW are harmonic, and since, consequently, by (221), the six rows of four points QRAU&oAPSAU', RPBV&ad QSBV, PQC IF and RSGW are harmonic, therefore, by the property (a) of Art. 259, the three pairs of points U and U\ V and F', Wand W lie on the three polar s of the three points A, B, C with respect to the circle; and therefore &c. (170). To prove a. If Z and X', Zand Y', Zand Z' be the three pairs of opposite intersections of the tetragram determined by the four tangents at the four points P, Q^ R^ S on the circle ; and A, B, C the three vertices of the triangle determined by their three lines of connection XX\ YY', ZZ' ; then since, by the fundamental property (a) of Art. 236, the three rows of four points BCXX\ CA YY\ ABZZ' are harmonic, and since, consequently, by (221), the six pencils of four rays X.QRAX' and X'.PSAX, Y.RPBY' and Y'.QSBY, Z.PQCZ' and Z'.BSCZ are harmonic, therefore, by the property {a) of Art. 259, the three pairs of lines XA and X'A, YB and Y'B, ZG and ZC pass through the three poles of the three lines XX"', YY', ZZ' with respect to the circle ; and therefore &c. (170). The reader will perceive immediately, that not only are the above properties reciprocals to their common circle in the figure, but that the demonstrations above given of them are reciprocals to it also. Cor. r. Since, in the former case, by (166), the four tangents to the circle at the four points P, Q, R, S intersect two and two in opposite pairs on the polars of the three points A, B, C with respect to the circle; and since, in the latter case, by (166), the four points of contact with the circle P, Q, R, S of the four tangents connect two and two in opposite pairs through the poles of the three lines BC, CA, AB with respect to the circle ; hence, from the above propeities a and a com- bined, it appears that — In the tetrastigm detei-mined by any four points on a circle, and in the tetragram determined hy the four corresponding tangents HARMONIC PROPERTIES OF THE CIRCLE. 85 to the circle, or conversely; the two self-reciprocal triangles de- termined hy ike vertices of the three angles of connection, in the former case, and hy the axes of the three chords of intersection, in the latter case, are identical. Cor. 2°. Again, from the harmonicism of the three pencils of four rays A.BCUU', B.GAVV, G.ABWW, with that of the several rows they determine on all axes, in the former case, and of that of the three rows of four points BCXX', CA YY', ABZZ', with that of the several pencils they determine at all vertices, in the latter case ; it appears from the same that— In the tetrastigm determined hy any four points on a circle, and in the tetragram determined by the four corresponding tangents to the, circle, or conversely — a. The three pairs of opposite connectors of the former divide harmonically the three angles of the triangle determined by the axes of the three chords of intersection of the latter. a. The three pairs of opposite intersections of the latter divide harmonically the three sides of the triangle determined by the vertices of the three angles of connection of the former. Cor. 3°. Again, the concurrence of the four triads of lines PA, PB, PC; QA, QB, QC; RA, SB, EC; SA, SB, 8C in- volving, by (243, a), the coUinearlty of the four triads of points U, V, W; V, W, U'; W, U', V ; U, V, W in the former case ; and the coUinearity of the four triads of points Y, Z, X' ; Z, X, Y' ; X, Y, Z' ; X', Y', Z Involving, by (243, a), the concurrence of the four triads of lines AX, BY', CZ' ; BY, CZ', AX'; CZ, AX', BY'; AX, BY, CZ in the latter case; it appears from the same that — In the tetrastigm determined hy any four points on a circle, and in the tetragram determined by the four corresponding tangents to ike circle, or conversely — a. The three pairs of opposite connectors of the former in- tersect with the axes of the three chords of intersection of the latter at six points lying three and three on four lines. a'. The three pairs of opposite intersections of the latter connect with the vertices of the three angles of connection of the former by six lines parsing three and three through four points. Cob. 4°. Again, as the four points P, Q, R, S on the circle. 86 HAEMOJJIC PROPEETrES OF THE CIECLE. taken in diiFerent orders, determine the three dlflferent inscribed quadrilaterals whose pairs of opposite vertices connect by the three pairs of lines QR and P/S, iZPand QS, PQ and RS; and as the four tangents at them to the circle, taken in different orders, determine the three corresponding exscribed quadri- laterals whose pairs of opposite sides intersect at the three pairs of points X and X', Y and Y\ Z and Z' ; it appears also from the same that — In each pair of corresponding quadrilaterals determined hy any four points mi a circle taken in any order and hy the four cor- responding tangents to the circle taken in the same order — a. The two pairs of intersections of opposite sides are collinear and harmonic. a!. The two pairs of connectors of opposite vertices are con- current and harmonic. CoE. 5°. Again, the three pairs of points B and (7, C and A, A and B being the three pairs of centres of perspective of the three pairs of opposite segments QR and P8, RP and QS, PQ and R8^ in the former case ; and the three pairs of lines YY' and ZZ\ ZZ' and XX', XX' and YY' being the three pairs of axes of perspective of the three pairs of opposite angles QXR and PX'S, RYP and QY'S, PZQ and RZ'S, in the latter case ; it appears also from the same that — a. The two centres of perspective of any two chords inscribed to a circle are conjugate pioints with respect to the circle, and connect harmonically with the intersection of the axes of the chords hy a pair of conjugate lines with respect to the circle. a. The two axes of perspective of any two angles exscribed to a cirde are conjugate lines with respect to the circle, and intersect harmonically with the connector of the vertices of the angles at a pair of conjugate points with respect to the circle. From these latter properties it may be easily shewn con- versely that — h. When the directions of any two chords inscribed to a circle divide harmonically the angle determined hy any two conjugate lines with respect to the circle, the two centres of perspective of the chords are the two poles of the lines. b'. When the vertices of any two angles exscribed to a circle divide harmonically the segment determined hy any two conjugate HARMONIC PROPERTIES OF THE {JIUCLE. 87 points with respect to the circle, the two axes of perspective of the angles are the two polars of the points. For, in the former case, if QB and PS be the two chords ; AB and A C the two conjugate lines ; B the pole of either of them A G with respect to the circle ; BQ and BR Its connectors with the extremities of either chord QR; and P'S' the con- nector of the two second intersections of BR and BQ with the circle; then since, by the above (a), the two lines QR and P'S' pass through the point A and divide harmonically the angle BA C, and since, by hypothesis, the two lines QR and PS do the same, therefore the two lines PS and P'S' coincide ; and therefore &c. And, in the latter case, if QXR and PX'S be the two angles ; B and C the two conjugate points ; A C the polar of either of them B with respect to the circle ; Y and Y' its intersections with the sides of either angle QXR ; and X" the intersection of the two second tangents from Y and Y' to the circle ; then since, by the above {a), the two points X and X" lie on the line BG and divide harmonically the segment BG, and since, by hypo- thesis, the two points X and X' do the same, therefore the two points X' and X" coincide ; and therefore &c. It is evident, from these latter properties, that for every tri- angle self-reciprocal with respect to a circle, an infinite number of tetrastigms could be inscribed to the circle ivhose pairs of opposite points would connect through the vertices of the triangle^ and an infinite number of tetragrams could be exscribed to the circle whose pairs of opposite lines would intersect on the sides of the ti-iangle. For, by those properties, every pair of lines dividing harmonically any angle of the triangle would determine four points on the circle fulfilling the former condition, and every pair of points dividing harmonically any side of the triangle would determine four tangents to the circle fulfilling the latter condition ; and therefore &c. CoE. 6°. Again, the three pairs of points X and X', Y and Y', Z and Z', and the three pairs of lines QR and PS, RP and QS, PQ and RS being pole and polar to each other with respect to the circle; therefore, from the harmonicism of the three rows of four points BCXX', CAYY', ABZZ', and of the three pencils of four rays A.BGUU', B.CAVV, C.ABWW, it appears from the same, as in Cors. 2° and 4°, that — 88 HARMONIC PEOFEKTIES OF THE CIRCLE. a. For every two chords inscribed to a circle, the two poles of their directions are collinear toithj and harmonic conjugates with respect to, their two centres of perspective. a'. For every two angles exscribed to a circle, the two polars of their vertices are concurrent with, and harmonic conjugates with respect to, their two axes of perspective. Cor. 7°. Again, since, by the second part of Art. 257, the three lines BG, CA, AB determine the three pairs of points on the circle, and the three points A, B, G determine the three pairs of tangents to the circle, which divide harmonically the three pairs of arcs QR and PS, RP and QS, PQ and RS; it appears also, from the same, that — a. The two centres of perspective of any two chords inscribed to a circle are collinear with the two points on the circle which divide harmonic-ally the two arcs intercepted by the chords. a'. The- two axes of perspective of any two angles exscribed to a circle are concurrent with the two tangents to the circle which divide harmonically the two arcs intercepted by the angles. Of these latter properties the first (a) supplies an obvious and very rapid method of determining by linear constructions only, without the aid of a circle, the two points on a given circle which divide two given arcs of it harmonically. See Arts. 230 and 257, Cor. 4°. Cor. 8°. Again, every three of the four points P, Q, R, 8 on the circle, in the former case, determining an inscribed tri- angle whose three sides pass through the three points A, B, G, every two of which are conjugates to each other and to the third with respect to the circle; and every three of the four tangents at the four points P, Q, R, S to the circle, in the latter case, determining an exscribed triangle whose three vertices lie on the three lines BG, GA, AB, every two of which are con- jugates to each other and to the third with respect to the circle ; it appears also, from the same, that — a. In every triangle inscribed to a circle, every two of the three sides intersect with every line conjugate to the third at a pair of conjugate points with respect to the circle. a'. In every triangle exscribed to a circle, every two of the three vertices connect with every point conjugate to the third by a pair of conjugate lines with respect to the circle. HARMONIC PEOPEETIES OP THE CIRCLE. 89 From these latter properties it may be easily shewn that conversely — b. When^ of a triangle inscribed to a circle^ two of the three sides pass through a pair of conjugate points with respect to the circle, the third passes through the pole of their line of connection. v. When, of a triangle exscribed to a circle, two of the three vertices lie on a pair of conjugate lines with respect to the circle, the third lies on the polar of their point of intersection. For, in the former case, if PQR be the inscribed triangle whose two sides PR and PQ pass through the two conjugate points B and C with respect to the circle ; A the pole of the line BC; and V and W the two points at which the two lines AG and AB, which, by (175, 5°), are the polars of the two points B and G, intersect with the aforesaid sides PR and PQ of the triangle; then, the two rows of four points PRBV and PQGW, having the common point P, being harmonic (259, a), therefore, by (222, 5°), the three lines BW, GV, and QR are concurrent ; and therefore &c. And, in the latter case, if XYZ be the exscribed triangle whose two vertices Y and Z lie on the two conjugate lines AC and AB with respect to the circle; BG the polar of the point A ; and BY and GZ the two lines by which the two points B and C, which, by (175, 5°), are the polars of the two lines A G and AB, connect with the aforesaid vertices Y and Z of the triangle ; then, the two pencils of four rays Y.ZXBG and Z. YXBG, having the common ray YZ, being harmonic (259, a'), therefore, by (222, 6°), the three points B, C, and X are colllnear ; and therefore &c. Cob. 9°. If, while the two triangles PQR and XYZ deter- mined by any three of the four points on the circle, in the former case, and by the corresponding three of the four tangents to the circle, in the latter case, with the circle to which they are respectively inscribed and exscribed, are supposed to remain fixed ; the triangle ABG, connected with them as above, be con- ceived to vary, in consequence of the simultaneous variation of the fourth point S and of the corresponding tangent X' Y'Z', on which, in that case, it of course depends; then since, by the above, the triangle ABG in every position is self-reciprocal with respect to the circle ; it appears, consequently, that — For every two triangles determined by any three points on a 90 HARMONIC PROPEKTIES OP THE CIRCLE. Circle and by the three corresponding tangents to the circle,^ or conversely, an infinite nuinber of triangles could he constructed, at once inscribed to the former and exscribed to ike latter, and all self-reciprocal with respect to the circle. It is evident, from this latter, that the solutions of the two reciprocal problems, "for a given circle to determine a self- reciprocal triangle either inscribed to any triangle inscribed to itself or exscribed to any triangle exscribed to itself," are both indeterminate. Cor. 10°. If, on the other hand, while the triangle ABC, with the circle to which it is self-recipi'ocal, are supposed to remain fixed ; the two triangles PQR and XYZ, connected with them as above, be conceived to vary simultaneously, in con- sequence of the simultaneous variation of the point 8 and of the tangent X'Y'Z', on which, in that case, they of course depend ; then since, by the above, the two triangles PQR and XYZ respectively inscribed and exscribed to the circle are re- spectively exscribed and inscribed to the triangle ABC; it appears, consequently, that — For every triangle self-reciprocal with respect to a circle, an infinite number of triangles could be constructed at once inscribed to the circle and exscribed to the triangle ; and, also, an infinite number of corresponding triangles at once exscribed to the circle and inscribed to tlie triangle. It is evident, from this latter, that the solutions of the two reciprocal problems, " for a given circle to determine either an inscribed triangle exscribed to, or an exscribed triangle in- scribed to, any self-reciprocal triangle with respect to itself," are both indeterminate. Cor, 11°. Of the three triangles PQR, XYZ, and ABC, thus constituting in every position a cycle in which each triangle is inscribed to one and exscribed to the other of the remaining two; the two first being in perspective, with the third by virtue of their relations of connexion with it, and with each other by virtue of the general property 1° of Art. 180; it appears consequently that — In every cycle of three triangles determined by any arbitrary triangle, any exscribed triangle inscribed to its polar circle, and HARM0l5lC PROPERTIES OF THE CIRCLE. 91 the corresponding inscribed triangle exscribed to its polar circle, evert/ two of the three are in perspective (140). It will be shewn, in another chapter, that for every cycle of three triangles, however originating, in which, as above, each triangle is inscribed to one and exscribed to the other of the remaining two, when any two of the three are in perspective every two of the three are in perspective. Cor. 12°. The centre of perspective S of the two triangles PQR and ABC being a point on the circle, and the axis of perspective X' Y'Z' of the two XYZ and ABG being a tan- gent to the circle ; it appears consequently also that — a. The centre of perspective of any triangle, with any ex- scribed triangle inscribed to its polar circle, is a point on the circle. o'. The axis of perspective of any triangle, with any in- scribed triangle exscribed to its polar circle, is a tangent to the circle. It is evident that when, for the same original triangle, the two derived triangles in those properties correspond, the point on and tangent to the polar circle correspond also. Cor. 13°. If be the point of concurrence of the three lines of connection PX, QY, RZ of the three pairs of corresponding vertices P and X, Q and Y, R and Z, and / the line of col- linearity of the three points of intersection P, Q', R' of the three pairs of corresponding sides QR and YZ, RP and ZX, PQ and XY, of the two triangles PQR and XYZ; then, from the harmonicism of the three pencils of four rays P.QRXX', Q.RPYY', R.PQZZ' (236, a), and consequently (221) of the two rows of four points determined by any two of them on the two non-corresponding sides of the triangle ABG through whose intersection their two vertices connect, which two rows have that intersection for a common point; therefore, by (222, 5°), the point 0, that is, the centre of perspective of the two tri- angles PQR and XYZ, is coUinear with the three points U, V, W, that is, with the axis of perspective of the two tri- angles PQR and ABC; and, from the harmonicism of the three rows of four points YZPP, ZXQQ, XYRE (236, o'), and consequently (221) of the two pencils of four rays deter- 92 HARMONIC PROPERTIES OF THE CIRCLE. mined by any two of them at the two non-corresponding vertices of the triangle ABC on whose connector their two axes intersect, which two pencils have that connector for a common ray ; therefore, by (222, 6°), the line /, that is, the axis of per- spective of the two triangles PQR and XYZ, is concurrent with the three lines AX, BY, CZ, that is, with the centre of perspective of the two triangles XYZ and ABC; hence it appears that — In every cycle of three triangles determined hy any arhitrary triangle, any exscribed triangle inscribed to its polar circle, and the corresponding inscribed triangle exscribed to its polar circle. a. The centre of perspective of the second and third lies on the axis of perspective of the first and second. a The axis of perspective of the second and third passes through the centre of perspective of the first and third. The centre of perspective S of the two triangles PQR and ABC lying also, evidently, on the axis of perspective X'Y'Z' of the two triangles XYZ and ABC; these properties for the whole three triangles may consequently be stated more sym- metrically as follows : — In every cycle of three triangles determined hy any arhitrary triangle, any exscribed triangle inscribed to its polar circle, and the corresponding inscribed triangle exscribed to its polar circle ; the centre of perspective of each with that to which it is inscribed lies on its axis of perspective with that to which it is exscribed. It will be seen, in another chapter, that this latter property is true generally of every cycle of three triangles, each inscribed to one and exscribed to the other of the remaining two and in perspective with either and consequently with both. Cor. 14°. Since, from the barmonicism of the three rows of four points QRAU, RPBV, PQCW, the line of coUinearity of the three points U, V, W is the polar of the point of con- currence of the three lines AP, BQ, CR with respect to the three sides of the triangle PQR (250, Cor. 2°) ; and since, from the barmonicism of the three pencils of four rays X. YZAX', Y.ZXBY, Z.XYCZ', the point of concurrence of the three lines AX, BY, CZ is the pole of the line of coUinearity of the three points X, Y', Z' with respect to the three vertices of the triangle XYZ (250, Cor. 2°) ; hence, conceiving the point 8 and HARMOlftc PROPERTIES OF THE CIRCLE. 93 the line X'Y'Z' to vary while the two triangles PQR and XYZ remain fixed, it follows from the two reciprocal properties a and a of the preceding corollary (12°), that — a. If a variable point describe a fixed circle, its polar with respect to the three sides of any inscribed triangle turns round a fixed pointy the centre of perspective of the inscribed with the cor- respondinrf exscribed triangle. a. If a variable line envelope a fixed circle, its pole with re- spect to the three vertices of any exscribed triangle moves upon a fixed line, the axis of perspective of the exscribed with the corre- sponding inscribed triangle. In the particular case when both triangles are equilateral, their centre and axis of perspective, in all cases evidently pole and polar to each other with respect to the circle, being then the centre of the circle and the line at infinity (142), it follows from the converses of the preceding properties a and «', that — b. If a variable line turn round a fixed point, its pole with respect to the three vertices of any equilateral triangle concentric with, the point describes the circle circumscribed to the triangle. b'. If a variable point describe tlie line at infinity, its polar with respect to the three sides of any equilateral triangle envelopes the circle inscribed to the triangle. The polar of a point at infinity, with respect to any system of lines, being the diameter, corresponding to its direction, of the polygram determined by the lines (248, c) ; this latter pro- perty b' is therefore identical with that stated in the concluding paragraph of Art. 126, viz., that in every equilateral triangle the several diameters of the figure envelope its inscribed circle. Cor. 15°. Again, since from the harmonicism of the three rows of four points BCUU\ CAW, ABWW, the line of coUinearity of the three points U, F, W is the polar of the point of concurrence of the three lines AU', BV, CW with respect to the three sides of the triangle ABC; and since, from the harmonicism of the three rows of four points BCXX', CAYY', ABZZ', the point of concurrence of the three lines AX, BY, CZ is the pole of the line of coUinearity of the three points X', Y', Z' with respect to the three vertices of the triangle ABC; hence, from the same, again, it appears also, that — 94 HARMONIC PROPERTIES OF THE CIRCLE. a. The pole, with respect to tJie three vertices of any triangle^ of its axis of perspective with any exscribed triangle inscribed to its polar circle, is a point on the circle. a. The polar, with respect to the three sides of any triangle, of its centre of perspective vnth any inscribed triangle exscribed to its polar circle, is a tangent to the circle. It is evident, as in Cor. 12°, that when, for the same original triangle, the two derived triangles In those properties corre- spond, the point ou and tangent to the polar circle corre- spond also. 262. From the two reciprocal properties of the preceding article, combined with the general property (180, 1°) that every two triangles reciprocal polars to each other with respect to a circle are in perspective, the two following reciprocal properties of a triangle with respect to an arbitrary circle can be readily inferred, viz. — a. The three angles, subtended at the vertices of a triangle by the three pairs of intersections of its opposite sides with an arbitrary circle, determine three second pairs of intersections leith the circle whose connectors intersect collinearly with the corresponding sides of the triangle. a'. The three segments, intercepted on the sides of a triangle by the three pairs of tangents from its opposite vertices to an arbitrary circle, determine three second pairs of tangents to the circle whose intersections connect concurrently with the corre- sponding vertices of the triangle. HARMftNlC PROPERTIES OF THE CIRCLE. 95 For, in the case of a, if A^ £, C be the three vertices of the triangle ; X and X', Y and Y', Z and Z' the three pairs of intersections of its opposite sides with the circle ; TJ and Z7', V and F', W and W the three second pairs of intersections of the three angles XAX\ YBY\ ZCZ with the circle; and P, C B. the three intersections of the three pairs of lines TJU' and XX', VV and YT, WW and ZZ' ; then since, by the general property (a) of the preceding article, the three points P, Q, B lie on the polars of the three points A, B, C with respect to the circle ; therefore, by the general property (180, 1°), they lie on the axis of perspective of the triangle ABC and its polar triangle A'B' C with respect to the circle ; and there- fore &c. And, in the case of a, if A, B, C, as before, be the three vertices of the triangle ; XX', YY', ZZ' the three seg- ments intercepted on its opposite sides by the three pairs of tangents from them to the circle ; and P, Q, R the three inter- sections of the three second pairs of tangents XU and X' U', YV and TV, ZW and Z'W from the three pairs of points Xand X', Fand Y', ^and Z' to the circle; then since, by the general property (a') of the preceding article, the three lines PA, QB, BC pass through the poles of the three lines BC, CA, AB with respect to the circle ; therefore, by the general property 96 HARMONIC PROPEKTIES OF THE CIRCLE. (180, 1°), they pass through the centre of perspective of the triangle ABC and its polar triangle A'B'C with respect to the circle ; and therefore &c. Cor. In the particular cases when the circle either passes through the three vertices or touches the three sides of the triangle ; since then, in either case, the three lines UU', W, WW, in a, are the polars of the three vertices, and the three points P, Q, B, in a, are the poles of the three sides, of the triangle ABC, with respect to the circle ; the above properties give consequently, in those cases, the two reciprocal properties, established originally on other principles in Examples 3° and 4°, Art. 137, and inferred subsequently, as particular cases, firstly, from the two reciprocal theorems of Pascal and Brianchon (148, a and b) respecting any hexagon inscribed and exscribed to a circle, and afterwards, from the general property of (180, 1°) respecting any two triangles reciprocal polars to each other with respect to a circle. 263. By virtue of the same general property (180, 1°) re- specting the perspective of every two triangles reciprocal polars to each other with respect to a circle, the two reciprocal pro- perties, b and b\ Cor. 5°, of the same article (261), supply the following very elegant reciprocal solutions of the two reciprocal problems — a. To construct a triangle at once exscribed to a given triangle and inscribed to a given circle. a!. To construct a triangle at once inscribed to a given triangle and exscribed to a given circle. In o, if ABC be the given triangle ; A'B'C its polar triangle with respect to the given circle ; and Z>, E, F the three points of intersection of the three concurrent connectors AA', BB', CC with the three corresponding sides B'C, C'A', A'B' of the latter triangle^ then, of the six intersections of the three lines EF, FD, DE with the circle, one set of three for diflferent lines determine one XYZ, and the other set of three the other X'Y'Z', of the two triangles required. For, the three pairs of lines B'C and AD, C'A and BE, A'B' and CF being conjugate pairs with respect to the circle (174), and the three angles they determine being cut bar- HARJiONIC PROPERTIES OF THE CIRCLE. 97 monically by the corresponding angles of the triangle DJEF determined by their three vertices (242, a) ; therefore, by pro- perty b, Cor. 5° of Art. 261, the three pairs of lines YZ and TZ; ZX and Z'X\ ZF and XT' pass through the three points A, B, C respectively ; and therefore &c. In a, if ABC (fig., page 98) be the given triangle; A'B'C its polar triangle with respect to the given circle ; L,M,N the three collinear intersections of BC and B'C, CA and C'A\ AB and A'B'; and P, Q, B the three vertices of the triangle determined by the three connectors of L, Jf, N with the corre- sponding vertices A\ B\ C of the polar triangle ; then, of the six tangents from the three points P, Q, B to the circle, one set of three for diflferent points determine one XYZ, and the other set of three the other X' Y'Z\ of the two triangles required. For, the three pairs of points A' and i, B' and M, C and N being conjugate pairs with respect to the circle (174), and the three segments they determine being cut harmonically by the corresponding sides of the triangle PQB determined by their three axes (242, o) ; therefore, by property J', Cor. 5° of Art. 261, the three pairs of points X and X\ Y and Y\ Z and Z' Ue on the three lines BC, CA, AB respectively; and therefore «!fec. Cob. In the particular case when the given triangle is self-reciprocal with respect to the given circle (170), the two tri- angles ABC and AB' C then coincide, and the two preceding VOL. II. H 98 HARMONIC PROPERTIES OF THE CIRCLE. constructions are consequently indeterminate ; hence, as shewn already in Cor. 10°, Art. 261, the solutions of the two reciprocal problems to construct a triangle either exscribed to a given tri- angle and inserted to its jpolar circle or inscribed to a given triangle and exscribed to its polar circle are both indeterminate. 264. For every triangle self-reciprocal with respect to a circle (170), the following metric relation results readily from the HABMONIC PROPERTIES OP THE CIRCLE. 99 general property a of Art, 259, combined with the property of every right-angled triangle given in 4°, Cor. 2°, Art. 83, viz. — For any triangle self-reciprocal with respect to a circle^ if A, 5, C be the three vertices, AB, B8, CT the three tangents from them to the circle, and P any arbitrary point on the latter, then always — AE\ {BPCy + BS\ { CPAf + CT. [APB]' = ; the quantities within the parentheses signifying the areas of the three triangles they respectively represent. For, if C be the vertex of the triangle internal to the circle, Q the second intersection of the line CP with the circle, D its in- tersection with the side AB of the triangle opposite to (7, and J)U the tangent from D to the circle ; then since, from the harmonicism of the system of four points CJDPQ (259, «.), CP^^_ CP.GQ _ _ CT^ DP'~ DP.DQ~ BIT' (Euc. III. 35, 36), therefore, at once, multiplying by the square of AB, CP\ AB\ I)U' + DP'.AB\ CT' = 0, from which, assuming for a moment that BBf\ AE' + Ajy. BS' = AB\D U\ it follows of course, at once, that CF'.BW.AW + GP'.AB^.BS^ + DP.AB\ CT^ = 0, which is manifestly equivalent to the above, the three parallelo- grams CP.BD, CP.AD, DP.AB (see figure) being the doubles of the three triangles BPG, CPA, APB respectively; and therefore &c. To prove the relation assumed in the above ; if F and F (see figure) be the two points inverse at once to the circle and to the line AB (149), which, by (156) and (177), lie both on the circle on AB as diameter ; then since, by (157), AB=AE=AF, BS=BE=BF, DU=DE=DF, and since by the relation 4°, h2 ^■^i-; 100 HARMONIC PROPERTIES OF THE CIRCLE. Cor. 2°, Art. 83, the two angles AEB and AFB being both right (Euc. III. 31), BJy'.AE' + A17.BIP = AB\DE% and BD\AF'' + AI^.BF' = AB\DF', therefore BE^.AE'+ AD'.BS'=^AB\DU'; and therefore &c. 265. The two following reciprocal pi-opertles, again of every tetrastigm inscribed and of every tetragram exscribed to a circle, result readily from those of Cor. 10°, Art. 62, and of 4°, Cor. 2°, Art. 179, combined with those of Cors. 3° and 6°, Art. 235, viz. — a. The four segments intercepted on any arhitrary line, hy any circle and by the three angles of connection of any inscribed tetrastigm, have a common segment of harmonic section, real or imaginary. a'. The four angles subtended at any arbitrary point, by any circle and by the three chords of intersection of any exscribed tetragram, have a common angle of harmonic section, real or imaginary. To prove a. If L and L', M and M', N and N' be the three pairs of opposite connectors of the tetrastigm ; X and X', Y and Y', Z and Z' their three pairs of intersections with the line ; and P and Q the two intersections of the latter with the circle ; then since, by the property. Cor. 10°, Art. 62, PL.PL' = PM.PM' ^PN.PN', QL.QL = QM. QM'= QN. QN' ; and, since evidently, by pairs of similar triangles, PL^PX PMPY PNPZ QL QX' QM~ QY' QN~~QZ' PL _ FX PW _ FT QL' ~ QX' ' QM' ~ ~QY' ' therefore, at once, by division, PX.PX _ FY. FY' QX. QX' ~ QY.QY'- QZ.QZ ' and therefore &c. ; the three segments XX', YY', ZZ' having consequently, by (235, Cor. ?i°), a common segment of harmonic section, real or imaginary, with the segment FQ. FN' QN'- FZ' QZ' PZ.PZ' HARMONIC PROPERTIES OF THE CIRCLE. 101 To prove a'. If P and P', Q and Q\ R and R be the three pairs of opposite intersections of the tetragram ; U and U\ V and F', W and W their three pairs of connectors with the point ; and L and M the two tangents from the latter to the circle; then since, by the property 4°, Cor. 2°, Art. 179, PL. PL QL.Q'L RL.RL PM.P'M ~ QM.Q'M~ RM.RM' and since, evidently, by (61), PL sinLU QL smLV RL sini W PM ~ sinMU ' QM ~ £nMV ' RM~ smMW P'L sinLU' Q'L smLV RL smLW FM ~ sinMU' ' Q'M sinJ/F' ' RM~&mMW' therefore, at once, by substitution, ainLU.smLV siuiF.sinLF' AnLW.AnLW BmMU.sinMU' smMV.sinMV' ~ smMW.^mMW and therefore &c. ; the three angles UU\ VV\ WW having consequently, by (235, Cor, 6°), a common angle of harmonic section, real or imaginary, with the angle LM. The above reciprocal properties evidently verify, for the particular cases of tetrastigms inscribed and of tetragrams ex- scribed to circles, the general properties established for all tetrastigms and tetragrams in Cor. 3°, Art. 245. COE. 1°. The extremities of the common segment of har- monic section, in the former case, being conjugate points, by (245, Cor. 1°, J'), with respect to the tetrastigm, and, by (259, a), with respect to the circle ; and the sides of the common angle of harmonic section, in the latter case, being conjugate lines, by (245, Cor. 1°, J), with respect to the tetragram, and, by (259, a'), with respect to the circle ; hence, from the above, it appears that — a. Every two conjugate points with respect to any tetrastigm inscribed to a circle are conjugate points with respect to the circle. a'. Every two conjugate lines with respect to any tetragram ex- scribed to a circle are conjugate lines toith respect to the circle. COK. 2°. Since, for every two points conjugates at once with respect to a circle and to any inscribed tetrastigm, the four polars of either, with respect to the circle and to the three angles of 102 HARMONIC PKOPERTIES OF THE CIRCLE. connection of the tetrastigin, pass through the other (174, and 245, Cor. -5°,/'); and since, for every two lines conjugates at once with respect to a circle and to any exscribed tetragram, the four poles of either, with respect to the circle and to the three chords of intersection of the tetragram, lie on the other (174, and 245, Cor. 5°, /) ; hence also, from the above, it appears that — a. The four polars of any pointy with respect to any circle and to the three angles of connection of any inscribed tetrastigni, are concurrent. a'. The four poles of any line^ with respect to any circle and to the three chords of intersection of any exscribed tetragram, are collinear. Cor. 3°. The centre of any circle and the line at infinity being pole and polar to each other with respect to the circle ; it appears, from the latter properties, for the particular cases when the point in a is the centre of the circle and when the line in a' is the line at infinity, that — a. The three polars of the centre of a circle, with respect to the three angles of connection of any inscribed tetrastigm, are con- current with the line at infinity. a. The three poles of the line at infinity, with respect to the. three chords of intersection of any tetragram exscribed to a circle, are collinear with the centre of the circle. The pole of any segment with respect to the line at infinity being the middle point of the segment (216, 3°) ; the latter property a may be stated, otherwise thus, as follows — In every tetrastigm determined by four tangents to a circle, the three middle points of the three chords of intersection of the figure are collinear with the centre of the circle. This property the reader may very easily verify, h priori, for himself. CoE. 4°. Again, every point at infinity and the diameter of any circle perpendicular to its direction being pole and polar to each other with respect to the circle ; it appears also, from the same properties, for the particular cases where the point in a is at infinity and where the line in a passes through the centre of the circle, that — HARMONIC PROPERTIES OF THE CIRCLE. 103 a. The three polars of any point at infinity^ with reject to the three angles of connection of any tetrastigm inscribed to a circle, are concurrent with the diameter of the circle perpendicular to the direction of the point. a. The three poles of any diameter of a circle., with reject to the three chords of intersection of any tetragram exscribed to the circle, are collinear with the point at infinity in the direction perpendicular to the diameter. Cor. 5°. Again, every point on a circle and the tangent at it to the circle being pole and polar to each other with respect to the circle ; it appears also, from the same properties, for the particular cases when the point in a is on the circle and when the line in a! touches the circle, that — a. The three polars of any point on a circle, with respect to the three angles of connection of any inscribed tetrastigm, are con- current with the tangent at the point. a. The three poles of any tangent to a circle, with respect to the three chords of intersection of any exscribed tetragram, are collinear with the point of contact of the tangent. Cor. 6*. In the particular cases, of the original properties of the present article, when the arbitrary line in a is the line at infinity and when the arbitrary point in a is the centre of the circle; since, by (260, 1°, a), every two points harmonic conjugates with respect to the two circular points at infinity subtend right angles at all points not at infinity, and since, by (216, 3°), the sides of all right angles are the bisectors of all angles they cut harmonically ; it appears, consequently, from them, for those particular cases, that — a. In every tetrastigm inscribed to a circle, the three segments intercepted by the three angles of connection on the line at infinity subtend at every point three angles having a common pair of bisectors. a'. In every tetragram exscribed to a circle, the three angles subtended by the three chords of intersection at the centre of the circle have a common pair of bisectors. These properties, like those of Cor. 3°, the reader may easily verify, h priori, for himself. 266. The two following reciprocal properties, analogous to those of Art. 246, are evident from those of Art. 259, viz. — 104 HARMONIC PROPERTIES OF THE CIRCLE. a. If on a variable line L, turning round ajixedpoint 0, and intersecting a fixed circle A^ at two variable points X, and Xj, a variable point P be taken so as to satvrfy the relation ike point P moves on a fixed line 1 ; the polar^ viz.^ of the point with respect to the circle. a. If through a variable point P, moving on a fixed line J, and subtending a fixed circle A^ by two variable tangents U^ and £^, a variable line be drawn so as to satisfy the relation sinlU, '^lmIU^~ ' the line L turns round a fixed point ; the pole^ viz., of the line 1 with respect to the circle. For, in the case of a, the two points and P, being har- monic conjugates with respect to the two JT, and X^ (214), are conjugate points with respect to the circle (259), and therefore &c. (175, 1°); and, in the case of a', the two lines / and L, being harmonic conjugates with respect to the two U^ and U^ (214), are conjugate lines with respect to the circle (259), and therefore &c. (175, 1°). 267. The two reciprocal properties of the preceding article, respecting a single circle, are evidently particular cases of the two following, respecting a system of any number of circles; which are analogous to those of Art. 248 respecting a system of any number of lines, and with the establishment of which we shall conclude the present long Chapter. a. If on a variable line L, turning round a fixed point 0, and intersecting any system affixed circles -4^, 5„, C7„, &c. at a system of pairs of variable points X, andX^, Y^ and Y^, Z^ andZ^^ (Use, a variable point Pbe taken so as to satisfy the relation -(§,+§)+'•(?$, + §!.)+«•(§.-§}+*-». a, J, c, d:c. being any system of finite multiples, positive or negative ; the point P moves on a fixed line 2, termed the polar of the point with respect to the system of circles A^, B^, C„, d;c. for the system of multiples o, b, c, tfcc. HARMONIC PROPERTIES OF THE CIRCLE. 105 a. If through a variable point P, moving on a fixed line /, and subtending any system of fixed circles A^, B^^ (7„, &c. by a system of pairs of variable tangents U^ and Z^^, F, and F^, W^ and W^j, cfcc, a variable line L be drawn so as to satisfy the relation \sm/C/j smlU^J \sinIV^ smlV^J /sin LW, sin LWa „ a, b, c, cfcc. being any system of finite multiples^ positive or negative ; the line L tarns round a fixed point 0, termed the pole of the line I with respect to the system of circles A^, B^, C„, : sin BD, =«, the two ratios m : n and n : m are what are termed anharmonic ratios of the section of the line, or angle, AB, by the two points, or lines, C and D ; for any line or angle, AB, every two points or lines of section, G and D, determine therefore two different anharmonic ratios, reciprocals of each other. The name " anharmonic" was given to this simple function of the section of a line or angle by Chasles, who was the first to perceive its utility and to apply it extensively in geometry ; because that in the particular case when m and n are equal in magnitude and opposite in sign, the section of the line or angle becomes what from ancient times had been familiarly known as " harmonic," and which from its special importance has been treated of separately in Chapter xiil. 269. The two anharmonic ratios of the section of a line or angle AB by any two points or lines of section C and D, like every other pair of magnitudes reciprocals to each other, have of course always the same sign, positive if C and D be both external or both internal, and negative if one be external and the other internal to AB; but the positions, absolute or relative, of G and D, being quite arbitrary, either may have any absolute magnitude from to ao , and the other the I'eciprocal of the same from x to 0. THE§KY OF ANHARMONIC SECTION. 109 When either anharmonic ratio = 0, the other = oo ; and, conversely, when either = x , the other = ; in both those extreme cases it is evident that one or other of the two points or lines of section, C and D, coincides with one or other of the two extremities of the line or angle, A and B. When either anharmonic ratio =+1, the other also =±1; these are the only two cases in which the two anharmonic ratios of the section of a line or angle are equal, + 1 and — 1 being the only two numbers which are equal to their reciprocals ; in the latter case the section of the line or angle is, as already noticed, harmonic; and in the former case it is evident that either the two points or lines of section, C and D, or the two extre- mities of the line or angle, A and B, coincide with each other. For the three particular values of either anharmonic ratio of the section of a line or angle AB by two points or lines C and i?, 0, cc , and + 1, some two of the four points or lines A, B, C, -D, therefore, coincide. For every other value of either, however, they are all four distinct from each other. 270. When for one of the two points or lines of section, J) suppose, the two simple ratios for the single section each = 1, that is, when Z> is the point or line of external bisection of the segment or angle AB ; then, whatever be the position of the other point or line of section (7, the two anharmonic ratios for the douile section hy C and D comhined, become in that case the two simple ratios for the single section by C alone. This particular case is deserving of special attention, not only on account of its comparative simplicity, but because, as we shall presently see, every other case of anharmonic section of a line or angle, whatever be the positions of the two points or lines of section, may be reduced to it. 271. When two segments, or angles, or a segment and an angle, AB and A'B', are cut in equal anharmonic ratios by two pairs of sectors, C and B, C and J)', they are said to be cut equianharmonically ; and so, also, is the same segment, or angle, AB, when cut in equal anharmonic ratios by two different pairs of sectors, C and B, C ' and D'. A more general definition of the relation of equianharmonicism will be given further on. 110 THEORY OF ANHABMONIC SECTION. Two pairs of lines, L and if, L' and J/', are said sometimes to intersect, and sometimes to divide, two segments, AB and A'B\ equianharmonically, when their pairs of intersections with their axes divide them equianharmonically; and, two pairs of points, P and Q, P' and Q\ are said sometimes to subtend, and sometimes to divide, two angles, AB and A'B', equianharmoni- cally, when their pairs of connectors with their vertices divide them equianharmonically. These modes of expression are frequently employed for shortness in the applications of the theory of anharmonic section. 272. When a segment or angle AB is cut equianharmonically hy the two pairs of sectors C and D, C and D\ it is also cut equianharmonically hy the two pairs C and C\ D and D' ; and conversely. For since, by hypothesis, AC AD_AC AB_ imAC ^ i\\\AD _ %\aAG ' sinJ^ . 1BG''BD~ BC' BU ' *'''' Sr»C ' sin 52) ~ sin^C" " iva.BD' ' Therefore, at once, by alternation, AG 4C^_AD AD_ wciAC sin^C _ imAD slnJ^i)' BG ' BG' ~ BD ■ BD' ' "''' sin^C ' ainBG' ~ ainBD ' einBD' ' and therefore «fec. In exactly the same manner it may be shown, in accordance with the mode of expression noticed at the close of the pre- ceding article, that when a segment AB is divided equianhar- monically by the two pairs of lines L and M, U and M\ it is also divided equianharmonically by the two pairs L and L', M andM' ; and that when an angle AB is subtended equianharmonically by the two pairs of points P and Q, P' and Q, it is also subtended equianharmonically by the two pairs P and P*, Q and Q. For since, in the two cases, respectively, AL AM_AL AM AP AQ _AP AQ_ BL '• BM ~ BL ' BM' ' ' BP' BQ~ BF ' BQ ' therefore, at once, by alternation, in the two, respectively, j_L 4l;_^ am^ AP at _aq aq BL ' BL' BM ' BM' ' *°"' BP' BF ~ BQ' BQ'' and therefore &c. TH^RY OF ANHARMONIC SECTION. Ill The following is an obvious corollary from the above : When a segment or angle AB is cut harmonically by the tioo segments or angles XY and X' Y\ it is cut equianharmoni- cally hy the two XX' and YY'. And also every two lines L and M and their two poles P and Q with respect to any segment AB, or any two points P and Q and their two polars L and M with respect to any angle AB, divide the segment or angle equi- anharmonically. 273. 77ie two anharmonic ratios of the section of any segment or angle AB by any two points or lines C and D are the same in magnitude and sign as those of the section of the segment or angle CD by the two points or lines A and B. For, by simple alternation, AC AD CA CB smAC _ smAD sin CA ^ sinCB smBC ' sinBD ~ sinDA ' aiaDB ' sin-4Z> sin AG _ ain CB _ ainCA am BD ' ainBG ~ aiaDB ' amDA ' and therefore &c. Of this general property of anharmonic section, from which it appears that every two segments having a common axis, or angles having a common vertex, AB and CD, cut each other equi- anharmonically, the property of harmonic section proved in Art. 218 is obviously a particular case. In the same manner exactly it may be shown, in accordance with the mode of expression noticed at the close of Art. 271, that the two anharmonic ratios of the section of any segment or angle AB by any two lines or points C and D are the same in magnitude and sign as those oj the angle or segment CD by the two points or lines A and B. For, since in either case, by simple alternation, AC AD _CA CB^ ^ :^ ^_CB CA BC'bD~ DA'DB' ' BD'- BC ~ DB' DA' therefore &c. This important property of anharmonic section, from which it appears that every segment and angle, AB and CD, hotvever circumstance as to magnitude or position, divide BC- BD DA DB' or. and AD BD AC CB • BG~ DB' CA DA' or, 112 THEOUY OF ANHAKMONIC SECTION. each other equianharmoniexilly, will be presently considered under another form. The sign common to the two reciprocal eqiiianharmonic ratios determined by the mutual section of two segments or angles, or of a segment and angle, AB and CD^ depends of course on the relative positions of their respective extremities, A and jB, C and D\ being obviously negative when those of one alternate with those of the other in the order of their occurrence, and positive when they do not. 274, As four points on a common axis or rays through a common vertex A^ B, G, D determine, whatever be their order and disposition, six different segments or angles corresponding to each other two and two in three sets of opposite pairs BG and AD, GA and BD, AB and CD ; and as, by the preceding, the two segments or angles constituting each pair of opposltes cut each other In the same two auharmonic ratios, reciprocals of each other ; it follows, therefore, that four points on a common axis or rays through a common vertex determine in general six different anharnionic ratios, in pairs reciprocals of each other. These three pairs of reciprocal ratios corresponding to the three pairs of opposite segments or angles BG and AD, GA and BD, AB and CD, are respectively as follows : For four points on a common axis, BA BD , GA CD BA.CD GA.BD GA'- GD ^""^ BA'- BD' "'■' CA.BD ^"'^ BA.CD' •■•'^^'^ CB CD AB AD GB.AD AB.GD AB- AD ^"'^ CB'- CD' °^' AB.GD ^""^ GB.AD'-'' ■ AG AD . BC^ BD AC.BD BG.A D BC'' BD AC'- AD' °''' BG.AD *°" 'AC:bD' ^^^^^'^ For four rays through a common vertex, smBA siu BD ,sm GA sinCD sin GA ■ sin 67? ^hTai ' sm BD ' amBA . sin GD , sin GA . sin BD or. and ■ p. . ^^ , ....:. ...(1'). Hin H4 am tlj) ' ^ I sin GA . sin BD sin BA . sin CD sin GB sin GD , sin ^5 sin^Z> muiB ■ smZD sinCS ' sinC^ ' THEORY OF ANHARMONIC SECTION. 113 ' sinAB.smCD smCB.sinAD' ^ '' ajnAC sin AD , sin^C smBD '^BG ' aiaBD imZC" mKZD ' sinACsinBD _. sin £C. sin -42) , „ °'*' smBCsinAJ) """^ sm A Cain BD' ^ '' which may, for convenience, be represented in either case by the abridged notation P and -p, Q and -^, B and -= respectively ; and which, as is evident from mere inspection of their values, are connected in either case by the relations P.(2.i? = -land^.^.^=-l. 275. In the case of points on a common axis, if any one of the four, D suppose, be at infinity, the six simple ratios BB : CB and CB : BB, CB : AB and AB : CB, AB : BB and BB : AB, into which that point enters, are all = 1 ; and the six anharmonic ratios for the entire four A, B, G, B become consequently the six simple ratios for the remaining three A, B, C; which, in the order above given in the general case, viz., for BC cut at A, for CA cut at B, and for AB cut at C, are respectively as follows, ba , ca cb , ab ac , bc cI*°*:bI' Zb^'^^cb' bC^^ac'^ and which are evidently connected by the same relations as in the general case. To this comparatively simple case it will appear in the sequel that every other case of anharmonic ratio, whether of points or rays, whatever be the order or disposition of either, may be reduced. 276. Whatever be the order and disposition of four points or rays constituting a row or pencil, A, B, C, B, it is evident that of the three pairs of reciprocal anharmonic ratios they deter- mine two are always positive and the third always negative; the negative corresponding to the pair of opposite segments or angles they determine whose extremities alternate with each VOL. II. 1 114 THEORY OP ANHAEMONIC SECTION. other in the order of their succession, and the two positive to the two pairs whose extremities do not so alternate. Hence, as seen above, for three of them P, Q, B, and for their three reciprocals, the product of any three of them of different pairs is always negative. When three points or rays of an anbarmonic system, any one of its six anharmonic ratios, and the order in which the four constituents enter in the formation of the ratio are given, the fourth point or ray is of coarse implicitly given also; its determination depending only on the section of a given segment or angle into two parts whose lengths or sines shall have a given magnitude and sign. 277. The six anharmonic ratios P and -pf Qandyi, Rand^ determined hy the same row of four points or pencil of four rays Af Bf C, D are connected two and two by the three relations ^+^=^' -^+1=^' «+i=i' tchatetxr he the order and disposition erf the constituents of either. For, since for every system of fonr points A^ B, C, D on a, common axis, whatever be their order and disposition (82), BC.A1)+ CA.BD + AB.CD = (a); and, since for every system of four rays A^B^ C^D through a com- mon vertex, whatever be their order and disposition (82, Cor. 3°), sin5C.sin^i) + 8inC4.8in52>+sin.4Asin<7Z> = 0...(a'); therefore, dividing each successively by each of its three com- ponents, the three relations above given result at once in each case. In the particular case of points on a common axis, when one of the four. D is at infinity, since then P= BA : CA, Q = CB : AB, B = AC: BC, the above relations are evident to mere inspection; and to this comparatively simple case, as already stated, all others may be reduced. From the above relations, combined with those already given, it appears that the six anharmonic ratios of the same row or pencil of four points or rays, though in general all different, are never independent of each other, but that, on the contrary, whatever be the order and disposition of the constituent points TUl^RY OF ANHARMONIC SECTION. 115 or rays, they are always so connected with each other that any one of the entire six determines the remaining five ; so that if any one of them be given or known, all the others may be regarded as implicitly given or known with it. P— 1 Thus, supposing P known, then, from the above, Q = — p— and B = pj and the three P, Q, B thus known, so of course are their three reciprocals, which are the remaining three ratios. As an example, let P= — 1, that is, let the row or pencil form an harmonic system (213)5 ^^^^ Q=^ Mid B=^. flence, when four points or rays form an harmonic row or pencil, and when therefore one pair of their reciprocal anharmonic ratios =—1, the other two pairs are 2 and ^, and ^ and 2 respectively; the same results obtained in a different manner in Art. 219. 278. When two rows of four points or pencils of four rays, or a row of four points and a pencil of four rays, A, P, (7, D and A', P*, C", P', are such that a single anharmonic ratio is the same for both systems^ the entire six anharmonic ratios are the same for ioth systems. For, denoting by P and p, Q and -j^, B and -^ the six for the system A, P, (7, P, and by P and „ , Q and ^ , P' and -pi the six for the system A\ P', G\ D' ; since, by the preceding article, P— 1 P' — 1 1 ,1 V = — p" > Q — pi J \ — P' ~ 1 — P" when P= P', then Q=Q and B = B', and therefore &c. the reciprocals of equal magnitudes being of course equal. Two rows of four points or pencils of four rays, or a row of four points and a pencil of four rays, thus related to each other that the six anharmonic ratios are the same for both systems, are said to be equianharmonic (271) ; and the pairs of constituents, A and A', B' and P', G and C", P and P', which enter similarly Into the several pairs of equal ratios, are said to be corresponding or homologous pairs. 12 116 THKORY OP ANHAEMONIC SECTION. In every case of equianharmonicism between two systems of four constituents, A, B, C, D and A', B', C, D", which cor- respond two and two in pairs, A and A', B and B', C and C\ D and D\ when three pairs of corresponding constituents and one constituent of the fourth pair are given, the second constituent of the fourth pair is of course implicitly given also. Various constructions for determining it will be given further on. Cor. 1°. Since for every two equianharmonic systems of four constituents A^ B, C, D and A', B, C\ D\ by the above, F^F", Q= Q',R = R', therefore for the same, by (277), relations often of much use in establishing the circumstance of equianharmonicism between two systems of four constituents when the simpler relations P= P", Q= Q, R — R' are not as readily applicable. CoE. 2°. Since two or more magnitudes of any kind when equal to the same magnitude are equal to each other, it follows evi- dently, from the nature of equianharmonicism, as above explained, that when two or more systems of four constittients A\ J5', C7', i?' ; A",B", G",D" ,- A"\B"', C"',D"' ; dx. are equianharmonic with the same system A., B, C, Z), they are equianharmonic with each other. 279. Dr. Salmon has employed the following very con- venient notation for expressing the equianharmonicism of two or more Bystems of four constituents A, 5, C, D ; A', B\ C", U ; A", B", C", B" ; &c. viz. : [ABCB] = {A'B' CD'] = {A"B' C"B"] = &c., where the symbol [ABCD] is regarded as the general represen- tative of the entire six anharmonic ratios for the system A^ B,C,D; [A'B' CD] as that of the entire six for the system A',B', C, 2)'; {A"B"C'D"] as that of the entire six for the system A"j B", C, D" ; &c. and where the letters representing corresponding constituents are invariably written in the same THEORY OF ANHAEMONIC SECTION. 117 order in all ; so that the corresponding groups of ratios, which alone are equal in the several systems, may be evident to in- spection, without the trouble of seeking for, or the danger of mistaking them. We shall employ the same notation generally in the comparison of equianharmonic systems. And in the same manner as the notation [ABCD] is to be regarded as the general symbolical representative of the entire six anharmonic ratios of the system of four constituents A, B, (7, jW, the precaution respecting similarity of order being in- variably observed in all cases of comparison with other systems, so the expression " anharmonic ratio of four points or rays" when used, as it constantly is, in the singular number, is to be regarded as the general nominal representative of the entire six for the system ; the same precaution respecting similarity of order being invariably attended to in all cases of comparison between two or more systems. As the symbol {ABCD] is employed to denote, in the sense above explained, the anharmonic ratio of the system of four points A, B, C, D when collinear, or of the system of four lines Af -B, C, D when concurrent; so, for shortness, the symbol [O.ABGD] is employed to represent, in the same sense, the anharmonic ratio of the pencil of four lines by which the system of four points A, B, C, J), whether collinear or not, connects with the vertex 0, or of the row of four points at which the system of four lines A, B, C, D, whether concurrent or not, intersects with the axis 0. 280. Any order of the four constituents of an anharmonic system of points or rays, A, B, (7, D suppose, may be altered in three different ways, viz. into B, A, D, C, or C, Z>, A, B, or D, C, B, A, without affecting, either in magnitude or sign, any of the six anharmonic ratios of the system, corresponding to that order. To prove this, or, which is the same thing, to shew that always {ABCD} = {BADC} = {CDAB} = {DCBA], it will only be necessary (278) to establish its truth for any one of the six ratios for the first order, compared with the three that correspond to it in the other three. 118 THEORY OF ANHARMONIC SECTION. , ^ ^ BA.CD Taking then arbitrarily any one for the nrst, ^^ ^j^ , sin5^.8inCZ> suppose, for the case of points, or its analogue ^j^^^ %mBD for the case of rays, and placing beside it its three correspondents in the other three ; we have, for the whole four, the system BA.CD AB.DC BG.AB CD.BA CA.BD' DB.AC AG.DB' BD.CA^ in the case of points, or the analogous system sin^^.singP sin^g.sinJC sinJCsin^-g sin CO. sin .g^ sinCM.sin^Z)' saiDB.&mAG' sinAC.imDB' BmBD.wa.CA in the case of rays ; which on mere inspection are seen, in either case, to be equal both in magnitude and sign. In comparing together the preceding, or any other four equivalent orders ; it appears that, to go from any order to an equivalent order, any two of the four constituents, may he inter- changed provided the remaining two he interchanged also. Hence the following simple rule for the formation from any given order of its three equivalents, viz. every interchange of two constituents is to he accompanied hy the interchange of the other two ; this is Chasles' rule, the reason of which is evident from the obvious signification of the double interchange as regards the three pairs c& opposite segments or angles determined by the four points or rays of the system. Cor. Since, from the above, for any row of four points or pencil of four rays A^ B, C, D, {ABCD} = {BADC} = {CDAB] = [DCBA] .... (a) ; and since again, for any other row of four points or pencil of four rays A\ £', C\ D', [A'B'C'D] = {B'A'D'C} = {C'D'A'B'}={D'C'B'A'}... {a') ; it follows therefore that the equality of any one of the four equivalents of group a to any one of the four of group a', whether the two compared correspond to each other or not, is sufficient to establish the equianharmouicism of the two systems, if so related to each other. This is an important consideration of which frequent use is made in the applications of the theory of anharmonic section. THEORT OP ANHARMONIC SECTION. 119 281. Thffre is one, and but one, case in which one pair of constituents of an anhartnonic row or pencil, when all Jbur dis- tinct from each other, may be interchanged, without requiring the simultaneotis interchange of the other pair in order to preserve the anharmonic equivalence of the changed to the original order ; VIZ. when the system is harmonic, and when the interchanged constituents are conjugates. For, if a system A, B, C, D be such that [ABCD] = {ABDC], then, according as it consists of points or rays, AC AD AD AG em AC mn AD am AD i\uAC BC' BD~ BD'BC "''' sin^C " sin BD ~ ainBD " sin^C ' and therefore, as the case may be, AC AD sin AC ainAD BG' BD^^^ sinBG ' SIbS^' ~ " ' but, for the positive sign the two points or rays C and D coincide (269), and, for the negative sign they are harmonic conjugates with respect to the two A and B (213) ; and there- fore &c. Of all criteria of the mutual harmonic section of two segments or angles AB and CD (218), the above relation {ABCD} = {ABDC} is probably the most universally applicable, especially in the higher departments of geometry. Several examples of its application will be given in the sequel ; and it will appear that, in some of the cases of harmonic section established in the two preceding chapters, it might, under a different order of treatment, have replaced with advantage the criteria employed. 282. When two systems of any common number of consti- tuents, both of points, or both of rays, or one of points and one of rays, A, B, C, D, E, F, G, &c. and A\ B', C, D', E', F, G', J:} = {FC'D'^'} and {CADi:} = {C'AD'E'} and [ABDE] = {A'B'UE'}. Since, by hypothesis, [ABCD] = {A'B'C'I)'}, therefore BA BD imBA smBB B'A' B'D' AnB'A' smB'U CA ' CD sin GA ' sin CD C'A' ' C'ly sin C'A' ' sin G'ly ' and sbce, by hypothesis, {ABCE} = [A'B'C'E'], therefore BA BE miBA sm.BE B'A' B'E' smB'A' BinB'E' CA ' CE BmCA ' sin CE~ C'A' ' C'E' sinCA' ' sin C'E' ' therefore, at once, by division of ratios, BDBE sinBD sinBE _ B'D' B'E' sinB'I)' sin B'E' CD ' CE *"■ sin CD ' sin CE ~ CD' ' C'E' °^ AnC'D' ' sin C'E' ' and therefore {BCDE} = {B'C'D'E'] ; and similarly [CADE] = [C'A'D'E'] and [ABDE] = {A'B'D'E'] ; and therefore &c. To prove 2°, or to shew that, when [ABCD] = {A'B'C'Df] and [ABCE] = [A'B'C'E'} and [ABCF] = {A'B' C'F'}, then {ADEF} = {A'D'E'F'} and [BDEF] = {B'D'E'F'} and { CDEF} = { C'D'E'F']. Since, by hypothesis, [ABCD] = {A'B' CD'} and {ABCE} = {A'B'C'E'} and {ABCF} = {A'B' C'F}, therefore by 1°, {ABDE} = {A'B'D'E'} and {ABDF} = {A'B'D'F'}, and therefore, by the same, {ADEF} = {A'D'E'F} ; and similarly {BDEF} = {B'D'E'F'} and {CDEF} = {C'D'EF} ; and therefore &c. THEORY OF ANHAEMONIC SECTION. 121 To prove 3°, or to shew that, when {ABCD} = {A'B'C'iy} and {ABCE} = {A'B'C'E'] and {ABCF} = {A'B'CT} and {ABCG} = [A'B'C'G'}, then {BEFG} = {D'EF' G'}. Since, by hypothesis, {ABCD} = {A'B'C'B'} and {ABGE\ = {A'B'C'E'} and {^5Ci?'} = {A'B' C'F'} and {^5C(?} = {A'B' C G'}, therefore, by 2°, {ABFF} = {^'i?'£'ii"} and {ABFG} = {^'i>'£'(?'}, and therefore, by 1°, {BFFG] = {B'E'F'G'}; and therefore &c. Two systems of any common number of constituents, A, B, C, B, F, F, G, &c. and A\ B', C", B\ E', F\ G', &c., thus corresponding in pairs, A and A\ B and B\ C and C", &c., and thus related to each other that every four pairs of cor- responding constituents form equianharmonic systems, have been termed by Chasles komographic, and will be treated of at length imder that denomination in another chapter. They occur very frequently in the applications of the theory of anharmonic section, and the relation between them may, when necessary, be repi'esented by the obvious extension of Dr. Salmon's nota- tion for simple equianharmonicism between two systems of four (279), viz. : {ABCBFFG, &c.} = {A'B'C'D'E'F'G', &c.} ; the same precaution respecting order among the representa- tives of corresponding constituents, so essential in the simpler, being, of course, not less indispensable in the more general case. See Art. 279. CoE. r. If, while three pairs of corresponding constituents, A and A', B and B", C and C, of two equianharmonic systems, both of points, or both of rays, or one of points and one of rays, A, B, G, B and -4', B\ C\ D\ are supposed to remain fixed, the fourth pair, D and 2>', be conceived to vary, preserving always, however, the equianharmonicism of the two systems; then, every two positions of the variable pair may be con- 122 THEORY OF ANHABMONIC SECTION. ceived to take the places of D and 2)', E and E', every three positions those of D and i)', E and E\ F and F, and every four positions those of D and D', E and E\ i?'and i?", G and (?', in the preceding ; hence, from the above, it appears that — When a variable pair of constituents, points, or rays, or a point and ray, D and D', form in every position equianharmonic systems with three fixed pairs, A and A', B and B', C and C, then — 1°. Every two positions of D form, with BC, with CA, and with AB, systems equianharmonic with those formed by the two corresponding positions of D' , with B'C, with CA', and with A'B'. 2°. Every three positions of D form, with A, with B, and with C, systems equianharmonic with those formed by the three corre- sponding positions of D', with A', with B', and with C 3°. Every four positions of D form a system equianharmonic with the four corresponding positions of D'. COE. 2°. Since, during the variation of D and Z)', in the above, the relation [ABCD] = {A'B' CD'} is constantly pre- served with A and A', B and B', G and C, therefore throughont the entire variation AD AC «m.AD sinAG A'D' A'C sin^'Z?' sin^'g ' BD ' BG ^^ sin BD ' slnBG ~ B'D" " B'G' ^^ ainB'D' ' imB'C' ' and therefore, by alternation, AD amAD A'D sinA'D" AG alaAC A'C amA'C BD °'' ain BD ' B'D °^ ainB'D' ~ BG °' aiuBC 'WC' °^ ainB'G' = a constant ratio, A, B, C and A', B', G' being fixed ; hence it appears that — When two fixed segments or angles, or a fixed segment and a fixed angle, AB and A'B', are cut by two variable sectors, D and D, so that, throvghout their variation, ^^ or —. — =^ : ^^mt o»* A'j>i BD aiuBD BD sin sin .,jy in any constant ratio, then — 1°. Every two positions of D fwm with A and B a system, equianharmonic with that formed by the two corresponding positions ofD' with A' and B'. 2°. Every three positions of D form with A and with B systems equianharmonic with those formed by the three corre- sponding positions ofD' with A' and with B'. THEORY OF ANHARMONIC SECTION. 123 3°. Every four positions of D form a system equiankarmonic with the four corresjponding positions of D'. CoK. 3°. In the particular case, of these latter properties, when the two segments or angles AB and A'B' coincide, so as to form but a single segment or angle AB, the constant ^. AD A'B' sin .42) sin^'D' , , , . J^atio -gyr : -fjTrr, or -: — 5^ : -. — —-— becomes then (smce JUt BD siaBB sm B JU ^ A = A' and B=B') the anharmonic ratio of the section of the segment or angle AB by the two points or lines of section I) and Z)' (268) ; hence it appears that — If a fxed segment or angle AB be cut in any constant an- harmonic ratio by a variable pair of sectors D and D' ; then — 1°. Eoery two positions of D and the two corresponding posi- tions of U form equianharmonic systems with A and B. 2°. Every three positions of D and the three corresponding positions of If form equianharmonic systems with A and with B. 3°. Every four positions of D and the four corresponding positions of D' form equianharmonic systems. COK. 4°. In the particular case, of these latter properties, when the constant anharmonic ratio of the section = — 1, that is, when the section of the fixed segment or angle AB by the variable pair of sectors I) and B' is constantly harmonic (213) ; since then, and then only, {ABBI)'} = {ABB'B} in every position of B and B' (281), that is, since then, and then only, the two points or lines of section are interchangeable in every position without violating the constant anharmonic ratio of their section of the fixed segment or angle AB ; hence it appears that — When a fixed segment or angle AB is cut harmonically by a variable pair of conjugates D and Z)', then — 1°. Every two positions of D and the two corresponding positions of U determine four constitttents, every two of which and their two conjugates form equianharmonic systems with A and B, 2°. Every three positions of B and the three corresponding positions of Z>' determine six constituents, every three of which and their three conjugates form equianharmonic systems with A and with B. 3°. Every four positions of D and the four corresponding 124 THEORY OP ANHARMONIC SECTION. positions of D' determine eight constituents ^ every four of which and their four conjugates form equianharmonic systems. COR. 5°. When, in these latter properties, the two conju- gates D and D' coincide, in one of their positions, with the two ly and D, in another of their positions ; the properties them- selves become evidently modified as follows — 1°. Every position of D and the corresponding position of U determine two constituents, which taken in both orders form equi- anharmonic systems with A and B. 2°. Every two positions of D and the two corresponding positions of D' determine four constituents^ every three of which and their three conjugates forra equianharmonic systems unth A and with B. 3°. Every three positions of D and the three corresponding positions of D' determine six constituents, every four of which and their four conjugates form equianharmonic systems. N.B. To the principles established in this article the im- portant modem theories, of Homographic Division, of Double Points and Bays in Homographic Division, and of Involution, may all be referred ; as will appear in the sequel in the chapters in which they are severally discussed. 283. When two triads of points on a common axis or rays through a common vertex, A, B, C and A', B\ C, which corre- spond in pairs, A and A', B and B', C and C, are such that any two systems determined hy four of the six constituents and their four correspondents are equianharmonic, then every two systems determined by four of them and their four correspondents are equianharmonic. For, in either case, the relation {BCAA'} = [B-CA'A], (l°), gives at once, by (272) and (280), the two equivalent relations {B'CAA'] = {BC'A'A} and {BC'AA'} = [B'CA'A}; ...{l'); the first of which, combined with the original, gives, by virtue of the general property 1° of the preceding article, the relation {CABB'} = {C'A'B'B], (2°), THEORY OF ANHAEMONIC SECTION. 125 and with it, by (272) and (280), the two equivalent relations { C'ABB'} = { CA'B'B] and { CA'BB'} = { C'AB'B} ; ... (2') ; and the second of which, combined with the original, gives, by virtue of the same general property, the relation {ABCC} = [A'B'O'C], (3°), and with it, by (272) and (280), the two equivalent relations {A'BCC] = {AB'C'G} and {AB'CC] = {A'BC'C}; ...(3'); and, each of the six cases of anharmonic equivalence (1°), (2°), (3°) and (1'), (2'), (3') thus involving the remaining five, there- fore &c. Cob. 1°. That, Jbr every two triads related as above to each other, the three segments or angles, AA', BB', CC, determined by the three pairs of corresponding constituents, A and A\ B and B', C and C", have a common segment or angle MN of harmonic section, real or imaginary, (see 3°, Cor. 5°, of the preceding article), may be easily shewn as follows : If MN be the common segment or angle of harmonic section, real or imaginary, of any two of them, AA' and BB' suppose, then since, by 2°, Cor. 5°, of the preceding article, {MBAA'] = {MB'A'B], or, {MABB'] = [MA'B'B], [NBAA'} = {NB'A'B], or, {NABB'] = [NA'B'B]; and since, by relations (1°) and (1'), or (2°) and (2'), of the above, { OBAA'] = [ C'B'A'A], or, { CABB'} = { C'A'B'B], {G'BAA'} = {CB'A!A], or, [CABB'] = {GA'B'B] ; therefore, at once, in either case, by virtue of the general property (3°) of the preceding article, {MNCC'} = [MNC'G], and therefore &c. ; JOT thus cutting GG' also harmonically (281). CoR. 2°. That every three lines through a point determine with the three perpendiculars to them through the point a system of six rays related as above to each other, is evident from the circumstance that every two pencils determined by four of the 126 THEORY OF ANHAEMONIC SECTION. six constituent rays and their four perpendiculars are similar, and therefore equianharmonic (278). And the general property of Cor. 1° is obviously verified for this particular case by that established on other principles in Art. 260, viz., that all right angles, having a common vertex, have a common imaginary angle of harmonic section, vis., that subtended at their common vertex hy the two cyclic points at infinity. N.B. The property of the present article has been made by Chasles the basis of the modern theory of Involution, and, as such, has been discussed by him at considerable length in his Chapter on that subject. 284. When two equianharmonic systems of points on a common axis or rays through a common vertex, A, B, C, D and A', B', C, D', are such that any two cf their corresponding con- stituents may be interchanged without violating their relation of equianharmonicism, then every two of their corresponding con- stituents may be interchanged without violating their relation of equianharmonicism. For, in either case, the two relations {ABCD] = {A'B'G'I)'} and {A'BCD] = {AB'C'D'] give, by virtue of the general property 1° of Art. 282, the three [AA'CD] = [A A C'jy}, {AA'BB] = {A'AUB'], {AA'BC} = {A'AB'C'}, and with them consequently, by (272) and (280), the equi- valent three {AA" CD'] = [A' A CD], [AA'DB'] = {A' ABB], {AA'BC] = {A'AB'C], of which, the first terms of the second, third, and first, com- bined respectively with the second terms of the third, first, and second, give, by virtue of the same general property 1° of Art. 282, the three {AB'CD] = {A'BC'D^], {ABC'D} = {A'B'CJy}, {ABCD'] = {A'B'C'D}, in each of which, since again, for the same reason, the original THffORY OF ANHABMONIC SECTION. 127 psur of constituents A and A' may be interchanged, thus giving the three {A'B' CD} = {ABG'B}, [A'BG'D] = {AB' CD'}, {A'BCD'} = {AB'C'B}, and so exhausting the entire number of different combinations of four and their four correspondents that could be formed from the four pairs of corresponding constituents A and A\ B and B', C and C, D and D' ; therefore &c. Cofi. 1°. That,/w every system of eight colUnear or concurrent constitiients, corresjiondmg in pairs^ which are thus related to each other that every two systems determined hy Jour of them and their four correspondents are equianharmonic^ the four segments or angles, AA', BB', CC, DD\ determined hy the four pairs of corresponding constituents, A and A\ B and B\ C and C\ D and 2)', have a common segment or angle MN of harmonic section, real or imaginary, (see 3°, Cor. 4°, Art. 282), may be shewn in precisely the same manner as for the particular case established in the corollary of the preceding article. If MN be the common segment or angle of harmonic section, real or imaginary, of any two of them, AA' and BB', then since, by 2°, Cor. 5°, Art. 282, [MBAA'] = {MB'AB], or, {MABB']==[MA'B'B], {NBAA'] = {NB'A:B}, or, {NABB'] = [NA'B'B]; and since, as shewn above, [CBAA]=^{C'B'A'B], or, [CABB-]^[C'A'B'B], {C'BAA'] = [CB'A'B], or, [C'ABB'\ = [CA'B'B]; with relations exactly similar in which G and C' are replaced by D and D'; therefore at once, in either case, by virtue of the general property 3° of Art. 282, [MNGC'] = {MNC'C], and, {MNDU\ = {MND'D], and therefore &c. ; MN thus cutting GG' and DD' also har- monically (281). Cob. 2°. That every four lines through a point determine with the four perpendiculars to them through the point a system of eight rays related as above to each other, is evident (as in Cor. 2° of the preceding article) from the consideration that every two pencils determined by four of the eight constituent 128 THEOKT OF ANHABMONIC SECTION. rays and their four perpendiculars are similar, and therefore equianharmonic (278). And (as in that same corollary) the general property of Cor. 1° is obviously verified for their parti- cular case by that established on other principles in Art. 260, respecting the harmonic section of every right angle by the two imaginary lines connecting its vertex with the two cyclic points at infinity. CoK. 3°. That, as above stated, the property of the preceding is a particular case of that of the present article, appears at once by supposing the fourth pair of corresponding constituents, D and D\ in the above, to coincide, successively, with the first, second, and third pairs, A and A\ B and B\ C and C" ; its un- accented taking the places of their accented constituents, and conversely; as the three groups of relations 1° and 1', 2° and 2', 3° and 3' of the preceding would then result evidently from those of the present article, therefore &c. N.B. The property of this article is also of considerable importance in the Theory of Involution, under which head it will again be referred to in a subsequent chapter. 285. Every pencil of four rays determines a row of four points equianharmonic with itself on every axis ; and, conversely, every row of four points determines a pencil of four rays equianr harmonic with itself at every vertex. Let, in either case, be the vertex of the pencil, and A, B, C, D the four points of the row ; then since, by (65), BABO smBOA 3 i^_BO ainBOD CA CO'&mCOA' ^'*' CD^Cd'^^GOD' therefore, at once, by division of ratios, BABD^ JinBOA sin BOD CA' CD smCOA' smCOD ^^^i , . ., , CB CD sin COB Bin COD and similarly Js'- 23 = ^AOB' ^^AOD (2), andfinaUy ^.^^ ^ ^nAOC ^sjnAOD ^ BC-BD ainBOC sinBOD ^^'' and therefore &c. (268). THEORY OF ANHAKMUNIC SECTION. 129 As for the corresponding property of harmonic section, proved in Art. 221, which is manifestly a particular case of the above, there is one case, and one only, in which the above demonstration fails, viz., when the vertex of the pencil is at an infinite distance; but in that case, as noticed before in the article referred to, the four rays of the pencil being parallel (16), the property is evident without any demon- stration (Euc. VI. 10). Of all properties of anharmonic section, the above, which shews that all anhartnonic ratios whether of rows of points or pencils of rays are preserved unchanged in perspective (130), is much the most important; as an abstract proposition, like its particular case already referred to, it was known to the Ancients^, but it was only in modern and comparatively recent times that its importance was perceived; it is to it indeed mainly that the theory of anharmonic section owes its utility and power as an instrument of investigation and proof in modem geo- metry. See Art. 221. CoK. 1°. When one of the four points of the row, D suppose, is at infinity, that is, when the axis of the row is parallel to the corresponding ray OD of the pencil, or conversely (16) ; since then the three ratios BD : CD, CB : AD, AD : BD and their three reciprocals are all = 1 , and since, therefore, the six an- harmonic ratios of the row are the three simple ratios BA : CA, CB: AB, AC: BC and their three reciprocals (275); hence, from the above — The six anharmonic ratios of any pencil of four rays are equal to the six simple ratios of the three segments, taken in pairs, intercepted by any three of them on any axis parallel to the fourth, CoR. 2°. As every row of four points determines six seg- ments, and ever}' pencil of four rays determines six angles, corresponding two and two in opposite pairs (274), the general property itself, as in fact it was proved above, may be stated otherwise thus, as follows — Every two angles having a common vertex and the two seg- ments they intercut on any axis, and conversely, every two segments having a common axis and the two angles they subtend at any vertex, cut each other equianharmonically. VOL. IX. K 130 THEORY OP ANHARMONIO SECTION. Cor. 3°. When, in Cor. 2°, the common axis of the segments is parallel to a side of one of the angles, it follows evidently, from Cor. 1°, that — For every two angles having a common vertex^ every chord of either parallel to a side of the other is cut by the second side of the latter in the two anharmonic ratios of their mutual section. N.B. Of this latter property that of Art. 224 is evidently a particular case. 286'. Among the immediate consequences from the general property of the preceding article may be noticed the following : 1°. The same pencil of four rays determines equianharmanic rows of four points on all axes / and, the same row of four points determines equianharmonic pencils of four rays at all vertices. For, the several rows, in the former case, are all equian- harmonic with the pencil, and therefore with each other ; and the several pencils, in the latter case, are all equianharmonic with the row, and therefore with each other. 2°. Every two rows of four points or pencils of four rays in perspective with each other are equianharmonic. For, the two rows, in the former case, subtend the centre of perspective by the same pencil of four rays ; and, the two pencils, in the latter case, intersect the axis of perspective at the same row of four points ; and therefore &c. 3°. Every two rows of four points or pencils of four rays in perspective with the same row or pencil are equianharmonic. For, they are both equianharmonic with the row or pencil with which they are both in perspective, and therefore with each other. 4°. Every two rows of three points in perspective with each other form equianharmonic systems with the intersection of their axes ; and, every two pencils of three rays in perspective with each other form equianharmonic systems with the connector of their vertices. For, the two rows, in the former case, combined each with the intersection of their axes, subtend the centre of perspective by the same pencil of four rays ; and, the two pencils, in the latter case, combined each with the connector of their vertices, intersect the axis of perspective at the same row of four points ; and therefore &c. THEORY OF ANHAEMONIC SECTION. 131 5°. A Jixed pencil of four rays determines on a variable line, moving according to any law^ a variable row of four points having a constant anharmonic ratio ; and^ a fixed row of four points determines at a variable point, moving according to any law, a variable pencil of four rays having a constant anhar- monic ratio. For, the variable row, in the former case, is equianharmoDic in every position with the fixed determining pencil; and, the variable pencil, in the latter case, is equianharmonic in eveiy position with the fixed determining row ; and therefore «S:c. 6°. A fixed pencil of three rays determines, on a variable line turning round a fixed point, a variable row of three points forming with the fixed point a system having a constant anhar- monic ratio ; and, a fixed row of three points determines, at a variable point moving on a fixed line, a variable pencil of three lines forming with the fixed line a system having a constant an- harmonic ratio. For, the variable row, in the former case, combined with the fixed point, is equianharmonic in every position with the fixed pencil, combined with the line common to its vertex and the fixed point; and, the variable pencil, in the latter case, combined with the fixed line, is equianharmonic in every position with the fixed row, combined with the point common to its axis and the fixed line ; and therefore &c. N.B. Of the above inferences, 4° is evidently a particular case of 2°, and 6° is evidently a particular case of 5°. 287. The same general property supplies obvious solutions of the two following reciprocal problems, to which, as will appear in the sequel, several others in the theory of anharmonic section may be reduced, viz. — a. Through a given point to draw the line whose intersections with three given lines, which are not concurrent, shall determine, with the given point, a system of four points having, in a given assigned order, a given anharmonic ratio. a'. On a given line to find the point whose connectors with three given points, which are not collinear, shall determine, with the given line, a system of four rays having, in a given assigned order, a given anharmonic ratio. k2 132 THEORY OF ANHAEMONTO SECTION. For, if, in the former case, (fig. a) be the given point ; X, F, Z the three intersections of the required with the three given lines; and A^ B, C the three vertices of the triangle determined by the latter ; then since, by the general property of Art. 285, the three pencils of four rays A.XYZO^ B.XYZO, C.XYZO, which have each three rays given, are all equian- harmonic with the row of four points XYZO; and since, by hypothesis, the anharmonic ratio of the latter is given ; therefore the anharmonic ratios of the former, and with them their fourth ra.ja AX, £Y, CZ are given; and therefore &c. And, if, in the latter case, (fig. a') be the given line ; X, Y, Z the three connectors of the required with the three given points; and A, B, C the three sides of the triangle determined by the latter; then since, by the general property of Art. 285, the three rows of four points A. XYZO, B.XYZO, C.XYZO, which have each three points given, are all equianharmonic with the pencil of four rays XYZO; and since, by hypothesis, the anharmonic ratio of the latter is given ; therefore the an- harmonic ratios of the former, and with them their fourth points AX, BY, CZ, are given ; and therefore &c. These two reciprocal solutions may be briefly summed up in one as follows: Since, in both cases alike, by the general property of Art. 285, the three systems A. XYZO, B.XYZO, C.XYZO, which have each three constituents given, are equi- anharmonic with the system XYZO, whose anharmonic ratio is, by hypothesis, given ; therefore their fourth constituents AX, BY, CZ are implicitly given; and therefore &c. N.B. The preceding problems are manifestly indeterminate or impossible, the former when the three given lines are con- current, and the latter when the three given points are coUinear; for, by 6° of the preceding article, the anharmonic ratio of the THEORY OF ANHAEMONIC SECTION. 133 system XYZO is then constant.^ whatever be the direction of its axis in the former case, or the position of its vertex in the latter case ; and therefore &c. (59). 288. The two following reciprocal inferences from the same general property are, evidently, the converses of the two em- bodied, in a single statement, in 2°, Art. 286, viz. — a. When two equianharmonic rows of points on different axes A, B^ C, D and A', B', C, D' are such that three of their pairs of corresponding points A and A\ B and B\ C and C connect by lines AA\ BB\ GC passing through a common point 0, the fourth pair D and D' connect also by a line DD' parsing through iJie same point 0. a'. When two equianharmonic pencils of rays through different vertices A, B, C, D and A', B\ C, B are such that three of their pairs of corresponding rays A and A\ B and B\ C and C intersect at points AA\ BB\ CO' lying on a common line 0, the fourth pair D and D' intersect also at a point DD' lying on the same line 0. For, in the former case (figs, a and /8), the two rows of points A, B, C, D and A', B', C", D' being, by hypothesis, equianharmonic, so therefore (285) are the two pencils of rays OA, OB, OC, OD, and 0A\ OB', OC, OD'-, but three pairs of corresponding rays of those two equianharmonic pencils OA and OA', OB and OB', OC and OC coincide; therefore the fourth pair OD and OD' coincide also; and therefore &c. And, in the latter case (figs, a' and ^'), the two pencils of rays A, B, 0, D and .^4', B'. C, !>' being, by hyno*'---'''. enuian- 134 THEORY OP ANHARMONIC SECTION. harmonic, so therefore (285) are the two rows of points OA, OB, OC, OD, and OA', OB', OC, OB' ; but three pairs of corre- sponding points of those two equianharmonic rows OA and OA, OB a,nd OB', OC and OC coincide; therefore the fourth pair OB and OB^ coincide also ; and therefore &c. The above reciprocal properties may be briefly summed up in one as follows — When, of two equianharmonic rows of four points or pencils of four rays A, B, C, B and A', B', C, B', three pairs of corre- sponding constituents A and A', B and B', G and C are in perspective (130], the fourth pair B and B' are in perspective tvith them. And so also may the reciprocal demonstrations above g^ven of them, as follows — Since, in both cases, by hypothesis, {ABCB} = {A'B'C'B'} ; therefore, in both cases, by (285), {O.ABCB} = {0. A'B'C'B'}, O being the centre (or axis) of perspective of A and A', B and B', C and C" ; but, in both cases, by hypothesis, OA = OA', OB=OB', OC=^OC'; therefore, in both cases, OB=OB'', and therefore &c. 289. The two following, again, are very important particular cases of those of the preceding article ; and are, also, evidently, the converses of the two combined, in a single statement, in 4°, Art. 286, viz.— a. When two equianharmonic rows of points on different axes B, C, B, E and B', C, U, E' are such thai a pair of their cor- responding points E and E' coincide at the intersection of the aaxs, the remaining three pairs B and B', C and C, B and Bf THEORY OF ANHAEMONIC SECTION. 135 connect hy three lines BB', CC, DD' passing throvgh a common point 0. a'. When two equianharmonic pencils of rays through different vertices B, C, Z>, E and B', C", D', E' are such that a pair of their corresponding rays E and E' coincide along the connector of the vertices, the remaining three pairs B and B\ C and C\ D and D' intersect at three points BB\ CC, DD' lying on a common line O. For, if, in the former case, (figa. a and yS of the preceding article), be the intersection of any two of them BB' and GC 5 then, the two rows of points B, C, B, E and B', C, £>', E' being, by hypothesis, equianharmonic, so therefore (285) are the two pencils of rays OB, OC, OB, OE and OB', OC, OB', OE'; but three pairs of their corresponding rays OB and OB', 00 and OC, OE and OE' coincide; therefore the fourth pair OD and OD' coincide also ; and therefore &c. And if, in the latter case, (figs, a and /3' of the preceding article), be the connector of any two of them BB' and CC ; then, the two pencils of raya B, C, D, E and B', C, D', E' being, by hypothesis, equian- harmonic, 80 therefore (285) are the two rows of points OB, 00, OD, OE and OB', OC, OD', OE' ; but three pairs of their corresponding points OB and OB', 00 and OC, OE and OE' coincide ; therefore the fourth pair OD and OD' coincide also ; and therefore &c. Like those of the preceding article, of which they are im- portant particular cases, the above reciprocal properties may be briefly summed up in one, as follows — When, of two equianharmonic rows of four points or pencils of four rays B, 0, D, E and B', C, D), E', one pair of corre~ spofiding constituents E and E' coincide, the remaining three pairs B and B', G and C, D and U are in perspective. And so, like those of the same, may the reciprocal demon- strations above given of them, as follows — Since, in both cases, by hypothesis, [BGDE] = [B'CD'E'] ; therefore, in both cases, by (285), [0 .BCDE] = [O.B' CUE'], being the point (or line) common to the two lines (or points) BB' and GG' ; but, in both cases, by hypothesis, OB=OB', OG=OC'j OE=OE'; therefore, in both cases, OD=OU', and therefore &c. 136 THEORY OF ANHARMONIC SECTION'. 290. The two following, again, are useful inferences from the important reciprocal properties of the preceding article, viz. — a. Every two equianharmonic rows of points on different axes A,B,C,D and A', B', C\ D" determine two pencils of rays in perspective at any two vertices and 0" lying on the line of connection AA' of any of their four pairs of corresponding points A and A'. a. Eeery two equianharmonic pencils of rays through different vertices A, B, C, D and A', B', C, D' determine two rows of points in perspective on any two axes and 0' passing through the point of intersection AA' of any of their four pairs of cor- responding rays A and A'. For, in the former case, the two rows of points A, B, C, D and A', B', C", J)' (fig. a) being, by hypothesis, equianharmonic, BO therefore (285) are the two pencils of rays OA, OB, OC, OD and O'A', O'B', 0'G\ O'D'; but, for the two vertices and 0', the pair of corresponding rays OA and OA', by hypothesis, coincide ; therefore (289, a') the three remaining pairs OB and 0'B\ OC and O'C", OD and O'J?', intersect at three points Q, R, S, lying on a common line /; and therefore &c. And, in the latter case, the two pencils of rays A, B, C, D and A', B', C", B' (fig. a') being, by hypothesis, equianharmonic, so therefore (285) are the two rows of points OA, OB, OC, OB and OA', O'B', O'C, O'D'-, but, for the two axes and 0', the pair of corresponding points OA and O'A', by hypothesis, coincide ; therefore (289, a) the three remaining pairs OB and O'B', OC and O'C, OD and O'D' connect by three lines Q, S, 8 passing through a common point /; and therefore &c. THEORY OF ANHARMONIC SECTION. 137 As, in the two preceding articles, these two reciprocal de- monstrations may be briefly summed up in one as follows — Since, in both cases, {ABCD} = {A'B'G'n'}; therefore, in both, {0. ABCD} = {0'.A'B'C'I)'} ; but, in both cases, OA and O'A' coincide; therefore, in both, the two systems OB, OC, OD and (yB', 0' C", O'D' are In perspective ; and therefore &c. N.B. In the above reciprocal properties, the two points (or lines) and 0' might, of course, coincide respectively with the two A' and A ; the properties themselves as above stated, and their demonstrations as above given, would remain unchanged ; but the axis (or centre) of perspective / of the two pencils (or rows) O.ABCD and O'.A'B'G'I)' would then have certain im- portant relations with respect to the two determining rows (or pencils) A, B, C, D and A', B', C", Z>', which will be considered at length in another chapter. 291. The two reciprocal properties of the preceding article supply ready solutions, by linear constructions only, without the aid of the circle, of the two following reciprocal problems, viz. — a. Given three pairs of corresponding constituents A and A\ B and B\ C and C of two equianharmonic systems of points A, B, C, D and A\ B', C, D' on different axes, and the fourth point D of eiih:r system; to d-termine the fourth point D' of the other system. a'. Given three pairs of coTvespondinff constituents A and A', B and B', C and C of two equianharrnonic systems of rays A, B, G, D and A', B', C", -D' through different vertices, and the fourth ray D of either system ; to determine the fourth ray D' of the other system. For, in the former case, from any two points and 0', taken arbitrarily on the line of connection AA' of any one of the three given pairs of corresponding points A and A' (fig. a of preceding Art.), drawing the two pairs of lines OB and O'B', OG and O'C intersecting at the two points Q and R; and from the point of intersection 8 of the two lines QR and OD drawing the line SO ; the latter line, by property (a) of the preceding article, intersects with the axis L' of the system Ay B\ C\ If at the required point B'. And, in the latter case, 138 THEOEY OF ANHAEMONIC SECTION. on any two lines and 0', drawn arbitrarilj through the point of intersection AA' of any one of the three given pairs of corre- sponding rays A and A' (fig, a' of preceding Art.), taking the two pairs of points O^and (JB', OC and O'C connecting by the two lines Q and B; and on the line of connection S of the two points QB and OD taking the point SCf ; the latter point, by property (a) of the preceding article, connects with the vertex L' of the system A', B', C\ U by the required ray D'. These two reciprocal constructions may be briefly summed up in one as follows : The two given points or rays A and A give the line or point AA' ; on or through which are taken or drawn arbitrarily the two points or lines and 0' ; which, with the two given pairs of points or lines B and B',, C and C, give the two pairs of lines or points OB and O'B', OC and O'C; which give the two points or lines Q and B ; which give the line or point QB; which, with the line or point OB, gives the point or line S; which, with the point or line 0', gives the line 80' ; which, with the axis or vertex L' of the system A', B', C", IX, gives the required fourth point or ray B'. N.B. As noticed at the close of the preceding article, the two assumed points or lines O and 0', in the two preceding re- ciprocal constructions, naight be taken to coincide with the given two A' and A respectively ; but no simplification worth mentioning would be obtained by so taking them. In the particular case when the three given pairs of corre- sponding constituents A and A', B and B', C and C are in per- spective (130), either constituent of the remaining pair D and ly is given immediately with the other, being, by (288), in perspective with it to the same centre or axis. 292. Every row of four points ia equianhamumic with the pencil of four rays determined ly their four polara with respect to any circle ; and, conversely, every pencil of four rays is equi- anharmonic with the row of four points determined by their four poles with respect to any circle (166, Cor. 1°). For, in either case, the pencil determined by the four rays being similar to that subtended by the four points at the centre of the circle (171, 2°); and the latter pencil, by virtue of the general property of Art. 285, being equianharmonic with the THEt5RY OF ANHARMONIC SECTION. 139 row determined by the four points ; therefore &c. The property of harmonic section established in Art. 223 is evidently a par- ticular case of this. In the applications of the theory of anharmonic section, the above property, from which it appears that all anharmonic ratios, whether of rows of points or pencils of rays, are preserved unchanged in reciprocation (172), ranks next in importance to that of Art. 285, from which, as above demonstrated, like its particular case already referred to, it is indeed an inference. Sy virtue of it all anharmonic properties of geometrical figures are in fact double, every anharmonic property of any figure being accompanied by a corresponding anharmonic property of its re- ciprocal J^ure to any circle (172), the estahlishment of eilJier of which involves that of the other without the necessity of any further demonstration (173). As, in the applications of the theory of harmonic section given in Chapters XIV. and XV., the principal anharmonic properties of figures consisting only of points and lines, and also of figures involving circles so far as their re- ciprocals are properties involving no higher figures (173), will be given in the next and following chapters, arranged for the most part in reciprocal pairs, placed in immediate connection with each other, and marked by corresponding letters, accented and unaccented, so as to keep the circumstance of this duality, which forms such a remarkable feature in modem geometry, continually present before the reader, and furnish him at the same time with numerous additional examples by which to exercise and perfect himself in the reciprocating process de- scribed in Art. 172, and already exemplified at some length in the chapters referred to. The five articles immediately preceding the present furnish obvious examples of this mode of arrange- ment; and, until the closing chapter, where It would be in- admissible for the reason mentioned in Art. 173, the same will be adhered to as systematically as possible throughout the remainder of the work. ( 140 ) CHAPTER XVII. ANHARMONIC PROPERTIES OF THE POINT AND LINE. 293. In the applications of the theory of anharmonlc section to the geometry of the Point and Line, the two following properties, reciprocals of each other, present themselves so frequently that we shall commence this chapter on the subject with their statement and proof. If A^ B^ C be any three points lying on a line [or lines passing through a point) 0, and A', B', C any three others lying on another line [or passing through another point) CX / t?ie three intersections [or connectors) A", B'\ C" of the three pairs of connectors [or intersections) BC and B'C, GA' and G'A, AB' and A'B lie on a third line [or pass through a third point) 0" ; which determines with and 0' a triangle OO'O", whose opposite vertices [or sides) I, I', I" are connected with the three collinear [or concurrent) triads A, B, G; A',B\ G' ; A!',B'\ G" by the three groups of equianharmonic relations {BGI'I") = {B'GT'I} = {B"G"IT]. [GAIT] = [G'ATI] = {G"A"Il'} [ (1), {ABI'I"] = {A'B'ri} = {A"B"IT} J and, as a consequence from them, aho by the two {ABGT } = {A'B' C'l"} = {A"B" G"I }] {ABGI"} = {A'B'C'I ] = [A"B"G"T}) ^^^' For, if 0" be the line of connection (fig. a) (or the point of intersection (fig, a')) of some two. A" and B" suppose, of the three points (or lines) A", B", G"; and /, J','^!" the three opposite vertices (or sides) of the triangle determined by the three lines (or points) 0, 0\ 0"; then, since the two triads of points (or lines) B, G, /' and G', B', 1 are in perspective, therefore, by (286, 4°) [BGri"} = [G'B'II"} = {B'G'T'I} (280) ; ANHAEMONTC PROPERTIES, &C. 141 and, since the two triads of points (or lines) C, A, T and A\ C, /are In perspective, therefore, by the same, {CAT I"} = {A'C'ir} = {C'ATI} (280) ; therefore, by the general property (1°), Art. (282), {ABIT} = [A'BTI} = {B'A'II"] (280) ; and therefore, by (289), the two triads of points (or lines) A, B, T and B', A', I are in perspective ; which proves the first parts of both properties, in which, as is evident from the figures and mode of establishment, any two of the three colliuear (or concurrent) triads A, B, C] A', B\ C ; A'\ JB", G" may be regarded as the original pair, and the third as that derived from them by the construction involved in the corresponding statement, whichever it be. To prove the second parts of both properties ; since, in either case, as shown above for the two original triads A, B, C and a;b;c; {BCri") = {£' CI" I], [ GAIT] = { CAT I], [ABIF] = [ABTI] ; therefore, by cyclic interchange between each and the derived triad A\ B", C", [BGIT] = [B'Cri] = {B"G"ir], [GAIT] = [CAT I] = [C'A'II'}, [ABir] = [ABTI] = [A"B"ir} ; which are the relations (I) as above stated ; and, as from them the relations (2) follow immediately in virtue of the general property (2°), Art. (282), therefore &c. 142 ANHAEMONIC PROPEETIES In the particular case when any two of the three triads, A, B, C and A', B\ C suppose, are in perspective ; it is evident, from (240), that the line of coUinearity (or point of concurrence) 0" of the third A", B", C" is the polar of their centre (or the pole of their axis) of perspective, with respect to the angle (or segment) 00' determined by their two lines of coUinearity (or points of concurrence) O and 0' (217). In that case the three lines (or points) 0, 0\ 0" being concurrent (or coUinear), and the three points (or lines) J, /', J" consequently coincident, therefore, by relations (2) above, {ABCI} = {A'B'C'I} = {A"B"C"I}, and therefore, by (289), the three triads A, B, C; A', B\ C; A" J B'\ C" are two and two in perspective. Hence — When, of three collinear [or concurrent) triads A, B, C/ A', B', C / A", B'\ 0" connected cyclically as above, any two are in perspective, then every two are in perspective ; and the axis {or vertex) of each is the polar of the centre [or the pole of the axis) of perspective of the other two, with respect to the angle [or segment) determined hy their axes [or vertices). 294. Among the numerous inferences from the two reci- procal properties of the preceding article, the following, in pairs reciprocals of each other, are deserving of attention. r. The three pairs of pomts (or lines) A and A', B and B', C and C may be regarded as determining three segments (or angles) AA', BB', CC, which, taken in pairs, have a common centre (or axis) of perspective /", and of which, taken in pairs, the three points (or lines) A", B", G" are the remaining three centres (or axes) of perspective; and similarly for the three pairs A and A", B' and 5", C and C", and for the three ^"and^, ^"and^, C" and G; hence, generally, from the first parts of the two reciprocal properties in question — a. When, of three segments taken in pairs, three of the six centres of perspective coincide, the remaining three are collinear, a'. When, of three angles taken in pairs, three of the six axes of perspective coincide, the remaining three are concurrent. 2°. The two triads of points (or lines) A, B, C and A', B' C' may be regarded as the two sets of three alternate vertices (or sides) of a hexagon AB'CA'BG', of which A and A', OP THE POINT AND LINE. 143 B and B\ G and C are the three pairs of opposite vertices (or sides), and A'\ B", C" the three intersections (or connectors) of the three pairs of opposite sides (or vertices) BC and B'C, CA' and O'A^ AB' and A'B; and similarly for the two triads A\ B\ C and A'\ B'% C", and for the two A', B", C" and A, B, C; hence, generally, from the same again — a. When, of a hexagon, both triads of alternate vertices are coUinear, the three intersections of opposite sides are coUinear. of. When, of a hexagon, both triads of alternate sides are concurrent, the three connectors of opposite vertices are concurrent. 3°. The three triads of points (or lines) A, B, C; A', B', C; A", B", C" may be regarded as determining six different cycles of three triangles, each inscribed to one and exscribed to the other of the remaining two, viz. — BA'C, B'A"C', B"AC" and BA"C, B'AC, B"A'C", CBA, G'B'A', C"BA" and CB"A, C'BA', G"BA% AG'B, A'C'B', A"GB" and AG"B, A'GB', A"G'B", for each of which the three points (or lines) /, F, I" are the points of intersection (or lines of connection) of the three pairs of corresponding sides (or vertices) 0" and 0", 0" and 0, O and 0' respectively; and similarly for the remaining pairs of corresponding sides (or vertices) ; hence, generally, from relations (1) and (2) of the preceding article, respectively — In every cycle of three, triangles each inscribed to one and exscribed to the other of the remaining two. a. The sides and opposite vertices of each divide equianhar- monically the corresponding sides of that to which it is inscribed. a'. The vertices and opposite sides of each divide equianhar- monically the corres^ponding angles of that to which it is exscribed. b. The pairs of corresponding sides of every two intersect equianharmonically the corresponding sides of the third. b'. 27ie pairs of corresponding vertices of every two subtend equianharmonically the corresponding angles of the third. 4°. When, for the system of three triangles constituting any one of the six cyelea, B A' G, B A" C', B" AC" suppose, in the preceding, the equlanharmonic section in properties (a) and (a) is harmonic; since then, by the two reciprocal properties (a) and (a) of Art. (243), the three triangles constituting the 144 ANHAEMONIC PKOPEETIES cycle are two and two in perspective, and conversely; and since always, by the first parts of the two reciprocal properties of the preceding article, the three triads of points of intersection (or lines of connexion) of BA' aniB'A", CA' ani C A", B'B" and C'C", B'A" and B"A, CA" and C"A, B"B and 0"C, B"A and BA\ C"A and CA\ BB' and CC, are collinear (or concurrent) ; hence generally, as noticed in Cors. (11°) and (13°), Art. (261)— In a cycle of three triangles each inscribed to one and exscribed to the other of the reraaining two. a. When any two of the three are in perspective, every two of the three are in perspective. b. For each triangle, its centre of perspective with that to which it is inscribed lies on its axis of perspective with that to which it is exscribed, and reciprocally, its axis of perspective with that to which it is exscribed passes through its centre of per- spective with that to which it is inscribed. 5°. In the two triads of points (or lines) A, B, and A, B', C, if, while the three constituents A, B, C of either and any two A' and B' of the other are supposed to remain fixed, the third constituent C ' of the latter be conceived to vary, causing of course the simultaneous variation of the two consti- tuents A" and B" of the third triad A!', B", C" ; since then, in every position of the variable triangle A" C'B", the three vertices (or sides) lie on the three fixed lines (or pass through the three fixed points) B'C, A'B', CA', while the three sides (or vertices) pass through the three fixed points (or lie on the three fixed lines) A, C", B ; hence again, generally, from the first parts of the reciprocal properties of the preceding article — a. When, of a variable triangle whose three vertices move on fixed lines, two of the sides turn round the corresponding vertices of any fixed triangle exscribed to that determined by the lines, the third turns round its third vertex. a'. When, of a variable triangle whose three sides turn round fixed points, two of the vertices move on the corresponding sides of any fixed triangle inscribed to that determined by the points, the third moves on its third side. 8f the point and line. 145 6 . When the two fixed triangles, in the two reciprocal properties a and a of the preceding (5°), are in perspective ; since then, by property a of (4°), the variable triangle, in every position, is in perspective with both ; and since also, by property h of the same, its centre of perspective with that to which it 18 exscribed lies, in every position, on the axis of perspective of the two, while its axis of perspective with that to which it is inscribed passes, in every position, through the centre of perspective of the two ; hence, generally, from those properties combined, it appears that — In a cycle of three triangles, each inscribed to one and ex- scribed to the other of the remaining two and in perspective with both ; if, while two of the three are supposed to remain fixed, the third be conceived to vary, then — a. The centre of perspective, of the variable with the fixed triangle to which it is exscribed, moves on the axis of perfective of the two fixed triangles. a'. The axis of perspective, of the variable with the fixed triangle to which it is inscribed, turns round the centre of per- spective of the two fixed triangles. 7°. Since, in every cycle of three triangles each inscribed to one and exscribed to the other of the remaining two; by virtue of the two reciprocal properties a and a' of (3°), or of either of them combined with the general property of Art. 285, the opposite vertices and sides of each divide in the same anharmonic ratios, the corresponding sides of that to which it is inscribed, and the corresponding angles of that to which it Is exscribed ; which three sets of equal anharmonic ratios are of course fixed when two of the three triangles of the cycle are fixed, however the third may vary; hence, generally, from those properties com- bined with those of (5°), it appears that — In any cycle of three triangles, each inserted to one and exscribed to the other of ike remaining two; if, while two of the three are supposed to remain fixed, the third be conceived to vary, then of the latter with respect to the two former — a. The opposite vertices and sides divide in the same constant anharmonic ratios, the corresponding sides of that to which it is inscribed, and the corresponding angles of that to lohich it is exscribed. VOL. II. L 146 ANHARMONIC PROPERTIES b. The opposite siiles and angles are divided in the same constant anharmonic ratios, the former by the corresponding vertices and sides of that to which it is exscribed, and the latter by the corresponding sides and vertices of that to which it is inscribed. 8°. Since again, by virtue of the same two reciprocal pro- perties, or of either of them combined with that of Art. 285, the opposite vertices and sides of every two triangles, either inscribed to a third and exscribed to a fourth exscribed to the third, or exscribed to a third and inscribed to a fourth inscribed to the third, divide, in equal anharmonic ratios, the correspond- ing sides of that to which they are inscribed, and the correspond- ing angles of that to which they are exscribed ; while their sides and angles are divided in the same equal anharmonic ratios, the former by the corresponding vertices and sides of that to which they are exscribed, and the latter by the corresponding sides and vertices of that to which they are inscribed ; hence, again, from the same, conversely, as may also be easily shewn directly, it appears that — a. When, of two triangles inscribed to a third, the opposite vertices and sides divide in any equal anharmonic ratios the cor- responding sides of the third / the intersections of their pairs of corresponding sides determine a fourth triangle, inscribed to each of themselves and exscribed to the third, whose opposite sides and vertices divide in the same equal anharmonic ratios their pairs of corresponding sides and the corresponding angles of the third. a'. When of tioo triangles exscribed to a third, the opposite sides and vertices divide in any eqyuil anharmonic ratios the cor- responding angles of the third; the connectors of their pairs of corresponding vertices determine a fourth triangle, exscribed to each of themselves and inscribed to the third, whose opposite vertices and sides divide in the same equal anharmonic ratios their pairs of corresponding angles and the corresponding sides of the third. 9°. In the particular case when, in the two reciprocal pro- perties a and a of the preceding (8°), the three seta of equal anharmonic ratios are all harmonic ; since then, by (243), the several pairs of corresponding triangles are in perspective, and OF THE POINT AND LINE. 147 conversely ; hence from those properties combined with those of the latter article, it appears that — a. When two triangles are each inscribed to a third and in perspective with it ; the intersections of their pairs of corresponding sides determine, a fourth triangle, inscribed to each of themselves, exscribed to the third, and in perspective with all three. a. When two triangles are each exscribed to a third and in perspective with it; the connectors of their pairs of corresponding vertices determine a fourth triangle, exscribed to each of themselves, inscribed to the third, and in perspective with all three. 10°. Since, when two triangles are each inscribed or exscribed to a third, the sides of the third are the connectors of their pairs of corresponding vertices in the former case, and the vertices of the third the intersections of their pairs of corres- ponding sides in the latter case ; it appears, consequently, from the two separate parts of either of the two reciprocal, and also converse, properties of the preceding (9°), that — a. Of two triangles whose vertices and sides correspond in pairs ; when the connectors of their pairs of corresponding vertices determine a common exscribed triangle in perspective with both, the intersections of their pairs of corresponding sides determine a common inscribed triangle in perspective with both ; and con- versely.* • The above is evidently a particular case of the following : Of two triangles whose vertices and sides correspond in pairs t when the connectors of their pairs of corresponding vertices determine a common exscribed triangle in perspective with either, the intersections of their pairs of corresponding sides determine a common inscribed triangle in perspective with the other : and conversely. Which may be proved readily, from the general relations a and 6' of Art. 134, as follows : If X, Y, Z be the three vertices of one of the original triangles ; X', Y' Z' those of the other ; A, B, C those of the common exscribed triangle determined by the three connectors j and A', B', C those of the common inscribed triangle determined by the three intersections; since then always, in virtue of relation a of the article in question, CX ' AY ' BZ ZA' ■ X'B' ' Y'C ^ '' *°^ BX' CY AZ' X^ ZB_ XC 'CX' 'AY'BZ'^ ZA ' XB' ' YC ^ ^' L2 148 ANHAEMONIC PEOPEBTIES b. Of the two derived thus each in perspective with both the original triangles; that exscribed to the original two is inscribed to the other of themselves, that inscribed to the original two is exscribed to the other of themselves, and they are also in perspec- tive with each other. 295. The two following properties, reciprocals of each other, form, as explained in Art. 140, the basis of the theory of per- spective in modern geometry; and establish, at the same time, the equianharmonic relations connecting the several pairs of corresponding points and lines of every two figures in perspec- tive with their centre and axis of perspective (141). If A,B, C be any three points on three concurrent lines (<»' lines through three collinear points) ; A', B', C any three other points on the same lines {or lines through the same points) ; and X, Y, Zthe three intersections [or connectors) of the three pairs of connectors {(yr intersections) BC and B'C, CA and C'A', AB and AB' ; the three points {or lines) X, Y, Z are collinear {or concurrent) ; and their line of collinearity {or point of con- currence) I intersects with the three lines {or connects with the three points) AA', BB', CC at three points {or by three lines) U, V, W which, with their point of concurrence {or line of col- linearity) 0, determine the group of equianharmonic relations [AA'OU] = [BB'OV] = [CC'OW]. For, if I be the line of connection (fig. a) (or the point of intersection (fig. a')) of any two, X and Y suppose, of the three points (or lines) X, Y, Z; and JJ, V, Wits three points of in- tersection (or lines of connection) with the three lines (or points) AA', BB', CC ; then, since the two triads of points (or lines) B, B', Fand 0, C, Ware in perspective, therefore, by (286,4°), {BB'OV} = [CC'OW}; and, since the two triads of points (or lines) C, C, W and A, A', U are in perspective, therefore, by the same, [CC'OW]=[AA'OU]; from which, since imme- coDsequently, when either equivalent of either relation = - 1, the other also c - 1 ; that is, in virtue of relation V of the same, when XYZ and ABC are in perspective, then X'Y'Z' and A' B'C axe in perspective, and con- versely, by (l)j and when X'Y'Z' and ABC are in perspective, then XYZ and A'B'C are in perspective, and conversely, by (2) ; and therefore &c. OF THE POINT AND LINE. 149 diately, [AA'OU} = {BB'OV}, therefore, by (289), the two triads of points (or lines) A, A\ U and J5, B', V are in perspective ; and therefore &c., the second parts of both properties having been established in the demonstrations of the first. COE. 1°. If X\ F', Z" be the three intersections (or con- nectors) of the three pairs of connectors (or intersections) BC and B'C, CA' and C'A, AB' and A'B-, it may, of course, be shewn, in precisely the same manner, that the three triads of points (or lines) Y\ Z\ Z; Z', X', Y; X', Y\ Z are also collinear (or concurrent) ; their three lines of coUinearity (or points of concurrence) determining, with that of the triad X, Yj Zj a tetragram (or tetrastigm), of which the three pairs of corresponding points (or lines) X and X', Y and Y\ Z and Z' are the three pairs of opposite intersections (or connectors) ; and each line (or point) determining, by its intersections (or connectors) with the original three, a group of equiauharmonic relations similar to the above, and differing only in the inter- change of the constituents of the corresponding reversed pair of the three A and A\ B and B\ C and C in the equivalent which contains them. COE. 2°. In the particular case when the three equianhar- monic systems of points (or rays) A, A\ 0, U; B, B', 0, F; C, C", 0, W are harmonic, that is, when the three segments (or angles) AA', BB', CO' are cut harmonically by the three OU, OV, OW; since then (281), [AA'OU] = [A-AOU], [BB-OV] = [B'BOV], {CC'OW]^{G'COW], 150 ANHARMONIC PROPERTIES therefore the three points (or lines) Z7, V, Ware the same for the three lines of coUinearity (or points of concurrence) of the three triads r, Z', X; Z\ A", T; X\ Y\ Z' as for that of the triad X, F, Z\ and therefore the whole six points (or lines) X and X', Y and F', Z and Z' lie on the same line (or pass through the same point) /; which, in that case, is consequently, the common polar of the point (or the common pole of the line) with respect at once to the six angles (or segments) determined hy the six pairs of opposite connectors (or intersections) BG and B' C\BG' and B'C; C^ and C'A\ CA' a.iiAC'A; AB and A'B', AB' and A'B (217); a property which is also evident from the two reciprocal properties 5° and 6° of Art. 222. The three pairs of points (or tangents) A and A', B and B', C and C determined by any circle on any three con- current lines (or at any three collinear points) 0Z7, OF, OW furnish an obvious and important example of the particular case in question. For, the three pairs of points (or lines) XandX', Fand F', Z aaA Z' lying on the polar (or passing through the pole) / of the point of concurrence (or line of col- linearity) with respect to the circle (261); and the three A and A', B and B\ C and C being consequently pairs of harmonic conjugates with respect to the three and Z7, and F, and W^ respectively (259) ; therefore &c. Cor. 3°. In the same case it is evident also, from the general property of Art. 285, that, of the six points (or lines) A and A', B and B\ C and C", every two conjugates connect (or intersect) with the remaining four equlanharmonically ; for since, by the general property in question, {A.BB'CC] = {ZZ'YY' } = {A'.BB'CC} (1°), {B. CC'AA'} = {XX'ZZ' } = {B. CC'AA'} (2°), {C.AA'BB'] = {YY'XX'} = {C',AA'BB'} (3°), therefore &c. That, in the same case, the same property is true of every two of the six points (or lines) A and A', B and B', C and C, whether conjugates or not, might be shewn without difficulty ; but a more general property, which will include it as a particular case, will form the subject of a subsequent article of the present chapter. OP THE POINT AND LINE. 151 Cor. 4°. That the triad of points (or rays) CT, V, W de- termines with the triad X, Y, Z in the general case, and therefore also with each of the three triads F,Z',Z; Z\X\ F; X\ Y\ Z in the particular case considered in the two preceding corollaries, a system of six constituents corresponding two and two in opposite pairs, every four of which are equianharmonic with their four opposites (283), may be easily shewn as follows : The three pencils of four rays (or rows of four points) O.VVWX, O.UVWY, O.UVWZ, (see figures) being in perspective with the three A.UZYX, B.ZVXY, C.YXWZ, on the three lines (or at the three points) BG, CA, AB respec- tively, or with the three A.UZYX, B'.ZVXY, C.YXWZ, on the three lines (or at the three points) B' C, CA', A'B' re- spectively ; therefore, from either set of perspectives, by the general property of Art. 285, { UVWX] = {UZYX } = [XYZU ] (280) (1), { UVWY] = [ZVXY j = {XYZV ] (280) (2), {UVWZ] ={YXWZ} = [XYZW] (280) (3), and therefore &c. (283). That the two triads X, Y, Z and X', Y', Z' when coUinear (or concurrent), Cors. 2° and 3°, are connected by the same relation, might be shewn, without diffi- culty, either from the above or independently ; but another more general property, under which it will come as a particular case, will form the subject of another subsequent article of the pre- sent chapter. COE. 5°. If E and E', F and F, G and G' be the three pairs of intersections (or connectors), with the three pairs of con- nectors (or intei-sections) BG and B'C', CA and CA\ AB and A!B' respectively, of any line passing through (or point lying on) 0', and K its intersection (or connector) with /; It may be easily shown, from the general property of Art. 285, that the three quartets of points or lines E, E', 0, K; F, F\ 0, K; Gf G', 0, K are equianharmonic with each other and with the 162 AN HARMONIC PBOPEKTIES three A, A', 0, U; B, B', 0, V; C, G\ 0, W. For since in either case, by the property in question, {EE'OK} = {BB'OV } = {CC'OW} (1), {FF'OK} = {CC'OW} = {AA'OU] (2), {GG'OK} = {AA'OU} = {BB'OV} (3), therefore &c. Three similar triads of equianharmonic quartets, result, of course, on the successive interchanges (as in Cor. 1°) of A and A', B and B', C and C in the constructions deter- mining ^ and ^', ^and^, G And G' as above given; all of which, in common with the original triad, are evidently alike harmonic in the particular case considered in Cor. 2°. The three pairs of points (or tangents) A and A', B and B\ C and C, determined by any circle on any three concurrent lines (or at any three collinear points) OU, OV, OW^ furnish, as observed at the close of that corollary, an important example of a case in which they are all thus harmonic. COE. 6°. In the same case it follows immediately, from the general property (2°) of Art. 282, that the two triads of points (or lines) E, F, G and E\ F, G' of the preceding corollary (5°) determine equianharmonic systems, both with the point (or line) 0, and with the point (or line) K. For since, by the property In question, the two equianharmonic relations of the preceding corollary, viz. : {EE'OK} = {FF'OK] = {GG'OK} (1), give immediately the two {EFGO] = [E'F'G'0] and [EFGK] = [E'F'G'K\ ...{2), therefore &c. When, as for the three pairs of points (or tangents) A and A\ B and B\ G and C" determined by any circle on any three concurrent lines (or at any three collinear points) OU^ OVj OW, the three equianharmonic systems in (1) are all harmonic ; it follows also, from (3°, Cor. 5°) of the same article (282), that the same two triads E, F, G and E', F, G' determine in three opposite pairs E and E', Fani F, G and G' a system of six collinear (or concurrent) constituents, every four of which are equianharmonic with their four opposites (283). OF THE POINT AND LINE. 153 296. Among the various inferences from the two reciprocal properties of the preceding article, the following, in pairs re- ciprocals of each other, are deserving of attention. 1°. The three pairs of points (or lines) A and ^', B and B\ C and C", may be regarded as determining three segments (or angles) AA\ BB\ CC; whose axes (or vertices) are concurrent (or coUinear), and of which, taken in pairs, the three points (or lines) X, F, Z are three of the six centres (or axes) of per- spective, every two of which become changed into their opposites by the interchange of extremities of one of the two determining segments (or angles), those of the other remaining unchanged ; hence, generally, from the two properties in question, as shewn in part on other principles in Art. 146, it appears that — a. When the axes of three segments are concurrent^ the six centres of perspective of the three pairs tliey determine lie, three and three, on four lines ; each of which, with the point of con- currence of the axes, divides the three segments equianhar- monically. a'. When the vertices of three angles are collinear, the six axes of perspective of the three pairs they df-termine pass, three and three, through four points ; each of which, with the line of collinearity of the vertices, divides the three angles equianhar- monically. 2°. The two triads of points (or lines) A, B, C and A', B', C may be regarded as determining two triangles ABC and A'B'C; the connectors (or intersections) AA', BB', CC of whose pairs of corresponding vertices (or sides) A and A', B and B', C and C are concurrent (or collinear) ; and so may also the three pairs of triads B, C, A' and B', C, A; C, A, B' and C, A', B; A, B, C and A', B', C resulting from them by the three different interchanges of corresponding constituents. Hence again from the same, and from the obvious inference from them contained in Cor. 5° of the preceding article, it appears generally, as shewn in part on other principles in Art. 140, that — JFor two triangles whose vertices and sides correspond in pairs— a. When the connectors of the three pairs of corresponding vertices are concurrent, the intersections of the three pairs of corresponding sides are collinear/ and, reciprocally, when the intersections of the three pairs of corresponding sides are coUinear, 154 ANHAEMONIC PEOPEETIES the connectors of the three pairs of corresponding vertices are con- current.* b. When thus related to each other, the line of collinearity and the point cf concurrence divide eqydanharmonically, at once the three segments determined hy the three pairs of corresponding vertices, and the three angles determined hy the three pairs of corresponding sides, c. In the same CMse, more generally, the same point and line divide equianharmonically , at once all segments determined by their pairs of corresponding sides on lines passing through the former, and all angles determined by their pairs of corresponding vertices at points lying on the latter. When two triangles, related as above to each other, are either both inscribed or both exscribed to the same circle ; they furnish, see Cor. 2° of the preceding article, an important example of the particular case in which the several equi- anharraonic sections in the two latter properties b and c are all harmonic. Hence, since, by 3°, Cor. 5°, Art. 282, every three pairs of points or lines, harmonic conjugates with respect to the same pair, determine a system of six colHnear or con- current constituents every four of which are equianharmonic with their four conjugates, it follows from the above (c), that — When two triangles either both inscribed or both exscribed to the same circle are in perspective. a. Every line passing through their centre of perspective in- tersects with their three pairs of coi'responding sides at six points, * This important property, which, as stated in Art. 140, is the basis of the theory of perspective in the geometry of plane figures, may be established, even more readily than either by the above or by the method employed in that article, as follows : If ABC and A'BC (fig. a. Art. 295) be the two triangles ; then in the quadrilateral CC'XY determined by any two pairs of their corresponding sides BC and B'C, CA and C'A', to which that determined by the two pairs of opposite vertices A and A', B and B is inscribed, the relation AC.BX.B'C'.A'Y= A'C'.BX.BC.AY, by virtue of the general relation a of Art. 134, being at once the criterion, that the two lines AA' and BB' should intersect at the same point O on the diagonal CO, and that the two AB and A'B' should intersect at the same point Z on the diagonal XY, therefore &c. OF THE POINT AND LINE. 155 in opposite pairs, every four of which are equianharmonic with their four opposites. a . Every point lying on their axis of perspective connects with their three pairs of corresponding vertices hy six lines, in opposite pairs, every four of which are equianharmonic with their four opposites. 3°. If, in the preceding (2°), while the point (or line) 0, one of the two triangles ABC, and two X and Y of the three points (or lines) X, Y, Z, are supposed to remain fixed, the other tri- angle A'B'C be conceived to vary consistently with the restriction of the fixity of X and Y; then, in every position of the variable triangle, since, by the first parts of the same pro- perties, the third point (or line) Z is also fixed, being the point of intersection (or line of connection) of the two fixed lines (or points) AB and XY, and since, by the two reciprocal properties Cor. 4° of the same article, the three points (or lines) X, Y, Z determine with the three U, V, W a. system of six coUinear (or concurrent) constituents, corresponding two and two in opposite pairs U and X, V and Y, W and Z, every four of which are equianharmonic with their four opposites ; hence, generally, — a. When, of a variable triangle whose three vertices move on three fixed concurrent lines, two of the sides turned round two fLced points, the third turns round a third fixed point collinear with the other two. a'. When, of a variable triangle whose three sides turn round three fixed collinear points, two of the vertices move on two fixed lines, the third moves on a third fixed line concurrent with the other two. b. The three fixed points, and the three intersections of their line of collinearity with the three fixed lines, in the fonner case, determine, in three opposite pairs, a row of six points every four of which are equianharmonic with their four opposites. b'. The three fixed lines, and the three connectors of their point of concurrence with the three fixed points, in the latter case, deter- mine, in three opposite pairs, a pencil of six rays every four of which are equianharmonic with their four opposites. 4°. Any two pairs of corresponding vertices (or sides) A and A', B and B' of the two triangles ABC and A'B'C, in 156 ANHARMONIC PROPERTIES the same (2°), may be regarded as the four points of a tetra- stigm (or the four lines of a tetragram) ; and the two pairs of opposite sides (or vertices) BC and B'C, CA and C'A' as the four lines of an exscribed tetragram (or the four points of an inscribed tetrastigm) ; hence, again, from the first parts of the same properties, it appears (see note to 2°, «, of the present article) that — a. For a tetrastigm inscribed to a tetragram^ when the inter- section of a pair of opposite connectors of the former is collinear with a pair of opposite intersections of the latter, the intersections of the remaining two pairs of opposite connectors of the former are collinear each with one of the two remaining pairs of opposite intersections of the latter. a'. For a tetragram exscribed to a tetrastigm, when the connector of a pair of opposite intersections cf the former is concurrent with a pair of opposite connectors of the latter, the connectors of the remaining two pairs of opposite intersections of the former are concurrent each with one of the two remaining pairs of opposite connectors of the latter. 297. As, from property a of inference 2° of the preceding article, it was shewn without difficulty in Art. 141 that the same property is true generally of every two figures in perspective with each other; so, from properties b and c of the same, it follows immediately, that, generally — For every two figures in perspective with each other, tlie centre and axis of perspective divide equianharmonically, at once all segments determined by pairs of corresponding points, and all angles determined by pairs of corresponding lines. And the same appears at once, h priori, from the considera- tion that the property being, by virtue of the general property of Art. 285, evidently true for every two is therefore true for all such segments or angles. This constant anharmonic ratio of section is termed, with respect to the figures, their anharmonic ratio of perspective ; and from the circumstance of its constancy in all cases of perspective, it follows that, for figures whose centre or axis of perspective is at infinity, all segments determined by pairs of corresponding points are cut in the same ratio by the one not at infinity (270). OF THE POINT AND LINE. 157 A property already, it will be remembered, established on other principles in Arts. 142 and 143. In the particular case, when, for two figures in perspective, the anharmonic ratio of perspective = — 1 ; all segments deter- mined by pairs of corresponding points, and all angles determined by pairs of corresponding lines, are cut harmonically, by the centre and axis of perspective ; the figures themselves are said to be in harmonic perspective; and when either their centre or axis of perspective is at infinity, all segments determined by pairs of corresponding points are bisected internally by the one not at infinity. Every two figures inserted, or exscribed, to the same circle furnish, when in perspective, an obvious and important example of two figures whose anharmonic ratio of perspective =—1. For, their centre and axis of perspective being then (167) pole and polar to each other with respect to the circle to which they are both inscribed or exscribed, and consequently (259) dividing harmonically at once all segments determined by their pairs of corresponding points, and all angles determined by their pairs of corresponding lines, therefore &c. When, in their case, either the centre or axis of perspective is at infinity, the internal bisection of all segments determined by their pairs of corresponding points by the one not at infinity is evident from 3° and 5°, Art. 165. Cor. As two circles, however circumstanced as to magnitude and position, are always doubly in perspective with respect to each centre of perspective (207) ; the line at infinity and their radical axis being the axes of their two perspectives for both (206) ; it appears consequently, from the above, that — For two circles, however circumstanced as to magnitude and position — a. The line at infinity and each centre of perspective divide in the same constant anharmonic ratio, at once all segments deter- mined by pairs of homologous points, and all angles determined by pairs of hor/iologous lines, with respect to that centre of per- spective. b. The radical axis and each centre of perfective divide in the same constant anharmonic ratio, at once all segments 158 ANHARMONIC PEOPEETIES determined by pairs of antihomologous points^ and all angles determined hy pairs of antihomologous lines, with respect to that centre of perspective. Since, for any two circles, the line at infinity and the radical axis bisect externally and internally the two segments, real or imaginary, intercepted between the two pairs of at once homo- logous and antihomologous points determined by the two common tangents through each centre of perspective (182) ; while the centre of perspective itself divides them in the positive or negative ratio of the similitude of the circles according as it is external or internal (198) ; it follows, consequently (268), that the two constant anharraonic ratios of perspective, both in a and h separately for the two centres of perspective, and also in a and b combined for each centre of perspective, are equal in magnitude and opposite in sign ; the absolute value common to the whole four being the constant ratio of simili- tude of the circles. 298. Since, for every two figures reciprocal polars to each other with respect to a circle (170), there correspond: 1°. To every point or line of either, a line or point of the other (170). 2°. To every connector of two points or intersection of two lines of either, the intersection of the two corresponding lines or the connector of the two corresponding points of the other (167). 3°. To every coUinear system of points or concurrent system of lines of either, the concurrent system of corresponding lines or the coUinear system of corresponding points of the other (166, Cor. 1°). 4°. To every anharmonic row of four points or pencil of four rays of either, the equianharmonic pencil of four corre- sponding rays or row of four corresponding points of the other (292). Hence, from the general property of the preceding article combined with those of Art. 141, the following important properties of figures in perspective, as regards reciprocation to an arbitrary circle (172) — a. Every two figures in perspective with each other reciprocate to any circle into two figures in perspective with each other. b. The centre of perspective of either pair of figures and the axis of perspective of the other pair are pole and polar to each other with respect to the circle. OF THE POINT AND LINE. 159 c. The anharmomc ratios of perspective of the two pairs, original and reciprocal, are equal. These properties are evident from the considerations, re- spectively: a. That, as in the original figures all pairs of corresponding points connect concurrently and all pairs of cor- responding lines intersect coUinearly, so in the reciprocal figures all pairs of corresponding lines intersect coUinearly and all pairs of corresponding points connect concurrently ; h. That the point of concurrence and the line of coUinearity for the original figures are the pole and polar respectively of the line of col- linearity and the point of concurrence for the reciprocal figures ; and, c. That all anharmonic ratios whether of rows or pencils are preserved unchanged in reciprocation to any arbitrary circle. Cor. Since, with respect to any circle, the polar of its centre is the line at infinity, and the pole of any line passing through its centre is the point at infinity in the direction perpendicular to the line (165, 3°, 5°) ; it follows, consequently, from the above, for the particular cases when the centre of the reciprocating circle is (1°) at the centre, and (2°) on the axis, of perspective of the original figures, that — 1°. Every two figures in perspective with each other reciprocate, to any circle whose centre is at their centre of perspective, into two similar and similarly [or oppositely) placed figures, whose ratio of similitude is equal to their anharmonic ratio of perspective (see Art, 142). 2°. Every two figures in perspective with each other reciprocate, to any circle whose centre is on their axis of perspective, into two figures consisting of pairs of points connecting by parallel lines all cut by the same line in their anharmonic ratio of per- spective (see Art. 143). In the particular case when the anharmonic ratio of perspec- tive of the original figures = — 1 ; the lines connecting the several pairs of corresponding points of the two reciprocal figures are all bisected internally, by their centre of perspective In the case of 1°, and by their axis of perspective in the case of 2°. As all anharmonic ratios whether of rows or pencils are preserved unchanged in reciprocation (293) ; it follows, conse- quently, from these latter properties 1° and 2°, that all anharmonic 160 ANHARMONIC PROPERTIES properties of pairs of figures in either of the two particular cases of perspective in which the axis or the centre of per- spective is at infinity are true generally of all pairs of figures in perspective with each other. 299. The two following properties, reciprocals of each other, are in the theory of anharmonic what those of Art. 236 are in that of harmonic section. a. In every tetrastigm, the three pairs of opposite connectors intersect with every line at three pairs of opposite points, every four of which are equianharmonic with their four opposites. a. In every tetragram, the three pairs of opposite intersections connect with every paint hy thre£ pairs of opposite rays, every four of which are equianharmonic with their four opposites. For, if A, B, C, D be the four points constituting the tetra- stigm (fig. a), or the four lines constituting the tetragram (fig. a'); BG and AD, CA and BD, AB and CD the three pairs of opposite connectors (or intersections) of the figure ; X and X', Y and Y', Z and Z' their three pairs of opposite intersections (or connectors) with any arbitrary line (or point) 0; and JJ, V, Wthe three points of intersection (or chords of connection) of the figure ; then — 1°. The two rows of four points (or pencils of four rays) Y, Z, X, X' and Z', Y\ X, X' being in perspective, at the points (or on the lines) A and D, with the row (or pencil) C, B, X, U; and the two F, Z, X', X and Z', Y, X', X being so, at the points (or on the lines) B and C, with the row (or pencil) D, A, X\ U; therefore, by (286, 3°), { YZXX'} = {Z' Y'XX'] = { Y'Z'XX] (280),) and { Y'ZXX} = {Z' YX'X} = { YZXX'} (280), I " " " ^" • OP THE POINT AND LINE. 161 2°. The two rows of four points (or pencils of four rays) ^1 ^1 ^i y and X\ Z', y, Y' being in perspective, at the points (or on the lines) B and i), with the row (or pencil) A, C, r, V; and the two Z', X, T, Y and X', Z, Y', Y being so, at the points (or on the lines) C and A, with the row (or pencil) I), B, T, V; therefore, by the same, {ZXYT} = {X'Z' YY'} = {ZX' Y' Y] (280),| and {Z'XY' Y} = {X'ZY' Y] = {ZX' YY'} (280), j "'^'' 3°. The two rows of four points (or pencils of four rays) X, Y, Z, Z and y, X' ^ Z^ Z' being in perspective, at the points (or on the lines) G and Z», with the row (or pencil) 5, A^ Z, W; and the two X', Y, Z, Z and F', X, Z', Z being so, at the points (or on the lines) A and B, with the row (or pencil) -D, (7, Z', W; therefore, by the same, {XYZZ] = { Y'X'ZZ] = {X' Y'Z'Z] (280),| and {X'YZZ]={Y'XZ'Z]={XY'ZZ'] (280),J ^"'^ and therefore &c. (283). COE. 1°. In the particular case when the line (or point) passes through (or lies on) any two of the three points of intersection (or lines of connection) U, V, W of the figure; since then, X=X'=Z7if it pass through (or lie on) U', Y=Y'=V if it pass through (or lie on) V; Z= Z =W \i it pass through (or lie on) W\ therefore, from 1°, 2°, 3°, respectively, of the above — 1°. If it pass through (or lie on) Fand TF, then {VWXX'\ = {VWX'X\ (1°). 2°. If it pass through (or lie on) TFand Z7, then {W'UYY'\ = {WTJY'Y] (2°). 3°, If It pass through (or lie on) V and F, then {UVZZ'} = {UVZ'Z} (3°), and therefore (281), as established before on other principles in (236)— a. In every tetrastigm^ the three pairs of opposite connectors divide harmonically, each the segment determined hy the inter- sections of the remaining two (107). VOL. II. M 162 ANHAEMONIC PROPERTIES a. In every tetragram, the three pairs of opposite intersections divide harmonically^ each the angle determined by the connectors of the remaining two (107). Cor. 2°. In the particular cases when the line (or point) is at infinity; the six intersections, in the former case, being the six points at infinity in the directions of the six connectors of the tetrastigin, and the six connectors, in the latter case, being the six parallels in the direction of through the six inter- sections of the tetragram ; while, in every case, every four of the former are equianharmonic with the pencil they determine at any point, and every four of the latter with the row they determine on any line. Hence, from the general properties applied to those cases, it appears that — a. The six parallels through any pointy to the six connectors of any tetraMigm, determine at tJie point, in three opposite pairs, a pencil of six rays every four of which are equianharmonic with their four opposites. a'. The six perpendiculars to any line, through the six inter- sections of any tetragram, determine on the line, in three opposite pairs, a row of six points every four of which are equianharmonic with their four opposites. N.B. The six parallels, in the former case, and the six perpendiculars, in the latter case, might evidently be turned in the same direction of rotation through any common angle, without affecting in either case the above relations between them. Cor. 3°. As the three segments (or angles) determined by three pairs of opposite constituents every four of which are equianharmonic with their four opposites, have in all cases a common segment (or angle) of harmonic section, real or imaginary (283, Cor. 1°); it follows consequently from the above, as shewn already on other principles in (245, Cor. 3°), that — a. The three s&jments intercepted on any line, by the three pairs of opposite connectors of any tetrastigm, have a common segm£nt of harmonic section, real or imaginary, a. The three angles subtended at any point, by the three pairs of opposite intersections of any tetragram, have a cwnmon angle of harmonic section, real or imaginary. OF THE POINT AND LINE. 163 Cor. 4°. As any three A, JB, C o{ the four points (or lines) A, B, C, D constituting the tetrastigm (or tetragram) may be regarded as the three vertices (or sides) of a triangle ABC, and the fourth Z) as the point of concurrence (or line of col- linearity) of any three concurrent lines through its three vertices (or coUinear points on its three sides) ; the two reciprocal pro- perties of the present article, respecting the tetrastigm and tetragram, may consequently be regarded as anharmonic pro- perties of the triangle, and stated accordingly as follows — a. The three sides of any triangle, and any three concurrent lines through the three 7)ertices, intersect with every line at six points, corresponding two and two in opposite pairs,' every four of which are equianharmonic with their four opposites. a'. The three vertices of any triangle, and any three collinear points on the three sides, connect with every point by six lines, corresponding two and two in opposite pairs, every four of which are equianharmonic with their four opposites. COE. 5°. In the particular cases of the latter properties, ■when the fourth point (or line) D is at infinity; then, since, in the former case, the three lines of connection AD, BD, CD are parallel, and since, in the latter case, the three points of intersection AD, BD, CD are the three at infinity in the directions of the three sides of the triangle ; hence, from those properties applied to these particular cases, it appears that — a. The three intersections with any line of the three sides of any triangle determine, with the three projections on the line of the three vertices of the triangle, a system of six points, cxyrre- spondinq two and two in opposite pairs, every four of which are equianharmonic with their four opposites. a'. The three connectors with any point of the three vertices of any triangle determine, with the three parallels through the point to the three sides of the triangle, a system of six rays, cor- responding two and two in opposite pairs, every four of which are equianharmonic with their four opposites. COE. 6°. In the particular case when the arbitrary point in property a of the preceding corollary (5°) is the polar centre of the triangle (168) ; since, then, each connector and the cor- responding parallel are perpendiculars to each other, it follows M2 1G4 ANHARMONIC PROPERTIES consequently, from that property applied to this particular case, as is a priori evident (283, Cor. 2°), that — Every three lines through a point determine, with the three perpendiculars to them, through the point, a system of six rays, corresponding two and two in opposite pairs, every four of which are equianharmonic with their four opposites. Cor. 7°. Since from the original property a of the present article, combined with that of the preceding corollary (6°), it follows that — when, of the three angles subtended at a point hy the three chords of intersection of any tetragram, two are right, the third is right also ; hence, from the familiar property (Euc. III. 31), that the vertices of all right angles subtending a common segment lie on the circle of which the segment is a diameter, it appears, as proved already more generally on other principles in Art. 189, Cor. 1°, thatn- The three circles on the three chords of intersection of any tetragram as diameters pass each through the two points of inter- section of the other two ; and have, tJierefore, all three, a common pair of points, real or imaginary. 300. The two reciprocal properties of the preceding article supply obvious solutions, by linear constructions only without the aid of the circle, of the two following reciprocal problems, viz. — Of two triads of collinear points {or concurrent lines), which correspond two and two in opposite pairs, and every four of which are equianharmonic with their four opposites ; given either triad and any two constituents of the other, to determine the third constituent of the latter. Thus, in the figures (a) and (a') of that article, given the triad of points (or rays) X, Y, Z, and any two constituents X' and Y' of the other X', Y, Z', to determine the third con- stituent Z' of the latter. In the former case, through the three given points X, Y, Z (fig. a) drawing arbitrarily any three non- concurrent lines; the three opposite vertices A, B, C of the triangle they determine, by property a of the preceding article, connect with their three opposites X', Y', Z' by three con- current lines AX', BY', GZ , two of which AX' and BY' being given determine the point of concurrence D, and there- OF THE POINT AND LINE. 165 fore the third CZ\ which intersects with the line at the required point Z'. And, in the latter case, on the three given lines X, Y^ Z (fig. a') taking arbitrarily any three non-coUinear points; the three opposite sides A^ B, C of the triangle they determine, by property a of the preceding article, intersect with their three opposites X', Y', Z' at three collinear points AX; BY', GZ, two of which AX' and BY' being given deter- mine the line of coUinearity Z>, and therefore the third CZ', which connects with the point by the required line Z'. 301. Of all anbarmonic properties of figures of points and lines, the two following, reciprocals of each other, lead to the greatest number of consequences in the theory of conic sec- tions, viz. — a. When, of six points, any four connect equianharmonically with the remaining two, then every four connect equianharmoni- cally with the remaining two. a'. When, of six lines, any four intersect equianharmonically with the rem,aining two, then every four intersect equianharmoni- cally with the remaining two. V Let A, B, C, D, JE, F be the six points (fig. a) or the six lines (fig. a) ; when any four of them C, D, E, F connect (or intersect) equianharmonically with the remaining two A and B, then any other four of them A, B, E, F connect (or intersect) equianharmonically with the remaining two C and D. For, O being the line of connection (or the point of intersection) of the two points (or lines) E and F which are common to the two systems of four A, B, E, F and C, D, E, F for which the relation is given and to be proved respectively, if X and X', Y and Y', Z and Z' be the three pairs of opposite intersections 166 ANHARMONIC PROPERTIES (or connectors) of with the three pairs of opposite connectors (or intersections) BC and AD^ CA and BD, AB and CD of the tetrastigra (or tetragram) determined by the four A,B,C,I> which are not common to the two systems; then since, by hypothesis, {A.CI)JSIF} = {B.CI>EF^s, and since, consequently, by (285), f YX'EF} = [XY'EF}, therefore, by (272), { YXEF} = {X'Y'EF}, and consequently, by (285), {C.ABEF} = {I).ABEF}; and therefore &c. The above demonstration, though apparently establishing the property only for the six cases in which the quartet for which the relation is given has but two constituents in common with that for which it is to be proved, in reality establishes it for the eight cases in which the two quartets have three constituents in common as well ; for establishing it, as shewn above, for every quartet having but two constituents in common with that for which it is given, it consequently establishes it at the same time for every quartet having but two constituents in common with each of the latter ; and therefore &c. Two coUinear triads of points on different axes (or con- current triads of lines through different vertices) (293) furnish an obvious, but veiy particular, example of a system of six points (or lines) every four of which connect (or intersect) equianharmonically with the remaining two. Every system of six points on (or tangents to) the same circle, as will be shewn at the opening of the next chapter, also comes under the same head, and possesses, in consequence, every property of the more general system depending only on the existence of the aforesaid equianharmonic relations between its constituent points (or lines). Cor. 1°. As the three pairs of points (or lines) A and B, C and B, E and F (or any other three pairs into which the six may be resolved) may be regarded as determining three seg- ments (or angles) AB, CD, EF, the extremities of some, and therefore of every, two of which connect (or intersect) with those of the third equianharmonically; the above reciprocal properties may, consequently, be stated (as indeed they were proved) in the following equivalent, but less general, forms, viz. — OF THE POINT AND LINE. 167 a. When, of three segments, the extremities of any two connect equianharmonically with those of the third, then the Kctremities of every two connect equianharmonically with those of the third.. a. When, of three angles, the sides of any two intersect equi- anharmonically with those of the third, then the sides of every two intersect equianharmonically with those of the third. Cor. 2°. As, in the tetrastigm (or tetragram) determined by the four points (or lines) A, B, C, D (or by any other four of the six), the three pairs of equianharmonic relations, for the three pairs of opposite segments (or angles) BC and AD, CA and BD, AB and CD, with each other, and with the seg- ment (or angle) EF determined by the remaining two E and F, viz. — {A.CBEF\ = [D.GBEF] and {B.DAEF\ = {C.DAEF]...{\°), {B.ACEF] = [D.ACEF] z.-o.i[G.DBEF] = [A.DBEF]...{2°), {C.BAEF}={D.BAEF} and {A.DGEF] = {B.DCEF}...{3°], and the three corresponding pairs for the three pairs of segments (or angles) XYaniX'Y', FiT and Y'Z', ZX and Z'X' they determine on the line (or at the point) EF, viz. — { YZEF } = { Y'Z'FE] and { Y'ZEF } = { YZFE \ (l'), {ZXEF\ = \ZX'FE\ and [Z XEF ] = [ZX FE\ (2'), {XYEF\ = {XY'FE\ and {X' YEF\ = {XY FE\ (.3'), by virtue of (285) and (280), mutually involve each other ; hence, see (299) and (284), it appears generally that — a. Every two points, which connect equianharmonically loith the four points of any tetrastigm, form, with the six determined on their line of connection hy the six connectors of the tetrastigm, a system of eight points, in four opposite pairs, every four of which are equianharmonic with their four opposites / and, conversely, every two points, which form such a system with the six determined on their line of connection by the six connectors of any tetrastigm, connect equianharmonically with the four points of the tetrastigm. a. Every two lines, which intersect equianharmonically with the four lines of any tetragram, form, with the six determined at their point of intersection by the six intersections of the tetragram, a system of eight rays, in four opposite pairs, every four of which are equianharmonic with their four opposites; and, conversely. 168 ANHAEMONIC PROPERTIES every two lines ^ which form such a system with the six determined at their point of intersection hy the six intersections of any tetragram,^ intersect equianharmonically with the four lines of the tetragram. Cor. 3°. As an example of the criterion of the above rela- tion of equianharmonicism for a system of six points or lines supplied by the second parts of the two reciprocal properties a and a' of the preceding corollary (Cor. 3°) ; suppose the two triangles ABC and DEF employed in its establishment were both self-reciprocal with respect to the same circle (168) ; since then evidently, by (167), the several pairs of points (or lines) X and X\ Y and Y\ Z and Z\ E and F would be pairs of conjugates with respect to the curcle (174), and consequently pairs of harmonic conjugates with respect to the two points (or tangents), real or imaginary, determined with the circle by the line or point EF (259) ; therefore, by (3°, Cor. 4°, Art. 282), they would satisfy the criterion expressed in the two properties ; and therefore — For every two sdf reciprocal triangles with respect to the same circle, every four of the six vertices connect, and every four of the six sides intersect, equianharmonically with the remaining two. 302. Reserving for the next chapter the principal conse- quences resulting, in the geometry of the circle, from the cir- cumstance of every six concyclic points or tangents being connected by the equianharmonic relations of the preceding article ; we shall conclude the present with the two following reciprocal properties of such systems in general, and with a few of the many consequences to which they lead in the geometry of the point and line. a. In a hexagon, when the intersections of the three pairs of opposite sides are collinear, every four of the six vertices con- nect equianharmonically with the remaining two ; and, c&nversely, when any four of the six vertices connect equianharmonically with the remaining two, the intersections of the three pairs of opposite sides are collinear. a'. In a hexagon, when the connectors of the three pairs of opposite vertices are concurrent, every four of the six sides inter- sect equianharmonically with the remaining two ; and, conversely, ■when any four of the six sides intersect equianharmonically with OF THE POINT AND LINE. 169 the remaining tioo, the connectors of the three j>airs of opposite vertices are concurrent. For, if Xand X', Fand F, Zand Z' be the three pairs of opposite vertices (fig. a), or sides (fig. a'), of the hexagon ; A and A\ B and 5', C and C the three pairs of corresponding vertices (or sides) of the two triangles determined by its two triads of alternate sides (or vertices) ; and V, F, W the three points of intersection (or lines of connection) of its three pairs of opposite sides (or vertices) ; then since, by the general pro- perty of Art. (285), the three pairs of equianharmonic relations {Y.X'Y'Z'X} = [Z.X'Y'Z'X] a.nA{Y.XYZX'} = [Z.XYZX'] (1°), {Z. Y'Z'X'Y] = [X.YZ'X'Y] and {Z'.YZXY'}={X'.YZXY'} (n {X.Z'X'Y'Z} = {Y.Z'X'Y'Z} and {X'.ZXYZ'} = {T.ZXYZ'\ (3"), and the three corresponding pairs [WY'BX] = {VC'Z'X } and {WYB'X'} = [VCZX' ] ... (1'), [UZGY ] = {WA'X-Y] and [UZC'Y' ] = [WAXY] ... (2'), [VXAZ } = [UB'Y'Z ] and [VXAZ' } = {UBYZ' } ... (3'), mutually involve each other ; and since, by (286, 4°), the latter are all involved in, while, by (289), any one of them involves, the collinearity (or concurrence) of the three points (or lines) Z7, F, W; therefore &c. The hexagons originally considered in the celebrated theo- rems of Pascal and Brianchon established on other principles in 170 ASHAEMONIC PROPERTIES Art. 148, coming under the second parts of the above reciprocal properties a and a' respectively ; the names " Pascal hexagon" and "Brianchon hexagon" are in consequence applied gene- rally, the former to all hexagons whose pairs of opposite sides intersect coUinearly, and the latter to all whose pairs of opposite vertices connect concurrently; the line of coUinearity in the former case, and the point of concurrence in the latter case being termed respectively the " Pascal line" and " Brianchon point" of the hexagon. For the same reason the names " Pascal hexastigm" and " Brianchon hexagram" are applied generally, the former to all systems of six points, every four of which connect equianharmonically with the remaining two, and the latter to all systems of six lines, every four of which intersect equianharmonically with the remaining two; all hexagons determined by such systems being, by virtue of the same properties, Pascal and Brianchon hexagons in the two cases respectively. Since for every two triangles in perspective, the three pairs of corresponding sides intersect coUinearly on the axis of per- spective, and the three pairs of corresponding vertices connect concurrently through the centre of perspective (140); it follows consequently, from the above, that for every two triangles in perspective; the four hexagons of which their pairs of correspond- ing are pairs of opposite sides are Pascal hexagons., of which their axis of perspective is the common Pascal line ; and the four of which their pairs of corresponding are pairs of opposite vertices are Brianchon hexagons^ of which their centre of perspective is the common Brianchon point. The same hexagon might be at once a Pascal and a Brianchon hexagon, and when such would of course in its double capacity combine the properties of both ; every hexagon at once in- scribed to one circle and exscribed to another circle furnishes an example of a hexagon of this nature. 303. From the two reciprocal properties of the preceding article, combined with the fundamental two of Art. 140 respect- ing triangles in perspective, the following consequences, in pairs reciprocals of each other, may be readily inferred, viz. — a. The intersections of the six pairs of alternate sides of a OF THE POINT AND LINE. 171 Pascal hexagon, taken in consecutive order, determine a Brian- clion hexagon. a. The connectors of the six pairs of alternate vertices of a Brianchon hexagon, taken in consecutive order, determine a Pascal hexagon. For, if (figures of last article) X and X', Y and Y', Z and Z' be the three pairs of opposite vertices (or sides) of the original hexagon, and A and A, B and B', G and C those of the de- rived hexagon ; then since, by hypothesis, the three pairs of corresponding sides (or vertices) BC ani B'C, CA and C'A', AB and A'B' of the two triangles ABC and A'B'C intersect coUinearly (or connect concurrently), therefore, by (140), their three pairs of corresponding vertices (or sides) A and A', B and B', C and C connect concurrently (or intersect col- linearly) ; and therefore &c. h. When the vertices of three angles are colli7iear, the twelve remaining intersections of their six determining lines may he divided, in four different ways, into two groups of six, determining one a Pascal and ike other a Brianchon hexagon. v. When the axes of three segments are concurrent, the twelve remaining connectors of their six determining points may he divided, in four different ways, into two groups of six, determining one a Brianchon and the other a Pascal hexagon. For, the four hexagons, of which the three pairs of lines (or points) determining the three angles (or segments) are the three pairs of opposite sides (or vertices), being, by the preceding article, Pascal (or Brianchon) hexagons ; and, the four hexagons determined by the intersections (or connectors) of their pairs of alternate sides (or vertices) being, by the two properties a and a' just proved, Brianchon (or Pascal) hexagons ; therefore &c. In the figures of the preceding article, the three coUinear points (or concurrent lines) U, F, W being the three vertices (or axes) of the three determining angles (or segments), the four sets of two complementary groups of six points (or lines) X,B', C; X',B, C and A, Y\Z; A', Y, Z (1), Y, C, A; T, C, A' and B, Z, X; B\ Z, T (2), Z, A', B; Z; A, B' and C, X, Y; C;X, Y' (3), X, Y, Z; X; Y',Z' and A, B, C; A', B', C (4), 172 ANHAEMONIC PROPERTIES are those determining the four Pascal and Brianchon (or Brian- chon and Pascal) hexagons in the four cases respectively. c. Of the, sixty liexagons determined by the same Pascal hexas- tigm, the sixty Pascal lines pass three and three through twenty points. c. Of the sixty hexagons determined hy the same Brianchon hexagram, the sixty Brianchon points lie three and three on twenty lines. For, of the four hexagons, of which the two triads of points (or lines) X, Y, Z and X, F, Z\ in the figures of the last article, are the two triads of alternate vertices (or sides), viz. : XY'YX'ZZ\ YZ'ZY'XX', ZXXZ'YY', XZ'YX'ZY\ while the Pascal line (or Brianchon point) of the fourth is the line (or point) WW, those of the three first are the three lines (or points) AA', BB' CC respectively, which, by the preceding a and a, or by the general property of triangles in perspective (140), are concurrent (or coUinear) ; and, the same being of course true for every other similarly circumstanced three of the entire sixty, therefore, &c. The above theorem c (and with it of course its reciprocal c) is due to M. Steiner, who was the first to direct the attention of geometers to the complete figure determined by a system of six points (or lines), every four of which connect (or intersect) equianharmonically with the remaining two. The subject has since, from time to time, engaged the attention of difi^erent eminent geometers, including M. Pliicker, Dr. Salmon, Professor Cayley, and Mr. Kirkman, by whom several other properties of the same nature have been discovered ; of the principal of which, an abstract will be found in Dr. Salmon's Conic Sections, Ed. 4, note 1, page 357, and further details in Mr. Kirkman's published paper, Cambridge and Dublin Mathematical Journal, Vol. v., p. 185. d. For each of the fifteen triads of non-conterminous segments determined hy the same Pascal hexastigm, the six centres of per- spective of the three pairs they determine lie three and three on four lines. d'. For each of the fifteen triads of non-conterminous angles determined by the same Pascal hexagram, the six axes of perspec- OP THE POINT AND LINE. 173 tive of the three pairs they determine pass three and three through four points. For if XX\ YY\ ZZ' (same figures) be any triad of non- conterminous segments (or angles) determined by the six points (or lines) of the hexastigm (or hexagram) ; TJ^V^W the three intersections (or connectors) of the three pairs of connectors (or intersections) YZ' and TZ, ZX' and Z'X, XY' and XY; and U, V, W those of the three pairs FZand Y'Z', ZX and Z'X', ^F and XT; then, in the four hexagons YXZTX'Z', ZYXZ'YX, XZYXZY\ YZ'XY'ZX, the four triads of points for lines) V TV' U; W V\ V; U', V, W; U, F, W, being the four triads of intersections (or connectors) of pairs of opposite sides (or vertices), are coUinear (or concurrent) ; and the same being of course true for each of the remaining fourteen triads of non-conterminous segments (or angles) determined by the hexastigm (or hexagram), therefore, &c. 304. The two following reciprocal criteria that six points lying in pairs on the three sides of a triangle should deter- mine a Pascal hexastigm, and that six lines passing in pairs through the three vertices of a triangle should determine a Brianchon hexagram, result immediately from the two of Art. 147 for the perspective of two triangles ; viz. — When three pairs of points {or lines) X and X\ Y and Y', Z and Z^ lying on the three sides [or passing through the three vertices) BG, CA, AB of a triangle whose three opposite vertices {or sides) are -4, B, C, determine a Pascal hexastigm {or Brianchon hexagra7/i), they satisfy — a. In the former case the general relation BX.BX CY.GY ' AZ.AZ _ CX.CX' • A Y.A Y' • BZ.BZ' ~ "*' ' a'. In the latter ca^e the reciprocal relation si ngX.sin-gJC' sinCF.sinCF smAZ.sm.AZ' ^ sin CX.smGX' ' sinA Y.smA Y' ' sin BZ.sinBZ' "^ ' and conversely, when of the above two reciprocal relations they satisfy the one corresponding to their case, they determine a Pascal hexastigm {or Brianchon hexagram). For, of those two reciprocal relations, that corresponding to 174 ANHARMONIC PROPERTIES the case being, by (147), the criterion that the triangle determined, by the three lines (or points) XX\ YY\ ZZ\ or the triangle ABC, should be in perspective with each of the eight triangles determined by the eight triads of points (or lines) YZ', ZX\ XY' ; YZ,ZX,XY; YZ,ZX,X'Y'', YZ, Z'X', XY ; Y'Z, Z'X, XY; Y'Z, ZX, XY; Y'Z, ZX, XY; YZ, ZX, XY; and, conversely, being, by the same, fulfilled when it is in perspective with any one of them ; therefore, &c. (302). Cor. 1°. Since, for the same triangle, by (65) ; every three pairs of points on the three sides which satisfy relation (a) connect with the opposite vertices by three pairs of lines satisfy- ing relation (a) ; while, conversely, every three pairs of lines through the three vertices which satisfy relation («') intersect with the opposite sides at three pairs of points satisfying relation (a) ; hence, from the above, it appears, generally, that — When three pairs of points on the three sides of a triangle determine a Pascal hexastigm, their three pairs of connectors with the opposite vertices determine a Brianchon hexagram ; and, con- versely, when three pairs of lines through the three vertices of a triangle determine a Brianchon hexagram, their three pairs of intersections with the opposite sides determine a Pascal hexastigm. Cor. 2°. Since, for any triangle, by (134); every two triads of points X, Y, Z, and X', T, Z' oa the three sides BG, GA, AB, which are both either coUinear or concurrently connectant with the opposite vertices, satisfy relation (a) ; while, reciprocally, every two triads of lines X, Y, Z and X', Y', Z' through the three vertices BC, GA, AB, which are both either concurrent or coUinearly intersectant with the opposite sides, satisfy relation (a'); hence, again, from the above, it appears that — When two triads of points on the three sides of a triangle are both either collinear or concurrently connectant with the opposite vertices they determine a Pascal hexastigm ; and, reciprocally, when two triads of lines through the three vertices of a triangle are loth either concurrent or coUinearly intersectant with the opposite sides they determine a Brianchon hexagram. Cor. 3°. Since again, conversely, for any triangle, by the same, if, of two triads of points X, Y, Zand X', T, Z on the three sides BC, CA, AB which satisfy relation (o), one be either •bp THE POINT AND LINE. 175 coUinear or concurrently connectant with the opposite vertices, so is the other also ; while, reciprocally, if, of two triads of lines X, F, Zand X', T, Z' through the three vertices BC, CA, AB which satisfy relation (a'), one be either concurrent or coUinearly intersectant with the opposite sides, so is the other also ; hence also, from the above, it appears, conversely, that — Of two triads of points on the three sides of a triangle which determine a Pascal hexastigm^ if one he either collinear or con- currently/ connectant with the opposite vertices, so is the other also ; and, reciprocally/, of two triads of lines through the three vertices of a triangle, which determine a Brianchon hexagratn, if one be either concurrent or coUinearly intersectant with the o^osite sides, so is the other also. N.B. — Of the reciprocal properties of this corollary, those established, on other considerations, in Examples 9°, 10°, 11°, 12°, Art 137, are evidently particular cases, CoK. 4°. Since, by the two reciprocal relations (a) and (a) of Art. 245, every three pairs of points (or lines) X and X', Y and Y', Z and Z', harmonic conjugates with respect to the three chords of intersection (or angles of connection) of any tetragram (or tetrastigm), divide the three sides (or angles) BC, CA, AB of the triangle determined by the axes (or vertices) of the three chords (or angles) so as to satisfy the above relations (a) and [a) respectively ; hence, again, from the above, it appears generally that— a. Every three pairs of points, harmonic conjugates with respect to the three chords of intersection of a tetragram, determine a Pascal kexastigm. a'. Every three pairs of lines, harmonic conjugates with respect to the three angles of connection of a tetrastigm, determine a Brianchon hescagram. N.B. — Of the two reciprocal properties of this corollary, the two (a) and (6) of the article referred to in their proof (245) are evidently particular cases. COE. 5°. Assuming, as will be shown in the next chapter, that when five of the six points (or lines) determining a Pascal hexastigm (or Brianchon hexagram) are points on (or tangents to) a common circle, the sixth also is a point on (or tangent to) 176 ANHARMONIC PROPERTIES, &C. the same circle; it follows evidently, from the two reciprocal properties a and a of the preceding corollary (4°), that — a. Every circle^ dividing two of the three chords of in- tersection of a tetragram harmonically^ divides the third also harmonically. a!. Every circle^ subtending two of the three angles of con- nection of a tetrastigm harmonically, subtends the third also harmonically. Properties, the first of which, it will be remembered, was proved before, on other principles, in Art. 228, Cor. 6°. ( 177 ) CHAPTER XVIII. ANHAKMONIC PROPERTIES OF THE CIRCLE. 305. Among the various anharmonic properties of the circle, the two following, reciprocals of each other, are those to which the designation is most commonly applied ; and they obviously include, as particular cases, the two already given in Art. 252 at the commencement of Chapter xv., viz. : a. Every system of four points on a circle determines equir anharmonic pencils of rays at every two^ and therefore at all, points on the circle. d. Every system of four tangents to a circle determines egui- anharmonic rows of points on every two, and therefore on all, tangents to tJie circle. For, in the former case, if A, B, C, D be any four points on a circle ; then since, for every two points E and F on the circle, the two pencils of four rays E.ABCD and F.ABCD arc similar (25, 1°), therefore &c. And, in the latter case, if A, B', C, B' be any four tangents to a circle; then since, for cveiy two tangents E' and F to the circle, the two rows of four points VOL. II. N 178 AN HARMONIC PROPERTIES OP THE CIRCLE. E'.AB'G'iy and J".^'-B'C'i)' determine similar pencils at the centre of the circle (25, 2°), therefore &c. (285). Since, by virtue of the above reciprocal properties, every six points A, B, C, D, E, Fon a. circle form a system of six points, every four of which connect equianharmonically with the re- maining two, and every six tangents A', B\ C", Z)', E\ F' to a circle form a system of six lines, every four of which intersect equianharmonically with the remaining two; all anharmonic properties, consequently, which are true, in general, of any system of six points or lines thus related to each other (301), are true, in particular, of every system of six points on or tangents to the same circle. See Arts. 301 to 304. Again, since, under the process of reciprocation to an arbi- trary circle (172), all systems of points and tangents of the original become transformed into systems of tangents and points of the reciprocal figure (159), and all anharmonic rows and pencils of the original into equianharmonic pencils and rows of the reciprocal figure (292) ; it follows, consequently, that the above reciprocal properties, with all the consequences to which they lead in the geometry of the circle, are true, more generally, not only of the circle, but also of every figure into which the circle can become transformed by reciprocation ; either in the original involving the other in the reciprocal figure, and conversely. See Art. 173. 306. If, in the preceding, while the four points A, B, C, D and one of the two E, F, or the four tangents A', B\ C, D' and one of the two E\ F, are supposed to remain fixed, the remain- ing point, or tangent, be conceived to vary, and, in the course of its variation, to go round the entire circle ; since then, in every position of the variable point, or tangent, {E.ABCD] = [F.ABCD], or, {E'.A'B'C'iy} = {F.A'B'C'D}, the two reciprocal properties of the preceding article may con- sequently be stated as follows : a. Every system of four fxed points on a circle determines, at a variable ffih point on the circle, a variable pencil of four rays having a constant anharmonic ratio. a. Every system of four fixed tangents to a circle determines, anhaeSonic peopeeties of the ciecle. 179 on a varialle fifth tangent to the circle^ a variable row of four points having a constant anharmonic ratio. When the variable point, or tangent, in the course of its variation, coincides with one of the four fixed points, or tangents ; the corresponding ray of the variable pencil, or point of the variable row, becomes then the tangent at the fixed point (19), or the point of contact of the fixed tangent (20) ; but the entire pencil, or row, has still the same constant anharmonic ratio as for eveiy other position of the variable point, or tangent. See Art. 255. This constant anharmonic ratio is commonly termed that of the four fixed points on the circle, in the former case, and that of the four fixed tangents to the circle, in the latter case; it being, of course, always Implicitly understood to mean, as above explained, that of the pencil determined by the four at any fifth point on the circle, in the former case, and that of the row determined by the four on any fifth tangent to the circle, in the latter case. See Art. 279. For the same reasons stated in the concluding paragraph of the preceding article, the above reciprocal properties are true generally, not only of the circle, but also of every figiu-e into which the circle can become transformed by reciprocation; either in the original involving the other in the reciprocal figure, and conversely. 307. The pencil of four rays determined by any system cf four points on a circle at any fifth point on the drcle, and the row of four points determined by the corresponding system of four tangents to the circle on any fifth tangent to the drchj are eguianharmonic. For, if A, B, C, D be any four points on a circle, and A', B\ G\ D' the four corresponding tangents to the circle ; E any fifth point on the circle, and E' any fifth tangent to the circle ; then since, for all positions of E and E\ by (25, 1° and 2°), the pencil of four rays E.ABGD is similar to that determined by the row of four points E'.A'B'G'D' at the centre of the circle, therefore &c. (285). In the particular case when the fifth tangent E' is that corresponding to the fifth point E; the four points E'.A'B'C'iy n2 180 ANHARMONIC PROrEETIES OF THE CIRCLE; are tbe four poles, with respect to the circle, of the four points E.ABCD (165, 6°) ; and their equianharmonicism follows, as a particular case, from the general property of Art. 292. Tlie above anharmonic equivalence is generally represented^ for shortness, by Dr. Salmon's Notation, Art. 279, viz. — {ABCD] = [A'B'C'D']; it being, of course, always understood that the two equivalents refer, respectively, to the pencil determined by the four points -4, B,' C, D at any fifth point on the circle, and to the row determined by the four corresponding tangents AB'O'D' on any fifth tangent to the circle. Again, for the reasons stated in the concluding paragraph of Art. 305, the above property is true generally, not only of the circle, but also of every figure into which the circle can become transformed by reciprocation. 308. The six anharmonic ratios P and •=, Q and -pr, R and ■= of the pencil determined by any four points A, B, C, D on a circle at any fifth point E on the circle, or by the four corres- ponding tangents A, B\ C", D' on any fifth tangent E' to the circle (307), may be expressed in terms of the six chords connecting the four points two and two, exactly as for four points on a line (274), viz. — BA BD , CA CD BA.CD ^ CA.BD ca'-gd''''^ba'-bd^'''^ -GA:m ^""^ mrm (^)' cb cd ,ab ad cb.ad , ab.cd ■ab'-ad ^"'^cb'-cI)^'''^ aKcd ^^cKad (2)' ac ad . bg bd ac.bd ^ bc.ad Bc'-rn'-''^ ag'-ad^""'^ bc:ad ''''^ acted (^)- For, in the three pairs of reciprocal ratios (1), (2), (3), dividing each chord involved by the diameter of the circle, and substitu- ting for the resulting quotient the sine of the angle subtended by. that chord at any point E on the circle (62) ; the three pairs of corresponding anharmonic ratios of the pencil E.ABCD determined by the four points A, B, C, D at the point E, viz. — ANHARMONIC PROPERTIES OP THE CIRCLE. 181 sin^^^ am BED sin CEA sin CE P sin CEA '• sin CED ^^°^ ainBEA ' ^^ED '■" ^^ '' sin CEB ^ sin CED sin AEB smAED sinAEB ' sinAED sin CEB ' sin CED ^ '' smAEC ^ am AED sin BEC sin g^Z> sin^^C ■ am BED Bin AEC' sinAED ^^ '' or their three corresponding equivalents (see (1'), (2'), (3') (Art. 274), are the immediate result ; and therefore &c. 309. Two diflferent systems of four points on, or tangents to, the same circle, or two different circles. A, B, C, D and A', B\ C, D', which correspond in pairs A to A', B to B', C to C\ D to D', are said to be equianharmonic, when the pencils of four rays, or the rows of four points, they determine at all points on, or on all tangents to, their circle, or circles, are equianharmonic. With the same understanding, as to meaning, as in the particular case considered in Art. 307, all such cases of anharmonic equivalence may in general be represented, for shortness, by Dr. Salmon's Notation (279), viz. — {ABCD} = {A'B'C'D'} ; the several pairs of corresponding constituents being, of course, invariably written in the same order in the two equivalents, in every case of its employment. Two similar systems of four points on the same circle, or on two diflFerent circles, as determining similar pencils of four rays at all points on their circle, or circles, furnish the most obvious as well as the simplest example of two equianliannonic systems of concyclic points; and the two systems of corresponding tangents to the circle, or circles, furnish the most obvious as well as the simplest example of two equianharmonic systems of concyclic tangents, in the sense above defined. Thus, for two circles, every two systems determined by four points or tangents of either, and by the four homologous points or tangents of the other with respect to either centre of similitude of the two (198), as being evidently similar, are equianharmonic in that sense. For the same circle, it is evident, from 2°, Art. 286, that every two systems of four points which determine pencils in perspec- tive at any two points on the circle, auJ every two systems 182 ANHAEMONIC PEOPEETIES OF THE CIECLE. of four tangents which determine rows in perspective on any two tangents to the circle, are equianharmonic in the same sense. 310. As, in Art. 290, for two equianharmonic rows of four points on different axes, or for two equianharmonic pencils of four rays through diflFerent vertices; so, for two concyclic systems of four points, or tangents, equianharmonic in the sense of the preceding article, It is evident, from the two reciprocal properties (a) and (a') of Art. 289, that — a. Every two equianharmonic systems of four points on the same circle determine two pencils of rays in perspective, either at any point of the other, and the latter at the corresponding point of the former. a'. Every two equianharmonic systems of four tangents to the same circle determine two rows cf points in perspective^ either on any tangent cf the other, and the latter on the corresponding tangent of the former. For, if ^, B, C, D and A', B', C, B be the two systems of points, or tangents ; then, each pair of pencils or rows A.A'B'C'B' and A'.ABCD, B.A'B'C'ir and B'.ABCD, C.A'B'Cn and C'.ABCI), D.A'B'C'D' and D'.ABCB, being equianharmonic, and having a common ray or point, therefore &c. (289). Cob. As, in Art. 291, for two equianharmonic rows of four points on different axes, or pencils of four rays through different vertices; so, for two concyclic systems of four points, or tangents, equianharmonic in the sense in question, the above reciprocal properties supply ready solutions, by linear constructions only, without the aid of a second circle, of the two following reciprocal problems, viz. — a. Given three pairs of corresponding constituents of two equianharmonic systems of points on the same circle, and the fourth point of either system, to determine the fourth point of the other system. a'. Given three pairs of corresponding constituents of two equianharmonic systems of tangents to the same circle, and the fourth tangent of either system, to determine the fourth tangent of the other system. ANHAEf ONIC PROPERTIES OF THE CIRCLE. 183 The two reciprocal coustmctlons given in the article referred to (291), modified in the manner mentioned in the note at its close, apply, word for word and letter for letter, to these problems also. 311. The two following reciprocal cases of anharmonic equi- valence, between concyclic systems of points and tangents, result immediately from the fundamental properties of Art. 305, and lead to several remarkable consequences in the modern geometry of the circle, viz. — If A^ B, C, D he any four points on [or tangents to) the same circle^ and X, F", Z the three points of intersection [or lines of connection) of the three pairs of connectors [or intersections) BG and AD, CA and BD, AB and CD (figs, a and a') ; then (see Art. 272). 1°. For every pair of concyclic points [or tangents) E and F which connect through [or intersect on) X, [ABEF\^[CDEF] and [ACEF\ = [BDEF] [a); 2°. For every pair G and H which connect through [or in- tersect on) Y, {BCGH] = [ADGE] and {BAGH] = {CDGH}...[b) ; 3°. For every pair K. and L which connect through [or in- tersect on) Z, [GAEL] = [BDKL] and {GBKL] = [ADKL] .... (c) ; and, conversely, all pairs of concyclic points [or tangents) which fulfil either, and therefore the other (272), of any of the three preceding pairs of eguianharmonic relations, connect through [or intersect on) the corresponding one of the three points [or lines) X, Y, Z. For, by the two converse pairs of reciprocal properties (286, 4°) and (289), the coUinearity (or concurrence) of the three points (or lines) E, F, and X, in the first case, involves, and is involved in either of, the two equianharmonic relations [B.ACEF] = [A.BDEF] and [G.ABEF] = [A.GDEF] ; that of the three G, H, and Y, in the second case, involves, and is involved in either of, the two {G.BAGH] = [B.CDGIl] and [A.BCGH] = {B.ADGH} ; 184 ANHAEMONIC PKOPERTIES OF THE CIRCLE. X / ^)r -.^X/y^ (-) \ 1 A 3/ { 1 y/^ \ \ /^^ >'>>. ^^- ^/ rr r^ z^.^ ■''?'• [^ ^ and that of the three K, Z/, and Z, in the third case, involves, and is involved in either of, the two [A.CBKL] = { C.ADKL] and {B. CAKL] = { C.BDKL] ; and therefore &c. (305). COE. 1°. The three points (or lines) X, Y, Z in the above, taken in pairs Y and Z, Z and X, X and Y^ being the three pairs of centres (or axes) of perspective of the three pairs of inscribed chords (or exscribed angles) BG and AD, GA and BD, AB and GD determined by the four concyclic points (or tangents) A, B, G, B; it follows consequently, from the three pairs of equianharmonic relations (a), [b), (c), that — a. Every two points on a circle, which connect through either centre of perspective of any two inscribed chords, divide equian- harmonically the two arcs of the circle intercepted by the chords ; and, conversely, every two points on a circle, which divide equi- anharmonically any two arcs of the circle, connect through one or other of the two centres of perspective of the two inscribed chords determined by the arcs. a'. Every two tangents to_ a circle, which intersect on either axis of perspective of any two exscribed angles, divide equian- harmonically the two arcs of the circle intercepted by the angles ; and, conversely, every two tangents to a circle, which divide equi- anharmonically any two arcs of the circle, intersect on one or other cf the two axes of perspective of the two eoescribed angles determined by the arcs. CoE. 2°. In the particular case where the two arcs of the circle, intercepted between two of the four points (or tangents). ANHAEMONIC PROPERTIES OF THE CIRCLE. 185 B and G suppose, and between the remaining two A and D, are equal ; since then, evidently, one of the two centres of per- spective Z (fig. a) of the two inscribed chords they determine ia at an infinite distance, while one of the two axes of perspective Z (fig. a') of the two exscribed angles they determine passes through the centre of the circle ; and since, consequently, the two circular points at infinity (260) connect through the former, ■while the two tangents from the centre of the circle intersect on the latter ; hence, from the first parts of the above, it appears that— a. Every two, and therefore all, equal arcs of tJie same circle are cut equianharmonically hy the two circular points at infinity, a'. Every two, and therefore all, equal arcs of the same circle are cut equianharmonically hy the two central tangents to the circle. Cor. 3°. In the particular case when, of the four points (or tangents) A, B, C, D, any two, B and G suppose, and also the remaining two, A and D, coincide ; and when, consequently (19 and 20), the point (or line) X is the intersection of the terminal tangents (or the connector of the terminal points) of the arc of the circle AB intercepted between the two pairs of coincident points (or tangents) B= G and A = D; since then, for every pair of coucyclic points (or tangents) E and F which connect through (or intersect on) X, by either relation (a) of the above, {ABEF] = [BAEE^ ; and since, by (281), every pair of concyclic points or tangents E and F which fulfil the latter relation are harmonic conjugates to each other with respect to the two A and B; hence also, from the above, as already established on other principles in Art. 257, it appears that— a. Every two points on a circle, which connect through the intersection of any two tangents to the circle, divide harmonically the arc of the circle intercepted by the tangents; and, conversely, every two points on a circle, which divide any arc of the circle harmonically, connect through the intersection of the terminal tangents of ike arc. a. Every tioo tangents to a circle, which intersect on the con- nector of any tioo points on the circle, divide harmonically the arc of the circle intercepted hy the points; and, conversely, 186 ANHARMONIC PEOPERTIES OF THE CIRCLE. every two tangents to a circle, which divide any arc of the circle harmonically, intersect on the connector of the terminal points of the arc. Cor. 4°. When, in the general case, the two points (or tangents) E and F connect through (or intersect on) not only X but also Y, or the two Q and H connect through (or inter- sect on) not only Y but also Z, or the two K and L connect through (or intersect on) not only Z but also X; since then, by the first and second of the general relations (a) and (J), (J) and (c), (c) and (a), respectively, combined — [ABEF] =[BAEF] and [CDEF] =[DCEF], {BCGH\ = [GBGE] and [ADGH\^{DAGB], {CAKL]=[ACKL} and [BDKL] ={DBKL]; and since, consequently, by (281), the several systems of four constituents are all harmonic; hence again, from the above, as already shewn on other principles in Art. 261, it appears that— a. The two points on a circle^ which are collinear with the two centres of perspective of any two inscribed chords, divide harmonically the two arcs of the circle intercepted by the chords ; and, conversely, the two points on a circle, which divide har- monically the two arcs of the circle intercepted by any two in- scribed chords, are collinear with the two centres of perspective of the chords. a . The two tangents to a circle, which are concurrent with ike two axes of perspective of any two exscribed angles, divide harmonically the two arcs of the circle intercepted by the angles / and, conversely, the two tangents to a circle, which divide har- monically the two arcs of the circle intercepted by any two ex- scribed angles, are concurrent with the two axes of perspective of the angles. Cor. 6°. Since, in the same case (see figures), by virtue of the general property of Art. 285 — {EFXY}=:{D.EFAB}={C.EFBA] = {B.EFCI)}=^{A.EFI>C}, {EFYX]=[aEFAB} = {D.EFBA} = {A.EFCB}^{B.EFnC}, with similar groups of relations for the system G, II, Y, Z, and ANHaSmONIC PEOPEETIES OF THE CIRCLE. 187 for the system Z, L, Z, JT, (which, it will be observed, prove directly the harmonic relations of the preceding corollary) ; it follows consequently, by (306), that — {EFXY}=[EFYX], {GHYZ]={GHZY], {KLZX]=[KLXZ]; and therefore, by (281), as established already on other con- siderations in Art. 261, that — a. The two centres of perspective of any two chords inscribed to a circle divide harmonically the segment, real or imaginary, intercepted between the two collinear points on the circle. a'. The two axes of perspective of any two angles easscribed to a circle divide harmonically the angle, real or imaginary, inter- cepted between the two concurrent tangents to the circle. Cor. 6°. In the particular case when the four points (or tangents) A, B, G, D are in pairs, A and B, V and D suppose, diametrically opposite to each other ; since then, evidently, the two centres of perspective Z and X (fig. a) of the two inscribed chords AB and CD, and with them of course all collinear points, are at infinity, while the two axes of perspective Z and X (fig. a') of the two exscribed angles AB and CD, and with them of course all concurrent lines, pass through the centre of the circle ; hence, from Cor. 4°, as established already on other considerations in Art. 260, it appears that — a. Every two, and therefore all, semicircular arcs of the same circle are cut harmonically by the two circular points at infinity. a'. Every two, and therefore all, semicircular arcs of the same circle are cut harmonically by the two central tangents to the circle. Cor. 7°. The two reciprocal properties of Cor. 1° supply obvious solutions of the three following pairs of reciprocal problems, viz. — a. To draw a line, 1°. passing through a given point and determining two points on a given circle dividing two given arcs of the circle equianharmonically ; 2°. touching one given circle and determining two points on another given circle dividing two given arcs of the latter circle equianharmonically; 3°. determining two points on each of two given circles dividing two given arcs of each circle equianharmonically. a'. To find a point, 1°. lying on a given line and determining two tangents to a given circle dividing two given arcs of the circle 188 ANHAEMONIC PEOPEKTIES OF THE CIECLE. equianharmonically ; 2°. lying on one given circle and determining two tangents to another given circle dividing two given arcs of the latter circle equianharmonically ; 3°. determining two tangents to each of two given circles dividing two given arcs of each circle equianharmonically. Since, by the corollary in question (Cor. 1°), every two points on (or tangents to) a circle, which connect through either centre of perspective of the two inscribed chords (or intersect on either axis of perspective of the two exscrlbed angles) determined by any two arcs of the circle, divide those arcs equianharmonically; it follows, consequently, that, of the above pairs of reciprocal problems, the first of each group admits of two, and the second and third of each admit of four, different solutions, the two points (or tangents) corresponding to any or all of which may, according to circumstances, be aa often imaginary as real. Cor. 8°. Since (156) every circle, whose chord of inter- section with either of two orthogonal circles passes through the centre of the other, is orthogonal to the latter; while, conversely, every circle orthogonal to one of two orthogonal circles determines a chord of the other passing through the centre of the former; it follows consequently, from the first and second parts of property (a) of the same corollary (Cor. 1°) respectively, that — Every circle orthogonal to either of two orthogcmal circles cuts equianharmonically every two arcs of the other intercepted between two diameters of the former ; and, conversely, every circle cutting any two arcs of another circle equianharmonically is orthogonal to one or other of the two circles ortlwgonal to the latter, and to each other, 176, 1°, whose centres are the two centres of perspective of the chords of the arcs. CoE. 9°. Since (187, 2°) a variable circle, whose chords of intersection, real or imaginary, with two fixed circles pass through two fixed points, describes the coaxal system orthogonal to the pair of circles concentric with the points and orthogonal to the circles ; it follows evidently, from the second part of the general propertj- of the preceding corollary, that — 1 . A variable circle, passing through a fixed -point and anhaTImonig properties of the circle. 189 cutting two fieed arcs of a fixed circle equianharmonically^ passes also through the inverse of the point with respect to one or other of the two circles orthogonal to the fixed circle and concentric with the two centres of perspective of the chords of its arcs. 2°. A variable circle^ cutting two fixed arcs of each of two fixed circles equianharmonically^ describes one or other of the four coaxal systems orthogonal to a circle of each pair orthogonal to one of the fixed circles and concentric with the two centres of per- spective of the chords of its arcs. Cor. 10°. Since (156, Cor. 4°) a circle may be described, 1°. passing through two given points and cutting a given circle orthogonally ; 2°. passing through a given point and cutting two given circles orthogonally ; 3°. cutting three given circles ortho- gonally ; the first part of the same general property (that of Cor. 8°) supplies obviously the two, four, and eight solutions, respectively, of the three following problems, viz. — To describe a circle^ 1°. passing through two given points and cutting two given arcs of a given circle equianharmunically ; 2°. passing through a given point and cutting two given arcs of each of two given circles equianharmonically ; 3°. cutting two given arcs of eacli of three given circles equianharmonically. 312. Since every two similar angles, however circumstanced as to position, absolute or relative, intercept equal arcs on every circle passing through their two vertices, and since every circle, whatever be its magnitude or position, passes through the two circular points at infinity (260) ; it follows consequently, from relations (a) of Cors. 2° and 6° of the preceding article, that — Every two, and therefore all, similar angles, however circum- stanced as to position, absolute or relative, are cut equianhar- monically, and if right angles harmonically, by the lines connecting their vertices with the two circular points at infinity ; the value of the common anharmonic ratio of their section depending on their common form, and being = — 1 when that form is rectangular. This remarkable result, which for the particular case of right angles has, it will be remembered, been already established on other principles in Art. 260, is of considerable importance in the higher departments of modern geometry; as bringing at once under the operation of all processes of geometrical trans- 190 ANHAEMONIC PROPERTIES OF THE CIRCLE. formation, such as reciprocation, projection, &e., under which anharmonic ratios remain unchanged, all properties of geo- metrical figures involving similar angles ; and shewing, in general, what such properties become by transformation when the angles themselves, as they generally do, lose by change of form their character of similarity under the process of transformation. 313. From the two reciprocal properties of Art. 311 the two following, also reciprocal, properties respecting concyclic triads of points and tangents in perspective may be immediately inferred, viz. — When two systems of three points on [or tangents to) the same circle A, B, C and A\ B\ C\ which correspond two and two in opposite pairs A and A', B and B\ C and C", are in perspective, every pair of systems determined hy four of the six constituents and their four opposites are equianharmonic ; and, conversely, when they are such that any pair of systems determined hy four of the six constituents and their four opposites are equianharmonic, they are in perspective. For, if be the point of concurrence (fig. a), or the line of collinearity (fig. a'), of two of the three lines of connection (or points of intersection) AA', BB', CC of the three pairs of opposite constituents A and A', B and B', C and C ; then since, by the two reciprocal properties of the article in question (311), the concurrence (or collinearity) with them of the third involves both, and is involved in either of, the two equianhar- monic relations, ANHA]ffi*ONIC PROPERTIES OP THE CIRCLE. 191 1°. If BB' and CC be the two, and AA' the third, [BCAA-] = { G'B'AA'] = [F C'A'A], (280)| and {BC'AA'\ = {GB'AA'} =[B'CA'A}^ (280)1**"'^"^' 2°. If CC and AA' be the two, and BB" the third, { CABB'} = {A' G'BB'} = { C'A'B'B}, (280)) „. . and {CA'BB'} = {AC'BB'] ={C'AB'B], (280)J '*" ^ ' ' 3°. If ^^' and BB' be the two, and CO' the third, {^jBCC} = [B'A' CC'} = [A'B' C C], (280)) , . and {AB-CG'] = [BACC'}=[ABC'G], (280)1 ^*'^' therefore &c. (283). Otherwise thus : Since, when the two triads are in per- spective, their three pairs of opposite constituents A and A', B and 5', G and G' divide harmonically the arc MN^ real or imaginary, intercepted between the two tangents to the circle from the centre of perspective (or the two intersections with the circle of the axis of perspective) (257), therefore, by 3°, Cor. .5, Art. 282, every four of the six constituents and their four opposites form equianharmonic systems. And since, conversely, when any four of the six constituents and their four opposites form equianharmonic systems, the three pairs of opposite constituents divide harmonically a common arc MN, real or imaginary (Cor. 1°, Art. 283), therefore, (by 257), the two triads are in perspective. This latter demonstration, though perhaps on the whole simpler, yet, as involving the contingent elements M and N^ is consequently less general than the former in which all the elements involved are permanent (21). In the particular cases when, in the first parts of the above reciprocal properties, the centre of perspective 0, in the former case, is the centre of the circle, and the axis of perspective 0, in the latter case, is the line at infinity; the three pairs of corresponding constituents A and A', B and -B', G and 6" being then diametrically opposite pairs with respect to the circle, the two properties are evident, h priori, from the obvious similarity and consequent equianharmonicism of every two systems deter- mined by four of them and their four opposites. See Cor. 2°, Art. 283, from which also, Euc. III. 31, the properties evidently follow in the same cases. 192 ANHAEMONIC PEOPERTIES OP THE CIEOLE. Cor. 1°. It follows, indirectly, from both parts of the above properties combined, that when a system of six points on [or tangents to) the same circle^ which correspond two and two in opposite pairs A and A\ B and B\ C and C\ is such that any two systems determined by four of the six constituents and their four opposites are equianharmonic, then every two systems deter- mined by four of them and their four opposites are equianharmonic. For, by the second parts of the above properties, the equian- harmonicism of any one of the six pairs of conjugate groups of four into which the system may be divided involves the perspective of the two conjugate triads A^ B, C and A\ B', C of which it consists, and consequently, by the first parts of the same properties, the equianharmonicism of the remaining five. This property, it will be remembered, was proved directly for coUinear points and concurrent lines, and with them, implicitly, for concyclic points and tangents, in Art. 283, and the above indirect verification of it for the latter may of course be re- garded as extending to the former also. CoE. 2°. The first parts of the above reciprocal properties supply obvious solutions of the two reciprocal problems, of two triads of concyclic points [or tangents) A, B, G and A', B\ C, which correspond two and two in opposite pairs, and every four of which are equianharinonic with their four opposites ; given any two pairs of corresponding constituents A and A', B and B', and either constituent C of the third pair C and C, to determine the second constituent C of that pair. For, the two lines of connection (or points of intersection) AA and BB' of the two given pairs A and A', B and B' determine, by their point of intersection (or line of connection), the centre (or axis) of perspective of the two triads, and with it, consequently, the conjugate C" to the given constituent C of the third pair C and C", 314. From the two reciprocal properties of the preceding article, respecting concyclic triads of points or tangents in perspective, it may be readily inferred, for concyclic quartets of points or tangents in perspective, that, more generally — When two systems of foxvr points on {or tangents to) the same circle A, 2?, C, D and A\ B', C", D\ which correspond two and ANHAuftoNIC PROPERTIES OP THE CIRCLE. 193 two in cpposite pairs A and A', B and B\ C and C, D and D', are in perspective^ every pair of systems determined hy four of the eight constituents and their four opposites are equianharmonic ; and, conversely, when two equianharmonic systems of four points on {or tangents to) the same circle A, B, C, D and A\ B\ C\ D' are such that a pair of their corresponding constituents may he interchanged without violating their relation of equianharmonicism, they are in perspective. For, from the general property (1°) Art. 282, see figures of preceding article. 1°. Any two of the three equianharmonic relations { CDAA'] = { CBfAA], [DBAA'} = [D'B'A'A], {BCAA'} = {B'C'A'A], involve and are involved in the two {BODA] = [RC'iyA} and {BGDA'] = {B'G'DA}...{a) ; 2°. Any two of the three equianharmonic relations [DABB'] = [D'AB'B], {AGBB'] = [AC'B'B], {CDBB'} = {C'1)'B'B}, involve and are involved in the two {CBAB} = {C'D'A'B'} and {GDAB'} = {C'D'A'B} ...{b); 3°. Any two of the three equianharmonic relations {ABGC'} = {A'B'C'G}, {BBCG'} = {B'B'G'G}, {I)AGG'} = {J!yA'G'G}, involve and are involved in the two {DABC} = {B'A'B'G'} and {BABC'} = {D'A'B'G\...{c); 4°. Any two of the three equianharmonic relations {BCBB'l = {B'G'B'B], {GADB'] = {CA'D'D}, {ABDD'] = [AB'UB], involve and are involved in the two [ABGD] = [AB'C'B'] and {ABCD'] = {AB'C'B]...[d) ; and as three similar quartets of equianharmonic relations result evidently from the interchange ; firstly, of B and B' in 1°, of C and G' in 2°, of D and U in 3°, and of A and A in 4°; secondly, of G and G' in 1°, of D and D' in 2°, of A and A in VOL. II. 194 ANHAEMONIC PROPEETIES OP THE CIRCLE. 3°, and of B and B' in 4°; and thirdly, of D and D' in 1°, of A and A in 2°, of B and B' in 3°, and of C and C in 4°; therefore &c. Otherwise thus, as for the properties of the preceding article, ■which are included in the above as particular cases. Since (same figures) when the two quartets are in perspective, their four pairs of opposite constituents A and -4', B and B\ C and C", X> and jy divide harmonically the arc MN, real or imaginary, intercepted between the two tangents to the circle from the centre of perspective (or the two intersections with the circle of the axis of perspective) (257) ; therefore, by 3°, Cor. 4°, Art. 282, every four of the eight constituents and their four opposites form equianharmonic systems ; and since, conversely, when the two quartets are equianharmonic, and preserve their equianharmonicism on the interchanges of a pair of their cor- responding constituents, the four pairs of corresponding consti- tuents divide harmonically a common arc MN, real or imaginary, (Cor. 1°, Art. 284) ; therefore, by (257), the two quartets are in perspective. This latter demonstration has the same advantages and disadvantages, compared with the former, as for the properties of the preceding article. In the particular cases when, in the first parts of the above reciprocal properties, the centre of perspective 0, in the former case, is the centre of the circle, and the axis of perspective O, in the latter case, is the line at infinity ; the four pairs of corres- ponding constituents A and A', B and B', C and C", I) and B' being then diametrically opposite pairs with respect to the circle, the two properties are evident, h priori, from the obvious similarity, and consequent equianharmonicism, of every two systems determined by four of them and their four opposites. See Cor. 2°, Art. 284, from which also, Euc. III. 31, the pro- perties evidently follow in the same cases. Cor. 1°. It follows, indirectly, from both parts of the above properties combined, that when two equianharmonic systems of four points on (or tangents to) the same circle A, B, C, D and A\ B\ C", D' are such that a pair of their corresponding consti- tuents may he interchanged without violating their relation of equianharmonicism, then every pair of their corresponding con- stituents may be interchanged without violating their relation of ANHAElfbNIC PROPERTIES OP THE CIRCLE. 195 equianharmonicism. For, by the second parts of the above pro- perties, the possibility of a single such interchange involves the perspective of the systems, and consequently, by the first parts of the same properties, the possibility of every such interchange. This property also, like that of the corollary of the preceding article, it wLU be remembered, was proved directly for coUinear points and concurrent lines, and therefore implicitly for concyclic points and tangents, in Art. 284, and the above may be regarded as an indirect verification of it for the former as well as for the latter. Cor. 2°. The first parts of the above reciprocal properties supply obvious solutions of the two reciprocal problems ; of two quartets of concyclic points {or tangents) A, B, (7, D and A', B\ C, D'y which correspond two and two in opposite pairs, and every four of which are equianharmonic with their four opposites ; given any two pairs of corresponding constituents A and A\ B and B\ and any two non-corresponding constituents of the remaining two pairs, to determine the remaining two con- stituents of those pairs. For, the two lines of connection (or points of intersection) AA' and BB' of the two given pairs of corresponding constituents A and A', B and B' determine, by their point of intersection (or line of connection), the centre (or axis) of perspective of the two quartets, and with it, conse- quently, the two correspondents to the two given non-corres- ponding constituents of the remaining two pairs. 315. The two reciprocal properties of the preceding article are sometimes enunciated as follows : When three pairs of points on {or tangents to) the same circle A and A', B and B', C and G' determine two triads in perspec- tive A, B, C and A', B', C ; every fourth pair 1) and Z>' deter- mines with them two quartets A, B, C, J) and A\ B\ C, D' which if in perspective are equianharmonic, and which if equian- harmonic are in perspective. And the following reciprocal demonstrations, based on the two reciprocal properties a and a of Art. 261, are sometimes given of them. If be the centre (fig. a), or the axis (fig. a'), of perspective of the two triads A, B, C and A!, B', C ; 1 its polar (fig. a), or its pole (fig. a'), with respect to the circle ; 02 lyo ANHARMONIC PROPEKTIES OF THE CIRCLE. E and E' any fifth pair of points (or tangents) connecting through (or intersecting on) 0; P, Q, R, 8 the four points of intersection (or lines of connection) of the four pairs of lines of connection (or points of intersection) EA and E'A', EB and E'B\ EG and E'C\ ED and E'D' ; and P', (?', R\ S' the four for the four pairs EA' and E'A, EB' and E'B, EC and E'C, ED' and E'D ; then since, by the properties in question, (a and a', Art. 261), the two triads of points (or lines) P, Q, R and P', Q'^ R' lie on the line (or pass through the point) /; and since on the coUinearity (or concurrence) of the two 8 and 8' with them depends, at once, the circumstance of the two points (or tangents) D and Z>' connecting through (or intersecting on) {a and a', Art. 261), and the circumstance of the two pairs of pencils (or rows) E.ABGD and E'.A'B'C'D', E.A'B'C'D and E'.ABCD, that is, of the two quarteta of points (or tan- gents) A, B, C, D and A', P', C", D', being equianharmonic (o and a, Art. 288) ; therefore &c. CoE. 1°. Since, when two systems of any common number of points on (or tangents to) the same circle A, B, C, D, &c. and A', P', C, D', &c, which correspond in pairs A and A', ANHARMONH; PKOPEBTIES OF THE CIECLE. 197 B and B\ G and C\ D and Z>', &c. are in perspective, any pair, or any number of pairs, of their corresponding constituents may evidently be interchanged without violating their rela- tion of perspective ; therefore, in the equianharmonic relation [ABGD] = {A' B' C D'], which connects, as above shewn, every two quartets A, B, C, D and A', B\ C", D' in perspective, the accented and unaccented constituents may be interchanged at pleasure without violating the relation of equianharmonicism ; thus, for the eight different combinations of four and their four correspondents that could be formed from the four pairs of cor- responding constituents A and A', B and B\ C and C, D and D' (314), giving rise (see figures), as observed in Arts. 284 and 314, to the eight following different cases of anharmonic equi- valence, viz. — {ABCD ] = [A'B'C'iy] = [PQRS ] = [FqR'8'], {A'BCD ] = {AB'C'B'} = {FQRS } = {PQ'B'S'], {AB'CD j = [A' BCD' ] = {PQ'ES } = [P'QES'], {ABG'D } = {A'B'GD' } = [PQE'S } = [P'Q'RS'], [ABGU ] = [A'B'G'D] = [PQR8' ] = [P'Q'R'S], [A'B'GD] = {ABG'D' } = {P'Q'RS} = {PQRS' ], {ABG'D] = {AB'GD' ] = {P'QR'S] = {PQ'RS' j, {A'BGD' ] = {AB' G'D } = {P QR8' } = {PQ'R'S ] ; for none of which, however, is it to be supposed, as is sometimes erroneously done by beginners, that the anharmonicism is that of tbe pencil (or row) determined, at the centre (or on the axis) of perspective 0, by the four lines of connection (or points of intersection) AA', BB', CC', DD', whose concurrence (or col- linearity) constitutes the common perspective of them all. Cor. 2°. If, while, of two concyclic quartets of points or tangents in perspective A, B, G, D and A', B', G', D', three pairs of corresponding constituents A and A', B and B', G and C" •are supposed to remain fixed, the fourth pair D and D' be conceived to vary; and, in the course of their variation, to coincide successively ; firstly, D with A' and Z>' with A ; secondly, D with B' and D' with B; thirdly, D with C' and D' with C; since then, for every position of D and D', by the 198 ANHARMONIC PKOPERTIES OP THE CIRCLE. above, {ABCD} = {A'B'G'D'], therefore for the particular posi- tions in question (see figures) — {BGAA'] = {B'C'A'A } = {QEPP'} = [Q'EPP] ... (1°), {CABB'] = [C'A'B'B] = [RPQQ] = {R'FQ'Q} ... (2°), {ABCC} = {A'B'C'C} = [PQBE'} = {FQ'E'B} ... (3°) ; and from them, by the interchange, as explained in Cor. 1°, of B and B' in (1°), of C and C in (2°), and of A and A' in (3°), {B'CAA'} = {BC'A'A} = [Q'RPF] = [QR'PP] .... (1'), [C'ABB'} ^[CA'FB] = {EPQQ] = [RFQQ] .... (2'), [A'BCG'] = [AB'C'C]==[FQBE] = [PQ'R'B] .... (3') ; and, conversely, if, for two concyclic triads of points or tangents A, B, C and A', B', C", any of the six. preceding relations (1°) or (1'), (2°) or (2'), (3°) or (3') exist, since then the necessary perspective of the repeated with each unrepeated pair of corres- ponding constituents involves, by the above, the perspective of the triads; therefore, see Cor, 3°, Art. 284, the two reciprocal properties respecting concyclic triads in perspective, established on other considerations in Art. 313, are particular cases of those respecting concyclic quartets in perspective, established by their aid in the subsequent article. 316. From the general property of the preceding article, that every two quartets of points or tangents of the same circle in perspective are equianharmonic, combined with the circum- stance of the evident equianharmonicism of every two similar quartets of points or tangents, either of the same or of different circles (309) ; it follows immediately that — For any two circles^ every two systems determined hy four points or tangents of either, and by the four antihomoloffous points or tangents of the other toith respect to either centre of perfective of the two (198), are equianharmonic. For, the system for either circle being similar to the homo- logous system for the other with respect to either centre of perspective of the two, and the latter being in perspective with the corresponding antihomologous system for the same circle (198), therefore &c. anhaeSonic properties of the circle, 199 The above property may obviously be stated otherwise (206), as follows — For any two circles, every two pairs of antihomologous arcs, with respect to eitJier centre of perspective, divide each other equi- anharmomcaUy. Cor. 1°. As every circle, intersectiDg two others at any equal or supplemental angles, intercepts on them a pair of antihomologous arcs with respect to their external or internal centre of perspective, according as the angles of intersection are equal or supplemental (211); hence, from the above, it appears that — a. Every circle intersecting two others at any equal, or supple- mental, angles divides equianharmonically all pairs of their antihomologous arcs with respect to their external, or internal, centre of perspective, b. When two circles each intersect two others at any equal or supplemental angles, the pairs of arcs they intercept on them divide each other equianharmonically. c. When two circles intersect two others, one at any angles and the other at the same or the supplemental angles, their pairs of arcs interested by them divide each other equianharmonically. d. When two circles intersect two others, both at the same equal or supplemental angles, the pairs of arcs they intercept on them, and their pairs of arcs intercepted by them, both divide each other equianharmonically. Cor. 2°, Since every circle orthogonal to two others in- tersects the two at once at equal and at supplemental angles, and since, of the entire system of circles orthogonal to the same two, one, viz. their common diameter, is a line; it fol- lows, consequently, from a and d of the preceding (Cor. 1°), that— a. Every circle orthogonal to two others divides equianhar- monically all pairs of their antihomologous arcs with respect to either of their centres of perspective. b. Every two circles determine on every circle orthogonal to them both a system equianharmom'c with that they determine on their common diameter. c. When two circles are orthogonal to two others, both pairs 200 AN HARMONIC PKOPEETIES OF THE CIKCLE. determine equianharmonic systems, each on the circles of the other and on their own common diameter. Cor. 3°. If, in property c of Cor. 1°, one of the two intersect- ing circles be conceived to vary while the other and the two Intersected circles remain fixed; since then, by virtue of the general property (193, Cor. 8°), the variable circle intersecting two fixed circles at two constant angles intersects at a third constant angle every third fixed circle coaxal with them, it follows consequently, from that property, that — When a variable circle intersects any two fixed circles at any two constant angles ; a. its arc intercepted by either is cut in a constant anharmonic ratio by the other ; b. its arcs intercepted by both are cut in constant anharmonic ratios by all fixed circles coaxal with both. 317. The two following reciprocal properties, respecting any two concyclic triads of points and tangents, are in the modem geometry of the circle what those of Art. (293) are in that of the point and line, and lead to as many and important conse- quences in the applications of the theory of anharmonic section. If A, B, G be any three points on [or tangents to) a circle^ A', B', C any other three points on [or tangents to) the same circle, and X, T, Z the three intersections [or connectors) of the three pairs of connectors [or intersections) BC and B'C, CA' and G'A, AB' and A'B; the three points [or lines) X, Y, Z are coUinear [or concurrent) ; and their line of collinearity [or point of concurrence) determines with the circle two points [or tangents) M and N connected with the two original triads A, B, C and A', B', C by the three groups of equianharmonic relations {BCMN} = {B'C^MN]^ {CAMN] = {C'A'MN}\ (i), {ABMN} = {A'B'MN}) {AA'MN}={BB'MN]=.{CC'MN} (2), [ABCM} = {A'B'C'M}] {ABCN} = {A'B'C'N]\ '^^' For, if M and N be the two points, fig. (a), (or tangents, fig- (»'))) determined with the circle by the line of connection ANHAEftoNIC PROPERTIES OF THE CIRCLE. 201 {or point of intersection) / of any two, X and Y suppose, of the three points (or lines) X, 7, Z; then since the two triads of concyclic points (or tangents) B, C, M and C\ B\ N are in perspective, therefore, by the first parts of the two reciprocal properties of Art. 3] 1, {BCMN] = {B'C'MN] and [BB'MN] = [CC'MN]... [a] ; and since the two triads (7, -4, M and A\ C, N are in perspec- tive, therefore, by the same, [CAMN] = [C'AMN] and [CCMN] = [AA'MN}... [b] ; therefore, by the general property (1°) Art. 282, or directly as regards the second equivalents, [ABMN] = {A:B'MN} and [AA'MN] = [BB'MN]... [c) ; and therefore, by the second parts of the two reciprocal proper- ties of Art. 311, the two triads A, B, M and jB', A', N are in perspective ; which proves the first parts of the above reciprocal properties, and with them the two groups of equianharmonic relations (1) and (2), from either of which the group (3) follows immediately by virtue of the general property 2°, Cor. 3°, Art. 282. Cor. 1°. If X', Y', Z' be the three intersections (or con- nectors) of the three pairs of connectors (or intersections) BG and B'C, CA and C'A!., AB and A'B' ; it may, of course, be shewn, in precisely the same manner, that the three triads of points (or lines) Y\ Z', X; Z\ X\ Y; X\ F', Z are also collinear (or concurrent); their three lines of collinearity (or points of concurrence) determining, with that of the triad X, y, Z, a tetragram (or tetrastigm), of which the three pairs of corresponding points (or lines) X and X', Y and Y\ Z and Z' 202 ANHAEMONIC PEOPEETIES OP THE CIECLE. are the three pairs of opposite intersections (or connectors) ; and each line (or point) determining two points on (or tangents to) the circle connected with the original six by three groups of equianharmonic relations similar to the above, and differing only in the interchange of the constituents of the two corres- ponding reversed pairs of the three A and A\ B and B'^ G and C in the several equivalents which contain them. COE, 2°. In the particular case when the three equian- harmonic systems of points (or tangents) A, A', 3f, N; B, B', M, N; (7, C, M, N of group (2) are harmonic, that is, when the three intercepted arcs^^', BB', CC are cut harmonically by the intercepted arc MN; since then (281) {AA'MN] = {A'AMN}, [BB'MN] = {B'BMN}, { CG'MN] = { C CMN], therefore the two points (or tangents) M and N are the same for the three lines of collinearity (or points of concurrence) of the three triads Y\ Z\ X\ Z\ X', Y; X', T, Z, as for that of the triad X, F, Z; and therefore the whole six points (or lines) X and X', JT and Y', Z and Z' lie on the same line (or pass through the same point) /. In this case it is evident, from (257), that the three lines of connection (or points of intersection) AA', BB', CC of the three pairs of corresponding points (or tangents) A and A', B and B', C and C are concurrent (or coUinear), and that the line (or point) / is the polar of their point of concurrence (or the pole of their line of collinearity) with respect to the circle ; a property, the converse of which, for two concyclic triads of points (or tangents) in perspective, is evident from Art. 261. COE. 3°. In the same case it is easily seen that, as the three pairs of concyclic points (or tangents) A and A', B and B', C and C divide harmonically the arc of the circle MN in- tercepted between the two points (or tangents) M and N, so the three pairs of coUinear points (or concurrent lines) X and X', Y and Y', Z and Z' divide hannonically the segment (or angle) MN intercepted between them. For, since, by the general pro- perty of Art. 285, {MNXX'} = {C'.3mBB'} or {B'.MNCC], {MNYY'] = IA'.MNCC'} or {C'.MNAA'}, {MNZZ' } = [B:MNAA'} or {A'.MNBB'} ; anhaSmonic properties of the circle. 203 and since, by hypothesis, the three concyclic systems of points (or tangents) M, N, A, A'; M, JSf, B, B'; M, N, C, C are harmonic ; therefore the three coUinear (or concurrent) systems of points (or lines) M, N, X, X'; M, N, F, Y'; M, N, Z, Z' are harmonic, and therefore &c. The converse of this property, for two concyclic triads of points (or tangents) in perspective, is evident from Cor. 5°, Art. 282. Cor. 4°. That, in the same case, the three pairs of collinear points (or concurrent lines) X and X\ Y and Y"', Z and Z' con- stitute a system of six constituents, corresponding two and two in opposite pairs, every four of which are equianharmonic with their four opposites, follows also immediately from the pre- ceding Cor. 3°. For, the three intercepted segments (or angles) XX\ YY\ ZZ' having a common segment (or angle) of har- monic section, real or imaginary, MN, therefore &c. The converse of this property also, for two concyclic triads of points (or tangents) in perspective, is, like the preceding, evident from Cor. 5°, Art. 282. 318. From the two reciprocal properties of the preceding article, the following inferences, in pairs reciprocals of each other, may be shewn in precisely the same manner as the cor- responding inferences of Art. (294) from those of its preceding article (293). 1°. The three pairs of concyclic points (or tangents) A and A\ B and B', C and C" may be regarded as determining three chords (or angles) AA\ BB\ CC inscribed (or exscribed) to the circle to which they belong, of which, taken in pairs, the three points (or lines) X, Y, Z are three of the six centres (or axes) of perspective ; every two of which evidently become changed into their two opposites by the interchange of extremities of one of the two determining chords (or angles), those of the other remaining unchanged ; hence, generally, from the first parts, and from the equianharmonic relations (1) of the second parts, of the two reciprocal properties in question. a. For every three chords inscribed to the same circle, taken in pairs, the six centres of perspective lie three and three on four lines ; eaxih of which determines two points on the circle which divide eguianharmonically the three arcs interested hy the chords. 204 ANHAEMONIC PEOPERTIKS OF THE CIRCLE. d. Far every three angles exscribed to the same circle^ taken in pairs, the six axes of perspective pass three and three through Jbur points ; each of which determines two tangents to the circle which divide equianharmonically the three arcs intercepted hy the angles. In the particular case when the directions of the three chords (or the vertices of the three angles) AA', BB\ CC are concurrent (or coUinear) ; the six centres (or axes) of perspective of the three pairs they determine, being then, by Art. 261, all collinear with the polar of their point of concurrence (or con- current with the pole of their line of collinearity) with respect to the circle, the four lines (or points) of the general case then coincide ; and the two points or tangents they determine with the circle, by Art. 257, divide harmonically the three area intercepted by the chords (or angles). 2°. The two concyclic triads of points (or tangents) A, B, G and A\ B\ C may be regarded as the two triads of alternate vertices (or sides) of a hexagon AB'CA'BC Inscribed (or ex- scribed) to the circle to which they belong, of which A and A\ B and B\ and C" are the three pairs of opposite vertices (or sides), and X, Y, Z the three intersections (or connectors) of the three pairs of opposite sides (or vertices) BC and B' G^ GAl and G'A, AB' and A'B; hence, generally, from the first parts, and from the equianharmonic relations (2) and (3) of the second parts of the same, respectively — a. In every hexagon inserted to a circle, the three intersec- tions of opposite sides are collinear ; and their line of collinearity determines two points on the circle which form equianharmonic systemSf separately with the two triads of alternate, and con- jointly with the three pairs of opposite, vertices of the hexagon. CL. In every hexagon exscribed to a circle, the three connectors of opposite vertices are concurrent ; and their point of concurrence determines two tangents to the circle which form equianharmonic systems, separately with the two triads of alternate, and conjointly with the three pairs of opposite, sides of the hexagon. By virtue of the fundamental property of triangles in per- spective (140), the first parts of these latter properties are evidently identical with the celebrated theorems of Pascal and Brianchon, established already, on other considerations. In Art. 148, and generalized subsequently, on principles Independent ANHARMONIC PROPERTIES OF THE CIRCLE. 205 of the circumstance as to whether the two points (or tangents) M and N are imaginary or real, in Art. 302. 3°. In the two concyclic triads of points (or tangents) A^ £, C7 and ^', B', C", if, while the three constituents A, £, C of either and any two A' and B' of the other are supposed to remain fixed, the third constituent C" of the latter be conceived to vary, causing of course the simultaneous variation of the two constituents X and Y of the coUinear (or concurrent) triad X, Y, Z; since then, of the variable triangle XYC, the three sides (or vertices) turn round the three fixed points (or move on the three fixed lines) A^ B, Z, and the two vertices (or sides) X and Y move on the two fixed lines (or turn round the two fixed points) CB' and CA', while the third vertex (or side) C" describes (or envelopes) the circle to which the concyclic points (or tangents) belong ; hence, conversely — a. When, of a variable triangle whose sides turn round ficed points, two of the vertices move on fixed lines whose intersections with each other, and with the corresponding sides of the fixed triangle determined by the points, form with the opposite vertices of that triangle a concyclic system of points ; the third vertex de- scribes the circle determined by the five points. a. When, of a variable triangle whose vertices move on fixed lines, two of the sides turn round fixed points whose connectors with each other, and with the corresponding vertices of the fixed triangle determined by the lines, form with the opposite sides of that triangle a concyclic system of tangents ; the third side en- velopes the circle determined by the five tangents. The locus and envelope of these latter properties, as well as those of 5°, Art. 294, are evidently particular cases of the more general " locus of the third vertex of a variable triangle whose remaining vertices move on fixed lines while its three sides turn roimd fixed points" and "envelope of the third side of a variable triangle whose remaining sides turn round fixed points while its three vertices move on fixed lines ;" which, in general, by reciprocation of the above to an arbitrary circle, are easily seen to be the more general figures into which the circle becomes transformed by reciprocation (173). 4°. In the two concyclic triads of points (or tangents) A, B, C and A', B', C, if, while two pairs of corresponding con- 206 ANHAEMONIC PEOPERTIES OF THE CIRCLE. stituents A and A', B and B' are supposed to remain fixed, the third pair be conceived to vary, causing of course the simul- taneous variation of the two non-corresponding constituents X and Y of the collinear (or concurrent) triad X, Y, Z\ since then, in every position of the variable tetragram (or tetrastigm) determined by the four lines (or points) AC and AG^ BC and B' G turning round the four fixed points (or moving on the four fixed tangents) A and A\ B and B', the pair of opposite intersections (or connectors) C and C" lie on (or touch) the circle to which the concyclic triads belong, while the remaining two pairs connect through (or intersect on) the two centres (or axes) of perspective of the two inscribed chords (or exscribed angles) AB and A'B' ; hence, generally — a. When, of a variable tetragram whose four lines turn round four fixed concyclic points^ a pair of opposite intersections de- scribe the circle determined by the points^ the two remaining pairs connect through the intersections of the two corresponding pairs of opposite connectors of the points. b. Wlien^ of a variable tetrastigm whose four points move on four fixed concyclic tangents j a pair of opposite connectors envelope the circle determined by the tangents, the two remaining pairs intersect on the connectors of the two corresponding pairs of opposite intersections of the tangents. 319. The two groups of equianharmonlc relations (a) and (a') of the same article (317) supply obvious and rapid solutions of the two following pairs of reciprocal problems, than which, as will appear in the sequel, none, perhaps, are of more importance in the applications of the theory of anharmonic section, viz. — Given two concyclic triads of points {or tangents) A, B, C and A', B', C whose constituents correspond in pairs A and A' B and B\ G and C ' ; to determine the two concyclic points (w tangents) M and N which form equianharmonic systems ; 1°, separately with the two triads; 2°, conjointly with their three pairs of corresponding constituents. For, constructing the hexagon AB'CA'BC' (see figures of Art. 317) of which the two given triads of points (or tangents) Af B, G and A', B\ C' are the two triads of alternate vertices (or sides), and their three pairs of corresponding constituents ANHAEftoNIC PROPERTIES OF THE CIRCLE. 207 A and A\ B and B\ G and C" the three pairs of opposite vertices (or sides) ; that is, the hexagon determined by the two lines of connection (or points of intersection) of each constituent A, J5, G of either triad with the two non-cor- responding constituents B' and G\ G' and A, A' and B' of the other triad ; then, by the relations in question, the line of coUinearity (or point of concun-ence) of the three intersections of its opposite sides (or the three connectors of its opposite vertices) X, Y", Z determines with the circle the two points (or tangents) M and N, real or imaginary, which (see 2°, of the preceding article) solve at once the two problems. In the particular case when the three lines of connection (or points of intersection) AA\ BB', GG' of the three pairs of corresponding constituents A and A\ B and B\ G and C are concurrent (or coUinear), that is, when the two given triads of points (or tangents) are in perspective ; then, as already noticed in Cor. 2°, of Art. 317, the polar of their point of concurrence (or the pole of their line of coUinearity) with respect to the circle is the line (or point) which determines with the latter the two points (or tangents) M and N, real or imaginary, which solve at once the two problems. 320. The two following reciprocal properties, respecting the two triangles determined by any two concyclic triads of points or tangents in perspective, follow also from the same, or from the reciprocal theorems of Pascal and Brianchon, Arts; (148) and (302), with which, as shewn in the preceding 318, 2°, their first parts are virtually identical, viz. — a. When two triangles inscribed to the same circle are in per- spective, the three lines of connection of the vertices of either with any point on the circle intersect with the corresponding sides of the other at three points coUinear with each other and with the centre of perspective. a. When two triangles exscribed to tJte same circle are in per- spective, the three points of intersection of the sides of either with any tangent to the circle connect with the corresponding vertices of the other by three li)ies concurrent with each other and with the aons of perfective. For, if -4, B, G and A\ B\ G' be the two triads of vertices 208 ANHARMONIC PROPERTIES OF THE CIRCLE. (or sides) of the two triangles, their centre (or axis) of perspective, D any arbitrary point on (or tangent to) the circle, and X, y, Z the three points of intersection (or lines of connec- tion) of the three lines (or points) DA\ DB', DC with the three BC, CA, AB respectively; then since, in the three Pascal (or Brianchon) hexagons whose vertices (or sides) in consecutive order are respectively DB'BACC\ DC CBAA', DA' A CBB', the three triads of points (or lines) YOZ, ZOX, XOY are those determining their three Pascal lines (or Brianchon points) re- spectively, therefore &c. — N. B. In the particular case when, in the first of the above pair of reciprocal properties (a), the centre of perspective of the two triangles ABC and A'B'C is the centre of the circle; the three lines DA', DB', DC being then perpendiculars to the three DA, DB, DC (Euc. iii. 31), the property consequently becomes that established on other principles in Ex. 6°, Art. 137. If D' be the point (or tangent) corresponding to D in the same perspective with the two inscribed (or exscribed) triangles ABC and A'B'C; and X', Y', Z' the three points of intersec- tion (or lines of connection) of the three lines (or points) D'A^ D'B, DC with the three B'C, CA', A'B' respectively; it is easy to shew, in the same manner precisely as above, that the three points [or lines) X', Y', Z', which hy the above are collinear [or concurrent) with each other and with ike point (or line) 0, are also collinear [or concurrent) with the three X, Y, Z, with which they consequently (313) determine, in three opposite pairs X andX.', Y and Y', Z and Z', a system of six constituents, every four of which are equianharmonic with their four opposites. For, in the three Pascal (or Brianchon) hexagons whose vertices (or sides) in consecutive order are respectively B'DD'CAA', C'DD'ABB', A'DD'BCC, the three triads of points (or lines) YOZ', ZOX', XOY' being those determining their three Pascal lines (or Brianchon points) respectively, therefore &c. In the same case, it is easy to shew also that the two triads of collinear points [or concurrent lines) X, Y, Zand X', Y', Z' determine equianharmonic systems with the centre [or axis) of perspective 0. For, since, by (285), {XYZO} = {D.A'B' CD] and [X'Y'Z'0}={D'.ABGD], and since, by (3U), {A'B'C'D'} = {ABCD], therefore {XYZO} = [X'Y'Z'O) ; and therefore &c. ANHAKMONIC PEOPEETIES OF THE CIRCLE. 209 This property is evidently a particular case of that estahlished on other principles for any two triangles in perspective in Art. 295, Cor. 6°. By reciprocation to an arbitrary circle, the above, as well as all the other pairs of reciprocal properties established in this chapter, with all the consequences to which they lead in the geometry of the circle, are seen at once to be true, not only of circles, but generally of all figures into which circles become transformed by reciprocation ; all such, as noticed in the opening article (305), possessing alike the two fundamental anharmonic properties a and a of that article, from which, as has been seen, all the others established in the chapter have been successively inferred. VOL. 11. ( 210 ) CHAPTER XIX. THEORY OF HOMO GRAPHIC DIVISION. 321. Two rows of points or pencils of rays, or a row of pomts and a pencil of rays, A, B, 0, D, JE, F^ &c. and A\ B\ G\ U^ E'^ F\ &c. whose constituents correspond in pairs A and ^', B and B\ G and C", D and D\ E and E', F sxiA F", &c. are said to be homograpMc (282) when every four con- stituents of one and the four corresponding constituents of the other are equianharmonic (278). Every two similar rows of points or pencils of rays (268) ; every row of points and pencil of rays determined by it, or pencil of rays and row of points determined by it (285) ; every row of points and pencil of rays reciprocal to each other with respect to any circle (202) ; are evidently thus related to each other. In accordance with the above definition of bomography be- tween two rows of points or pencils of rays, or a row of points and a pencil of rays, whose constituents correspond in pairs ; two variable points or lines, or a variable point and line, dividing two fixed segments or angles, or a fixed segment and angle, so that every four positions of one and the four corresponding positions of the other are equianharmonic, are said to divide JuymograpTiically the two segments or angles, or the segment and angle ; the two systems of constituents determined by their several pairs of corresponding positions being, as above defined, homographic. Hence the meaning and origin of the name Jwmographic division as applied, by Chasles, to this the process by which homographic systems are most frequently generated in modem geometry. 322. Two systems of points on or tangents to, or a system of points on and a system of tangents to, the same circle, or two Tril:ORT OP HOMOQEAPHIC DIVISION. 211 different circles, A, B, C, D, E, F, &c. and A', B', C, D', E', I", &c. whose constituents correspond in pairs A and A', B and B\ C and C\ D and D\ E and E\ F a.nA F\ &c. are also said to be homographic under the same circumstances as rows of points and pencils of rays ; viz., when every four constituents of one and the four corresponding constituents of the other are equianharmonic (309). Every two similar systems of points on or tangents to the same circle or two different circles (305) ; every system of points on and the corresponding system of tangents to the same circle (307) ; every two systems of points on or tangents to the same circle in perspective with each other to any centre or axis (315) ; are evidently thus related to each other. It will appear in the sequel that homographic systems of points on, or of tangents to, the same circle possess not unfre- quently comparative facilities of management in the general case when the radius of the circle is finite, which are altogether lost in the two extreme cases when it is either evanescent or infinite; and when, consequently, the two systems of points are coUinear in the one case, and the two systems of tangents concurrent in the other. 323. As two or more magnitudes of any kind when equal to a common magnitude are equal to each other; it is evident, from the conditions of homography as stated in the two preced- ing articles (see Cor. 2°, Art. 278), that voh&n, two or more systems of any species are homographic with a common system^ they are homographic with each other ; and their several pairs or groups of constitiienfs which correspond to the same constituent of the camm^m system correspond to each other. All rows of points or pencils of rays in perspective with the same row or pencil (285) ; all rows or points or pencils of rays reciprocals to the same pencil or row with respect to different circles (292) ; all pencils of rays determined by the same system of points on a circle at different points on the circle, and all rows of points determined by the same system of tangents to a circle on different tangents to the circle (305) ; all systems of points determined by the same pencil of rays on different circles passing through its vertex, and all systems of tangents determined by the same row of points p2 212 THEOET OF HOMOGEAPHIC DIVISION. to different circles touching its axis (309) ; are thus homographic with each other. And as, again, two or more magnitudes of any kind when equal, not all as above to a common magnitude, but each instead to a different one of as many equal magnitudes, are also equal to each other; it follows consequently, as evidently, from the same conditions, that when two or more systems of any species are homographic each with a different one of as mxmy homographic systems, they are homographic with each other / and their several pairs or groups of constituents which correspond to corresponding pairs or groups of the homographic systems correspond to each other. All rows of points or pencils of rays determined by homographic pencils of rays or rows of points (285) ; all rows of points or pencils of rays in perspective with homographic rows or pencils (285) ; all rows of points or pencils of rays reciprocals to homographic pencils of rays or rows of points with respect to circles (292) ; all systems of points determined by homographic pencils of rays on circles passing through their vertices, or systems of tangents determined by homo- graphic rows of points to circles touching their axes (305) ; aU systems of points on or tangents to common circles in per- spective with homographic systems of points on or tangents to the same circles (315) ; are thus homographic with each other. 324. The relation of homography between two or more systems of any species, whose constituents correspond in pairs or groups A, A', A", &c.; B, B', B'\ &c.; C, 6", C", &c.; B, B', Bl\ &c.; E, E, E\ &c.; F, F, F", &c.; &c., may be always symbolically represented, as observed in Art. 282, by the obvious extension of Dr. Salmon's very convenient notation for the equianharmonlcism of any groups of their corresponding quartets, viz. — {ABCBEF&c.} = [A'B'C'B'E'F &c.} = {A"B"C"B"E"F" &c.} = &c. The essential precaution, respecting uniformity of order among the corresponding constituents in the several groups, being of course invariably attended to in every case of its employment (see 279.) THEORY OF HOMOGRAPHIC DIVISION. 213 325. The following are some fundamental examples of cases of homographic division, grouped in reciprocal pairs, in all of which the relation of homographj between the generated systems appears from the nature of the law connecting the several pairs, or groups, of corresponding constituents, which is ^ven in each. £x. a. Two variable points on a fixed line or circle, either separated hy a constant interval, or hav'ng a fijced middle point, determine two homo- graphic systems of points on the line or circle. Ex. a'. Two variable tangents to a fixed point or circle, either inclined at a constant angle, or having a fixed middle tangent, determine two homo- graphic systems of tangents to tite point or circle. For, in each of the eight cases alike, the two generated systems are evidently similar, and therefore homogiaphic by the simplest criterion of the relation (321). Ex. h. A variable line, intersecting a fixed circle at any constant angle, determines two homographic systems of points on every concentric circle. Ex. &'. A variable point, subtending a fixed circle at any constant angle, determines two homographic systems of tangents to every concentric circle. Here again, in both cases alike, the tiro generated systems are evidently similar, and therefore, as in the preceding examples, homographic by the simplest criterion of the relation (321). Ex. c. A variable line, enveloping a fixed circle, determines homographic systems of points on all fixed tangents to the circle. Ex. c'. A variable point, describing a fixed circle, determines homographic tgstems of rays at all fixed points on the circle. For, all the pencils in the latter case being similar (25, 1°), and all the rows in the former case determining similar pencils at the centre of the circle (25, 2°)j therefore &c. (285). Ex. d. A variable line, turning round a fixed point, determines homo- graphic systems of points, on all fixed lines, and on all fixed circles passi7ig through the point. Ex. d". A variable point, moving on a fixed line, determines homographic systems of tangents, to all fixed points, and to all fixed circles touching the line. For, all the generated systems being, in each case, homographic with the determining pencil or row (285 and 306); therefore &c. (323). Ex. e. Two variable lines, turning round a fixed point, and either in- clined at a constant angle or having a fixed middle line, determine homo- graphic systems of points, on all fixed lines, and o. aU fixed circles passing through the point. Ex. «'. Two variaible points, moving on a fixed line, and either separated hy a constant intervul or having a fixed middle point, determine homographic systems of tangents, to all fixed points, and to all fixed circles touching the line. 214 THEORY OF HOMOGEAPHIC DIVISION. For, in both cases of each, as in the preceding examples, all the generated systems being homographic with their determining pencils or rows (286 and 306); and the latter, by examples o and o', being homo- graphic with each other ; therefore &c. (323). Ex. /. A variable line, turning round a fixed point, determines two homographic systems of points on any fixed circle. 'EiX.f. A variable point, moving on a fixed line, determines two homo- graphic systems of tangents to any fixed circle. "Eat, in both cases alike, every two quartets of corresponding constituents of the two generated systems are equianharmonic (315, a and a'); and therefore &c. (321). Ex. g. A variable line, turning round either centre of perspective of two fixed circles (207), determines four homographic systems of points on the two circles. Ex. jf'. A variable point, moving on either axis of perspective of two fixed circles (207), determines four homographic systems of tangents to the two circles. For, in both cases, each system for either circle being homologous or antihomologous (198 and 204), and therefore homographic (316), with one of the two for the other circle; and the two for the same circle being homographic, by the preceding examplesy and/' ; therefore &c. (323). Ex. h. Two variable points on a fixed line or circle, dividing harmoni- cally a fixed segment of the line or arc of the circle, determine two homographic systems of points on the line or circle. Ex. /<'. Two variable tangents to a fixed point or circle, dividing harmonically a fixed angle at the point or arc of the circle, determine two homographic systems of tangents to the point or circle. For, in the latter cases of both, the two systems, being in perspective (257), are consequently, as in examples / and /', homographic by (315, a and a') ; and they evidently involve the former (309) ; which however follow at once directly from (3°, Cor. 4°, Art. 282). Ex. i. Two variable points on a fixed line or circle, dividing eguian- harmonicaHy two fixed segments of the line or arcs of the circle, determine two homographic systems of points on the line or circle. Ex. i'. Two variable tangents to a fixed point or circle, dividing equi- anharmonically two fixed angles at the point or arcs of the circle, determine two homographic systems of tangents to the point or circle. For, in the latter cases of both again, the two systems, being in perspec- tive (313), are consequently, as in the preceding examples, homographic by (315, a and a'); and they also evidently involve the former (309); which however are reduced at once to those of the preceding examples by (283, Cor. 1°), from which it appears that any two sectors C and C, which cut AB' and BA' equianharmonically, are harmonic conjugates with respect to the two M and N which cut AA' and BB' harmonically ; and therefore &c. Ex.j. Two variable lines, dividing a fixed angle hamtonicaUy, or two fixed angles having a common vertex equumharmonieally, determine homo- THBORY OF HOMOGKAPHIC DIVISION. 215 graphic systems of points, on all fixed lines, and on all fixed circles passing through the vertex of the angle or angles. Ex.y. Two variable points, dividing a fixed segment harmomeaUy, or two fixed segments having a common axis equianharmonically, determine homographic systems of tangents, to all fixed points, and to all fixed circles touching the axis of the segment or segments. For, in both cases of each, as in examples e and e', all the generated systems are homographic with their determining pencils or rows (285 and 306) ; and the latter, by examples h and h', or i and >', are homographic with each other ; and therefore &c. (323). Ex. k. When, of a variable polygon of any order inscribed to a fixed circle, all the sides but one turn round fixed points, or envelope fixed circles concentric with the original; the several vertices determine so many homo- graphic systems of points on the circle. Ex. h'. When, of a varicAle polygon of any order exscribed to a fixed circle, all the vertices but one move on fixed lines, or describe fixed circles concentric unth the original; the several sides determine so many homo- graphic systems of tangents to the circle. These follow imm' \ ^^ \s.mB'G''BinB'I)T therefore, as in the preceding article, {ABCD] = {A'B'C'I)'}', and therefore &c. Conversely, For any two homographic systems, both collinear or concurrent, or one collinear and one concurrent ; if A and A', B and B' be any two pairs of corresponding constituents, then, for every other pair C and C', the ratio (AC smAG \ ( A'C sin^'C' N \BG "^ &mBC)''\B'G' °^ amB'G') is constant, both in magnitude and sign. For, since for every other two pairs G and C", I> and D', by the homography of the systems, {ABGD}={A'B'G'iy}, and since, consequently, (AG AD\ f sin AC sinAD ^ _ (A'C AU\ \BC '■ Bd) """ K^^BC • ^^Bd) ~ \B'C ' B'jyj f smA'C sin^'i?' \ *''' [smB'C'smB'J)')' (AD saiAD\ (A'D' saiA'D'\ 222 THEORY OF HOMOGRAPHIC DIVISION. therefore at once, by alternation, 7\ ( A'C smAC' \ IJ'KB'C ^^ sin B'C) ) j • \B'D' ^'^ smB'D') ' and therefore &c. It follows evidently, from this latter property, that the criterion of homography furnished by the above is, as regards coUinear and concurrent systems, as general as that of the pre- ceding article; and applies, equally with it, to every case, without exception, of the generation of two homographic systems of either species by the simultaneous variation of a pair of con- nected constituents C and C Cor. From the first part of the above general property, it follows, immediately, that — When two variable lines or points, or a variable line and point, I and T, are connected in every position with two fixed pairs of points or lines, or with a fi^xedpair of points and a fixed pair of lines, A and B, A' and B', by the constant relation fAl A'r\ . . - ,'AI BI\ \BI ' WI'j *"" '^ ^g^^^'^'^nt [ ^Tj, : -gT^, 1 in any constant ratio, positive or negative ; they divide the two segments or angles, or the segment and angle, AB and A'B', homograpTiically ; and the two pairs of corresponding constituents in the two ratios, A and A', B and B', are two pairs of corre- sponding constituents in the two divisions. For, if C and C be the two points of intersection or lines of connection, or the point of intersection and line of connection, of /and /' with AB a.ni A'B' ; then, since, according to the case, evidently, /AC Bin AC \ _AI (A'C AnA'G' \ _ AT \BC *"' sin^C; ~ BI ^"'^ \WC' """ siaB'c) ~ BT ' therefore &c. These properties are useful in the modem theories of homographic and of correlative transformation, as will appear in the sequel in the chapters in which they are respectively discussed. 329. Two variable sectors, C and C, dividing a fixed segment or angle, AB, in any constant anharmonic ratio, positive or THEORY OF HOMOGRAPHIC DIVISION. 223 negative, determine two homograpMc systems of points or rays ; of which the two extremities, A and B, of the fixed segment or angle constitute each a pair of corresponding constituents coin- ciding with each other. For, as in the more general property of the preceding article (under the first part of which, as observed in Art. 282, Cor. 3°, this manifestly comes as a particular case), if any one position G and C of the variable pair be regarded as fixed; then since for every other position D and Z>', by hypothesis, {ABGC] = {ABDD'], therefore, by (272), {ABCD] = {ABG'D% and therefore, by (327), D and D' determine two homographic systems, of which A (= A') and B (= B') constitute each a pair of corresponding constituents coinciding with each other; as it is evident h priori they ought, two variable magnitudes of any kind having a constant ratio to each other (268), whatever be its magnitude or sign, provided only it be finite, necessarily vanishing, becoming infinite, and changing sign together. Conversely, For any two homographic rows of points or pencils of rays having a common axis or vertex, if A=A' and B = B' he two pairs of corresponding constituents which coincide with each other ; then, for every other pair G and G' of their corresponding constittients, the anharmonic ratio of section of the intercepted segment or angle AB is constant both in magnitude and sign. For, since, for every other two pairs G and G', D and -0',by the homography of the systems and the hypothesis that A' and B' coincide with A and B respectively, {ABCD] = [ABG'B'} ; there- fore, at once, by (272), [ABCG'} = [ABBD'} ; and therefore &c. It will be shewn in the next chapter that, for every two homographic rows of points or pencils of rays having a common axis or vertex, there exist always two pairs of corresponding points or rays, real or imaginary, which thus coincide with each other, and which have been termed in consequence, by Chasles, the double points or rays of the systems. Their properties and uses are among the most interesting and important in the whole theory of homographic division, and will form the entire subject of the chapter. 330. Two variable points B and B' on two fixed lines, the ratio of whose distances from, two fixed points A and A' on the 224 THEORY OP HOMOGRAPHIC DIVISION. Unes is constant both in magnitude and sign, determine two homO' graphic systems, of which A and A', and the two points at infinity on the lines, are two pairs of corresponding constituents. For, the systems being similar are therefore homographicj and, whatever be the magnitude and sign of the ratio, provided only it be finite, the two variable distances AB and A'B' vanish and become infinite together ; and therefore &c. Conversely, when two homographic rows of points are such that the two points at infinity oti their axes, co and co ', are corre- sponding constituents of the systems, they are similar. For, if A and A', B and B', C and C be any other three pairs of their corresponding constituents, then since, by hypo- thesis, {ABCao ] = {A'B'C'ao '}, therefore (275) BC-.CA: AB= B'C : G'A' : A'B' ; and therefore &c. The criterion of similitude between two homographic rows of points on any axes, supplied by the second part of the above, viz. the correspondence of the point at infinity of one to the point at infinity of the other, is in all cases very readily applicable ; depending, as there shewn, on the circumstance of similitude only, and being independent, as shewn in the first part, of the magnitude and sign of the ratio of similitude, provided only it be finite. COR. In the particular case when the two points A and A' coincide at the intersection / of the lines; the species of the variable triangle BIB' being then evidently constant, it follows consequently from the first part of the above, (as is otherwise evident) that a variable line, determining with two fiaxd lines a triangle of constant species, divides the lines homographically ; the common point and the point at infinity corresponding in each division to the common point and the point at infinity in the other. 331. Two variable points C and C on two fixed lines^ the rectangle under whose distances form two fiaxd points A and B' on the lines is constant both in magnitude and sign, determine two homographic systems ; of which A and B' correspond to the points at infinity on the lines. For, if any one position C and C of the variable pair be TfiEOET OP HOMOGRAPHIC DIVISION. 225 regarded as fixed, then since, for every other posftion D and D', by hypothesis, AC.B'C = AD.B'D\ and since consequently CA : DA = D'B' : C'B', therefore, by (275), {CDAI25, Ex. d), in order that the two homo- graphic pencils (323) they determine at Q and P respectively should be in perspective, a pair of their corresponding positions should be collinear with those points, which could be the case only when the latter are collinear cither with ij or with the intersection of L and M. And, in the latter case, the two pencils determined by E and Fat P and Q being in all cases homographic (325, Ex. d'), in order that the two homographic rows (323) they determine on M and X respectively should be in perspective, a pair of their corresponding positions should be concurrent with those lines; 'which could be the case only when the latter are concurrent either with If or with the connector of P and Q. See the pairs of reciprocal properties {296, 3°) and (294, 5°) where, on other principles, the perspectives were shewn to exist in those cases respectively. Ex. b. Two vertices of a variable triangle A and B move on two fixed lines X and M, the two opposite sides E and F turn round two fixed points P and Q, and the third side G envelopes a fixed circle touching L and M ; required the condition that the third vertex C should move on a third fij^ed line Iff. Ex. v. Two sides of a variable triangle E and F turn round two fixed points P and Q, the two opposite vertices A and B move on two fixed lines L and M, and the third vertex C describes a fixed circle passing through P and Q : required the condition that the third side G should turn round a third fixed point S. In the former case, as in Ex. a, the two rows determined by A and JB on X and Jf being in all cases homographic (325, Ex. e), in order that the two homographic pencils (323) they determine at Q and P respectively should be in perspective, a pair of their corresponding positions should be collinear with those points, which could be the ease only when the latter connect by a tangent to the circle. And, in the latter case, as in Ex. a' the two pencils determined by E and FatP and Q being in all cases homo- graphic (325, Ex. &), in order that the two homographic rows (323) they determine on M and X respectively should be in perspective, a pair of their corresponding positions should be concurrent with those lines ; which could be the case only when the latter intersect at a point on the circle. That THft»RT OF HOMOGRAPHIC DIVISION. 233 the perspectives exist in those cases, the reader can find no difficulty in shewing independently. Ex. c. If a variable point P move on a fixed line I, required the con- dition that its polar L, with respect to the three sides of a fired triangle, for any triad of constant multiples, should turn round a fined point. Ex. c. If a variable line L turn round a fixed point O, required tJte condition that its pole P, with Tespeet to the three vertices of a fixed triangle, for any triad of constant multiples, should move on a fixed line. In the former case, if A, B, C be the three vertices of the triangle, and .3l, y, Z the three intersections of X with its opposite sides .B, F, G respectively; then, the anharmonic ratios of the three pencils A .BCPX, B . CAPY, C.ABPZ being in all cases constant (2oO, a), and the three rows determined by X, T, Z on E, F, G respectively being, consequently, in all cases homographic with that determined by P on 7 (329), and therefore with each other (323); in order that they should be in perspective, a pair of coiresponding positions should coincide, of J' and Zat A, or of Zand X At B, or of X and yat C; consequently, of jBl^and CZ with BA and CA, or of CZ and AX with CB and AB, or of ^X and BYviXh -^Cand BC; and consequently (329) of BP and CP with BA and CA, or of CP and AP with CB and AB, or oi AP and BP with ^ Cand BC; .which could be the case only when a position of P coincides with, and when consequently I passes through, one of the three points A, B, C. And, in the latter case, if E, F, G be the three sides of the triangle, and U, V, W^'the three connectors of P with its opposite vertices A, B, Crespec- tively ; then, the anharmonic ratios of the three rows E.FGLU, F.GELV, G. EFL W being in all eases constant (250, a"), and the three pencils de- termined by U, V, W at A, B, C respectively being, consequently, in all cases homographic with that determined by Z at O (329), and therefore with each other (323) ; in order that they should be in perspective, a pair of corresponding positions should coincide, of V and W along E, or of JFand ?7 along P", or of {/and F along G; consequently, of PF and GW with PjBand GE, or of G^andPf with GPand PP.or oiEUaaAFV .with EG and FG ; and consequently (329), of FL and GL with FE and GE, or of GL and EL with GF and EF, or of EL and FL with EG and FG ; which could be the case only when a position of L coincides with, and when consequently O lies on, one of the three lines E, F, G. That, in the former case, when I passes through any vertex A of the triangle, then L turns round a fixed point X on the opposite side BC, and that, in the latter case, when O lies on any side E of the triangle, then P moves on a fixed line U through the opposite vertex FG, is evident from the constancy of the anharmonic ratio, of the pencil A. BCPX in the former case (250, o), and of the row E. FGLU iu the latter case (2o(>, o'). Ex. d.Ifa variable point P describe a fixed circle, its polar L, with respect to the three rides of any fixed inscribed triangle, for any triad of constant multiples, turns round afixedpoint O : the pole, viz., with respect to the three vertices of the triangle, of its axis of perspective I with the cor- responding exscribed triangle, for the reciprocal triad of multiples. 234 THEOEY OF HOMOGEAPHIC DIVISION. Ex. d. If a variable line L envelope a fixed circle, its pole P, vAth respect to the three vertices of any fixed exscribed triangle, for any triad of constant multiples, moves on a fixed line I; the polar, viz., with respect to the three sides of the triangle, of its centre of perspective O with the corre- sponding inscribed triangle, for the reciprocal triad of multiples. For, in the former case (employing the reasoning and notation of Ex. c) the three rows determined hj' X, T, Z on E, F, G respectively, are homo- graphic with the system determined by P on the circle, and therefore with each other j and they are always in perspective, because that, as P in the course of its revolution passes successively through every point on the circle, r and Z coincide with it and with each other as it passes through A, Z and X with it and with each other as it passes through B, X and I' with it and with each other as it passes through C; and therefore &c. And, in the latter case (employing the reasoning and notation of Ex. d) the three pencils determined by TJ, V,Wa.t A, B, C respectively, are homographic with the system determined by L to the circle, and therefore with each other; and they are always in perspective, because that, as X in the course of its revolution passes successively through every tangent to the circle, V and W coincide with it and with each other as it passes through M, ^and 17 with it and with each other as it passes through JP, UaxiA. V with it and with each other as it passes through 6; and therefore &c. That the common centre of perspective O in the former case, and the common axis of perspective I in the latter case, are the particular point and line stated in the above enunciations respectively; is evident from the general relations of Art. 251, by taking the three particular positions of L corresponding to the three passages of P through A, B, and C in the former case, and the three of P corresponding to the three of X through E, F, and G in the latter case. And from the properties themselves, thus or in any other manner obtained, inferences exactly analogous to those of Cor. 14°, Art. 261, for the particular cases there established on other prin- ciples, may of course be drawn in precisely the same manner. 337. From the same criteria of perspective between homo- graphic rows and pencils, combined with the reciprocal pro- perties of Arts. 293 and 317 respecting arbitrary pairs of corresponding triads, the four following general properties of homographic systems, in pairs reciprocals of each other, may be readily inferred ; viz. — a. For any two homographic rows of points on different axes, or systems of points on the same circle ; all pairs of cor- respondinff connectors of pairs of non-corresponding constituents (such as AB' and A'JB, AC and A'C, BC and B'C, &c.) intersect on the same fixed line ; termed the directive axis of the systems. TH^ET OP HOMOQEAPHIC DITISION. 235 a. For any two Tiomographic pencils oj rays through different vertices^ or systems of tangents to the same circle ; all pairs of cor- responding intersections of pairs of non-corresponding constituents {such asAB' and A'B, AC and A'C, BC and B'C, &c.) connect through the same fixed point 0; termed the directive centre of the systems. For, firstly, since, for each separate pair of corresponding constituents A and A', the two homographic pencils (or rows) A.A'B'C'D'E'F' &c. and A'.ABCDEF &c., they determine with the opposite systems, being, by the criteria of Art. 334, in perspective; therefore, for each separate pair A and A', the several points of intersection (or lines of connection) of AB' and A'B, AC and A'G, AD' and A'D^ &c., are coUinear (or con- current). And, secondly, since for every three pairs A and A'^ B and 5', G and C, the three points of intersection (or lines of connexion) oiBC and B' C, of CA' and CA, and ofAB' and A'B, being, by the general properties of Arts. 293 and 317, collinear (or concurrent) ; therefore, for every three A and A', B and B', C and C, the three axes (or centres) of perspective of the three pairs of homographic pencils (or rows) A.A'B'CD'E'F &c. and A'.ABCDEF &c.,B. A'B' CB'F'F" &c. and B'ABCDEF&c, C.A'B'CD'E'F' &c. and C.ABCDEF&c, coincide. And, as their coincidence for any two arbitrary pairs involves evidently their coincidence for all pairs, therefore &c. Oiven, in any of the four cases, thre^ pairs of corresponding constituents A and -4', B and B', C and C of the ttco systems, to determine the line {or point) 0. The three given pairs of corresponding constituents A and A', B and B', C and C give at once the three pairs of corresponding connectors (or intersec- tions) of pairs of non-corresponding constituents BC and B'C, CA' and CA, AB' and A'B; any two of whose three collinear points of intersection (or concurrent lines of connexion) deter- mine, by the above, the required line (or point) 0. By aid of the line (or point) 0, thus determined from three pairs of corresponding constituents A and A', 5 and B', C and C of the systems, any number of other pairs D and D', E and E', F and F may be obtained at pleasure. For, in the former case, two variable lines turning round A and A, or B and B', or C7and C, and intersecting in every position on 0, determine 236 THEORY OF HOMOGKAPHIC DIVISION. successively all other pairs D and D\ E and E\ F and F, &c. ; and, in the latter case, two variable points, moving on A and A, or 5 and 5', or G and C", and connecting in every position through 0, determine successively all other pairs D and D\ E and E\ J?" and F, &c. ; (see Arts. 291, Note, and 310, Cor.), Hence the name directive as applied, in either case, to ; the line or point so designated being the axis or centre that directs the moment of the two variable lines or points which, giving in every position a pair of corresponding constituents, thus by their variation generate the systems. If if and iV be the two points (or lines) of the systems which lie on (or pass through) 0, and / the point (or ray) common to the axes (or vertices) in the case of the two rows (or pencils) ; then since, in their case, by relations (2) Art. 293, {ABCM}^{A'B'C'I} and [AB'C'N] = {ABCI}', and since, in the case of the two concyclic systems of points (or tangents), by relations (3) Art. 317, {ABCM} = {A'B'C'M} and {ABC]S[} = {A'B'C'N]; the two points or lines M and N are, therefore, in the former case, the two constituents of the two systems corresponding to the point (or ray) / common to their axes (or vertices) ; and, in the latter case, the two double points (or lines), as they are termed, of the systems, that is, the two points on (or tangents to) their common circle at each of which a pair of their correspond- ing constituents coincide (see Art. 329). When, in any of the four cases, the two systems are in per- spective; their directive axis (or centre) is then evidently (240 and 261) the polar of the centre (or the pole of the axis) of perspective, with respect to the angle (or segment) determined by the two axes (or vertices) in the case of the two rows (or pencils), or with respect to the common circle in the case of the two concyclic systems of points (or tangents). In the former case, therefore, the two points (or rays) M and N coindde, as they ought (334), with the point (or ray) / common to the two axes (or vertices). 338. Again, from the same criteria of perspective, combined with the particular case of homographic division considered in THEORY OF HOMOGRAPHIC DIVISION. 237 Art. 329, and with the reciprocal properties of Arts. 240 and 261 respecting poles and polars to angles or segments and to circles, the four following general properties of homographic systems, in pairs reciprocals of each other, may as readily he inferred; viz. — a. A variable Une^ determmvng two TiomograpMc rows of points on different axes, or systems of points on a common circle — 1°. Intersects with any four positions of itself at a quartet of points equianhairmonic with the two corresponding quartets of the generated systems. 2°. Determines on all positions of itself rows of points homo- graphic with each other and with the original systems. a. A variable point, determining two hom.ographic pencils of rays at different vertices, or systems of tangents to a common circle — 1°. Connects with any four positions of itself by a quartet of rays equianharmonic with the two corresponding quartets of the generated systems. 2°. Determines at all positions of itself pencils of rays homo- graphic with each other and with the original systems. For, in the former case, if A and A', B and B', G and C", D and D be the four pairs of corresponding constituents deter- mined on the axes (fig. a) or circle (fig. /S) by any four positions AA\ BB', CC, DD' of the variable line ; E and E' the fifth pair corresponding to any fifth positions EE' ; A", B", C", D" the four intersections of the four positions with the fifth ; /the intersection of the axes (fig. a) or of the tangents at E and E' to the circle (fig. /8) ; and P, Q, B, 8 the four intersections of the four pairs of connectors EA' and E'A, EB' and E'B, 238 THEOET OF HOMOGRAPHIC DIVISION. EC and E'C, EU and E'D, which, by the preceding article, are coUInear, and lie on the directive axis of the systems; then, since, by Arts. 240 and 261, the four lines IP, IQ, IB, IS are the four polars of the four points A", B", C", D" with respect to the axes (fig. a) or circle (fig. )S) ; and since, con- sequently, in either case, the four pairs of lines lA' and IP, IB" and IQ, 10" and IE, ID" and 18 are pairs of harmonic conjugates with respect to the two IE and IE' ; therefore, in either case, by Art. 329, {I.A"B"C"D"} = [LPQB8]; and therefore, in either, by Art. 285, {A"B"C"B"} = {PQBS} = {ABCD} = [A'B'C'iy] ; which being true for the intersections of every four with all fifth positions of the variable line, therefore &c. And, In the latter case, if A and A', B and B', C and C, D and B' be the four pairs of corresponding constituents determined at the vertices (fig. a') or to the circle (fig. ^') by any four positions AA', BB', CC, DBf of the variable point; ^and E' the fifth pair corresponding to any fifth position EE' ; A\ B", C", 2>" the four connectors (not drawn in the figures) of the four positions with the fifth ; /the connector of the vertices (fig. a) or of the points of contact of E and E' with the circle (fig. /S') ; and P, Q, R, S the four connectors of the four pairs of intersections EA' and E'A, EB' and E'B, EC and E'C, ED' and E'D, which, by the preceding article, are concnrrent, and pass through theOky of homographic division. 239 the directive centre of the systems ; then, since, by Arts. 240 and 261, the four points IP^ IQ, IB, IS are the four poles of the four lines A'\ B", C", B" with respect to the vertices (fig. a') or circle (fig. /S') ; and since, consequently, in either case, the four pairs of points lA" and IP, IB" and IQ, IC" and IB, IB" and IS are pairs of harmonic conjugates with respect to the two IE and IE' ; therefore, in either case, by Art. 329, {I.A"B"C"B"} = {I.PQBS} ; and therefore, in either, by Art. 285, {A"B"C"B"} = {PQBS} = [ABOB] = {A'B'C'B} ; which being true for the connectors of every four with all fifth positions of the variable point, therefore &c. The above reciprocal demonstrations (which it may be ob- served would be simplified for the concyclic systems in both cases by the general property of Art. 292) may be briefly summed up in one as follows. Since, in all four cases alike, by Arts. 240 and 261, 74" and /P, IB" smd IQ, IC" smd IB, ID" and 75 are pairs of harmonic conjugates with respect to IE and IE' ; therefore, in all four alike, by Art. 329, {I.A"B"C"B"] = {I.PQBS} ; and therefore, in all alike, by Art. 285, [A"B"C"iy'} = {PQBS} = {ABCB} = {A'B'C'B'} ; which, for all alike, establishes at once the two properties in question. Cor. 1°. It follows of course immediately, from the first parts of the above reciprocal properties, that when a variable line, which does not turn round a fixed point, determines homo- graphic rows on any two fixed lines, it coincides, once in the course of its entire variation, with each of the lines; and, reciprocally, that when a variable point, which does not move on a fixed line, determines homographic pencils at any two fixed points, it coin- cides, once in the course of its entire variation, with each of the points. Which are also evident, a priori, from the consideration that, when the variable line (or point) passes, in the course of its variation, through the intersection of the fixed lines (or over the connector of the fixed points), if the corresponding constituents of the two homographic systems do not then coincide, as they do 240 THEORY OP HOMOGKAPHIC DIVISION. not when the systems are not in perspective (334), it must itself necessarily coincide with one or other fixed line (or point). Cor. 2°. It follows also immediately, from the same, that a variable line^ determining homographic rows of points on any two fixed lines, intersects with every four fixed positions cf itself at a variable quartet of points having a constant anharmonic ratio, and determines with every five positions of itself an equianharmonic hexa- gram (301)/ and, reciprocally, that a variable point, determining homographic pencils of rays at any two fixed points, connects with every four fixed positions of itself by a variable quartet of rays having a constant anharmonic ratio, and determines with every five positions of itself an equianharmonic hexastigm (301). These are the general properties of homographic rows and pencils, whose converses were given in examples e and e', g and g' of Art. 333. 339. As every two homographic pencils of rays through any vertices determine on every axis two homographic rows of points whose constituents at infinity correspond to those of the determining pencils to whose directions the axis is parallel (16) ; it follows, from the criterion of similitude between homographic rows given in Art. 330, that — When two homographic pencils of rays through any vertices have a pair of corresponding constituents whose directions are parallel ; they determine on every axis parallel to those directions two similar rows ; whose ratio of similitude, evidently constant both in magnitude and sign when the two constituents coincide, varies when they do not with every position of the axis ; changes sign, passing through 0, as it passes in either direction through the vertex of the pencil of antecedents ; again changes sign, passing through IX , as it passes in either direction through the vertex cf the pencil of consequents ; and passes without change of sign through every intermediate absolute magnitude, in continuous increase from to 00 during its passage in either direction frvm the former to the latter, and in continuous decrease from, oo to during its passage in either direction from the latter to the former. Given the parallel pair of corresponding constituents, / and F, and any other two pairs, A and A', B and B', of the two pencils, the particular axis L parallel to / and /' for which the ratio of similitude shall have any given value, positive or negative, THflDRT OF HOMOGEAPHIC DIVISION. 241 may be readily determined aa follows : Denoting by and 0' the vertices of the two pencils, by X and X', Y and Y' the four intersections of L with A and A', B and B' respectively, and by Z its intersection with the line 0' ; then, the ratios of OZ and 0' Z io XY &nA. X' Y' respectively being evidently given with the direction of 2/, when the ratio of XY to X'Y\ which is that of the similitude of the systems, is also given, that of OZ : O'Z, and with it of course the position of Z, is consequently given ; and therefore &c. The particular cases when the given ratio has, as of course it may have, the particular values + 1 (and when consequently the several segments intercepted on L by the several pairs of corresponding rays A and A\ B and B', C and C", Z> and D', &c. of the two pencils, have a common magnitude or middle point) differ in no respect from the general case when it has any value positive or negative. Since, for every two homographic pencils of rays through any vertices, there exist, as will be shewn in the next chapter, two pairs of corresponding constituents, real or imaginary, whose directions are parallel ; it follows consequently, from the above, that — For every two TiomograpMc pencils of rays through any vertices, there exist two directions, real or imaginary, on all lines parallel to either of which they determine similar rows of points ; and, of the two systems of parallels determined hy those directions, two particular lines on each of which their several pairs of corresponding rays intercept equal segments, and two others on each of which the several intercepted segments have a common middle point. In the particular case when the two pencils are in perspec- tive, it is evident, from Art. 334, or independently, that the two directions in question are parallel, one to the connector of their vertices and the other to their axis of perspective ; on the former of which lines the several pairs of corresponding rays intercept evidently a common segment, and on the latter of which the several segments they intercept are of course all evanescent. 340. As, from the general property of Art. 330, it was shewn, at the close of the preceding article, that — a. For every two homographic pencils of rays through different VOL. II. E 242 THEORY OP HOMOGRAPHIC DIVISION. vertices^ there exist two lines, real or imaginary, on each of which the several pairs of corresponding rays intercept equal segments ; and also two others, real or imaginary, for each of which the several intercepted segments have a common middle point. So, from the general property of Art. 331, it may be shewn that, reciprocally — a'. For every two homographic rows of points on different axes, there exist two points, always real, at each of which the several pairs of corresponding points subtend equal angles ; and also two others, sometimes imxiginary, for each of which the several sub- tended angles have a common middle ray. For, if/ (figs, a and /3) be the intersection of the axes ; Pand Q the points of their rows whose correspondents are at infinity ; A and A' a variable pair of corresponding points of the systems ; and E and i^the two fixed points, real or imaginary, for which the two rectangles PE. QE and PF. QF are equal to the constant rectangle PA.QA' (331), and for which the two pairs of angles IPE and IQE, IPF and IQF, measured in opposite directions of rotation (fig. a] and in the same direction of rotation (fig. /3), are equal; then, since, from the evident similarity in either case of the two pairs of triangles APE and EQA', APF and FQA' in every position of A and A, the two pairs of angles PAE and QEA', PAF and QFA' (or the two pairs PEA and QA'E, PFA and QA'F) are always equal; therefore, as A and A' vary, the two pairs of lines EA and EA', FA and FA revolve always through equal angles, in the same direction of rotation (fig. a), and in opposite directions of rotation TdteoRY OP HOMOGEAPHIC DIVISION. 243 (fig- /3)> round the two fixed points E and F; and there- fore &c.* Of the two (evidently similar and equal) triangles PEQ and PFQ^ whose two vertices E and F are the two points involved in the above properties, and which combined form evidently a paral- lelogram in the former case (fig. a) ; the common base PQ, the rectangle under the sides, and the diflference (fig. a) or sum (fig. /3) of the base angles, being known, the triangles are conse- quently completely determined ; and, while evidently always real in the former case, are imaginary in the latter when the rect- angle PE. QE or PF. QF is greater than for any point on the known circle PIQ^ which in that case evidently passes always through E and F. In the particular case when the two rows are in perspective, like the analogous case in the preceding article, it is evident, from Art. 334, or independently, that, of the two points E and F in the former property, one coincides with the intersection of the axes and the other with the centre of perspec- tive ; the constant angle for the foi-mer being evidently the fixed angle between the axes, and for the latter being of course evanescent. COE. 1°. The three pairs of lines EA and FA, EA' and FA\ EI a,ni Fl, in (fig. a), being evidently equally inclined to the bisectors of the three corresponding angles of the triangle AIA' ; and the three rectangles under the three pairs of perpen- diculars from E and F upon the three sides of that triangle being consequently equal in magnitude and sign ; hence, from the above, it appears that — a. When a variable line intersects with two fixed lines homo- graphically, the rectangle under its distances from the tico fixed points, at which the several pairs of corresponding intersections subtend constant angles, is constant both in magnitude and sign. And from the analogous property of the preceding article, respecting homographic pencils of rays, it may be shewn that reciprocally — a'. When a variable point cotinects with two fixed points homographically, the rectangle under its distances from the two • The above demonstration was communicated to the author by Mr. Casey. e2 244 THEORY OF HOMOGRAPHIC DIVISION. fixed lines, on which the several pairs of corresponding connectors intercept constant segments, is constant both in magnitude and sign. For, if and 0' be the two fixed points ; P any position of the variable point ; OE and 0'E\ OF and O'F' the two pairs of corresponding rays, of the two horaographic pencils, whose directions are parallel ; L and M the two parallels to them, intersect- ing at /, on each of which the several pairs of corre- sponding rays intercept con- stant segments, andon which consequently PO and PO' intercept segments AA' and BB' which are equal to EE' and FF' respectively; then, PX and PY being parallels through P to L and If, since, by pairs of similar triangles, AX : PX =OF:BF and AX : PX = OF : B'F, and since AX- A X= OF- O'F, and BF=B'F, therefore AX= OF and AX=^ O'F; from which, since AP= OB and AP= OB', therefore PX.PY= OE. 0F= O'E'. O'F, or, PL.PM= OL. 0M== O'L. O'M; and therefore &c. CoR. 2°. Since, by (338, a and a, 2°), a variable line (or point), intersecting (or connecting) homographically with two fixed lines (or points), intersects (or connects) homographically with all positions of itself; it follows, consequently, from the two reciprocal properties a and a' of the preceding. Cor. 1°, that — a. A variable line, the rectangle under whose distances from two fixed points is constant in magnitude and sign, intersects homographically with all positions of itself. a'. A variable point, the rectangle under whose distances from two fixed lines is constant in magnitude and sign, connects homo- graphically with all positions of itself. THEORY OF HOMOGRAPHIC DIVISION. 245 COE. 3°. If X and Y be the feet of the perpendiculars from E and F on AA' in (fig. a), Z the intersection with AA' of the position BB' which intersects it at right angles, and the middle point of EF; then since (49) OX^ = 0Y' = EX. Fr+ HEF)', and since (Euc. I. 48, and li. 5, 6), OZ' = 2EX.FY+ iiEF)' ; it follows again, consequently, from a Cor. 1°, that — When a variable line intersects hornographically with two fixed lines — a. In every position^ the feet of the two perpendiculars on it, from the two fixed points for which their rectangle is constant^ lie on a fixed circle ; whose centre bisects the interval between the points^ and the square of whose radius = the constant rectangle + the square of the semi-interval. b. Every two of its positions which intersect at right angles intersect on another fixed circle ; concentric with the former, the square of whose radius = twice the same constant rectangle + the square of the same semi-interval. N.B. The several properties of homographic divisions con- tained in this and the two preceding articles are of very familiar occurrence in the Theory of Conic Sections, where alone indeed the subject has scope for the full and adequate developemect due to its importance. ( 246 ) CHAPTER XX. ON THE DOUBLE POINTS AND LINES OF HOMOGRAPHIC SYS1"EMS. 341. When the axes of two homographic rows of points or the vertices of two homographic pencils of rays coincide, there exist always two pairs of corresponding constituents, real or imaginary, whose positions coincide, and which accordingly have been termed by Chasles the double points or rays of the si/stems (329). Assuming for the present the existence of such points or rays, the following properties are evident from their mere definition. 1°, No more than two double points or rays could exist unless the systems altogether coincided. For, since, for every four pairs of corresponding constituents A and A', B and B\ C and C, D and i>', the systems being homographic, [ABCD] = [A'B'C'D'}^ li A = A', and B=B', and C= C, then necessarily D = D' \ and, as for the same reason E=E\ F=F', G= G', H=H\ &c., therefore &c. 2°. Every three pairs of corresponding constituents A and A\ B and B', G and G' are connected with the two double points or rays M{=M') and N{=N') taken separately hy the relations [ABGM] ={A'B'G'M] and [ABGN] = [A'B'G'N] ; and every two pairs A and A, B and B' with both combined by the relation {MNAB} = {MNA'B'}. These relations, which are evident from the horaography of the systems and the hypothesis respecting ilf and N, are characteristic of the double points or rays, and sufficient in all cases to identify and distinguish them, 3°. Each double point or ray being of course equivalent to a pair of corresponding constitaents, and three pairs of cor- ON THE D(TbBLE POINTS OF HOMOGEAPHIC SYSTEMS. 247 responding constituents being sufficient to determine any two homographic systems (327) ; one double point or ray with two pairs of corresponding constituents, or both double points or rays with a single pair of corresponding constituents, are there- fore sufficient to determine two homographic rows or pencils whose axes or vertices coincide. 4°. For the same reason, three pairs of corresponding con- stituents, given, taken, or known in any manner, being sufficient in all cases to determine every thing connected with the two homographic systems to which they belong (327), are therefore sufficient to determine the two double points or rays of two homographic rows or pencils whose axes or vertices coincide. (See Art. 348). 5°. As two homographic rows of points on any axis de- termine two homographic pencils of rays at any vertex, and conversely, the two double points of one correspond always to the two double rays of the other, and conversely. 342. Every two corresponding constituents of two homographic rows or pencils, whose axes or vertices coincide, divide in the same constant anharmonic ratio the segment or angle determined hy the two double points or rays of the systems. For, since, for every two pairs of corresponding constituents A and A\ B and B\ by 2° of the preceding article, [MNAB] = [MNA'B'], therefore, by (272), [MNAA] = [MNBB'] ; and, since, for the same reason, {MNBB'] = [MNCC], {MNCC} = [MNDB"}, &c. ; therefore &c. The particular case when the constant anharmonic ratio of section = — 1 , that is, when the several pairs of corresponding constituents A and A\ B and B', G and C", D and Z)', &c. divide harmonically the segment or angle MN, will be con- sidered in connexion with the Theory of Involution in the next Chapter. 343. Every two constituents of the two systems, corresponding to the same point on the common axis or ray throttgh the common 248 ON THE DOUBLE POINTS AND LINES vertexj divide in the sqtmre of the above constant anharmonic ratio the segment or angle determined hy the two double points or rays of the systems. For, every point on the common axis or ray through the common vertex belonging of course indifferently to both systems, if P be the correspondent of any point or ray / regarded as belonging to one system, and Q the correspondent of the same point or ray /regarded as belonging to the other system, then since, by the preceding article, (MP MI\ /sin MP sm MI\ ^ ,,, [-Np'm) '' liki^ = sb^j = ^°^^* ^'^' and (MI MQ\ fmxiMI smMQ\ , ,„, KwrNQ) "■•lib^ = sb^j="=°°^* ^'^' therefore at once, by composition of ratios, (MP MQ\ (smMP smMQ\ ^ ,,, [np'-nqj °' [^^NP'-^^mr ''"'"' ^'^' and therefore &c. (268). The particular case where this constant anharmonic ratio of section = 1, that is, when the two constituents of the two systems corresponding to the same point on their common axis or ray through their common vertex always coincide, will also be con- sidered in reference to the Theory of Involution in the next Chapter. CoK. 1°. When, in the above, the point or line / is either point or line of bisection, external or internal, of the segment or angle MN, then — Uie segment or angle PQ has the same points or lines of bisection as the segment or angle MN. For, since, in that case, [MI: NT) or (sinMI : ainNI) =±1, therefore, in the same case, by (1) and (2) above, [MP: NP).{MQ : NQ) or (sinJI/P: BmNP).{e,inMQ: smNQ)=+] • and therefore &c. Cor. 2°, The point of external bisection of every segment of a line being the point at infinity on the line, it follows imme- diately, from the preceding Cor, 1°, that — In the case of two homographic rows of points on a common OF HOMOGKAPHIC SYSTEMS. 249 axtfi^ the segment MN intercepted between the two double points is concentric with the segment PQ intercepted between the two con- stituents of the systems corresponding to the jpoint at infinity on the axis. A consequence which, by virtue of the general property of Art. 331, may also be proved otherwise as follows: Since, for every two pairs of corresponding constituents A and A\ B and^' of the systems, by the property in question, PA.QA =PB.QB\ when A = A' = M, and B=B' = N, then PM.QM=PN.QN; and therefore &c. (Euc. ii. 5, 6). Cor. 3°. If, in the above, the point or line / be conceived to vary, causing of course the simultaneous variation of the two Pand Q^ then — The two points or rays P and Q determine two homographic rows or pencils having the same double points or rays M and N with the original systems. This follows immediately from relation (3) by virtue of the general property of Art. 329 ; and the same is evident also from the consideration that the systems determined by P and Q are both homographic with that determined by 7, with which, com- bined separately, they constitute in fact the original systems. N.B. From the general, or any derived, property of either this or the preceding article, it is evident that every two homo- graphic rows of points or pencils of rays, whose axes or vertices coincide, are symmetrically disposed on opposite sides of each point or line of bisection, external and internal, of the segment or angle MN determined by the two double points or rays M and N of the systems ; which two points or lines of bisection, always real, are therefore the two points or lines of symmetry of the systems. 344. Of two homographic rows of points on a common axis ; when one double point is at infinity^ the rows are similar, and have the other double point for their centre of similitude ; and, when both double points are at infinity, the rows are similar, similarly placed, and equal. For, since, by (342) , whatever be the positions of the two double points, [MNAA] = {MNBB'} = [MNCC] = {MNDU} = &e. 250 ON THE DOUBLE POINTS AND LINES = a constant ; when one of them N is at infinity, then, for the other M, by (275), MA :MA' = MB: MB' = MC : MC = MB : MD' = &c. = a constant ; and when the other of them M is also at infinity, then, by (15), the constant = + I ; and therefore &c. (41). In the particular case when the constant = — 1, then 3IA = - MA\ MB = - MB', MC=- MC, MB = - MU, &c. and the systems, as in the preceding case, are similar and equal, but are oppositely, in place of similarity, placed on the axis (33). In this case, also, the several segments AA', BB', CC, DD', &c., intercepted between the several pairs of corresponding con- stituents of the rows, are evidently concentric, being all bisected internally by the double point M not at infinity. The converse of the above, viz., that, when two homographic rows of points on a common axis are similar, then, whatever he the magnitude and sign of their ratio of similitude, provided only it he finite, one double point is their centre of similitude and the other the point at infinity on their axis, is evident from the constancy of the ratio MA : MA', from which it follows at once that the two variable distances MA and MA become necessarily evanes- cent and infinite together. In the solitary and exceptional case of entire coincidence between two similar, similarly placed, and equal rows of points on a common axis, every point on the axis is of course indif- ferently a double point. (See 341, 1°). 345. From the general property of the preceding article, the two following results may be immediately inferred, viz. — 1°. Any two homographic rows of points on a common aayis, whose double points are real, may he regarded as the perspective, to any arbitrary centre, of two similar rows on a common axis depending in direction on the position of the centre. 2°. Any two homographic rows of points on a common axis, whose double points coincide, may be regarded as the perspective, to any arbitrary centre, of the two similar, similarly placed, and equal rows on a common axis depending in direction on the position of the centre. For, as the two homographic rows of points on the common axis determine in all cases two homographic pencils of rays at OP HOMOGRAPHIC SYSTEMS. 251 every vertex, whose double rays correspond to the double points on the axis (341, 5°) ; any axis parallel to the direction of either double ray if they be distinct, or to the common direction of the two if they coincide, would intersect the pencils in two homo- graphic rows of points, having one double point at infinity in the former case, and both double points at infinity in the latter case ; and therefore &c. (344). .346. The two following general properties of two homo- graphic rows or pencils, whose double points or rays are imaginary, have been given by Chasles, viz. — 1°. Any two homographic rows of points on a common axis, whose doMe points are imaginary, may be regarded as generated hy the revolution of a variable angle of constant magnitude round one or other of two fixed vertices, reflexions of each other with respect to the axis, 2°. Any two homographic pencils of rays through a common certex, whose double rays are imaginary, may be regarded as the perspective to any arbitrary axis of a pencil generated by the revolution of a variable angle of constant magnitude round a fixed vertex. To prove the first of these properties (which evidently in- volves the second), it is only necessary to shew that, under the circumstances of the case, a real point E (and with it, of course, its reflexion F with respect to the axis) can always be found, at which some three of the segments AA, BB', GO' intercepted between pairs of corresponding points shall subtend equal angles ; for, if three of them subtend equal angles at any point, it follows necessarily, from the homography of the rows, that they must all subtend equal angles at the same point. And that two such points, reflexions of each other with respect to the axis, exist always in this case, is evident from Art. 161 ; for, the three circles, loci of points at which the three pairs of segments BB' and CO', CC and AA', AA and BB' subtend equal angles, are then (see the article in question) all i-eal, and intersect at two real points E and F, which (the centres of the three circles being all on the axis) are of course reflexions of each other with respect to the axis. The second property follows at once, as above observed, from 252 ON THE DOUBLE POINTS AND LINES the first; for, as the two homographic pencils, whose double rays are by hypothesis imaginary, intersect with every axis in two homographic rows whose double points are imaginary; and, as there always exist, by the above 1°, two real points E and F^ with respect to each of which the latter may be regarded as generated by the revolution of a variable angle of constant magnitude revolving round it as a fixed vertex ; therefore &c. 347. The entire preceding theoiy applies of course, in its main features, as well to two homographic systems of points on a common circle or of tangents to a common circle (322), as to two systems of points on a common axis or of rays through a common vertex; and every two such systems have accord- ingly, for every as well as for either limiting magnitude of the common circle, two pairs of corresponding constituents, real or imaginary, whose positions coincide, and which are therefore termed the double points or tangents of the systems. That every two corresponding constituents A and A' of the systems divide in the same constant anharmonic ratio the arc of the common circle intercepted between the two double points or tangents if and N; that every two constituents Pand Q of the systems corresponding to the same point or tangent / divide in the square of the same constant anharmonic ratio the same intercepted arc ; that, as / varies, P and Q determine two homo- graphic systems having the same double points or tangents with the original systems ; that when I is equidistant from or equi- inclined to M and N then is it also equidistant from or equi- inclined to P and Q; and that the systems themselves are always symmetrically disposed on opposite sides of each of the two points or tangents equidistant from or equi-inclined to M and N, which two points or tangents are therefore the two points or lines of symmetry of the systems ; appear all in precisely the same manner as for the two extreme states of the circle in Arts. 342 and 343. As every two homographic systems of points on a common circle determine two homographic pencils of rays at any point on the circle, and conversely ; and as every two homographic systems of tangents to a common circle determine two homo- graphic rows of points on any tangent to the circle, and con- OF HOMOGRAPHIC SYSTEMS. 253 Tersely ; It is evident that the double points of one correspond always to the double rays of the other, and conversely, in the former case; and that the double tangents of one correspond always to the double points of the other, and conversely, in the latter case. 348. Given three pairs of corresponding constituents^ A and A' J B and B\ C and 0", of two hotnograpJiic systems, of points on a common axis, or of rays through a common vertex, or of points on a common circle, or of tangents to a common circle ; to construct the two double points or lines, M and N, of the systems. 1°. In the case of points on a common circle. Drawing any two of the three pairs of corresponding connectors of pairs of non-corresponding constituents, BC and B' C, CA' and C'A, AB' and A'B, (see fig. a. Art. 317) ; the line of connection XY of their two points of intersection X and Y (which by 337, a, is the directive axis of the systems) will pass through the intersec- tion -2' of the third pair (317), and will determine on the circle two points M and N, real or imaginary, which satisfy (317) the equianharmonic relations of 2°, Art. 341, and which are consequently the two double points of the systems. 2°. In the case of tangents to a common circle. Taking any two of the three pairs of corresponding intersections of pairs of non-corresponding constituents, BC and B'C, CA' and C'A, AB' and A'B, (see fig. a', Art. 317) ; the point of intersection XY of their two lines of connection X and Y (which, by 337, a', is the directive centre of the systems) will lie on the connector Z of the third pair (317), and will determine to the circle two tangents M and N, real or imaginary, which satisfy (317) the equianharmonic relations of 2°, Art. 341, and which are con- sequently the two double lines of the systems. 3°. In the case of rays through a common vertex. Describing arbitrarily any circle passing through the common vertex, and taking on it its three pairs of second intersections with the three pairs of rays; the two double points, found by 1°, of the two homographic systems determined by the latter on the circle will connect with the common vertex by the required double rays (347). 254 ON THE DOUBLE POINTS AND LINES 4°. In the case of points on a common axis. Describing arbitrarily any circle touching the common axis, and drawing to it Its three pairs of second tangents through the three pairs of points; the two double lines, found by 2°, of the two homo- graphic systems determined by the latter to the circle will in- tersect with the common axis at the required double points (347). These several constructions are all perfectly general, and applicable with equal facility to every variety of disposition of the three given pairs of corresponding constituents. Various direct constructions may also be given for the extreme cases of coUInear and concurrent systems (4° and 3°) without reducing them, as above, to the general cases of concycllc systems (2° and 1°) ; but the above, though Indirect, are on the whole the simplest of which they are susceptible. 349. The following construction for determining directly the two double points, in the case of collinear systems on a common axis, has been given by Chasles. Assuming arbitrarily any point not on the common axis, and describing through it any two of the three pairs of corre- sponding circles passing through pairs of non-corresponding constituents, BOG' and B'OG, COA' and C'OA, AOB' and A' OB, which Intersect again respectively at three second points P, Q, R] the circle passing through 0, and through the second intersections P and Q of the two described pairs, will pass through the second Intersection R of the third pair, and will intersect with the common axis at the two double points M and N of the systems. For, since the three circles BOC, B'OC, and Jf OiV pass through the two common points and P, therefore, by similar pencils at O and P, {MNBB'] = [MNCC] ; and since the three COA', COA, and MON pass through the two common points and Q, therefore, by similar pencils at and Q, {MNGC'] = {MNAA']; consequently at once [MNAA'} = {MNBB'] ; and therefore, by similar pencils at and JB, the three circles AOB', A' OB, and J/OJVpass through the two common points and R ; and since, as just shewn, {MNAA'] = {MNBB'] = {MNCC'], OP HOMOGRAPHIC SYSTEMS. 255 therefore (342) M and N are the two double points of the systems. N.B. It will appear in the sequel that this construction is the transformation, by inversion from any point on the circle, of that given in 1° of the preceding article for concyclic systems on a common circle. 350. The following again, derived from the general property of Art. 332, is another construction for the direct determination of the two double points in the same case of coUinear systems on a common axis. Taking the two points P and Q corresponding in the two systems to the point at infinity on the common axis, and dividing their intercepted interval PQ at the two points M and iV" for which the two rectangles PM. QM and PN. QN are each equal in magnitude and sign to the common value of the three equal rectangles PA.QA', PJB.QB', PC.QC (332) ; the two points of section M and N are evidently the required double points. See also Cor. 2°, Art. 343. In any case of the construction of the double points or lines of two homographic systems by means of three pairs of corre- sponding constituents ; if the two constituents A and A' of any pair happened to coincide, the point or line A (= A') would itself be one of the required double points or lines, and the general construction for the other would be much simplified; and if, moreover, the two B and B' of either remaining pair happened also to coincide, the point or line B (= B') would itself be the other, and all construction would be dispensed with. In this last case, the third pair C and C would furnish the value of the constant anharmonic ratio {ABCV'} distinctive of the particular pair of homographic systems determined by the three pairs of corresponding constituents (342). 351. The general constructions 1° and 2° of Art. 348, for the double points and lines in the cases of concyclic systems of points and tangents, lead each to a remarkable result when applied to the particular case of similar and similarly ranged systems separated from each other by an interval of any finite magnitude. 256 ON THE DOUBLE POINTS AND LINES For, in that case, the several arcs AA', BB\ CC, DD', &c., intercepted between the several pairs of corresponding con- stituents A and A', B and B', G and C", D and D', &c., being all equal and cyclically co-directional, the three points (or lines) X, F, Z^ and with them of course the two M and N^ deter- mined by the construction in question (1° or 2°, Art. 348), consequently (see Art. 312) lie on the line at infinity (or pass through the centre of the circle) ; which in that case is ac- cordingly the directive axis (or centre) of the systems (337, a or a). Hence the remarkable results, that — a. Every two similar and similarly ranged systems of points on a comvion circle have the same two {imaginary) double points, whatever he their interval of separation from each other ; viz. ilie fixed two lying on the line at infinity (260, 2°, h). a'. Every two similar and similarly ranged systems of tangents to a common circle have the same tioo [imaginary) double lines, whatever be their interval of separation from each other ; viz. the fixed tico passing through the centre of the circle (260, 2°, b'). In the special case when the interval of separation between the systems is nothing, that is, when the systems altogether coincide; since then A = A', B=B', C=C\ D = I)', &c., the three points (or lines) X, Y, Z, and with them of course the two 2f and N, determined by the same constructions, become, as they ought, indeterminate ; every point on (or tangent to) the common circle being then of course indifferently a double point (or line) of the systems. 352. As the two homographic systems of rays, generated by the sides of a variable angle of any invariable foi-m re- volving round a fixed vertex, determine two similar and similarly ranged systems of points on any circle passing through the fixed vertex, the double points of which correspond to the double rays of the determining pencils ; hence, from property a of the preceding article, the remarkable result, that — Every two homographic pencik of rays, determined by the sides of a variable angle of invariable form revolving round a fixed vertex, have the same {imaginary) double rays, whatever be the form of the angle ; the connectors, viz. of the fixed vertex with the two fixed circular points at infinity. • OF HOMOGRAPH IC SYSTEMS. 257 A result from which it follows at once, by Art. 342, as shewn already on other principles in Art. 312, that the sides of a variable angle of invariable form revolving round a fixed vertex divide in a constant anharmonic ratio the angle sub- tended at the fixed vertex by the two fixed circular points at infinity ; the value of the anharmonic ratio of section depending of course on the particular figure of the angle. A property which, established independently as in the article referred to or otherwise, involves evidently the above conversely, by virtue of Art. 329. 353. There is probably in the entire range of modem geometry no problem to some case or other of which a greater number and variety of others, admitting of two solutions, are reducible than that of the construction of the double points or lines of two homographic systems by means of three pairs of corresponding constituents ; some connected directly with the subject of homographic division, but far the greater number having no apparent connexion with it. Of the former class, the following are a few, the applications of which are extremely numerous and varied. Ex. 1°. Given three pairs of corresponding constituents A and A', S and B, C and C of two homographic systems of points on a common line or circle, or of tangents to a common point or circle ; to determine the pair M and M' for which AM ^ ± AM.'. Taking the three points or lines A", B", C" connected with the given point or line A by the relations AA" = 0, AB" = ± A'B', AC" = ± A'C; and constructing the second double point or line M= M" of the two homo- graphic systems determined by the three pairs of corresponding constituents A and A", B and B", Cand C" ; the point or line Mis that which in the system A, B, C, &c. is connected with its correspondent M' in the system A', B', C, &c. by the required relation AM= ± A'M'. For, since, by construction, A"B" = ± A'B' and A"C" --- + A'C, the two homographic systems A", B", C", &o. and A'. B', C, &c. are similar, and their ratio of similitude = ± 1 ; therefore A"M" or AM = + A'M' ; and therefore &c. N.B. In the case of points on a common axis, the more general problem " Given three pairs of corresponding constituents A and A', B and B', C and C, to determine the pair M and M' for which the ratio AM : A'M' shall have any given magnitude and sign," may evidently be solved in precisely the same manner. VOL. II. S 258 ON THE DOUBLE POINTS AND LINES Ex. 2°. Given three pairs of corre^ionding constituents A and A', B and B', C and C" of two homographic systems of points on a common line or circle, or of tangents to a common point or circle; to determine the ttoo pairs M and M', N and N' whose intercepted segments or angles MM', NN' shall have a given magnitude and sign. Taking the three points or lines A", B", C" connected with the given three A, B, C hy the common relation AA" = BB" = CC" = the given segment or angle; constructing then the two double points or lines M' = M" and N' = N" of the two homographic systems determined by the three pairs of corresponding constituents .4' and.il", B' axAB", C and C"; and taking finally the two points or lines M and N connected with the two M' and N' by the common relation MM' = NN' = the given segment or angle ; the two pairs of constituents M and M', N and N', are those required. For, since, by construction, AA" = BB" = CC" = MM" = JVJV" = the given segment or angle, therefore {ABCMN} = [A"B"C"M"N"}; and since again, by construction, M' = M" and N' = JV" are the two double points or lines of the two homographic systems A', B', C, &c. and A", B", C", &c., therefore {^'.B'C'Jtf'iV'l = U".B"C"Jlf"iV"}; consequently therefore {ABCMN} = {A'B'C'M'N'}; or, the two pairs of constituents M and M', N and N', which by construction intercept the required seg- ment or angle, are pairs of corresponding constituents of the two homo- graphic systems A, B, C, &c. and A', B', C, &c. ; and therefore &c. N.B. The constructions in the present and preceding examples are both based on the obvious consideration that when two homographic systems of points, rays, or tangents have a common axis, vertex, or circle, a movement of either along the common axis, or round the common vertex or circle, the other remaining fixed, would alter (increase or diminish as the case might be) the distances between the several pairs of corresponding constituents by the amount of the movement ; so that those correspondents which coincided before would be separated after by that amount, and conversely. Ex. 3°. Given three pairs of corresponding constituents A and A', B and B', C and C of two homographic systems of points on a common line or circle, or of tangents to a common point or circle : to determine the two pairs MandM', N andN' whose intercepted segments or angles MM', NN' shall have a given middle point or line O. Taking the three points or lines A", B", O" connected with the given three A, B, C by the common relation OA" = - OA, OB" = - OB, OC" = - OC; constructing then the two double points or lines M' = M" and N' = N" of the two homographic systems determined by the three pairs of corresponding constituents A' and A!', B' and B", C and C" ; and taking finally the two points or lines M and N connected with the two M' and N' by the common relation 0M= - OM', 0N=- ON' ; the two pairs of constituents Ma.nd M', JVand N' are those required. For, since, by construction, AA", BB", CC", MM", NN" are all OF HOMOGKAPHIC SYSTEMS. 259 bisected by O, iheTehtB [ASCSIN] = {A"S"C"M"N"}; and since again, by construction, M' = M" and N' = N" are the two double points or lines of the two homographic systems A', S', C, &c. and A", B", CC", &c., therefore {A'B'C'M'N'] = {A"B"C"M"N"}; consequently therefore {ABCMN} = [A'B'C'M'N'}; or, the two pairs of constituents Jlf and M', N and N', which by construction have the required middle point or line O, are pairs of corresponding constituents of the two homographic systems A, B, C, &c. and A', B', C", &c. j and therefore &c. N.B. The problems of the present and preceding examples are mani- festly equivalent to the following, viz. : " Given three pairs of corresponding constituents A and A', B and B', C and C of two homographic systems of points on u common line or circle, or of tangents to a common point or circle; to determine the two pairs ilfand M', JVand JV', for which the sum or diiference PM± PM', PN± PN', P being a given point or tangent, shall have a given magnitude and sign." Ex. 4°. Oiven three pairs of corresponding constituents A and A', B and B', CandC' of tuio homoffraphic systems of points on a common line or circle, or of tangents to a common point or circle ; to determine the two pairs Mand M', N and N' tehich shall form with two given points or tangents P and Q a system having a given anliarmonic ratio. Taking the three points or lines A", B", C" connected with the given three A, B, C by the common relation {PQAA"} = {PQBB"] = {PQCC"} = the given anharmonic ratio ; constructing then the two double points or lines M' = M" and N' = iV" of the two homographic systems determined by the three pairs of corresponding constituents A' and A", S' and B", C and C"; and taking finally the two points or lines M and iV connected with the two M' and N' by the common relation [PQMM'} = {PQNN'} == the given anharmonic ratio; the two pairs of constituents ilfand JIf', i\rand N' are those required. For, since, by construction, {PQAA"}^PQBB"}={PQCC"MPQ^J^"] = {PQKN"} = the given anharmonic ratio, therefore (329) {ABCMN^} = {A"B"C"M"N"}; and, since again, by construction, M'=^M" and N' = N" are the two double points or lines of the two homographic systems A', B', C, &c. and A", B", C", &c. therefore {A-B-CJU'lf} = [A"B"C"M"N"i ; consequently therefore {ABCMN} = {A'B'C'M'N') ; or, the two pairs of constituents M and M', JV and N', which by con- struction form with P and Q the given anharmonic ratio, are pairs of cor- responding constituents of the two homographic systems A, B, C, &c. and A', B; C', &c. ; and therefore &c. Ex. 5°. Cfiven three paia of corresponding constituents A and A', B and B', C and C of one pair of homographic systems of points on a common line or circle, or of tangents to a common point or circle; and also three pairs P and P ", Q and Q", K and It' of another pair of homographic systems of points on the same line or circle, or of tangents to the same point S2 260 ON THE DOUBLE POINTS AND LINES or circle : to determine the two pairs M and M' or M", IT and N' or N" common to both pairs of systems. Taking the three points or lines A", B", C" connected with the given three A, B, C hy the relations {PQRA} = {P"Q'R"A"), [PQSB] = {P"Q"R"B"], {PQBC} = {P"Q"i2"C") ; constructing then the two double points or lines M' = M" and N' = N" of the two homographic systems determined by the three pairs of corresponding constituents A' and A", B'a.TiAB", C'and C" ; and taking finally the two points or lines M and N connected with the two M' = M" anA N' = N" by the relations \ABCM) = {A'B-C'M-} = {A"B"C"M"i and {ABCN] = {A'B'C'N'} = {A"B"C"N"}; the two pairs of constituents Jf and M' or M", N and N' or JV" are those required. For, since, by virtue of the preceding relations, {ABCMN} = {A'B'C'M'N'}, and also (327) {PQRMN} = {P"Q"R"M"N"}; and since by construction M' = M" and. N' = N"; therefore M and M' or M", N and N' or N" are pairs of corresponding constituents of both pairs of homographic systems ; and therefore &c, N.B. This latter problem evidently comprehends the three preceding as particular cases; and with them a variety of others of the same nature corresponding to the variety of other ways in which homographic systems may be generated. See the various articles of the preceding chapter in which the principal of them are given. Ex. 6°. Given three pairs of corresponding constituents A and A', B and B', C and C of two homographic systems of points on a common circle, or of tangents to a common circle ; to determine the two pairs M and M', N and N'. a. Whose lines of connexion, in the former ease, shall pass through a given point P. a'. Whose points of intersection, in the latter case, shall lie on a given line L. In the former case. Taking the three second intersections A", B", C" with the circle of the three lines PA, PB,PC; constructing then the two double points M' = M" and JV' = N" of the two homographic systems determined by the three pairs of corresponding constituents A' and A", B' and B", C and C"; and taking finally the two second intersections ilfand JTwith the circle of the two lines PM' and PN' ; the two pairs of points Jtf and M', N aaA N' are those required. In the latter case. Drawing the three second tangents A", B", C" to the circle through the three points LA, LB, LC\ constructing then the two double tangents M' = M" and N' = N" of the two homographic systems determined by the three pairs of corresponding constituents A' and A", B' and B", C and C" ; and drawing finally the two second tangents M and N to the circle through the two points LM' and LN'; the two pairs of tangents ilf and M', N axiA. N' are those required. OF HOMOGRAPHIC SYSTEMS. 261 For, in both cases, since, by construction, {A'S'C'M'N'} ^iA^'S^'CM-'N"], and since, by (313, a and a), [A"B"C"M"N") = {ASCMN); therefore {ABCMN] = [A'B'C'M'N'] ; and therefore &c. £x. 7°. Given three pairs of corresponditiff constituents A and A', JB and £', C and C of two homographic systems of points or. any two axes, or of rays through any two vertices ; to determine the two pairs M and M', NandN'. a. Whose lines of connexion, in the former case, shall pass throtiffh u given point, or touch a given circle tangent to the two axes. a'. Whose points of intersection, in the latter case, shall lie on a given line, or on a given circle passing through the two vertices. Here, evidently, the three given pairs of corresponding constituents determine ; in the former case, the corresponding three of two homo- graphic systems of rays through the given point, or of tangents to the given circle, whose double lines intersect with the given axes at the required pairs of constituents ; and, in the latter case, the corresponding three of two homographic systems of points on the given line, or circle, whose double points connect with the given vertices by the required pairs of constituents. N.B. In the case of the latter property a'; the two rays, constituting each required pair of corresponding constituents, being parallel when the given line is at infinity (16), and intersecting at a given angle for every given circle passing through the two vertices (£uc. ill. 21,22); the two solutions, real or imaginary, (see Art. 339) of the problem " Given three pairs of corresponding constituents A and A', B and B', Cand C of two homographic systems of rays through different vertices, to determine the two pairs jifandilf', iVandiV' whose directions are parallel, or, more generally, intersect at any given angle," are consequently given by it for every form of the angle. Ex. 8. Given three pairs of corresponding constituents A and A', B and B', C and C of two homographic systems of points on any two axes, or of rays through any two vertices: to determine the two pairs M and M', NandN'. a. Whose lines of connection with a given point P, in the former ease, shall, 1°, coincide ; 2°, contain a given angle ; 3°, make equal angles with a given line through the point ; 4°, divide in a given anharmonic ratio a given angle at the point. a'. Whose points of intersection with a given line L, in the latter case, shall, 1°, coincide ; 2°, intercept a given segment ; 3°, moke equal segments with a given point on the line : i°, divide in a given anharmonic ratio a given segment of the line. Here, in the former case ; the required pairs of connectors PJf and PM', PN and PN ' are evidently, with respect to the two homographic pencils of rays determined at P by the three given pairs of connectors PA and PA', PB and PB', PC and PC; in 1°, the double rays; in 2°. 262 ON THE DOUBLE POINTS AND LINES the pain containing the given angle; in 3°, the pairs equally inclined to the g^ven line ; in 4°, the pairs dividing in the given anharmonic ratio the given angle. And, in the latter case; the required pairs of intersections LM and LM', LN and LN' are evidently, with respect to the two homographic rows of points determined on L by the three given pairs of intersections LA and LA', LB and LB', ZCand LC ; in 1°, the double points; in 2°, the pairs intercepting the given segment ; in 3°, the pairs equidistant from the given point; in 4°, the pairs dividing in the given anharmonic ratio the given segment. In both cases, consequently, while the solution of problem 1° is reduced at once to the corresponding case of Art. 348, those of problems 2°, 3", 4° are reduced to those of examples 2°, 3°, 4° of the present article. N.B. The above reciprocal solutions would all manifestly remain unchanged, if the point P were replaced by a circle touching the two axes, in the former case, and the line X by a circle containing the two vertices, in the latter case. Ex. 9°. Given two triads of corresponding contlituents A, A', A" and B, B', B" of three homographic systems of points on a common line or circle, or of tangents to u common point or circle; to determine the three systems which shall have a pair of triple points or lines M= M' = M" and If= N' = N"; with the positions of the two triple points or lines M and N. Constructing the two double points or lines Jlf and N of the two homo- graphic systems determined by the three pairs of corresponding constituents A and B, A' and B', A" and ^"; the two points or lines itf and N are those required; and the position of either, as supplying a third triad of corresponding constituents in addition to the given two, determines, of course, the required systems (327). For, since, by (342), {MNAB} = {MNA' B'] ={MNA"B"]; therefore &c. COE. 1°. The following property of two conjugate triads of homographic systems of points on a common line or circle, or of tangents to a common point or circle, follows immediately from the above. If the three systems determined by the three triads of corresponding constituents A, A', A" ; B, B, E' ; C, C, C" have a pair of triple points or lines M and N: the three determined by the three triads A, B, C; A', B", C ; A", B", C" have also a pair of triple points or lines ; and the triple points or lines are the same for both triads. For, as the relations [MNABC] = [MNA'B'C] = {MNA"B"C"\ (1), involve reciprocally, by (272), the relations {MNAA'A") = {MNBB'B") = [MNCC'C"} (2), and conversely ; therefore &c. CoE. 2°. The comparison of both groups of relations (1) and (2) of the OP HOMOGHAPHIC SYSTEMS. 263 preceding (Cor. 1°) gives immediately (282) the three following groups of relations among the nine constituent points or lines themselves, viz. — {A'A"SC\ = {A"AS'C']^{AA'£"C"] (1), {B'B"CA} = IS"SC'A'} = {SS'C'A"} ... (2), {C'C'AS} = {C'CA'B'] = {CC'A"B "} (3), which are therefore the conditions, necessary and sufficient, that either (and therefore the other) of the two triads of homographic systems, deter- mined by the two conjugate triads of corresponding constituents, should have a pair of triple points or lines. Ex. 10°, a. Given two triadt of corresponding constituents A, A', A' and B, B', B" of three homographic rows of points on any three axes, and one of the three lines L which intersect with the three axes at a triad of corresponding points P, B', B " ; to determine the other two M and N which intersect with them also at triads of corresponding points Q, Q', Q" and J£, jKf R"t Ex. 10°. a'. Given two triads of corresponding constituents A, A', A" and B, B',B" of three homographic pencils of rays through any three vertices, and one of the three points P will connect with the three vertices by a triad of corresponding rays L, L', L" ; to determine the other two Q and JR which connect with them also by triads of corresponding rays M, M', M" and JV, N', N"; In the former case. Taking the two triads of points X, X', X" and Y, Y', Y" at which the sides of the two triangles AA'A" and BB'B" intersect with the given line L ; and constructing the two double points £ and F of the two homographic rows determined on L by the three pairs of corresponding constituents X and Y, X' and Y', X" and Y" ; the required lines M and N pass through £ and F respectively, and may therefore be determined by the first case of Ex. 7°, a. For, if F and F be the two points at which Af and JV intersect with X ; then since, by the first case of (338, a), {FFXY] = [Q'It'A'B' } = {Q'R'A"B"] (1), {EFX'Y' 1 = {Q"Ii"A"B"] = {QRAB } (2), {FFX"Y"} = {QSAB ]={Q'£:a'B' } (3)j therefore at once {EFXYi = {FFX'Y') = {EFX"Y"}i and therefore &c. (342). In the latter case. Taking the two triads of lines X, X', X" and Y, Y', Y" by which the vertices of the two triangles A, A', A" and B, B', B" connect with the given point P; and constructing the two double rays JBand f of the two homographic pencils determined at Pby the three pairs of corresponding constituents .y and Y, X' and Y', X" and Y" ; the required points Q and R lie on E and F respectively, and may therefore be determined by the first case of Ex. 7°, a'. 264 ON THE DOUBLE POINTS AND LINES For, if E and F be the two lines by which Q and M connect with P ; then since, by the first case of (338, a'), {EFXr} = {M'N'A'B' } = {M"N"A"B"} (!'). {EFX'Y' ] = {M"j!f"A"B"] = {MNA B ) (2'), {EFX"Y"} = [MNA B } = {M'N'A'B' } (3') ; therefore at once {EFXT} = {EFX'Y'\ = {EFX"Y"]; and therefore &c. (342). CoE. 1°. If C, C, C" be any third triad of corresponding constituents of the three systems, in either case ; and Z, Z', Z" the three pointe or lines at which the three sides or by which the three vertices of the triangle C, C, C" intersect with the line X or connect with the point P ; then since, in either case, for the same reason as above, {EFXYZ) = (EFX'Y'Z'] = {EFX"Y"Z"}, and similarly for all triads; therefore the three rows or pencils X, Y, Z, &c; X', Y', Z', &c.; X", Y", Z", &c. are homographic, and have E and F for triple points or rays. Hence the following general properties of the three lines L, M, N'm. the former case, and of the three points P, Q, R in the latter case. a. The tides of the system of triangles formed by the several triads of corresponding points determine on each line, in ihe former case, three homographic rows having a pair of triple points ; and the triple points on each line are its intersections with the other two. a'. The vertices of the systems of triangles formed by the several triads cf corresponding rays determine at each point, in the latter case, three homographic pencils having a pair of triple rays; and the triple rays at each point are its connectors with the other two. K.B. That, in both cases, the three triads of systems are homographic with each other and with the systems of the original triad, follows also im- mediately from the first parts of (338, b and V) ; and that for each triad the points or lines in question are triple, is evident also from the obvious consideration, that, of a triangle, when the three vertices are coUinear, the three sides intersect with every line at a triad of coincident points, and, when the three sides are concurrent, the three vertices connect with every point by a triad of coincident lines. Cor. 2°. From the reciprocal properties of the preceding corollary it follows immediately, by virtue of (341, 1°), that — For three homographic rows of points on different axes, or pencils cf rays through different vertices, no more thnn three triads of corresponding constituents could he coUinear in the former ease, or concurrent in the latter case, unless all triads of corresponding constituents were coUinear in the former case, or concurrent in the latter ease. • OF HOMOGRAPHIC SYSTEMS. 265 For, if four collinear or concurrent triads existed, then, of the four lines of collinearity or points of concurrence, every three, by the properties in question, would intersect or connect with the fourth at three triple points or by three triple rays of the three homographic rows or pencils determined on or at it by the sides or veriices of the system of triangles formed by the several other triads; which three homographic rows or pencils should therefore (341, 1°) entirely coincide; and therefore &c. N.B. Between the three lines X, M, iVand the three axes of the rows in the ibrmer case, or between the three points P, Q, R, and the three vertices of the pencils in the latter case, no relation of connexion necessarily exists ; both triads in either case may be given or taken arbitrarily ; and give rise in all cases to two conjugate triads of homographic rows or pencils, determined ; in the former case, by the three triads of corresponding constituents P, P, P" ; Q, Q', Q" ; S, S', M" on the three axes, and by the three F, Q, Jt; P, Q', P; P', Q", -R" on the three lines; and, in the latter case, by the three triads of corresponding constituents L, L', L" ; M, M', M"; N, N', N" at the three vertices, and by the three L, M, JV'; i', M', If'; L", M", N" at the three points; between which there exist several interesting relations of connexion, though the two triads of lines or points which determine them are entirely arbitrary. When, of six lines or points given or taken arbitrarily, any (and therefore every) four intersect or connect equianharmonically with the remaining two (301, a and a') ; then (338), of the three homographic rows or pencils determined by any three of them on or at the remaining three, all triads of corresponding constituents are collinear or concurrent; and, conversely, when, of three homographic rows or pencils given or taken arbitrarily, any four (and therefore all) triads of corresponding constituents are collinear or concurrent (Cor. 2° above) ; then (338), of the three axes and any three lines of collinearity, or of the three vertices and any three points of concurrence, every four of the six intersect or connect equianharmonically with the remaining two. Ex. 11°. a. Given two triads of corresponding constituents A, A', A" and B, B', B" of three homographic systems, one A, B, C, ij-c. of points on an axis, and two A', B', C, S^e. and A", B", C", Sj-c. of points on a circle ; and one of three lines L which determine a collinear triad P, P ', P' ; to construct the other two M and N which determine collinear triads Q, Q', Q " and B, B', B". Ex. 11°. a'. Given two triads of corresponding constituents A, A', A" and B, B', B' of three homographic ^sterns, one A,B, C, Sj-c. of rags through a vertex, and two A', B', C, ISfc. and A", B", C", SfC. of tangents to a circle ; and one of the three points P which determine a concurj-ent triad L, L', L" ; to construct the other two R and. S which determine con- current triads M, M',M" and N, N', N". Here, since in the former case, by the second case of (338, a), the several 266 ON THE DOUBLE POINTS AND tlNES lines of connexion A' A", B'B", C'C", Sic. of the several pairs of corre- sponding points on the circle determine on any two of themselves H'S' and £'E" two coUinear systems A^, B,, C,, &c., and A„, B,„ C„, &C., homo- graphic with the two concyclic systems A,' B', C, &c., and A", B", C", &c. and therefore with the coUinear system A, B, C, &c. ; and, since in the latter case, by the second case of (338, a'), the several points of intersection A' A", B'B", C'C", &c. of the several pairs of corresponding tangents to the circle determine at any two of themselves S'H" and K'K" two concur- rent systems A,, B,, C,, &c., and A,,, 5„, C„, &c., homographic with the two concyclic systems A, B', C, &c., and A", B", C", &c., and therefore with the concurrent system A, B, C, &c. ; the two reciprocal problems of the present are consequently reducible at once to those of the preceding example ; and the various inferences there drawn are accordingly applicable here also. 354, In all the examples of the preceding article, the two homographic systems, whose double points or lines were the object of enquiry, direct or indirect, were supposed to have been given by means of three pairs of corresponding con- stituents A and A', B and B\ C and C ; which, in all cases, as shewn in Art. 327, implicitly determine the systems, and all particulars connected with them. In the applications of the theory, however, it is the law connecting the several pairs of corresponding constituents, whatever it be, and not the actual triad of constituents themselves, which is generally given; and, should the law of connexion not be such as to furnish the required double points or lines directly by a simpler construction, a certain preliminary process is consequently necessary before the particular construction corresponding to the case, as already described, can be applied. This preliminary process is however uniformly the same in all cases, and consists simply in taking arbitrarily any three constituents A^ B, C of either system, and constructing their three correspondents A', B', C of the other, in accordance with the given law of connexion, whatever it be. The three pairs of corresponding constituents A and A', B and B', C and C necessary and sufficient to determine the two homographic systems, whose double points or lines give the two solutions of the proposed problem, are thus obtained ; and the subsequent process is that already described and exemplified at some length in the preceding article. • OP HOMOGEAPHIC SYSTEMS. 267 355. If, in the performance of the preliminary process de- scribed in the preceding article, two of the three coincidences A = A\ B—B\ C=C' should happen to result, the required double points or lines, and therefore the two solutions of the proposed problem, would of course be obtained without the necessity of any further construction. In the performance of the preliminary process, therefore, each arbitrary assumption of a point or line A of either system, from which to construct the corresponding point or line A' of the other system by application of the given law of connection between them, may be regarded as an attempt to solve the proposed problem by the method of trial; which would be successful if A! = A'^ but which of course results generally in a failure^ of which A A' re- presents the amount of error both in magnitude and sign. And it is by a simple and uniform process, based on the data resulting from three such attempts and their failures, that, as in the method of false position in Arithmetic, the true solutions of the proposed problem are by this method eventually obtained. 356. With a few examples of problems solved by the above method of trial, and coming under the second class (353) of those reducible to the determination of the double points or lines of two homographic systems, we shall conclude the present chapter. £z. 1°. To divide a given segment or angle EF in a given anharmonic ratio, by a segment or angle MM', or NN', of given magnitude, or having a given point or line of bisection. Assuming arbitrarily any three points on the axis of the segment or rays through the vertex of the angle A, B, C; and constructing the three A', S, C for which {EFAA'} = {EFBB'\ = {FFCC} = the given anhar- monic ratio, and also the three A", B", C" for which the three segments or angles AA,BB!, CC have the given magnitude or bisector; if, having proceeded so far, two of the three coincidences A' = A", B" = B", C = C" happen to result, the problem is solved ; if not, the two systems of points or rays A', B', C", &c. and A", B", C", &c., being both homographic with the system A, B, C, &c. (329), and therefore with each other (323), the two double points or rays M' = M" and 2/ ' = JV" of the two former, with their two correspondents 3f and Nia the latter, give the two segments or angles MM' and JS'If' which satisfy its two conditions. 268 ON THE DOUBLE POINTS AND LINES N.B. Of the above problems (which evidently include as particular cases those of 1° and 2°, Cor. 3°, Art. 227) the first may obviously be stated otherwise as follows : " To place two segments or angles of given magnitude so as to cut each other in a given anharmonic ratio." Ex. 2°. To divide two given segments or angles EF and GH, having a common axis or vertex, in two given anharnumic ratios, by a common segment or angle MM', or NN'. Assuming arbitrarily any three points on the axis or rays through the vertex A, B, C; and constructing the three A', B', C for which {EFAA'} = {EFBB") = {FFCC} = the given anharmonic ratio for EF, and also the three A", B", C" for which {GHAA"} = {GEBB'\ = {GHCC"] = the given anharmonic ratio for GJI; if, having proceeded so far, two of the three coincidences A' = A", B! = B", C = C" happen to result, the problem is solved; if not, the two systems of points or rays A',B', C, &c. and A", B", C", &c. being homographic with the system A, S, C, &c. (329), and therefore with each other (323), the two double points OTxay&M' = M" and N' = N" of the two former, with their two correspondents M and N in the latter, give the two segments or angles MM' and NN' which satisfy its two conditions. N.B. To the first of the above problems (which evidently include those of Art. 230 as particular cases) the following, by virtue of the general property of Art. 332, may obviously be reduced : " Given four points P, Q, iJ, (S on a common axis, to determine the two Maxid M', or N and N', on the axis, for which the two rectangles PM. QM' and MM. SM', or PN. QN' and EN. SN', shall be given in magnitude and sign." Ex. 3°. Given two points on or tangents to a circle E and F, to divide their intercepted arc EF in a given anfiarmonic ratio by two others M and M', or Nand N' ; a. Connecting, in the former case, through a given point P. a'. Intersecting, in the latter case, on a given line L. Assuming arbitrarily any three points on or tangents to the circle A, B, C; and constructing the three A', B', C for which [EFAA'] = {EFBB'} = {EFCC] = the given anharmonic ratio, and also the three A", B", C which connect with A, B, C in the former case through the given point P, or intersect with .4, B, Cin the latter case on the given line L ; if, having proceeded so far, two of the three coincidences A' = A", B' = B", C = C" happen to result, the problem is solved; if not, the two systems of points or tangents A', B', C, &c. and A", B ", C", &c. being both homographic with the system A, B, C, &c. (329 and 315), and therefore with each other (323), the two double points or tangents M' = M" and N' = N" of the two former, with their two correspondents .af and Nin the latter, give the two pairs of points or tangents M and M', J^Tand N' which fulfil botii required conditions. • OF HOMOGRAPHIC SYSTEMS. 269 N.B. To the above problems (which evidently include those of Cor. 4°, Art. 257, as particular cases) the following, by virtue of the general pro- perty of Art. 257, may obviously be reduced ; viz. " To divide two given arcs of two given circles, one harmonically, and the other in any given anharmonic ratio, by four collinear points on, or by four concurrent tangents to, the circles." Ex. 4°. a. On two given lines L and L' to find two points M and M', or N and N ', whose lines of connection with each of two given points P and P' shall; V, contain a given angle; 2°, make eqval angles with a given line through the point ; 3^ divide in a given anharmonic ratio a given angle at the point. Ex. 4°. a'. Through two given points P and P' to draw two lines M and M', or N and N', whose points of intersection with each of two given lines L and L' shall; 1°, intercept a given segment ; 2', make equal segments with a given point on the Une ; 3°, divide in a given anharmonic ratio a given segment of the line. In the former case; taking arbitrarily, on either line X, any three points A, S, C; and constructing, on the other X', the three A', B', C for which the three angles APA', BPB', CPC fulfil the required condition for the point P, and also the three A'', B", C" for which the three AP'A", BP'S", CP'C" fulfil that for the point P'. And, in the latter case; drawing arbitrarily, through either point P, any three rays A, B, C; and constructing, through the other P', the three A', B', C for which the three segments ALA', BLB', CLC fulfil the required condition for the line X, and also the three A", B", C" for wliich the three AL'A", BL'B", CL'C" fulfil that for the line X'. If, in either case, having proceeded so far, two of the three coincidences A' - A", B' = B", C = C" happen to result; the problem is solved ; if not, the two systems of points or rays A', B ', C, &c. and A",B", C", &c. being homographic with the system A, B, C, &c. (see Ex. 8° of preceding Art.), and therefore with each other (323), the two double points or rays M' = M" and N' = N'' of the two former, with their two correspondents M and A'' in the latter, are the two pairs of points or rays Jlf and M', iVand N' which fulfil the required conditions. Ex. 5°. a. Through a given point P to draw a line intersecting with four given lines Xj, X,, L^ X, at a system of four points Mi, M-i, M^, Mt, or N„ iVj, JV3, JV4, having a given anharmonic ratio. Ex. 5°. a'. On a given line X to find a point connecting with four given points P„ Pj, P3, Pi by a system of four rays JIf „ M„ M„ M„ or iVi, Nr^, N„ iV„ having a given anharmonic ratio. In the former case. Taking arbitrarily any three points A^, B„ C, on any one of the four given lines X. ; and drawing through them the three lines intersecting with the remaining three X^, X3, X, at the three triads of points A^, Aj, Ai-, B^ B3, B,; C„ C,, C, determining with A^, B„ C, the given anharmonic ratio (287, a) ; if, having proceeded so far, two of the three lines so drawn happen to pass through the given point P, the problem 270 ON THE DOUBLE POINTS AND LINES is solved j if not, the four systems of points A^, S„ C„ &c. j A^, B„ Cj, &o. ; A„ B„ C3, &c. ; A^, B„ C„ &c., on the four given lines i„ L„ Z3, L^, heing homographic (333, Ex. e), the two double rays of the tvro homo- graphic pencils determined by any two of them at P are the two lines that solve it. In the latter case. Drawing arbitrarily any three rays A^, S^, C, through any one of the four given points P, ; and taking on them the three points connecting with the remaining three Pa P3, P* by the three triads of rays A„ Ay, A,; B^, B„ B,; Cj, C3, C, determining with A^, B^, C, the given anharmonic ratio (287, a') ; if, having proceeded so far, two of the three points so taken happen to lie on the ^ven line L, the problem is solved; if not, the four systems of rays A^, B^, C,, &c.; A^ B„ C„ &c. ; A^ J9„ C,, &c. ; A„ B^, C„ &e., through the four given points P^ Pj, P„ P,, being homographic (333, £x, e'), the two double points of the two homographic rows determined by any two of them on L are the two points that solve it. Cor. 1°. Regarding the four lines in a, or the four points, in a', as grouped in two pairs determining two angles in the former case, or two segments in the latter case ; the above reciprocal problems may be stated otherwise as follows : a. Through a given point to draw a line the tegmenta intercepted on which hy two given angles shall divide each other in a given anharmonic ratio. a'. On a given line to find a point the angles subtended at which hy two given segments shall divide each other in a given anharmonic ratio. Cob, 2°. If any one of the four lines, in a, or of the four points, in a', be at infinity ; the problems for the remaining three (See Cor. 3°, Art. 285) become modified as follows : u. Through a given point to draw a line intersecting with three given lines at three points the ratios of whose three intercepted segments shall he given. a'. On a given line to find a point connecting with three given points hy three lines the ratios of the three segments intercepted hy which on a second given line shall be given. N.B. Of these latter problems the first (a) is obviously a very particular case of that proposed, with others, for solution, on other principles, In 3°, Cor. 1°, Art. 66. Ex. 6°. a. To draw a line intersecting with six given lines L^, X,, L^ X„ L„ X, {or five if any two coincide) so that four points of intersection Mi, M^, itf„ Mi, or J\r„ N,, iVj, JV4, shall have one given anharmonic ratio, and four more Jf,, M^, M„ 3/,, or N^, N„ N^ N^ another given anharmonic ratio. Ex. 6°. a'. To find a point connecting with six given points Pj, Pa, Pj, P4, P5, P, {or five if any two coincide) so that four rays of connexion Mi, M„ Mg, Mf, or Ni, JVj, JVj, iV^, shall have one given anharmonic ratio, and four more M^, M,, M„ M„ or 2f„ N„ JV5, N„ another given anharmonic ratio. •of homographic systems. 271 In the former case. Taking arbitrarily any three points ul,, S^ CI on Z,; and drawing through them the three lines intersecting with L^, X3, X, at the three triads of points A^, A,, ^, ; S^i ^i> -^«i ^v ^3< ^i determining with A-s,, B^, C, the first given anharmonic ratio, and also the three inter- secting with X,, X5, X, at the three triads of points A\, A„ A^; B\, B„ £,; C'j, Cj, C, determining with A^, £,, Cj the second given anharmonic ratio (287, a) ; if, having proceeded so far, two of the three coincidences A^ = A'„ £2 = B^, C, = C,' happen to result, the problem is solved ; if not, the two systems of points A^ B^, C^ &c. and A^, S^, C,', &c. on Xj being both homographic with the systems Ai, B^, C,, &c. on Xj (333, Ex. e), and therefore with each other (323), the two double points M^ = 3f,' and N^ = A",' of the two former connect with their two correspondents Mi and JT, in the latter by the two lines which solve it. In the latter case. Drawing arbitrarily any three rays A^, B^, C^ through P| ; and taking on them the three points connecting with P,, P,, P4 by the three triads of rays A„ A,, A^ j P„ B^ B^ ; C„ C^, C, determin- ing with A^, Bj, C\ the first given anharmonic ratio, and also the three connecting with P„ P„ P, by the three triads of rays A^', A„ A, ; B,', B„ B,; C^', C„ C, determining with A^, B„ C, the second given anharmonic ratio (287, a!) ; if, having proceeded so far, two of the three coincidences, A^ - A^', B2 = B3', C, = Cj happen to result, the problem is solved ; if not, the two systems of rays A^, B„ Cj, &c. and A^, B{, C{, &c. through Pj being both homographic with the system A^, S^, C„ &c. through P^ (333, Ex. e'), and therefore with each other (323), the two double rays M, = Jifj' and JV, = JVj' of the two former intersect with their two correspon- dents Ml and JV, in the latter at the two points which solve it. Cor. 1°. Regarding the six lines, in a, or the six points, in a', as grouped in three pairs determining three angles in the former case or three segments in the latter case ; the above reciprocal problems, like those of the preceding example, may be stated otherwise as follows : a. To draw a line the segments intercepted on which by three given angles shall divide each other two and two in given anharmonic ratios. a'. Tojlnd a point the angles subtended at which by three given segments sJtall divide each other two and two in given anharmonic ratios. CoE. 2°. If any two of the four lines, in a, or of the four points, in a', which do not enter into both anharmonic ratios, coincide at infinity; the problems for the remaining four (see Cor. 3°, Art. 285) become modified as follows: — a. To draw a line two of whose intersections with four given lines shall divide in given ratios its segment intercepted by the remaining two. a'. To find a point two of whose connectors with four given points shall divide in given ratios the segment interested on a given line by the remaining two. N.B. Of these latter problems, the first (n), it will be remembered, was already proposed for solution, on other principles, at the close of Art. 65. 272 ON THE DOUBLE POINTS AND LINES Ex. 7°. a. To a given circle to inscribe a polygon of any order, whose several sides shall pass through given points, or any or all of them touch instead given circles concentric with the original. Ex. 7°. a'. To a given circle to ezscribe a polygon of any order, whose several vertices shall lie on given lines, or any or all of them lie instead on given circles concentric with the ordinal. In the former case. Taking arbitrarily any three points Ai, 5„ CI on the given circle ; and constructing successively the several triads of inscribed chords A,A^, B,B^, Cfi^; A,A„ B^B„ C.C^; A^A„ B^B^, C,C,; &c.; A^,^y, B,,B„,^, C„C^, passing through (or touching) the several given points (or concentric circles) corresponding respectively to the several successive sides of the polygon ; if, having proceeded so far, two of the three coincidences A„^y = Ai, B,,^^ = B,, <7„,, = C, happen to result, the problem is solved ; if not, the several systems of points A^, Bi, C„ &c. ; A„ B„ C„ &c ; ^3, B„ C3, &c., &c.; ^,„„ 5,,^, C,^„ &c., being all homo- graphic (315), the two double points M^ = Jlf„, and JV, = N„,^ of the first and last give the first vertices of the two polygons that solve it. In the latter case. Drawing arbitrarily any three tangents A„ B-^, C, to the given circle ; and constructing successively the several triads of exscribed angles A^A^, B,B^, C^C^; A^A,, B^B^ C^C^; A^A^ B^B^, C^Ct; &c.; ■^n-^„ti> -S„-B,^„ C„C„,^ having their vertices on the several given lines (or concentric circles) corresponding respectively to the several successive vertices of the polygon; if , having proceeded so far, two of the three coincidences .4„i = A^, B,u.i = B^, C^i = Ci happen to result, the problem is solved ; if not, the several systems of tangents A^, B^, C„ &c.; A^ B,, C^, &c.; ^3,^3, C3, &c., &c.; ^^i, B„^„ C,^„ &c. being all homographic (315), the two double lines Mi = M^^i and JV, = N',^^i of the first and last give the first sides of the two polygons that solve it. N.B. Of the above reciprocal problems, those of Art. 263, solved there on other principles, are evidently particular cases. Ex. 8^. a. To a given circle to inscribe a polygon of any order, whose several sides shall divide in given anharmonic ratios given arcs of the circle. Ex. 8°. a*. To a given circle to exscribe a polygon of any order, whose several angles shall divide in given anharmonic ratios given arcs of the circle. In the former case. Taking arbitrarily any three points Ai, B^, C^ on the given circle; and constructing successively the several triads of in- scribed chords AiA,, B^B^, C^C,; A,A^ B^B^, C^C,; A^A^, B,B,, CjC^; &c. A„A,u.ii -Sb-B^i, C„C„^.i dividing in the several given anharmonic ratios the several given arcs of the circle corresponding respectively to the several successive sides of the polygon ; if, having proceeded so far, two of the three coincidences .il„„ = Ai, B^^ = £,, C„^^ = Cj happen to result, the problem is solved; if not, the several systems of points Ai, Bi, C,, &c. ; A^, B,, Cj, &c. ; A„ B^ C„ &c. i A„^i, B„^j, C^i, &c. being all homographic (329), the two double points Jf, = J!f„i and JVi = JV,^( of the first and last give the first vertices of the two polygons that solve it. • OF HOMOGKAPHIC SYSTEMS. 273 In the latter case. Drawing arbitrarily any three tangents Ai, Bi, C^ to the given circle; and constructing successively the several triads of exscribed angles A^A^, B,B„ C^C.,; A,A,, S.B^, CM,; A^A„ B^St, C^C^ &c. ; A,A^^, B„B^i, C^C^i dividing in the several given anharmonic ratios the several given arcs of the circle corresponding respectively to the several successive angles of the polygon ; If, having proceeded so far, two of the three coincidences A^^ = Ay £„, = B^, C„, -- C, happen to result, the problem is solved ; if not, the several systems of tangents A^, B^, C,, &c. ; A^, Bj, Cj, &c. ; A.^, B^ C3, &c. ; A^j, B^., Cm, &c. being all homo- graphic (329), the two double lines JIf, = M^-^ and iV, = N,^ of the first and last give the first sides of the two polygons that solve it. N.B. That the above reciprocal problems involve, as particular cases, those of the preceding example, is evident from Arts. 267 and 311, Cor. 3°. Ex. 9°. a. To construct a polygon of any order, whose several vertices shall lie on given lines, and whose several sides ihnll pass through given points, or any or all of them touch instead given circles tangent to the pairs of lines on which the adjacent vertices lie. Ex. 9°. a: To construct a polygon of any order, whose several tides shall pass through given points, and whose several vertices shall lie on given lines, or any or all of them lie instead on given circles passing through the pairs of points through which the adjacent sides pass. In the former case. On any one of the given lines Zj taking arbi- trarily any three points Ai, .B,, C, ; and on the several others L„ L,, i,, &c. L„, taken in the order of the several successive vertices of the polygon, and finally on the original X, itself, constructing successively the several triads of points A„ B^, C,; A^ B^, C,; A^, B„ C^^, &c...A„ B„, C„; and .<[S. 275 A„^i = A,, B„^^ = B^, C„_, = C, happen to result, the problem is solved; if not, the several systems of rays Aj, 5„ C'l, &c. ; A2, S,, C,, &c. ; A^, S^, C3, &c. &c. ; yi„_,, S,„„ C„.,, &c. being all homographic (329), the two double lines Mi = J/„_, and iV, = N,,^.^ of the first and last give the sides through Pj of the two polygons that solve it. N.B. The above reciprocal constructions would evidently remain un- altered, if any line in the former case, or any point in the latter case, were replaced by a circle, containing in the former case the two points, or touching in the latter case the two lines, between which it lies in the order of the several successive vertices or sides of the polygon. •r-2 ( 276 ) CHAPTER XXI. ON THE RELATION OF INVOLUTION BETWEEN HOMOGRAPHIC SYSTEMS. 357. When the axes of two homograpbic rows of points or the vertices of two homograpbic pencils of rays coincide, every point on the common axis or ray through the common vertex belongs of course indiflFerently to both systems, and has in general two diiFerent correspondents, one as belonging to one system, and the other as belonging to the other system ; it some- times happens, however, that these two correspondents always coincide, as appears from the following fundamental theorem : When two homographic rows of points on a common axis, or pencils of rays through a common vertex, are such that any one point on the axis, or ray through the vertex, has the same cor- respondent to whichever system it be regarded as belonging, then every point on the axis, or ray through the vertex, possesses the samt property. Let A,B,C, D, E, F, &c. and A\ B', C, If, E, F, &e. be the two systems ; and let any one point or ray P, denoted by A or B' according to its system, have in both cases the same cor- respondent Q, denoted by A or B according to its system; then every other point or ray R, denoted by G or U according to its system, has in both cases the same correspondent 8, denoted by C or D according to its system. For, the two systems being homographic, [ABCD] = [A'B'G'B'}; but, by hypothesis, A = B'=P, B = A'=Q, C=D' = R, therefore [PQRD} = {QPC'R]=-[PQRG} (280); therefore, at once, I)= C— 8] and therefore &c. The same theorem may also be stated in the somewhat different, but obviously equivalent form, as follows : For two homographic rows of points on a common axis, or pencils of rays through a common vertex, the interchangeability of ON THE RELATION OP INVOLUTION. 277 a single pair of corresponding constituents involves that of every pair. (See ^t. 284). 358. Two homograpblc rows of points on a common axis, or pencils of rays through a common vertex, related as above to each other, that every point on the common axis, or ray through the common vertex, has the same correspondent to whichever system it be regarded as belonging, are said tc be in involution with each other. In the same case, their common axis or vertex is termed the axis or vertex of the involution ; their two double points or rays (341) are termed the double points or rays of the involntion ; and their several pairs of corresponding constituents, from their property of interchangeability, are termed conjugate points or rays of the involution. Every two conjugate groups of two homographic rows or pencils in involution are said also to be in involution with each other, provided they contain at least three points or rays each ; that number of pairs of corresponding constituents of any two homographic systems being requisite (327) to determine the systems. Hence, two triads of corresponding points or rays, having a common axis or vertex, are said to be in involution, when the two homographic rows or pencils they determine are in involution with each other. 359. Two pairs of corresponding constituents are sufficient to determine two homographic rows of points on a common axis, or pencils of rays through a common vertex, when in involution with each other. For, the relation of involution between the two systems requiring (357) that every pair of corresponding consti- tuents should be interchangeable, the interchange of the two con- stituents of either pair, when two are given or known, would supply the third pair necessary and sufficient to determine the systems (327). From the nature of the relation of involution between two homo»raphic rows or pencils (357), it is evident (285) that every two rows in involution on any axis determine two pencils in involution at every vertex, and, conversely, that every two pencils in involution at any vertex determine two rows in invo- lution on every axis. 278 ON THE EELATION OF INVOLUTION 360. The fundamental theorem of Art. 357 applies, of course, as well to two homographic systems of points on a common circle, or of tangents to a common circle, as to two rows of points on a common axis, or pencils of rays through a common vertex ; and two such systems accordingly, or any two conjugate groups of two such systems, containing at least three constituents each, are also said to be in involution with each other under the same circumstances exactly as if the circle were a line in the former case or a point in the latter case. It is evident that systems of points in involution on any circle determine pencils of rays in involution at every point on the circle; and, conversely, that pencils of rays in involu- tion at any vertex determine systems of points in involution on every circle passing through the vertex. Also, that systems of tangents in involution to any circle determine rows of points in involution on every tangent to the circle ; and, conversely, that rows of points in involution on any axis determine systems of tangents In involution to every circle touching the axis. 361. The following are a few fundamental examples of two homographic systems in involution with each other; from which it will be seen that the relation, when existing between two systems otherwise known to be homographic, is generally apparent of itself when the law connecting the several pairs of con-esponding constituents in the generation of the systems is given or known. Ex. 1°. A fixed segment or angle is cat harmonically by a variable pair of conjugates ; the two homographic rows or pencils determined by the two points or lines of section (329) are in involution. For, each point or line of section has in every position the other for its correspondent to whichever system it be regarded as belonging; and therefore (Sro. (358). Ex. 2°. A variable segment or angle has a fixed pair of points or lines of bisection ; the two similar and therefore homographic rows or pencils deter- mined by its two bounding points or lines are in involution. For, each bounding point or line has in every position the other for its correspondent to whichever system it be regarded as belonging; and therefore &c. (358). Ex. 3". Two variable points on a fixed line have a constant product of distances from a fixed point on the line ; the two hotnographie rows they determine on the line (331) are in involution. :between hojiographic systems. 279 For, each variable point has in every position the other for its corres- pondent to whichever system it be regarded as belonging; and therefore &c. (358). Ex. 4°. Tteo rariable lines through a fixed point intersect constantly at right angles ; the two similar and there/ore homographic pencils they deter- mine at the point are in involution. For, each variable line has in every position the other for its corres- pondent to whichever system it be regarded as belonging ; and therefore &c. (358). Ex. 5^. Two variable points on a fixed circle connect constantly through a fixed point; the two homographic systems they determine on the circle (^315) are in involution. For, each variable point has in every position the other for its corres- pondent to whichever system it be regarded as belonging ; and therefore &c. (358). Ex. 6°. Two variable tangents to a fixed circle intersect constantly on a fixed line ; the two homographic systems they determine to the circle (315) are in involution. For, each variable tangent has in every position the other for its corres- pondent to whichever system it be regarded as belonging ; and therefore &c. (358). N.B. To the first of the above examples, which the reader will readily perceive involves the remaining five, it will appear in the sequel that every case of involution between two homographic systems, of points on a common line or circle, or of tangents to a common point or circle, may be reduced. 362. The following additional examples of homographic systems in involution, all reducible to some or other of the preceding, and all of the same class with them, the law con- necting the several pairs of corresponding constituents in their generation being given in all, are left as exercises to the reader. Ex. 1°. A variable circle, passing through two fixed points, determines two systems of points in involution on any fixed line or circle. Ex. 2^. A variable circle, coaxal with two fixed points, determines two systems of points in involution on any fixed line or circle. Ex. 3°. A variable circle, of any coaxal system, determines two systems of points in involution on any fixed line or circle. Ex. 4°. A variable circle, passing through a fixed point and intersecting a fixed line or circle at riglit angles, determines two systems of points in involution on the line or circle, Ex. 5°. A variable circle, intersecting two fixed lines or circles (or a fixed line and circle) at right angles, determines two systems of points iu involution on each line or circle. Ex. 6^ A variable circle, passing through a fixed point and intersecting 280 ON THE RELATION OF INVOLUTION two fixed circles at equal (or supplemental) angles, determines two systems of points in involution on each circle. Ex. 7°. A variable circle, intersecting three fixed circles at equal (or at any invariable combination of equal and supplemental) angles, determines two systems of points in involution on each circle. Ex. 8°. A variable circle, having a fixed pole and polar, determines two systems; o. of points in involution on every line through the pole; a', of rays in involution at every point on the polar. Ex. 9°. Any number of circles, having a common pair of conjugate points or lines (174), determine two systems; a. in the former case, of points in involution on the comiector of the points ; a', in the latter case, of rays in involution at the intersection of the lines. Ex. 10°. Any number of circles, orthogonal to a common circle, deter- mine two rows of points in involution on every diameter of the circle (156). Ex. 11°. Any number of circles, intersecting two common circles at equal (or supplemental) angles, determine two rows of points in involution on every line passing through the external (or internal) centre of perspective of the circles (211, Cor. 1°.) Ex. 12°. Any number of circles, intersecting two common circles at angles whose cosines have any constant ratio, determine two rows of points in involution on every line dividing the interval between their centres in the compound ratio of their radii and of the cosines of the angles. (193, Cor. 1°.) N.B. In the first eight of the above examples, the relation of homo- graphy between the two systems has been already established in Arl. 336, and in the remaining four it follows at once from examples 1° and 4° of the preceding article, by virtue of the properties referred to in their state- ments ; the additional relation of involution between them appears in all from the same consideration (3J8) thai, of every pair of constituents deter- mined by the same circle, each has the other for its correspondent to which- ever system it be regarded as belonging. 363. Any two homograpkic systems of points on a common line or circle, or of tangents to a common point or circle, fnay, if not already in involution, be brought into the particular relative position constituting that relation, by the absolute movement of either, or both, on the common line or circle in the former case, or round the common point or circle in the latter case. For, taking arbitrarily any pair of corresponding constituents A and A' of the two systems, and determining, by Ex. 1°, Art. 353, the pair 5 and B' for which AB = -A'B'; a move- ment of either or both of the systems which would bring A to coincide with B' and B to coincide with A' would then, by the fundamental theorem of Art. 357, place them in involution IfeTWEEN HOMOGBAPHIC SYSTEMS, 281 ■with each other; and that without altering the relative direc- tions of succession of the several constituents of one and of the corresponding constituents of the other. Since, by the example in question (Ex. 1°, Art. 353), for each pair of corresponding constituents A and A of the original systems, there exists, not only a pair B and B' for which AB = - A'B\ but also a pair C and C for which AC=^ + A'C'; a movement of either or both of the systems which would bring A to coincide with C" and C to coincide with A' would also, by the same theorem of Art. 357, place them in involution with each other; but, of course, not without altering the direction of succession of the several constituents of one of them, that of the corresponding constituents of the other remaining un- changed. Hence, For every two homographic systems of points on a common line or circle^ or of tangents to a common point or circle^ there emst two different relative positions of involution with each other ; the relative directions of succession of the several consti- tuents of one and of the corresponding constituents of the other being opposites in the two positions. 364. For every two homographic systems in involution with each other, every three pairs of corresponding constituents deter- mine a system of six points or lines, every four of which are equianharmonic with their four correspondents (see Art. 283). For, every point on the common line or circle, or tangent to the common point or circle, having the same correspondent to whichever system it be regarded as belonging (358), every two conjugate quartets determined by any three pairs of their corresponding constituents (283) are consequently conjugate quartets of the two systems, and as such are of course equi- anharmonic, the systems being homographic. Conversely, When, of two homographic systems of points on a common line or circle, or of tangents to a common point or circle, any two conjugate quartets determined by any three pairs of their corresponding constituents are equianharmonic, the two systems are in involution with each other. For, one of the three pairs of corresponding constituents being necessarily common to the two conjugate quartets (see 282 ON THE RELATION OF INVOLUTION Art. 283), the equianharmonlcism of the latter involves conse- quently the interchangeability of the former, and with it there- fore, by the fundamental theorem of Art. 357, the involution of the systems. CoK. 1°. As three pairs of corresponding constituents, of points on a common line or circle, or of tangents to a common point or circle, determine six different pairs of conjugate quartets (283), it follows, indirectly, from the above con- verse properties combined, that the anharmonic equivalence of any one of the six pairs involves the anharmonic equivalence of each of the remaining five. It was upon this property as basis (which it will be remembered was proved directly for collinear and concurrent systems in Art. 283, and otherwise indirectly for concyclic systems in Art. 313, Cor. 1°) that M. Chasles originally founded the whole theory of Involution ; because that by means of it the relation is generally perceived to exist in cases (many of considerable interest which he was himself the first to investigate) where but three pairs of con- jugates are given. CoK. 2°. It is evident also from the same properties that when any number of pairs of corresponding constituents^ of points on a common line or circle, or of tangents to a common 2>o{nt or circle, form each an involution with the same tico pairs, they form involutions three and three with each other; or, to express the same thing differently, lohen a variable pair of corresponding constituents, of points on a common line or circle, or of tatigents to a common point or circle, form in every position an involution with two fixed pairs, they determine two homographic systems in involution with each other. 365. The following are a few fundamental examples, grouped in reciprocal pairs, of cases of three pairs of corresponding con- stituents satisfying the criterion of the preceding article, and therefore in involution with each other. They were among the first originally given by Chasles, and have been shown to satisfy the criterion in the articles referred to with their state- ments respectively : Ex. a. The three pairs of opposite connectors of every tetrastigm de- termine on every line a system of six points in involution (299, a). BETWEEN HOMOGRAPHIC SYSTEMS. 283 Ex. a'. The three pairs of opposite intersections of every tetragram determine at every point a system of six rays in involution (299, a'), Ex. h. The six parallels through any point to the three pairs of opposite connectors of any tetrastigm form a system of six rays in involution (299, Cor. 2°, a). Ex. 6'. The six projections on any line of the three pairs of opposite intersections of any tetragram form a system of six points in involution (299, Cor. 2°, a'). Ex. e. The three sides of any triangle, and any three concurrent lines through the three vertices, determine on every line a system of six points in involution (299, Cor. 4°, a). Ex. c'. The three vertices of any triangle, and any three collinear points on the three sides, determine at every point a system of six rays in involu- tion (299, Cor. 4°, a'). Ex. d. The three intersections with any line of the three sides of any triangle determine, with the three projections on the line of the three vertices of the triangle, a system of six points in involution (299, Cor. 5°, a). Ex. d'. The three connectors with any point of the three vertices of any triangle determine, with the three parallels through the point to the three sides of the triangle, a system of six rays In involution (299, Cor. 5°, a'.). Ex. e. Every circle, and any two of the three pairs of opposite connectors of any inscribed tetrastigm, determine on every line a system of six points in involution (301, Cor. 2°, a.). Ex. «'. Every circle, and any two of the three pairs of opposite intersec- tions of any exscribed tetragram, subtend at every point a system of six rays in involution (301, Cor. 2°, a'.). Ex.y. Every three pairs of points on a circle which connect by con- current lines form a system of six points in involution (313.). Ex./'. Every three pairs of tangents to a circle which intersect at collinear points form a system of six tangents in involution (313.). Ex. g. Every three pairs of points on a line or circle, harmonic con- jugates to each other with respect to the same two points on the line or circle, form a system of six points in involution (282, Cor. 5°, 3°.). Ex. g'. Every three pairs of tangents to a point or circle, harmonic conjugates to each other with respect to the same two tangents to the point or circle, form a system of six tangents in involution (282, Cor. 5°, 3°.). 366. To the preceding fundamental cases of involution be- tween three pairs of corresponding constituents, several others, involving like them but three pairs of conjugates, are reducible ; the following are some examples, grouped in reciprocal pairs, the reductions of which are left as exercises to the reader. Ex. «. When the directions of three segments are concurrent, the six centres of perspective of their three groups of two determine at every point a system of six rays in involution (295, Cor. 1°.). 284 ON THE RELATION OP INVOLUTION Ex. a'. When the vertices of three angles are collinear, the six axes of perspective of their three groups of two determine on every line a system of six points in involution (295, Cor. 1°.). Ex. h. When the extremities of three segments form an equianharmonic hexastigm, the six centres of perspective of their three groups of two deter- mine at every point a system of six rays in involution (303, d.). Ex. v. When the sides of three angles form an equianharmonic hexa- gram, the six axes of perspective of their three groups of two determine on every line a system of six points in involution (303, d',). Ex. c. When three segments combine the characteristics of examples a and I, the six centres of perspective of their three groups of two are collinear and in involution (295, Cor. 4°.). Ex. c'. When three angles combine the characteristics of examples a' and b', the six axes of perspective of their three groups of two are con- current and in involution ('2.9b, Cor. 4°.). Ex. d. The six centres of perspective, of any three chords inscribed to a circle taken in pairs, determine at every point a system of six rays in invo- lution (.JIT, Cor. 1°.). Ex. d'. The six axes of perspective, of any three angles exscribed to a circle taken in pairs, determine on every line a system of six points in involution (317, Cor. 1°.). Ex. e. When the directions of three chords inscribed to a circle are concurrent, the six centers of perspective of their three groups of two are collinear and in involution (317, Cor. 4°.). Ex. e'. When the vertices of three angles exscribed to a circle are collinear, the six axes of perspective of their three groups of two are con- current and in involution (317, Cor. 4°.). Ex./. When two triangles either inscribed or exscribed to the same circle are in perspective, their three pairs of corresponding sides determine six points in involution on every line through the centre of perspective (320). Ex. /'. When two triangles either exscribed or inscribed to the same circle are in perspective, their three pairs of corresponding vertices deter- mine six rays in involution at every point on the axis of perspective (320^ £x. g. When four circles pass through a common point, the six axes of perspective through the point of their six groups of two form a system of six rays in involution. (See Ex. a' of preceding Art.) Ex. jf'. When four circles touch a common line, the six centres of per- spective on the line of their six groups of two form a system of six points in involution. (See Ex. a of preceding Art.) Ex. h. Every two circles and their two centres of perspective subtend at every point a system of six rays in involution. Ex. h'. Every two circles and their two axes of perspective determine on every line a system of six points in involution. N.B. In the reduction of these two last examples to examples e' and e of the preceding article respectively, it is to be remembered, with respect to any two circles, that the two centres of perspective are a pair of opposite BETWEEN HOMOGRAPHIC SYSTEMS. 285 intersections of the tetragram exscribed to both determined by their four common tangents, and that the two axes of perspective are a pair of opposite connectors of the tetrastlgm inscribed to both determined by their four common points. (See Art. 207.) 367. When two homograpMc rows of points on a common axis, or pencils of rays through a com,mon vertex, are in invo- lution ; every three pairs of corresponding constituents A and A\ S and J5', G and C are connected. a. In the former case, hy the symmetrical relation BA' CR AC CA' • A£' • £C' ~ ' or, which is the same thing, by the equivalent relation £C'.CA'.AB' + B' G. G'A .A'B = 0; a'. In the latter case, by the corresponding relation BiaBA ' ain CB' aln AG' _ smSZ' ■ aiuAB' " ain BG' ~ ' or, which is the same thing, by the equivalent relation siDBG'.siaGA'.aiDAB'+smB'G.siQGA.amA'B=^0; every constituent being interchangeable with its conjugate in each (357). And, conversely, when of two homographic rows of points on a common axis, or pencils of rays through a common vertex, any three pairs of corresponding constituents A and A', B and B\ G and G' are connected by relation (a) in the former case, or by relation («') in the latter case ; the two systems are in involution. For, taking any four of the six points or rays, B, G, A', C" suppose, and equating any one of their six anharmonic ratios, BA'.CG' rr CA'.BC' or siaBA'.siaGG' -^smCA'.smBG' suppose, to the corresponding anharmonic ratio B'A.C'C-ir G'A.B'G or siaB'A.sinG'G-ramG'A.ainB'C of their four correspondents B', G', A, G (364), the relation a or a' immediately results ; from which again, conversely, the anharmonic equivalence of the two conjugate quartets .B, G,A', G' and B", G', A, G (or, from its symmetry, of each of the three pairs of conjugate quartets B, G, A', G' and B', G\ A, G; C, A, B', A' and C", A', B, A; A, B, G', B' and A", B\ C, B) reciprocally results ; and therefore &c. (364). 286 ON THE RELATION OF INVOLUTION The above relation, a or a as the case may be, (or any of the three others of similar form resulting from it by the three interchanges (357) of the three involved pairs of conjugates) being characteristic of the involution of the three pairs of col- linear points or concurrent rays A and A!^ B and J5', C and C", is termed accordingly their equation of involution / and, geome- trically interpreted, it expresses, in a form at once concise and symmetrical, the anharmonic equivalence (364) of every two conjugate quartets of their six constituent points or rays, 368. The following are a few examples of the application of the preceding relation as a criterion of involution between three pairs of coUinear points or concurrent rays; in some of which the equianharmonic relations of Art. 364, previously established on other principles, may be regarded as thus verified at the same time : Ex. 1°. Tlie three intersections toith any line of the three sides of any triangle determine, toith the three projections on the line of the three vertices of the triangle, a system of six points in involution (Ex. d. Art. 365). For, if P, Q, It be the three vertices of the triangle ; A, B, C their three projections on the line ; and A', B' , C the three intersections of the opposite sides with the same ; then, since (Euc. vi. 4.) B£^BQ CB^CR AC^ AP CA' CJt' AB'AP' BC'^BQ' therefore at once, by composition of ratios, BA- CB ' ^C' CA' ■ AB' ' BC ~ ' and therefore &o., by relation (o). Ex. 2°. The three connectors with any point of the three vertices of any triangle determine, toith the three parallels through the point to the three sides of the triangle, a system of six rays in involution (Ex. d', Art. 365.). For, if A, B, C be the three vertices of the triangle ; O the point ; and OA', OB', OC the three parallels through it to the opposite sides; then, since (63) sin BOA' ^ 0£ sinCOB' OA sin AOC _ OB sinCOA' OB' sia AOB'" OC sm BOC'~ OA' therefore at once, by composition of ratios, siaBOA' sin COB' sin .4 OC" sinCO.4' ■ saiAOB' ' sin BOC and therefore &c., by relation (a'). 1 :^TWEEN HOMOGRAPHIC SYSTEMS, 281 Ex. 3°. The three sides of any triangle, and any three concurrent lines through the three vertices, determine on every line a system of six points in involution (Ex. c, Art. 365). For, if P, Q, li be the three vertices of the triangle ; A, S, C their three perspectives on the line from any arbitrary point O; and A', B', C the three intersections of the opposite sides with the same; then, since (134, a) ^— = ^99 9^ _CB ^OR AC^ _ AP OP CA' CR' OR' AB~ AP' OP' lbC''~ BQ'OQ' therefore at once, by composition of ratios, BA^ CB^ AC^ ^ CA' ■ AB' ■ BC ~ ' and therefore &c., by relation (o). Ex. 4^. The three vertices of any triangle, and any three collinear points o/i the three sides, determine at every point a system of six rays in involution (Ex. c. Art. 365). For, if A, B, C be the three vertices of the triangle ; A', B", C the three collinear points on the opposite sides; and O any arbitrary point; then, since (65) sin BO A' ^ BjV BO sin CO.B' _ CB^ CO^ siaAOC _ AC^ AO Bin COA' ~ CA '' CO ' siii^OB' ~ AB^ ' AO ' sin BOC' ~ BC '' BO ' therefore at once, by composition of ratios, sin BOA - sin CO Jg' sin ^ PC BA' CB' AC sinCOA' ' sin AOB' ' sin BOC ^ CA' ' AB' ' BC ' which latter being = 1 (134, a), therefore &c., by relation (a'). Ex. 5°. The six perpendiculars to any line through the three vertices, and through any three collinear points on the three sides, of any triangle determine on the line a system of six points in involution (Ex. b'. Art. 365). For, if P, Q, R be the three vertices of the triangle ; A, B, C their three projections on the line; P', Q', R' the three collinear points on the opposite sides; and A', £', C' their three projections on the line; then, since (Euc. vi. 10.) BA' _ QP^ CB- _ RQ AC _ PR VA'~ MP" AB~ PQ" BC^ QR" therefore at once, by composition of ratios, BA' CB^^ AC _ OF R& PR CA' ' AB' ' BC ~ RP' ' PQ ' QR ' which latter being = 1 (134, a), therefore &c., by relation (o). Ex. 6°. The six perpendiculars through any point to the three sides, and to any three concurrent lines through the three vertices, of any triangle determine at the point a system of six rays in involution (Ex. b, Art. 365). 288 ON TH£ EELATION OP INVOLUTION For, if P, Q,S be the three vertices of the triangle; I the point of concurrence of the three lines passing through them ; O the point through which the six perpendiculars pass; OA, OB, OC the three of them to the three lines IP, IQ, IR; and OA', OB', OC the three of them to the three opposite sides QR, RP, PQ; then since ifi'S) sin BOA' _ IR ainCOB' ^ IP sin A OC ^ IQ BiaCOA' ~ IQ' BinAOB' IR' ainBOC IP' therefore at once, by composition of ratios, sin BOA' sin CO-P' sin-iJOC" _ . sin CO^' ■ an A OB' ' sin BOC ~ ' and therefore &c., by relation (o'). Ex. 7°. When three circles of a coaxal tyttem touch the three sides of a triangle at three points which are either collinear or concurrently coiinectant Kith the opposite vertices ; their three centres form, with those of the three circles of the system which pass through the three vertices of the triangle, a system of six points in involution,* For, if P, Q, R be the three yertices of the triangle ; P', Q', R' the three points of contact on the opposite sides ; A, B, C the centres of the three circles passing through P, Q, R; and A', B', C those of the three touching at P', Q*, .B'; then since (192, Cor. 1°.) PQ" _ AB^ Q^ ^ BC Rr^ ^ CA^ PR"^ AC' QP" BA" RQ'' CB" therefore at once, by composition of ratios, f Q" QR' RP'^ ^ AB' BC a£ PR'' ■ QP" ' RQ'^~ AC' BA' ' CB' '' the former of which being = 1 (134, a or b'), therefore &c., by relation (a). 369. The following additional examples of the application of the same relation, as a criterion of involution between three pairs of collinear points or concurrent rays, are left as exercises to the reader. Ex. 1°. If a segment or angle A A' be cut harmonirally by any two pairs of conjugates B and C, B' and C ; the three pairs of collinear points or concurrent rays A and A', B and B', C'and C are in involution. Ex. 2°. The two pairs of conjugates A and A', B and B' of any harmonic system are in involution with the two harmonic conjugates C and C of every collinear point or concurrent ray with respect to themselves. Ex. 3°. H A, B, C be any three collinear points or concurrent rays, A' any fourth collinear point or concurrent ray, and B', C the two har- * This property was communicated to the Author by Mr. Casey. •ETWEEN HOMOGRAPHIC SYSTEMS. 289 monic conjugates of A with respect to C and A, A and B respectively ; the three pairs A and A', B and B', Cand C are in involution. Ex. 4°. \i A, B, Cbe any three coUinear points or concurrent rays, and A', B', C the three harmonic conjugates of any fourth collinear point or concurrent ray D with respect to B and C, C and A, A and B respectively ; the three triads of pairs £ and B', C and C, A andi>; Cand C, A and^', J? and D ; A and A', B and B', C and D are each in involution. Ex. 5°. If A, B, C be any three coUinear points or concurrent rays, A',B', C any other three collinear or concurrent with them, and A", B", C" the three harmonic conj ugates of A', B', C with respect to B and C, C and A, A and£ respectively; the three triads A', B", C"; A", B', C" ; A",B", C' are each in involution with the triad A, B, C. 370. Every two conjugate points or lines of two homograpTiio systems in involution are harmonic conjugates with respect to the two double points or lines, real or imaginary, of the systems. (See Art. 342). For, since, for any three pairs of conjugates A and A', B and B\ C and C in involution, [ABCC] = [A'B'C'G] (.364) ; li A=A' = M and B = B' = N, which is the characteristic of the two double points or lines (341), then {MNCC'} = {MNG'C}, whatever be the third pair Cand C; and therefore &c. (281). This very simple law connecting the several pairs of cor- responding constituents, in every case of involution between two homographic systems of any common species, would also have followed at once negatively from its converse shewn already (361, Ex. 1°) to result directly from the fundamental definition of involution (358). And, while confirming the state- ment in the note at the close of the same article (361), it evidently comprehends in a form at once simple and complete every other law connected with the subject. 371. The following are immediate consequences from the general property of the preceding article. 1°. In every involution of points on, or tangents to, a common circle. a. The several pairs of conjugate points, in the former case, connect through a common point (257). a. The several pairs of conjugate lines, in the latter case, intersect on a common line (257). J). The two dovhh points, in the former case, lie on the polar of the common point with respect to the circle (165, 6°.). VOL. II. U 290 ON THE RELATION OP INVOLUTION v. The two double lines, in the latter case, pass through the pole of the common line with respect to the circle (165, 6°.). These properties, which shew in fact that every two homo- graphic systems of points on or tangents to a common circle in involution are in perspective, and that the two double points or lines lie on the polar of the centre or pass through the pole of the axis of perspective with respect to the circle, would also have followed at once negatively from their converses shewn already (361, Ex. 5° and 6°) to result immediately from the fundamental definition of involution (358) ; or, they would have followed directly from the second part of the general property of Art. 313, by virtue of the equianharmonic relations of Art. 364. 2°. In every involution of points on a common axis. a. The several circles passing through the several pairs of conjugates, and any common point not on the axis, pass all through a second common point not on the axis (226, 2°.). b. The line connecting the two common points through which they all pass bisects the interval, real or imaginary, between the two double points of the systems (226, 2°.). c. The rectangle under the distances of the several pairs of conjugates from the point of bisection is constant, and equal in magnitude and sign to the square of the semi-interval between the doMe points (225). From the third of these properties (which, like that from which it results (370), would also have followed at once nega- tively from its converse shown already (361, Ex. 3°) to result from the fundamental definition of involution) , it appears that — For every two homographic rows of points in involution on a common axis, there exists a point [always real and evidently con- jugate to that at infinity on the axis), the rectangle under whose distances from the several pairs of conjugates is constant, in mag- nitude and sign, and equal to the square of the semi-interval, real or imaginary, between their two doublepoints. The point possessing this property is termed the centre of the involution ; and the involu- tion itself is said to be positive or negative according as the sign of the constant rectangle is positive or negative, or, which is the same thing, according as the two double points of the systems are real or imaginary. Between homographic systems. 291 COE. 1°. The above properties 1° and 2° supply obvious and rapid solutions ; the former of the following problems — Given two pairs of conjugate points or tangents of two homo- graphic systems in involution on a common circle; to determine, a. the centre or axis of perspective of the systems ; b. the two double points or tangents, real or imaginary, of the systems; c. the conjugate to any third point or tangent of either system ; d. the pair of conjugates having a given middle point or tangent ; e. the two pairs of conjugates intercepting a chord or angle of given magnitude ; f. the two pairs of conjugates determining with two given points or tangents a given anharmonic ratio. Given two pairs of conjugate points or tangents of each of two different involutions on the same common circle ; to determine the pair of conjugates comraon to both involutions. And the latter of the corresponding problems — Given two pairs of conjugates of two homographic rows of points in involution on a common axis; to determine, a. the centre of the involution ; b. the two double points, real or ima- ginary, of the systems ; c. the conjugate to any third point of either system ; d. the pair of conjugates having a given middle point; e. the two pairs of conjugates intercepting a segment of given length; f. the two pairs of conjugates dividing a given segment in a given anharmonic ratio. Given two pairs of conjugates of each of two different involu- tions of points on the same common axis ; to determine the pair of conjugates common to both involutions. The corresponding problems for homographic pencils of rays in involution through a common vertex are not included directly in any of the above ; but they are evidently reducible immediately to those for the two homographic systems of points determined by the pencils on any circle passing through their common vertex (306), or on any line not passing through it (285). CoK. 2°, Since, for two homographic systems of points in involution on a common circle, there exists always one, and in general but one, pair of corresponding constituents diametrically opposite to each other, viz. those determined by the diameter of the circle which passes through their centre of perspective ; and since when two then all pairs are diametrically opposite, the centre of the circle being in that case the centre of perspective U2 292 ON THE RELATION OF INVOLUTION of the systems. Hence, conceiving an arbitrary circle passing through the common vertex of any two homographic pencils of rays in involution, it follows at once (Euc. III. 31) that — For two homograpMc j^encils of rays in involution through a common vertex. a. There exists always one, and in general hut one, pair of conjugate rays which intersect at right angles. b. When two pairs of conjugates intersect at right angles, then all pairs of conjugates intersect at right angles. N.B. These latter properties, which admit also of easy direct demonstration, are often useful in the higher departments of geometry. 372. The property of the centre (371, 2°, c) in the case of two homographic rows of points in involution on a common axis, viz. that the rectangle under its distances from every pair of conjugates is constant in magnitude and sign, follows also immediately from the general property (331) of the two corre- spondents of the point at infinity of any two homographic rows of points on a common axis. For, if P and Q be the two correspondents to the point at infinity on the common axis regarded as belonging first to one and then to the other of the two rows ; since then always, for every pair of corresponding constituents A and A', by the property in question, the rectangle PA.QA' is constant in magnitude and sign ; therefore, when the rows are in involution, and when consequently (358) P= Q= (the point at infinity like every other point on the common axis having the same correspondent to whichever system it be regarded as belonging), the rectangle OA.OA' is constant in magnitude and sign; and therefore &c. This very simple property of involution might have been made the basis of the entire theory; but, as it belongs only to coUinear systems of points on a common axis, that actuaUy employed (358), being applicable alike to involutions of all species without exception, has been adopted in preference. 373. The property (363) that any two homographic rows of points on a common axis, may, if not already in involution, Between homogbaphic systems. 293 be brought in two different ways into tie particular relative position to be so, follows also as an easy consequence from the same general property (331). For, if the two correspondents P and Q of the point at infinity on the common axis, regarded as belonging first to one and then to the other system, do not already coincide; they may first be brought together to a common point by the absolute moment of one or both of the systems along the common axis, thus giving one position of involution (361, Ex. 3°) ; and then, when together, the axis may be turned round the point as centre, carrying with it one system but not the other, and brought again to coincide with its original position in the opposite direction, thus giving another and opposite position of invo- lution (361, Ex. 3°). From this way of regarding the question, it appears that the constant rectangle OA.OA' is the same in magnitude, but opposite in sign, in the two positions of involution of the two rows; hence, the value of that rectangle being of course the constant by which alone any one involution of points on a common axis differs from any other, all such involutions being evidently similar in figure and differing only in magnitude, it appears that — In the two positions of involution of the same two Tiomographic rows of points on a common axis, the two const-ants of involution are always equal in magnitude and opposite in sign. 374. The property of the centre supplies in many cases a very simple criterion of the relation of involution between three or more pairs of corresponding points on a common axis ; as, for instance, in the three following examples : Ex. 1°. Every line passing tkrougli their radical centre intersects with any three circles at three pairs of points in involution. For, if A and A', B and B', C and C he the three pairs of intersec- tions, real or imaginary, and the radical centre ; then, since (183) OA . OA' = OB. OB' = OC. OC, therefore &c. ConTersely, Every line intersecting with any three circles at three pairs of points in involution passes through their radical centre. For, if A and A', B and B", C and C be the three pairs of intersections, and O the centre of their involution ; then, since OA . OA' = OB . OB = OC.OC, therefore &c. (183). 294 ON THE RELATION OF INVOLUTION Ex. 2°. When a number of circles have a common radical centre, every line passing through it intersects with them at as many pairs of points in involution. For, if A and A', B and B', Cand C, D and D', &c. be the several pairs of intersections, real or imaginary, and O the common radical centre ; then since, as before, OA . OA' = OB . OB' = 00. OC = OD. OD' = &c. therefore &c. Conversely, When a number of circles intersect tvith a line at as many pairs of points in involution, they have a common radical centre through which the line passes. For, if A and A', B and B', Cand C, D and JD', &c. be the several pairs of intersections, and O the centre of their involution ; then, since OA . OA' = OB. OB' = OC.OC'= OD. OD' = &c. therefore &c. Ex, 3°. When a number of circles have a common radical axis, every line intersects with them at as many pairs of points in involution. For, if A and A', B and B', C and C, D and D' be the several pairs of intersections, real or imaginary, with the several circles, and O the intersec- tion with the radical axis; then, since (187, 1°) OA.OA'= OB. OB' = OC. OC = OD. OD' = &c. therefore &c. Conversely, When a number of circles intersect with three different lines, which are not concurrent, at as many pairs of points in involution, they have a common radical axis. For, at the centre of each involution they have a common radical centre ; and as the three lines by hypothesis, are not concurrent, two at least of the three centres must necessarily be different ; and therefore &c. (187, 1°).) 375. For any two homographlc systems of points on a common line or circle, or of tangents to a common point or circle, from the general properties of Arts. 364 and 370, it may be shown immediately that — 1°. Every two corresponding pairs of non-corresponding con- stituents A and B\ A' and B are in involution with the two double points or lines M and N, real or imaginary, of the systems. (See Arts. 349 and 371, 2°, a). For, since, for every four pairs of con*esponding constituents A and A', B and B\ C and C', D and Z>', by the homography of the systems, {ABCD] = [A'B'C'D'}; \iC=C' = M and D = D' = N, which is the characteristic of the two double points or lines (341), then {ABMN} = [A'B'MN] = {B'A'NM] (280) ; and therefore &c. (364). Or thus, for either case of the circle, to which the others are of course reducible. The three lines of connection (or points of intersection) AB', A'B, MN being concurrent (or coUincar) BfiTWEEN HOMOGRAPHIC SYSTEMS. 295 (337) ; therefore {ABMN} = [B'A'MN] (313) ; and therefore &c. (364). 2 . IfP^ and Pj, he the two correspondents of any constituent P, regarded as belonging first to erne and then to the other system, and P, the harmonic conjugate of P with respect to P, and P^ ; ihen^ as P varies — a. The two systems determined by P, and P^ are homographic, and have the same double points or lines with the original systems. h. The two systems determined by P and P, are in involution, and have also the same double points or lines with the original systems. c. In every position, P and P, are harmonic conjugates with respect to the two double points or lines of the original systems. Of these properties; the first (a) is evident from the con- sideration that the two systems determined by P, and P^ are homographic with that determined by P and therefore with each other (323), and that when P in the course of its variation coincides with either double point or line M or N of the original systems, its two correspondents P, and Pj coincide with the same double point or line, and therefore with each other; and the second (5) follows immediately (370) from the third (c), which may be proved as follows : The pair of points or lines P, and Pj, the pair M and N, and the coincident pair P and P, being in involution, by the preceding 1°, have therefore a common pair of harmonic con- jugates (370) ; one of which being, of course, the double point or line P, Its harmonic conjugate P, with respect to P, and P^ is therefore its harmonic conjugate with respect to M and N also ; and therefore &c. Or thus, for either case of the circle, to which the others are of course reducible. The three lines of connexion (or points of intersection) PtP^, PP, MN being concurrent (or coUinear) (337) ; and the three lines of connexion (or points of intersection) P^P^i PPi P^P^ being also concurrent (or collinear) (257) ; therefore the three lines of connexion (or points of intersection) PP, P3P,, ilfiV are also concurrent (or coUinear) ; and therefore &c. (257). 296 ON THE RELATION OF INVOLUTION 376. The general property of Art. 370 supplies obvious and rapid solutions of the two following pairs of reciprocal pro- blems, viz. — a. Through a given point to draw a line intersecting two given angles, or circles, so that the point shall he a double point of the involution determined hy the two pairs of intersections. a'. On a given line to find a point sitbtending two given segments, or circles, so that the line shall be a double line of the involution determined hy the two pairs of subtenders. For, since, by the property of that article (370), the two double points (or lines) of the involution in question are har- monic conjugates with respect to the two pairs of intersections (or subtenders), and therefore conjugate points (or lines) with respect to the two angles (or segments) (217), or to the two circles (259, a or al) ; therefore, the two polars of the given point (or poles of the given line) with respect to the two given angles (or segments) (217), or to the two given circles (174), determine, by their point of intersection (or line of connexion), the second double point (or line) of that involution ; and the two double points (or lines) being thus known, their line of connexion (or point of intersection) is of course the required line (or point). N.B. When, in the former case, the given point has the same polar with respect to the two given angles or circles, and when, in the latter case, the g;iven line has the same pole with respect to the two given segments or circles, the above reciprocal constructions become, as they ought, indeterminate ; every line through the given point in the former case, and every point on the given line in the latter case, then evidently satisfy- ing the conditions of the problem. 377. The equianharmonic relations of Art. 364, combined with the general property of Art. 327, reduce also the solutions of the two following reciprocal problems to those of the first parts of the two of Ex. 7°, Art. 353 ; viz. — a. Through a given point to draw a line intersecting with five given lines, so that any two assigned pairs of the five inter- sections shall be in involution with the point and fifth. a'. On a given line to find a point connecting with five given JBETWEEN HOMOGEAPHIC SYSTEMS. 297 points, so that any two assigned pairs of the five connectors shall he in involution urith the line and fifth. For, denoting by C the given point (or line), by A and A', B and B' the two assigned pairs of intersections (or connectors) for any line drawn through (or point taken on) (7, by C" the conjugate of C in the involution determined by A and \A\ B and B' on (or at) that line (or point), by and 0' the vertices of the two angles (or axes of the two segments) determined by the two pairs of the given lines (or points) corresponding to A and J5, A' and B' respectively, and by / the fifth given line (or point) on (or through) which C" is to lie (or pass) in the required involution ; then, whatever be the position of the line drawn through (or of the point taken on) C, since {ABCC} = {B'A'CC} (364), therefore [0. ABCC] ={0'.B'A'CG'} (285) ; and since the three pairs of coiresponding rays (or points) OA and O'B', 05 and O'A', OCand ffC are fixed, therefore the pair OC and O'C, which vary with the position of that line (or point), determine two homographic systems (327) whose two pairs of corresponding constituents intersecting on (or con- necting through) I (353, Ex. 7°, a' or a) determine the two positions of C" whose connectors (or intersections) with C give the two solutions of the problem a (or a). N.B. When the given line (or point) I coincides with one, and the given point (or line) C lies on (or passes through) the other, of the two axes (or centers) of perspective of the two angles (or segments) determined by the two pairs of given lines (or points) corresponding to the two pairs of conjugates A and A', B and B' of the involution ; the two positions of C, given by the above, become, as they ought to be, inde- terminate ; every line passing through (or point lying on) C then evidently determining the required involution. (See ex- amples a and a', Art. 365). Cob. When, in the former problem, the line / is at infinity ; then the point C, being the conjugate of the point at infinity on its axis, is consequently the centre of the involution determined on the required line by the two pairs of intersections A and A', B and B' (372) ; hence by the above are given the two solu- tions, real or imaginary, of the problem. 298 ON THE KELATION OF INVOLUTION Through a given point to draw a line intersecting two given angles, so that the point shall he the centre of the involution determined by the two pairs of intersections. When the two given angles are two opposite angles of a parallelogram, and when the given point is on the diagonal not passing through their vertices, this problem is indeter- minate for the same reason as in the general case ; the point being then evidently (Euc. vi. 16) the centre of the involution for every line passing through it. 378. From the two reciprocal properties of Art. 337, re- specting the directive axis of two homographic rows of points on diflferent axes, and the directive centre of two homographic pencils of rays through different vertices, the two following reciprocal properties of involution, with respect to such systems, may be immediately inferred ; viz. — a. Every line intersecting two homographic pencils of rays through different vertices in two homographic rows of points in involution passes through their directive centre ; and, conversely, every line passing through the directive centre of two homographic pencils of rays through different vertices intersects them in two homographic rows of points in involution. a'. Every point subtending two homographic rows of points on different axes by two homographic pencils of rays in involution lies on their directive axis ; and, conversely, every point lying on the directive axis of two homographic i-ows of points on dif- ferent axes subtends them by two homographic pencils of rays in involution. For, if a line intersect (or a point connect) with . the two systems of rays (or points) in two homographic rows (or by two homographic pencils) in involution, the two correspondents A' and B of every two rays (or points) A and B' which intersect on (or connect through) it, must also intersect on (or connect through) it (358), and therefore &c. (337) ; and, conversely, if a line pass through the directive centre (or a point lie on the directive axis) of the two systems of rays (or points), the two correspondents A' and B of every two rays (or points) A and B' which intersect on (or connect through) it, must also intersect on (or connect through) it (337), and therefore &c. (358). BETWEEN HOMOGEAPHIC SYSTEMS. 299 By virtue of the fundamental theorem of Art. 357 the same results may be arrived at, without the aid of the reciprocal properties of Art. 337, from the consideration that when the point on the line at which it intersects with the ray common to the two pencils (or the line through the point by which it connects with the point common to the two rows) has the same correspondent to whichever of the two rows of intersection (or pencils of connection) it be regarded as belonging (357), the line (or point) itself passes through the intersection (or lies on the connector) of the two correspondents of the common ray (or point) ; and, conversely, when the line (or point) passes through that intersection (or lies on that connector), its two rows of intersection (or pencils of connection) with the two systems of rays (or points) have, in that intersection (or con- nector) and in its own intersection (or connector) with the ray (or point) common to the two systems, a pair of interchange- able correspondents ; and therefore &c. (357). Cor. 1°. When, in the former case, the ray common to the two pencils is at infinity ; that is, when the pencils consist each of parallel lines; their directive centre being then the conjugate to the point at infinity, and therefore the centre, of the involution they determine on every line passing through it (337), It appears consequently, from the above, that— When two homographic pencils consist each of parallel lines, their directive centre is the centre of the involution they determine on every line passing through it. CoE. 2°. The above reciprocal properties supply, in the general case, obvious and rapid solutions of the following re- ciprocal problems ; viz. — a. To draw a tangent to a given point or circle., whose two triads of intersections with two given triads of concurrent lines through different vertices shall he in involution with any assigned correspondence of pairs of constituents. a'. To find a point on a given line or circle, whose two triads of connectors with two given triads of collinear points on different axes shall be in involution with any assigned correspondence of pairs of constituents. 300 ON THE RELATION OF INVOLUTION. 379. The two reciprocal properties of Art. 337, supply abo solutions of the two following reciprocal problems ; viz. — a. Given three pairs of corresponding constituents of two Twmographic pencils of rays through different vertices ; to describe a circle passing through their two vertices, and determining with them two concyclic triads of points in involution, ci. Given three pairs of corresponding constituents of two homographic rows of points on different axes ; to describe a circle touching their two axes., and determining with them two concyclic triads of tangents in involution. For, as the two points (or tangents) P and Q, determined with the required circle by any pair of corresponding rays (or points) A and A' of the two given triads, must, by the properties in question, connect through the directive centre (or intersect on the directive axis) of the two systems, which is given with the two triads (337) ; and as, in addition, the direc- tion of their line of connection (or the sum or difference of the distances from A and A' of their point of intersection) FQ is given, being manifestly the same for every circle passing through the two vertices (or touching the two axes) ; the solution of the problem is therefore evident in the former case, and reducible to that of Art. 54 in the latter case ; and therefore &c. N.B. Since, for every pair of corresponding constituents A and A' of the two homographic pencils (or rows), the line (or point) PQ passes through (or lies on) their directive centre (or axis) 0, when they determine with the circle two systems in involu- tion (337) ; it follows, consequently, that the two determined systems, when in involution, are in perspective. A property which, it will be remembered, was proved for every two con- cyclic systems of points (or tangents) in involution, in 1°, Art. 371. ( 301 ) CHAPTER XXII. METHODS OF GEOMETRICAL TRANSFORMATION. THEORY OF HOMOGRAPHIC FIGURES. 380. Two figures of any kind, F and F\ in which corre- spond, to every point of either a point of the other, to every line of either a line of the other, to every connector of two points of either the connector of the two corresponding points of the other, and to every intersection of two lines of either the intersection of the two corresponding lines of the other, are said to be hmno- grapMc when every two of their corresponding quartets whether of coUinear points or of concurrent lines are equianharmonic. Every two figures in perspective with each other to any centre and axis (141) are evidently thus related to each other (286, 2°). As two anharmonic quartets of any kind, when each equi- anharmonic with a common quartet, are equianharmonic with each other ; it follows at once, from the above definition, that when two figures of any kind F and F" are each homographic with a common figure F, they are homographic with each other. 381. Every two figures i^and F satisfying the four prelimi- nary conditions, whether homographic or not, possess evidently the following properties in relation to each other. 1°. To every coUinear system of points or concurrent system of lines of either, corresponds a coUinear system of points or con- current system of lines of the other. For, every connector of two points (or intersection of two lines) of either corresponding to the connector of the two corre- sponding points (or the intersection of the two corresponding lines) of the other, when, for any system of the points (or lines) of either, every two connect by a common line (or intersect at a common point), then, for the corresponding system of the points (or lines) of the other, every two connect by the corresponding line (or intersect at the corresponding point) ; and therefore, &c. 302 METHODS OF GEOMETRICAL TRANSFORMATION. 2°. To every two collinear systems of points or concurrent systems of lines of either in perspective with each other, correspond two collinear systems of points or concurrent systems of lines of the other in perspective with each other. For, the concurrence (or collinearity) of the several lines of connection (or points of intersection) of the several pairs of corre- sponding constituents of the two systems, for either, involves, by 1°, a similar concurrence of connectors (or collinearity of inter- sections) of pairs of corresponding constituents of the two cor- responding systems, for the other; and therefore &c. (130). 3°. To every two figures of the points and lines of either in perspective with each other, correspond two figures of tlie points and lines of the other in perspective with each other. For, the concurrence of the several lines of connection of the several pairs of con-esponding points, and the collinearity of the several points of intersection of the several pairs of corresponding lines, of the two figures, for either, involve, by 1°, a similar concurrence of connectors and collinearity of intersections of pairs of corresponding constituents of the two corresponding figures, for the other; and therefore &c. (141). 4°. To a variable point moving on a fixed line or a variable line turning round a fixed point of eitlier, corresponds a variable point moving on the corresponding fixed line or a variable line turning round the corresponding fi/xed point of the other. For, since every two positions of the variable point (or line) ■connect by the same fixed line (or intersect at the same fixed point) for the former; therefore by 1°, every two positions of the variable point (or line) connect by the corresponding fixed line (or intersect at the corresponding fixed point) for the latter ; and therefore &c. 5°. To a variable point or line of either the ratio of whose distances from two fixed lines or points is constant, corresponds a variable point or line of the other the ratio of whose distances from the two corresponding fixed lines or points is constant. For, since the variable point (or line) evidently moves on a line concurrent with the two fixed lines (or turns round a point collinear with the two fixed points) for the former ; therefore, by the preceding property 4°, the variable point (or line) moves on a line concurrent with the two corresponding fixed lines (or THEORY OP HOMOGEAPHIC FIGURES. 303 turns round a point collinear with the two corresponding fixed points) for the latter ; and therefore &c. 6 . To a variable polygon of either all whose vertices move onjhxd lines and all whose sides hut one turn round fixed points, or conversely, corresponds a variable polygon of the other all whose vertices move on the corresponding fixed lines and all whose sides hut one turn round the corresponding fixed points, or conversely. For, since, by 4°, to every variable point moving on a fixed line (or variable line turning round a fixed point) of either, corresponds a variable point moving on the corresponding fixed line (or a variable line turning round the corresponding fixed point) of the other ; therefore &c. 7°. To every harmonic row of four points or pencil of four rays of either, corresponds an harmonic row of four points or pencil of four rays of the other. For, as every harmonic row (or pencil) may he regarded as determined by two angles and their two axes of perspective on the connector of their vertices (or by two segments and their two centres of perspective at the intersection of their axes) (241) ; and, as to the vertices and axes of perspective of any two angles (or the axes and centres of perspective of any two segments) of either correspond the vertices or axes of perspective of the two corresponding angles (or the axes and centres of perspective of the two corresponding segments) of the other ; therefore &c. 8°. To every pair of lines or points conjugate to each other toith respect to any segment or angle of either, correspond a pair of lines or points conjugate to each other with respect to the corresponding segment or angle of the other. For, as every two lines (or points) conjugate to each other with respect to any segment (or angle) intersect with the axis of the segment (or connect with the vertex of the angle) at two points (or by two lines) which divide the segment (or angle) harmonically (217) ; therefore &c. by the preceding property 7°. 9°. To every point and line pole and polar to each other with respect to any triangle of either, correspond a point and line pole and polar to each other with respect to the corresponding triangle of the other. For, as every point, and the intersection of its polar with each 304 METHODS OF GEOMETRICAL TKANSFOEMATION. side (or every line, and the connector of its pole with each vertex) of any triangle, are conjugate to each other with respect to the opposite angle (or side) of the triangle (250, Cor. 2°) ; therefore &c. hy the preceding property 8°. 10°. To a variable point or line of either determining with four fixed points or lines an harmonic pencil or row, corresponds a variable point or line of the other determining with the four corresponding fixed points or lines an harmonic pencil or row. For, the harmonicism of the quartet of variable rays (or points), in every position of the variable point (or line) for either, involving, by 7°, the harmonicism of the corresponding quartet of variable rays (or points), in every position of the variable point (or line) for the other ; therefore &c. 11°. To every two equianharmonic rows of four points or pencils of four rays of either, correspond two equianharmonic rows of four points or pencils of four rays of the other. For, as every two equianharmonic rows of four points (or pencils of four rays) may be regarded as determined, on their respective axes (or at their respective vertices), by two quartets of rays (or points) in perspective with each other (290) ; and, as to every two quartets of rays (or points) in perspective for either correspond two quartets of rays (or points) in perspective for the other (property 2° above) ; therefore &c. 12°. To every equianharmonic hexastigm or hexagram of either, corresponds an equianharmonic hexastigm or hexagram of the other. For, the equianharmoniclsm of the two pencils of connection (or rows of intersection) of any two with the remaining four of the six points (or lines) for either hexastigm (or hexagram) involving, by the preceding property 11°, the equianharmoni- clsm of the two corresponding pencils (or rows) for the other hexastigm (or hexagram); therefore &c. (301). The same result follows also from the reciprocal properties of Art. 302, by virtue of the preceding property 1°. 13°. To a variable point or line of either determining with four fixed points or lines a pencil or row having a constant anharmonic ratio, corresponds a variahle point or line of the other determining with the four corresponding fixed points or lines a pencil or row having a constant anharmonic ratio. THftOEY OF HOMOGKAPHIC FIGUKES. 305 For, the equianharmonicism of the two quartets of rays (or points), in every two positions of the variable point (or line) for either, involving, by property 11°, the equianharmonicism of the two corresponding quartets of rays (or points), in every two positions of the variable point (or line) for the other; therefore &c. 14 . To every two homographic rotos of points or pencils of rays of either, correspond two homographic rows of points or pencils of rays of the other. For, the equianharmonicism of every two quartets of cor- responding constituents of the two rows (or pencils) for either involving, by property 11°, the equianharmonicism of every two quartets of corresponding constituents of the two corresponding rows (or pencils) for the other; therefore &c. (321). 15°. To two homographic coaxal rows or concentric pencils of either in involution with each other, correspond two homographic coaasal rows or conc^entric pencils of the other in involution with each other. For, every interchange of corresponding constituents of the two rows (or pencils) for either Involving evidently a corre- sponding interchange of corresponding constituents of the two corresponding rows (or pencils) for the other ; the interchange- ability of every pair of corresponding constituents for either involves consequently the interchangeability of every pair of corresponding constituents for the other; and therefore &c. (357). The same result follows also from the general property of Art. 370, by virtue of the preceding property 7°. 16°. To the double points or rays of any two homographic coaxal rows or concentric pencils of either, correspond the double points or rays of the two corresponding coaxal rows or concentric pencils of the other. For, every coincidence of corresponding constituents of the two rows (or pencils) for either involving evidently a corre- sponding coincidence of corresponding constituents of the two corresponding rows (or pencils) for the other ; the two coincl- dencea, real or imaginary, of pairs of corresponding constituents, which constitute the two double points (or rays) for either, cor- respond consequently to the two coincidences, real or imaginary, of pairs of corresponding constituents, which constitute th6 VOL. II. X 306 METHODS OF GEOMETEICAL TKANSFOEMATION. two douWe points (or rays) for the other; and therefore &c. (341). 17°. To a variable point or line of either connecting or intersecting with two fixed points or lines honnographically^ corresponds a variable point or line of the other connecting or intersecting zoith the two corresponding fixed points or lines homograph ically. For, the equianharmonicism of every two quartets of corre- sponding connectors (or intersections) of the variable with the two fixed points (or lines), for either, involving, by property 1 r, the equianharmonicism of every two quartets of correspond- ing connectors (or intersections) of the variable with the two corresponding fixed points (or lines), for the other; therefore &c. (321). 1 8°. To a variable point or line of either the rectangle under whose distances from two fixed lines or points is constant^ corre- sponds a variable point or line of the other tlie rectangle under whose distances from two {not necessarily corresponding) fixed lines or points is constant. For, the variable line (or point) of the former intersecting (or connecting) with every two fixed positions of itself homo- graphically (340, Cor. 2°) ; and the variable line (or point) of the latter consequently, by the preceding property 17°, inter- secting (or connecting) with every two fixed positions of itself bomographically ; therefore &c. (340, Cor. 1°). 19°. To a variable point or line of either whose angle of connection with two fixed points or chord of intersection with two fixed lines intercepts on a fixed line or subtends at a fixed point a segment or angle of constant magnitude, corresponds a variable point or line of the other whose angle of connection with the two corresponding fixed points or chord of intersection with the two corresponding fixed lines intercepts on a {not necessarily cor- responding) fixed line or subtends at a {not nece,ssarily cor- responding) fixed point a segment or angle of constant mag- nitude. For, the variable line (or point) of the former intersecting (or connecting) with the two fixed lines (or points) bomographically (325, a and a) ; and the variable line (or point) of the latter consequently, by property 17°, intersecting (or connecting) with TBEORT OF HOMOGRAPHIC FIGCKES. 307 the two corresponding fixed lines (or points) homographically ; therefore &c. (339 and 340). 20°. For continuous figures^ all pairs of corresponding points determine pairs of corresponding tangents, and all pairs of cor- responding tangents determine pairs of corresponding points. For, every connector of two points of either corresponding to the connector of the two corresponding points of the other, and every intersection of two lines of either corresponding to the intersection of the two corresponding lines of the other ; and the coincidence of any two points or lines of either involving the coincidence of the two corresponding points or lines of the other; therefore &c. (19 and 20). 382. From the fundamental definition of Art. 380, the fol- lowing general property of homographic figures may be readily inferred; viz. — If A and A\ B and B' be any two fixed pairs of correspond- ing points [or lines) of any two homograpMc figures F and F\ and I and F any variable pair of corresponding lines {or points') of the figures ; then^for every position of I and i', the ratio (4I.^\or its eauivalent (^ ■ -^) \BI' BT) equivalent \^^,j., . ^,^j ts constant, hoih in magnitude and sign. For, if Z and Z' be the two variable points of intersection (or lines of connection) of the two variable lines (or points) I and /' with the two fixed lines (or points) AB and AF respectively ; then, since, by hypothesis, ^and Z' determine two homographic rows (or pencils) of which A and A\ B and B' are two pairs of corresponding constituents (380), therefore, by (828), the ratio (AZ A'Z' \ (am AZ^ sm A'Z' \ \BZ' WZ') *"" VmBZ ' smB'Z'J ' to which, in the corresponding case, the above is manifestly equivalent, is constant both in magnitude and sign; and therefore &c. Cor. 1°. If ^ and B' be the two lines of the two figures whose two correspondents A' and B coincide at infinity ; since then, for every two pairs of corresponding points P and P', X2 308 MKTHODS OF GEOMETRICAL TRANSFORMATION. Q and Q' of the figures, the two ratios PB : QB and FA' : Q'A' each = 1 (15), and since, for all cases, by the above, fPA FA\ _ (QA QA\ [fbtb-J \QB- Q'BT therefore, for the case in question, PA.P'B'= QA.Q'B\ and therefore — For any two homograpliic figures F and F\ if A and B' be the two lines whose two correspondents A' and B coincide at infinity, then, for every pair of corresponding points P and P' of the figures, the rectangle PA.P'B' is constant in magnitude and sign. CoR. 2°. From the simple relation of the preceding corollary, the following properties of any two homographic figures F and F", with respect to their two lines A and B' whose correspon- dents A' and B coincide at infinity, may be immediately in- ferred ; viz. — 1°, Every two corresponding segments PQ and P'Q of any tioo corresponding lines L and U are cut in reciprocal ratios by the two lines A and B' respectively. For, since, by the relation, PA.PB' = QA.Q'B'; therefore, at once, PA: QA= QB' : FB' ; and therefore &c. (Euc. vi. 4). 2°. For a variable pair of corresponding points P and P' on any fixed pair of corresponding lines L and II, if H and K' he the intersections of the latter with A and B' resijectively, the rect- angle HP.KF' is constant in magnitude and sign. For, the two ratios PA : Pif and FB' : P'K' being both constant, by hypothesis, and the rectangle PA.P'B' being con- stant in magnitude and sign, by the relation ; therefore &c. 3°. For every pair of corresponding points P and P', if L and L' be any fixed pair of corresponding lines, the ratio PU ■¥ PA : P'U' -i-P'B' is constant in magnitude and sign. For, since, by the general property of the present article, the two ratios PL-^PA : FL' -i-FA' and PL^ PB : PL ^FB' are constant in magnitude and sign, and since the ratio PB : FA' = 1 ; therefore &c. 4°. To every line L of F parallel to A, corresponds a line L' of F' parallel to B' ; and conversely. For, since, for every two pairs of corresponding points P and F, Q and Q',on any pair of corresponding lines i and X', by THEORY OF HOMOGEAPHIC FIGURES, 309 1 , PA : QA = QB' : P'B' ; consequently when either equiva- .lent = + 1 so is the other also ; and therefore &c. (15). 5°. For every pair of corresponding lines L and L parallel to A and B' respectively^ the rectangle AL.B'L' is constant in magnitude and sign. For, since, for any pair of corresponding points P and P on L and L' respectively, the rectangle PA. P'B' is constant in magnitude and sign, by the relation ; therefore &c. 6°. Every two corresponding lines L and L' parallel to A and B' respectively are divided similarly hy the several pairs of coi-responding points that lie on them. For, since, for any number of pairs of corresponding points P and P', Q and Q\ B and B\ &c. on L and L respectively, if M and M' be any other pair of corresponding lines not parallel to A and B', by 3°, PM' -. PA : QM'' -h QA : BAP -^ BA, &c. = P'3r'---P'B' : g.V- Q'B' : B'M"- : B'B, &c. ; and since, by hypothesis, PA: QA: BA, &c. = P'B' : Q'B' : B'B', &c. = I ; therefore PM' : QAP : BM% &c. = P'M" : Q'W' : B'M'\ &c.; and therefore &c. (Euc. vi. 4). N.B. Of these several results, the second, fourth, and sixth are also evident h priori from the fundamental definition of Art. 380 ; the fourth from (16), from the consideration that to every line L of i^ passing through the point AB, corresponds a line U of F' passing through the corresponding point A'B' ; "■flie sixth form (330), from the consideration that to the point AB on L, corresponds the point A'B" on L'; and the second from (331), from the consideration that to the two points I£ and K' at which L and L' intersect with A and B' respectively, correspond the two H' and K at which L' and L intersect with 'A' and B respectively. CoR. 3°. When the figures are such that a pair of their corresponding lines A and A' coincide at infinity ; then, since for every pair of corresponding points P and P' the ratio PA : P'A' = ], therefore for every other fixed pair of correspond- ing lines B and B', by the above, the ratio PB : P'B' is constant in magnitude and sign, however P and P' vary, and therefore — When two homographic figures F and F' have a pair of corresponding lines A and A' coinciding at infinity, the distance 310 METHODS OF GEOMETEICAL TRANSFORMATION. of a variable point P from any fixed line 5, of either F^ is to the distance of the corresponding variable point P' from the corre- sponding fixed line B', of the other F\ in a ratio constant in magnitude and sign. COE. 4°. From the general property of the preceding corol- lary, the following consequences, respecting two homographic figures F and F' having a pair of corresponding lines A and A' coinciding at infinity, may be readily inferred. 1°. To every two parallel lines L and M of either F, correspond two parallel lines 1! and M' of the other F'. For, since, for every two pairs of corresponding points P and P\ Q and Q on either pair of corresponding lines L and L', by the preceding, PM : P'M' = QM : Q'M' ; consequently, when always PM= QM, then always P'M' =■ QM ; and there- fore &c. 2°. For every two parallel lines L and M of either F having any fixed direction, and for the two corresponding parallel lines L' and M' of the other F^ the ratio LM : L'M' is constant in magnitude and sign. For, since, for a variable pair of corresponding points P and P on either pair of corresponding lines L and L', by the same, the ratio PM : P'M' is then constant in magnitude and sign ; therefore &c. 3°. Every two of their corresponding lines L and L' are divided similarly by their several pairs of corresponding points Pand P', Q and Q, B and B', dsc. For, if M and M' be any other pair of corresponding lines not parallel to L and Z', and and 0' the two points of inter- section LM and L'M' ; then since, by the same, PM : P'M' = QM: Q'M' = BM'. EM', &c. therefore, by Euc. VI. 4, PO '. PO=Q0: Q(y = BO : Eff, &c. ; and therefore &c. 4°. For every two points P and Q of either F whose line of connection is parallel to any fixed direction L, and for the two corresponding points P and Q' of the other F', the ratio PQ : P'Q' is constant in magnitude and sign. For, if M and M' be a pair of corresponding Imes passing through either pair of corresponding points P and P' and parallel to any pair of corresponding fixed directions not coin- ciding with L and L' ; then, since by the same, the ratio THEOET OF HOMOGEAPHIC FIQUEES. 311 MQ'.M'Q is constant in magnitude and sign, therefore, by Euc. VI. 4, so is also the ratio PQ : P'Q ] and therefore &c. 5°. For every three points P, Q, R of either F^ and for the three corresponding points P', Q\ K of the other F\ the area of the triangle PQB is to the area of the triangle P'Q'E in a constant ratio. For, if L and L' be a pair of corresponding lines passing through any pair P and P' of the corresponding points, and parallel to any fixed pair of corresponding directions of the figures ; and S and 8' their pair of Intersections with the pair of opposite sides QR and Q'R' of the triangles ; then since, by the preceding properties 4° and 2°, the ratio PS : P'S' is constant, and the two ratios QL : Q'L' and RL : R'L' are constant and equal, therefore, the difference of the two areas PQS and PRS, or the area PQR (75), is to the diflference of the two areas P'Q'8' and PR'S', or the area P'Q'R' (75), in a constant ratio; and therefore «S;c. 6°. For every system of points P, Q, i?, ;S^, T, &c. of either F, and for the corresponding system of points P', Q', R, S\ T\ &c. of the other F', the area of every polygon determined by the former (108;) is to that of the corresponding polygon determined by the latter (108) in the same constant ratio. For, if PQR ST, &c. and P'Q'R ST, &c. be any pair of corresponding polygons determined by the two systems of points, and and 0' any independent pair of corresponding points of the figures ; then since, by the preceding property 5°, the several triangular areas POQ, QOR, ROS, SOT &c. are to the several corresponding areas PffQ', QO'R, EOS', S'OT &c. in the same constant ratio, therefore the sura of the former, or the area of the polygon PQR ST, &c. (118) is to the sum of the latter, or the area of the polygon P'QR'S'T' &c. (118), in the same constant ratio ; and therefore &c. N.B. Of these several properties, the first and third are also evident ft ^>«brt from (16) and (330), from the obvious conside- ration that when two homographic figures have a pair of cor- respond ing lines coinciding at infinity, then to every point at infinity of either corresponds, by their fundamental definition, a point at infinity of the other. 312 METHODS OF GEOMETRICAL TRANSFORMATION. 383. From the same fundamental definition of Art. 380, it follows, precisely in the same manner as the general property of the preceding article, that — If A andA\ Band B\ Cand C be any three fixed pairs of cor- responding points [or lines) of any two Jiomographic figures F and F\ and I and I' any variable ijair of corresponding lines {or points) of the figures ; then^for every position of I and /', the three ratios fBI BT\ (CI CT\ (AI AI\ \ CI ' Cr) ' \AI ' A'l'J ' \BI • B'i) ' or their three equivalents f BI CI \ / CI AI\ /AI BI\ \B'I' ' C'l'J ' V cr ' at) ' Ut ■ BT'J ' every two of which manifestly involve the third, are constant, both in magnitude and sign. For, as in the preceding article, if X and X', V and Y', Z and Z' be the three pairs of intersections (or connectors) of /and/' with BCandB'C, CA and C'A', AB a.ai A' B' re- spectively ; then since, for the same reason as in the preceding article, the three ratios f BX B'X' \ f smBX amB'X' \ \CX' C'X'J °^ [ainCX'smCX')' fCY C'T\ fsjnCY sinC'r'N [ay'-A'Y'J ""^ KsinAY'- ^S^A^YT /AZ AyZ\ f smAZ BmA'Z '\ \BZ- B'Z') "'■ KsinBZ'fivaB'z}' to which, in the corresponding cases, the above are manifestly equivalent, are constant both in magnitude and sign; there- fore &c. CoR. 1°. The above supplies obvious solutions of the two following problems : Given, of two homographic figures F and F\ three pairs of corresponding points [or lines) A and A\ B and B\ C and C, and a pair of corresponding lines [or points) D and U, to determine the. line [or point) E of either of them F correspond- ing to any assumed line (or point) E' of the other F'. For, since, by the above — THEORY OF HOMOGEAPHIC FIGUKER. 313 BE B'E' _BD B'U CE''~C'E'~~CD'' cTy^ CE CE _CD CD' ae'' A E' ~ Id'' 'ah ^ AE A'E' _ AD AD^ BE ' B'E' ~ BD • WD' ' the three ratios BE : CE, CE : AE, AE : BE, which mani- festly determine the position of the required line (or point) E, are consequently given ; and therefore &c. The pai'ticular cases where the given line (or point) E' is at infinity present no special peculiarity ; the three ratios B'E' : CE', CE' : AE', AE' : B'E' being simply all = 1 in the former case, and having for values sin^'i' : smCL', smCL' : sin^'Z', sin^'i' : sinB'L' respectively, where L' is any line parallel to the direction of E', in the latter case. COE. 2°. As three pairs of corresponding points (or lines) A and A', B and B', C and C of two homographic figures F and F' determine (380) three pairs of corresponding lines (or points) ^Cand^'C, CA and CA, ^J? and ^'5' of the figures; the solutions of the two problems : Given, of two homographic figures F and F', four pairs of corresponding points {or lines) A and A, B and B', C and C, D and D' ; to determine the point [or line) E of either of them F corresponding to any assumed point [or line) E' of the other F' ; may consequently be regarded as included in those of the above ; the particular cases where the given point (or line) E' is at infinity, presenting, as above obsen'ed, no exceptional or special peculiarity. CoE. 3°. It appears also immediately from the above, that •when, for two homographic figures F and F", three pairs of cor- responding points [or lines) A and A, B and B', C and C coin- cide, the coincidence of any independent pair of corresponding lines [or points) D and D' involves the coincidence of every other pair E and E', and therefore of the figures themselves F and F'. For, when, in the three relations of Cor. 1°, which as there shewn result immediately from it, A =A', B=B', C=C', if, in addi- tion, D = D', then necessarily E=E'; and therefore &c. Cor. 4°. For the same reason as in Cor. 2°, it follows of 314 METHODS OF QEOMETBIOAL TEANSFOEMATION. course from the preceding, Cor. 3°, that when, for two homo- graphic figures F and F\four independent pairs of corresponding points [or lines) A and A\ B and B\ G and C\ D and D' coincide, then all pairs of corresponding points (or lines) E and E', and consequently the figures themselves F and F' coincide. Which is also evident h priori from the fundamental charac- teristic of homographic figures (380) that, for every two cor- responding quintets A, B, C, D, E and A', B', C", Z>', E' of their points (or lines), the five relations {A.BCDE} = [A'.B'C'D'E'], {B. CDEA] = {B'.C'D'E'A'}, {C.DEAB] = [C'.D'E'A'B'], [D.EABC] = {D'.E'A'B'C'}, [E.ABCD] = {E'.A'B'C'D} must in all cases exist together; which, when A=A', B=B', C= C, D = iy, would be manifestly impossible unless also E=E' ] and therefore &c. N.B. It will appear in the sequel that, for every pair of homographic figures F and F\ there exists a unique triangle A, whose three elements of either species A, B, C, regarded as be- longing to either figure, coincide, as supposed in the two latter corollaries 3° and 4°, with their three correspondents of the same species A', B', C in the other figure. Of the triangle A, thus related to the two figures i^'and F', two pairs of opposite elements (vertices and sides) may be imaginary, but the third pair are always real. 384. On the converse of the property of the preceding Article, the following general construction for the double gene- ration (26) of a pair of homographic figures, by the simultaneous variation of a pair of connected points, or lines, has been based by Chasles, the originator of the general theory. If A and A\ B and B', C and C be the three pairs of cor- responding sides [or vertices) of any two arbitrary fixed triangles ABC and A'B'C, and I and T a pair of variable points {or lines) so connected that, in every position, any two of the three ratios (BI BT\ (CI CT\ (AI A'r\ \ CI '• Ct) ' \AI '• A T) ' \BI '• WTJ ' THKORT OP HOMOGEAPHIC FIGURES. 315 OfT of their three equivalents \B'i' '■ c'l'j ' V cr • A-r) ' Ut ■• B'l'J ' and with them of course the third, are constant in magnititde and sign ; the two variable points (or lines) I and T generate two homographic figures F and F\ of which A and A\ B and B\ C and C are three pairs of corresponding lines [or points). That the two figures F and F' resulting from either mode of generation are thus homographic, follows of course conversely from the property of the preceding article ; but it may be easily shewn directly that they fulfil all the conditions of connection of the fundamental definition of Art. 380 ; for — 1°. To every point [or line) of either corresponds a point [or line) of the other. This is evident from the law of their genera- tion ; every two points (or lines) / and T connected by the above relations, whether generating pairs or not, thus corresponding with respect to them. 2°. To every line {or point) of either corresponds a line (or point) of the other. For, when a variable point (or line) f of the former is connected, in every position, with the three fixed lines (or points) A^ B, (7 by a relation of the form a.AI+h.BI+c.CI=0 (a), where a, h, c are any three constant multiples ; then, by virtue of the above relations, the corresponding point (or line) T of tlie latter Is connected with the three fixed lines (or points) A', B', C by a corresponding relation of similar form a'.A'T + b'.BT + c'.C'I'=0 (a'), where a', b\ c are three other constant multiples whose ratios to a, i, c respectively depend on and are given with those of the same relations ; but, by the general property of Art. 120 (or 85), the former relation (a) is the condition that the variable point (or line) / should move on a fixed line (or turn round a fixed point) 0, and the latter (a) is the condition that the coiTe- sponding point (or line) T should move on a corresponding fixed line (or turn round a corresponding fixed point) 0' ; and there- fore &c. 3°. To the connector of any two points (or the intersection of 316 METHODS OF GE05IETEICAL TKANSFOEMATION. any two lines) of either, corresponds the connector of the two cor-r responding points {or the intersection of the two corresponding lines) of the other. For, since, to a line passing through any two points (or a point lying on any two lines) of eitlier, corresponds, by the preceding property 2°, a line passing through the two corresponding points (or a point lying on the two corresponding lines) of the other ; therefore &c. 4". To the intersection of any two lines [or the connector oj any two points) of eith r, corresjionds the intersection of the two cor- responding lines {or the connector of the two corresponding points) of the other. For, since, to two lines passing through any point (or two points lying on any line) of either, coirespond, by the same property 2°, two lines passing through the correspond- ing point (or two points lying on the corresponding line) of the other ; therefore &c. 5°. Every two of their corresponding quartets ofcollinear points {or concurrent lines) are equianharmonic. For, the four con- nectors (or intersections) of any quartet 7„ ij,, I^, ^ of the points (or lines) of the former, whether collinear (or concurrent) or not, with any vertex (or side) ^C or (7-4 or ^B of the triangle ABC being (by Cor. Art. 328) equianharmonic with the four connectors (or intersections) of the corresponding quartet /,', 7^', Jj', 7/ of the points (or lines) of the latter with the correspond- ing vertex (or side) B' C or C'A' or AB' of the triangle A'B'C; therefore &c. (285). 6°. Every two of their corresponding quartets of concurrent lines {or collinear points) are equianharmonic. For, the four intersections (or connectors) of any quartet 0,, 0,, 0,,, 0^ of the lines (or points) of the former, whether concurrent (or collinear) or not, with any fifth line (or point) 0^ of the figure being (by the preceding properties 4° and 5°) equianharmonic with the four intersections (or connectors) of the corresponding quartet 0/, Oj', 0^, 0^ of the lines (or points) of the latter with the corresponding fifth line (or point) 0^ of the figure ; therefore .&c. (285). That, for either mode of generation, the three pairs of cor- responding vertices and sides of the two fixed triangles ABG and A'B' C are pairs of corresponding points and lines of the two resulting figures i^'and F\ is evident from the relations of fHEOEY OP HOMOGKAPHIC FIGURES. 317 generation ; from which it follows immediately, in either case, that the evanescence of any one or two of the three distances AI^ £1, CI, for the former, involves necessarily the simultaneous evanescence of the corresponding one or two of the three corre- sponding distances AT, B'T, C'F, for the latter ; and there- fore &c. N.B. When, of the two arbitrary triangles of construction ABC And A'B'C in either of the above modes of generation, the three pairs of corresponding elements A and A', B and B', C and C coincide, the triangle ABC is then, with respect to the two resulting figures F and F', that to which allusion was made in the note at the close of the preceding article (383). 385. From the general constructions of the preceding article the following consequences respecting the liomographic transfor- mation of figures may be readily inferred, viz. — 1°. Any figure F may he transformed homograpliicaUy into another F, in which any four points [or lines), given or taken arbitrarily, shall correspond to any assigned four points [or lines) of the original figure. For, of the four given pairs of corresponding points or lines, any three determine the two fixed triangles of construction ^^Cand A'B'C, and the fourth give the values of the three constant ratios of construction (BI BT\ (CI CT\ (AI A^^ \CI '■ CI') ' \aI • AI) ' \BI ■ B'T) ' and therefore &c. See Cors. 1° and 2°, Art. 283. The obvious conditions, that when, for either of two homo- graphic figures F and F', three points are coUinear or three lines concurrent, then, for the other, the three corresponding points must also be coUinear or the three corresponding lines concurrent, and that when, for either, four points by their coUinearity or four lines by their concurrence form an anhar- monic quartet, then, for the other, the four corresponding points by their coUinearity or the four corresponding lines by their concurrence must form an equianharmonic quartet, are the only restrictions on the perfect generality of the above. The former condition may indeed be violated, but, when it is, it is easy to 318 METHODS OP GEOMETEICAL TRANSFORMATION. see, from the general process of construction, that the figure for which the three points are collinear or the three lines con- current, when their three correspondents in the other are not, must (except for the fourth point or line of the other) have all its points collinear or all its lines concurrent with the three. For, if, in any position of 1 and J', any one, AI suppose, of the six distances AI and A'I\ BI and J5T, CI and G'T be evanes- cent when its correspondent A'T is not, then, in every position of / and /', from the constancy of the three ratios of construc- tion, either the same distance AI, or each of the two non- corresponding distances B'T and G'T is evanescent ; and there- fore &c. See the general remark 2° of Art. 31, an illustration of which is supplied by the above. 2°. In ike Jiomographic transformation of any figure F into another F', the line {or any point) at infinity, regarded as belong- ing to either, may he made to correspond to any assigned line [or point), regarded as belonging to the oilier. This follows at once from the preceding property 1°; the three ratios of construction (BI BJ\ (CI CT\ (AI AT\ V CI ' CI' J ' \Af A'T J ' [bI '• B'lj being given as definitely (see Gors. 1° and 2°, Art. 383) when one of the two given points or lines / and /' is at infinity, as when both are at a finite distance ; and therefore &c. By virtue of the above general property 1°, combined with its particular case 2°, the tetrastigm or tetragram determined by any four points or lines of F may be transformed homo- graphically into another of any simpler or more convenient form for F' ; such, for instance, as the four vertices or sides of a parallelogram of any form, or, more generally, the three vertices or sides of a triangle of any form combined with any remarkable or convenient point or line connected with its figure. By this means, the demonstration of a property, or the solution of a problem, when such property or problem admits of homographic transformation, may frequently be much simplified; as, for instance, in the pairs of reciprocal properties a and a' of Art. 236, a and a' of Art. 245, a and a of Art. 299, the demonstrations of which are comparatively easy (239) when, in the first case of THEORY OP HOMOQKAPHIC FIQUEES. 319 eacli, one of the four lines of the tetragram is at infinity, and when, in the second case of each, one of the four points of the tetrastigm is the polar centre of the triangle determined by the remaining three ; positions into which, if not originally in them, they may at once be thrown by homographic transformation, and so placed in the circumstances most favourable to their estab- lishment. 3°. In the homographic transformation of any figure F into another i*", the correspondents to any assigned five points [or lines) of the original^no three of which are collinear [or concurrent)^ may be made to lie on [or touch) a circle^ given or taken arbitrarily. To prove this, it is only necessary (380) to shew that, for every quintet of points (or lines) A, B, C, Z), U, no three of which are collinear (or concurrent), a corresponding quintet of concyclic points (or tangents) A', B', C, D', E' may be found on (or to) any given circle, satisfying the five conditions {A'.B'C'D'E'] = [A.BCDE], {B'.C'B'E'A'} = {B.CBEA}, {C'.D'E'AB'] = [C.DEAB], {D'.E'A'B'C] = {D.EABC}, [E'.A'B'C'D'] = [E.ABCD] ; which will manifestly be the case if any collinear (or concurrent) quintet A'\ B", C", D", E", can be found satisfying the five corresponding conditions {B"C"D"E"] = [A.BCDE], {C"D"E"A"] = {B.CBEA], [B"E"A"B"} = {G.BEAB], {E"A"B"G"} = [B.EABG}, {A"B" C"B"] = [E.ABCB] ; inasmuch as their five connectors (or intersections) with any arbitraiy point on (or tangent to) the circle, will of course de- termine, by their five second intersections with (or tangents to) the circle, a concyclic quintet of points (or tangents) A\ B', C", B\ E' satisfying the required conditions. And that every two D and E of the five points (or lines) A, B, C, D, E determine such a collinear (or concurrent) quintet with the three intersec- tions (or connectors) of their line of connection (or point of in- tersection) BE with the three sides (or vertices) BG, GA, AB of the triangle determined by the remaining three A, B, C; may be readily shown as follows : Denoting by X, Y, Z the three intersections (or connectors) 320 METHODS OF GEOMETlilCAL TEANSFOKMATION. of the three lines (or points) BC^ CA, AB with the line (or point) DE; then since, immediately, by the general property of Art. 285, {ZYDE} = {A.BCDE], {XZDE} = {B.CADE}, [YXDE] = [G.ABDE]\ and since, from the perspective of A.XYZD or B.XYZD or C.XYZD with D.ABCE, and of A.XYZE or B.XYZE or C.XYZE with E.ABCD, by 4°, Art. 286, {XYZI)] = {D.ABGE} and [XYZE} = {E.ABGI)\; therefore &c. It follows immediately, from this latter property, that every figure, locus of a variable point every six of whose positions form an equianharmonic hexastigm (301, a), or envelope of a variable line every six of whose positions form an equianharmonic hexa- gram (301, a'), mMy be transformed homographically into a circle; for, if transformed, by the above, so that the correspondents to any five of its points (or tangents) shall lie on (or touch) a circle, the correspondent of every sixth point (or tangent) must, by virtue of its connection with the five, lie on (or touch) the same circle (305); and therefore «&c. Thus: 1°. Every figure, locus of a variable point determining in every position an equianharmonic hexastigm with five fixed points, or envelope of a variable line determining in every position an equianharmonic hexagram with five fixed lines (301, a and a!) ; 2°. Every figure, locus of a variable point connecting with four fixed points by four lines, or envelope of a variable line intersecting with four fixed lines at four points, having any constant anhannonic ratio (333, e and «') ; 3°. Every figure, locus of a variable point con- necting homographically with two fixed points, or envelope of a variable line intersecting homographically with two fixed lines (338, Cor. 2°) ; 4°. Every figure, locus of a variable point the rectangle under whose distances form two fixed lines, or envelope of a variable line the rectangle under whose distances form two fixed points, is constant in magnitude and sign (340, Cor. 2°, a and a) ; may be transformed homographically into a circle; and all their properties admitting of homographic transformation, such as their harmonic and anharmonic proper- ties, consequently inferred from the comparatively simple and familiar properties of the circle. See Chapters xv. and xvni. ; all the properties of which, not involving the magnitudes of angles, are consequently true, not only of the circle, but of all the figures above enumerated also. TH?ORY OF HOMOGRAPHIC FIGURES. 321 It follows also, from the same property, that five points [or tangents), given or taken arbitrarily, completely determine any figure homographic to a circle ; for, if transformed, by the above, so that the correspondents of the five points (or tangents) shall lie on (or touch) a circle, all the other points (or tangents) of the figure are then implicitly given, as the correspondents to the several other points (or tangents) of the circle ; and there- fore &c. Given five points {or tangents) A, B, C, D, E of a figure homographic to a circle, the five corresponding tangents [or points) AA, BB, GO, DD, EE of the figure are given implicitly with them ; for, since, for the five corresponding points (or tangents) A', B\ C, jy, E' of the circle, by (306), {A\A'B'C'D'E'] = [B'.A'B'C'B'E'} = {C'.A'B'C'D'E'] = [D'.A'B'C'D'E'} = {E'.A'B'C'D'E], therefore, for the five given points (or tangents) A, B, C, Z>,-E; of the figure, by (380) and (381,20°), [A. ABODE] = \B.ABODE} = {O.ABODE} = {I).ABOI)E}={E. ABODE]; and, since, of each of these five latter homographic pencils (or rows), four rays (or points) are actually given, therefore, of each, the fifth ray (or point) is implicitly given ; and therefore &c. 386. Of the numerous properties of the interesting and important class of figures into which the circle may be trans- formed homographically, the few following, derived on the preceding principles from those of the circle, may be taken as so many examples illustrative of the utility of the process of homo- graphic transformation in modern geometry. Ex. 1°. No figure homographic to a circle could have either three col- linear points or three concurrent tangents. For, if, of a figure homographic to a circle, three points were coUinear, or three tangents concurrent, then, of the circle itself, by (381, 1°), the three corresponding points should be coUinear, or the three corresponding tangents concurrent; and therefore &c. N.B. The only exception to this fundamental property occurs in the cases noticed in connection with property 1° of the preceding article (385), when the figure is in one or other limiting state of its general form, and has either an infinite number of coUinear points lying on one or other of two definite lines, or an infinite number of concurrent tangents passing through one or other of two definite points. See the general remark 2° of Art. 31 ; of which the above and all similar exceptional cases supply so many illustrations. VOL. II. Y 822 METHODS OP GKOMETRICAL TRANSFORMATION. Ex. 2°. No figure homographic to a circle could have either three points at infinity or three parallel tangente. This is manifestly a particular case of the general property of the pre- ceding article ; all points at infinity being collinear, and all parallel lines concurrent (136) ; and therefore &c. N.B. As, in the process of homographic transformation of one figure into another, the correspondent to any line of the original may be thrown to infinity in the transformed figure, (See 2° of the preceding article), a circle will eomequently he transformed homographically into a figure hating two distinct, coincident, or imaginary points at infinity, according as the line whose correspondent is thrown to infinity in the transformation intersects it at two distinct, coincident, or imaginary points (21). Since, in the parti- cular ease of coincidence, the original and transformed lines are tangents to the original and transformed figures (19), a circle may consequently he trans- formed homographically into a figure having a tangent at infinity, by merely throwing to infinity, in the transformation, the correspondent to any tangent to itself. The transformed figure possesses in this latter case, as will be seen in the sequel, some special properties peculiar to the case. Ex. 3°. In every figure homographic to a circle, every three points {or tangents) and the three corresponding tangents {or points) determine two triangles in perspective (140). For, both properties, by examples 3° and 4° of Art. 137, being true of the circle itself, are consequently, by properties 1° and 20° of Art. 381, true of every figure homographic to it ; and therefore &c. N.B. The consequences resulting from this general property applied to the particular cases, when two of the three points are at infinity, the third being arbitrary, and when one of the three tangents is at infinity, the re- maining two being arbitrary, are left as exercises to the reader. Cor. The above, in the general case, supplies obvious solutions of the two following problems : given, of a figure homographic to a circle, any three points (or tangents) and two of the three corresponding tangents {or points), to determine the third corresponding tangent {or point). Ex. 4°. In every figure homographic to a circle, every six points {or tangents) determine an equianharmonie hexastigm {or hexagram) (301). For, both properties, by a and a' of Art. 305, being true of the circle itself, are consequently by 12° of Art. 381, true of every figure homographic to it; and therefore &c. N.B. By virtue of this general property, every system of six points on (or tangents to) any figure homographic to a circle possesses all the pro- perties of a system of six points (or lines) determining an equianharmonie hexastigm (or hexagram). See Arts. 301 to 304. Cor. If, in the above, while any five of the six points (or tangents) are supposed to remain fixed, the sixth be conceived to vary, and in the course of its variation to coincide successively with each of the five that remain TrffcORY OF HOMOGEAPHIC FIGURES. 323 fixed; the theorems of Pascal and Brianchon (302, a and a') applied to the five cases of coincidence, supply ready solutions by linear constructions only, without the aid of the circle, of the two following problems : given of a figure homographic to a circle, any five points (or tangents), to determine the five corresponding tangents {or points). For, of the three collinear in- tersections of pairs of opposite sides (or concurrent connectors of pairs of opposite vertices) of any one of the sixty hexagons determined by the six points (or tangents), in the general position of the variable point (or tangent), that corresponding to the opposite side (or vertex) of the pentagon they determine in any position of coincidence, gives at once the line of connection (or point of intersection) of the two coincident points (or lines), that is (19 and 20) the tangent (or point) corresponding to that position j and therefore &c. See also 3° of the preceding article. Ex. 5°. In every figure homographic to a circle, a variable point {or tangent) determines with every four fixed points {or tangents) a variable quartet of rays {or points) having a constant anharmonic ratio. For, both properties, by a and a' of Art. 306, being true of the circle itself, are consequently, by 13° of Art. 381, true of every figure homo- graphic to it; and therefore &c. Cor. 1°. From the first part of the above, applied to the particular case when two of the four fixed points are the two, real or imaginary, at which, the figure intersects infinity (see Ex. 2°, note) ; it appears at once, as shown already in Art. 312 for the particular case of the circle, that — In every figure homographic to a circle, the angle connecting a variabh with any two fixed points of the figure is cat in a constant anhannonic ratio by the angle connecting it with the two points, real or imaginary, at which the figure intersects infinity. Cob. 2°. From the second part of the same, applied to the particular case when, for a figure having a tangent at infinity (see Ex. 2°, note), one of the four fixed tangents, whatever be the positions of the remaining three, is the tangent at infinity, it follows at once, by virtue of the general property of Art. 275, that — When a figure homographic to a circle has a tangent at infinity, the segment of a variable intercepted by any two fixed tangents is cut in a con- stant ratio by every third fixed tangent to the figure. Cor. 3°. Since, in the same case, by Art. 55, Cor. 3°, 6, the variable circle circumscribing the triangle determined by the variable with any two of the three fixed tangents, in the preceding corollary, passes in every position through a fixed point on the circle circumscribing the triangle deter- mined by the three fixed tangents; it follows consequently from the same, by virtue of the property referred to, that — When a figure homographic to a circle has a tangent at infinity, the variable circle circumscribing the triangle determined by a variable with any two fixed tangents to tlte figure, passes through a fixed point. t2 324 METHODS OF GEOMETRICAL TRANSFORMATIOX. Cor. 4°. From both parts of the above, applied to the case when, for nny figure, the constant anharmonic ratio of the quartet of rays (or points) determined by the four fixed points (or tangents) with the variable fifth point (or tangent) = - 1, in which case the four former are said to form an harmonic system, it follows, precisely as shewn for the circle itself in Art. 311, Cor. 3°, a and a!, that— In every figure homographic to « circle, when four points {or tangents) form an hnrmonic system, the pair of tangents {or paints) corresponding to either pair of conjugates are concurrent (or collinear) with the connector (or intersection) of the other pair. CoK. 5°. From the converses of the two preceding properties, shewn with themselves for the circle in the place above referred to, and also on other principles in Art. 257, it follows immediately, as shewn for the circle itself in Art. 258, that— a. In every figure homographic to a circle, the segment intercepted on a variable by any two fixed tangents is cut harmonically at the corresponding variable point and at its intersection with the connector of the two correspond- ing filed points. a'. In every figure homographic to a circle, the angle subtended at a variable by any two fixed points is cut harmonically by the corresponding variable tangent and by its connector with the intersection of the two corre- sponding fixed tangents. Cor. 6°. From the first of the two preceding properties, applied to the particular case when the two fixed tangents are the two, real or imaginary, whose points of contact are at infinity (see Ex. 2°, note), since then the second point of harmonic sectiuu in every pusition of the variable tangent is at infinity, it follows consequently, by virtue of .3°, Art. 216, that — In every figure homographic to a circle, the segment intercepted on a variable by the two fixed tangents, real or imaginary, whose points of contact are at infinity, is bisected at its point of contact with the figure. N.B. With respect to the numerous inferences from the two very fertile properties of the present example, it may be observed generally that, by virtue of them, all the properties established for the circle in Chapters XV. and XVIII. not involving directly the magnitudes of angles, are true generally of all figures into which circles may be transformed homcgra- phically ; the circumstance that such figures may have real points at infinity (Ex. 2°, note) giving rise sometimes, as in Cors. 2°, 3°, 6° above, to im- portant modifications not occurring, from the absence of that circumstance, in the case of the circle itself. Ex. 6°. In every figure homographic to a circle, a variable point (or tangent) connects {or intersects) homographically with every two fixed points (or tangents). For, both properties, by examples c and if of Art. 325, being true of the circle itself, are consequently, by property 11° of Art. 381, true of every figure homographic to it ; and therefore &c. THEORY OF HOMOGRAPHIC FIGURES. 325 Cor. 1°. From the second part of the above, applied to the case of a figure having a tangent at infinity (Ex. 2°, note), since then, whatever be the positions of the two fixed tangents, their two points at infinity are cor- responding points of the two homographic divisions determined on them by the variable tangent, it follows consequently, by virtue of the general property of Art. 330, that— mien u figure homographic to a circle has a tangent at infinity, a variable tangent divides every two, and therefore all, fixed tangents similarly. Cor. 2°. From the same, applied to the particular case when, for any figure, the two fixed tangents are the two, real or imaginary, whose points of contact are at infinity (Ex. 2°, note), since then their common intersec- tion is the point on each coiresponding to that at infinity on the other, it follows consequently, by virtue of the particular property of Art. 331, Cor. that— In every figure homographic to a circle, a variable determines with the tico fixed tangents, real or imaginary, whose points of contact are at irtfinity, a triangle of constant area. Cor, 3^. From the same again, applied to the case when the two fixed tangents, whatever be their common absolute direction, are parallel, since then the point on each corresponding to that at infinity on the other is its point of contact with the figure, it follows consequently, by virtue of the general property of Art. 331, that — In every figure homographic to a circle, a variable intersects with every two fixed tangents, whose directions are parallel, at two variable points, the rectatigle under whose distances from their two fixed points of contact is constant in magnitude and sign. Cor. 4°. Since, for any two fixed tangents, the point on each correspond- ing to that at infinity on the other, in their homographic division hy a variable tangent, is that of its intersection with the second fixed tangent parallel to the other (Ex. 2^), it follows also from the same, by virtue of the same general property of Art. 331, that — In every figure homographic to a circle, a variable tangent intersects with each pair of adjacent sides of any fixed parallehgram exscribed to the figure, at a pair of variable points, the rectangle under whose distances from their pair of non-conterminous extremities is constant in magnitude and sign. Cor. 3°. From the two parts of the above, by virtue of the two general properties a and a' of Art. 340, it appears that — ci. In every figure Itmnographic to a circle, the angle subtended by a vari- able at any two fixed points of the figure intercepts segments of constant magnitude on each of two fixed lines, and also segments having fixed middle points on each of two other fixed lines. a'. In every figure homographic to a circle, the segment intercepted on a variable hy any two fixed tangents to the figure subtends angles of constant magnitude at each of two fixed points, and also angles having fixed middle lines at each ofixoo other fixed points. 326 METHODS OF GEOMETKICAL TRANSFORMATION. CoE. 6°. And, from both parts, again, by virtue of the two general pro- perties a and a' of Cor. 1° of the same article (340), that — a. For every figure homographic to a circle, there exist two point* (always real) the rectangle under whose distances from a variable tangent to the figure is constant in magnitude and sign. a'. For every figure homographic to a circle, there exist two lines {sometimes imaginary) the rectangle under whose distances from a variable point on the figure is constant in magnitude and sign. From the first of these latter properties it follows, as shewn in Art. 340, Cor. 3°, b, that, in every figure homographic to a circle, the locus of the inter- sections of all pairs of rectangular tangents is a circle, which of course opens out into a line in the particular case where the figure Itas a tangent at irfinity. That the two lines in the second property are the two tangents to the figure whose points of contact are at infinity (Ex. 2°, note), is evident from Cor. 6° of the preceding combined with Cor. 2° of the present article ; and the same may also be shewn directly in a variety of ways. Ex. 7°. When, of any figure homographic to a circle, two variable points {or tangents) connect through {or intersect on) a fixed point {or line), the two corresponding tangents {or points) intersect on {or connect through) a fixed line {or point). For, both properties, by Cor. 3° of Art. 166, being true of the circle itself, are consequently, by 1° and 20° of Art. 381, true of every figure homographic to it ; and therefore &c, N.B. Every point and line related to each other, with respect to any figure homographic to a circle, as in both parts of the above general pro- perty, are said, as in the case of the circle itself, to be pole and polar to each other with respect to the figure ; and the entire nomenclature connected with the subject of poles and polars, as employed in Chapter X. with respect to the circle, being extended in the same manner to every figure possessing the corresponding properties, it follows evidently, for the same reason as above, from the nature of those properties as given in that chapter, and from the fundamental relations of homographic figures as stated in articles 380 and 381, that generally— In every case of tlie transformation of a circle into any figure homo- graphic to it, every point and line pole and polar to each other (165), every two points or lines conjugate to each other (174), every two triangles or other figures reciprocal polars to each other (170), every triangle or other figure reciprocal to itself (170), Sj-c. with respect to the circle, are trans- formed into correspondents of the same nature similarly related to each other with respect ta the figure. CoE. From the second and first parts of the above, applied respectively to the particular cases when the fixed line, or polar, is at infinity, and when the fixed point, or pole, is at infinity in any direction, it follows at once that— THEORY OP HOMOGRAPHIC FIGURES. 327 a. In every figure homoffraphic to a circle, all pairs of points at which the corresponding tangents are parallel connect through a fixed point, the pole with respect to the figure of the line at infinity. a'. In every figure homographic to a circle, all pairs of tangents whose chords of contact are parallel intersect on a fixed line, the polar with re- ject to the figure of the point at infinity in the direction of the chords. N.B. The pole of the line at infinity and the several polars passing through it of the several points at infinity, possesses some remarkable pro- perties with respect to the figure ; the principal of which will appear from the general property of the next example, to which we now proceed. Ex. 8^. When, of any figure homographic to a circle, two variable points [or tangents) connect through {or intersect on) afi^edpoint (or line), the hamtonic conjugate with respect to them of the fixed point (or line) moves on (or turns round) a fixed line (or point). For, both properties, by a and a' of Art. 259, being true of the circle itself, are consequently, by 10^ and 20° of Art. 381, true of every figure homographic to it ; and therefore &c. Cor. From the first part of the above, applied to the particular cases when the fixed line, or polar, is at infinity, and when the fixed point, or pole, is at infinity in any direction, it follows immediately, by virtue of 3°, Art. 216, that— a. In every figure homographic to a circle, every chord of the figure which passes through the pole of the line at infinity is bisected at that point. a'. In every figure homographic to a circle, every chord of the figure whose direction passes through any point at infinity is bisected by the polar of that point. By virtue of these two important properties of every figure homographic to a circle, the pole of the line at infinity, as bisecting every chord passing through it, is termed the centre of the figure, and the polar of every point at infinity, as bisecting every chord parallel to the direction of the point, is termed a diameter of the figure. That every diameter passes through the centre, and thus derives its name, is evident, either generally from the consideration that, as its pole lies on the line at infinity, it consequently passes itself through the pole of that line, or particularly, from the consideration that, as bisecting every chord whose direction passes through its pole, it consequently bisects the particular one which passes through the centre. In the particular case when the figure has a tangent at infinity (Ex. 2°, note), the pole of the line at infinity being then its point of contact with the figure (165, 4°), the centre consequently is at infinity, and every diameter consequently parallel to its direction. In every figure not having a tangent at infinity, every two diameters whose directions pass each through the pole of the other (174) are termed conjugate diameters of the figure; they, evidently, bisect each all chords parallel to the other, pass each through the points of contact of the two tan- genU parallel to the other, and, like all other conjugate lines of the figure 328 METUODS OF GEOMETKICAL TRANSFORMATION. (259), are haimonic conjugates to each other with respect to the two central tangents of the figure, that is (165, 6°) to the two tangents, real or imaginary, whose points of contact are at infinity. The two lines bisecting the two pairs of opposite sides of any inscribed parallelogram, as bisecting each a pair of chords parallel to the other, are consequently a pair of conjugate diameters of the figure ; and so, for the same reason, by virtue of the property a! Cor. 1°, of the preceding example, are the two connecting the two pairs of opposite vertices of any exscribed parallelogram also. In the particular case when the figure is itself a circle, all pairs of diameters intersecting at right angles are evidently pairs of conjugate diameters, and conversely all pairs of conjugate diameters evidently intersect at right angles. Of the diflerent paiis of conjugate diameters, all of which, as just observed, divide harmonically the angle, real or imaginary, determined by the two central tangents to the figure, the particular pair which bisect that angle, externally and internally, and which are consequently at right angles to each other, are termed the axes of the figure ; they evidently bisect each all chords perpendicular to itself, and consequently divide each the entire figure into two similar, equal, and symmetrical halves, reflexions of each other with respect to itself (50). For figures having a tangent at infinity (Ex. 2°, note) two axes also exist, but, for such figures, the centre being at infinity, one of the two axes is consequently also at infinity, and but one therefore at a finite distance. For the ciixle itself, evei-y diameter is evidently an axis. For a given figure homngraphic to a circle the centre and axes may he readily determined as follows ; drawing any two pairs of parallel chords in different directions, the two connectors of their two pairs of middle points are two diameters of the figure which by their intersection consequently determine the centre ; should the centre thus determined be at infinity, any two parallel cliords perpendicular to its direction determine evidently, by their two middle points, the axis not at infinity; and should it not, any circle concentric with it, and intersecting the figure, determines evidently, by its four intersections (see 50), an inscribed rectangle whose two pairs of opposite sides determine, by their two pairs of middle points, the two axes. In the particular case when the figure is itself a circle, the directions of the axes determined by this latter part of the construction become, as they ought, indeterminate. Ex. 9°. Eveiy two triangles reciprocal polars to each other with respect to any figure homographic to a circle are in perspective ; and their centre and axis of perspective are pole and polar to each other with respect to the figure. For, the first part of the property, by 1°, Art. 180, being true of the circle itself, is consequently, by Ex. 7°, note, and 1°, Art. 381, true of every figure homographic to it; and the second part, by virtue of the general property of Art. 167, being evident alike for circle and figure, therefore &c. This general property evidently includes as particular cases those given in Ex. 3° of the present article. Conversely, every two triangles in perspective are, with their centre and '^EORY OF HOMOGEAPHIC FIGURES. 329 axis o/perspeclive, recipfoeal polar$ to each other with respect to a unique Jlffure homographic to a circle, which is implicitly given tehen the triangles them- selves are given. For, if A and A', B and £', C and C (see figs, a and a'. Art. 295) be the three pairs of corresponding vertices (fig. a), or sides (fig. <»')i of the two triangles ; O and I their centre and axis (or axis and centre) of perspective ; U, V, fFthe three points of intersection (or lines of connection) of the three lines (or points) AA', BB', CC with the line (or point) I; B and If, B and E', J" and F' their three pairs of intersec- tions (or connectors) with the three pairs of lines (or points) 3Cand B'C, CA and C'A', AB and A'B'; and, G and G\ H anA H', K ani K' the three pairs of collinear points (or concurrent lines) which divide harmoni- cally the three pairs of segments (or angles) AD" and A'D, BE' and B'B, CF' and C'F (230) ; then, since, from the involution of the three triads of segments or angles AD', AD, and OU; BE', B'E, and OV; CF', C'F, and OW (299, a and a'), and the consequent harmonicism of the three quartets of points (or rays) O, G', O, V; H, H', O, V; K, K', O, W (370), the six points (or lines) G and G', H and H', K and K' determine an equianharmonic hexastigm (or hexagram) (296, Cor. 3°), they are con- sequently six points on (or tangents to) the same figure homographic to a circle (Ex. 5°) ; which figure being determined by any five of them (3°, Art. 385), and being such that the nine pairs of points (or lines) A and ly, B and E; Cand F'; A' and D, B' and E, C and F; O and U, O and V, O and W are pairs of conjugates with respect to it (Ex. 8°), therefore &c. Ex. 10°. Every figure homographic to a circle intersects (or subtends) harmonically the three sides (or angles) of every triangle self-reciprocal with respect to itself; and, conversely, every triangle whose three sides (or angles) are intersected (or subtended) harmonically by any figure homo- graphic to a circle is self-reciprocal with respect to the figure. For, both properties, by a and a' of Art. 259, being true of the circle itself, are consequently, by 10° and 20° of Art. 381, true of every figure homographic to it ; and therefore &c. Given, of a figure homographic to a circle, u self reciprocal triangle ABC, and the position of the centre O, the directions of the two central tangents (real or imaginary) and of the two axes (always real) may be readily deter- mined as fijllows : The three lines OA, OB, OC, connecting the centre with the three vertices, and the three parallels OA', OB', OC through the centre to the three opposite sides, of the triangle, determining (by Ex. 8") three pairs of conjugate diameters OA and OA', OB and OB', OCand OC' of the figure; the two double rays (real or imaginary) Oilfand OiV(370), and the two rectangular rays (always real) O/and 02' 1 371, Cor. 2°), of the involution they determine (368, Ex. 2°), are respectively the two pairs of lines in question (Ex. 8°). In the particular case when O is the polar centre of the triangle ABC(\Q&), the three pairs of lines OA and OA', OB and OB; OCand OC' being rectangular, so therefore are all pairs of conjugates of the involution they determine, and the directions of the axes are conse- quently indeterminate; as they ought, the figure being then a circle (168). 330 METHODS OF GEOMETRICAL TRANSFORMATION. That, in the same case, the figure itself is implicitly given, may also be readily sheten as follows : If P and P', Q and Q', R and R' be its three pairs of intersections with the three diameters OA, OB, OC, and X, Y, Z the three intersections of the latter with the three sides BC, CA, AB of the triangle; then, the three central chords PP", QQ,', RR' being all bisected at O, and cut harmonically at the three pairs of points A and X, B and Y, C and Z respectively (Ex. 8°), therefore (225) OP* = OP'* = OA . OX, OCe= 0Q''= OB.OY, ORr = OR" = OC.OZ; relations which give at once the six points P and P', Q and Q', R and R', and therefore the figure itself (385, 3°). In the particular case when O is the polar centre of the triangle ABC (168), the three rectangles OA.OX, OB.OY, 00. OZ being equal in magnitude and sign ; so therefore are the six semi-diameters OP and OP', OQ and OQ', OR and OR'; as they ought, the figure being then a circle (168). More generally, given, of a figure homographic to a circle, a self -reci- procal triangle, and a point and line pole and polar to each other; the two tangents, real or imaginary, to the figure through the former, and the two intersections, real or imaginary, of the figure with the latter, may be readily determined as follows; It A, B, C be the three vertices (or sides) of the triangle, O and I the point and line (or line and point), and A', B', C the three intersections (or connectors) of the latter with the three opposite sides (or vertices) BC, CA, AB of the triangle; then, the three pairs of lines (or points) OA and OA', OB and Ob', OC and OC being (174 and Ex. 7°) pairs of conjugates with respect to the figure, the two double rays (or points), real or imaginary, 03I and ON of the involution they determine (299) are (370) the two tangents (or points) in question (Ex. 8°), and their two intersections (or connectors) with /, are the two corresponding points (or tangents) (Ex. 7°). That, in the same ease, the figure itself is implicitly given, may also be readily shewn as follows: If G and G', ^and H', JTand K' he its three pairs of intersections with (or tangents through) the thvee lines (or points) OA, OB, OC; X, Y, Z the three intersections (or connectors) of the latter with the three opposite sides BC, CA, AB of the triangle ; and U, V, W their three intersections (or connectors) with the line (or point) /; then, the three pairs of points (or tangents) G and G', ^Tand H', JTand K', dividing, as they do, harmonically the three pairs of segments (or angles) AX and OU, BY and OV, CZ and OW respectively (Ex. 8°), are con- sequently given (230), and with them of course the figure itself (385, 3°). Ex. 11°. In every tetrastigm (or tetragram) determined by four points on (or tangents to) any figure Iwmographic to a circle, the three intersec- tions (or connectors) of the three pairs of opposite connectors [or intersec- tions) determine a self-reciprocal triangle with reipect to the figure. For, both properties, by a and a' of Art. 261, being true of the circle itself, are consequently, by Ex. 7°, note, true of every figure homographic to it ; and therefore &c. N.B. By virtue of the above, all the properties established for the circle •AeOKY op HOMOGRAPHIC FI6UKES. 331 in the several corollaries of the article referred to in their proof (261), with the applications of them given in the two succeeding articles (262 and 263), are seen at once to hold, without modification of any kind, not only for the circle, but for every figure into which the circle may be transformed homo- graphically also. That, for every figure homographie to a circle, the six vertices (or sides) of every two self reciprocal triangles determine an equianliarmonie hexastigm (or hexagram) ; appears, in precisely the same manner as for the circle itself. See Art. 301, Cor. 3°. Ex. 12^. In every figure homographie to a circle, if A and A', B and B', C and C' be the three pairs of opjMsite connectors (or intersections) of the tetrastigm (or tetragram) determined by any four fixed points on (or tangents to) the figure, and I a variable point (or tangent) of the figure ; then, in every position of J, the three rectangles lA. lA', IB. IB', IC.IC are to each other, two and two, in constant ratios. For, since, by Ex. 6°, the variable point (or tangent) /, in the course of its variation, divides homographically the three pairs of angles (or segments) iC'and CB', C4'and^C', AB' and £.4', therefore, by Cor. Art. 328, the three ratios / IB IC\ IIC^ . IA\ tlA^ , IB\ KuT ■ is)' \IA' '' 10')' [iB ■ lA') are constant in magnitude and sign; and therefore &c. N.B. That, for the circle itself, the three rectangles, in the former case are all equal, and in the latter case are proportional to the three OA . OA', OB . OB', OC . OC, where O is the centre of the circle, has been shewn in Cor. 10=, Art. 62, and in 4°, Cor. 2°, Art. 179; from which, of course, the above would have followed also, by virtue of the general relations of con- struction given in Art. 3S4, but by a process on the whole less simple and instructive than that actually employed for their establishment. Cob. From the above, applied to the particular case when two of the four fixed points (or tangents) coincide with the remaining two, since then, of the six lines of connection (or points of intersection) of the tetrastigm (or tetragram) they determine, four necessarily coincide, while the remain- ing two are the two tangents (or points) corresponding to the two coincident pairs of points (or tangents), it follows consequently that — In every figure homographie to a circle, if A and B be any two fixed tangents (or points), C the connector (or intersection) of the two corre- sponding points (or tangents), and I a variable point (or tangent) ; then, in every position of I, the ratio I A. IB : IC is constant in magnitude and sign. N.B. That, for the circle itself, the constant ratio, in the former case = 1, and in the latter case = OA . OB : OC, where O is the centre of the circle, has been shewn in Art. 48, Ex. 9°, and in Art. 179, Cor. 1°; from which, as above observed for the general properties, those of the corollary 332 METHODS OF GEOMETKICAL TRANSFORMATION. itself would have followed by virtue of the general relations of construction given in Art. 384. Ex. 13^ In every figure homographic to a circle, if A, B, C be the three sides {or vertices) of any fixed triangle inscribed {or exscribed) to tlie figure, and I a variable point (or tangent) of the figure ; then, in every position of I, ah c „ — + — + — = 0, lA IB IC ' where a, b, c are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. For, if A', £', C" be the three lines of connection (or points of intersec- tion) of the three fixed vertices (or sides) of the triangle ABC with any fourth fixed point on (or tangent to) the figure ; then, in every position of I, since, by the preceding example, the three rectangles lA . lA', IB . IB', IC. IC are to each other two and two in constant ratios, and since, by Cor. 6° (or 4°) of Art, 82, from the concurrence (or collinearity) of A', B\ C, a'.IAih'.IB' + c'.IC' = 0, where a', b', c' are three multiples, whose ratios to each other two and two are constant in magnitude and sign; therefore &c. N.B. That, for the circle itself, the three constant multiples are propor- tional, in the former case to the three sides a, b, c, and in the latter case to the three differences s - a, s - b, s - c between the semi-perimeter and the three sides, of the triangle ABC; maybe readily seen from Cor. 3°, Art. 64, and from Cor. 1°, Art. 179; or the reader may easily prove the same inde- pendently for himself. Ex. 14°. In every figure homographic to a circle, if A, B, C be the three sides (or vertices) of any fixed triangle exscribed (or inscribed) to the figure, and I a variable point (or tangent) of the figure ; then, in every position of I, , a.IA^ + b.IBUc.IC'' = 0, where a, h, e are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. ¥oi,iS A', B', Che the three sides (or vertices) of the corresponding fixed triangle inscribed (or exscribed) to the figure; then, in every position of J, since, by Ex. 10°, Cor. the three rectangles IB . IC, IC. I A, I A . IB are to the three squares lA", IB'', IC" respectively in constant ratios, and since, by the preceding example, a b' C . lA-'W'lC'-''' where a', V, C are three multiples, whose ratios to each other two and two are constant in magnitude and sign ; therefore &c. N.B. That, for the circle itself, the three constant multiples are propor- tional, in the former case to the three sides a', V, d, and in the latter case ■Aeoet of homographic figures. 333 to the three differences s' - a', s' - V, s' - c' between the semi-perimeter and the three sides, of the triangle A'B'C, appears at Oiice, by virtue of the above demonstration, from the note to the preceding example ; or, ps there observed, the reader may easily prove the same independently for himself. Ex. 15°. In every figure Iwmographic to a circle, if A, B, C be the three vertices (or aides) of any fixed triangle self reciprocal with respect to the figure, and I a variable tangent (or point) of the figure ; then, in every position of I, a.IA' ^b.IB''^c.IC' = where a, b, c are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. For, if ^ and Y be the two fixed points (or tangents) of the figure which are collinear (or concurrent) with any two, A and B, of the three vertices (or sides) of the fixed triangle ABC, and with which they conse- quently determine, by Ex. 10°, the harmonic row (or pencil) A, B, X, Y; then, since, by Ex. 7°, the two corresponding fixed tangents (or points) pass through (or lie on) the third vertex (or side) C, therefore, in every position of I, by Ex. 12°, Cor. the rectangle IX, 71' is to the square of IC in a constant ratio j and it remains only to shew that, in every position of /, the same rectangle is connected with the squares oi lA and IB by a relation of the above form. The four points (or lines) A, B, X, Y being collinear (or concurrent) therefore, by Cors. 4° and 6° of Art. 82, whatever be the position of the line (or point) I, (AB or sin AB) IX = - (BX or sin BX) lA + {AX or sin AX) IB, (AB or &\nAB) IY= - [BY or sin BY) lA + {AY oi sin^I') IB, from which, multiplying, remembering that, from the harmonicism of the system A, B, X, Y, by relations (1) and (1') Art. 219. (^.yor Bin^2r).(.Bror smBY)^{AYaT &mAY).{BXoT smBX) = 0, it follows, immediately, that {AS or sin AB)'. IX . IY= {BX or sin BX ).{BY or sin BY) . lA' + {AX or sin AX ) . (AY oi sin AY) . IB" ; which being, in either case, of the form in question, therefore &c. N.B. For the circle itself, the quantities, to which the three constant multiples a, b, c are proportional, are evidently given for the latter case in the relation of Art. 264, and can from the same, of course, be at once inferred for the former case by Dr. Salmon's Theorem., given in Art. 179. The reader however can have no difiBculty in determining them directly in either case for himself. 387. With the four following general properties of any two homographic figures we shall conclude the present chapter. 1°, For any two homographic fiijures F and F', the two 334 METHODS OF GEOMETRICAL TRANSFORM AT £0N. correspondents 7, and /,, in the two figures^ of a variable imnt or line /, moving according to any law^ generate two homographic figures <7, and G^^ in which all pairs of corresponding elements^ whether points or lines, which coincide with each other are the same as in the original figures. For, the two figures G^ and G.^, generated by the two variable points (or lines) 7, and 7„ being each homographic with the figure G, generated by the variable point (or line) 7 (384), and there- fore homographic with each other (380), therefore &c. as regards the first part ; and since, when any two points (or lines) A^ and A^ of G^ and G^ coincide, then evidently the point (or line) A of G, to which they correspond in the original figures T' and F', coincides with both, therefore &c. as regards the second part. N.B. As, for two homographic rows of points or pencils of rays having a common axis or vertex (341), so, for two homo- graphic figures of any kind F and F, every point or line at or along which a pair of corresponding elements coincide, is termed a double point or line of the figures. It was shewn in Cor. 4°, Art. 383, that, for two homographic figures of any kind, no more than three independent double points or lines could exist unless the figures altogether coincided; and it will be shewn in the next general property (2°) that, for every two homographic figures, three double elements of each species (two of which however may be and often are imaginary) do always exist, and constitute in fact the three elements of each species (vertices and sides) of the same triangle A. 2°. For every two homographic figures F and F, however situated, there exists a triangle [unique or indeterminate) A ; whose three elements of either species [sides or vertices) constitute each a pair of corresponding elements [lines or points) of the ^figures coinciding with each other. For, if Xj, 1\, Z^ and X^, 1^, Z^ be the two triads of colli- near points (or concurrent lines) corresponding in the two figures to the same arbitrary triad of collinear points (or concurrent lines) X, F, Z regarded as belonging first to one and then to the other figure; and A, B, C the three lines (or points) which intersect with the three axes (or connect with the three vertices) of the three homographic rows (or pencils) determined T%EOET OP HOMOGKAPHIC FIGURES. 335 bj the three triads of corresponding points (or rays) X„ JT, X^ ; I^,, Y, Y^ ; ^j, Z^ Z^ at (or by) three triads of corresponding points (or rays) CTj, TJ, U^; F., F, F,; TF„ W'', TF, (Ex. 10°, Art. 353) ; the three lines (or points) A^ B, G, as determining three pairs of corresponding lines (or points) of the figures Z7, U and UU^, F,F and VV^, TF,TFand TFTF, (380) which coincide with each other, determine consequently a triangle A whose three sides (or vertices), and therefore whose three vertices (or sides) also (380), fulfil the conditions in question ; and there- fore &c. N.B. The triangle A, when none of its elements are known, cannot in general be constructed by elementary geometry ; but if one of its sides (or vertices) A be known, the remaining two £ and C, which may be real or imaginary, are given implicitly with them, and may in all cases be determined immediately by the corresponding construction a or a of the problem Ex. 10°, of Art. 353, applied to the arbitrary triad of points (or lines) X, Y, Z and the two X„ F,, Z^ and X^, F^, Z^ determinable from them by Cor. 1° of Art. 383. 3°. When, fir two homograpMc figures F and F, the three double lines [or points) A, S, C are concurrent [or collinear), the figures themselves are in perspective ; and the point cf concurrence [or line of collinearity) is their centre [or axis) of perspective. For, if U and Z7', F and F', IF and W be any three pairs of corresponding points on (or lines through) the three double lines (orpoints)^=^', 5=5', C=C'; and Z and Z', Fand Y\ Z and Z\ &c. any number of other pairs of corresponding points (or lines) of the figures; then since, by 380, being evidently a double point (or line), [0. UVWXYZ &c.} = [O.U'V'W'X'Y'Z' &c.}, and since, by hypothesis, 0U= OU', OV=OV', OTF=OTF', therefore (268) OX=OX', 0Y=0Y; 0Z= OZ', &c.; and therefore &c. (141). N.B. Since, for every two homographic figures F and F, every pair of corresponding lines L and L' parallel to the two parti- cular lines A and B' whose two correspondents .4' and B coincide at infinity are divided similarly by the several pairs of corre- sponding points F and P', Q and Q', R and E, 8 and S', &c. that lie on them (6°, Cor. 2°, Art. 382) ; and since, for every two 336 METHODS OF GEOMETRICAL TRANSFORMATION. figures F and F' in perspective with each other, the several lines of connection PP\ QQ', RB', 88', &c. of all pairs of correspond- ing points P and P', Q and Q\ R and R\ 8 and 8\ &c. are concurrent (141) ; it follows consequently that when two homo- graphic figures F and F' are in perspective, their two lines A and B' whose two correspondents A' and B coincide at infinity are parallel (Euc. VI. 2). In the particular case when the centre of perspective is at infinity, since then all pairs of corresponding points connect by parallel lines, and since consequently all pairs of corresponding lines however situated are divided simi- larly by the several pairs of corresponding points that lie on them, the figures themselves consequently have a pair of corre- sponding lines coinciding at infinity (3°, Cor. 4°, Art. 382). 4°. Every two homographia figures F and F\ which have not a double line at infinity, may he placed, in two pairs of different and opposite positions relatively to each other, so as to he in perspective with each other. For this the following (always possible and determinate) construction, based on the note to the preceding property 3°, has been given by Chasles. On any pair of corresponding lines L and L' of the figures parallel to the two A and B' whose two correspondents A' and B coincide at infinity (4°, Cor. 2°, Art. 382) taking arbitrarily any two pairs of corresponding points P and F, Q and Q' ; drawing through them (by Ex. 1°, Art. 353) either of the two pairs (always real) of corresponding lines PO and P'O', QO and Q'(y for which the two pairs of angles OPQ and O'P'Q, OQPa.nA. O'Q'P are equal in absolute mag- nitude, and for which consequently the pair of coiTesponding triangles POQ and F'O'Q' are similar ; and placing the figures, in either case, in either of the two opposite positions, relatively to each other, in which the pair of corresponding points and 0', and the two pairs of corresponding lines OP and O'P', OQ and 0'^' shall coincide; the four resulting positions thus obtained are positions of perspective. For, since, in each, for all pairs of corresponding points P and F, Q and (^, B and R', 8 and 8', &c. on the pair of (then parallel) corresponding lines L and L' (by 6°, Cor. 2°, Art. 382) PQ :P'Q' = PR: FR' = P8:F8'= &c. therefore (by Euc. VI. 4) their several lines of connection PF, QQ, RR', •THEORY OF HOMOGRAPHIC FIGURES. 337 SS\ &c. all concur to the double point 0=0'; and therefore &c. See note to the preceding property 3°. In the particular case when the two figures F and F' have a double line at infinity, the above construction of Chasles' be- comes, as observed by himself, and as it ought, indeterminate ; tJie fgures havinrj then, in fact, not two [ahcays real), but an infinite number {real or imaginary), of pairs of opposite positions cf perspective icith each other. This is evident in the case of two similar figures, for which, when both right or left, all pairs of similar and opposite positions (33) are positions of perspective, the double line at infinity being for all alike the common axis of perspective (142); and, for any other two figures of the same class, it appears readily from the consideration that, if ani/ two pairs of corresponding points F and P'j Q and Q', not at infinity, can be found for which in absolute magnitude FQ = F Q, the placing of the figures in either of the two positions in which F shall coincide with F, and Q with Q, will place them in a position of perspective, by virtue of the general property 3° of the pi'esent article ; the point at infinity on the double line FQ being then, by hypothesis, a third double point on that line. And that, for every pair of coiTCsponding points F and F of the figures, four different second pairs Q aud Q, on two pairs of corresponding lines (real or imaginary) passing through F and F', can be found satisfying the required condition, may be readily shown as follows : Drawing arbitrarily any two particular pairs of corresponding lines L and L', 21 and M' through any particular pair of corre- sponding points F and F ; then, since, for every other pair of corresponding points Q and Q\ the two ratios QL : QL' and QM : Q'M' are given in magnitude and sign (Cor. 3°, Art. 382), if, in addition, the ratio FQ : F' Q' be also given in absolute mag- nitude (as it is in the case in question), the two pairs of corre- sponding directions (real or imaginary) of the two lines FQ and F Q' (which when superposed, as above described, constitute the axis of perspective of the figures) are manifestly given with it ; and therefore &c. N.B. In every case when two homographic figures of any kind are brought by any means into any position of perspective with each other, it is evident, from the general property of VOL. II. z 338 METHODS OF GEOMETRICAL TEANSFORMATIOJT. Art. 141, that, if either figure be turned through two right angles, either in its plane round the centre or with its plane round the axis of perspective, the other remaining unmoved, it will be in perspective, in its new as well as in its original position, with the other. From this, combined with the property, above established in the general case, that two homographic figures have in general but four different relative positions of perspective with each other, it follows indirectly that if, in any position of perspective of any two homographic figures F and F\ either figure receive both the above movements in succession, the other the while remaining unmoved, its ultimate position as regards the other is independent of the order in which the movements take place. A property of figures in perspective which the reader may easily verify directly for himself. ( 339 ) CHAPTER XXIII. METHODS OP GEOMETRICAL TRANSFORMATION, THEORY OF CORRELATIVE FIGURES. 388. Two figures of any kind, F and F\ in which corre- spond, to every point of either a line of the other, to every line of either a point of the other, to every connector of two points of either the intersection of the two corresponding lines of the other, and to every intersection of two lines of either the con- nector of the two corresponding points of the other, are said to he correlative when every quartet of collinear points or con- current lines of either and the corresponding quartet of concurrent lines or collinear points of the other are equianharmonic. Every two figures reciprocal polars to each other with respect to any circle (170) are evidently thus related to each other (292). As two anharmonic quartets of any kind, when each equi- anharmonic with a common quartet, are equianharmonic with each other; it follows at once, from the above definition, that when two figures of any kind F' and F' are each correlative with a common figure F^ they are homographic icitk each other. (See Art. 380). 389. Every two figures F and F' satisfying the four pre- liminary conditions, whether correlative or not, possess evidently the following properties in relation to each other. 1°, lo every collinear system of points or concurrent system of lines of either , corresponds a concurrent system of lines or colli- near system of points of the other. For, every connector of two points (or intersection of two lines) of either corresponding to the intersection of the two cor- responding lines (or the connector of the two corresponding points) of the other; when, for any system of the points (or lines) of either, every two connect by a common line (or inter- z2 340 METHODS OF GEOMETRICAL TRANSFORMATION. sect at a common point), then, for the corresponding system of the lines (or points) of the other, every two intersect at the cor- responding point (or connect by the corresponding line) ; and therefore &c. 2°. To every two collimar systems of points or concurrent systems of lines of either in perspective with each other ^ correspond two concurrent systems of lines or collinear systems of points of the otlier iH perspective with each other. For, the concurrence (or coUiuearity) of the several lines of connection (or points of intersection) of tlic several pairs of cor- responding constituents of the two systems, for either, involves, by 1°, the coUinearity (or concurrence) of the several points of intersection (or lines of connection) of the several pairs of corre- sponding constituents of the two corresponding systems, for the other ; and therefore &c. (130). 3°. To every two figures of the points and lines of either in perspective with each other ^ correspond two fgures of the lines and points of the other in perspective with each other. For, the concun-ence of the several lines of connection of the several pairs of coi'responding points, and the coUinearity of the several points of intersection of the several pairs of corre- sponding lines, of the two figures, for either, involve, by 1°, the coUinearity of the several points of intersection of the several pairs of conxsponding lines, and the concurrence of the several lines of connection of the several pairs of corresponding points, of the two corresponding figures, for the other; and therefore &c. (Ul). 4°. To a variable point moving on a fixed line or a variable line turning round a fixed point of either., corresponds a variable line turning round the corresponding fixed point or a variable pioint moving on the corresponding fixed line of the, other. For, since every two positions of the variable point (or line) connect by the same fixed line (or intersect at the same fixed point) for the former ; therefore, by 1°, every two positions of the variable line (or point) intersect at the corresponding fixed point (or connect by the corresponding fixed line) for the latter ; and therefore &c- y'. To a variable p)oint or line of either the ratio of whose distances from two fived lines or points is constant, corresponds a THEORY OF CORRELATIVE FIGURES. 341 variable line or point of the other the ratio of whose distances from the two corresponding fixed points or lines is constant. For, since the variable point (or line) evidently moves on a line concurrent -with the two fixed lines (or turns round a point collinear with the two fixed points) for the former; therefore, by the preceding property 4°, the variable line (or point) turns round a point collinear with the two corresponding fixed points (or moves on a line concurrent with the two corresponding fixed lines) for the latter ; and therefore &c. 6°. To a variable polygon of either all ichose vertices move on fixed lines and all whose sides hut one turn round fixed points^ or conversely^ corresponds a variable polygon of the other all whose sides turn round the corresponding fixed points and all whose vertices hut one move on the corresponding fixed lines^ or con- versely. For, since, by 4°, to every variable point moving on a fixed line (or variable line turning round a fixed point) of either, cor- responds a variable line turning round the corresponding fixed point (or a variable point moving on the corresponding fixed line) of the other ; therefore &c. 7°. To every harmonic row of four points or pencil of four rays of either^ corresponds an harmonic pencil of four rays or row of four points of the other. For, as every harmonic row (or pencil) may be regarded as determined by two angles and their two axes of perspective on the connector of their vertices (or by two segments and their two centres of perepective at the intersection of their axes) (241) ; and as, to the vertices and axes of perspective of any two angles (or the axes and centres of perspective of any two segments) of either, correspond the axes and centres of perspec- tive of the tvvo corresponding segments (or the vertices and axes of perspective of the two coi-responding angles) of the other; therefore &c. 8°. To every pair of lines or p)oints conjugate to each other with respect to any segment or angle of either, correspond a pair of points or lines conjugate to each other with respect to the corre- sponding angle or segment of the other. For, as every two lines (or points) conjugate to each other with respect to any segment (or angle) intersect with the axis 342 MLTUODS OF GEOMETRICAL TEANSFORifATION. of the segment (or connect with the vertex of the angle) at two points (or by two lines) which divide the segment (or angle) harmonically (217) ; therefore &c. by the preceding property 7°. 9°. To every point and line pole and polar to each other with respect to any triangle of either ^ correspond a line and point polar and pole to each oilier with respect to the corresponding triangle of the other. For, as every point and the intersection of its polar with each side (or every line and the connector of its pole with each vertex) of any triangle are conjugate to each other with respect to the opposite angle (or side) of the triangle (250, Cor. 2°) ; therefore &c. by the preceding property 8°. 10°. To a variable point or line of either determining with four fixed points or lines an harmonic pencil or row, corresponds a variable line or point of the other determining with the four corre- sponding fixed lines or points an harmonic row or pencil. For, the harmonicism of the quartet of variable rays (or points) in every position of the variable point (or line), for either, involving, by 7°, the harmonicism of the corresponding quartet of variable points (or rays) in every position of the variable line (or point), for the other ; therefore &c. 11°. To every two equianharmonic rows of four points or pencils of four rays of either, correspond two equianharmonic pencils of four rays or rows of four points of the other. For, as every two equianharmonic rows of four points (or pencils of four rays) may be regarded as determined, on their respective axes (or at their respective vertices), by two quartets of rays (or points) in perspective with each other (290) ; and as, to every two quartets of rays (or points) in perspective, for either, correspond two quartets of points (or rays) in perspective, for the other (property 2° above) ; therefore &c. 12°. To every equianharinonic hexastigm or hexagram of either, corresponds an equianharmonic hexagram or hetcastigm of the other. For, the equianharmonicism of the two pencils of connection (or rows of intersection) of any two with the remaining four of the six points (or lines) of the hexastigm (or hexagram) for the former, involving, by the preceding property 11°, the equianhar- monicism of the two corresponding rows (or pencils) of the cor- •theory op cokrelative figures. 343 responding hexagram (or hexastigm) for the latter; therefore &c. (301). The same result follows also from the reciprocal properties of Art. 302, by virtue of the preceding property 1°. 13 . To a variable point or line of either determining with four ^fixed points or lines a pencil or row having a constant anhar- monic ratio, corresponds a variable line or point of the other determining with the four corresponding fxed lines or points a row or pencil having a constant anharmonic ratio. For, the equianharmonicism of the two quartets of rays (or points), in every two positions of the variable point (or line), for the former, involving, by 11°, the equianharmonicism of the two corresponding quartets of points (or rays), in every two positions of the variable line (or point), for the latter; therefore &c. 14 . To every two homographic rows of points or pencils of rays of either, correspond two homographic pencils of rays or roios of points of the other. For, the equianharmonicism of every two quartets of corre- sponding constituents of the two rows (or pencils), for the former, involving, by 11°, the equianharmonicism of every two quartets of corresponding constituents of the two corresponding pencils (or rows), for the latter ; therefore &c. (321). 15°. To two homographic coaxal rows or concentric pencils of either in involution with each other, correspond two homographic concentric pencils or coaxal rows of ike other in involution with each other. For, every interchange of corresponding constituents of the two systems, for either, involving evidently a corresponding in- terchange of corresponding constituents of the two coiTesponding systems, for the other; the interchangeability of every pair of corresponding constituents, for either, involves consequently the interchangeability of every pair of corresponding constituents, for the other ; and therefore &c. (357). The same result follows also from the general property of Art. 370, by virtue of the pre- ceding property 7°. 16°. To the double points or rays of any two homographic coaxal rows or concentric pencils of either, correspond the double rays or points of the two corresponding concentric pencils or coaxal rows of the other. For, every coincidence of corresponding constituents of the 344 METHODS OF GEOMETBICAL TRANSFOEMATION. two systems, for either, involving, evidently, a corresponding coincidence of corresponding constituents of the two correspond- ing systems, for the other ; the two coincidences, real or imagi- nary, of pairs of corresponding constituents, which constitute the two double points (or rays) for the former, correspond, consequently, to the two coincidences, real or imaginary, of pairs of corresponding constituents, which constitute the two double rays (or points) for the latter ; and therefore &c. (341). 1 7°. To a variable point or line of either connecting or inter- secting with two fixed points or lines homographicalli/, corresponds a variable line or point of the other intersecting or connecting with the two corresponding foeed lines or points homographicalli/. For, the equianbarmonicism of every two quartets of corre- sponding connectors (or intersections) of the variable with the two fixed points (or lines), for the former, involving, by 11°, the equianbarmonicism of every two quartets of corresponding intersections (or connectors) of the variable with the two corre- sponding fixed lines (or points), for the latter ; therefore &c. (321).^ 18°. To a variable point or line of either the rectangle under whose distances from two fixed lines or 2wints is constant^ corre- sponds a variable line or point of the other the rectangle under whose distances from two {not iiecessarily corresponding^ fixed points or lines is constant. For, the variable point (or line) of the former connecting (or intersecting) with every two fixed positions of itself homo- graphically (340, Cor. 2°) ; and the variable line (or point) of the latter consequently, by the preceding property 17°, intersect- ing or connecting with every two fixed positions of itself homo- graphically ; therefore &c. (340, Cor. 1°). 1 9°. To a variable point or line of either whose angle of con- nection with two fixed points or chord of intersection with two fixed lines intercepts on a fixed line or subtends at a fixed point a segment or angle of constant magnitude, corresponds a variable line or point of the other whose chord of intersection with tJie two corresponding fixed lines or angle of connection with the two cor- responding fixed points subtends at a {jiot necessarily correspond- ing) fixed point or intercepts on a {not necessarily corresponding) fixed line an angle or segment of constant magnitude. ItUEOEY OF COUJiELATIVE FIGURES. 345 For, the variable point (or line) of the former connecting (or intersecting) with the two fixed points (or lines) homographically (325, a and a) ; and the variable line (or point) of the latter consequently, by 17°, intersecting (or connecting) with the two con-esponding fixed lines (or points) homographically; there- fore &c. (339 and 340). 20°. For continuous firjures^ tlie tangr-nt at any jioint of either corresponds to the point of contact of the corresponding tangent to the other, and the point of contact of any tangent to eitlier corre- sponds to the tangent at the corresponding point of the other. For, every connector of two points of either corresponding to the intersection of the two corresponding lines of the other, and every intersection of two lines of either corresponding to the connector of the two corresponding points of the other ; and the coincidence of any two points or lines of either involving the coincidence of the two corresponding lines or points of the other; therefore &c. (19 and 20). 390. From the fundamental definition of Art. 388, the fol- lowing general property of any two correlative figures F and F' may be readily inferred ; viz. — If A and B he any two fixed j>oints [or lines) of either figure, A! andB' the two corresponding lines {or points) of the other, I any variable line [or point) of the former, and I' the corresponding point [or line) of the latter ; then, for every position of land T, the ratio (AI AT\ . (AIBI\ is constant, both in magnitude and sign. For, if Z be the variable point of intersection (or line of con- nection) of the variable line (or point) / with the fixed line (or point) AB, and Z' the variable line of connection (or point of intersection) of the variable point (or line) T with the fixed point (or line) A'B' ; then, since, by hypothesis, Z and Z' deter- mine a homographic row and pencil (or pencil and row) of which A and A', B and B' are two pairs of corresponding con- stituents (388), therefore, by (328), the ratio fAZ sm A'Z '\ f smAZ AZ'\ \BZ '■ smBZ') °'' KsinBZ ' B'Z'j ' 346 METHODS OF &E0METE1CAL TRANSFORMATION. to which, in the corresponding case, the above is manifestly equivalent, is constant both in magnitude and sign ; and therefore &c. 391. From the same fundamental definition, it follows, pre- cisely iu the same manner as the general property of the pre- ceding article, that, for any two correlative figures F and F, Jf A, Bj Che any three fixed points [or lines) of either f^ure^ A', B\ C the three corresponding lines [or points) of the other, I any variable line [or point) of the former, and T the corresponding point [or line) of the latter ; then, for every position of I and I\ the three ratios fBI BT\ /CI CT\ (AI J//'\ \ CI '■ CI') ' \AI • A'l'J ' [bI '• B'I'J ' or their three equivalents \B'I' • C'l'j ' V Cr ' A'TJ ' [at ' BT'J ' any two of which manifestly involve the third, are constant, both in magnitude and sign. For, as in the preceding article, if A', Y, Z be the three in- tersections (or connectors) of / with BC, CA, AB respectively, and X', Y', Z' the three connectors (or intersections) of T with B'C, C'A, AB' respectively ; then, since, for the same reason as in the preceding article, the three ratios IBX ^mBX\ f smBX B'X'\ \CX'- ^mC'X'j °'' UnCJV' CA-'j' (CY sinC'r '\ (mnCY C'Y' Ur'sin^'FV "'' VmAY- AY'. lAZ imAZ \ f smAZ A'Z'\ [bz' sm B'Z'J ""^ [sin BZ'-Wz)' to which, in the coiTesponding cases, the above are manifestly equivalent, are constant both in magnitude and sign; there- fore &c. Cor. 1°. The above supplies obvious solutions of the two following problems : given, of either of two comlative figures F, three points {or lines) A, B, C and a line [or point) D, and, of the ■ftlEOEY OF CORRELATIVE FIGURES. 347 Other F\ the three corresponding lines [or points) A', B', C and the corresponding point {or line) D ; to determine the line [or point) E of the former F correspionding to any assumed point {or line) E' of the latter F'. For, since, by the above, BE EE' _BD B'D' CE'''G'E'~CD' cTy^ CE C'E' _ CD CD' AE''jJE'~A])''AD' AE AE^_AD ATD^ BE • B'E' ~ BD • WD' ' the three ratios BE : CE, CE : AE, AE : BE, which manifestly determine the position of the required line (or point) E, are consequently given ; and therefore &c. As already observed for homographic figures (Cor. 1°, Art. 383), the particular cases where the given point (or line) E' is at infinity present no special peculiarity; the three ratios B'E' : C'E', CE' : A'E', A'E : BE' having "the values sin^'i' : sinCX', ainCL' : smA'L', sin^'Z' : sin B'L' respectively, where L' is anv line parallel to the direction of E', in the former case, and being simply all = 1, in the latter case. Cor. 2°. As three points (or lines) A, B, C of either of two correlative figures F, and the three corresponding lines (or points) A', B', C of the other F, determine (888) three lines (or points) BC, CA, AB of the former, and the three correspond- ing points (or lines) B' C, CA', AB' of the latter ; the solu- tions of the two problems : given, of either of two correlative figures F, four jpoints {or lines) A, B, C, D, and, of the other F', the four corresponding lines {or points) A', B', C, D' ; to deter- mine the point {or line) E of the former F corresponding to any assumed line [or point) E' of the latter F ; may consequently be regarded as Included in those of the above ; the particular cases where the given line (or point) E' is at infinity, presenting, as above observed, no exceptional or special peculiarity. Cor. 3°. It appears also immediately from the above, that vihen, for two correlative figures F and F', three points {or lines) A, B, C of either F are interchangeable icith the three correspond- ing lines {or points) A', B', C of the other F', the interchange- 348 METHODS OF GEOJtETEICAL TRANSFORMATION. abilitif of any independent line [or 2}o{nt) D of the former with the corresponding point [or line) D' of the latter involves the interchangeahilitij of every other line {or point) E of the former with the corresjjonding point [or line) E' of the latter. For, when, in the three relations of Cor. 1°, which as there shewn result immediately from it, A and A', B and B\ C and C" are inter- changeable, if, in addition, Z> and D' are interchangeable, then necessarily, E and E' are interchangeable ; and therefore &c. COE. 4°. For the same reason as in Cor. 2°, it follows of course from the preceding, Cor. 3°, that when, for two correlative figures F and F\ four independent jwints [or lines) A, B, C, I) of either Fare interchangeable with the four corresponding lines [or points) A', B', C, 1)' of the other F, then every point [or line) E of the former is interchangeable with the corresponding line [or point) E' of the latter. Which is also evident a priori from the fundamental characteristic of correlative figures (388) that, for every quintet A, B, C, I), E of the points (or lines) of either F and the corresponding quintet A', B', C\ i>', E' of the lines (or points) of the other F\ the five relations [A.BCDE] = [A'.B'G'D'E'], {B.CDEA} = {B.C'B'E'A'}, [C.DEAB] = {C'.D'E'A'B'}, [D:EABC] = {D .E' A B' G'\, [E.ABCD] = [E' .A' B' C D'] must in all cases exist together; which, when A and A\ B and B\ C and C\ D and D' are interchangeable, would be manifestly impossible unless also E and E' were in- terchangeable ; and therefore &c. N.B. It will appear in the sequel that, for every pair of cor- relative figures F and F', there exists a unique pair of corre- sponding triangles A and A', for which the three elements of either species A, B, C of either, regarded as belonging to either figure, correspond interchangeably, as supposed in the two latter corollaries 3° and 4°, to the three of the other species A', B', C of the other, regarded as belonging to the other figure ; and of which, as in the corresponding property of homographic figures (Note, Art. 383), though two pairs of corresponding elements may be imaginary, the third pair are always real. When the two triangles A and A', thus related to the two figures 2^ and F', coincide, that is when the three pairs of interchangeable elements A and A\ B and B', G and G' which determine them are the three pairs of opposite elements (vertices and sides) of the same THEORY OP COiiiUELATIVE FIGURES. 349 triangle A ; then, as will appear also in the sequel, all pairs of corresponding elements D and D\ E and E\ &c. of the figures are interchangeable as ■well. 392. On the converse of the property of the preceding article, the following general construction for the double genersition (26) of a pair of correlative figures, by the simultaneous variation of a connected point and line, or line and point, has been based by Chasles, the originator of the genei-al theory. If A^B^ Che the three sides {or vertices) and A\ B\ C the three corresponding vertices (or sides) of any two arbitrary fixed triangles ABC and A' B' C, and I and T a variahle poiiit and line [or line and point) so connected that, in every position, any two of the three ratios (BI BT\ (CI CT\ (AI A'I' \ \ CI • ai'j ' \AI ■AT)' \BI ' B'l;) ' or of their three equivalents (BI CI \ (CI AI\ (AI BI\ \bt' viT \C'I''a'IT [a'I''B'it and with them of course the third, are constant in magnitude and sign ; the variable j)oint and line [or line cmd point) I and I' generate two correlative figures F and F', in which A and A', B and B\ C and C correspond in pairs as line and point [or point and line). That the two figures F and F' resulting from either mode of generation are thus correlative, follows of couree convereely from the property of the preceding article ; but, as in the corre- sponding case of homographic figures (Art. 384), it may be easily shewn directly that they fulfil all the conditions of con- nection of the fundamental definition of Art. 388 ; for — 1°, To every point [or line) of the former corresponds a line [or point) of the latter. This is evident from the law of their generation ; every point and line (or line and point) / and /' connected by the above relations, whether generating pairs or not, thus corresponding with respect to them. 2°. To every line [or point) of the former corresponds a point [or line) ftfthe latter. For, when a variable point (or line) /of 350 METHODS OF GEOMETRICAL TRANSFORMATION. the former is connected, in every position, with the three fixed lines (or points) A, B, Chy a, relation of the form a.AI+b.JBI+c.CI = (a), where a, &, c are any three constant multiples, then, by virtue of the above relations, the corresponding line (or point) T of the latter is connected, in every position, with the three fixed points (or lines) A', B\ C by a corresponding relation of similar form a'.A'I' + b'.B'I' + c'.C'I'=0 (a'), where a, h', c are three other constant multiples whose ratios to o, 6, c respectively depend on and are given with those of the same relations; but, by the general properties of Arts. 120 and 83, the former relation (a) is the condition that the variable point (or line) / should move on a fixed line (or turn round a fixed point) 0, and the latter (a) is the condition that the corre- sponding line (or point) 7' should turn round a corresponding fixed point (or move on a corresponding fixed line) 0'; and therefore &c. 3°, To the connector of any two points [or the intersection of any two lines) of the former corresponds the intersection of the two corresponding lines (or the connector of the two corresponding points) of the latter. For, since, to a line passing through any two points (or a point lying on any two lines) of the former cor- responds, by the preceding property 2°, a point lying on the two corresponding lines (or a line passing through the two cor- responding points) of the latter ; therefore &c. 4°. To the intersection of any two lines {or the connector of any two points) of the former corresponds the connector of the two corresponding points [or the intersection of the two corresponding lines) of the latter. For, since, to two lines passing through any point (or two points lying on any line) of the former, correspond, by the same property 2°, two points lying on the corresponding line (or two lines passing through the corresponding point) of the latter ; therefore &c. 5°. Every quartet of collinear points [or concurrent lines) of the former is equianharmonic with the corresponding quartet of concurrent lines [or collinear points) of the latter. For, the four connectors (or intersections) of any quartet /,, /,, J„ 7^ of the fkEORY OF COERELATIVE FIGURES. 351 points (or lines) of the former, whether coUinear (or concurrent) or not, with any vertex (or side) BC or CA or AB of the triangle ABC being (by Cor. Art. 328) equianharmonic with the four intersections (or connectors) of the corresponding quartet //, 7^', 7,', 7^' of the lines (or points) of the latter with the corresponding side (or vertex) B'C or CA' or A'B' of the triangle A'B'C; therefore &c. (285). 6°. Every quartet of concurrent lines [or coUinear points) of the former is equianharmonic with the corresj)onding quartet of coUinear points [or concurrent lines) of the latter. For, the four intersections (or connectors) of any quartet 0„ 0^, O,, 0^ of the lines (or points) of the former, whether concurrent (or coUinear) or not, with any iifth line (or point) 0. of the figure, being (by the preceding properties 4° and 5°) equianharmonic with the four connectors (or intersections) of the corresponding quartet O,', OJ, 0,', 0^' of the points (or lines) of the latter with the corresponding fifth point (or line) 0^ of the figure; therefore &c. (285). That, for either mode of generation, the three vertices and sides of one correspond respectively to the three corresponding sides and vertices of the other of the two fixed triangles ABC and A'B'C, as point and line and as line and point, in the two resulting figures F and 7", is evident from the relations of generation ; from which, as in the corresponding case of homo- graphic figures (Art. 384), it follows immediately, in either case, that the evanescence of any one or two of the three distances AIj BI, CI, for the former, involves necessarily the simultaneous evanescence of the corresponding one or two of the three corre- sponding distances A'l', B'F, CI', for the latter ; and there- fore &c. N.B. When, of the two arbitrary triangles of construction ABC a.ni A'B'C in either of the above modes of generation, the three pairs of corresponding elements ^1 and A', B and B', C and C are the three pairs of opposite elements (vertices and sides) of a common triangle A ; the triangle A is then, with respect to the two resulting figures F and F', that to which allusion was made in the note at the close of the preceding article (391). 352 METHODS OF GEOMETRICAL TRANSFORMATION. 393. From the general constnjctions of tbe preceding article the following consequences respecting the correlative transforma- tion of figure,^ may be immediately inferred, viz. — 1°. Any fi-gure F may be transformed correlatively into another F'^in which any four lines {or points)^ given or taken arbitrarily, shall correspond to any assigned four points [or lines) of the original figure. For, of the four given pairs of coiTCsponding points and lines or lines and points, any three determine the two fixed triangles of construction ABC and AB'G\ and the fourth gives the values of the three constant ratios of construction (BI BT\ (CI C'J'\ (AI A^\ KcrC'lT \Af AIT \Bf B'lT and therefore &c. See Cors. 1° and 2°, Art. 391, The obvious conditions, that when, for either of two correla- tive figures F and i^', three points are coUinear or three lines concurrent, then, for the other, the three corresponding lines must be concurrent or the three corresponding points coUinear, and that when, for either, four points by their coUinearity or four lines by their concurrence form an anharmonic quartet, then, for the other, the four corresponding lines by their concurrence or the four corresponding points by their coUinearity must form an equi- anharmonic quartet, are the only restrictions on the perfect generality of the above. The former condition may indeed be violated, but, when it is, as in the corresponding case for homo- graphic figures (385, ]°), it is easy to see, from the general process of construction, that the figure for which the three points are coUinear, or the three lines concurrent, when their three corre- spondents in the other are not concurrent, or coUinear, must (except for the fourth line or point of the other) have all its points coUinear, or all its lines concurrent, with the three. For, if, in any position of / and 7', any one, AI suppose, of the six distances .47 and AI\ 7?7and B'I\ (77 and CT be evanescent when Its correspondent A'T is not, then in every position of 7 and i', from the constancy of the three ratios of construction, either the same distance AI, or each of the two non-correspond- ing distances B'T and CT, is evanescent; and therefore &c. iiee the general remark 2° of Art. 31, an Illustration of which, 'ftlEORY OF CORRELATIVE FIGURES. 353 as in the corresponding case of homographic figures above referred to, is supplied by the above. 2 . In the correlative transformation of any figure F into another F", the line [or any iwint) at infinity, regarded as belong- ing to either, may be mnde to corresjjond to any assigned point [or line), regarded as belonging to the other. This follows at once from the preceding property 1°; the three ratios of construction (BI B'I' \ /CI CT\ (AI AI\ [or CI) ' \Af at) ' [bi • B'r) being given as definitely (see Cors. Tand 2°, Art. 391) when, of the given point and line, or line and point, / and T, one is at infinity, as when both are at a finite distance ; and therefore &c. By virtue of the above general property 1°, combined with its particular case 2°, the tetrastigm or tetragram determined by any four points or lines of F may be transformed correlatively into a tetragram or tetrastigm of any arbitrary or convenient form for F ; such for instance (see 2°, Art. 385) as the four sides or vertices of a parallelogram of any form, or, more generally, the three sides or vertices of a triangle of any form, combined with any remarkable or convenient line or point connected with its figure. By this means, as in the corresponding case for homographic figures (2°, Art. 385), the demonstration of a pro- perty, or the solution of a problem, wlien such property or problem admits of correlative transformation, may frequently be much simplified ; as, for instance, in the three pairs of reciprocal properties there referred to (2°, Art. 385), whose direct demon- strations are comparatively easy under the circumstances there stated, and which under any other circumstances may be trans- formed correlatively, each into the other, and brought by the transformation under the circumstances most favourable to their establishment. 3°. In the correlative transformation of atiy figure F into another F', the correspondents to any assigned five points [or lines) of the original, no three of which are collinear [or concurrent), may be mude to touch [or lie on) a circle, given or taken arbitrarily. From this property, which may be proved in precisely the same manner as the corresponding property of homographic figures VOL. II. A A 354 METHODS OF GEOMETRICAL TRANSFOKMATION. given in 3°, Art. 385, it follows immediately that every Jlgure^ locue of a variable point every six of xchose positions form an equianhar- monic hexastigm (301, a), or envelope of a variable line every six of whose positions form an equianharmonic hexagram (301, a'), may be transformed correlafively into a circle ; for, if transformed, by the above, so that the correspondents to any five of its points (or tangents) shall touch (or lie on) a circle, the correspondent to every sixth point (or tangent) must, by virtue of its connec- tion with the five, touch (or lie on) the same circle (305) ; and therefore &c. Thus, the four classes of loci and envelopes enumerated in the article above referred to (385, 3°), may be transformed, not only homographically, as there shewn, but also correlatively, into circles, and all their properties admitting of correlative transformation, such as their harmonic and anhar- monic properties, consequently inferred from the comparatively simple and familiar properties of the circle. See chapters XV. and XVIII. ; all the properties of which, not involving the magni- tudes of angles, are consequently true not only of circles, but of the sevei-al classes of figures there enumerated also. It follows also from the same, as in the corresponding case for homographic figures (385, 3°), that fve points (or tangents), given or taken arbitrarily, completely determine any figure corre- lative to a circle ; for, if transformed, by the above, so that the correspondents of the five points (or tangents) shall touch (or lie on) a circle, all the otlier points (or tangents) of the figure are then implicitly given as the correspondents to the several other 'angents (or points) of the circle; and therefore &c. Given five points [or tangents) A, B, C, 2>, E of a figure cor- relative to a circle, the five corresponding t-angents [or points), A A, BB, CC, DD, EE of the figure are given implicitly with them ; for, since, for the five corresponding tangents (or points) A', B', C, D', E' of the circle, by (306,) {A'.AB' C'D'E'] = {B'.A'B'C'D'E'} = {C'.A'B'C'JJ'E'] = {D'. A' B' C'D'E'} = {E'. A' B' C'D'E'}, therefore, for the five given points (or tangents) A, B, C, D, E of the figure, by (388) and (389, 20°), [A.ABCDE} = [B.ABCDE] = {C.ABCDE}^[D.ABCDE}^[E.ABCDE} ; and since, of each of these five latter homographic pencils (or rows), four rays (or points) are actually given, therefore, of each, the fiftli ray (or point) is implicitly given ; and therefore &c. THEORY OF CORRELATIVE FIGURES. 355 394. Of the numerous properties of the interesting and important class of figures into which the circle may be trans- formed correlatively, the few following, derived on the preceding principles from those of the circle, and identical with those already derived from the same by homographic transformation in Art. 386, may be taken as so many examples illustrative of the utility of the process of correlative transformation in modern geometry. Ex. 1°. No figure correlative to a circle could have either three coUinear points or three concurrent tangents. For, if, of a figure correlative to a circle, either three points were colUnear or three tangents concurrent, then, of the circle itself, by (389, 1°) the three corresponding tangents should be concurrent, or the three corresponding points coUinear ; and therefore &c. N.B. As in the corresponding property of figures homographic to the circle (386, Ex. 1°), the only exception to this fundamental property occurs in the cases noticed in connection with property 1° of the preceding article (393), where the figure is in one or other limiting state of its general form, and has either an infinite number of collinear points lying on one or other of two definite lines, or an infinite number of concurrent tangents passing through one or other of two definite points. See the general remark 2" of Art. 31 ; of which the above and all similar exceptional cases supply so many illustrations. Ex. 2°. No figure correlative to a circle could have either three points at infinity or three parallel tangents. This, as in the corresponding property of figures homographic to the circle (386, Ex. 2°), is manifestly a particular case of the general property of the preceding article ; all points at infinity being collinear and all parallel lines concurrent (136); and therefore &c. N.B. As, in the process of correlative transformation of one figure into another, the line corresponding to any point of the original may be thrown to infinity in the transformed figure (see 2°, of the preceding article), a circle will consequently he transformed correlatively into a figure having two distinct, coincident, or imaginary points at infinity, according as the poiiit whose correspondent is thrown to infinity in the transformation subtends it by two distinct, coincident, or imaginary tangents (21). Since, in the parti- cular case of coincidence, the original point and correspondent line are a point on and a tangent to the original and transformed figures respectively (20), a circle may consequently be transformed correlatively into a figure having a tangent at itifinity, by merely throwing to infinity in the transfor- mation the correspondent to any point on itself. As in the corresponding case of figures homographic to the circle, the transformed figure possesses in this latter case, as will be seen in the sequel, some special propovies peculiar to the case. X k2 356 METHODS OF GEOMETRICAL TRANSFORMATION. Ex. 3°. In every figure correlative to a circle, every three points [or tangents) and the three corresponding tangents (or points) determine two triangles in perspective (140). For, each property, by examples 3° and 4° of Art. 137, being true of the circle itself, the other consequently, by properties 1° and 20° of Art. 389, is true of every figure correlative to it; and therefore &c. (For consequences, see same Ex. Art. 386). Ex. 4°. In every Jigwe correlative to a circle, every six points (or tangents) determine an equianharmonic hexastigm (or hexagram) (301). For, each property, by a and a' of Art. o05, being true of the circle itself, the other consequently, by 12° of Art. 389, is true of every figure correlative to it ; and therefore &c. (For consequences, see same Ex. Art. 386.) Ex. 5°. In every figure correlative to a circle, a variable point (or tangent) determines with every four fixed points (or tangents) a variable quartet of rays (or points) having a constant anharmonic ratio. For, each property, by a and a' of Art. 306, being true of the circle itself, the other consequently, by 13° of Art. 389, is true of every figure correlative to it; and therefore &c. (For consequences, see same Ex. Art. 386). Ex. 6°. In every figure correlative to a circle, a variable point (or tangent) connects (or intersects) homograjjhicully with every two fixed points (or tangents). For, each property, by examples c and c of Art. 325, being true of the circle itself, the other consequently, by property 14° of Art. 389, is true of every figure correlative to it ; and therefore &c. (For consequences, see same Ex. Art. 386.) Ex. 7°. When, of any figure correlative to a circle, two variable points (or tangents) connect through (or intersect on) a fixed point (or line), the two correspnnding tangents [or points) intersect on (or connect through) a fixed line (or point). For, each property, by Cor. 3° of Art. 166, being true of the circle itself, the other consequently, by 1° and 20° of Art. 389, is true of every figure correlative to it; and therefore &o. (For consequences and resulting defi- nitions, see same Ex. Art. 386), Ex. 8°. Hlxen, of any figure correlative to a circle, two variable points (or tangents) connect through {or intersect on) a fixed point [or line), the har- monic conjugate with respect to them of the fixed point [or line) moves on [or turns round) a fixed Uue [or point). For, each property, by a and a' of Art. 259, being true of the circle itself, the other consequently, by 10° and 20° of Art. 389, is true of every figure correlative to it ; and therefore &c. (For consequences and resulting defi- nitions, see same Ex. Art. 386). Ex. 9°. livery two triangles reciprocal polars to each other with respect to ang figure correlative to a circle are in perspective ; and their centre and axis of perspective are jtole and polar to each other with respect to the figure. tHEORY OF CORRELATIVE FIGURES. 357 For, the first part of the property, by 1° Art. 180, being true of the circle itself, is consequently, by 1° Art. 389, true of every figure correlative to it; and the second part, by virtue of the general property of Art. 167, being evident alike for circle and figure; therefore &c. (For consequences, see same Ex. Art. 286). Ex. IdP. Every figure correlative to a circle intersects (or subtends) har- monically the three sides {or angles) of evert/ triangle self-reciprocal with respect to itself; and conversely, every triangle whose three sides (or angles) are intersected (or subtended) harmonically by any figure correlative to a circle is self-reciprocal with respect to the figure. For, each property, by a and a' of Art. 259, being true of the circle itself, the other consequently, by 10° and 20° of Art. 389, is true of every figure correlative to it j and therefore &c. (For consequences, see same Ex. Art. 386). Ex. 11°. In every tetrastigm (or tetragram) determined by four points on [or tangents to) any figure correlative to a circle, the three intersections (or connectors) of the three pairs of opposite connectors (or intersections') deter- mine a self-reciprocal triangle with respect to the figure. For, each property, by a and a' of Art. 261, being true of the circle itself, the other consequently, by Ex. 7°, is true of every figure correlative to it; and therefore &c. (For consequences, see Arts. 261, 262, 263). Ex. 12°. In every figure correlative to a circle, if A and A', B and B', C and C be the three pairs of opposite connectors (or intersections) of the telrastiffm {or tetragram) determined by any four fixed points on (or tangents to) the figure, and la variable point (or tangent) of the figure ; then, in every position of I, the three rectangles I A . IA\ IB . IB', IC. IC are to each other, tmo and two, in constant ratios. These properties follow from Ex. 6°, precisely in the same manner as for figures homographic to the circle in the corresponding example of Art. 386, and lead precisely to the same consequences ; see note and corollary to that example in the article referred to. Ex. 13°. In every figure correlative to a circle, if A, B, C be the three rides {or vertices) of any fixed triangle inscribed [or exscribed) to the figure, and I a variable point {or tangent) of the figure ; then, in every position of I, a.IB.IC*b.IC.IA-^a.IB.IC=0, where a, b, e are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. These properties follow from those of the preceding example (12°), pre- cisely as for ^^res homographic to the circle in the corresponding example of Art. 386 ; and the three multiples a, b, c have for the circle itself the same values given for those figures in the note to that example. Ex. 14°. In every figure correlative to a circle, if A, B, C be the three ndtt {or vertices) of any fired triangle exscribed {or inscribed) to the figure. 358 METHODS OF GEOMETUICAL TUANSFORMATION. and I a variable point (or tangent) of the figure ; then, in every potitiim of I, where a, h, c are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. These properties follow from those of the preceding example (13°), pre- cisely as for figures homographic to the circle in the corresponding example of Art. 386; and the three multiples a, h, c have for the circle itself the same values given for those figures in the note to that example. Ex. 15°. In every figure correlative to a circle, if A, S, Cbe the three vertices {or sides) of any fixed triangle self-reciprocal with respect to the figure, and I a variable tangent (or point) of the figure ; then, in every position of I, a.IA'+b.IB' + c.IC'=0, where a, b, c are three multiples, whose ratios to each other, two and two, are constant in magnitude and sign. These properties follow from examples 10° and 12°, precisely in the same manner as for figures homographic to the circle in the corresponding example of Art. :{S6 ; and the three multiples a, b, c have for the circle itself the same values as for those figures. (See note to the example in question.) N.B. From the properties of figures correlative to a circle given in the examples of the present article, compared with those of figures homo- graphic to a circle given in the corresponding examples of Art. 386, the reader will at once perceive the complete identity existing between both classes of figures ; and can consequently, in investigating their properties from those of the circle, employ in every case whichever mode of transfor- mation appears best adapted to the cose. 395. With the four following general properties of any two correlative figures we shall conclude the present chapter : r. For any two correlative figures F and F', the two corre- spondents /, and /„ in tJie two figures, of a variable line or point /, moving according to any law, generate two homographic figures G^ and G^, in which all pairs of corresponding elements, whether points or lines, which coincide with each other, have the same cor. respondents in the two original figures. For, the two figures G^ and G^ generated by the two variable points (or lines) /, and I^ being each correlative with the figure G generated by the variable line (or point) I (392), and there- fore homographic with each other (388), therefore &c. as regards the first part; and since, when any two points (or lines) A and A^ of G^ and G, coincide, then evidently the line (or point) A of G, towhich they correspond, corresponds in the two original THEORY OF CORRELATIVE FIGURES. 359 figures F and F' to the point (or line) of coincidence, therefore &c. as regards the second part. 2 . For every pair of correlative figures F and F\ however situated^ there exists a pair of corresponding triangles [unique or indeterminate) A and A', for which the three elements of either species {vertices or sides) of either are interchangeable^ as regards the figures^ with the three corresponding elements of the other species (sides or vertices) of the other. This follows from the preceding property 1°, by virtue of the general property 2° of Art. 387 ; for, since, by the preceding property 1°, the two correspondents I^ and I^ of a variable point (or line) /, moving according to any law, generate two homo- graphic figures ^ ; 1 1 k- M c'JaW A \ L the line, A the foot of the perpendicular upon it from 0, and A' the inverse of ^ with respect to the circle of inversion^; the circle on OA' as diameter is the required inverse of L. For, drawing any line through intersecting i at P and the circle in question at P', and joining A'P' ; then, since, by similar right-angled triangles AOP and P'OA', the two rectangles OP. OP" and OA.OA' are always equal in magnitude and sign, therefore &c. (149). The diameter MN of the circle of inversion perpendicular to L being bisected at and cut harmonically at A and A' (225), therefore (231, Cor. 3°) the two rectangles^^'. JO and A3I.AN are always equal in magnitude and sign, and therefore (181) the line L is always the radical axis of its inverse and the circle of inversion. A property which, when (as In fig. a') it Inter- VOL. II. B B 370 METHODS OF GEOMETRICAL TRANSFORMATION. sects the latter at real points, is evident h priori from the general property 10° of Art. 401. To prove 5°. If (same figures) be the centre of inversion, A' the diametrically opposite point of the circle passing through it, and A the inverse of A' with respect to the circle of inversion K; the line L passing through A perpendicular to A A' is the inverse required. For, since (as in the preceding case), for every line passing through and intersecting the circle and L at P' and F respec- tively, the two rectangles OP. OP" and OA.OA' are always equal in magnitude and sign; therefore &c. (149). That, in all cases in which the original circle passes, as above, through the centre of inversion 0, the line L inverse to it is the radical axis of itself and the circle of inversion K, appears of course in precisely the same manner as for the preceding pro- perty 4°. To prove 6°. If (figs. /8 and /8', 7 and y) be the centre of Inversion, C that of the circle whose inverse is required, A and B the extremities of its diameter passing through 0, and A' and B' the two inverses of A and B with respect to the circle •theouy of invekse figures. 371 of inversion iT; the circle on A'B' as diameter is the inverse required. For, being, by virtue of the relation OA.OA=OB.OB' (149), one of the two centres of perspective (the external in figs. ^ and /3', the internal in figs. 7 and 7') of the two circles on AB and AB' as diameters (199), and every line passing through it consequently intersecting them at two pairs of antihomologous points P and P", Q and Q with respect to it, for which the four rectangles OP.OP, OQ.OQ\ OA.OA', OjB. 05' are all equal in magnitude and sign (198); therefore &c. (149). The diameter MN of the circle of inversion passing through the centres G and C of the original and inverse circles being cut harmonically by each pair of inverse points A and A', B and B' (225), the three circles on AB^ A'B', and MN as diameters, that is the original and inverse circles and the circle of inversion, are consequently always coaxal (229). A property which, when (as in figs. yS' and 7') the former and latter intersect at real points, is evident a priori from the general property 10" of Art. 401. N.B. The several figures (a and a', ^ and /3', 7 and 7') given with the above proofs, though applicable as they stand only to the case in which the circle of inversion K is real, may be adapted immediately to that in which it is imaglnar}', by simply turning in each the inverse figure round the centre of inversion through two right angles into the opposite position, the original figure remaining unchanged ; the changed and original figures will then be inverse to each other with respect not to the real circle K but to the concentric imaginary circle K' the negative square of whose radius is equal in absolute magnitude to the positive square of the radius of K (149). From a comparison of the two figures l3 and /3' with the two 7 and 7', both in their original and changed positions, it is evident that, while, for a real circle of inversion A', the centre of inversion is the external or the internal centre of perspec- tive of the original and inverse circles according as it is external or internal to both, the reverse exactly is the case for an imagi- nary circle of inversion K', the centre of inversion being then their internal or external centre of perspective according as B li2 372 METHODS OF GEOMETKICAL TRANSFORMATION. it is external or internal to both. The consideration of this diiFerence is important whenever it is necessary, as it sometimes is, to compare as to magnitude, in accordance with the conven- tion of Art. 23, the angles of intersection of two known circles and of their two inverses with respect to a known circle of inversion, real or imaginary. 404. The centi-e and radius of the circle inverse to a given line or circle with respect to a given circle of inversion JBT, real or imaginary, may be found immediately, on the principles just stated, as follows : In the case of the line. If (figs. a. and a' of preceding article) be the centre and OR the radius of inversion, L the line, A the foot of the perpendicular OA on it from 0, A' the inverse of A with respect to K, and C" the middle point of 0A\ which, by 4° of the preceding article, is the centre of the circle inverse to L; then, since OA.OA' = OR" (149), and since 0A'=20C', therefore 0C'= OR' -i-2 0A= OR' -^^OL; which is conse- quently the formula by which to calculate in numbers the posi- tion of the centre C" and the magnitude of the radius OC of the circle inverse to L with respect to K. In the case of the circle. If, as before, and OR (figs, ff and ^', 7 and y of the preceding article) be the centre and radius of inversion, r and r the radii of the original and inverse circles, d and d' the distances OC and OC of their centres C and C" from 0, and t and t' the lengths (real or imaginary) of the tangents OS and 05', or OTanA 0T\ to them from 0; then since, being one of their two centres of similitude (199), d'^d=r'^r = t'^t=^t't-he=OR'^ [d' - r"), therefore ^' = {d^ -^^ ^"^' *■' = (^) •' ' which are consequently the formulae by which to calculate in numbers the position of the centre C and the length of the radius r' of the inverse circle, when, with the circle of inversion K, the centre C and radius r of the original circle are given. The same formulae may also be obtained easily without the aid of the two tangents ( and t' which are as often imaginary (figs. 7 and 7') as real (figs. ^ and /8') ; for If Pand P', Q and ^, THEORY OF INVERSE FIGURES. 373 be the two pairs of inverse points at which any line passing through O intersects the original and inverse circles; then since (199) d'^d = r ^r= OF ^ OQ or OQ'-^ OP=[OP. OF or OQ.OQ')^OP.OQ= OE'^ (d'-r'), therefore &c. 405. The centre of the circle inverse to a given line or circle with respect to a given circle of inversion K, real or imaginary, is given also in every case by the general property, that — TTie inverse of ike centre of inversion with respect to any line or circle inverts into the centre of the circle inverse to the line or circle. For, if and OR (same figures and notation as before) be the centre and radius of inversion, / the inverse of with respect to the original line or circle, and C the centre of the inverse circle or line ; then, since, in the case of the line (figs. a and a'), 0I=20A (150), therefore OC'.OI= OA'.OA=OR\ and therefore &c. (149) ; and, since, in the case of the circle (figs. y9 and ^, y and 7'), 00. 01= OF.OQ (149), therefore (199) 0C.01=0F.0F or OQ.OQ'= OIP; and therefore &c. (149). Cob. 1°. If T (same figures) be the inverse of with respect to the inverse circle or line ; then, since, for precisely the same reasons as above, 00.01'= OS', therefore reciprocally — The centre of any circle inverts into the inverse of the centre of inversion with respect to the inverse circle. COE. 2°. The two perpendiculars at the two points I and F to the line //' (same figures) being (165) the two polars of the point with respect to the original and inverse circles, and inverting with respect to K into the two circles on the two intervals OC and OC as diameters (4°, Art. 403); it follows consequently that — The polar with respect to any circle of the centre of inversion inverts info the circle whose diameter is the interval between that point and the centre of the inverse circle. The circle on the interval between the centres of inversion and of any circle as diameter inverts into the polar of the centre of inversion with respect to the inverse circle. N.B. A different, and more general, proof of these useful properties, on a principle common to the line and circle m- differently, will be presently given. 374 METHODS OF GEOAIETRICAL TKAXSFOEMATION. 406. As reffards the effect of inversion on the several anhar- raonic ratios of coUinear and concyclic quartets of points (274 and 308), it may be easily shewn that in all cases — Every four points on any line or circle and the four correspond- ing points on its inverse with respect to any circle of inversion^ real or imaginary^ are equianharmonic. For, of the two inverse figures, when one is a line, and the other consequently a circle passing through the centre of inver- sion (4° and 5°, Art. 403), the two quartets are then equianhar- monic with the common pencil they determine at that point (28.5 and 306), and therefore with each other; and when both are circles, and neither consequently passing through the centre of inversion, the two quartets are then antihomologous with respect to the centre of perspective of the circles which coincides with that point (403, 6°), and therefore &c. (316). In the parti- cular case of a line or circle intersecting the circle of inversion at right angles, and consequently inverting into itself (403, 3°), all pairs of corresponding points are then harmonic conjugates with respect to the two points of intersection (149 and 257), and therefore &c. (282, Cor. 4°). N.B. Next to the general property 3° of Art. 402, the above, from w^hich it appears that all anharmom'c ratios ofcollinear and concyclic quartets of points are preserved unchanged in inversion, is the most important connected with the subject of the present chapter. 407. As regards the general property of intersecting figures and their inverses, just adverted to in the note at the close of the preceding article, when applied to the case of intersecting circles and their inverses, it may be stated, in accordance with the convention of Art. 23, more definitely as follows : When two circles intersect, their angle of intersection is equal or supplemental to that of their two inverses with respect to any circle of inversion, real or imaginary, according as the centre of inversion is external or internal to both, or external to one and in- ternal to the other. For, the centre of inversion being the external or the internal centre of perspective of both pairs of inverse circles in the former case, and the external of one pair and the internal of the other THEORY OP INVERSE FIGURES, 375 pair in the latter case (403, note) ; and every circle passing through either pair of inverse points of intersection (402, 1°) consequently intersecting both pairs at equal or at supplemental angles in the former case, and one pair at equal and the other pair at supplemental angles in the latter case (211) ; there- fore &c. N.B, It follows of course from the above that, though, in accordance with the general property 3° of Art. 402, the angle of intersection of two circles undergoes, as a figure, no change of form under the process of inversion, yet, in accordance with the convention of Art. 23, it may, and often does, as a magnitude, change into its supplement under that process (24). In the applications of the theory of inversion to the geometry of the circle, this circumstance must of course, when necessary, be always attended to. The two cases of contact, external and internal, come of course under it as particular cases (23) ; and in hut one case alone, that of orthogonal intersection, which presents no ambiguity, can the precaution ever be entirely dispensed with. 408. Between the squares of the common tangents and the rectangles under the radii of any two intersecting circles and of their two inverses with respect to any circle of inversion, real or imaginary, the following metric relation results immediately from the general property of the preceding article, combined with that of Euc. II. 12, 13 ; viz.— When two circles inter sect, the squares [disregarding signs) of their two common tangents are to the rectangle under their radii, as the squares {disregarding signs) of the two common tangents are to the rectangle under the radii of their two inverses with re- spect to any circle of inversion, real or imaginary ; squares having similar or opposite signs corresponding in the two proportions, according as the angles of intersection of the two pairs of circles, original and inverse, are equal or supplemental* * The above, established generally for any two circles and their two inverses to any circle of inversion, has been applied with considerable suc- cess by Mr. Casey to the investigation of some interesting properties of circles, which, but for the length to which the present volume has extended, would have been noticed here. The same geometer has also obtained by inversion an indirect but general proof of the first part of Dr. Hart's 376 METHODS OF GEOMETEICAL TRANSFOEMATION. For if a, J, c be the three sides of the triangle determined by the two radii to either point of intersection and by the in- terval between the centres of the original pair of circles, and a', 5', c' those of the corresponding triangle for the inverse pair ; then, since, by the general property of the preceding article, the two angles of those triangles opposite to c and c' are either equal or supplemental, therefore, by Euc. II. 12 and 13, the two differences of squares c" — (o — J)° and (a + J)' — c', which (disregarding signs) are the squares of the two common tangents to the original pair, are to the two corresponding or non-corre- sponding differences of squares c'* - [a' — b'Y and {a' + 6')" — c'*, which (disregarding signs) are the squares of the two common tangents to the inverse pair, as the rectangle oJ, which is that under the radii of the original pair, is to the rectangle |a'6', which is that under the radii of the inverse pair ; and there- fore &c. N.B. In the two extreme cases of real Intersection, viz. contact, external and internal, the above ratios have in magni- tude and sign the extreme values + 4 and 0, and — 4 respec- tively ; and in the mean case of real intersection, viz. orthogonal, they have in magnitude and sign the two intermediate mean values + 2 and — 2. The property however being true generally for every two circles and their two inverses to any circle of inversion, they may, for imaginary intersection, have in magni- tude and sign, any value from to + cc , according as the distance between their centres is greater than the sum or less than the difference of their radii. 409. Intersecting circles possess also the following evident properties with respect to inversion : 1°. When two circles intersect, every circle passing through their two points of intersection inverts into a circle p)assing trough the two points of intersection of their two inverse circles. ' Theorem, respecting the eight circles of contact of three arbitrary circles, stated at the close of Art. 212. His paper on the latter subject in the Quarterly Journal of Pure and Applied Mathematics (Vol. t. page 318) had been in fact published when that article was written, but the author was unaware at the time that any demonstration, direct or indirect, had been obtained of it by Elementary Geometry. THEORY OF INVERSE FIGURES. 377 For, since, to every circle of any figure corresponds a circle of the inverse figure (403, 6°), and since to every point of inter- section of any two figures corresponds an inverse point of inter- section of the two inverse figures (402, 1°) ; therefore &c. 2°. When two circles intersect, the circle passing through the two points of intersection and through the centre of inversion in- verts into the radical aans of the two inverse circles. For, that circle, passing through the centre of inversion, in- verts into a line (403, 5°), and, passing through the two points of intersection of the two original circles, the line into which it inverts passes through the two points of intersection of the two inverse circles (402, 1°) ; and therefore &c. 3°. Every two intersecting circles and their two circles of inver- sion (398) invert, to every circle, into two intersecting circles and their two circles of inversion (398). For, since, for the two original circles, the two circles of antisimilitude, or inversion (398), pass through the two points, and bisect, one externally and the other internally, the two angles, of intersection (201) ; therefore, for the two circles inverse to the two former, the two circles inverse to the two latter pass through the two points (402, 1°), and bisect, one externally and the other internally, the two angles (402, 3°), of intersection ; and therefore &c. (201). 4°. Every -two intersecting circles invert into equal circles to every circle having its centre on either of their two circles of inver- sion, external or internal. For, that circle of inversion of the original circles then inverting into the radical axis of the two inverse circles (2° above), and two intersecting circles being evidently equal when their radical axis bisects their two angles of intersection (3° above) ; therefore &c. 5°. Every two intersecting circles invert into circles whose radii have a constant ratio to every circle having its centre on the same circle passing throvgh their two points of intersection. For, the latter circle inverting to every such circle into the radical axis of the two inverse circles (2° above), and the radii of any two intersecting circles being evidently in the inverse ratio of the sines of the segments into which their radical axis divides, externally or internally, their two angles of intersection ; therefore &c. (402, 3°). 378 METHODS OP GEOMETRICAL TRANSFORMATION. N.B. It will be presently shewn that the several properties above given are all general, and true, with obvious modifications, of any two circles whether intersecting or not. 410. If C be a circle of any radius, finite or infinitely great or small, and C its inverse with respect to any arbitrary circle of inversion K^ of which the centre is ; then always — 1°. Every circle D orthogonal to C inverts into a circle D' orthogonal to C. 2°. Every two points P and Q inverse to each other with re- spect to C invert into two points P' and Q inverse to each other with respect to C. 3°. Every diameter of C inverts into a circle through ortho- gonal to C ; and^ conversely, every circle, through orthogonal to C inverts into a diameter of C. 4°. The centre of G inverts into the inverse of with respect to C ; and, conversely, the inverse of with respect to C inverts into the centre of C. Of these very useful properties as regards inversion ; the first follows Immediately from the general property (407) that the angle of intersection of any two circles is equal or supple- mental to that of their two inverses with respect to any circle of inversion K; the second from the first, from the consideration that as every circle D passing through P and Q intersects G at right angles (156), therefore, by 1°, every circle D' passing through P' and Q' intersects G' at right angles, and therefore &c. (156) ; the third also from the first, of which it is evidently a particular case, from the consideration that, by virtue of it, every line orthogonal to G inverts Into a circle through orthogonal to C" (403, 4°), and that, conversely, every circle through orthogonal to G inverts into a line orthogonal to C' (403, 5°) ; and the fourth ; either from the second, of which it is evidently a particular case, from the consideration that, by virtue of it, when P Is at infinity then Q and P" are the centres of G and K respectively (149, 3°) and Q' consequently the inverse of the latter with respect to G', and when conversely F is at infinity then Q' and Pare the centers of G' and K respectively (149, 3°) and Q consequently the inverse of the latter with respect to C; or from the third, more readily perhaps, from the consideration THEORY OF INVERSE FIGURES. 379 that, by virtue of it, evt^ry line passing through the centre of C (22, 1°) inverts into a circle passing through and through its inverse with respect to C (156), and, conversely, every circle passing through and through its inverse with respect to G (156) inverts into a line passing through the centre of C' (22, 1°) ; and therefore &c. (402, 1°). See also Art. 405, where this last property was established on other principles for the line and circle separately. Cue. r. From properties 1° and 3° of the above, the follow- ing consequences are at once evident, viz. — a. Every circle orthogonal to two circles inverts^ to every circle, into a circle orthogonal to the two inverse ch'cles. b. The particular circle orthogonal to two circles, which passes through the centre of inversion, inverts into the line orthogonal, to the two inverse circles. c. The line orthogonal to two circles inverts into the particular circle orthogonal to the two inverse circles which passes through the centre of inversion. d. The circle orthogonal to three circles inverts, to every circle, into the circle orthogonal to the three inverse circles, e. The circle orthogonal to three circles inverts, to every circle through whose centre it passes, into a line orthogonal to the three inverse circles. f. Evf.ry system, of circles having a common orthogonal circle in- verts, to every circle, into a system Jiaving a common orthogonal circle. g. And, to every circle having its centre on the common ortho- gonal circle, into a system having a common orthogonal line. h. Every system of circles having a common pair of orthogonal circJes inverts, to every circle, into a system having a common pair of orthogonal circles. i. And, to every circle having its centre at either point of in- tersection of the common orthogonal pair, into a system having a common pair of orthogonal lines. COE. 2°. From properties 2° and 4° of the above, the follow- ing also are at once evident, viz. : a. Every two non-intersecting circles and their common pair of inverse points invert, to every circle, into two non-intersecting circles and their common pair of inverse points. 380 METHODS OF GEOMETRICAL TRANSFOEMATION. h. Every two non-intersecting circles invert., to every circle having its centre at either of their common pair of inverse points.) into two circles having a common centre^ the inverse of the other. c. Every two circles having a common centre invert., to every circle., into two non-intersecting circles whose common pair of in- verse points are the centre of inversion and the inverse of the common centre. d. Every system of circles having a common pair of inverse points inverts, to every circle, into a system having a common pair of inverse points, inverse to the original pair. e. Every system of circles having a common pair of inverse points inverts, to every circle having its centre at either point, into a system having a common centre, the inverse of the other. f. Every system of circles having a common centre inverts, to every circle, into a system having a common pair of inverse pointSy the centre of inversion and the inverse of the common centre. 411. Every two figures E and E, inverse to each other with respect to any line or circle C, being composed of pairs of points P and Q, S and 8, &c. inverse to each other with respect to the line or circle C (396) ; it follows, consequently, from the general property 2° of the preceding article, that — Every two figures E and F, inverse to each other with respect to any line or circle C, invert — a. To every circle whose centre is not on the line or circle, into two figures inverse to each other with respect to the inverse circle. h. To every circle whose centre is on the line or circle, into two figures reflexions of each other with respect to the inverse line. CoE. Every two circles, however circumstanced as to mag- nitude and position, being figures inverse to each other with respect to each of their two circles of anti»militude (398) ; it follows, consequently, from the above, as established already on other principles for the particular case of intersecting circles in 3° and 4° of Art. 409, that— a. Every two circles and their two circles of inversion invert, to every circle, into two circles and their two circles of inversion. h. Every two circles invert into equal circles to every circle having its centre on either of their two circles of inversion. THEORY OF INVEESE FIGURES. 381 412. Coaxal circles (184) possess, with respect to inversion, the following among other important properties, viz. — 1 . Every system of coaxal circles of the common points species inverts, from either common point as centre, into a system of con- current lines, whose vertex is the inverse of the other common point. 2 . Every system of coaxal circles of the limiting points species inverts, from either limiting point as centre, into a system of con- centric circles, whose centre is the inverse of the other limiting point. 3 . Every system of coaxal circles of either species, whose com- mon or limiting points coincide, inverts, from the point of coin- cidence as centre, into a system of parallel lines, whose direction is that of their common tangent at the point. 4°. Every system of coaxal circles of either species inverts, from every centre, into a coaxal system of the same species, whose common or limiting points, distinct or coincident, are the inverses of those of the original system. Firstly, for a system having real common points M and N, distinct or coincident. Since, by hypothesis, the component figures of the system are all circles passing through M and N, therefore, by (403, 6° and 5°), those of the inverse system, for every centre not coinciding with either M or N, are all circles passing through M' and N' the inverses, distinct or coincident, of M and N, and, for each centre coinciding with either M or N, are all lines passing through the inverse N' or M' of the other N or M; and therefore &c. as regards 1° and the first part of 4°. And, since, when M and N coincide at 0, then M' and N' coincide at infinity on the line MN (401, 14°) ; there- fore &c. as regards the first part of 3°. Secondly, for a system having real limiting points E and F, distinct or coincident. Since, by hypothesis, the component figures of the system are all circles intersecting at right angles every circle passing through E and i^ (188, 5°), therefore, by (403, 6° and 5°, and 407) those of the inverse system, for every centre not coinciding with either E or F, are all circles intersecting at right angles every circle passing through E' and F' the inverses, distinct or coincident, of E and F, and, for each centre coin- ciding with either E or F, are all circles intersecting at right 382 METHODS OF GEOMETRICAL TKANSFOKMATION. angles every line passing through the inverse F' or E' of the other ForE; and therefore &c. as regards 2° and the second part of 4°. And, since, when E and F coincide at 0, then E' and F' coincide at infinity on the line EF (401 , 14°) ; therefore «S:c. as regards the second part of 3°. In every case, the particular circles of the original and ortho- gonal systems (185) which pass through the centre of inversion invert evidently into the radical and central axes of the inverse system. For, as passing both tlirough the centre of inversion, they invert both into lines (403, 5°), of which one (the former) is a particular circle of the inverse system, and the other (the latter) intersects all its circles at right angles (407) ; and therefore &c. And, conversely, for the same reason, the radical and central axes of the original system invert respectively, in all cases, into the particular circles of the inverse system and of its ortliogonal system which pass through the centre of inversion. N.B. That the two parts of property 3° above are in fact identical, and express a common property which appears at once from (403, 5°), is evident h priori from (184), the coaxal system consisting in both cases of circles having a common tangent at a common point. The consideration that a system of lines passing through a common point at infinity in any direction, and a system of circles having a common centre at infinity in the perpendicular direction, are in fact identical (16 and 18) explains h posteriori the reason why properties so different as 1° and 2° above lead to the common result they do in the par- ticular case of 3°. 413. From properties 1° and 2° of the preceding article, in- cluding of course their common particular case 3°, it follows of course, conversely, that — 1°. Every system of lines passing through a common point {whether at a finite distance or at infinity) inverts^ from any centre^ into a system of coaxal circles of the common points species; the centre of inversion and the inverse of the common point being the two common points of the inverse system. 2°. Every system of circles having a common centre {whether at a finite distance or at infinity) inverts, from any centre, into a "THEORY OF INVERSE FIGURES. 383 system of coaxal circles of the limiting points species ; the centre of inversion and the inverse of the common centre being the two limiting points of the inverse system. Both of which are also evident directly ; the first from the consideration that every line of the system inverts into a circle passing through the centre of inversion and through the inverse of the common point (403, 4°) ; and the second from the con- sideration that, as every line passing through the common centre intersects every circle of the system at right angles, therefore every circle passing through the centre of inversion and through the inverse of the common centre (403, 4°) intersects every circle of the inverse system at right angles (407), and therefore &c. (188, 5°). 414. By virtue of the results established in the two preced- ing articles, every property of coaxal circles, involving only considerations of contact, or intersection at constant angles, with lines or circles, or anharmonic equivalence of quartets of points, or homographic division by lines or circles, which (406 and 407) undergo no change in the process of inversion, may be reduced, according to the nature of the system, to one or other of the comparatively simple cases of concurrent lines or concentric circles (including of course the common case of parallel lines), for which it is sometimes evident to mere perception. Thus for instance, the two general properties, established directly on other principles in (193, Cor. 8°) and in (326, Exs. h and i), that " a variable circle intersecting any two fxed circles at any two constant angles intersects at constant angles all circles of the system coaxal with them, touches two particular circles of the system, and cuts a third at right angles /" and that " a variable circle intersecting any two fixed circles at any tivo constant angles deter- mines four homographic systems of points on the circles them- selves, and, generally, 2n homographic systems of points on every n circles of the system coaxal with them y" are both evident to mere perception for the particular cases of a system of concurrent lines and of a system of concentric circles, to one or other of which every other case, by virtue of the preceding properties, may be reduced by inversion. 384 METHODS OP GEOMETRICAL TRANSFORMATION. 415. Every problem again connected with a system of coaxal circles, not involving other than similar considerations, may, by virtue of the same results, be reduced, according to the nature of the system, to the corresponding problem for a system of con- current lines or of concentric circles ; for which, as might natu- rally be expected, the solution, if not evident to mere inspection, is generally simpler than for the original system. Thus for instance the two following problems, " To describe a circle of a coaxal system/ 1°, intersecting a given circle at a given angle; 2°, dividing a given arc of a given circle in a given anharmonic ratio /" are reducible to one or other pair, as the case may be, of the corresponding simpler problems : ^'^ to draw a line passing through a given pointy or to describe a circle having a given centre^ and fulfilling the required condition (1° or 2°) for the inverse circle or arc /" the solutions of which, for condition 1° are evident in either case, and for condition 2° have been given, actually in the former case and virtually in the latter case, in Ex. 3°, a, Art. 356. And the solution of every such problem, once obtained, after inversion, for the corresponding simplified form, gives of course, by inversion back again, the solution of the original in its general form. 416. Coaxal circles possess also, with respect to inversion, the following important property, particular cases of which have been already established on other principles in Art. 409, viz. — Every two circles invert into two whose radii have a constant ratio from every point on any third circle coaxal with themselves. For, if O and k be the centre and radius of inversion, A and B the centres of the two original circles, a and b their two radii, « and V the two tangents to them from 0, a' and V the radii of the two inverse circles, and Z the centre of the coaxal circle on ifc* /fc" which lies ; then, since (404) a =—^ a and 6' = — j- J, therefore jT = T • -i = 7 . -jy (192, Cor. 1°), which being of course con- stant when Z is fixed, whatever be the position of on the coaxal circle of which it is the centre, therefore &c. Coft. r. As the radius of a circle may have either sign in- differently, it appears from the above that, for every pair of THEOKTf OF INVERSE FIGURES. 385 original circles, two different coaxal circles loci of correspond to each particular value of the constant ratio of the inverse radii ; that, when the ratio is given, their two centres Z and Z' , ^, , .. AZ ah' , AZ' _ a V , are given by the relations ^^= ± 7 .— and ^7^, = + ^ .-•. and, J3// a HZ ha as already shewn on other principles in (409, 5°), that, when the original circles intersect, they divide their angles of intersec- tion, externally and internally, into parts whose sines are in the inverse of the ratio. COE. 2°. In the particular case when ^^ = + t and -=7= = + 7 , Ji/i b rSZ that is, when Z and Z' are the two centres of perspective of the two original circles (195) ; then a = b\ and therefore, as already shewn on other principles in (411, Cor. b). Every two circles invert into equal circles from every point on either of the two coaxal circles whose centres are their two centres of perspective. COE. 3°. HA, £, C be any three given circles, and X and X', Fand y, Z and Z' the three pairs of circles coaxal with B and 6', C and A, A and B respectively for whose several points, as centres of inversion, the three pairs of radii h' and c', c and a', a and h' of the three corresponding pairs of inverse circles B' and C, C" and A', A' and B' have the three pairs of ratios determined bv any three given lines taken two and two ; since then (the compound of the three ratios being necessarily = 1) the six centres of the three pairs of circles X and X\ Y and Y\ Z and Z' (determinable, when the three ratios a -.h'-.c are given, by the general relations of Cor. 1°) lie three and three on four lines (which when d = V = c are, by Cor. 2°, the four axes of simili- tude (197) of the three original circles A, B, C), the six circles themselves pass consequently three and three through four pairs of conjugate points P and P', Q and Q\ R and R', S and S' reflexions of each other with respect to the four lines, and real or imaginary according to circumstances (190). Hence, from Cor. 1°, it appears that there exist in general eight different points, corresponding to each other two and two in four conjugate 2>airs, real or imaginary according to circumstances, and determinahle in every case by the intersections three and three of six determinahle VOL. II. ^' ^ 386 METHODS OF GEOMETRICAL TRANSFOKMATIOX. circleSj for which three given circles invert into circles whose radii are proportional to three given lines. Cor. 4°. By virtue of the preceding, many problems involv- ing only contacts, or intersections at angles of prescribed forms, which undergo no change by inversion (407), may be transformed from one system of thi-ee circles to another, the relative magni- tudes of whose radii may be more convenient for their solutions, and the solutions thus obtained then transformed back again by inversion to the original S3'stera. Thus, for instance, the two solutions, real or imaginary, of the particular problem, " To describe a circle having contacts of the same species with three given circles, or, more generally, intersecting three given circles at combinations of the same affection of three given angles and their thre^ supplements;^'' are evident, respectively, for three equal circles, and for three whose radii are inversely as the cosines of the three con-esponding angles, the radical centre of the three being tlicn, in either case, the common centre of the required pair of circles (2, Cor. 1°, Art. 183). And, since to such a system every given system of three may be transformed by in- version from any one of eight different centres, corresponding two and two in four conjugate pairs, real or imaginary, accord- ing to circumstances (Cor. 3°), the four pairs of conjugate solu- tions, real or imaginary, of the general problem, " To describe a circle touching three given circles, or, more generally, intersecting with three given circles at three angles of given forms /" may consequently be obtained by four different inversions of the three given circles from any four, no two of which are conjugates, of the aforesaid eight centres, and by the four inversions back again of the four pairs of solutions thus obtained for their four corresponding triads of inverse circles, 417. In the applications of the theory of inversion, the dis- tances, absolute and relative, between the inverses of pairs of points have occasionally to be considered. The following are the principal relations to be used in such cases : If A, B, C, D, E, &c. be any number of points, and A', B', C\ D', E', (fee. their several inverses with respect to any centre and radius of inversion. and OR ; then — • THEORY OF INVERSE FIGURES. 387 1°. For every two points A, B, and their two inverses A', B', A'B' :AB=OB': OA.OB. 2°. For every three points A, B, C, and their three inverses a; B', C, B'C : C'A' : A'B'=OA.BC: OB.CA : OC.AB. 3°. For every four points A, B, C, B, and their four inverses a; b\ c\ b\ B-C:A'B':C'A:B'B':A'B'.C'B'^BG.AB:CA.BB:AB.CB. Of these relations, the second and third are both evident from the first, which may be easily proved as follows : Since OA . 0A'=0B.0B'=0R\ the two triangles AOB and B'OA' are similar ; therefore A'B' : AB= OA' : 0B= OA.OA' : OA.OB = B^: OA.OB; and therefore &c. Cor. r. It is evident, from the first of the preceding rela- tions, that, for a given radius of inversion, the absolute distance A'B' between the inverses of any two points A and B varies directly as the distance AB and inversely as the rectangle OA.OB; from the second, that, for any radius of inversion, tlio ratio A'C : B'C of the distances of the inverses of any two points A and B from that of any third C varies directly as the ratio AG : BO and inversely as the ratio OA : OB; and from the third, that, for any centre and radius of inversion, the three rectangles B' C'.A'B', C'A'.B'B', A'B'. CD' for the four inverses -4', B'^ C", D' of any four points A, B, C, I) are proportional to, and connected consequently by every linear relation connecting, the three corresponding rectangles BO.AB, C'A.BD, AB.CD for the four points themselves. Cor. 2°. Any three points A, B, C being given, it is evident, from 2°, that, when the ratio A' C : B' C is given, then lies on a given circle coaxal with A and B (152), that, viz. for every point of which OA : OB=AC-^BC: A'C ^ B'C (1.58) ; and that, when the three ratios B'C : CA : A'B', and with them of course the species of the triangle A'B'C, are given, then is one or other of the two points common to the three given circles coaxal with B and C, C and A, A and B respec- tively (152), and with each other (190), for each point common CC2 388 METHODS OF GEOMETRICAL TRAXSFOEMATIOX. to which OA: OB: OC=B'C'^BC: C A' j CA : A'B' -^AB (158). Cor. 3". When, for four points A, B, C, Z>, one of the three rectangles BC.AD, CA.BDj AB.GD is equal in absolute value to the sum of the other two, it is evident, from 3°, that, for the four inverse points A\ B\ C, D', the corresponding rectangle is equal in absolute value to the sum of the other two ; hence (82), when four points are either colUnear or concyclic^ tlieir four in- verses with respect to any circle of inversion are also either concyclic or collinear. A property verifying indirectly the results estab- lished in 4°, 5°, G" of Art. 403. 418. From the principles of inversion estfiblished in the preceding articles, it is evident that every property in the geometry of the line and circle, involving only the equality or constancy of angles of intersection or of anharmonic quartets of points collinear or concyclic, may be transformed by inversion into another of the same character in which any circle will be changed into a line, or circle having any required centre or radius; any two circles into two lines, two concentric circles, two equal circles, or two circles whose radii shall have any required ratio to each other; or any three circles into three others, the distances of whose centres or the magnitudes of whose radii shall have any required ratios two and two to each other ; and thus its establishment may be, and often is, rendered much simpler than in the original form. Thus, for instance, the pro- perties (established in 326, Ex. /, and 193, Cor. 4°) that "a variable circle passing through a fixed point and intersecting a fixed circle at a constant angle divides the latter homograr>hically and envelopes another coaxal with it and the point " become trans- formed by inversion, from the point as centre, into the self- evident properties that "a variable line intersecting a fixed circle at a cx>nstant angle divides the circle homographically and envelopes a concentric circle;^'' the properties (established in 326, Ex. h. and 143, Cor. 6°) that " a variable circle intersecting two fixed circles at two constant angles divides both homographically and envelopes two others coaxal with both " become transformed by inversion, from either point common to both circles when they intersect, or from either point Inverse to both when they do not, THEORY OF INVERSE FIGURES. 389 into the self-evident properties that " a variaUe circle intersecting two fixed lines, or concentric circles, at tico constant angles divides hoth homographically and envelopes two others concurrent, or con- centric, with loth;" and the two properties (established in 326, Ex. h, and 211, Cor. G°, I) that "a variable circle intersecting three fixed circles at equal [or at any invariable combination of equal and supplemental) angles divides all three homographically and determines a coaxal system" become transformed by inversion, from either of the corresponding pair of conjugates of the eight points into which the three circles invert into three of equal radii (416, Cor. 3°), into the self-evident properties that '^^ a variable circle intersecting three fixed circles of equal radii at equal angles divides all three homographically and determines a concentric system." And similarly for all properties of the same nature ; any of which, when seen or proved to be true in their trans- formed, may always, by virtue of the general properties of Arts. 406 and 407, be regarded as established in their original forms. 419. Every problem too, in the geometry of the line and circle, involving only considerations of the same nature, may be transformed in the same manner by inversion into another of the same character, in which, while all other essential elements remain unchanged, any one, two, or three, of the circles involved shall be modified in any of the ways above enumerated ; and its solution thus rendered, in many cases, much simpler than in its original form. Thus, for instance, the problem " To describe a circle passing through two given p)oints and intersecting a given circle at a given angle" becomes transformed by inversion, from either point as centre, into the very simple problem " to draw a line passing through a given point and interse