Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/elementsofgeometOOhalsrich o', > > > J 1 ,» • • • THE ELEMENTS OF GEOMETRY. BY GEORGE BRUCE HALSTED, A.B., A.M., AND EX-FELLOW OF PRINCETON COLLEGE; PH.D. AND EX-FELLOW OF JOHNS HOPKINS university; instructor in POST-GRADUATE MATHEMATICS, PRINCETON college; PROFESSOR OF PURE AND APPLIED MATHEMATICS, UNIVERSITY OF TEXAS. NEW YORK : JOHN WILEY & SONS, 15 ASTOR PLACE. 1885. . Copyright, 1885, By JOHN WILEY & SONS. CAJORI ELECTROTVPED AND PRINTED BY RAND, AVERY, AND COMPANY, (by permission) TO J. J. SYLVESTER, A.M., CAM.; F.R.S., L. AND E.; CORRESPONDING MEMBER INSTITUTE OF FRANCE; MEMBER ACADEMY OF SCIENCES IN BERLIN, GOTTINGEN, NAPLES, MILAN, ST. PETERSBURG, ETC.; LL.D., UNIV. OF DUBLIN, AND U. OF E.; D.C.L., OXFORD; HON. FELLOW OF ST. JOHN'S COL., CAM.; SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD; In (Grateful B£membrance OF BENEFITS CONFERRED THROUGHOUT TWO FORMATIVE YEARS. 918182 PREFACE. In America the geometries most in vogue at present are vitiated by the immediate assumption and misuse of that subtile term " direction ; " and teachers who know something of the Non-Euclidian, or even the modern synthetic geom- etries, are seeing the evils of this superficial " directional " method. Moreover the attempt, in these books, to take away by definition from the familiar word "distance" its abstract character and connection with length-units, only confuses the ordinary student. A reference to the article Measurement in the " Encyclopaedia Britannica " will show that around the word "distance" centers the most abstruse advance in pure science and philosophy. An elementary geometry has no need of the words direction and distance. The present work, composed with special reference to use in teaching, yet strives to present the Elements of Geometry in a way so absolutely logical and compact, that they may be ready as rock-foundation for more advanced study. Besides the acquirement of facts, there properly belongs to Geometry an educational value beyond any other element- VI PREFACE. ary subject. In it the mind first finds logic a practical instrument of real power. The method published in my Mensuration for the treat- ment of solid angles, with my words steregon and sieradian, having been adopted by such eminent authorities, may I venture to recommend the use of the word sect suggested in the same volume } From 1877 I regularly gave my classes the method of Book IX. In 1883 my pupil, H. B. Fine, at my suggestion, wrote out a Syllabus of Spherical Geometry on the lines of my teaching, which I have followed in Book IX. The figures, which I think give this geometry a special advantage, owe all their beauty to my colleague. Professor A. V. Lane, who has given them the benefit of his artistic skill and mastery of graphics. The whole work is greatly indebted to my pupil and friend, Dr. F. A. C. Perrine. We have striven after accuracy. Any corrections or suggestions relating to the book will be thankfully received. GEORGE BRUCE HALSTED. 2004 Matilde Street, Austin, Texas. CONTENTS. THE ELEMENTS OF GEOMETRY. BOOK I. CHAPTER I. ON LOGIC. I. Definitions. — Statements. ARTICLE PAGE I. A statement defined i Immediate inference defined ... i Equivalent statements defined . . i A statement implied i Statements reduced to two terms and a copula 2 Logically simple sentence .... 2 Logically composite sentence . . 2 Conditional statements reduce to the typical form 2 9. Subject and predicate 2 II. Definitions. — Classes. 10. Individual terms defined .... 2 11. Class terms defined 2 12. How a class is defined 3 13. Contradictory of a class .... 3 14. Contradictories include the universe, 3 III. The Universe of Discourse. 15. Universe of discourse defined . . 3 16. The classes x and non-x still con- tradictory 3 17. Different divisions into contradic- tories 3 IV. Contranominal, Converse, Inverse, Obverse. ARTICLE PAGE 18. Contranominal defined 19. Converse defined 20. Inverse defined 21. Obverse defined 22. Two theorems prove four .... 23. Classes proved identical .... 24. 25- 26. 27. 28. 29. 30- 31- 32. V. On Theorems. A theorem defined .... Demonstration defined . , Corollary defined ..... Hypothesis and conclusion , A geometric theorem defined Use of type in this work . , Inverses of theorem with composite hypothesis 6 VI. On Proving Inverses. Inverses often proved geometrically, 6 Rule of identity 6 Rule of inversion 7 vii Vlll CONTENTS. CHAPTER II. THE PRIMARY CONCEPTS OF GEOMETRY. I. Definitions of Geometric Magnitudes. ARTICLE PAGE 34. Geometry defined 8 A solid defined 8 A surface defined 8 A line defined 8 A point defined 8 A magnitude defined 8 35- 36. yi- 38. 39. 40-43 44. 45- 46. 47. 48. 49. 50. 51- 52. 53- 54. 55- 56. 57- 58. 59- 60. 61. 62. 63. 64. 65. Point, line, surface, solid, de- fined, using motion .... 8 Why space is called tri-dimensional, 9 Straight line defined ..... 9 A curve defined 9 Line henceforth means straight line 9 A sect defined 9 A plane defined 9 Plane defined, using motion ... 9 A figure defined 9 A plane figure defined 9 Circle and center defined .... 10 Radius defined 10 Diameter defined 10 Arc defined 10 Parallel lines defined 10 Angle, vertex, arms, defined ... 10 Explement defined 11 Equal angles defined 11 Adjacent angles and their sum de- fined II Straight angle defined 12 When two adjacent angles equal a straight angle 12 Supplement defined 12 Supplemental adjacent angles . . 12 66. Right angle defined 13 67. A perpendicular defined .... 13 68. Complement defined 13 69. Acute angle defined 13 70. Obtuse angle defined 13 71. Perigon defined 13 72. Sum of angles about a point is a perigon 14 73. Reflex angle defined 14 74. Oblique angles and lines defined . 14 75. Vertical angles defined 14 76. Bisector defined 14 Tj. A trace defined 14 •jS. A triangle defined 15 79. Vertices defined 15 80. Sides defined 15 81. Interior angle defined 15 82. Exterior angle defined 15 83. Congruent defined 15 84. Equivalent defined 15 II. Properties of Distinct Things. 85-93. The so-called axioms of arith- metic I5> 16 III. Some Geometrical Assumptions about Euclid'' s Space. 94-99. The so-called geometric axioms, 16 IV. The Assumed Constructions. 100-103. The so-called postulates . . . 17 Table of symbols used 17 CHAPTER III. PRIMARY RELATIONS OF LINES, ANGLES, AND TRIANGLES, I. Angles about a Point. ARTICLE PAGE 104-105. Right and straight angles . 18, 19 106-107. Perigons 19 1 08-1 10. Vertical angles 20, 21 II. Angles about Two Points. ARTICLE PAGE III. Transversal defined .... 22 1 1 2-1 13. Corresponding and alternate angles 22 CONTENTS. IX III. Triangles. ARTICLE PAGE 114. Equilateral triangle defined ... 24 Isosceles triangle defined .... 24 Scalene triangle defined .... 24 Base and vertex defined .... 24 Angles of triangle mean interior angles 24 [15. 16. :i7. ARTICLE PAGE 119. Right-angled triangle and hypoth- enuse defined 120. Obtuse-angled triangle defined 121. Acute-angled triangle defined 122. Equiangular triangle defined Homologous defined . . . 123. 124-129, Congruent triangles CHAPTER IV. PROBLEMS. ARTICLE PAGE 130. Problem defined 30 131. Solution defined 30 ARTICLE 132-40. Deduced constructions PAGE 30-35 CHAPTER V. JNEQUALITIES. ARTICLE PAGE 141. Sign of inequality explained . 36 142-150. Theorems on one triangle . 36-39 ARTICLE PAGE 151. Oblique sect defined .... 39 152-164. Obliques and triangles . . 40-47 CHAPTER VI. PARALLELS. ARTICLE 165-172. Fundamental theorems PAGE ARTICLE PAGE 48-51 I 173-175. Application to the triangle . 51,52 CHAPTER VII. TRIANGLES. PAGE 176-179. Congruent triangles . . . 53-55 180. Conditions of congruence of triangles 55 181. Locus defined 56 ARTICLE PAGE 182. How to prove a locus ... 56 183-186. Loci 57,58 187-191. Intersection of loci . . . 58-60 CHAPTER VIII. POLYGONS. I. Definitions. ARTICLE PAGE 192. A polygon defined 61 193. Its vertices defined 61 ARTICLE 194. Its sides defined .... 195. Perimeter defined . . . 196. Interior angles of a polygon PAGE . 61 . 61 . 62 CONTENTS. ARTICLE PAGE 197. Convex defined 62 198. 199. 200. 201. Diagonals defined 62 An equilateral polygon defined . . 62 An equiangular polygon defined . 62 A regular polygon defined ... 63 Mutually equilateral defined ... 63 Mutually equiangular defined . . 63 204, 205. Polygons may be either ... 63 206. Trigon, tetragon or quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, do- decagon, quindecagon .... 64 The surface of a polygon defined . 64 Parallelogram defined 64 Trapezoid defined 64 203. 207. 208. 209. II. General Properties. 210-215. Congruence, angles . 216-220. 221. 222. 223, 224, 225. 226. III. Parallelograms. Sides and angles . A rectangle defined A rhombus defined A square defined . Properties . . . 64-66 67-69 • 70 • 70 . 70 70,71 Triangle altitudes concurrent . 72 227-2: 232. 233. 234. 235- 236. 237. 238-240. 241-244. 245-255. 256, 257. 258-264. PAGE Intercepts 73> 74 Medial defined 74 Concurrent defined .... 75 Collinear defined 75 Medials concurrent • • . . 75 Centroid, orthocenter, circum- center, defined 75 IV. Equivalence. Equivalence, how proved . . 76 Base and altitude 76 Squares 77-79 Rectangles 80-86 Complements of parallelo- grams 87 V. Problems. Parallelograms equivalent to polygons 88-90 VI. Axial and Central Symmetry. 265. Corresponding points defined . 91 266. Axial symmetry defined . . 91 267. Central symmetry defined . . 92 268. 269. Symmetry in one figiire . . 92, 93 BOOK II RECTANGLES. 270. A continuous aggregate defined . 95 271. A discrete aggregate defined ... 95 272-279. Counting and number . . 95, 96 280. The commutative law for addi- tion 96 281. The sum of two sects defined . . 97 282. The commutative law holds for sects 97 283. The sum of two rectangles de- fined q8 ARTICLE PAGE 284. The commutative law holds for rectangles 98 285. The associative law 99 286. The associative law holds for sects, 99 287. The associative law holds for rect- angles 99 288. The commutative law for multipli- cation 100 289. The commutative law for rectangle of sects 100 CONTENTS, XI ARTICLE 290. 291-293. 294, 295. 296-302. PAGE The distributive law . . .100 The distributive law holds for sects and rectangles . 100-102 The square on the sum of two sects 102, 103 Rectangles and squares . 103-106 ARTICLE PAGE 303. The projection of a point de- fined 106 304. The projection of a sect de- fined 107 305-311. Squares on sides . , . 107-111 312-314. Problems 111-113 BOOK III THE CIRCLE. ARTICLE 315-319. 320,321. 322, 323. 324- 325- 326. 327. 328. 329- 330-332. 333» 334. 335» 336. 337-340. 341-343. 344-348. 349-353. 354-356. 357. 358, 359. 360. 361, 362. 363- 364. 365. I. Primary Properties. PAGE Center, radius, surface of circle, 115 Point and circle . . 16, 17 A secant A chord defined . . . A segment defined . . Expleraental arcs defined Semicircle defined ..... Major arc and minor arc de- fined Major segment and minor seg- ment defined 117 Circle and center 118 Diameter 118 Congruent circles 119 Concentric circles 120 Symmetry in the circle . 120, 121 Chords 122-124 Sects 125, 126 Circumscribed circle and in- scribed polygon 126 Coney clic defined 126 Perpendiculars on chords . .127 II. Angles at the Center. Angles subtended by arcs . .128 Sector 128 Angle subtended by a sect . .128 Inscribed angle 128 Corresponding arcs, angles, sectors, chords 129 PAGE 129, 130 130-132 132, 133 ARTICLE 366-368. Sum of two minor arcs . 369-374. Unequal arcs .... 375-378. Inscribed angles . , . III. Angles in Segments. 379-386. Angles made by chords and secants 134-138 IV. Tangents. 387, 388. Tangent and point of contact defined 138 389-402. Properties of tangents . 139-143 V. Two Circles. 403, 404. Common chord 144 405. Tangent circles defined . . . 144 406. Common axis of symmetry . 144 407-411. Relations between the center- sect, radii, and relative posi- tion 145-147 VI. Problems. 412. To bisect a given arc. . . . 148 413. Circumscribed polygon and in- scribed circle 148 414. Escribed circle defined . . .148 415-417. Problems 149, 150 Xll CONTENTS. BOOK IV REGULAR POLYGONS. I. Partition of a Perigon. ARTICLE PAGE 418,419. Successive bisection .... 151 420-422. Trisection 152 423-426. Five and fifteen . . . 153-154 II. Regular Polygons and Circles. 427-435. To inscribe and circum- scribe 154-159 436. Radius and apothem de- fined 159 ARTICLE PAGE 437. Side of inscribed hexagon equals radius 159 III. Least Perimeter in Equivalent Figures. — Greatest Surface in Isoperimetric Fig- ures. 438. Isoperimetric defined. . . .159 439-443. Equivalent and isoperimetric triangles 160-162 444-446. The circle 162-164 447-450. Regular polygons . . . 165-167 BOOK V. RATIO AND PROPORTION. ARTICLE 451-453- 454- 455- 456. 457. 458-460. 461, 462. 463- Multiples 169,170 Commensurable defined . . . Incommensurable defined . . Commensurability exceptional. Greatest common submulti- ple Incommensurable magnitudes. The side and diagonal of a square are incommensur- able 172,173 Scales of multiples defined PAGE ARTICLE ?, 170 464-472. . 170 473-476. . 170 477,478 , 170 479- . 170 480-482 , 171 483. 484-491. 2,173 492-496. . ^n PAGE Multiples 173-175 Scale of relation . . . 175, 176 Ratio, antecedent, consequent, 176 Proportion, extremes, means, fourth proportional . . .177 Equality of ratios defined . .177 Third proportional, mean pro- portional 177 Ratios compared . . . 178, 179 Ways of changing a propor- tion 180-182 BOOK VI RATIO APPLIED. I. Fundamental Geometric Proportions. ARTICLE PAGE 497-500. Intercepts by parallels . 183, 184 501-503. Segments of a sect . . 184-186 ARTICLE PAGE 504, 505. Rectangles are as their bases, j86, 187 506. Angles are as arcs 187 CONTENTS. Xlll II. Similar Figures. PAGE Similar figures defined . . .188 Similar triangles . . . 188-192 Homologous, ratio of simili- tude, similarly placed . .192 515,516. Similar polygons similarly placed 192, 193 517,518. Center of similitude .... 193 519-522. The right triangle .... 194 507. 508-513. 514. ARTICLE PAGE 523, 524. The bisector of an angle, 195, 196 525-528. Harmonic division 196 III. Rectangles and Polygons, 529-539. Rectangle of extremes equiv- alent to rectangle of means 197-204 540-547. Composition of ratios . 204-208 548-552. Theorems and problems, 208-212 GEOMETRY OF THREE DIMENSIONS. BOOK VII. OF PLANES AND LINES. ARTICLE PAGE 553. The plane defined 213 554-556. Assumptions about the plane .213 557-562. Determination of a plane, 214-216 563-565. Intersection of planes . 216,217 566. Principle of duality . .. . .217 567-578. Perpendicular to plane . 218-224 ARTICLE PAGE 579-581. Projection on plane . . 224, 225 582. Inclination defined . . . . 226 583-593- Parallel planes .... 226-231 594, 595- Smallest sect between lines ,232,233 596-604. Loci and angles made by planes 233-238 BOOK VIII TRI-DIMENSIONAL SPHERICS. ARTICLE 605-609. 610. 611,612. 613,614. 615,616. 617,618. 619. 620-624. 625. PAGE Sphere, center, radius, defined, 239 Opposite points and diameter defined 240 Point and sphere 240 Plane and sphere . . . 240, 241 Great circle and small circle . 241 Poles and axis 241 Hemispheres 241 Great circles and small cir- cles 242,243 Zone and its bases defined . . 243 ARTICLE PAGE 626-630. Tangency 243,244 631, 632. Intersecting spheres .... 245 633-639. Sphere and tetrahedron . 246-248 640, 641. Globe and globe-segment de- fined 248 642, 643. Central and planar sym- metry 248, 249 644. Globe equivalent to tetrahe- dron 250 645,646. Chords and arcs 251 647-649. Quadrant 252 XIV CONTENTS. 650-652. Spherical angles and perpen- dicular arcs .... 252, 253 ARTICLE 653, 654. Smallest line between two points 254,255 BOOK IX. TWO-DIMENSIONAL SPHERICS. ARTICLE PAGE 655. Introduction 257 656. Two-dimensional definition of the sphere 258 657-662. Fundamental properties, 258,259 663-666. Definitions 259 667-675. Assumptions with corolla- ries 259-261 676. All spherical lines intersect . 261 677, 678. A lune and its angle .... 262 679-684. Definitions 263, 264 685, 686. Perimeter of polygon . 264, 265 687. Symmetry without congru- ence 265 688-691. Congruent spherical tri- angles 266-268 692-697. Problems 268-270 ARTICLE PAGE 698-703. The line and its pole . . 270, 271 704. Bi-rectangular triangle . . . 272 705-707. Polar triangles .... 272, 273 708-719. Spherical triangles . . 274-280 720, 721. Loci 280, 281 722, 723. Symmetrical triangles equiva- lent 281, 282 725. Spherical excess defined . . 282 724-728. Triangles equivalent to lunes 282,283 729-731. Polygons equivalent to lunes 283, 284 732. Equivalent triangles on same base 284 IZZ^TA' Tangency 285 735-739. Theorems on circles . . 286, 287 BOOK X. POLYHEDRONS. ARTICLE PAGE 740-744. Polyhedron, faces, edges, sum- mits, section .... 289, 290 745-747- Pyramid, apex, base, lateral faces and edges .... 290 748. Parallel polygons 290 749-757. Prism, altitude, right prism, oblique 290, 291 758-760. Parallelopiped, quader, cube . 291 761. All summits joined by one line, 292 ARTICLE PAGE 762-764. Descartes' theorem .... 293 765. Euler's theorem 294 766-768. Ratio between quaders . 295-297 769. Parallelopiped equivalent to quader 298 770, 771. Triangular prism 299 n^^ 773- Pyramid sections 300 774. Equivalent tetrahedra . . . 301 775. Triangular pyramid and prism, 302 CONTENTS. XV BOOK XL MENSURATION, OR METRICAL GEOMETRY. CHAPTER I. THE METRIC SYSTEM. — LENGTH, AREA. ARnCLK PAGE 776-778. Measurement 303 779-782. Meter and metric system, 303, 304 783,784. Length 304>305 785-787. Area 305 ARTICLE PAGE 788. Area of rectangle 305 789, 790. Area of square 307 791, 792. Metric units of surface . . . 308 793. Given hypothenuse and side . 308 CHAPTER II. RATIO OF ANY CIRCLE TO ITS DIAMETER. ARTICLE PAGE 794. To compute perimeters .... 310 795. Length of circle 312 796. Variable defined 313 797. Limit defined 313 ARTICLE PAGB 798. Principle of limits . . . .313 799-802. Value of ir 315? 316 803,804. Circular measure of an angle 316,317 CHAPTER III. MEASUREMENT OF SURFACES. ARTICLE PAGE 805, 806. Area of parallelogram . . .318 807, 808. Area of triangle . . . 318, 319 809, 810. Area of regular polygon and circle 320 811-813. Area of sector and sector of annulus 321,322 814. Lateral area of prism . . . 322 815-822. Cylinder 323-325 823. Mantel defined 325 824. Conical surface 326 825-832. Cone and frustum . . . 326-329 Z-},1. Area of sphere 330 834, 835. Zone 331 CHAPTER IV. SPACE ANGLES. ARTICLE PAGE ^Z^i^yi' Plane and space angles de- fined 332 838. Symmetrical space angles . . . 332 839. Steregon 332 840. Steradian 333 841. Space angle made by two planes . 333 842. Spherical pyramid ...... 333 ARTICLE PAGE 843. Space angles proportional . . . 333 844. To transform any polyhedral angle, 334 845. 846. Area of a lune 334 847. Plane angle as measure of dihedral, 335 848. Properties of space angles . . .335 849-851. Area of spherical triangle or polygon 336 XVI CONTENTS. ARTICLE 852-854. 855,856. 857. 858, 859. 860-862. 863-865. 866-868. CHAPTER V. THE MEASUREMENT OF VOLUMES. Volume and metric units . .337 Volume of quader and cube . 338 Volume of parallelopiped . . 338 Volume of prism or cyl- inder 338,339 Volume of pyramid or cone, 339-341 Prismatoid defined . . 341, 342 Prismoid 343 ARTICLE PAGE 869,870. Tetrahedron and wedge are prismatoids 343 871, 872. Altitude and cross-section of a prismatoid 343 873, Volume of any prismatoid . . 344 874, 875. Frustum of pyramid or cone . 346 876. Volume of globe 347 Z^^, 878. similar solids 347 Direction 347 Principle of Duality 349 Linkage 350 Cross-ratio 352 Exercises 357-366 THE ELEMENTS OF GEOMETRY. BOOK I. CHAPTER I. ON LOGIC. I. Definitions. — Statements. 1. A statement is any combination of words that is either true or false ; e.g. {exempli gratia^ " for example "), All x^s are fs, 2. To pass from one statement to another, with a conscious- ness that belief in the first warrants belief in the second, is to infer. 3. Two statements are equivalent when one asserts just as much as the other, neither more nor less ; e.g., A equals By and B equals A. 4. A statement is implied in a previous statement when its truth follows of necessity from the truth of the previous state^ ment; e.g.. All x's are ys implies some fs are x*s. THE ELEMENTS OF GEOMETRY. 5. Any declarative sentence can be reduced to one or more simple statements, each consisting of three parts, namely, two terms or classes, and a copula, or relation, connecting them. A equals B, x is 7, are simple examples of this typical form of all statements. Here x and y each stand for any word or group of words that may have the force of a substantive in naming a class ; e.g., x\^y may mean all the x^^ are y^ ; man is mortal, means, all men are mortals. 6. A sentence containing only one such statement is a logically simple sentence ; e.g., Man is the only picture-making animal. 7. A sentence that contains more than one such statement is a logically composite sentence ; e.g., A and B are respectively equal to C and D, is composite, containing the two statements A equals C, and B equals D. 8. Statements that are expressly conditional, such as, if A is B, then C is D, reduce to the typical form as soon as we see that they mean {The consequence of M) is N. 9. In the typical form, bird is biped, x is y, we call x and y the terms of the statement; the first being called the subject, and the last the predicate. II. Definitions. — Classes. 10. Terms that name a single object are called individual terms ; e.g., Newton. 11. Terms that name any one of a group of objects are called class terms ; e.g., mafti crystal. The name stands for any object that has certain properties, and these properties are possessed in common by the whole group for which the name stands. ON LOGIC, 12. A class is defined by stating enough properties to decide whether a thing belongs to it or not. Thus, "rational animal" was given as a definition of "man." 13. If we denote by x the class possessing any given prop- erty, all things not possessing this property form another class, which is called the contradictory of the first, and is denoted by non-;ir, meaning " not x ; " e.g., the contradictory of animate is inanimate. 14. Any one thing belongs either to the class x or to the class non-;r, but no thing belongs to both. It follows that X is just as much the contradictory of nonvr as non-;r is of x. So any class y and the class non-7 ^^^ mutu- ally the contradictories of each other, and both together include all things in the universe ; e.g., unconscious and conscious. III. The Universe of Discourse. 15. In most investigations, we are not really considering all things in the world, but only the collection of all objects which are contemplated as objects about which assertion or denial may take place in the particular discourse. This collection we call our universe of discourse, leaving out of consideration, for the time, every thing not belonging to it. Thus, in talking of geometry, our terms have no reference to perfumes. 16. Within the universe of discourse, whether large or small, the classes x and non-;ir are still mutually contradictory, and every thing is in one or the other ; e.g., within the universe mammals, every thing is man or brute. 17. The exhaustive division into x and non-;ir is applicable to any universe, and so is of particular importance in logic. But a special universe of discourse may be capable of some entirely different division into contradictories, equally exhaus- tive. Thus, with reference to any particular magnitude, all THE ELEMENTS OF GEOMETRY. magnitudes of that kind may be exhaustively divided into the contradictories, Greater than, equal to, less than. IV. Contranominal, Converse, Inverse, Obverse. i8. If X and y are classes, our typical statement x \s y means, if a thing belongs to class x, then it also belongs to class y\ e.g., Man is mortal, means, to be in the class men, is to be in the class mortals. If the typical statement is true, then every individual x belongs to the class y : hence no x belongs to the class non-y, or no thing not y is, 3. thing x ; that is, every non-y is nonvt; : e.g., the immortals are not-human. The statements x is y, and non-^ is non-;ir, are called each the contranominal form of the other. Though both forms express the same fact, it is, nevertheless, often of importance to consider both. One form may more naturally connect the fact with others already in our mind, and so show us an unexpected depth and importance of meaning. 19. Since x is y means all the x's are j/'s, the class y thus contains all the individuals of the class x, and may contain others, besides. Some of the ys, then, must be ^s. Thus, from **a crystal is solid " we infer "some solids are crystals." This guarded statement, some y is Xy is called the logical converse of x is y. It is of no importance in geometry. 20. If, in the true statement x is y, we simply interchange the subject and predicate, without any restriction, we get the inverse statement y is x, which may be false. In geometry it often happens that inverses are true and important. When the inverse is not true, this arises from the circumstance that the subject of the direct statement has been more closely limited than was requisite for the truth of the statement. ON LOGIC. 21. The contranominal of the inverse, namely, non-;tr is non-^, is called the obverse of the original proposition. Of course, if the inverse is true, the obverse is true, and vice versa. To prove the obverse, amounts to the same thing as proving the inverse. They are the same statement, but may put the meaning expressed, in a different light to our minds. 22. If the original statement is x is j/, its contranominal is noii-y is non-Xy its inverse is y is Xy its obverse is non-x is non-y. The first two are equivalent, and the last two are equivalent. Thus, of four such associated theorems it will never be necessary to demonstrate more than two, care being taken that the two selected are not contranominal. 23. From the truth of either of two inverses, that of the other cannot be inferred. If, however, we can prove them both, then the classes x and y are identical. A perfect definition is always invertable. V. On Theorems. 24. A theorem is a statement usually capable of being inferred from other statements previously accepted as true. 25. The process by which we show that it may be so in- ferred is called the proof or the demonstration of the theorem. 26. A corollary to a theorem is a statement whose truth follows at once from that of the theorem, or from what has been given in the demonstration of the theorem. 27. A theorem consists of two parts, — the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. 28. A geometric theorem usually relates to some figure, and says that a figure which has a certain property has of necessity also another property ; or, stating it in our typical form, X is j, "a figure which has a certain property" is "a figure having another specified property.'* THE ELEMENTS OF GEOMETRY. The first part names or defines the figure to which the theorem relates ; the last part contains an additional property. The theorem is first stated in general terms, but in the proof we usually help the mind by a particular figure actually drawn on the page ; so that, before beginning the demonstration, the the- orem is restated with special reference to the figure to be used. 29. Type. — Beginners in geometry sometimes find it diffi- cult to distinguish clearly between what is assumed and what has to be proved in a theorem. It has been found to help them here, if the special enuncia- tion of what is given is printed in one kind of type; the special statement of what is required, in another sort of type ; and the demonstration, in still another. In the course of the proof, the reason for any step may be indicated in smaller type between that step and the next. 30. When the hypothesis of a theorem is composite, that is, consists of several distinct hypotheses, every theorem formed by interchanging the conclusion and one of the hypotheses is an inverse of the original theorem. VI. On Proving Inverses. 31. Often in geometry when the inverse, or its equivalent, the obverse, of a theorem, is true, it has to be proved geomet- rically quite apart from the original theorem. But if we have proved that every x is y, and also that there is but one individ- ual in the class y^ then we infer that y is x. The extra-logical proof required to establish an inverse is here contained in the proof that there is but one y. Rule of Identity. 32. If it has been proved that x is j, that no two ;tr's are the same /, and that there are as many individuals in class x as in class /, then we infer y is x. ON LOGIC, Rule of Inversion. 33. If the hypotheses of a group of demonstrated theorems exhaustively divide the universe of discourse into contradic- tories, so that one must be true, though we do not know which, and the conclusions are also contradictories, then the inverse of every theorem of the group will necessarily be true. Examples occur in geometry of the following type : — If a is greater than by then c is greater than d. If a is equal to by then c is equal to d. If a is less than by then c is less than d. Three such theorems having been demonstrated geometri- cally, the inverse of each is always and necessarily true. Take, for instance, the inverse of the first ; namely, when c is greater than dy then a is greater than b. This must be true ; for the second and third theorems imply that' if a is not greater than by then c is not greater than d. THE ELEMENTS OF GEOMETRY. CHAPTER IL THE PRIMARY CONCEPTS OF GEOMETRY. I. Definitions of Geometric Magnitudes. 34. Geometry is the science which treats of the properties of space. 35. A part of space occupied by a physical body, or marked out in any other way, is called a Solid. 3 D 36. The common boundary of two parts of a solid, or of a solid and the remainder of space, is a Surface. 37. The common boundary of two parts of a surface is a Line. 38. The common boundary of two parts of a line is a Point. 39. A Magnitude is any thing which can be added to itself so as to double. 40. A point has position without magnitude. 41. A line may be conceived of as traced or generated by a point on a moving body. The intersection of two lines is a point. 42. A line on a moving body may generate a surface. The intersection of two surfaces is a line. THE PRIMARY CONCEPTS OF GEOMETRY. 9 43. A surface on a moving body may generate a solid. 44. We cannot picture any motion of a solid which will generate any thing else than a solid. Thus, in our space experience, we have three steps down from a solid to a point which has no magnitude, or three steps up from a point to a solid ; so our space is said to have three dimensions. 45. A Straight Line is a line which pierces space evenly, so that a piece of space from along one side of it ^ will fit any side of any other portion. 46. A Curve is a line no part of which is ^^ "^>. straight. a curve. 47. Take notice: the word "line," unqualified, will hence- forth mean " straight line." 48. A Sect is the part of a line between two definite points. 49. A Plane Surfacey or a Plane, is a surface which divides space evenly, so that a piece of space from along one side of it will fit either side of any other portion. 50. A plane is generated by the motion of a line always passing through a fixed point and leaning on a fixed line. 51. A Figure is any definite combination of points, lines, curves, or surfaces. 52. A Plane Figure is in one plane. 10 THE ELEMENTS OF GEOMETRY, 53. If a sect turns about one of its end points, the other end point describes a curve which is called a Circle, The fixed end point is called the Center of the Circle, c 54. The Radius of a Circle is a sect drawn from the center to the circle. 55. A Diameter of a Circle is a sect drawn through the center, and terminated both ways by the circle. 56. An Arc is a. part of a circle. 57. Parallel Lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. a J) 58. When two lines are drawn from the same point, they are said to contain, or to make with each other, an Angle. THE PRIMARY CONCEPTS OF GEOMETRY. II The point is called the Vertex, and the lines are called the Arms, of the angle. A line drawn from the vertex, and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other, is said to turn through the angle ; and the angle is greater as the quantity of turning is greater. 59. Since the line can turn from the one position to the other in either of two ways, two angles are formed by two lines drawn from a point. Each of these angles is called the Explement of the other. If we say two lines going out from a point form an angle, we \^ are fixing the attention upon one of the two explemental angles which they really form ; and usually we mean the smaller angle. 60. Two angles are called Equal if they can be placed so that their arms coincide, and that both are described simul- taneously by the turning of the same line about their common vertex. 61. If two angles have the vertex and an arm in common, and do not overlap, they are called Adjacent Angles ; and the angle made by the other two arms on the side toward the com- mon arm is called the Sum of the Adjacent Angles, Thus, 12 THE ELEMENTS OF GEOMETRY. using the sign "4- for the word "angle," the sign of equality (=), and the sign of addition (+, plus), %. AOC = ^ AOB + t BOC. 62. A Straight Angle has its arms in the same line, and on different sides of the vertex. 63. The sum of two adjacent angles which have their exterior arms in the same line on different sides of the vertex is a straight angle. 64. When the sum of any two angles is a straight angle, each is said to be the Supplement of the other. O O2 Oi it 65. If two supplemental angles be added, their exterior THE PRIMARY CONCEPTS OF GEOMETRY. 13 arms will form one line ; and then the two angles are called Supplemental Adjacent Angles, 66. A Right Angle is half a straight angle. 67. A Perpendicular X,o a line is a line that makes a right angle with it. 68. When the sum of two angles is a right angle, each is said to be the Complement of the other. 69. An Acute Angle is one which is less than a right angle. 70. An Obtuse Angle is one which is greater than a right angle, but less than a straight angle. 71. The whole angle which a sect must turn through, about one of its end points, to take it all around into its first position, 14 THE ELEMENTS OF GEOMETRY. or, in one plane, the whole amount of angle round a point, is called a Perigon. J ^ 72. Since the angular magnitude about a point is neither increased nor diminished by the number of lines which radiate from the point, the sum of all the angles about a point in a plane is a perigon. 73. A Reflex Angle is one which is greater than a straight angle, but less than a perigon. 74. Acute, obtuse, and reflex angles, in distinction from right angles, straight angles, and perigons, are called Oblique Angles ; and intersecting lines which are not perpendicular to each other are called Oblique Lines. 75. When two lines intersect, a pair of angles contained by the same two lines on different sides of the vertex, having no arm in common, are called Vertical Angles. 76. That which divides a magnitude into two equal parts is said to halve or bisect the magnitude, and is called a Bisector. 77. If we imagine a figure moved, we may also suppose it to leave its outline, or Trace^ in the first position. THE PRIMARY CONCEPTS OF GEOMETRY. IS 78. A Triangle is a figure formed by three lines, each inter- secting the other two. 79. The three points of intersection are the three Vertices of the triangle. 80. The three sects joining the vertices are the Sides of the triangle. The side opposite A is named a ; the side opposite 81. An Interior Angle oi a triangle is one between two of the sects. 82. An Exterior Angle of a triangle is one between either sect and a line which is a continuation of another side. 83. Magnitudes which are identical in every respect except the place in space where they may be, are called Congruent. 84. Two magnitudes are Equivalettt which can be cut into parts congruent in pairs. II. Properties of Distinct Things. 85. The whole is greater than its part. 86. The whole is equal to the sum of all its parts. 87. Things which are equal to the same thing are equal to one another. 1 6 THE ELEMENTS OF GEOMETRY, 88. If equals be added to equals, the sums are equal. 89. If equals be taken from equals, the remainders are equal. 90. If to equals unequals are added, the sums are unequal, and the greater sum comes from adding the greater magnitude. 91. If equals are taken from unequals, the remainders are unequal, and the greater remainder is obtained from the greater magnitude. 92. Things that are double of the same thing, or of equal things, are equal to one another. 93. Things which are halves of the same thing, or of equal things, are equal to one another. III. Some Geometrical Assumptions about Euclid's Space. 94. A solid or a figure may be imagined to move about in space without any other change. Magnitudes which will coin- cide with one another after any motion in space, are congruent ; and congruent magnitudes can, after proper turning, be made to coincide, point for point, by superposition. 95. Two lines cannot meet twice ; that is, if two lines have two points in common, the two sects between those points coincide. 96. If two lines have a common sect, they coincide through- out. Therefore through two points, only one distinct line can pass. 97. If two points of a line are in a plane, the line lies wholly in the plane. 98. All straight angles are equal to one another. 99. Two lines which intersect one another cannot both be parallel to the same line. TABLE OF SYMBOLS. 17 IV. The Assumed Constructions. 100. Let it be granted that a line may be drawn from any- one point to any other point. loi. Let it be granted that a sect or a terminated line may be produced indefinitely in a line. 102. Let it be granted that a circle may be described around any point as a center, with a radius equal to a given sect. 103. Remark. — Here we are allowed the use of a straight edge, not marked with divisions, and the use of a pair of com- passes ; the edge being used for drawing and producing lines, the compasses for describing circles and for the transferrence of sects. But it is more important to note the implied restriction, namely, that no construction is allowable in elementary geom- etry which cannot be effected by combinations of these primary constructions. SYMBOLS USED. ~ similar. = equivalent. ^ congruent. II parallel to. _L perpendicular to. -i- plus. .'. therefore. ^ angle. < less than. > greater than-. A triangle. £y parallelogram* O circle, rt. right. St. straight. i8 THE ELEMENTS OF GEOMETRY. CHAPTER III. PRIMARY RELATIONS OF LINES, ANGLES, AND TRIANGLES. I. Angles about a Point. Theorem I. 104. All right angles are equal. Such a proposition is proved in geometry by showing it to be true of any two right angles we choose to take. ^ 3 G J£ D C O The hypothesis will be, that we have any two right angles ; as, for instance, right 4. A OB and right i. FHG. By the definition of a right angle (^6)^ the hypothesis will mean, :i A OB is half a straight angle at O, and ^ FHG is half a straight angle at H. The conclusion is, that ^ AOB and ^ FHG are equal. The proof consists in stating the equality of the two straight angles of which the angles AOB and FHG are halves, referring to the assumption (in 98) that all straight angles are equal, and then stating that therefore the right ^ AOB equals the right 4 FHG, because, by 93, things which are halves of equal things are equal. RELATIONS OF LINES, ANGLES, AND TRIANGLES. 1 9 Now, we may restate and condense this as follows, using the abbreviations rt. for " right," and st. for " straight," and the symbol .'. for the word "therefore" : Hypothesis. -4. A OB is rt ^ ; also ^ FUG is rt ^, Conclusion. ^ AOB = ^ FHG. Proof. ^ AOB \s half a st. ^ ; also ^ FBG is half a st. ^. (66. A right angle is half a straight angle.) By 98, all straight angles are equal, .-. ^AOB = ^FHG. (93. Halves of equals are equal.) 105. Corollary. From a point on a line, there cannot be more than one perpendicular to that line. Theorem II. 106. All perigons are equal. Xb^ JC i< ^ Hypothesis. 4- ^ ^^ '-^ ^ perigon ; also ^ DHD is a perigon. Conclusion. 4A0A= 4. DHD. Proof. Any line through the vertex of a perigon divides it into two straight angles. By 98, all straight angles are equal, .*. Perigon at O equals perigon at H. {92. Doubles of equals are equal.) 107. Corollary. From the preceding demonstration, it follows that half a perigon is a straight angle. 20 THE ELEMENTS OF GEOMETRY. Theorem III. 1 08. If two lines intersect each other ^ the vertical angles are equal. Hypothesis. AC and BD are two lines intersecting at O. Conclusions. ^ AOB = ^ COD. ^BOC ^ y^DOA. Proof. Because the sum of the angles AOB and BOC is the angle AOC, and by hypothesis AO and DC are in one line, .-. ^ AOB + 4. BOC = St. 4. In the same way, ^BOC + 4COD = St. 2^, .-. i.AOB^-4 BOC = i. BOC + 4. COD. Take away from these equal sums the common angle BOC^ and we have 4. AOB = 4COD. (89. If equals be taken from equals, the remainders are equal.) In the same way we may prove 4 BOC = 4DOA. Exercises, i. Two angles are formed at a point on one side of a line. Show that the lines which bisect these angles contain a right angle. RELATIONS OF LINES, ANGLES, AND TRIANGLES. 21 Theorem IV. 109. If four lines go out from a point so as to make each angle equal to the one not adjacent to it, the four lines will form only two intersecting li7ies. Hypothesis. Let OA, OB, OC, OD, be four lines, with the common point O ; and let 4. AOB = ^ COD, and 4.BOC = ^ DOA. Conclusions. AO and OC are in one line. BO and OD are in one line. Proof. ^ AOB + 4. BOC -f 4 COD + 4. DOA = a perigon, (71. The sum of all the angles about a point in a plane is a perigon.) By hypothesis, 4 AOB = 4 COD, and 4 BOC = 4 DOA, .*. twice 4 AOB ■+■ twice 4 BOC = a perigon; .-. 4 AOB + 4 BOC = a st. 4. (107. Half a perigon is a straight angle.) .'. AOand OC form one line. (65. If two supplemental angles be added, their exterior arms will form one line.) In the same way we may prove that 4 AOB and 4 DOA are sup- plemental adjacent angles, and so that BO and OD form one Hne. no. Corollary. The four bisectors of the four angles formed by two lines intersecting, form a pair of lines perpen- dicular to each other. 22 THE ELEMENTS OF GEOMETRY. II. Angles about Two Points. 111. A line cutting across other lines is called a Transversal. 112. If in a plane two lines are cut in two distinct points by a transversal, at each of the points four angles are deter- mined. Of these eight angles, four are between the two given lines (namely, ^, J, «, ^), and are called Interior Angles ; the other four lie outside the two lines, and are called Exterior Angles. Angles, one at each point, which lie on the same side of the transversal, the one exterior, the other interior, like / and a, are called Corresponding Angles. Two angles on opposite sides of the transversal, and both interior or both exterior, like J and a, are called Alternate Angles. Theorem V. 113. If two corresponding or two alternate angles are equal y or if two interior or two exterior angles on the same side of the transversal are supplemental, then every angle is equal to its, cor- respondijtg and to its alternate angle, aiid is supplemental to the angle on the same side of the transversal which is interior or exterior according as the first is interior or exterior. RELATIONS OF LINES, ANGLES^ AND TRIANGLES. 23 First Case. — Hypothesis, -^.a — 4. i. Conclusions. 2\ Given, AB, a lino ; a, a sect Required, to mark off a sect on AB equal to a. Construction. From any point (7 as a center, on AB, describe the arc of a circle with a radius equal to a. If D be the point in which the arc intersects AB, then CD will be the required sect. Proof. All the radii of the circle around O are, by construction, equal to a. OD is one of these radii, therefore it is equal to a. THE ELEMENTS OF GEOMETRY. Problem III. 134. To bisect a given angle. Given, ihe angle AOD. Required, to bisect it. Construction. From (9 as a center, with any radius, OA, describe the arc of a circle, cutting the arms of the angle in the points A and B. Join AB, On AB, on the side remote from (9, describe, by 132, an equilateral triangle, ABC. Join OC. The line OC will bisect the given angle AOD. Proof. In the triangles CAO and CBO we have OA = OB, CA = CB by construction, and the side CO common. .-. A CAO ^ A CBO, (129. Triangles with the three sides respectively equal are congruent.) .-. ^ COA = ^ COB, .'. CO is ihe Usector of ^ A OB. Exercises, i. Describe an isosceles triangle having each of the equal sides double the base. 2. Having given the hypothenuse and one of the sides of a right-angled triangle, construct the triangle. PROBLEMS, 33 Problem IV. 135. Through a given point on a given line, to draw a perpen- dicular to this line. ^\ C i :b Given, ihe line AB, and the point C in it Required, to draw from C a perpendicular to AB. Solution. By 1 34, bisect the st. 2^ A CB, Exercises. 5. Solve 134 without an equilateral triangle. 6. Divide a given angle into four equal parts. 3. Find a point in a line at a given distance from a givea point. When is the problem impossible } 4. Find a point in a line equally distant from two given points without the line. 5. From a given point without a given line, draw a line making with the given line an angle equal to one-half a right angle. 6. y4 is a point without a given line, EC. Construct an isosceles triangle, having A as vertex, whose base shall lie along EC, and be equal to a given sect. II. If a right-angle triangle have one acute angle double the other, the hypothenuse is double one side. 34 THE ELEMENTS OF GEOMETRY. Problem V. 136. To bisect a given sect. A # I * / • *« / I « / ' • * ' \ c : 1 Given, the sect AB. Required, to bisect it. Construction. On AB describe an equilateral triangle ABCj by 132. By 134, bisect the angle ACB by the line meeting AB in D, Then AB shall be bisected in Z>. Proof. In the triangles A CD and BCD ^ACD= ^BCD, and AC = BC by construction, and CD is common. .-. AD = BD, (124. Triangles are congruent if two sides and the included angle are equal in each.) AB is bisected at D. 137. Corollary I. The line drawn to bisect the angle at the vertex of an isosceles triangle, also bisects the base, and is perpendicular to it. 138. Corollary II. The line drawn from the vertex of an isosceles triangle to bisect the base, is perpendicular to it, and also bisects the vertical angle. Exercises. 12. Construct a right-angled isosceles triangle on a given sect as hypothenuse. PROBLEMS. 35 Problem VI. i39» From a point without a given line^ to drop a perpendicu- lar upon the line. ?€ i^ Given, Me line AB, and the point C without it Required, to drop from C a perpendicular upon AB. Construction. Take any point, Z>, on the other side of AB from C, and, by 102, from the center C, with radius CDy describe the arc FDG, meeting AB at F and G. By 136, bisect FG at H. Join CH, C^ shall be ±^^ (using ± for the words "perpendicular to "). Proof. Because in the triangles CHF and CHGy by construction, CF = CG, and HF = HG, and CH is common, .-. ^ CBF = ^ C^6J. (129. Triangles with three sides respectively equal are congruent.) .-. 4. FHC, being half the st. 4. FHG, is a rt. ^ ; and CH is perpendicular to AB. 140. Corollary. The line drawn from the vertex of an isosceles triangle perpendicular to the base, bisects it, and also bisects the vertical angle. Exercises. 13. Instead of bisecting FGy would it do to bisect the angle FCG } 36 THE ELEMENTS OF GEOMETRY. CHAPTER V. INEQUALITIES. 141' The symbol > is called the Sign of Inequality y and a> b means that a is greater than b\ so a ^ BCA. 4.CBD> ^BAC. Proof. By 136, bisect BC in H. By 100 and 10 1, join AH^ and produce it to F', making, by 133, HF = AH. By 100, join BF, INEQUALITIES. 37 Then, in the triangles AHC and FHB, by construction, CH = HB, AH = HF, and 4. AHC = ^ FHB ; (108. Vertical angles are equal.) /. A AHC ^ A FHB, (124. Triangles are congruent if two sides and the included angle are equal in each.) .-. 4 HCA = 4 HBF. Now, 4. CBD > 4 HBF, (85. The whole is greater than its part.) .-. 4 CBD > 4. HCA, Similarly, if CB be produced to G, it may be shown that 4ABG>4 BAC, .-. the vertical 4 CBD > 4 BAC. Theorem XL 143. Any two angles of a triangle are together less than a straight angle, jb Hypothesis. Let ABC be any A. Conclusion. 4 A 4- ^ ^ < st. ^ . Proof. Produce CA to D. Then 4 BAD >4B, (142. An exterior angle of a triangle is greater than either opposite interior angle.) .-. 4 BAD + 4 BAC>4 B -{- 4 BAC, .', st.4>4B-\-4 A, 144. Corollary I. No triangle can have more than one right angle or obtuse angle. 145. Corollary II. There can be only one perpendicular from a point to a line. 38 THE ELEMENTS OF GEOMETRY, Theorem XII. 146. If one side of a triangle be greater than a second^ the angle opposite the first must be greater than the angle opposite^ the second. ^ Hypothesis, a ABC, with side a > side c. Conclusion. ^ BA C> 4.C. Proof. By 133, from a cut off BD = c. By 100, join AD. Then, because BD = c, :. ^ BDA = ^ BAD. (126. In an isosceles triangle the angles opposite the equal sides are equal.) And, by 142, the exterior ^ BDA of A CD A > the opposite inte- rior ^ C, .-. also ^ BAD >^C. Still more \s^BAC>:^C. 147. Corollary. If one side of a triangle be less than the second, the angle opposite the first will be less than the angle opposite the second. For, if ^ < ^, :. b > a\ :. by 146, 4. B>4.A, .\ '4.A<:4.B. 148. Inverses. We have now proved, By 146, \i a> b, .'. 4. A > 4. B \ Byi26, if« = ^, .-. ^A—^B'j By 147, ii a < b, .*. :^ A < ^ B. Therefore, by 33, the inverses are necessarily true, namely, If :^A > ^B, .'. a > b; If ^A = ^B, .-. a =: b; If 4 A < 4.B, .-. a < b. 149. Corollary. An equiangular triangle is also equi- lateral. INEQUALITIES. 39 Theorem XIII. 150. The perpendicular is the least sect between a given point and a given line. Ji ^ Hypothesis, lei A he a given point, and BC a given line. Construction. By 139, from A drop AD A.BC. By 100, join A to Fy any point oi BC except D. Conclusion. AD < AF. Proof. Since, by construction, ^ ADF is rt. 4. , :. -4. ADF > i. AFD, (143. Any two angles of a triangle are together less than a straight angle.) .-. AF>AD. (148. If angle ADF is greater than angle AFD, therefore side AF is greater than side AD.) 151. Except the perpendicular, any sect from a point to a line is called an Oblique. Exercises. 14. Determine the least path from one point to another, subject to the condition that it shall meet a given straight line. 15. The four sides of a quadrilateral figure are together greater than its two diagonals. 40 THE ELEMENTS OF GEOMETRY. Theorem XIV. 152. Two obliques from a point, making equal sects from the foot of the i)erpendiculary are equal, and make equal angles with the line. Hypothesis. AC X. BD. BC = DC. Conclusions. AB = AD. 4. ABC = 4. ADC. Proof. AACB^t:^ACD. (124. Triangles are congruent if two sides and the included angle are equal in each.) 153- Corollary. For every oblique, there can be drawn one equal, and on the other side of the perpendicular. Exercises. 16. ABC is a triangle whose angle A is bisected by a line meeting BC at D. Prove AB greater than BD, and A C greater than CD. 17. Prove that the sum of the sects from any point to the three angles of a triangle is greater than one-half the perime- ter of the triangle. 18. The two sides of a triangle are together greater than twice the line drawn from the vertical angle to the middle point of the base. INEQUALITIES. 41 Theorem XV. 154. Of any two obliques betwee^t a given point and line^ that which makes the greater sect from the foot of the perpen- dicular is the greater. Hypothesis. AC 1. BD, CB > CF, Conclusion. AF < AB. Proof. By 153, take F on the same side of the perpendicular as B 'j then, in A AFCy because, by hypothesis, ^ ACF is rt., /. ^ AFC < rt. ^, (143. Any two angles of a triangle are together less than a straight angle.) .-. ^ AFB > rt. ^, since ^ AFC + ^ AFB = st. ^. /. ^ ABF < i AFB, (143. Any two angles of a triangle are together less than a straight angle.) /. AFc. a -{- ob. b -\- oa. Proof. By loi, produce BC to D; making, by 133, CD = b. By 100, join AD. Then, because AC = CD, ■ .'. t CDA = :^ CAD, Now, ^ BAD > 4. CAD, (85. The whole is greater than its part.) .*. also ^ BAD > 4. BDA. By 148, .-. BD>BA, But BD = a + b, and BA is c, .*. a -{- b>c. Similarly it may be shown that a -j- ob, and that b + oa, 157. Corollary. Any side of a triangle is greater than the difference between the other two sides. INEQUALITIES. 43 Theorem XVII. 158. If from the ends of one of the sides of a triangle two sects be drawn to a point within the triangle^ these will be together less than the other two sides of the triangle^ but will contain a greater angle. Hypothesis. D is a point within A ABC. Conclusions. (I.) AB + BC>AD + DC. (II.) 4. ADC >B. Proof. (I.) Join AD and CD. Produce CD to meet AB in F. Then {CB + BF) > {CF), and \_DF + FA;\ > [DA-]. (156. In a triangle any two sides are together greater than the third.) /. CB -h BA = (CB + BF) ^ FA> {CF) + FA =^ CD + IDF + FA'\ >CD + IDA']. (II.) Next, in A ^i^Z? :^ADC>:^AFD. (142. Exterior angle of a triangle is greater than opposite interior angle.) And for the same reason, in A CFB 4AFD>4.B, /. i.ADC>4.B. Exercises. 19. The perimeter of a triangle is greater than the sum of sects from a point within to the vertices. 44 THE ELEMENTS OF GEOMETRY. Theorem XVIII. 159. If two triangles have two sides of the one equal respec- tively to two sides of the other y but the included angle of the first greater than the included angle of the second^ then the third side of the first will be greater than the third side of the second. Hypothesis. ABC and DEF are two fr/ang/es, in which AB = DE, BC = EF, ^ ABC > 4. DEF, Conclusion. AC> DF. Proof. Place the triangles so that EF shall coincide with BC, and the point D fall on the same side of j5C as A. By 134, bisect ^ ABD, and let G be the point in which the bisector meets A C. By 100, join DG. Then, in the triangles ABG and BDG, by construction, ^ ABG = -4. DBG'i by hypothesis, AB = BD; and BG is common. .-. A ABG ^ A DBG, (124. Two triangles are congruent if two sides and the included angle are equal in each.) .-. AG = DG, .-. CA = CG i- GA = CG -{- GD. INEQUALITIES. 45 But CG -\- GD> CD, (156. Any two sides of a triangle are together greater than the third.) .-. CA>FD, 160. Corollary. U c = f^ a = dy and i.B<^ E, then i.E>'4.B\ and so, by 1 59, e>by .'. b 4.E, By 124, if ^^ = ;^^, By 160, if ^B < 4.E, Therefore, by 33, Rule of Inversion, 3 > e, :. ^B > ^E] If d = e, .-. ^B = ^E; If b < e, .'. ^B < ^E, Exercises. 20. In a ABC, AB < AC. D is the middle point of BC. Prove the angle ADB acute. 46 THE ELEMENTS OF GEOMETRY. Problem VI I. 162. To construct a triangle of which the sides shall be equal to three given sects, but any two whatever of these sects must be greater than the third. \ ^ ' \ » :d\ \ : a Z-^" 3F ^ » ' '•• *m^m' Tr c - Given, fhe three sects a, b, c, any two whatever greater than the third. Required, to make a triangle having its sides equal respectively to a, b, c. Construction. On a line DF, by 133, take a sect OG equal to one of the given sects, as b. From O as a, center, with a radius equal to one of the remaining sects, as ^, describe, by 102, a circle. From (9 as a center, with a radius equal to the remaining sect a, describe an arc intersecting the former circle at X. Join XO and XG, KGO will be the triangle required. Proof. By the construction and the equality of all radii of the same circle, the three sides GK^ KO, and OG are equal respectively to a, b, c. Limitation. It is necessary that any two of the sects should be greater than the third, or, what amounts to the same, the difference of any two sides less than the third. For, if a and c were together less than b, the circles in the figure would not meet ; and, if they were together equal to b, the point K INEQUALITIES. 47 would be on OG, and the triangle would become a sect. But as we have proved, in 156, that any two sides of a triangle are together greater than the third side, our solution of Problem VII will apply to any three sects that could possibly form a triangle. 163. Corollary. If a tri- angle be made of hinged rods, though the hinges be entirely free, the rods cannot turn upon them ; the triangle is rigid. Problem VIII. 164. At a given point in a given liney to make an angle equal to a given angle. Given, the point A, and the line AB and 4. C, Required, at A, with arm AB, to niake an angle = ^C Construction. By 100, join any two points in the arms of ^ C, thus forming a A DCE, By 162, make a triangle whose three sides shall be equal to the three sides oi DCE] making the sides equal to CD and CE intersect at A. .-. -2^ FAG ^ i. DCE, (zag. Triangles with three sides respectively equal are congruent.) 48 THE ELEMENTS OF GEOMETRY. CHAPTER VI. PARALLELS. Theorem XIX. ^^S* -^ t'^o lines cut by a transversal make alternate angles egualy the lines are parallel. / 7 This is the contranominal of 142, part of which may be stated thus : If two lines which meet are cut by a transversal, their alternate angles are unequal. 166. Corollary. If two lines cut by a transversal make corresponding angles equal, or interior or exterior angles on the same side of the transversal supplemental, the lines are parallel. For we know, by 113, that either of these suppositions makes also the alternate angles equal. Exercises. 21. From a given point without a given line, draw a line making an angle with the given line equal to a given angle. PARALLELS. 49 Problem IX. 167. Through agivenpomtf to draw a line parallel to a given line. /-? Given, a line AB and a point P. Required, to draw a line through P || AB (using II for the words "parallel to"). Construction. In AB take any point, as C. By 100, join PC, At P in the Une CP, by 164, make 4. CPD = 4. PCB. PD is II AB. Proof. By construction, the transversal PC makes alternate angles equal ; 4. CPD = 4 PCB, .', by 165, PD is II AB. Exercises. 22. Draw, through a given point between two^ lines which are not parallel, a sect, which shall be terminated by the given lines, and bisected at the given point. 23. If a line bisecting the exterior angle of a triangle be- parallel to the base, show that the triangle is isosceles. 24. Sects from the middle point of the hypothenuse of a/, right-angled triangle to the three vertices are equal. 25. Through two given points draw two lines, forming,, with a given line, a triangle equiangular with a given triangle. 26. AC, BD, are equal sects, drawn from the extremities of. the sect AB on opposite sides of it, such that the angles BAC„ ABD, are together equal to a straight angle. Prove that AB bisects CD. 50 THE ELEMENTS OF GEOMETRY, Theorem XX. 1 68. If a transversal cuts two parallels y the alternate angles are equal. /L Hypothesis. AB H CD cuf at H and K by transversal FG. Conclusion. 4. HKD = ^ KHB. Proof. A line through H, making alternate angles equal, is paral- lel to CD, by 165. By our assumption, 99, two different Imes through H cannot both be parallel to CD ; .*• by 31, the line through H |I CD is identical with the line which makes alternate angles equal. But, by hypothesis, AB is parallel to CD through H, .-. i. KHB = ^ HKD. 169. Corollary I. If a transversal cuts two parallels, it makes the alternate angles equal; and therefore, by 113, the corresponding angles are equal, and the two interior or two exterior angles on the same side of the transversal are supple- mental. 170. Corollary II. If a line be perpendicular to one of two parallels, it will be perpendicular to the other also. 171. CoNTRANOMiNAL OF i68. If alternate angles are un- equal, the two lines meet. PARALLELS. 51 So, if the interior angles on the same side of the transversal are not supplemental, the two lines meet ; and as, by 143, they cannot meet on that side of the transversal where the two interior angles are greater than a straight angle, therefore they must meet on the side where the two interior angles are together less than a straight angle. 172. CoNTRANOMiNAL OF 99. Lines in the same plane par- allel to the same line cannot intersect, and so are parallel to one another. Theorem XXI. 173. Each exterior angle of a triafigle is equal to the sum of the two interior opposite angles. Hypothesis. ABC any a, wiih side AB produced to D, Conclusion. ^ CBD = ^A -^ ^C. Proof. From B, by 167, draw BF \\ AC. Then, by 168, ^ C ^ CBF, and by 169, :^ A = ^ DBF\ :. by adding, 4. A ^ t^C = 4. FBD + ^ CBF = 4. CBD. Exercises. 2y. Each angle of an equilateral triangle is two-thirds of a right angle. Hence show how to trisect a right angle. 28. If any of the angles of an isosceles triangle be two- thirds of a right angle, the triangle must be equilateral. 52 THE ELEMENTS OF GEOMETRY. Theorem XXII. 174. The sum of the three interior angles of a triangle is equal to a straight angle. Hypothesis. ABC any A. Conclusion. ^ CAB + 4. B -^r ^C=st. ^. Proof. Produce a side, as CAy to D. By 173, tB ^-tC = 4 DAB. Add to both sides 4. BAC. .-. 4 CAB + ^^ + ^C = ^ DAB + 4. BAC = st. ^. 175. Corollary. In any right-angled triangle the two acute angles are complemental. TRIANGLES. 53 CHAPTER VII. triangles. Theorem XXIII. 176. Two triangles are congruent if two angles and an oppO' site side in the one are equal respectively to two angles and the corresp07iding side in the other. Hypothesis. ABC and DFG As, with ^A= tD, ^C = ^G, AB = DF. Conclusion. t^ABC^ t. DFG, Proof. Byi74, 2^^-f-^^H-^C' = st. :^ = ^ D '\- -4. F -\- ^G, By hypothesis, ^ A -[- :^C = 4. D -\- ^Gy (89. If equals be taken from equals, the remainders are equal.) .-. A ABC ^ A DFG, (128. Triangles are congruent if two angles and the included side are equal in each.) 54 THE ELEMENTS OF GEOMETRY, Theorem XXIV. "^11' If two triangles have two sides of the one equal respec- tively to two sides of the other^ and the angles opposite to one pair of equal sides equals theti the angles opposite to the other pair of equal sides are either equal or supplemental. The angles included by the equal sides must be either equal or unequal. Case I. If they are equal, the third angles are equal. (174. The sum of the angles of a triangle is a straight angle.) Case II. If the angles included by the equal sides are unequal, one must be the greater. Hypothesis. ABC and FGH as, with ¥^ =^^, AB = FG, BC = GH, ^ABC > i.G. Conclusion. ^ C -f ^ -^ = st. 4^ . Proof. By 164, make the 4. ABD = :^Gy .-. A ABD ^ A FGH, (128. Triangles are congruent if two angles and the included side are equal in each.) .-. ^ BDA ^ tH> and BD = GH, TRIANGLES. 55 But, by hypothesis, BC =^ GH, BC = BD, :, i. BDC = 4.C. (126. In an isosceles triangle the angles opposite equal sides are equal.) But 4. BDA + t BDC = St. ^, /. ^H ^ 4.C ^ St. ^. 178. Corollary I. If two triangles have two sides of the one respectively equal to two sides of the other, and the angles opposite to one pair of equal sides equal, then, if one of the angles opposite the other pair of equal sides is a right angle, or if they are oblique but not supplemental, or if the side opposite the given angle is not less than the other given side, the tri- angles are congruent. 179. Corollary II. Two right-angled triangles are con- gruent if the hypothenuse and one side of the one are equal respectively to the hypothenuse and one side of the other. On the Conditions of Congruence of Two Triangles. 180. A triangle has three sides and three angles. The three angles are not all independent, since, whenever two of them are given, the third may be determined by taking their sum from a straight angle. In four cases we have proved, that, if three independent parts of a triangle are given, the other parts are determined ; in other words, that there is only one triangle having those parts : — (124) Two sides and the angle between them. (128) Two angles and the side between them. (129) The three sides. (176) Two angles and the side opposite one of them. In the only other case, two sides and the angle opposite one 56 THE ELEMENTS OF GEOMETRY. of them, if the side opposite the given angle is shorter than the other given side, two different triangles may be formed, each of which will have the given parts. This is called the ambiguous case. Suppose that the side a and side r, and 4. C, are given. If a> c, then, making the 2^ Cy and cutting off CB = a, taking B as center, and describing an arc with radius equal to c, it may cut CD in two points ; and the two unequal triangles ABC and ABC will satisfy the required conditions. In these the angles opposite the side BC are supplemental, by 177. Loci. 181. The aggregate of all points and only those points which satisfy a given condition, is called the Loctis of points satisfying that condition. 182. Hence, in order that an aggregate of points, L, may be properly termed the locus of a point satisfying an assigned con- dition, d it is necessary and sufficient to demonstrate the fol- lowing pair of inverses : — (i) If a point satisfies C, it is upon L. (2) If a point is upon L, it satisfies C We know from 18, that, instead of (i), we may prove its contranominal : — (3) If a point is not upon Z, it does not satisfy C TRIANGLES. 57 Also, that, instead of (2), we may prove the obverse of (I):- (4) If a point does not satisfy C it is not upon L, Theorem XXV. 183. The locus of the point to which sects from two give7t points are equal is the perpendicular bisector of the sect joining them. T 1 \ \ , ,' *\ i ; C :b Hypothesis. AB a sect, C its mid-point PA = PB, Conclusion. PC J_ AB. Proof. Join PC. Then A ACP is ^ A BCP, (129. Triangles with the three sides respectively equal are congruent.) .-. i.ACP = 4.BCP, CP is the perpendicular bisector of AB. 184. Inverse. Hypothesis. P, any point on CP ± AB at its mid-point C. Conclusion. PA = PB. Proof, t^ ACP ^ aBCP. (124. Two triangles are congruent if two sides and the included angle are equal in each.) 58 THE ELEMENTS OF GEOMETRY, Theorem XXVI. 185. The locus of points from which perpendiculars on two given intersecting lines are equal, consists of the two bisectors of the angles betwee7i the given lines. Hypothesis. Given AB and CD intersecting at O, and P any point such that PM 1. AB ^ PN 1. CD, Conclusion. ^ POM = 4. PON. Proof, a POM ^ A PON, (179. Right triangles having hypothenuse and one side equal are congruent.) Therefore all points from which perpendiculars on two intersecting lines are equal, lie on the bisectors of the angles between them. 186. Inverse. Hypothesis. P, any point on bisector. Conclusion. PM = PN. Proof, a POM ^ PON. (176. Triangles having two angles and a corresponding side equal in each are congruent.) Hence, by 110, the two bisectors of the four angles between AB and CD are the locus of a point from which perpendiculars on AB and CD are equal. Intersection of Loci. 187. Where it is required to find points satisfying two con- ditions, if we leave out one condition, we may find a locus of points satisfying the other condition. TRIANGLES. 59 Thus, for each condition we may construct the correspond- ing locus ; and, if these two loci have points in common, these points, and these only, satisfy both conditions. Theorem XXVII. 188. There is one^ and only one, point from which sects to three given points not on a line are equal. o ...-V ^ Hypothesis. Given A, B, and C, not in a fine. Construction and Proof. By 183, every point to which sects from A and B are equal, lies on Z>iy, the perpendicular bisector of AB. Again : the locus of points to which sects from B and C are equal is the perpendicular bisector J^J^\ Now, Z>Zy and FF' intersect, since, if they were parallel, AB, which is perpendicular to one of them, would be perpendicular to the other also, by 170, and ABC would be one line. Let them intersect in O. By 95, they cannot intersect again. 189. Thus, OB = OA, and OC = OA, .-. OB = OC, Therefore, by 183, O is on the perpendicular bisector of y^ C There- fore Corollary. The three perpendicular bisectors of the sides of a triangle meet in a point from which sects to the three ver- tices of the triangle are equal. 60 THE ELEMENTS OF GEOMETRY. Problem X. 190. To Ji7td points from which perpendiculars on three given lines which form a triangle are equal. Given, a, b, r, ihres lines inierseciing in the three disiinci points Af By C. Solution. The locus of points from which perpendiculars on the two lines a and b are equal consists of the two bisectors of the angles between the lines. The locus of points from which perpendiculars on the two lines b and c are equal, similarly consists of the two lines bisecting the angles between b and c. Our two loci consist thus of two pairs of Hnes. Each line of one pair cuts each Hne of the second pair in one point ; so that we get four points, O, O^, O2, O^, common to the two loci. 191. By 185, the four intersection points of the bisectors of the angles between a and by and between b and Cy lie on the bisectors of the angles between a and c. Therefore Corollary. The six bisectors of the interior and exterior angles of a triangle meet four times, by threes, in a point. POLYGONS. 6 1 CHAPTER VIII. POLYGONS. I. Definitions. 192. A Polygon is a figure formed by a number of lines of which each is cut by the following one, and the last by the first. 193. The common points of the consecutive lines are called the Vertices of the polygon. 194. The sects between the consecutive vertices are called the Sides of the polygon. 195. The sum of the sides, a broken line, makes the Perim- eter of the polygon. 62 THE ELEMENTS OF GEOMETRY. 196. The angles between the consecutive sides and towards the enclosed surface are called the Interior Angles of the poly- gon. Every polygon has as many interior angles as sides. 197. A polygon is said to be Convex when no one of its interior angles is reflex. 198. The sects joining the vertices not consecutive are called Diagonals of the polygon. 199. When the sides of a polygon are all equal to one another, it is called Equilateral, 200. When the angles of a polygon are all equal to one another, it is called Equiangular, POL YGONS. 63 201. A polygon which is both equilateral and equiangular is called Regular, 202. Two polygons are Mutually Equilateral if the sides of the one are equal respectively to the sides of the other taken m the same order. 203. Two polygons are Mutually Equiangular if the angles of the one are equal respectively to the angles of the other taken in the same order. 204. Two polygons may be mutually equiangular without being mutually equilateral. 205. Except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular. 64 THE ELEMENTS OF GEOMETRY. ' 206. A polygon of three sides is a Trigon or Triangle ; one of four sides is a Tetragon or Quadrilateral ; one of five sides is a Pentagon ; one of six sides is a Hexagon; one of seven sides is a Heptagon ; one of eight, an Octagon ; of nine, a Nonagon ; of ten, a Decagon; of twelve, a Dodecagon ; of fif- teen, a Quindecagon. 207. The Surface of a polygon is that part of the plane enclosed by its perimeter. 208. A Parallelogram is a quadrilateral whose opposite sides are parallel. / 209. A Trapezoid is a quadrilateral with two sides parallel. II. General Properties. Theorem XXVIII. 210. If two polygons be mutually equilateral and mutually equiangular^ they are congruent. Proof. Superposition : they may be applied, the one to the other, so as to coincide. Exercises. 29. Is a parallelogram a trapezoid ? How could a triangle be considered a trapezoid t POL YGONS. 65 Theorem XXIX. 211. The sum of the interior angles of a polygon is two less straight angles than it has sides. Hypothesis. A polygon of n sides. Conclusion. Sum of ^^s = (n — 2) st. 2^'s. Proof. If we can draw all the diagonals from any one vertex with out cutting the perimeter, then we have a triangle for every side of the polygon, except the two which make our chosen vertex. Thus, we have (« — 2) triangles, whose angles make the interior angles of the polygon. But, by 174, the sum of the angles in each triangle is a straight angle, .'. Sum of ^*s in polygon = (n — 2) st. ^'s. 212. Corollary I. From each vertex of a polygon of n sides are (u — 3) diagonals. 213. Corollary II. The sum of the angles in a quadrilat- eral is a perigon. 214 gon of 71 sides is Corollary III. Each angle of an equiangular poly- (n — 2) St. ^'s 66 THE ELEMENTS OF GEOMETRY, Theorem XXX. 215. In a convex poly go7t^ the sum of the exterior angles, one at each vertex, made by producing each side in order, is a perigon. Hypothesis. A convex polygon of n sides, each in order pro- duced at one end. Conclusion. Sum of exterior 2(^ 's = perigon. Proof. Every interior angle, as ^ A, and its adjacent exterior angle, as :^ x, together = st. ^ . .*. all interior 2^ 's + all exterior ^ 's = n St. ^ 's. But, by 211, all interior ^'s = {n — 2) st. ^'s. .*. sum of all exterior 2^'s = a perigon. Exercises. 30. How many diagonals can be drawn in a polygon of n sides } 31. The exterior angle of a regular polygon is one-third of a right angle : find the number of sides in the polygon. 32. The four bisectors of the angles of a quadrilateral form a second quadrilateral whose opposite angles are supplemental. 33. Divide a right-angled triangle into two isosceles tri- angles. 34. In a right-angled triangle, the sect from the mid-point of the hypothenuse to the right angle is half the hypothenuse. POL YGONS. 67 III. Parallelograms. Theorem XXXI. 216. If two opposite sides of a quadrilateral are equal and parallel^ it is a parallelogram. Hypothesis. ABCD a quadrilateral, with AB = and \\ CD, Conclusion. AD \\ BC. Proof. Join AC. Then ^BAC = ^ A CD. (168. If a transversal cuts two parallels, the alternate angles are equal.) By hypothesis, AB = CD, and ^ C is common, /. ^ BCA = ^ CAD, (124. Triangles are congruent if two sides and the included angle are equal in each.) .-. BC I! AD. (165. Lines making alternate angles equal are parallel.) Exercises. 35. Find the number of elements required to determine a parallelogram. 36. The four sects which connect the mid-points of the consecutive sides of any quadrilateral, form a parallelogram. 37. The perpendiculars let fall from the extremities of the base of an isosceles triangle on the opposite sides will include an angle supplemental to the vertical angle of the triangle. 38. If BE bisects the angle ^ of a triangle ABC, and CE bisects the exterior angle A CD, the angle E is equal to one-half the angle A. 6S THE ELEMENTS OF GEOMETRY. Theorem XXXII. 217. The opposite sides and angles of a parallelogram are equal to one another^ and each diagonal bisects it. Hypothesis. Let AC be a diagonal of /z? ABCD (using the sign £y for the word " parallelogram "). Conclusions. AB = CD. BC = DA. t DAB = ^ BCD. ^B =: 4.D. A ABC ^ A CDA. Proof. ^ BAC = ^ ACD, and ^ CAD= ■4.BCA) (168. If a transversal cuts parallels, the alternate angles are equal.) therefore, adding, 4. BAC + 4. CAD = 4. BAD = 4 ACD + 4 ACB = 4 BCD. Again, the side A C included between the equal angles is common, .-. aABC^ a CDA, (128. Triangles are congruent if two angles and the included side are equal in each.) .-. AB = CD, BC = DA, and 4 B = 4 D. Exercises. 39. If one angle of a parallelogram be right, all the angles are right. 40. If two parallelograms have one angle of the one equal to one angle of the other, the parallelograms are mutually equi- angular. POLYGONS. 69 218. Corollary I. Any pair of parallels intercept equal sects of parallel transversals. 219. Corollary II. If two lines be respectively parallel to two other lines, any angle made by the first pair is equal or supplemental to any angle made by the second pair. 220. Corollary III. If two angles have their arms respec- tively perpendicular, they are equal or supplemental. For, revolving one of the angles through a right angle around its vertex, its arms become perpendicular to their traces, and therefore parallel to the arms of the other given angle. 70 THE ELEMENTS OF GEOMETRY. 221. Corollary IV. If one of the angles of a parallelo- gram is a right angle, all its angles are right angles, and it is called a Rectangle, Corollary V. If two consecutive sides of a parallelo- equal, all its sides are equal, and it is called a Rhombus. gram are 223. A Square is a rectangle having consecutive sides equal. 224. Both diagonals, AC SiVid BD, being drawn, it may with a few exceptions be proved that a quadrilateral which has any two of the following properties will also have the others : 1. AB II CD. 2. BC II DA. C ^ ^777^ 3. AB = CD. BC = DA. i. DAB = 4. BCD. 4. ABC = 4 CDA. The bisection of AC hy BD. The bisection of BD by AC. The bisection of the njhy AC. The bisection of the £17 by BD. 4. 5- 6. 7- 8. 9- 10. POLYGONS. 71 These ten combined in pairs will give forty-five pairs ; with each of these pairs it may be required to establish any of the eight other properties, and thus three hundred and sixty ques- tions respecting such quadrilaterals may be raised. For example, from i and 2, 217 proves 3, /. A ABE ^ A CDE, (128. Two triangles are congruent if two angles and the included side are equal in each.) .*. the diagonals of a parallelogram bisect each other. The inverse is to prove i and 2 from 7 and 8 : If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 225. Since by 170 two lines perpendicular to one of two parallels are perpendicular to the other, and by 166 are parallel, and so by 218 the sects intercepted on them are equal, there- fore all perpendicular sects between two parallels are equal. Exercises. 41. If the mid-points of the three sides of a triangle be joined, the four resulting triangles are equal. 42. If the diagonals of a parallelogram be equal, the paral- lelogram is a rectangle. 43. If the diagonals of a parallelogram cut at right angles, it is a rhombus. 44. If the diagonals of a parallelogram bisect the angles, it is a rhombus. 45. If the diagonals of a rectangle cut at right angles, it is a square. 72 THE ELEMENTS OF GEOMETRY. Theorem XXXIII. 226. The three perpendiculars froin the vertices of a triangle to the opposite sides ineet in a point. Hypothesis. Lei Djy, EE\ FF\ be the three perpendiculars from the vertices D, E, F, io the opposite sides of A DEF. Conclusion. Dlf , EE' , FF' , meet in a point O. Proof. By 167, through D, E, F, draw AB, BC, CA \\ FE, DF, DE. Then the figures ADFE, DBFE, are parallelograms. By 217, .♦. AD = FE ^ DBy :. Z> is mid-point of AB. In same way, E and F are mid-points oi AC and BC. But, since DIf , EE, FF', are respectively ± EF, FD, DEy .'. they must he ±AB, AC, and BC, (170. A line perpendicular to one of two parallels is perpendicular to the other.) .'. by 189, they meet in a point. Exercises. 46. The intersection of the sects joining the mid-points of opposite sides of any quadrilateral is the mid- point of the sect joining the mid-points of the diagonals. POL YGONS. 73 Theorem XXXIV. 227. If three or more parallels intercept equal sects on one transversal^ they intercept equal sects on every transversal. Hypothesis. AB = BC = CD =, etc., are sects on the trans- versa/ AC, intercepted by the parallels a, b, c, d, etc. FGj GHy BKy etc., are corresponding sects on any other trans- versal. Conclusion. FG = GH = HK =, etc. Proof. From F, G, and B, draw, by 167, FL, GM, and UN, all parallel to AD ; .-. they are all equal, because AB z=z BC = CD =, etc. (2x7. Opposite sides of a parallelogram are equal.) But ^ GFL = 4. HGM = 4. KHN, and 4. FGL = 4 GHM = 4 HKN, (169. If a transversal cuts two parallels, the corresponding angles are equal.) /. A FGL ^ A GHM ^ A HKN, (176. Triangles are congruent if they have two angles and a corresponding side equal in each.) /. FG= GH =: HK. 228. Corollary. The intercepted part of each parallel will differ from the neighboring intercepts by equal sects. 74 THE ELEMENTS OF GEOMETRY. Theorem XXXV. 229. The line drawn through the mid-point of one of the non- parallel sides of a trapezoid parallel to the parallel sides^ bisects the remaining side. Hypothesis. In the trapezoid ABCD, BF = FC, and AB \\ CD WFG. Conclusion. AG ^ GD. Proof. By 227, if parallels intercept equal sects on any transversal, they intercept equal sects on every transversal. 230. Corollary I. The line joining the mid-points of the non-parallel sides of a trapezoid is parallel to the parallel sides, for we have just proved it identical with a line drawn parallel to them through one mid-point. 231. Corollary II. If through the mid-point of one side of a triangle a line be drawn parallel to a second side, it will bisect the third side ; and, inversely, the sect joining the mid- points of any two sides of a triangle is parallel to the third side and equal to half of it. 232. A sect from any vertex of a triangle to the mid-point of the opposite side is called a Medial of the triangle. POL YGONS. 75 233. Three or more lines which intersect in the same point are said to be Concurrent. 234. Three or more points which lie on the same line are said to be Collinear, Theorem XXXVI. 235. The three medials of a triangle are concurrent in a trisection point of each. Proof. Let two medials, AD and BE, meet in O, Take G the mid-point of OA, and Hoi OB, Join GH, HD, DE, EG. By 231, in A ABO, GH is equal and parallel to \AB \ so is DE in A ABC', .*. by 216, GHDE is a parallelogram. But, by 224, the diagonals of a parallelogram bisect each other, /. AG ^ GO ^ on, and BH ^ HO = OE. So any medial cuts any other at its point of trisection remote from its vertex, /, the three are concurrent. 76 THE ELEMENTS OF GEOMETRY. 236. The intersection point of medials is called the Centroid of the triangle. The intersection point of perpendiculars from the vertices to the opposite sides is called the Orthocenter of the triangle. The intersection point of perpendiculars erected at the mid-points of the sides is called the Circumcenter of the triangle. IV. Equivalence. 237. Two plane figures are called equivalent if we can prove that they must contain the same extent of surface^ even if we do not show how to cut them into parts congruent in pairs. 238. The base of a figure is that one of its sides on which we imagine it to rest. Any side of a figure may be taken as its base. 239. The altitude of a figure is the perpendicular from its highest point to the line of its base. So the altitude of a parallelogram is the perpendicular dropped from any point of one side to the line of the opposite side. 240. The words " altitude " and " base " are correlative ; thus, a triangle may have three different altitudes. POL YGONS. 77 Problem XI. 241. To describe a square upon a given sect. Given, Me SGci AB. Required, to describe a square on AB. Construction. By 135, from A draw AC 1. AB. By 133, in AC make AB> = AB. By 167, through D draw DI^ || AB. By 167, through B draw BF \\ AD. AF will be the required square. Proof. By construction, AF is a parallelogram, AB = FD, and AD = BF, 4.F = ^Ay and ^B = ^ D. (217. In a parallelogram the opposite sides and angles are equal.) But, by construction, AB = AD, .'. AF is equilateral. Again, ^ A -^ :i^ B — ?X, ■^. (169. If two parallel lines are cut by a transversal, the two interior angles are supplemental.) But, by construction, ^ A\s rt., .-. 4.A = 4.B, and all four angles are rt. 242. Corollary. it is a rectangle. If a parallelogram have one angle right, 78 THE ELEMENTS OF GEOMETRY. Theorem XXXVII. 243. In any right-angled triangle y the square on the hypothe- nuse is equivalent to the sum of the squares of the other two sides. \3t ya Hypothesis, a ABC, right-angled at B. Conclusion. Square on AB -\- square oTi BC = square on AC. Proof. By 241, on hypothenuse A C, on the side toward the A ABC, describe the square ADFC. On the greater of the other two sides, as BC, by 133, lay off CG = AB. Join FG. Then, by construction, CA = FC, and AB = CG, and ^ CAB = •4. FCG, since each is the complement of A CB \ .'. A ABC ^ A CGF. Translate the A ABC upward, keeping point A on sect AD, and point C on sect CF, until A is on D, and C on F; then call B' the position of B. Likewise translate CGF to the right until C is on A, and F on Z> j then call G^ the position of G. POL YGONS. 79 The resulting figure, AG'DB'FGBA^ will be the squares on the other two sides, AB and BC. For, since the sum of the angles at 2) = st. 2>^, .*. G'D and DB' are in one line. Produce GB to meet this line at H. Then GF = FB' = BC, and t\. 4. B' = ^ GFB' = ^ FGH, .'. GFB'H equals square on BC, Again, G^'^ = AB, and rt. ^ G^' = ^ G'^^ = ^ ^^^y, /. AG'HB is the square on ^^, .-. sq. oiAC= sq. of ^^ + sq. of ^C 244. Corollary. Given any two squares placed side by side, with bases AB and BC in line ; to cut this figure into three pieces, two of which being translated without rotation, the figure shall be transformed into one square. In AC take AD = BC, and join D to the corners of the squares opposite B. Two right-angled triangles are thus pro- duced, with hypothenuses perpendicular to one another. Trans- late each triangle along the hypothenuse of the other. 80 THE ELEMENTS OF GEOMETRY. Theorem XXXVIII. 245. Two triangles are congruent if they are mutually equi- angular^ and have corresponding altitudes equal. Hypothesis. As ABC and BDF mutually equiangular, and alti- tude BH = altitude GB. Conclusion, a ABC ^ A BDF. Proof. Rt. a BDG ^ rt. a BCH, (176. Triangles are congruent when two angles and a side in one are equal to two angles and a corresponding side in the other.) BD = BC, .: A ABC ^ A BDF. (124. Triangles are congruent when two angles and the included side are equal in each.) Exercises. 47. If from the vertex of any triangle a per- pendicular be drawn to the base, the difference of the squares on the two sides of the triangle is equal to the difference of the squares on the parts of the base. 48. Show how to find a square triple a given square. 49. Five times the square on the hypothenuse of a right- angled triangle is equivalent to four times the sum of the squares on the mcdials to the other two sides. POL YGONS. 8l Theorem XXXIX. 246. If two consecutive sides of one rectangle he respectively equal to two consecutive sides of anothery the rectangles are con- gruent. X zrC Hypothesis. T\no ^s with equal bases and equal aliiiudes. Conclusion. They can be cut into parts congruent in pairs. Construction. Place the parallelograms on opposite sides of their coincident equal bases, AB. Produce a side, as FB, which when con- tinued will enter the other parallelogram. If it cuts out of the parallel- ogram at H before reaching the side CD opposite AB, then will the other cutting side, as CB, when produced, also leave the ^ ABFG before reaching the side FG opposite AB ; that is, the point, K, where CB cuts the line A G will be on the sect A G. For, through H and K draw BL and KM \\ AB, Then, in As ABB and ABK ^1 = ^1, ^2 = ^2, (i68. If a transversal cuts two parallels, the alternate angles are equal.) and side AB is common ; .-. A ABB ^ A ABK, POL YGONS. 85 But altitude of A AHB < altitude of £jABCD, /. altitude of A ABK < altitude of £y ABFG, and K is on sect A G. Now, A ABH ^ A BHL, and A ABK ^ A BKM, (217. The diagonal of a parallelogram makes two congruent triangles.) Taking away these four congruent triangles, we have left two paral- lelograms, HLCD and KMFG, with equal bases, and^ equal but dimin- ished altitudes. Treat these in the same way as the parallelograms first given ; and so continue until a produced side, as FRQ,2irvd so the other, CRN, also, reaches the side opposite the base before leaving the paral- lelogram. Then, as before, the As FRN and QRC are mutually equiangular; but now we know their corresponding altitudes are equal. .'. by 241, they are congruent, .-. FN =. CQ, .-. GN = DQ, Also, QR ^ RF = PG, and NR ^ RC ^ PD, 4.G = ^ FRP = 4. RQD, 4.D =^4. CRP = 4. RNG, 4. DPR = 4. PRN, 4GPR = 4.PRQ', and therefore the remaining trapezoids, PRQD ^ PRNG. (210. If two polygons be mutually equilateral and mutually equiangular, they are congruent.) 251. Corollary. Since a rectangle is a parallelogram, therefore a parallelogram is equivalent to the rectangle of its base and altitude. Exercises. 65. How do you know that HLCD and KMFG have equal altitudes "i 66. How do you know, that, if Q is on the sect CD, N is on the sect FG ? S6 THE ELEMENTS OF GEOMETRY. Theorem XLII. 252. A triangle is equivalent to half the rectangle of its base and altitude. Hypothesis, a ABC, with base AC and altiiude BD. Conclusion, a AB C — half the rectangle oi AC and BD. Proof. Through A draw AF \\ CB, and through B draw BF \\ CA, meeting AF in F; .'. L^ACB ^ ^BFA. (217. The diagonal of a parallelogram bisects it.) But £jAFBC = rectangle oi AC and BD, (248. A parallelogram is equivalent to the rectangle of its base and altitude.) .-. t.ABC=^\ ^AFBC = J rectangle oi AC and BD. 253. Corollary I. All triangles on the same base having their vertices in the same line parallel to the base, are equivalent. 254. Corollary II. Triangles having their vertices in the same point, and for their bases equal sects of the same line, are equivalent. 255. Corollary III. If a parallelogram and a triangle be upon the same base and between the same parallels, the paral- lelogram is double the triangle. Exercises. 6j. Equivalent triangles on equal bases have equal altitudes. POL YGONS. Zj Theorem XLIII. 256. If through any point on the diagonal of a parallelogram two lines be drawn parallel to the sideSy the two parallelograms y one on each side of the diagonal y will be equivalejit. Hypothesis. P any point on diagonal BD of /=; ABCD\ FG and HK lines ih rough P || AB and BC respectively, and meeting the four sides in the points F, G, Hy K, Conclusion. ^AKPG=^PFCH. Proof, a ABD = a BCD. A KBP = A BFP, A GPD = A PBD. (217. The diagonal of a parallelogram bisects it.) From A ABD take away A KBP and A GPD, and we have lisft C7 AKPG. From the equal A BCD take away the equal As BFP and PHDy and we have left the £=7 PFCH, .-. ^AKPG=^PFCH, (89. If equals be taken from equals, the remainders will be equal.) 257. In figures like the preceding, parallelograms like KBFP and GPHD are called parallelograms about the diagonal BD ; while Z3^s AKPG and PFCH are called complements of par- allelograms about the diagonal BD. Exercises. 68. When are the complements of the paral- lelograms about a diagonal of a parallelogram congruent ? 88 THE ELEMENTS OF GEOMETRY. V. Problems. Problem XII. 258. To describe a parallelogram equivalent to a given tri- angle, and having an angle equal to a given angle. M JU Given, a ABC and ^ G. Required, to describe a parallelogram — A ABC, while having an angle = 2^ G. Construction. Bisect ACmD. At D, by 164, make ^ ADF = 4. G, By 167, through A draw AH \\ DF, By 167, through B draw BFH \\ CA. AHFD will be the parallelogram required. Proof. Join DB. Then A ABD = aBCD, (254. Triangles having their vertices in same point, and for bases equal sects of the same line, are equivalent.) /. A ABC is double A AB£>. But /I7AHFD is also double A ABD, (255. If a parallelogram and a triangle be upon the same base and between the same parallels, the parallelogram is double the triangle.) /. npAHFD = A ABC, and, by construction, ^ ADF = ^ G, POLYGONS. 89 Problem XIII. 259. On a given sect as base, to describe a parallelogram equivalent to a given triangle^ and having an angle equal to a given angle, X ^^ Q /V Given, Mo seci AB, a CDF, 4. G. Required, to describe on AB a parallelogram = A CDF, while having an angle = ^ G. Construction. By 258, make ^ BHKL — A CDF, and having ^ HBL = 7^ G, and place it so that BH is on line AB produced. Produce KL. Draw AM \\ BL. Join MB. 4. HKM + i- KMA = St. ^, (169. If a transversal cuts two parallels, it makes the two interior angles supple- mental.) /. ^HKM ^ 4KMB<%\..4., Therefore KH and MB meet if produced through H and B. (171. If a transversal cuts two lines, and the interior angles are not supplemental, the lines meet.) Let them meet in Q. Through Q draw QN || HA, and produce LB and MA to meet QN'm P and N. :. ^ NPBA = C7 BHLK, (256. Complements of parallelograms about the diagonal are equivalent.) /. £yNPBA = A CFD, and 4. ABP = 4 G. 260. Corollary. Thus we can describe on a given base a rectangle equivalent to a given triangle. 90 THE ELEMENTS OF GEOMETRY. Problem XIV. 261. To describe a triangle equivalent to a given polygon. J) Given, a polygon ABCDFG, Required, to constntct an equivalent triangle. Construction. Join the ends of any pair of adjacent sides, as AB and BC, by the sect CA. Through the intermediate vertex, B, draw a hne H CA^ meeting GA produced in H. Join HC. Polygon ABCDFGA = polygon BCDFGB, and we have obtained an equivalent polygon having fewer sides. Proof. /\ABC= a ABC. (253. Triangles having the same base and equal altitudes are equivalent.) Add to each the polygon A CDFGA, .'. ABCDFGA = BCDFGB. In the same way the number of sides may be still further diminished by one until reduced to three. 262. Corollary I. Hence we can describe on a given base a parallelogram equivalent to a given polygon, and having an angle equal to a given angle. 263. Corollary II. Thus we can describe on a given base a rectangle equivalent to a given polygon. 264. Remark. To compare the surfaces of different poly- gons, we need only to construct rectangles equivalent to the given polygons, and all on the same base. Then, by comparing the altitudes, we are enabled to judge of the surfaces. POL YGONS. 91 VI. Axial and Central Symmetry. 265. If two figures coincide, every point A in the one coin- cides with a point A' in the other. These points are said to correspond. Hence to every point in one of two congruent figures there corresponds one, and only one, point in the other ; those points being called " corresponding " which coincide if one of the two figures is superimposed upon the other. Hence, calling those parts corresponding which coincide if the whole figures are made to coincide, it follows, that corresponding parts of con- gruent figures are themselves congruent. Symmetry with Regard to an Axis. 266. If we start with two figures in the position of coinci- dence, and take in the common plane any line or, we may turn the plane of one figure about this line x until its plane, after half a revolution, coincides again with the plane of the other figure. The two figures themselves will then have distinct positions in the same plane ; but they will have this property, that they THE ELEMENTS OF GEOMETRY. can be made to coincide by folding the plane over along the line X, Two figures in the same plane which have this property are said to be symmetrical with regard to the line x as an axis of symmetry. Symmetry with Regard to a Center. 267. If we take in the common plane of two coincident figures any point Xy we may turn the one figure about this point so that its plane slides over the plane of the other figure without ever separating from it. Let this turning be continued until one line to Jf, and therefore the whole moving figure, has been turned through a straight angle about X. Then the two congruent figures still lie in the same plane, and have such positions that one can be made to coincide with the other by turning it in the plane through a straight angle about the fixed point X. Two figures which have this property are said to be sym- metrical with regard to the point X as center of symmetry. 268. Any single figure has axial symmetry when it can be divided by an axis into two figures symmetrical with respect to that axis, or has central symmetry when it has a center such that every line drawn through it cuts the figure in two points symmetrical with respect to this center. POL YGONS. 93 Theorem XLIV. 269. If a figure has two axes of symmetry perpendiciUar to each othery then their intersection is a center of symmetry. : ; ^^. _\;>1 .'""'" : -V-l^ - >\ - ^ '^i J For, if X and y be two axes at right angles, then to a point A will correspond a point A' with regard to x as axis. To these will correspond points A^ and A/ with regard to j as axis. These points A^ and A/, will correspond to each other with regard to x. To see this, let us first fold over along y ; then A falls on A,y and A' on A/. If we now, without folding back, fold over along x. A, and with it Ai, will fall on A\ which coincides \vith A^. At the same time OA and OA^ coincide, so that the angles A Ox and AiOx' are equal, where x' denotes the continuation of ^ beyond O. It follows, that AOAl are in a line, and that the sect AA^ is bisected at O, or (P is a center of symmetry for AAlj and similarly for Aj_ and A' , BOOK 11. RECTANGLES. 270. A Contimious Aggregate is an assemblage in which two adjacent parts have the same boundary. 271. A Discrete or Discontinuous Aggregate is one in which two adjacent parts have different boundaries. A pile of cannon balls is a discrete aggregate. We know that any adjacent two could be painted different colors, and so they have direct independent boundaries. Our fingers are a discontinuous aggregate. 272. All counting belongs first to the fingers. 273. There is implied and bound up in the word " number " the assumption that a group of things comes ultimately to the same finger, in whatever order they are counted. This proposition is involved in the meaning of the phrase *• distinct things." Any one and any other of them make two. If they are attached to two of my fingers in a certain order, they can also be attached to the same fingers in the other order. Thus, one order of a group of three distinct things can be changed into any other order while using the same fingers, and so on with a group of four, etc. 95 96 THE ELEMENTS OF GEOMETRY. 274. By generalizing the use of the fingers in counting, man has made for himself a counting apparatus, which each one carries around in his mind. This counting apparatus, the natural series of numbers, was made from a discrete aggregate, and so will only correspond exactly to discrete aggregates. 275. In a row of shot, we can find between any two, only a finite number of others, and sometimes none at all. Just so in regard to any two numbers. A row of six shot can be divided into two equal parts ; but the half, which is three, we cannot divide into two equal parts : and so in a series of numbers. 276. But in 136 we have shown how any sect whatever may be bisected, and the bisection point is the boundary of both parts. So a line is not a discrete aggregate of points. It is something totally different in kind from the natural series of numbers. 277. The science of numbers is founded on the hypothesis of the distinctness of things. The science of space is founded on the entirely different hypothesis of continuity. 278. Numbers are essentially discontinuous, and therefore unsuited to express the relations of continuous magnitudes. 279. In arithmetic we are taught to add and multiply num- bers : we will now show how the laws for the addition and multiplication of these discrete aggregates are applicable to sects, which are continuous aggregates. THE COMMUTATIVE LAW FOR ADDITION. 280. In a sum of numbers we may change the order in which the numbers are added. If X and y represent numbers, this law is expressed by the equation X -\- y = y ■\' X, RECTANGLES. g/ It depends entirely on the interchangeability of any pair of the units of numeration. 281. The sum of two sects is the sect obtained by placing them on the same line so as not to overlap, with one end point in common. Thus, the sum of the sects a and b means the sect AC^ which can be divided into two parts, AB = a, and BC = b, 282. The commutative law holds for the summation of sects. a ■\- b = b -^ a. M AC= CA') for AC revolved through a straight angle may be superimposed upon C^A\ and will coincide point for point. The more general case, where three or more sects are added, follows from a repetition of the above. Thus, the commutative law for addition in geometry depends entirely on the possibility of motion without deformation. . 98 THE ELEMENTS OF GEOMETRY. 283. The sum of two rectangles is the hexagon formed by superimposing two sides, and bringing the bases into the same line. Thus, if two adjacent sides of one are a and b, and of the other c and d^ the sum of the rectangles ab and cd is ABCDFGA. 284. The commutative law holds for the addition of rect- angles ; that is, the sum is independent of the order of summa- tion. ab •\- cd ^ cd -\- ab\ for ABCDFGA turned over may be superimposed upon AB'CD'FCA\ and will coincide with it. Since, by 263, we can describe on a given base a rectangle equivalent to a given polygon, the more general case, where three or more rectangles are added, follows from a repetition of the above. RECTANGLES. 99 THE ASSOCIATIVE LAW. 285. In getting a sum of numbers, we may add the num- bers together in groups, and then add these groups. If we use parentheses to mean that the terms enclosed have been added together before they are added to another term, this law may be expressed symbolically by the equation X -{■ {y ■\- z) = X ^ y -\- z. 286. The associative law holds for the summation of sects. a + {3+c) = a + d-\-c = AD. J£ 2 '' » a{b -\- c ■\- a)— ab -^ ac ^ adfj\ f,, \ >'"{'' *'* ;'.; We may state this in words as follows : If there be any two sects one of which is divided into any number of partSy the rectaitgle contained by the two sects is equivalent to the recta^tgles contained by the undivided sect and the several parts of the divided sect. 292. \i b '\- c =. a, then ab -^ ac ■=. a{b + c) = aa == a^. Therefore If a sect be divided into any two parts^ the rectangles con- tained by the whole and each of the parts are together equivalent to the square on the whole sect. 102 THE ELEMENTS OF GEOMETRY. 293. If r = ^, then a{b -^ c) — a{b -\- a) — ab ■\- aa ^=- ab -\- a^. amC Therefore If a sect he divided into any two parts^ the rectangle con- tained by the whole and one of the parts is equivalent to the rectangle contained by the two parts ^ together with the square on the aforesaid part. 294. The rectangle of two equal sects is a square, and {a + by is only a condensed way of writing {a + b){a -\- b). But, by the distributive law, {a + b){a + b) = {a -\- b)a -\- {a + b)b. By the commutative law, (a + b)a = a{a + b). By the distributive law, a{a -\- b) = aa -\- ab = a^ -\- ab. In the same way, ^ g (a + b)b = ab -\- b^, {a + by = a^ + ab -^ {ab + b^). RECTANGLES. 103 By the associative law, a^ -ir ab -^ {ab + <^) = ^' + {ab + ab) + b^ = a^ -^ 2ab + b', ,', {a + by = a" -\- 2ab + ^. Therefore If a sect be divided into a7iy two parts, the square on the whole sect is equivalent to the squares on the two parts y together with twice the rectangle contained by the two parts. 295. Corollary. The square on a sect is four times the square on half the sect. a :-»■ «. c z. J) a, :b I 296. By the distributive law, and 294, a{a -{- b -\- b) -\- b^ = a"" -h ab -^ ab + b^ = (a ■}- b)'. From this, if we bisect the sect AB at C, and divide it into two unequal parts at D, and for BD put a, and for DC put b, we get the theorem : If a sect is divided into two equal parts, and also into two unequal parts, the rectangle contained by the uneqtial parts, together with the square oji the line between the points of section, is equivalent to the square on half the sect. 297. If CD = b, and AC = a + b, then their sum AD = a •}- b -\- b, and their difference is a ; therefore the rectangle con- tained by their sum and difference equals a(a + 2b), which, by 296, is the difference between (a + by and b^. 104- THE ELEMENTS OF GEOMETRY. Therefore The rectangle contained by the sum and difference of any two sects is equivalent to the difference between the squares on those sects. 298. If we bisect the sect AB at C, and produce it to Z?, and for BD put a, and for BC put 3, then the above equation, a{a ■\- b '\- b) -\- b"^ =i {a •\- by, gives us the theorem : If a sect be bisected^ and.produced to any pointy the rectangle contained by the whole line thus produced and the part of it pro- duced, together with the square on half the sect bisected, is equiv- alent to the square on the line which is made up of the half and the part produced. 299. By 294, {a + by -{- a^ = a^ + 2ab + ^^ + cC, By the associative law, a" ^ 2ab -^ b" -^ a^ =^ 20" + 2ab + ^^ By the distributive law, 2a^ 4- 2ab -{- b^ = 2a{a -^ b) -\- b*. By the commutative law, 2a(a -\- b) -{- b'' = 2{a + b")a + ^, /. {a -\- by -^ a^ = 2(a -{- b)a + b^. Therefore If a sect be divided into any two parts, the squares on the whole line and on one of the parts are equivalent to twice the RECTANGLES. •105 rectangle co7ttained by the whole and that party together with the square on the other part. Z- « 300. By the commutative and distributive laws, and 294, 4(« + b)a + b^ = 4a^ + 4ab + ^^ = (2a + by. By the associative and commutative laws, (2a -{-by=^(a + a + by = (la + ^] + «)% • .-. 4{a 4- b)a 4- ^^ = {[a 4- ^] + ^y. a 4t ^ ^ Therefore 7/" a sect be divided into any two parts, four times the rect- angle contained by the whole sect and one of the parts, together a a I I06 THE ELEMENTS OF GEOMETRY. with the square on the other part ^ is equivalent to the square on the line which is made up of the whole sect and the first part. 301. By the associative and commutative laws, and 294, {\_a -f /5] + ay + ^^2 = (<5 4- ^ay + ^' = /^ 4- /i^ab + 4^^ + ^* = 20" + A,ab + 2<^ + 2«^ By the distributive law, and 294, 20" 4- Adb + 2b'- 4- 2a^ — 2{a^ -\- 2ab + b^) ■\- 20" = 2{a -\- by + 2 rt. ^, .'. a^ > b^ •{- c^ ; By 306, if ^A < rt. ^, /. a^ < b^ + c^. Therefore, by 33, Rule of Inversion, If a"" = b^ -{- f% .-. ^ ^ == rt. ^ j If a' > b^ + ^, .-. ^A> rt. ^ ; If a^ < b^ + ^, /. ^A< rt. i. RECTANGLES. 109 Theorem III, 308. In any triangle, if a medial be drawn from the vertex to base^ the sum of the sqicares on the two sides is equivalent to twice the square on half the base, together with twice the square on the medial. Hypothesis, a ABC, with medial BM = i. Conclusion, a"^ + c^ =. 2{^by + 21^. Proof. — Case I. If ^ BMA = ^ BMC, then a^ = {\by + /% and c^ = {^by + /^ (243. In a right triangle, the square of the hypothenuse equals the sum of the squares of the other sides.) Case II. If ^ BMA does not equal ^ BMC, one of them must be the greater. Call the greater BMC. Then, in the obtuse-angled triangle BMC, by 305, a-^ {^dy + i^+ 2{\bj). In A BMA, by 306, ^ + 2(|^y) = {iby + i\ Adding, «- + ^ 4- 2{\bj) = 2{\by + 2/« + 2{\bj), ,\ a^ + c" = 2{iby 4- 2i^. 309. Corollary. The difference of the squares on the two sides is equivalent to twice the rectangle of the base and the projection of the medial on it. no THE ELEMENTS OF GEOMETRY. Theorem IV. 310. The sum of the squares on the four sides of any quad- rilateral is greater than the sum of the squares on the diagonals by four times the square on the sect joining the mid-points of the diagonals. ^ -q Hypothesis. EF is the sect joining fhe mid-points of the diag- onals AC and BD of the quadrilateral ABCD. Conclusion. ~A& + 'BC^ + C^' -f 'DA' = ^C" + BL^ -\-4£F\ Proof. Draw BE and DE. By 308, _ ,.r\2 AB" + BC' = 2\—-] + 2^^, and C&-h^'= 2{^ + 2DE'. Adding, AB" -^ BC^ -\- CD" -\- DA" =^ ^(^T + ^^^ + ^^^' But, by 308, BE'+ DE'^: 4^y + 2KF^, = AC"" + B& + 4^^^- RECTANGLES. Ill 311. Corollary. The sum of the squares on the four sides of a parallelogram is equivalent to the sum of the squares on the diagonals, because the diagonals of a parallelogram bisect each other. Problem I. 312. To square atty polygon. . H r^- Given, any polygon N. Required, to describe a square equivalent to N. Construction. Describe, by 263, the rectangle ABCD = N. Then, if AB — BC^ the required square is ABCD. If AB be not equal to BC, produce BA^ and cut off AF = AD, Bisect BF in G, and, with center G and radius GB, describe FLB. Produce DA to meet the circle in H. The square on AH shall be equal to N. Proof. Join GH. Then, by 296, because the sect BF is divided equally in G and unequally in A^ rectangle BA,AF + AC^ ^ GF" ^ GH"" = AH"" + AC^, by 243 ; AH^ = rectangle BA,AF = rectangle BA,AD = N. 112 THE ELEMENTS OF GEOMETRY. Problem II. 313. To divide a given sect into two parts so that the rect- angle contained by the whole and one of the parts shall be equal to the square on the other part, Jfr ,X Given, ihe sect AB. Required, to find a point C such that rectangle AB,BC = AC*. Construction. On AB^ by 241, describe the square ABDF, By 136, bisect AF in G. Join BG, Produce FA, and make GH = GB. On AH describe the square AHKC. Then AB,BC = ZC^ Proof. Produce KC to Z. Then, by 298, because FA is bisected in G and produced to H, rectangle FH,BA + 'A& = 'GH^ = 'GB" = GA' + AB\ .-. FB,HA = AB", But, since HK = HA, .-. FH,HK = AB", Take from each the common part AL, .', HC== CD; that is, AC^ = CB,BD = CB,BA, RECTANGLES. 1 13 Problem III. 314. To describe an isosceles triangle having each of the angles at the base double the ajigle at the vertex. Construction. Take any sect, AB, and, by 313, divide it in Cso that rectangle AB,AC = ~^^. With center C and radius CB, describe a circle. With same radius, but center A, describe a circle intersecting the preceding circle in D. Join AD, BD, CD. ABD will be the triangle required. Proof. By construction, CB = CD = DA, By 134, bisect ^ ADC by DF. By 137, DF \% Jl AC, and bisects AC. Then 'AB' -f Z^ = 'BD' + 2AB,AF. (306. The square on a side opposite an acute angle is less than the squares on the other two sides by twice the rectangle of either and the projection of the other on it.) But ^C = 2AFy .'. AB,AC == AB,2AF - 2AB,AF\ .'. by our initial construction, BC^ == 2AB,AF, .-. AB" + AD" = Bif + BC\ But, by construction, AD ^ BC, .'. AB" = BD', .-. AB = BD, .'. ^ ADB = ^ DAB = ^ ACD = ^ B -^ ^ BDC = 2 ^ B. (173. The exterior angle of a triangle equals the sum of the two opposite interior angles.) BOOK III. THE CIRCLE. I. Primary Properties. 315. If a sect turns about one of its end points, the other end point describes a curve called the Circle. 316. The fixed end point is called the Center of the circle. 317. The moving sect in any position is called a Radius of the circle. 318. As the motion of a sect does not enlarge or diminish it, all radii are equal. 319. Since the moving sect, after revolving through a peri- gon, returns to its original position, therefore the moving end point describes a closed curve. This divides the plane into two surfaces, one of which is swept over by the moving sect. This finite plane surface is called the surface of the circle. Any part of the circle is called an Arc, SS5 Il6 THE ELEMENTS OF GEOMETRY. Theorem I. 320. The sect to a point fivm the ce^iter of a circle is less tha?t, equal to, or greater than, the radius, according as the point is within, on, or without the circle. Proof. If a point is on the circle, the sect drawn to it from the center is a radius, for it is one of the positions of the describing sect. Any point, Q, within the circle lies on some radius, OQR, /. OQ < OR. If ^S* is without the circle, then the sect OS contains a radius OR, :. OS > OR. 321. By 33, Rule of Inversion, a point is within, on, or with- out the circle according as its sect from the center is less than, equal to, or greater than, the radius. 322. A Secant is a line which passes through two points on the circle. THE CIRCLE. 11/ Theqrem II. 323. A secant can meet the circle in only two points. Proof. By definition, all sects joining the center to points on the circle are equal, but from a point to a line there can be only two equal sects. (155. No more than two equal sects can be drawn from a point to a line.) 324. A Chord is the part of a secant between the two points where it intersects the circle. 325. A Segment of a circle is the figure made by a chord and one of the two arcs into which the chord divides the circle. 326. When two arcs together make an entire circle, each is said to be the Expleinejit of the other. 327. When two explemental arcs are equal, each is a Semicircle. 328. When two explemental arcs are unequal, the lesser is called the Minor Arc, and the greater is called the Major Arc. 329. A segment is called a Major or Minor Segment ac- cording as its arc is a major or minor arc. Il8 THE ELEMENTS OF GEOMETRY. Theorem III. 330. A circle has only one center. Hypothesis. Lei F, G, and H be points on a O. Conclusion. O FGH has only one center. Proof. Join FG and GH. Since, by definition, a center is a point from which all sects to the circle are equal, therefore any center of a circle through F and G is in the perpendicular bisector of FG, and any center of a circle through G and H is in the perpendicular bisector of GH. (183. The locus of the point to which sects from two given points are equal is the perpendicular bisector of the sect joining them.) But these two perpendicular bisectors can intersect in only one point, /. O FGH has only one center. 331. Corollary I. The perpendicular bisector of any chord passes through the center. 332. Corollary II. To find the center of any given cir- cle, or of any given arc of a circle, draw two non-parallel chords and their perpendicular bisectors. The center is the point where these bisectors intersect. 333. A Diameter is a chord through the center. 334. A diameter is equal to two radii : so all diameters are bisected by the center of the circle, and are equal. THE CIRCLE. II9 Theorem IV. 335. Circles of equal radii are congruent. Hypothesis. Two circles of which C and O are the centers, and radius CD = radius OP. Conclusion. The circles are congruent. Proof. Apply one circle to the other so that the center O shall coincide with center C, and sect OF fall upon line CD. Then, because OF = CD, the point F will coincide with the point D. Then every particular point in the one circle must coincide with some point in the other circle, because of the equality of radii. (321. A point is on the circle when its sect from the center is equal to the radius.) /. C ^ O 6>. 336. Corollary. After being applied, as above, the second circle may be turned about its center ; and still it will coincide with the first, though the point P no longer falls upon D. Hence, considering one circle as the trace of the other, — A circle can be made to slide along itself by being turned about its center. This fundamental property of this curve allows us to turn any figure connected with the circle about the center without changing its relation to the circle. 120 THE ELEMENTS OF GEOMETRY. 337. Circles which have the same center are called Con- centric. Theorem V. 338. Different concentric circles cannot have a point in com- mon. Proof. The points of the circle with the lesser radius are all within the larger circle. (321. A point is within the circle if its distance from the center is less than the radius.) 339. First Contranominal of 338. Two different circles with a point in common are not concentric. 340. Second Contranominal of 338. Two concentric cir- cles with a point in common coincide. 341. The center of a circle is a Center of Symmetry^ the end points of any diameter being corresponding points. This follows at once from the definition of Central Sym- metry, and the fundamental property that the circle slides along itself when turned about its center, and so coincides with itself after turning about the center through any angle. The circle is the only closed curve which will slide upon its trace. THE CIRCLE. 121 Theorem VI. 342. The perpendicular from the center of a circle to a secant bisects the chord ; and, if a line thivngh the center bisect a chord not passing through the center^ it cuts, it at right angles. Proof. For any chord, there is only one perpendicular front the center, only one line through the mid-point and center, only one per- pendicular bisector; and these, by 331, are identical. (331. The perpendicular bisector of any chord passes through the center.) 343. Every diafneter is an Axis of Symmetry, For if we fold over along a diameter, every point on the part of the circle turned over must fall on some point on the other part, since its sect from the center, which remains fixed, is a radius. Inverse. Every line which is an axis of symmetry of a circle contains a diameter. For, if not, there would be a point symmetrical to the center, and this, too, would again be a center : the circle would thus have two centers. (330. A circle has only one center.) Each circle has therefore, besides its center of symmetry^ an infinite number of axes of symmetry. 122 THE ELEMENTS OF GEOMETRY. Theorem VII. 344. Every chord lies wholly within the circle. Hypothesis. Lei A and B be any two points in ABC. Conclusion. Every point on chord AB between A and B is within the O ABC. Proof. Take any point, B>, in chord AB. By 332, find O, the center of the circle. . Join OA, 0Z>, OB. Then OB makes a greater sect than OD from the foot of the per- pendicular from O on the line AB, (342. The perpendicular from center on line AB bisects chord AB.) .-. OB > OD, (r54. The oblique which makes the greater sect from the foot of the perpendicular is the greater.) /. D is within the circle. .<320. A point is within the circle if its sect from the center is less than the radius.) 345' Corollary. If a line has a point within a circle, it is a secant, for the radius is greater than the perpendicular from the center to this line : so there will be two sects from the center to the line, each equal to the radius ; that is, the line will pass through two points on the circle. Thus, again, the circle is a closed curve. THE CIRCLE. 1 23 Theorem VIII. 346. In a circle, two chords which are not both diameters do not mutually bisect each other. Hypothesis. Lei ihe chords AB, CD, which do not both pass through the center, cut one another in the point F, in the O A CBD. Conclusion. AB and CD do not mutually bisect each other. Proof. If one of them pass through the center, it is not bisected by the other, which does not pass through the center. If neither pass through the center, find the center (9, and join OF, If F is the bisection point of one of the chords, as AB, then 4. OFB = rt. 4., (342. If a line through the center bisect a chord not passing through the center, it cuts it at right angles.) .*. "4- OFD is oblique, .-. OF does not bisect CD. Exercises. 69. What is the locus of mid points of parallel chords } 70. Prove by symmetry that the diameter perpendicular to a chord bisects that chord, bisects the two arcs into which this chord divides the circle, and bisects the angles at the center subtended by these arcs. 124 THE ELEMENTS OF GEOMETRY. Theorem IX. 347. If from any point not the center^ sects he drawn to the different points of a circle, the greatest is that which meets the circle after passing through the center ; the least is part of the same line. Hypothesis. From any point A, sects are drawn to a Q DBHK, whose center is C. First Conclusion. A CB > AD. Proof. Join CD. Then, because CB = CD, .: AB = AC + CB = AC -j- CD > AD. (156. Any two sides of a triangle are greater than the third.) Second Conclusion. AH < AK. Proof. When A is within the circle, BC = KC ^>x. c Given, ihree points, A, B, and C, not in t/ie same fine. Required, to describe a circle which shall pass through all of them. Construction. By 188, find the point 6>, such that OA = OB = OC, 356. Corollary. To circumscribe a circle about any given triangle, by 355, pass a circle through its three vertices. 357. Four or more points which lie on the same circle are called Concyclic, THE CIRCLE. 1 27 Theorem XI. 358. Perpendiculars from the center on equal chords are equal ; andy on unequal chords ^ that on the greater is the lesser. C Hypothesis. From center O, OH, OK, OL, J. chords AB = CD> FG, Conclusion. OH = OK < OL. Proof. Draw the radii OA, OB, OC, OD, OF. By hypothesis, AB = CD, ,\ A OAB ^ A OCjD, (129. Triangles with three sides respectively equal are congruent.) .*. the altitude OH = corresponding altitude OK. Again, because CZ> > FG, .'. CK > FL, But 'CK' + KD* = ~0C' = OF" = FL' + T&. But CX' > 'FL\ _ .-. KO^KLCf", /. OK < OL, 359* By 33, Rule of Inversion, Chords having equal perpendiculars from the center are equal ; and, of chords having unequal perpendiculars, the one with the lesser is the greater. 128 THE ELEMENTS OF GEOMETRY. II. Angles at the Center. 360. The explemental angles at the center of a circle, whose arms are the same radii, are said to be subtended by^ or to stand upon^ the explemental arcs opposite them intercepted by the radii, the reflex angle upon the major arc. 361. A Sector is the figure formed by two radii and the arc included between them. 362. The angle of the sector is the angle at the center which stands upon the arc of the sector. 363. A given sect is said to subtend a certain angle from a given point when the lines drawn from the point to the ends of the sect form that angle. 364. An inscribed angle is formed by two chords from the same point on the circle, and is said to stand upon the arc between its arms. j<5t THE CIRCLE. 1 29 Theorem XII. 365. In the same or equal circles ^ equal arcs are congruent^ subtend equal angles at the center^ determine equal sectors^ and are stibtended by equal chords. Hypothesis. Radius OA — radius CF, and arc AB = arc FG. Conclusions. I. ^ AOB = 4. FCG. II. Sector AOB = sector FCG. III. Chord AB = chord FG. Proof. Place the sector A OB over the sector FCG so that the center O shall fall on C, and the radius OA on the line CF. Then, because OA = CF^ .*. point A falls on point F. Again, because the radii are equal, every point of the arc AB will fall on some part of the circle FG. But arc AB = arc FG, .'. point B falls on G, .'. chord AB = chord FG, i.AOB ^ 4 FCG, and sector AOB = sector FCG. 366. In the same or equal circles the stim of two minor arcs is the arc obtained by placing them on the same circle so as not to overlap, with one end point in common. I30 THE ELEMENTS OF GEOMETRY, Theorem XIII. 367. A sum of two arcs of the same or equal circles subtends an angle at the center equal to the sum of the angles which each arc subtends separately. o Proof. Placing the arcs with two end points in common at B, join B and the other end points, A and C, to the center O, Then the angles are in such a position, that, by definition 61, ^ AOB + -4. BOC = i. AOC. But 4 AOC is subtended by arc AC, which is the sum of arc AB subtending ^ AOB, and arc BC subtending ^ BOC. 368. Corollary. Two sectors of the same or equal circles may be so placed as to form a sector whose arc is the sum of their arcs, whose angle is the sum of their angles, and whose surface is the sum of their surfaces. Theorem XIV. 369. In the same or equal circles, of two uneqtial arcs, ths greater subtends the greater angle at the center, and determines the greater sector. Proof. If the first arc is greater than the second, it is equal to the second plus a third arc; and so, by 367, the angle which the first sub- tends is greater than the angle which the second subtends by the angle which the third arc subtends at the center. And like is true of the sectors. 370. From 365 and 369, by 33, Rule of Inversion, THE CIRCLE. I31 In the same or equal circles, equal angles at the center intercept equal arcs, and determine equal sectors ; and, of two unequal angles at the center, the greater intercepts the greater arc, and determines the greater sector. 371. Corollary. A diameter of a circle divides the en- closed surface into two equal parts. 372. Again, from 365 and 369, by 33, Rule of Inversion, In the same or equal circles, equal sectors have equal arcs and equal angles ; and, of two unequal sectors, the greater has the greater arc and the greater angle. Theorem XV. 373' I''^ t^e same or equal circles y of two unequal minor arcSy the greater is subtended by the greater chord ; of two unequal major arcSy the greater is subtended by the lesser chord. J9. D Hypothesis. Minor arc AB > arc CD, Conclusion. Chord AB > chord C£>. Proof. 4. AOB > ^ COD, (369. The greater arc subtends the greater angle at the center.) .*. in As A OB and COD, side AB > side CD. (159. Two triangles with two sides equal, but the included angle greater in the first, have the third side greater in the first.) Secondly, because the minor arc with its major arc together make up the entire circle, therefore to a greater major arc will correspond a lesser minor arc, and therefore a lesser chord. 132 THE ELEMENTS OF GEOMETRY. 374. From 365 and 373, by 33, Rule of Inversion, In the same or equal circles, equal chords subtend equal major and minor arcs; and, of two unequal chords, the greater subtends the greater minor arc and the lesser major arc. Theorem XVI. 375. An angle at the center of a circle is double the inscribed angle standing up07t the same arc. Fig. I. Fig. 2. Hypothesis. Lei AB be any arc, O the center, C any point on the circle not on arc AB. Conclusion. ^ A OB = 2^ A CB. Proof. Join CO, and produce it to Z>. Because, being radii, OA = OC, .-. ^ OCA = ^ OAC, (126. The angles at the base of an isosceles triangle are equal.) .-. ^ AOD = 4. OCA 4- 4. OAC = 2^ OCA. (173. The exterior angle of a triangle equals the sum of the interior opposite angles.) Similarly, ^ DOB = 2^ OCB. Hence (in Fig. i) the sum or (in Fig. 2) the difference of the angles AOD, DOB, is double the sum or difference of OCA and OCB\ that is, ^ AOB = 2^ ACB. THE CIRCLE. 133 III. Angles in Segments. 376. An angle made by two lines drawn from a point in the arc of a segment to the extremities of the chord is said to be inscribed in the segment, and is called the angle in the seg- ment. 377. Corollary I. Angles inscribed in the same segment of a circle are equal. For each of the angles ACBy ADB is half of the angle subtended at the center by the arc ARB, 378. Corollary II. If a circle is divided into two seg- ments by a chord, any pair of angles, one in each segment, will be supplemental. For they are halves of the explemental angles at the center standing on the same explemental arcs. 134 THE ELEMENTS OF GEOMETRY. 379. Corollary III. The opposite angles of every quad- rilateral inscribed in a circle are supplemental. For they stand on explemental arcs, and so are halves of explemental angles at the center. Theorem XVII. -380. From a point on the side toward the segment ^ its chord subtends an angle less than^ equal tOy or greater than, the angle in the segment^ according as the point is without^ on^ or within the arc of the segment. Hypothesis. AC the chord of any segment, ABC; P any point without the segment on the same side of AC as B ; Q any point within the segment. Conclusions. Since, by 377, we know all angles inscribed in the segment are equal, we have only to prove I. ^ AFC < 4. ABC. II. 4AQC> 4 ABC, THE CIRCLE. 1 35 Proof. I. Let R be the point where PC meets the circle, and join RA, making A APR) then 4. APC <4.ARC, (142. An exterior angle of a triangle is greater than either interior opposite angle.) II. For the same reason, if ^^ is produced to meet the circumfer- 'ence at S, and SC joined, making A CSQ, 4- ^Q.^ > 4- CSA. 381. By 33, Rule of Inversion, On the side toward a segment, the vertex of a triangle, with its chord as base, will lie without, on, or within the arc of the segment, according as the vertical angle is less than, equal to, or greater than, an angle in the segment. Theorem XVIII. 382. If two opposite angles of a quadrilateral are supple- mental, a circle passing through any three of its vertices will contaifz the fourth. Hypothesis. ABCD is a quadrilateral, with ^ A -\- ^ C = st. ^. Conclusion. The four vertices lie on the circle determined by any three of them. Proof. By 355, pass a circle through the three points B, C, D. Take any point, F, on the arc DFB of the segment, on the same side of DB as A. Join FB,FD. Then^ 7^ + ^ C = st. ^. {379. The opposite angles of an inscribed quadrilateral are supplemental.) From hypothesis, .*. ^ A = :4 F, .*. ^ is on the arc DFB. (381. On the side toward a segment, the vertex of a triangle, with its chord as base, will lie on its arc if the vertical angle is equal to an angle in the segment.) 136 THE ELEMENTS OF GEOMETRY. Theorem XIX. 383. The angle in a segment is greater than^ equal to^ or less than, a right angle, accordijtg as the arc of the segment is less than, equal to, or greater than, a semicircle. Hypothesis. Lei AD be a diameter of a circle whose center is O. Take B and C points on the same circle. Conclusions. I. ^ ACD = rt. 2^, being half the straight angle A on. (375. An angle at the center is double the inscribed angle on the same arc.) II. .-. ^ADCx\..4., since 4.ADC -\- }^ABC = st. 4.. (379. Opposite angles of an inscribed quadrilateral are supplemental.) 384. By 33, Rule of Inversion, A segment is less than, equal to, or greater than, a semi- circle, according as the angle in it is greater than, equal to, or less than, a right angle. THE CIRCLE. 1 37 Theorem XX. 385. If two chords intersect within a circle y an angle formed and its vertical are each equal to half the angle at the center standing on the sum of the arcs they intercept. Hypothesis. Let the chords AC,BD intersect at F within the circle. Conclusion. ^ BFC = half ^ at center standing on (arc BC + arc DA) . Proof. Join CB>. 4. BFC = %. BDC + 4. DC A, (173. The exterior angle of a triangle is equal to the sum of the opposite interior angles.) But 2 4. BDC = ^ at center on BC, and 2 4 DCA = 4 at cen- ter on DA ; (375. An angle at the center is double the inscribed angle upon the same arc.) .-. 2 4 BFC — 4a.t center on arc (BC + DA). (367. A sum of two arcs subtends an angle at the center equal to the sum of the angles subtended by the arcs.) Exercises. 71. The end points of two equal chords of a circle are the vertices of a symmetrical trapezoid. 72. Every trapezoid inscribed in a circle is symmetrical. 138 THE ELEMENTS OF GEOMETRY. Theorem XXI. 386. An angle formed by two secants is half the angle at center standing on the difference of the arcs they intercept. Hypothesis. Lei two lines from F cut the circle whose center is O, in the points A, B, C, and D. Conclusion. ^ ^ = | 2j^ at (9 on difference between arc AB and arc CD. Proof. Join^C ^ CAD = 4.F^- 4.C. Doubling both sides, :^ at 6> on arc CD =2^i^-f^at^0n arc AB ; .*. twice ^ F = difference of 2j^ s at O on arcs CD and AB. IV. Tangents. 387. A line which will meet the circle in one point only is said to be a Tangent to the circle. 388. The point at which a tangent touches the circle is called the Point of Contact. THE CIRCLE. 1 39 Theorem XXII. 389. Of lines passiiig through the e7id of any radius, the perpendicular is a tangent to the circley and every other line is a secant. Hypothesis. A radius OP perpendicular fo PB, oblique to PC. Conclusions. I. PB is a tangent at P. II. PC is a secant. Proof. (I.) The sect from O to any point on PB, except P, is > OP, (150. The perpendicular is the least sect between a point and a line.) .*. every point of PB except P is outside the circle. {321. A point is without a circle if its sect from the center is greater than the radius.) Proof. (II.) A sect perpendicular to PC is less than the oblique OP, .*. a point of PC is within the circle ; /. PC is a secant. 390. Corollary I. One and only one tangent can be drawn to a circle at a given point on the circle. 391. Corollary II. To draw a tangent to a circle at a point on the circle, draw the perpendicular to the radius at the point. 140 THE ELEMENTS OF GEOMETRY. 392. Corollary III. The radius to the point of contact of any tangent is perpendicular to the tangent. 393* Corollary IV. The perpendicular to a tangent from the point of tangency passes through the center of the circle. 394. Corollary V. The perpendicular drawn from the center to the tangent passes through the point of contact. On the Three Relative Positions of a Line and a Circle. 395' Corollary VI. A line will be a secant, a tangent, or not meet the circle, according as its perpendicular from the center is less than, equal to, or greater than, the radius. 396. By 33, Rule of Inversion, The perpendicular on a line from the center will be less than, equal to, or greater than, the radius, according as it is a secant, tangent, or non-meeter. Theorem XXIII. 397. An angle formed by a tangent and a chord from the point of contact is half the angle at the center standing on the intercepted arc. y Htpothesis, AB is tangeni at C, and CD is a chord of O wUh center at O. THE CIRCLE. 141 Conclusions. ^DCB = ^^DOC, 4.DCA = |explementZ>(9C Proof. At C erect chord CF Jl AB. CF is a diameter of the circle. (393. The perpendicular to a tangent from the point of contact passes through the center of the circle.) Join OD. Then rt. ^ OCA = l^st. ^at O, also ^ OCD = i^ FOB, (375. The angle at the center is double the inscribed angle on the same arc.) Therefore, adding ^ £>CA = ^reflex ^ Z>OC, therefore the supplement of ^ VGA, which is ^ DCB, is half the exple- ment of reflex ^ DOC, which is 4. DOC, 398. Inverse. If the angle at the center standing on the arc intercepted by a chord equals twice the angle made by that chord and a line from its extremity on the same side as the arc, this line is a tangent. Proof. There is but one line which will make this angle ; and we already know, from 397, that a tangent makes it. Exercises. 73. The chord which joins the points of con- tact of parallel tangents to a circle is a diameter. 74. How may 397 be considered as a special case of 375 1 75. A parallelogram inscribed in a circle must be a rect- angle. y6. If a series of circles touch a given line at a given point, where will their centers all lie } 'jj. The angle of two tangents is double that of the chord of contact and the diameter through either point of contact. 142 THE ELEMENTS OF GEOMETRY. Theorem XXIV. 399. The angle formed by a tangent and a secant is half the angle at the center standing on the difference of the intercepted arcs. Hypothesis. Of iwo lines from Z>, one cuts at B and C the O whose center is O, the other is tangent to the same O at A. Conclusion. 2 ^ D — difference between ^ AOC and ^ AOB. Proof. Join AB. ^ ABC = ^D + ^ DAB, (173. The exterior angle of a triangle is equal to the two interior opposite angles.) .-. 4. AOC = 24.D + :^ AOB, (375. An angle at the center is double the inscribed angle on the same arc.) and (397. An angle formed by a tangent and chord is half the angle at the center on the intercepted arc.) .-. twice ^ Dis the difference between ^ AOC and ^ AOB. 400. Corollary. The angle formed by two tangents is half the angle at the center standing on the difference of the intercepted arcs. THE CIRCLE. I43 Problem II. 401. From a given point without a circle to draw a tangent to the circle. Given, a O ly/M center O, and a point P outside it Required, to draw through P a tangent to the circle. Construction. Join OP. Bisect OP in C. With center C and radius CP describe a circle cutting the given circle in F and G. Join PF and PG. These are tangents to O FGB. Proof. Join OF. ^ OFF is a rt. ^, (383. The angle in a semicircle is a right angle.) .-. PF is tangent at i^ to O BFG. (389. A line perpendicular to a radius at its extremity is a tangent to the circle.) 402. Corollary. Two tangents drawn to a circle from the same external point are equal, and make equal angles with the line joining that point to the center. 144 THE ELEMENTS OF GEOMETRY. V. Two Circles. Theorem XXV. 403. The line joining the centers of two circles which meet in two points is identical with the perpendicular bisector of the common chord. Proof. For the perpendicular bisector of the common chord must pass through the centers of the two circles. (331. The perpendicular bisector of any chord passes through the center.) 404. Corollary. Since any common chord is bisected by the line joining the centers, therefore if the two circles meet at a point on the line of centers, there is no common chord, and these circles have no second point in common. 405. Two circles which meet in one point only are said to touch each other, or to be tangent to one another, and the point at which they meet is called their point of contact. 406. By 343, Two circles^ not concentric^ have always one, and only one^ common axis of symmetry ; namely, their line of centers. For this is the only line which contains a diameter of each. THE CIRCLE. 1 45 Theorem XXVI. Obverse of 404. 407* Jf t'^0 circles have one common point not on the line through their centers^ they have also another common point. Hypothesis. O with center O and O with center C, having a common point B not on OC. Conclusion. They have another common point. Proof. Join OC, and from B drop a line perpendicular to OC 2X D, and prolong it, making DF = BD. F is the second common point. For A ODB ^ A OFD, and A CBD ^ A CFD-, (124. Triangles having two sides and the included angle equal in each are con- gruent.) .-. OF = OB, and CF =^ CB -, .'. F is on both Os. {321. A point is on the circle if its distance from the center is equal to the radius.) 408. CONTRANOMINAL OF 40/. If tWO circIcS tOUCh OHC another, the line through their centers passes through the point of contact. 409. Corollary. Two circles which touch one another have a common tangent at their point of contact ; namely, the perpendicular through that point to the line joining their centers. 146 THE ELEMENTS OF GEOMETRY, 410. Calling the sect joining the centers of two circles their center-sect c, and calling their radii r^ and rj, we have, in regard to the relative positions of two circles, — I. If ^>ri + ^2, therefore the Os are wholly exterior. 2. If ^ = /'j + r^y therefore the Os touch externally. 3. \i c the difference of radii, therefore the Os cut each other. THE CIRCLE. 1 47 4. If ^ = the difference of radii, therefore the Os touch internally. 5. If ^ < the difference of radii, therefore one is wholly interior to the other. 411. By 33, Rule of Inversion, the five inverses to the above are true. Exercises. 78. How must a line through one of the com- mon points of two intersecting circles be drawn in order that the two circles may intercept equal chords on it ? 79. Through one of the points of intersection of two circles draw the line on which the two circles intercept the greatest sect. 80. If any two lines be drawn through the point of contact of two circles, the lines joining their second intersections with each circle will be parallel. 148 THE ELEMENTS OF GEOMETRY, VI. Problems. Problem III. 412. To bisect a given arc. Given, the arc BD. Required, to bisect it. Construction. Join BD, and bisect the sect BD m F; 2X F erect a perpendicular cutting the arc in C. C is the mid point of the arc. Proof. Join BC, CD. aBCF^ aDFC, (124. Triangles having two sides and the included angle in each equal are con- gruent.) .-, chord BC = chord CD, a.TcBC = arc CD. (374. In the same circle, equal chords subtend equal minor arcs.) 413. A polygon is said to be circumscribed about a circle when all its sides are tangents to the circle. The circle is then said to be inscribed in the polygon. 414. A circle which touches one side of a triangle and the other two sides produced is called an Escribed Circle. THE CIRCLE. 149 Problem IV. 415. To describe a circle touching three given lines which are not all parallel^ and do not all pass through the same point. Given, three lines interseciing in A^ B, and C, Required, to describe a circle touching them. Construction. Draw the bisectors of the angles at A and C. These four bisectors will intersect in four points, (9, (9i, O^, Oy A circle described with any one of these points as center, and its perpendicular on any one of the three given lines as radius, will touch all three. Proof. By 186, every point in the bisector of an angle is equally distant from its arms ; Therefore, since O is on the bisector of ^ A, the perpendicular from O on AB equals the perpendicular from O on AC, which also equals the perpendicular from O on CB, since O is also on the bisector of ^ a 150 THE ELEMENTS OF GEOMETRY. 416. The four tangents common to two circles occur in two pairs intersecting on the common axis of symmetry. Problem V. 417. In a given circle to inscribe a triangle equiangular to a given triangle. Given, a and A ABC. Required, to describe in the (d a t^ equiangular to A ABC. Construction. Draw a tangent GH touching the circle at the point D. Make 4. HDK = 2^: C, and 4. GDL = ^ A. K and L being on the circumference, join KL. DKL is the required triangle. Proof. ^ KDL = ^ B. (174. The three angles of a triangle are equal to a straight angle.) 4K = 7^A, and ^ L = }^ C. (375' An inscribed angle is half the angle at the center on the intercepted arc.) (397. An angle formed by a tangent and a chord is half the angle at the center on the intercepted arc.) BOOK IV. REGULAR POLYGONS. I. Partition of a Perigon. Problem I. 418. To bisect a perigon. Solution. To bisect the perigon at the point (9, draw any line through O. This divides the perigon into two straight angles, and all straight angles are equal. 419. Corollary. By drawing a second line through O at right angles to the first, we cut the perigon into four equal parts ; and as we can bisect any angle, so we can cut the peri- gon into 8, 16, 32, 64, etc., equal parts. 152 THE ELEMENTS OF GEOMETRY. Problem II. 420. To trisect a perigon. Solution. To trisect the perigon at the point O, to O draw any line BO; on BO produced take a sect 0C\ on OC construct, by 132, an equilateral triangle CDO. ^ DOB is one-third of a perigon. For ^ DOC i?, one-third of a st. :^, (174. The three angles of a triangle are equal to a straight angle.) .*. ^ DOB is two-thirds of a st. 2(^. 421. Corollary. Since we can bisect any angle, so we may cut the perigon into 6, 12, 24, 48, etc., equal parts. 422. Remark. To trisect a7ty given angle is a problem beyond the power of strict Elementary Geometry, which allows the use of only the compasses and an unmarked ruler. There is an easy solution of it, which oversteps these limits only by using two marks on the straight-edge. The trisection of the angle, the duplication of the cube, and the quadrature of the circle, are the three famous problems of antiquity. REGULAR POLYGONS: 1 53 Problem III. 423. To cut a perigon mto five equal parts. Solution. By 314, describe an isosceles triangle -^4-5C having ^A = 4.C^24.B, Then ^ ^ is two-fifths of a st. ^ . (174. The three angles of any triangle are equal to a straight angle.) Therefore, to get a fifth of a perigon at a point O, construct, by 164, 4. GOB = ^A. 424. Corollary. Since we can bisect any angle, we may- cut a perigon into 10, 20, 40, 80, etc., equal parts. Problem IV. 425. To cut a perigon into fifteen equal parts, 'J3 Solution. At the perigon point (9, by 420, construct the ^ AOC = one-third of a perigon. By 423, make 2^ AOB = one-fifth of a perigon. Then of such parts, as a perigon contains fifteen, -i^ AOC contains five, and ^ AOB contains three, therefore ^ BOC contains two. So bisecting ^ BOC giwts one-fifteenth of a perigon. 154 THE ELEMENTS OF GEOMETRY, 426. Corollary. Hence a perigon may be divided into 30, 60, 120, etc., equal parts. II. Regular Polygons and Circles. Problem V. 427. To mscribe i?t a circle a regular polygon having a given number of sides. c This problem can be solved if a perigon can be divided into the given number of equal parts. For let the perigon at O, the center of the circle, be divided into a number of equal parts, and extend their arms to meet the circle in Ay By C, D, etc. Draw the chords, AB, BC, CD, etc. Then shall ABCD, etc., be a regular polygon. For if the figure be turned about its center O, until OA coincides with the trace of OB, therefore, because the angles are all equal, OB will coincide with the trace of OC, and 6>C with the trace of OD, etc. ; then AB will coincide with the trace of BC, and BC with the trace of CD, etc. ; .-. AB = BC == CD = etc., .•4 the polygon is equilateral. Moreover, since then ABC will coincide with the trace of BDCy .-. t ^BC = 4. BCD = etc., .*. the polygon is equiangular. Therefore ABCD, etc., is a regular polygon, and it is inscribed in the given circle. REGULAR POLYGONS. 1 55 428. Remark. From the time of Euclid, about 300 B.C., no advance was made in the inscription of regular polygons until Gauss, in 1796, found that a regular polygon of 17 sides was inscriptible, and in his abstruse Arithmetic, published in 1 80 1, gave the following : — In order that the geometric division of the circle into n parts may be possible, n must be 2, or a higher power of 2, or else a prime number of the form 2"^ + i, or a product of two or more different prime numbers of that form, or else the product of a power of 2 by one or more different prime num- bers of that form. In other words, it is necessary that n should contain no odd divisor not of the form 2"' + i, nor contain the same divisor of that form more than once. Below 300, the following 38 are the only possible values of /^: 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 6Z, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272. Exercises. 81. The square inscribed in a circle is double the square on the radius, and half the square on the diame- ter. ^2, Prove that each diagonal is parallel to a side of the regular pentagon. 83. An inscribed equilateral triangle is equivalent to half a regular hexagon inscribed in the same circle. 84. An equilateral triangle described on a given sect is equivalent to one-sixth of a regular hexagon described on the same sect. 85. If a triangle is equilateral, show that the radius of the circumscribed circle is double that of the inscribed ; and the radius of an escribed, triple. %6. The end points of a sect slide on two lines at right angles : find the locus of its mid-point. 156 THE ELEMENTS OF GEOMETRY. Problem VI. 429. To circumscribe about a given circle a regular polygon having a given number of sides. JW C X' This problem can be solved if a perigon can be divided into the given number of equal parts. For let the perigon at O, the center of the circle, be divided into a number of equal angles, and extend their arms to meet the circle in A^ Bj C, D, etc. Draw perpendiculars to these arms at A, B^ C, D, etc. These will be tangents. Call their points of intersection K, L, M, etc. Then shall KLM, etc., be a regular polygon. For if the figure be turned about its center O until OA coincides with the trace of OB, then, because the angles are all equal, OB will coincide with the trace of OC, and 6>Cwith the trace of OD, etc. Therefore the tangents at A, B, C, etc., will coincide with the traces of the tangents at B, C, D, etc. Hence the polygon will coincide with its trace ; also .-. KL = LM = etc., ^K = :^L = :^M = etc. ; therefore the polygon is regular, and it is circumscribed about the given circle. REGULAR POLYGONS. 1 5/ 430. Corollary. Hence we can circumscribe about a cir- cle regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, etc., sides. Problem VII. 431. To circumscribe a circle about a given regular polygon. Given, a regular polygon, as ABCDE, Required, to describe a circle about it. Construction. Bisect -4. EAB and 4- ABC by lines intersecting in O. With center O and radius OA describe a circle. This shall be the required circle. Proof. Join OC. Then A OBC'^ A OBAy (124. Triangles having two sides and the included angle in each equal are con- gruent.) .*. ^ OCB = 4 OAB = half one ^ of the regular polygon, .-. 4 B CI? is bisected. Similarly prove each ^ of the polygon bisected, .-. OA = OB = oc = on = OE, (148. In a triangle, sides opposite equal angles are equal.) therefore a circle with radius OA passes through B^ C, D, Ey and is circumscribed about the given polygon. 158 THE ELEMENTS OF GEOMETRY, Problem VIII. 432. To inscribe a circle in a given regular polygon. T c Given, a regular polygon, ABODE, Required, to inscribe a circle in it. Construction. Bisect 4. EAB and ^ ABC by lines intersecting in O. From O drop OP perpendicular to AB. The O with center O and radius OP shall be the O required. Proof. Join 00, OD, OE, and draw 0Q1.B0, OR A. CD, OS ± DEy and OT J. EA, Then A OBC^ A OBA, (124. Triangles having two sides and the included angle in each equal are con- gruent.) .*. ^ OCB = 2^ OAB = half one ^ of the regular polygon, .•; ^ BCD is bisected. Similarly prove each ^ of the polygon bisected. Again, A OBP ^ A OBQ, (176. Triangles having two angles and a corresponding side in each equal are con- gruent.) /. OP = OQ, REGULAR POLYGONS. 1 59 Similarly, OQ =^ OR = OS = OT, therefore a circle with radius OF will touch AB, BC, CD, DE, EA, at points P, Q, R, S, T; .'. it is inscribed in the given polygon. 433* Corollary I. The inscribed and circumscribed cir- cles of a regular polygon are concentric. 434. Corollary II. The bisectors of the angles of a reg- ular polygon all meet in a point which is the center both of the circumscribed and inscribed circles, and is called the center of the regular polygon. 435' Corollary III. The perpendicular bisectors of the sides of a regular polygon all pass through its center. 436. The radius of its circumscribed circle is called the radius of a regular polygon. The radius of its inscribed circle is called its apothem. 437. The side of a regular hexagon inscribed in a circle is equal to the radius. For the sects from the center to the ends of a side make an isosceles triangle, one of whose angles is one-third a straight angle ; therefore it is equilateral. III. Least Perimeter in Equivalent Figures. — Greatest Surface in Isoperimetric Figures. 438. Any two figures are called Isoperimetric when their perimeters are equal. l60 THE ELEMENTS OF GEOMETRY. Theorem I. 439' Of all equivalent triangles having the same base^ that which is isosceles has the least perimeter. Hypothesis. Lei ABC be an isosceles iriangle, and ABC any equivalent iriangle having ihe same base. Conclusion. AB + AC a A'BC. Proof. The vertex A' must fall between BC and the parallel AN\ since, if it fell upon AN, by the preceding proof, A'B •\- A' C > AB + AC\ and, if it fell beyond AN, the sum A'B + A'C would be still greater. Therefore the altitude of A ABC is greater than the altitude of A A'BC, and hence also its surface. 442. Corollary. Of all isoperimetric triangles, that which is equilateral is the greatest. For the greatest triangle having a given perimeter must be isosceles whichever side is taken as the base. 1 62 THE ELEMENTS OF GEOMETRY. Theorem III. 443. Of all triangles formed with the same two given sides ^ that in which these sides are perpendicular to each other has the greatest surface. Hypothesis. Lei ABC, A'BCy be two triangles having the sides AB, BC, respectively equal to A'B, BC\ and let ^ ABC be right Conclusion, a ABC > A A'BC. Proof. Taking BC sls the common base, the altitude AB of A ABC is greater than the altitude A'B> of A A'BC. Theorem IV. 444. Of all isoperimetric plane figures^ the circle contains the greatest surface. Proof. With a given perimeter, there may be an indefinite number of figures differing in form and size. The surface may be as small as we please, but cannot be increased indefinitely. Therefore, among all the figures of the same perimeter, there must be one greatest figure, or several equivalent greatest figures of different forms. REGULAR POLYGONS. 1 63 Every closed figure, if the greatest of a given perimeter, must be convex; that is, such that any sect joining two points of the perimeter Hes wholly within the figure. For let ACBNA be a non-convex figure, the sect AB, joining two of the points in its perimeter, lying without the figure ; then if the re-entrant portion A CB be revolved about the line AB into the position AC'B, the figure AC'BNA has the same perimeter as the first figure, but a greater surface. Now let AFBCA be a figure of greatest surface formed with a given perimeter ; then, taking any point A in its perimeter, and drawing AB to bisect the perimeter, it also bisects the surface. For if the sur- face of one of the parts, as AFB, were greater than that of the other part, A CB, then if the part AFB were revolved upon the line AB into the position AF'B, the surface of the figure AF'BFA would be greater than that of the figure AFBCA, and yet would have the same perim- eter. Now the angles AFB and AF'B must be right angles, else the tri- angles AFB and AF'B could be increased, by 443, without varying the chords AF,FB, AF',F'B, and then (the segments AGF, FEB, AG'F' , F'E'B, still standing on these chords) the whole figure would have increased without changing its perimeter. But F is any point in the curve AFB \ therefore this curve is a semicircle. (384. The arc of a segment is a semicircle if the angle in it is right.) Therefore the whole figure is a circle. 164 THE ELEMENTS OF GEOMETRY. Theorem V. 445* Of all equivalent plane figures^ the circle has the least peri7neter. Hypothesis. Lei C be a circle, and A any other figure having the same surface as C. Conclusion. The perimeter of C is less than that of A. Proof. Suppose B a circle with the same perimeter as the figure A ; then, by 444, A P'. Proof. Upon the sides a^ b, c, etc., of the polygon P' construct circular segments equal to those standing on the corresponding sides of P. The whole figure S' thus formed has the same perimeter as the circle Sy therefore, by 444, surface oi S > S' ; subtracting the circular segments from both, we have P>P\ Theorem VII. 447. Of all isoperimetric polygons having the sami nttmber of sides y the regular polygon is the greatest. Proof. I. The greatest polygon P, of all the isoperimetric poly- gons of the same number of sides, must have its sides equal ; for if two of its sides, as AB' , B'C, were unequal, we could, by 439, increase its surface by substituting for the triangle AB'C the isoperimetric isosceles triangle ABC. II. The greatest polygon constructed with the same number of equal sides must, by 446, be inscriptible in a circle. Therefore it is a regular polygon. Exercises, ^y. Of all triangles that can be inscribed in a given triangle, that v^hose vertices are the feet of the altitudes of the original triangle has the least perimeter. i66 THE ELEMENTS OF GEOMETRY. Theorem VIII. 448. Of all eqiiivalefit polygons having the same number of sides ^ the regular polyg07i has the least perimeter. Hypothesis. Let P be a regular polygon, and M any equiva- lent irregular polygon having the same number of sides as P. Conclusion. The perimeter of P is less than that of M. Proof. Let A" be a regular polygon having the same perimeter and the same number of sides as M \ then, by 447, M P, or P>P; therefore the perimeter of P is greater than that of P or of Q, BOOK V. RATIO AND PROPORTION. Multiples. 451. Notation. In Book V., capital letters denote magni- tudes. Magnitudes which are or may be of different kinds are denoted by letters taken from different alphabets. .^ + ^ is abbreviated into 2A. A + A + A = sA. The small Italic letters w, «, /, ^, denote whole numbers. mA means A taken m times ; nA means A taken n times j .*. mA + nA = {m + n)A. nA taken m times is mnA. 452. A greater magnitude is said to be a Multiple of a lesser magnitude when the greater is the sum of a number of parts each equal to the less ; that is, when the greater contains the less an exact number of times. Z69 I/O THE ELEMENTS OF GEOMETRY. 453. A lesser magnitude is a Submultiple^ or Aliquot Party of a greater magnitude when the less is contained an exact number of times in the greater. 454. When each of two magnitudes is a multiple of, or exactly contains, a third magnitude, they are said to be Com- mensurable. 455. If there is no magnitude which each of two given mag- nitudes will contain an exact number of times, they are called Incommensurable. 456. Remark. It is important the student should know, that of two magnitudes of the same kind taken at hazard, or one being given, and the other deduced by a geometrical con- struction, it is very much more likely that the two should be incommensurable than that they should be commensurable. To treat continuous magnitudes as commensurable would be to omit the normal, and give only the exceptional case. This makes the arithmetical treatment of ratio and proportion radically incomplete and inadequate for geometry. Problem I. 457. To find the greatest common submultiple or greatest common divisor of two given magnitudeSy if any exists. I 1 1 ^ 1 1 1 ^ J£JJ Let AB and CD be the two magnitudes. From AB, the greater, cut off as many parts as possible, each equal to CZ>y the less. If there be a remainder PB, set it off in like manner as often as possible upon CZ>. Should there be a second remainder HD, set it off in like manner upon the first remainder, and so on. RATIO AND proportion: 171 The process will terminate only if a remainder is obtained which is an aliquot part of the preceding one ; and, should it so terminate, the two given magnitudes will be commensurable, and have the last remainder for their greatest common divisor. For suppose HD the last remainder. Then HD is an aliquot part of FB, and so of CH, and therefore of CD, and therefore of AF. Thus being a submultiple of AF and FB, it is contained exactly in AB. And, moreover, it is the greatest common divisor of AB and CD. For since every divisor of CD and AB must divide AF, it must divide FB or CH, and therefore also HD. Hence the common divisor cannot be greater than HD. 458. Inverse of 457. If two magnitudes be commensura- ble, the above process will terminate. For now, by hypothesis, we have a greatest common divisor G. But G is contained exactly in every remainder. For Gy being a submultiple of CD, is also an aliquot part of AF, a multiple of CD ; and therefore, to be a submultiple of AB, it must be an aliquot part of FB the first remainder. Sim- ilarly, CD and the first remainder FB being divisible by G, the second remainder HD must be so, and in the same way the third and every subsequent remainder. But the alternate remainders decrease by more than half, and so the process must terminate at G ; for otherwise a remainder would be reached which, being less than G, could not be divisible by G. 459. CoNTRANOMiNAL OF 457. The above process applied to incommensurable magnitudes is interminable. 460. Obverse of 457. If the above process be intermin- able, the magnitudes are incommensurable. On this depends the demonstration, given in 461, of a remarkable theorem proved in the tenth book of Euclid's " Elements." 1/2 THE ELEMENTS OF GEOMETRY. Theorem I. 461. The side and diagonal of a square are incommensurable. F , Hypothesis. Lei ABCD be a square; AB, a side; AC, a diagonal. Conclusion. Then will AB and AChQ incommensurable. Proof. AC> AB, but < 2AB, Therefore a first remainder ^C is obtained by setting off on AC di part AE = AB. Erect EF perpendicular to ^C and meeting CD in F. Join AF. A ADF ^ A AEF, (179. Right triangles are congruent when the hypothenuse and one side are equal respectively in each.) .\ DF = FE. Again, rt. A CEF is isosceles, because one of the complemental angles, ECF, is half a rt. ^, .;. CE = EF = FD. Hence a common divisor oi EC and DC would be also a common divisor oi EC and EC. But EC and EC are again the side and diagonal of a square, there- fore the process is interminable. RATIO AND PROPORTION. 1 73 462. Another demonstration that the side and diagonal of any sqiiare are incommensurable. If you suppose them commensurable, let P represent their common measure ; that is, the aliquot part which is contained an exact number of times in the side, and also an exact number of times in the diagonal. Let VI represent the number of times P is contained in the side 6", and n the number of times P is contained in the diagonal D ; so that S = mP, and D = nP, where m and n are whole numbers. Then, by 243, the square on D = 2 sq. on S, and therefore n^ is an even number, therefore n is an even number, since the square of an odd number is odd j therefore n may be represented by 2^, .*. 2^^ = m^, therefore m must be even, and can be represented by 2t, and therefore ^ must be even, and so on forever. That is, m and n must be such whole numbers that they will divide by 2, and give quotients which will again divide by z, and so on forever. This is impossible ; for even if m and n were high powers of 2, they would by constant division reduce to i, an odd number. Scales of Multiples. 463. By taking magnitudes each equal to A, one, and two, three, four, etc., of them together we obtain a set of magni- tudes depending upon A, and all known when A is known;' namely. A, 2^4, $A, 4A, sA, 6A, . . . and so on, each being obtained by putting A to the preceding one. This we shall call the scale of multiples of ^. 464. If in be a whole number, 7nA and mBare called equi- multiples of A and P, or the same multiples of ^and B. 1/4 THE ELEMENTS OF GEOMETRY. 465. We assume: As A is greater than, equal to, or less than By so is mA greater than, equal to, or less than mB respectively. 466. By 33, Rule of Inversion, As mA is greater than, equal to, or less than mB^ so is A greater than, equal to, or less than B respectively. Theorem II. 467. Commensurable magnitudes have also a common mul- tiple. If A and B are commensurable magnitudes, there is some multiple of A which is also a multiple of B. Proof. Let C be a common divisor of A and B. The scale of multiples of C is C, 2C, 3C .... Now, by hypothesis, one of the multiples of this scale, suppose pC, is equal to A, and one, suppose qC, is equal to B. Hence, by 465, the multiple pqC is equal to qAy and the same multiple is equal to pB \ therefore, qA = pB. 468. Inverse of 467. Magnitudes which have a common multiple are commensurable. Proof. If pA = qB, then — will go p times into B, and q times into A. 469. Any whole number or fraction is commensurable with every whole number and fraction, being each divisible by unity over the product of the denominators. To find a common multiple, we have only to multiply together the whole numbers and the numerators of the frac- tions. RATIO AND PROPORTION. 1 75 470. Multiplication by a whole number or fraction is dis- tributive, m{A ■\- B -\- . . , ) =^ mA -^ vtB -^ , . . 471. Multiplication being commutative for whole numbers or fractions, .*. m{nA) = mnA — nmA = n{mA). 472. Magnitudes which are of the same kind can, being multiplied, exceed each the other. Scale of Relation. 473. The Scale of Relation of two magnitudes of the same kind is a list of the multiples of both, all arranged in ascending order of magnitude ; so that, any multiple of either magnitude being assigned, the scale of relation points out between which multiples of the other it lies. 474. If we call the side of any square 5, and its diagonal D, their scale of relation will commence thus, — S, D, 2S, 2D, zS, 46", 3A 5^,4A 65, 76*, 5 A ZS, 6D, gS, jD, loS, iiS, SZ>, 12S, gD, 135, 14S, loDj 156", iiX>, 166*, 12I?, 176", 186", i^D, igS, i^D, 20S, 21S, !$£>, etc. 141^, looD, 142S, etc. 14142136", iooooooZ>, 14142146, etc. 1414213566', iooooooooZ>, 1414213576, etc. And we have proved that no multiple of 5 will ever equal any multiple of D. 475. If A be less than B, one multiple at least of the scale of A will lie between each two consecutive multiples of the scale of B. Moreover, if A and B are two finite magnitudes of the same 1/6 THE ELEMENTS OF GEOMETRY. kind, however small A may be, we may, by continuing the scale of multiples of A sufficiently far, at length obtain a multiple of A greater than B. So we are justified in saying, I. We can always take mA greater than B or pB. II. We can always take nA such that it is greater than/^, but not greater than qB, provided that A is less than B, and / than q. 476. The scale of relation of two magnitudes will be changed if one is altered in size ever so little ; for some multiple of the altered magnitude can be found which will exceed, or fall short of, the same multiple of it before alteration, by more than the other original magnitude, and consequently the interdistribu- tion of the multiples of the two original magnitudes will differ from the interdistribution of the multiples of one of the original magnitudes and the second altered. Hence, when two magnitudes are known, the order of their multiples is fixed and known. Inversely, if by any means the order of the multiples is known, and also one of the original magnitudes, the other is of fixed size, even though we may not yet be in a condition to find it. 477' The order in which the multiples of A lie among the multiples of B, when all are arranged in ascending order of magnitude, and the series of multiples continued indefinitely, determines what is called the Ratio of A to B, and written A :B,in which the first magnitude is called the Antecedent, and the second the Consequent. 478. Hitherto we have compared one magnitude to another, with respect to quantity, only in general, according to the logical division greater than, equal to, less than. But double, triple, quadruple, quintuple, sextuple, are spe- cial cases of a more subtile relation which exists between every two magnitudes of the same kind. RATIO AND proportion: 1 7/ Ratio is the relation of one magnitude to another with respect to quantuplicity. 479. If the multiples of A and B interlie in the same order as the multiples of C and D, then the ratio ^ to ^ is the same as the ratio C to Dy and the four magnitudes are said to be Proportional^ or to form a Proportion. This is written A : B \: C : D, and read " ^ to ^ is the same as ^'to />." The second pair of magnitudes may be of a different kind from the first pair. A and D are called the Extrejues, B and C the Means ^ and D is said to be a Fourth Proportional to A^ By and C. 480. In the special case when A and B are commensurable, we can estimate their quantuple relation by considering what multiples they are of some common standard ; and so we can get two numbers whose ratio will be the same as the ratio of A to B. 481. The ratio of two magnitudes is the same as the ratio of two other magnitudes, when, any equimultiples whatsoever of the antecedents being taken, and likewise any equimultiples whatsoever of the consequents, the multiple of one antecedent is greater than, equal to, or less than, that of its consequent, according as the multiple of the other antecedent is greater than, equal to, or less than, that of its consequent. 482. Thus, the ratio oi A \.o B is the same as the ratio of C to D when mA is greater than, equal to, or less than nBy according as mC is greater than, equal to, or less than nBy whatever whole numbers m and n may be. 483. Three magnitudes (Ay By C) of the same kind are said to be proportionals when the ratio of the first to the second is the same as the ratio of the second to the third ; that is, when A \ B '.'. B \ C. In this case, C is said to be the Third Proportional to A and By and B the Mean Proportional between A and C. 178 THE ELEMENTS OF GEOMETRY. Theorem III. 484. Ratios which are the same as the same ratio are the same as 07te ajiother. Hypothesis. A \ B w C \ D^'axA A \ B w X -. V. Conclusion. C \ D w X \ V. Proof. By the inverse of 479, the multiples of C and D have the same interorder as those of A and B ; and, the same being true of the multiples of X and V, therefore the multiples of X and V have the same interorder as those of C and Z). 485. Remark. Thus ratios come under the statement, "Things equal to the same thing are equal to each other;" and a proportion may be spoken of as an equality of ratios. 486. Since the inverse of a complete definition is true, therefore if a pair of numbers / and ^ can be found such that pA is greater than, equal to, or less than gB when pC is not respectively greater than, equal to, or less than gDy then the ratio oi A to B is unequal to the ratio of C to D. 487. The first is to be to the second in a greater (or less) ratio than the third to the fourth when a multiple of the first takes a more (or less) advanced position among the multiples of the second than the same multiple of the third takes among those of the fourth. 488. The ratio of ^ to ^ is greater than that oi C to D when two whole numbers m and n can be found such that mA is greater than nB, while 7nC is not greater than nD ; or such that mA is equal to nB, while mCis less than nD. Theorem IV. 489. Equal magnitudes have the same ratio to the same magnitude^ and the same has the same ratio to equal magni- tudes. RATIO AND proportion: 1 79 For if A = B, .*. mA = mB. So if mA > nC, .*. mB > nC; and if equal, equal ; if less, less ; /. A : C :: B : a And also if ^T > mA, .'. pT> mB; and if equal, equal ; if less, less ; /. T : A :: T ', B, Theorem V. 490. Of two uneqtial magnitudes, the greater has a greater ratio to any other magnitude than the less has ; and the same magnitude has a greater ratio to the less, of two other magni- tudes, tlian it has to the greater, \i A > B,m can be found such that mB is less than mA by a greater magnitude than C. Hence, if mA — nC, or if mA is between wCand {n -f- 1)6', mB will be less than nC \ therefore, by 488, A '. C > B : C, Also, since nC > mB, while « C is not > mA, /. C : B > C : A, 491. From 489 and 490, by 33, Rule of Inversion, If ^ : C :: ^ : C, or if T : A :: T : B, :. A = B. li A :C>B:C, or a T :A B. If A : CT:B, /. A < B. l80 THE ELEMENTS OF GEOMETRY, Theorem VI. 492. If any number of magnitudes he proportionals y as one of the antecedents is to its consequent^ so will all the antecedents taken together be to all the consequents. Let A ', B ',', C ', D w E '. F, then A \ B '.'. A -ir C ^ E ', B -\- D -ir F, For as mA >, =, or < nB^ so, by inverse of 482, is »?C >, =, or < nD \ and so also is mE >, =, or < nF; .', so also is mA + mC -{• mE >, =, or < nB + nD + nF, and therefore so is m{A -\- C + E) >, —, or < n{B + D ■}- F); whence A : B :: A + C + E : B -\- Z> -i- F. Theorem VII. 493- The ratio of the equimultiples of two magnitudes is the same as the ratio of the magnitudes themselves, mA \ mB w A \ B, For as pA >, =, or < qB, so is mpA >, =, or < mqB ', but mpA = pmA, and mqB = qmBy .*. as pA >, =, or < qB, so is pmA >, =, or < qmB, whatever be the values of p and q; .-. A : B :: mA : mB. RATIO AND proportion: i8i Theorem VIII. 494. If four magnitudes be proportionals y then^ if the first be greater than the third, the second will be greater than the fourth; and if equal, equal ; and if less, less. Given, A \ B \\ C \ D. \i A > C, /. A '. B > C : B; therefore, from our hypothesis, C I r> > C '. B; therefore, by 491, If ^ = C, If ^ < C, B > D. ,'. A : B :i C : B, ,\ C : V :: C : B, .-. B =^n, /. A : B < C : By /. C '. D < C ; B, /. B < D, Theorem IX. 495' If four magnitudes of the same kind be proportionals, they will also be proportionals when taken alternately. Let A \ B w C : /?, the four magnitudes being of the same kind, then alternately, A '. C :. B \ D. For, by 493, mA\ mB \\ A \ B '.: C . D \\ nC ', nD, .*. mA : mB \\ nC \ nD \ therefore, by 494, mA >, =, or < «C, as mB >, =, or < nD', and, this being true for all values of m and «, .-. A \ C ','. B \ D, 1 82 THE ELEMENTS OF GEOMETRY, Theorem X. 496. If two ratios are equals the sum of the antecedent and consequent of the first has to the consequent the same ratio as the sum of the antecedent and consequent of the other has to its consequent. li A : B :: C '. D, then A ^ B '. B ','. C + D : D, Proof. Whatever multiple of ^ + ^ we choose to examine, take the same multiple of A, say 1 7^, and let it lie between some two mul- tiples of Bf say 23^ and 24^ ; then, by hypothesis, 1 7 C lies between 23Z) and 24D. Add I "jB to all the first, and i "jD to all the second ; then 1 7 (^ + B) lies between 40^ and 41^, and i7(C + D) between 40Z) and 412? ; and in the same manner for any other multiples. BOOK VI. RATIO APPLIED. I. Fundamental Geometric Proportions. Theorem I. 497* J^f i'^o ^i^^s ^^^ ^^^ by three parallel lines ^ the intercepts on the one are to 07ie aiiother in the same ratio as the correspond- ing intercepts on the other. Hypothesis. Lei the three parallels AA', BB\ CC, cut two other lines in A^ B, C, and A', B\ C , respectively. 183 1 84 THE ELEMENTS OF GEOMETRY. Conclusion. AB : BC :: A'B' : B' C , Proof. On the line ABC, by laying off m sects = AB, take BM = mAB, and, in the same way, BN = n.BC, taking J/ and N on the same side of B. From J/ and N draw lines 1| AA', cutting A'B'C in J/' and iV^'. .-. B'M' =^ m,A'B\ and B'N'^n.B'C. (227. If three or more parallels intercept equal sects on one transversal, they intercept equal sects on every transversal. ) But whatever be the numbers m and n, as BM (or m . AB) is >, = , or , =, or < B'N" (or n . ^'C') ; .-. AB : BC :: A'B' : ^'6". 498. Remark. Observe that the reasoning holds good, whether B is between A and C, or beyond A^ or beyond C, 499. Corollary I. If the points A and ^4' coincide, the figure ACC will be a triangle ; therefore a line parallel to one side of a triangle divides the other two sides proportionally. 500. Corollary II. If two lines are cut by four parallel lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other. 501. If a sect AB is produced, and the line cut at a point P outside the sect AB, the sect AB is said to be divided exter- nally at jP, and AP and BP are called External Segments of AB, RATIO APPLIED. 1 85 In distinction, if the point P is on the sect AB^ it is said to be divided ifiternally. Theorem II. 502. A given sect can be divided internally into two seg- ments having the same ratio as any two given sectSy and also externally unless the ratio be 07te of equality ; aftdj in each case, there is only o?te such point of division. I jr G :b -N M \ ^^4 \\ w \ % ^. 1 \ \ \ \ \ \l \ JC^ Given, ihe sect AB. On a line from A making any angle with AB, take A C and CD equal to the two given sects. Join BD. Draw CF \\ DB and meeting AB in F. By 497, AB is divided internally at F in the given ratio. If it could be divided internally at G in the same ratio, BH being drawn || CG to meet AD in H, A G would be to GB as ^ C to CH, and therefore not as ^ C to CD. (476. The scale of relation of two magnitudes will be changed if one is altered in size ever so little.) Hence F is the only point which divides AB internally in the given ratio. i86 THE ELEMENTS OF GEOMETRY, Again, if CD be taken so that A and D are on the same side of C, the like construction will determine the external point of division. In this case the construction will fail if CD = AC, for D would coincide with A. As above, we may also prove that there can be only one point of external division in the given ratio. C\ 503. Inverse of 499. A line which divides two sides of a triangle proportionally is parallel to the third. For a parallel from one of the points would divide the second side in the same ratio, but there is only one point of division of a given sect in a given ratio. Theorem III. 504. Rectangles of equal altitude are^ to one another in the same ratio as their bases. : 1 B M j)r Let AC, BC, be two rectangles having the common side OC, and their bases OA, OB, on the same side of OC. In the line OAB take OM = m .OA, and ON = n.OB, and complete the rectangles MC and NC. RATIO APPLIED. 1 87 Then MC = m.AC, and JVC = n.BC; and as OM is >, =, or < ON, so is MC respectively >, =, or < NC ; .*. rectangle ^C : rectangle ^C :: base OA : base OB. 505. Corollary. Parallelograms or triangles of equal alti- tude are to one another as their bases. Theorem IV. 506. In the same circle, or in equal circles, angles at the cen- ter and sectors are to one another as the arcs on which they stand. Let O and C be the centers of two equal circles ; AB, KL, any two arcs in them. Take an arc AM = m . AB ; then the angle or the sector between OA and OM equals m . A OB. (365. In equal circles, equal arcs subtend equal angles at the center.) Also take an arc KN = n . KL ; then the angle or sector between CK and CN equals n . KCL. But as AOM>, =, or < KCN, so respectively is arc AM >, =, or < arc KN) (370 and 372. In equal circles, equal angles at the center or equal sectors intercept equal arcs, and of two unequal angles or sectors the greater has the greater arc.) .-. AOB : KCL :: aicAB : aic KL. 1 88 THE ELEMENTS OF GEOMETRY, II. Similar Figures. 507. Similar figures are those of which the angles taken in the same order are equals and the sides between the equal angles proportional The figure ABCD is similar to A'B'C'jy \i 4. A ^ 4. A\ 4. B ^ 4. B\ etc., and also AB : A'B' :: BC : B' C x\ CD \ CU : : etc. Theorem V. 508. Mutually equiangular triangles are similar. Hypothesis, t^s ABC, FGH, having ^.s at A, B, and C, = ^.s at F, G, and H. Conclusion. AB \ FG w BC \ GH .. CA -. HF, Proof. Apply A FGH to A ABC so that the point G coincides with B, and GH falls on BC Then, because ^ 6^ = ^ ^, .-. GF faUs on BA. RATIO APPLIED. 1 89 Now, since 2< BF'H' = ^ A by hypothesis, .-. J^'H' II AC, (166. If corresponding angles are equal, the lines are parallel.) therefore, by 498, AB : BF' -.: CB : BH\ In the same way, by applying the A FGH so that the ^ s at ZT and C coincide, we may prove that BC \ GH : : CA : HF, 509. Corollary. A triangle is similar to any triangle cut off by a line parallel to one of its sides. Theorem VI. 510. Triangles having their sides taken in order proportional are similar. T J£ Hypothesis. AB : FG :: BC -. GH -,: CA : HF, Conclusion. :^ C = ^ H, and -4. A — -i^ F. Proof. On BA take BF" = GF, and draw FH' \\ CA ; .-. AB : F'B '.'. BC '. BH' : : CA : HF'. (508. Mutually equiangular triangles are similar.) Since FG = F'B, :, BC -. GH :: BC -. BH\ (484. Ratios equal to the same ratio are equal.) therefore, by 491, GH = BH'. In the same way HF = H'F', .-. L.FGH^ t.F'BH'. But F'BH' is similar to ABC. 1 90 THE ELEMENTS OF GEOMETRY. Theorem VII. 511. Two triangles having one angle of the one equal to one angle of the othevy and the sides about these angles proportional, are similar. Hypothesis. ^ B = ^ G, and AB : BC .. FG : GH, Conclusion, a AB C ~ a FGH (using ^ for the word " similar ") . Proof. In BA take BF' = GF, and draw F'L \\ AC, .'. A F'BL - A ABC, (508. Equiangular triangles are similar.) .-. AB : BC :: F'B : BL. But F'B — FG by construction ; therefore, from our hypothesis, AB ', BC '.'. F'B : GH, /. FB X BL i: F'B : GH, therefore, by 491, BL = GH, .-. A F'BL ^ A FGH, (124. Triangles having two sides and the included angle respectively equal are congruent.) .-. A FGH - A ABC. RATIO APPLIED. IQI Theorem VIII. 512. If two triangles have two sides of the 07ie proportional to two sides of the other^ and an angle in each opposite one cor- responding pair of these sides equals the angles opposite the other pair are either equal or supplemental. The angles included by the proportional sides are either equal or unequal. Case I. If they are equal, then the third angles are equal. (174. The sum of the angles of a triangle is a straight angle.) Case II. If the angles included by the proportional sides are unequal, one must be the greater. Hypothesis. ABC and FGH as with AB \ BC w FG \ GH, ^A==^F, ^B>^G, Conclusion. ^ C -f- ^ ^ = st. ^ . Proof. Make ^ ABV = ^ 6^ ; .♦. A ABD - A FGH; (508. Equiangular triangles are similar.) .-. AB : BD :: FG : GH; therefore, from our hypothesis, AB : BV :: AB : BC] therefore, by 491, BD = BC, .*. ^ C = ^ BJDC. But ^ BDC + ^ BDA = st. ^, .-. ^ C + ^ Zr = St. ^. 192 THE ELEMENTS OF GEOMETRY. 513. Corollary. If two triangles have two sides of the one proportional to two sides of the other, and an angle in each opposite one corresponding pair of these sides equal, then if one of the angles opposite the other pair is right, or if they are oblique, but not supplemental, or if the side opposite the given angle in each triangle is not less than the other proportional side, the triangles are similar. 514. In similar figures, sides between equal angles are called Homologous^ or corresponding. The ratio of a side of one polygon to its homologous side in a similar polygon is called the Ratio of Similitude of the polygons. Similar figures are said to be similarly placed vAv^n each side of the one is parallel to the corresponding side of the other. Theorem IX. 515. If two unequal similar figures are similarly placed^ all lines joining a vertex of one to the corresponding vertex of the other are concurrent. Hypothesis. ABCD and A'B'C'iy two similar figures simi- larly placed. Conclusion. The lines AA\ BB\ CC, etc., meet in a point P. Proof. Since AB and A'B' are unequal, .-. AA' and BB' are not |1 . Call their point of intersection F, Then AF : A'F '.: BF '. B'F :: AB : A'B, (508. Equiangular triangles are similar.) RATIO APPLIED. 1 93 In the same way, if BB' and CC meet in Q, then BQ : B'Q :: BC : B' C\ But, by hypothesis, AB : A'B' '.', BC '. B' C\ .-. BP : B'P :: BQ : B' Q, .'. the point Q coincides with P. (502. A line can be cut only at a single point into external segments having a given ratio.) 516. Corollary. Similar polygons may be divided into the same number of triangles similar and similarly placed. For if, with their corresponding sides parallel, one of the polygons were placed inside the other, the lines joining cor- responding vertices would so divide them. 517. The point of concurrence of the lines joining the equal angles of two similar and similarly placed figures is called the Center of Similitude of the two figures. 518. Corollary. The sects from the center of similitude along any line to the points where it meets corresponding sides of the similar figures are in the ratio of those sides. Exercises, %'j. Construct a polygon similar to a given polygon, the ratio of similitude of the two polygons being given. 194 THE ELEMENTS OF GEOMETRY. Theorem X. 519. y4 perpendicular from the right angle to the hypothenuse divides a right-angled triangle into two other triangles similar to the whole and to one another. Hypothesis, a ABC right-angled at C. CD ± AB. Conclusion, t. ACD ^ t. ABC -^ t. CBD. Proof. ^ CAD = ^BAC, and rt. ^ ADC = rt. ^ ACB, .-. ^ACD = ^ABC, (174. The three angles of any triangle are equal to a straight angle.) .-. aACD ~ hABC. In the same way we may prove A CBD ~ A ABC; .-. aACD ~ A CBD. 520. Corollary I. Each side of the right triangle is a mean proportional between the hypothenuse and its adjacent segment. For since A A CD ~ A ABC, .-. AB : AC :: AC : AD. 521. Corollary II. The perpendicular is a mean propor- tional between the segments of the hypothenuse. 522. Corollary III. Since, by 383, lines from any point in a circle to the ends of a diameter form a right angle, there- fore, if from any point of a circle a perpendicular be dropped RATIO APPLIED. 195 upon a diameter, it will be a mean proportional between the segments of the diameter. Theorem XL 523. The bisector of an interior or exterior angle of a triangle divides the opposite side internally or externally in the ratio of the other two sides of the triangle. Hypothesis. ABC any A. BD ihe bisector of 4. at B. Conclusion. AB : BC .'. AD \ DC. Proof. Draw AF \\ BD. Then, of the two angles at B given equal by hypothesis, one equals the corresponding interior angle at F, and the other the corresponding alternate angle at A, (168. A line cutting two parallels makes alternate angles equal, and (169) corre- sponding angles equal.) .-. AB = BF. (126. If two angles of a triangle are equal, the sides opposite are equal.) But BF : BC :-. AD -. DC, (499. If a line be parallel to a side of a triangle, it cuts the other sides propor- tionally.) /. AB :BC::ADi DC, 196 THE ELEMENTS OF GEOMETRY. 524. Inverse. Since, by 502, a sect can be cut at only one point internally in a given ratio, and at only one point exter- nally in a given ratio, therefore, by 32, Rule of Identity, if one side of a triangle is divided internally or externally in the ratio of the other sides, the line drawn from the point of division to the opposite vertex bisects the interior or exterior angle. 525. When a sect is divided internally and externally into segments having the same ratio, it is said to be divided har- monically. 526. Corollary. The bisectors of an interior and exterior angle at one vertex of a triangle divide the opposite side har- monically. Theorem XII. 527. If a secty AB, is divided harmonically at the points P and Q, the sect PQ will be divided harmonically at the points A and B. I ! , , 3L ^ 7» a Hypothesis. THq sect AB divided internally at F, and exter- nally at Q, so that AP '. BP '.'. AQ : BQ', therefore, by inversion, BP '. AP '.'. BQ '. AQ; therefore, by alternation, BP : BQ '.: AP : AQ. 528. The points A, B, and P, Q, of which each pair divide harmonically the sect terminated by the other pair, are called four Harmonic Points. RATIO APPLIED, 197 III. Rectangles and Polygons. Theorem XIII. 529. If four sects are proportionaly the rectangle contained by the extremes is equivalent to the rectangle contained by the means. ac ad ^ Let the four sects a^ b, c, ^/, be proportional. Then rectangle ad — be. Proof. On a and on b construct rectangles with altitude = c. On c and on d construct rectangles of altitude a. Then a : b :: ac : bCy and c : d :-, ac -, ad, (504. Rectangles of equal altitudes are to each other as their bases.) But, by hypothesis, a '. b '.: c ', d, therefore, by 491, .*. ac : be : : ae : ad; be = ad. 198 THE ELEMENTS OF GEOMETRY. 530. Inverse. If two rectangles are equivalent, the sides of the one will form the extremes, and the sides of the other the means, of a proportion. ^■c a^. Hypothesis. Reciangle ad = be. Conclusion, a \ b w c \ d. Proof. Since ad = be, therefore, by 489, ac : be but ae : be :: a : bj and ae : ad '.'. e : d, .'. a : b we : d. ae : ad\ 531. Corollary. If three sects are proportional, the rect- angle of the extremes is equivalent to the square on the mean. Theorem XIV. 532' If two chords intersect either within or without the circle, the rectangle contained by the segments of the ofie is equiv- alent to the rectangle contained by the segments of the other. RATIO APPLIED. 199 Hypothesis. Let the chords AB and CD intersect in P. Conclusion. Rectangle AP . PB = rectangle CP . PD. Proof. ^PAC == ^ PDB, (376. Angles in the same segment of a circle are equal.) and ^ APC = ^ BPD, .-. A APC - A BPD, (508. Equiangular triangles are similar.) /. AP : CP ::PI> : PB, .-. AP.PB = CP.PD, 533. Corollary. Let the point P be without the circle, and suppose DCP to revolve about P until C and D coincide ; then the secant DCP becomes a tangent, and the rectangle CP . PD becomes the square on PC. Therefore, if from a point without a circle a secant and tangent be drawn, the rectangle of the whole secant and part outside the circle is equivalent to the square of the tangent. 200 THE ELEMENTS OF GEOMETRY. Theorem XV. 534. The rectangle of two sides of a triangle is equivalent to the rectangle of two sects drawn from that vertex so as to make equal angles with the two sides ^ and produced^ one to the base, the other to the circle circumscribing the triangle. Hypothesis. ^ ABE = ^ CBD. Conclusion. Rectangle AB .BC — DB . BE, Proof. Join AE. Then 4-C^i.E, (376. Angles in the same segment of a circle are equal.) ^ CBD = 4. ABE, by hypothesis, .-. A CBD ~ A ABEy .-. AB : BE :: DB '. BC, .-. AB.BC ^ DB.BE. 535- Corollary I. If BD and BE coincide, they bisect the angle B ; therefore rectangle AB . BC ^ DB . BE = DB{BD + DE) = BD^ + BD . DE = BD^ + CD . DA (by 532). Therefore, when the bisector of an angle of a triangle RATIO APPLIED. 201 meets the base, the rectangle of the two sides is equivalent to the rectangle of the segments of the base, together with the square of the bisector. 536. Corollary II. If BD be a perpendicular, BE is a diameter, for angle BAE is then right ; therefore in any triangle -2> C the rectangle of two sides is equivalent to the rectangle of the diameter of the circumscribed circle by the perpendicular to the base from the vertex. Exercises. %Z. Prove that the inverse of 535 does not hold when AB — BC, 89. Prove 535 when it is an exterior angle which is bi- sected. 202 THE ELEMENTS OF GEOMETRY, Theorem XVI. 537. The rectangle of the diagonals of a quadrilateral in- scribed in a circle is equivalent to the sum of ihe two rectangles of its opposite sides. Hypothesis. ABCD an inscribed quadrilaieral. Conclusion. Rectangle A C . BD =^ AB . CD ■\- BC . DA, Proof. By 164, make ^ DAF = ^ BAG. To each add ^i^^C. Then, in A s ^ CZ? and ABF, ^DAC = ^BAF', also 4.ACD = ^ABD, (376. Angles inscribed in the same segment are equal.) .-. AACD'^ ^ABF, (508. Equiangular triangles are similar.) /. AC : AB :: CD i BF; therefore, by 529, AC .BF= AB. CD, Again, by construction, ^ DAF = ^ BAC, RATIO APPLIED. 203 Moreover, ^ ADF = ^ ACB, (376. Angles inscribed in the same segment are equal.) (508. Equiangular triangles are similar.) .-. AC : AD :: CB : DF'y therefore, by 529, AC ,FD = AD.BC, .-. AB .CD + BC ,DA = AC .BF+ AC .FD = AC{BF + FD) = AC . ^Z>. Problem I. 538. To alter a given sect in a given ratio. ^ 3 / k ^.^ . * Given, Me raf/b as ihat of sect a to sect b^ and given the sect AB. Required, to find a sect to which AB shall have the same ratio as a to b. Construction. Make any angle C, On one arm cut off CD = AB. On the other arm cut off CF = a, and FG = b. Join DFy and through G draw GH \\ DF, .-. AB : DH '.: a : b. 204 ^-^^ ELEMENTS OF GEOMETRY. 539. This is the same as finding a fourth proportional to three given sects. To find a third proportional to a and by make AB = ^ in the above construction. 540. Every alteration of a magnitude is an alteration in some ratio. Two or more alterations are jointly equivalent to some one alteration, and then this single alteration which produces the joint effect of two is said to be compounded of those two. The composition of the ratios of a\,Q b and (7 to Z> is per- formed by assuming F, altering it into G, so that F : G w a : by then altering G into 77, so that G : H '.: C : D. The joint effect turns F into H) and the ratio oi F to H is the ratio compounded of the two ratios, a : b and C : D. 541. A ratio arising from the composition of two equal ratios is called the Duplicate Ratio of either. Theorem XVII. 542. Mutually equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Hypothesis. In /=? ACy ^ BCD = ^ HCF of £y CG. Conclusion. /^ACi^yCG^ ratio compounded oi DC : CF, and^C : CH, RATIO APPLIED. 205 Proof. Place the /u^ so that HC and CB are in one line ; then, by log, DC and CF are in one line. Complete the £j BF. Then CD AC : C7BF .i DC : CF, and £yBF '. n7CG \\ BC '. CH, (505. Parallelograms of equal altitude are as their bases.) .'. c7 AC has to z:7 CG the ratio compounded oi DC : CF and BC : CH, 543. Corollary I. Triangles which have one angle of the one equal or supplemental to one angle of the other, being halves of equiangular parallelograms, are to one another in the ratio compounded of the ratios of the sides about those angles. 544. Corollary II. Since all rectangles are equiangular parallelograms, therefore the ratio compounded of two ratios between sects is the same as the ratio of the rectangle con- tained by the antecedents to the rectangle of the consequents. If the ratio compounded of a : a\ and d : d\ be written — . — , this corollary proves ^ . — = -^ ; and the composition ad a ab of ratios obeys the same laws as the multiplication of fractions. Thus ^ . ^ =^, and so the duplicate ratio of two sects is the 00 b^ same as the ratio of the squares on those sects. It will be seen hereafter that the special case of 542, when the parallelograms are rectangles, is made the foundation of all mensuration of surfaces. Exercises. 90. If mutually equiangular parallelograms are equivalent, so are rectangles with the same sides. 91. Equivalent parallelograms having the same sides as equivalent rectangles are mutually equiangular. 206 THE ELEMENTS OF GEOMETRY. Theorem XVIII. 545. Similar triangles are to one another as the squares on their corresponding sides. Let the similar triangles ABC^ AHK, be placed so as to have the sides AB, AC, along the corresponding sides AH, AK, and therefore BC II HK. Join CH, Since, by 505, triangles of equal altitude are as their bases, .-. A ABC : A AHC w AB \ AH, and A AHC : A AHK :: AC : AK = AB : AH, by hypothesis ; AB AB AB^ A ABC : A AHK = AH AH AH^ Exercises. 92. The ratio of the surfaces of two similar triangles is the square of the ratio of similitude of the tri- angles. 93. If the bisector of an angle of a triangle also bisect a side, the triangle is isosceles. 94. In every quadrilateral which cannot be inscribed in a circle, the rectangle contained by the diagonals is less than the sum of the two rectangles contained by the opposite sides. 95. The rectangles contained by any twp sides of triangles inscribed in equal circles are proportional to the perpendiculars on the third sides. 96. Squares are to one another in the duplicate ratio of their sides. RATIO APPLIED. 207 Theorem XIX. 546. Similar polygons are to each other as the squares on their corresponding" sides, ^ -g- Hypothesis. ABCD and A'B'Clf two similar polygons, of which BC and B' C are corresponding sides. Conclusion. ABCD : A'B'C'iy : : AB^ : A'B'\ Proof. By 516, the polygons may be divided into similar triangles. By 545, any pair of corresponding triangles are as the squares on corresponding sides, t^ABD BJy /\BCD BC^ AA'B'iy B'D^ AB'Cjy B'C'^ therefore, by 496, A ABD + A BCD BC A A'B'D^ + A B'C'D^ B'C'^ In the same way for a third pair of similar triangles, etc. 208 THE ELEMENTS OF GEOMETRY. 547. Corollary. If three sects form a proportion, a poly- gon on the first is to a similar polygon similarly described on the second as the first sect is to the third. Theorem XX. 548. The perimeters of any two regular polygons of the same number of sides have the same ratio as the radii of their circum- scribed circles. 7) Proof. The angles of two regular polygons of the same number of sides are all equal, and the ratio between any pair of sides is the same, therefore the polygons are similar. But lines drawn from the center to the extremities of any pair of sides are radii of the circumscribed circles, and make similar triangles, therefore, by 496, AB BC r " A'B'~ B'C ■?' AB + BC r CD A'B' 4- B'C r' c'jy' AB + BC -{- CD r A'B' ^B'C ^ C'jy r' RATIO APPLIED. 209 Problem II. 549. To find a mean proportional between two giveji sects. 3) Let AB, BC, be the two given sects. Place AB, BC, in the same line, and on AC describe the semicircle ADC. From B draw BD perpendicular to ^C BD is the mean proportional. (383. An angle inscribed in a semicircle is a right angle.) (521. If from the right angle a perpendicular be drawn to the hypothenuse, it will be a mean proportional between the segments of the hypothenuse.) Exercises. 97. If the given sects were ^C and ^C placed as in the above figure, how would you find a mean proportional between them 1 98. Half the sum of two sects is greater than the mean proportional between them. 99. The rectangle contained by two sects is a mean propor- tional between their squares. 100. The sum of perpendiculars drawn from any point within an equilateral triangle to the three sides equals its altitude. loi. The bisector of an angle of a triangle divides the triangle into two others, which are proportional to the sides of the bisected angle. 102. Lines which trisect a side of a triangle do not trisect the opposite angle. 103. In the above figure, if AF ± AD meet DB produced at F, then a ABD — a FOB. 2IO THE ELEMENTS OF GEOMETRY, Problem III. 550. On a given sect to describe a polygon similar to a given polygon. Let ABCDE be the given polygon, and A'B' the given sect. ^OYCiAD, AC, 4. A'B'C = 4. ABC, ^B'A'C = 4BAC, ■ .-. tB'C'A' = 4BCA, Make and Then make A A'B'C h.ABC, and ^.A'C'jy = 4ACD, 4. C'A'jy = 4 CAD, :, 4 CD A' = 4 CDA, t^AC'iy ~ A^CA etc. Therefore, from the first pair of similar triangles, A'B' : B'C :: AB : BC, and B'C : CA' :: BC : CA. From the second pair of similar triangles, CA' : C'ly :: CA : CD, ,', B'C : CD' :: BC : CD, and so on. Thus all the ^^s in one polygon are = the corresponding 2, and having ^ AEM = ^ CAE. By 549, between AC and AF find a mean proportional GH. ^y 55°> o^ ^^ construct the figure KGH similar and similarly described to the figure ABC. KGH is the figure required. It may be proved, as in 542, that ^C and AF are in one line, and also LE and EM-, therefore, by 505, AC '. AF '.'. n7 CE '. n7 AM : : ABC : £>, But AC : AF :: ABC : KGH, (547. If three sects form a proportion, a polygon on the first is to a similar polygon on the second as the first sect is to the third.) .-. ABC \ D .'. ABC : KGH-, therefore, by 491, KGH = D, GEOMETRY OF THREE DIMENSIONS. BOOK VII. OF PLANES AND LINES. 553. Already, in 50, a plane has been defined as the surface generated by the motion of a line always passing through a fixed point while it slides along a fixed line. 554. Already, in 97, the theorem has been assumed, that, if two points of a line are in a plane, the whole line lies in that plane. Two other assumptions will now be made : — 555. Any number of planes may be passed through any line. z 2 ^ 556. A plane may be revolved on any line lying in it. ax3 214 THE ELEMENTS OF GEOMETRY. Theorem I. 557. Through two intersecting lines y one plane y and only one^ passes. a Hypothesis. Two lines AB, BC, meeting in B. Conclusion. One plane, and only one, passes through them. Proof. By 555, let any plane FG be passed through AB, and, by 556, be revolved around on AB as an axis until it meets any point C of the line BC, The line BC then has two points in the plane FG'j and therefore, by 554, the whole line BC'i^m this plane. AlsOy any plane containing AB and BC must coincide with FG. For let Q be any point in a plane containing AB and BC. Draw QMN in this plane to cut AB, BC, inM and JV. Then, since M and JV are points in the plane FF, therefore, by 554, Q is a point in the plane FF. Similarly, any point in a plane containing AB, BC, must lie in FF; therefore any plane containing AB, BC, must coincide with FF. 558- Corollary I. Two lines which intersect lie in one plane, and a plane is completely determined by the condition that it passes through two intersecting lines. 559* Corollary II. Any number of lines, each of which intersects all the others at different points, lie in the same plane ; but a line may pass through the intersection of two others without being in their plane. OF PLANES AND LINES. 21 5 560. Corollary III. A line y and a point without that line, determine a plane, s Proof. Suppose AB the line, and C the point without AB. Draw the hne CD to any point D in AB. Then one plane con- tains AB and CD, therefore one plane contains AB and C. Again, any plane containing AB must contain D ; therefore, any plane containing AB and C must contain CD also. But there is only one plane that can contain AB and CD. Therefore there is only one plane that can contain AB and C. Hence the plane is completely determined. 561. Corollary IV. Three points not in the same line determine a plane. 57 For let A, B, C be three such points. Draw the line AB. Then a plane which contains A, B, and C must contain AB and C\ and a plane which contains AB and C must contain A, B, and C. Now, AB and C are contained by one plane, and one only ; .therefore A, B, and C are contained by one plane, and one only. Hence the plane is completely determined. 2l6 THE ELEMENTS OF GEOMETRY. 562. Two parallel lines determine a plane. For, by the definition of parallel lines, the two lines are in the same plane ; and, as only one plane can be drawn to con- tain one of the lines and any point in the other line, it follows that only one plane can be drawn to contain both lines. Theorem II. 563. If two planes cut one another^ their common section must be a straight line, JL Hypothesis. Lei AB and CD be two intersecting planes. Conclusion. Their common section is a straight line. Proof. Let M and N be two points common to both planes. Draw the straight line MN. Therefore, by 554, since M and N are in both planes, the straight line MN lies in both planes. And no point out of this line can be in both planes ; because then two planes would each contain the same line and the same point without it, which, by 560, is impossible. Hence every point in the common section of the planes lies in the straight line MN. OF PLANES AND LINES. 21/ 564. CoNTRANOMiNAL OF 554. A line which does not lie altogether in a plane may have no point, and cannot have more than one point, in common with the plane. Therefore three planes which do not pass through the same line cannot have more than one point in common ; for, by 563, the points common to two planes lie on a line, and this line can have only one point in common with the third plane. 565. All planes are congruent ; hence properties proved for one plane hold for all. A plane will slide upon its trace. Principle of Duality. 566. When any figure is given, we may construct a reciprocal figure by taking planes instead of points, and points instead of planes, but lines where we had lines. The figure reciprocal to four points which do not lie in a plane will consist of four planes which do not meet in a point. From any theorem we may infer a reciprocal theorem. Two points determine a line. Three points which are not in a line determine a plane. A line and a point without it determine a plane. Two lines in a plane determine a point. Two planes determine a line. Three planes which do not pass through a line determine a point. A line and a plane not through it determine a point. Two lines through a point de- termine a plane. There is also a more special principle of duality, which, in the plane, takes points and lines as reciprocal elements ; for they have this fundamental property in common, that two ele- ments of one kind determine one of the other. Thus, from a proposition relating to lines or angles in axial symmetry, we get a proposition relating to points or sects in central symmetry. The angle between two corre- sponding lines is bisected by the axis. The sect between tvvo corre- sponding points is bisected by the center. 2l8 THE ELEMENTS OF GEOMETRY. Theorem III. 567* If ^ ^^^^ ^^ perpendicular to two lines lying in a plane y it will be perpendicjilar to every other li7te lying in the plane and passing through its foot. Hypothesis. Let the line EF be perpendicular to each of the lines AB, CD, at E, the point of their intersection. Conclusion. EF _L GH, any other line lying in the plane ABCD, and passing through E. Proof. Take AE = BE, and CE = DE. Join AD and BC, and let G and H be the points in which the joining lines intersect the third line GH. Take any point F in EF. Join FA, FB, EC, FD, EG, FH. Then A AED ^ A BEC, (124. Triangles having two sides and the included angle equal are congruent.) .-. AD = BC, and ^ DAE = ^ CBE. Then A AEG ^ A BEH, (128. Triangles having two angles and the included side equal are congruent.) /. GE = HE, and AG = BH. OF PLANES AND LINES. 219 Then A AEF ^ A BEF, (124. Triangles having two sides and the included angle equal are congruent.) AF = BF. In the same way CF = DF, Then t.ADF^ t.BCF, (129. Triangles having three sides in each respectively equal are congruent.) /. 4. DAF = 4. CBF, Then A AFG ^ A BFH, (124. Triangles having two sides and the included angle equal are congruent.) FG = FH, Then A FEG ^ A FEH, (129. Triangles having three sides in each respectively equal are congruent.) /. 4 FEG = i. FEB, FE A. GH. As GH is any line whatever lying in the plane ABCDy and passing through E, /. EF J_ every such line. 568. A line meeting a plane so as to be perpendicular to every line lying in the plane and passing through the point of intersection, is said to be perpendicular to the plane. Then also the plane is said to be perpendicular to the line. 569. Corollary. At a given point in a plane, only one perpendicular to the plane can be erected ; and, from a point without a plane, only one perpendicular can be drawn to the plane. 220 THE ELEMENTS OF GEOMETRY. 570. Inverse of 567. All lines perpendicular to another line at the same point lie in the same plane. Hypothesis. AB any line ± BD and BE, BC any other line _L AB. Conclusion. ^C is in the plane BDE, Proof. For if not, let the plane passing through AB, BC, cut the plane BDE in the line BF. Then AB, BC, and BE are all in one plane ; and because AB J_ BD and BE, therefore, by 567, AB J. BF, But, by hypothesis, AB 1.BC; therefore, in the plane ABE we have two lines BC, BF, both ± AB at B, which, by 105, is impossible ; .-. BC lies in the plane DBE. 571. Corollary. If a right angle be turned round one of its arms as an axis, the other arm will generate a plane ; and when this second arm has described a perigon, it will have passed through every point of this plane. 572. Through any point D without a given line AB to pass a plane perpendicular to AB. In the plane determined by the line AB and the point D, draw DB J_ AB ; then revolve the rt. 4. ABD about AB. OF PLANES AND LINES. 221 Theorem IV. 573. Lines perpendicular to the same plane are parallel to each other. Hypothesis. AB, CD JL plane MN at the points B, D. Conclusion. AB \\ CD. Proof. Join BD, and draw DE i. BD in the plane MN, Make DE = AB, and join BE, AD, AE. Then A ABD ^ A EDB, (124. Triangles having two sides and the included angle equal are congruent.) /. AD = BE, Then A ABE ^ A EDA, (129. Triangles having three sides in each respectively equal are congruent.) .-. 4. ABE = 4. EDA, But, by hypothesis, ABE is a rt. 4, :. EDA is a rt. ^, .'. AD, BD, CD, are all in one plane. (570. All lines perpendicular to another at the same point lie in the same plane.) But AB is in this plane, since the points A and B are in it, .'. AB and CD He in the same plane ; and they are both X BD, ,'. AB II CD. (166. If two interior angles are supplemental, the lines are parallel.) 222 THE ELEMENTS OF GEOMETRY. 574. Inverse of 573. If one of two parallels is perpen- dicular to a plane, the other is also perpendicular to that plane. 7 Hypothesis. AB \\ CD, CD ± plane MN. Conclusion. AB _L plane MN. Proof. For if AB^ meeting the plane MN at B^ is not ± it, let BF \>Q l.iX.'y therefore, by 573, BF \\ CD, and, by hypothesis, BA \\ CD, But this is impossible, (99. Two intersecting lines cannot both be parallel to the same line.) .-. ^^ ± plane JfiV. 575. Corollary I. If one plane be perpendicular to one of two intersecting lines, and a second plane perpendicular to the second, their intersection is perpendicular to the plane of the two lines. - For their intersection DF is perpendicular to a line through D parallel to ABy and also perpendicular to a line through D parallel to BC. (574. If, of two parallels, one be perpendicular to a plane, the other is also.) 576. Corollary II. Two lines, each parallel to the same line, are parallel to each other, even though the three be not in OF PLANES AND LINES. • 223 one plane. For a plane perpendicular to the third line will, by 574, be perpendicular to each of the others; and therefore, by 573, they are parallel. Problem I. 577. To draw a line perpendicular to a given plane from a given point without it. GrvTEN, ihe plane BH and the point A without it. Required, to draw from A a line ± plane BH. Construction, In the plane draw any line BC, and, by 139, from A drsLW AD ±BC. In the plane, by 135, draw Z>jF ± BC; and from A, draw, by 139, AF ± DF. AF will be the required perpendicular to the plane. Proof. Through F draw GH \\ BC. Then, since BC 1. the plane ADFy (567. A line perpendicular to any two lines of a plane at their intersection is perpendicular to the plane.) .\ GH ± the plane ADF, (574. If two lines are parallel, and one is perpendicular to a plane, the other is also.) /. GH ± the line AF of the plane ADF, :. AF1.GH. But, by construction, AF _L DF, ,', AF J_ the plane passing through GH, DF. 224 THE ELEMENTS OF GEOMETRY. Problem II. 578. To erect a perpendicular to a given plane from a given point in the plane. 1 Let A be the given point. From any point B, without the plane, draw, by 577, -5 C ± the plane, and from A, by 167, draw AD \\ BC, /. AD ± the plane. {574. If one of two parallels is perpendicular to a given plane, the other is also.) 579. The projection of a point upon a plane is the foot of the perpendicular drawn from the point to the plane. 580. The projection of a line upon a plane is the locus of the feet of the perpendiculars dropped from every point of the line upon the plane. OF PLANES AND LINES. 225 These perpendiculars are all in the same plane, since any two are parallel, and any third is parallel to either, and has a point in their common plane, and therefore lies wholly in that plane ; therefore the projection of a line is a line. Theorem V. 581. A line makes with its own projectiojt upon a plane a less angle than with any other line ijt the plane. Hypothesis. Lq^ BA meet the plane MN at B, and let BA' be its project/on upon the plane MN^ and BC any other line drawn through B in the plane MN. Conclusion. ^ ABA' < ^ ABC. Proof. Take BC ^ BA. Join A C and A' C. Then, in As ABA' and ABC, AB = AB, BA' = BC, AA' < AC, (150. The perpendicular is the least sect from a point to a line.) .-. ^ ABA' <^ABC. (161. In two triangles, ii a = a', 6 = y, c Kc*, therefore C< C.) 226 THE ELEMENTS OF GEOMETRY, 582. The angle between a line and its projection on a plane is called the Inclinatioji of the line to the plane. 583. Parallel planes are such as never meet, how far soever they may be produced. 584. A line is parallel to a plane when they never meet, how far soever they may be produced. Theorem VI. 585. Planes to which the same line is perpendicular are par- allel. For, if not, they intersect. Call any point of their intersection X. Draw from X a line in each plane to the foot of the common perpen- dicular. Then from this point X we would have two perpendiculars to the same line, which is impossible, (145. There can be only one perpendicular from a point to a line.) .*. the planes cannot intersect, and .*. are parallel. 586. If two lines are parallel, every plane through one of them, except the plane of the parallels, is parallel to the other. Let AB and GH be the parallels, and DEF any plane through GH\ then the line AB and the plane DEF are par- allel. For the plane of the parallels ABHG intersects the plane DEF in the line GH\ and, if AB could meet the plane DEFy it could meet it only in some point of GH\ but AB can- not meet GH, since they are parallel by hypothesis. Therefore AB cannot meet the plane DEF. Applications, (i) Through any given line a plane can be passed parallel to any other given line. (2) Through any given point a plane can be passed parallel to any two given lines in space. OF PLANES AND LINES. 227 Theorem VII. 587- If ci pair of ijttersecting lines be parallel to another pair, bnt not in the same plane with them, the plane of the first pair is parallel to the plane of the secojtd pair. Hypothesis. AB \\ DE, and BC \\ EF. Conclusion. Plane ABC II plane DEF. Proof. From B draw BH J_ plane DEF, meeting it in H, Through B draw GB \\ DE, and BK \\ EF, .-. GB II AB, and ' BK \\ BC, (576. Two lines each parallel to the same line are parallel to each other.) /. -2^ ABB + ^ BBG = St. ^. But because BB was drawn J_ plane DEF, /. ^BBG^xt.^. So '^, and, in same way, :^ABB= rt. ^; ^ CBB = rt. ^ ; .-. BB A_rg\^nQABC, .♦. plane ABC \\ plane DEF. (585. Planes to which the same line is perpendicular are parallel.) 228 THE ELEMENTS OF GEOMETRY. Theorem VIII. 588. If a pair of intersecting lines be parallel to another pair, any angle made by the first pair is equal ,or supplemental to any angle made by the second pair. Hypothesis. AB \\ DE, and BC II EF. Conclusion. 2^ ABC = or supplemental to ^ DEF, Proof. Join BE. Since AB \\ DE, .'. AB, BE, and ED, are in one plane. In this plane, from A draw a line I| BE. It must meet the line DE. Call the intersection point D. .'. ABED is a ,C7 ; .-. AB = DE, and AD = BE. In same way, BC = EF, and CF = BE, .-. AD = CF. But since, by construction, AD \\ BE, and CF \\ BE, therefore, hysi6, AD II CF, /. ACFD isa,/=7, (216. If any two opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.) OF PLANES AND LINES. 229 .-. AC = DF, ,\ A ABC ^ A £>£F, (129. Triangles with three sides of the one equal to three of the other are con- gruent.) .-. ^ ABC = ^ JDjEK Theorem IX. 589. If two parallel planes be cut by another plane y their common sections with it are parallel. Call the parallel planes A and B, and the third plane X. Then the lines of intersection are in one plane, since they both lie in the plane X. Again, because one of these lines is in the plane A^ and the other in the parallel plane B, they can never meet. Therefore the two lines are in one plane, and can never meet ; that is, they aire parallel. 590. Corollary. Parallel sects included between two par- allel planes are equal. 230 THE ELEMENTS OF GEOME^TRY. Theorem X. 591. Parallel lines intersecting the same plane are equally inclined to it. Hypothesis. Conclusion. AB II CD, BB' ± plane ACF, niy A. plane ACF. ^ BAB' = V Dcjy, Proof. ^ ABB' = or supplemental to ^ CDL/, (588. If a pair of intersecting lines be parallel to another pair, any angle made by the first pair is equal or supplemental to any angle made by the second pair.) But 2< ABB' cannot be supplemental to ^ CDL/ , since each is acute, .-. -4. ABB' = 4. CDjy-, and therefore their complements are equal. 592. Two lines not in the same plane are regarded as mak- ing with one another the angles included by two intersecting lines drawn parallel respectively to them. We know, from 588, that these angles will always be the same, whatever the position of the point of intersection. OF PLANES AND LINES, 23 1 Theorem XL 593* V ^^^ //«^J be cut by three parallel planes, the corre- spondi7ig sects are proporiio7ial. Let the lines AB, CD, be cut by the || planes MN, FQ, RSy in the points Aj E, By and C, F, D. Conclusion. AE : EB :: CF : FD. Join AD, cutting the plane FQ in G, Join A C, BD, EG, EG. Then EG II BD. (589. If parallel planes be cut by a third plane, their common sections with it are parallel.) In the same way, AC \\ GF\ ,\ AE : EB :: AG : GD, and AG '. GD :: CF ', FD', (499. A line parallel to the base of a triangle divides the sides proportionally.) /. AE \ EB \\ CF '. FD. 232 THE ELEMENTS OF GEOMETRY. Theorem XII. 594. Two lines not in tJu same plane have one, and only oite^ common j)erpendicular. 3L T Hypothesis. AB and CD, two lines not in tha same plane, and therefore neither parallel nor intersecting each other. Conclusion. There is one line, and no more, perpendicular to both AB and CD. Proof. Through one of the lines, as CD, pass a plane, and let it revolve on CD as axis until it is parallel to AB. Call this plane MN. Let A'B' be the projection of AB on the plane MN, and let O be the point in which this projection intersects CD. Then O, like every point in the projection, is the foot of a perpendicular to the plane from some point of AB. Call this point P. Then, since FO is perpendicular to the plane MN, it is perpendicular to CD and A'B'. But A'B' is par- allel to AB, because, being in a plane parallel to AB, it can never meet AB ; and, being the projection of AB, it is, by 580, in the same plane with AB. .'. -since FO _L A'B', it is also ± AB, (170. A line perpendicular to one of two parallels is perpendicular to the other also.) ,\ OF ± both AB and CD. If there could be any other common perpendicular, call it F'Q. OF PLANES AND LINES, 233 Through Q draw in the plane MN, QR \\ AB. Since FQ ± AB, .'. FQ J_ QR-, but, by hypothesis, F Q 1. CD, :. i^'Q ± plane J/W; /. ^ is a point in A'B' , the projection of AB, .'. Q is O, the only point common to A'B' and CD, .-. jP' ^ coincides with /'C?. (569. At a given point in a given plane, only one perpendicular to the plane can be erected.) Theorem XIII. 595. The smallest sect between two lines not i7t the same plane is their common perpeitdictdar. For if any sect drawn from one to the other is not perpendicular to both, by dropping a perpendicular to the hne it cuts obliquely from the point where it meets the other line, we get a smaller sect. (150. The perpendicular is the smallest sect from a point to a line.) 596. Remark. If two planes intersect, and two intersecting lines are drawn, one in each plane, these lines may, for the same two planes, make an acute, right, or obtuse angle, according to their relation to the line of intersection of the planes. 234 THE ELEMENTS OF GEOMETRY. Theorem XIV. 597. The smallest sect from a point to a plane is the perpen- dicidar. Hypothesis. AD ± plane MN. B any other point of MN. Conclusion. AD < AB. Proof. Join AB, BD. In A ABDyX ^^^ is a rt. ^, .-. AD < AB. (150. The perpendicular is the least sect from a point to a line.) Theorem XV. 598. Equal obliques from a point to a plane m,eet the pla7te in a circle whose center is the foot of the perpendicular from the point to the plane. OF PLANES AND LINES. 235 Hypothesis. AB, AC, equal sects from A to the plane MN. AD ± plane MN, Conclusion. DB = DC. Proof. Rt. A ABD ^ rt. A A CD. (179. If two right triangles have the hypothenuse and one side respectively equal, they are congruent.) 599. Corollary I. To draw a perpendicular from a point to a plane, draw any oblique from the point to the plane ; revolve this sect about the point, tracing a circle on the plane ; find the center of this circle, and join it to the point. 600. Corollary II. If through the center of a circle a line be passed perpendicular to its plane, the sects from any point of this line to points on the circle are equal. Theorem XVI. 601. The locus of all points from which the two sects drawn to two fixed poiftts are equal, is the pla7ie bisecting at right angles the sect joining the two given points. X For the line from any such point to the mid point of the joining sect is perpendicular to that sect ; therefore all such lines form a plane perpendicular to that sect at its mid point. (570. All lines perpendicular to another line at the same point lie in the same plane.) 2^6 THE ELEMENTS OF GEOMETRY. Theorem XVII. 602. The locus of all points from which the two perpendicu- lars onto the same sides of two fixed planes are equal, is a plane determined by one such point and the intersection line of the tzvo given planes. Hypothesis. Lei AB and CD be the two given planes, and K one such point. Conclusion. Perpendiculars dropped on to AB and CD from any point in the plane determined by X and the intersection line BC are equal. Proof. Take P any point in the plane XCB. Draw PH ± plane AB, and BP _L plane CB>. Call G the point where the plane FBI/ cuts the line BC. Join FG, BG, GH. Draw KC || BG. Because the perpendiculars from K are given equal, therefore KC is equally inclined to the two planes. But BG has the same inclination to each as KC ; (591. Parallels intersecting the same plane are equally inclined to it.) /. ^ BGF = ^ BGH, /. A BGF ^ A BGH, BF = BH, OF PLANES AND LINES. 237 Theorem XVIII. 603 • If three lines not in the same plane meet at one pointy any two of the angles formed are together greater than the third. Proof. The theorem requires proof only when the third angle considered is greater than each of the others. In the plane of the greatest 4. BAC, make 4. BAF = 4 BAD. Make AF = AD. Through F draw a line BFC cutting AB in B^ and AC m C. Join DB and DC. A BAD ^ A BAF, (124. Triangles having two sides and the included angle respectively equal are congruent.) .-. BD = BF. But from A BCD we have BD + DC> BC. {156. Any two sides of a triangle are together greater than the third.) And, taking away the equals BD and BF, DC > FC, .-. in As CAD and CAF, we have 4 CAD > 4 CAR (161. If two triangles have two sides respectively equal, but the third side greater in the first, its opposite angle is greater in the first.) Adding the equal 4 s BAD and BAF gives 4 BAD + 4 CAD > 4 BAC. 238 THR ELEMENTS OF GEOMETRY. Theokem XIX. 604. If the vertices of a convex polygon be joined to a point not in its plane ^ the snin of the vertical angles of the triangles so made is less than a perigon. s Proof. The sum of the angles of the triangles which have the common vertex S is equal to the sum of the angles of the same number of triangles having their vertices 2X O in the plane of the polygon. But %. SAB + ^ SAE > ^ BAE, %. SBA 4- ^ SBC > t ABC, etc. (603. If three Hnes not in a plane meet at a point, any two of the angles formed are together greater than the third.) Hence, summing all these inequalities, the sum of the angles at the bases of the triangles whose vertex is S, is greater than the sum of the angles at the bases of the triangles whose vertex is O \ therefore the sum of the angles at ^ is less than the sum of the angles at O, that is, less than a perigon. BOOK VIII. TRI-DIMENSIONAL SPHERICS. 605. If one end point of a sect is fixed, the locus of the other end point is a Sphere. 606. The fixed end point is called the Center of the sphere. 607. The moving sect in any position is called the Radius of the sphere. 608. As the motion of a sect does not change it, all radii are equal. 609. The sphere is a closed surface ; for it has two points on every line passing through the center, and the center is midway between them. •39 240 THE ELEMENTS OF GEOMETRY. 6io. Two such points are called Opposite Points of the sphere, and the sect between them is called a Diameter. 6ii. The sect from a point to the center is less than, equal to, or greater than, the radius, according as the point is within, on, or without, the sphere. For, if a point is on the sphere, the sect drawn to it from the center is a radius ; if the point is within the sphere, it lies on some radius ; if without, it lies on the extension of some radius. 612. By 33, Rule of Inversion, a point is within, on, or with- out, the sphere, according as the sect to it from the center is less than, equal to, or greater than, the radius. Theorem I. 613. The common section of a sphere and a plane is a ciirle. Take any sphere with center O. Let A, B, C, etc., be points common to the sphere and a plane, and OD the perpendicular from O to the plane. Then OA = OB = OC =^ etc., being radii of the sphere, .*. ABC, etc., is a circle with center D. (598. Equal obliques from a point to a plane meet the plane in a circle whose center is the foot of the perpendicular from the point to the plane.) TRI-DIMENSIONAL SPHERICS. 24 1 614. Corollary. The line through the center of any circle of a sphere, perpendicular to its plane, passes through the cen- ter of the sphere. 615. A Great Circle of a sphere is any section of the sphere made by a plane which passes through the center. All other circles on the sphere are called Small Circles. 616. Corollary. All great circles of the sphere are equal, since each has for its radius the radius of the sphere. 617. The two points in which a perpendicular to its plane, through the center of a great or small circle of the sphere, intersects the sphere, are called the Poles of that circle. 618. Corollary. Since the perpendicular passes through the center of the sphere, the two poles of any circle are oppo- site points, and the diameter between them is called the Axis of that circle. Theorem II. 619. Every great circle divides the sphere into two congruent hemispheres. For if one hemisphere be turned about the fixed center of the sphere so that its plane returns to its former position, but inverted, the great circle will coincide with its own trace, and the two hemispheres will coincide. 242 THE ELEMENTS OF GEOMETRY. 620. Any two great circles of a sphere bisect each other. Since the planes of these circles both pass through the center of the sphere, their line of intersection is a diameter of the sphere, and therefore of each circle. 621. If any number of great circles pass through a point, they will also pass through the opposite point. 622. Through any two points in a sphere, not the extremi- ties of a diameter, one, and only one, great circle can be passed ; for the two given points and the center of the sphere determine TRI-DIMENSIONAL SPHERICS. 243 its plane. Through opposite points, an indefinite number of great circles can be passed. 623. Through any three points in a sphere, a plane can be passed, and but one ; therefore three points in a sphere deter- mine a circle of the sphere. 624. A small circle is the less the greater the sect from its center to the center of the sphere. For, with the same hypoth- enuse, one side of a right-angled triangle decreases as the other increases. ^««.^ 625. A Zone is a portion of a sphere included between two parallel planes. The circles made by the parallel planes are the Bases of the zone. 626. A line or plane is tangent to a sphere when it has one point, and only one, in common with the sphere. 627. Two spheres are tangent to each other when they have one point, and only one, in common. Exercises. 105. If through a fixed point, within or with- out a sphere, three lines are drawn perpendicular to each other, intersecting the sphere, the sum of the squares of the three intercepted chords is constant. Also the sum of the squares of the six segments of these chords is constant. 106. If three radii of a sphere, perpendicular to each other, are projected upon any plane, the sum of the squares of the three projections is equal to twice the square of the radius of the sphere. 244 ^-^^ ELEMENTS OF GEOMETRY. Theorem III. 628. A plane perpendicular to a radius of a sphere at its extremity is tangent to the sphere. For, by 597, this radius, being perpendicular to the plane, is the smallest sect from the center to the plane ; therefore every point of the plane is without the sphere except the foot of this radius. 629. Corollary. Every line perpendicular to a radius at its extremity is tangent to the sphere. 630. Inverse of 628. Every plane or line tangent to the sphere is perpendicular to the radius drawn to the point of contact. For since every point of the plane or line, except the point of contact, is without the sphere, the radius drawn to the point of contact is the smallest sect from the center of the sphere to the plane or line ; therefore, by 597, it is per- pendicular. Exercises. 107. Cut a given sphere by a plane passing through a given line, so that the section shall have a given radius. 108. Find the locus of points whose sect from point A is ^, and from point B \s b. TRI-DIMENSIONAL SPHERICS. 245 Theorem IV. 631. If two spheres cut one another ^ their miersection is a cir- cle whose plane is perpendicnlar to the line Joining the centers of the spheres^ and whose center is in that line. Hypothesis. Let C and O be the centers of the spheres, A and B any two points in their intersection. Conclusion. A and B are on a circle having its center on the line OC, and its plane perpendicular to that line. Proof. Join CA, CB, OA, OB. Then A CAO^ A CBO, because they have OC common, CA = CB, and OA = OBj radii of the same sphere. Since these As are =, therefore perpendiculars from A and B upon OC are equal, and meet OC at the same point, Z>. Then AB> and DB are in a plane ± OC ; and, being equal sects, their extremities A and B axe in a circle having its center at Z>. 632. Corollary. By moving the centers of the two inter- secting spheres toward or away from each other, we can make their circle of intersection decrease indefinitely toward its center ; therefore, if two spheres are tangent, either internally or externally, their centers and point of contact lie in the same line. 246 THE ELEMENTS OF GEOMETRY. Theorem V. 633. Through any four pohits not in the same plane, one spJiere, and only one, can be passed. Let A^ B, C, D, be the four points. Join them by any three sects, as AB, BC, CD. Bisect each at right angles by a plane. The plane bisecting BC has the line EH'm. common with the plane bisecting AB, and has the line FO in common with the plane bisecting CD. Moreover, EH i. plane ABC, and FO ± plane BCD, (575. If one plane be perpendicular to one of two intersecting lines, and a second plane perpendicular to the second, their intersection is perpendicular to the plane of the two lines.) .-. EH LEG, and FO JL FG, ,'. EH and FO meet, since, by hypothesis, ^ at 6^ > o and < st. ^. Call their point of meeting O. O shall be the center of the sphere containing A, B, C, and D, TRI-DIMENSIONAL SPHERICS. 247 For O is in three planes bisecting at right angles the sects AB, BC, CD, (601. The locus of all points from which the two sects drawn to two fixed points are equal, is the plane bisecting at right angles the sect joining the two given points.) 634. A Tetrahedron is a solid bounded by four triangular plane surfaces called its faces. The sides of the triangles are called its edges. 635. Corollary I. A sphere may be circumscribed about any tetrahedron. 636. Corollary II. The lines perpendicular to the faces of a tetrahedron through their circumcenters, intersect at a common point. 637. Corollary III. The six planes which bisect at right angles the six edges of a tetrahedron all pass through a com- mon point. 248 THE ELEMENTS OF GEOMETRY. Problem I. 638. To inscribe a sphere in a given tetrahedron. Construction. Through any edge and any point from which per- pendiculars to its two faces are equal, pass a plane. Do the same with two other edges in the same face as the first, and let the three planes so determined intersect at O. O shall be the center of the required sphere. (602. The locus of all points from which the perpendiculars on the same sides of two planes are equal, is a plane determined by one such point and the intersec- tion line of the given planes.) 639. Corollary. In the same way, four spheres may be escribed, each touching a face of the tetrahedron externally, and the other three faces produced. 640. The solid bounded by a sphere is called a Globe. 641. A globe-segment is a portion of a globe included be- tween two parallel planes. The sections of the globe made by the parallel planes are the bases of the segment. Any sect perpendicular to, and terminated by, the bases, is the altitude of the segment. 642. A figure with two points fixed can still be moved by revolving it about the line determined by the two points. TR /-DIMENSIONAL SPHERICS. 249 This revolution can be performed in either of two senses, and continued until the figure returns to its original position. The fixed line is called the Axis of Revohction, If the axis of revolution is any line passing through the center, a sphere slides upon its trace. This is because every section of a sphere by a plane is a circle. Any figure has central symmetry if it has a center which bisects all sects through it terminated by the surface. The sphere has central symmetry, and coincides with its trace throughout any motion during which the center remains fixed. Thus any figure drawn on a sphere may be moved about on the sphere without deformation. But, unlike planes, all spheres are not congruent. Only those with equal radii will coincide. In general, a figure drawn upon one sphere will not fit upon another. So we cannot apply the test of superposition, except on the same sphere or spheres whose radii are equal. Again, if we wish the angles of a figure on a sphere to remain the same while the sides increase, we must magnify the whole sphere : on the same sphere similar figures cannot exist. 643. Two points are symmetrical with respect to a plane when this plane bisects at right angles the sect joining them. Two figures are symmetrical with respect to a plane when every point of one figure has its symmetrical point in the other. Any figure has planar symmetry if it can be divided by a plane into two figures symmetrical with respect to that plane. The sphere is symmetrical with respect to every plane through its center. Any two spheres are symmetrical with respect to every plane through their line of centers. Every such plane cuts the spheres in two great circles ; and the five relations between the center-sect, radii, and relative position of these circles given in 410, with their inverses, hold for the two spheres. Any three spheres are symmetrical with respect to the plane determined by their centers. 250 THE ELEMENTS OF GEOMETRY, Theorem VI. 644. If a sphere he tangent to the parallel plafzes containing opposite edges of a tetrahedron^ and sections made in the globe and tetrahedron by one plane parallel to these are equivalenty sections made by any parallel plaiie are equivalent. K K \^. ij Jo L\ /^ ^^ Hypothesis. Let KJ be the sect J_ the edges EF and GH in the II tangent planes. Then Kf = DT, the diameter. Let sections made by plane ± Kf at R and ± DT at I, where KR = DI, be equivalent; that is, ^ MO = O PQ. Draw any parallel plane ACBLSN, Conclusion, ^l^ LN — O AB. Proof. Since A LEU ^ h. MEW, and A LHV ^ A MHZ, :. MW \ LU w EM -. EL :: fR : fS; MZ '. LV -.'. HM '. HL '.: KR : KS; (593. If lines be cut by three parallel planes, the corresponding sects are propor- tional.) .-. WM. MZ : C/L.LF :: fR .RK : fS . SK ; :, ^MO : ^ LN :: fR .RK : fS .SK (i) {542. Mutually equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.) TRI-DIMENSIONAL SPHERICS. 251 But OPQ \ Q)AB '.'. PP '. AC- :: TI . ID : TC . CD, (2) {546. Similar figures are to each other as the squares on their corresponding sects.) (522. If from any point in a circle a perpendicular be dropped upon a diameter, it will be a mean proportional between the segments of the diameter.) By hypothesis and construction, in proportions (i) and (2), the first, third, and fourth terms are respectively equal, Theorem VII. 645. The sects Joining its pole to points on any circle of a sphere are equal. Proof. (600. If through the center of a circle a line be passed perpendicular to its plane, the sects from any point of this line to points on the circle are equal.) 646. Corollary. Since chord PA equals chord PB, there- fore the arc subtended by chord PA in the great circle PA equals the arc subtended by chord PB in the great circle PB. Hence the great-circle-arcs joining a pole to points on its circle are equal. 252 THE ELEMENTS OF GEOMETRY. So, if an arc of a great circle be revolved in a sphere about one of its extremities, its other extremity will describe a circle of the sphere. 647. One-fourth a great circle is called a Quadrant. 648. The great-circle-arc joining any point in a great circle with its pole is a quadrant. p ^ o\ "'^N / 'N \3j :/ ::^ 649. If a point P be a quadrant from two points, A^ B, which are not opposite, it is the pole of the great circle through A, B ; for each of the angles POA, FOB, is right, and therefore PO is perpendicular to the plane OAB. 650. The angle between two intersecting curves is the angle between their tangents, at the point of intersection. When the curves are arcs of great circles of the same sphere, the angle is called a Spherical Angle. TRI-DIMENSIONAL SPHERICS, 253 651. If from the vertices, A and F, of any two angles in a sphere, as poles, great circles, BC and GH^ be described, the angles will be to one another in the ratio of the arcs of these circles intercepted between their sides (produced if necessary). For the angles A and F are equal respectively to the angles BOC and GOH. (591. Parallels intersecting the same plane are equally inclined to it.) But BOC and GOH are angles at the centers of equal cir- cles, and therefore, by 506, are to one another in the ratio of the arcs BC and GH. 652. Corollary. Any great-circle-arc drawn through the pole of a given great circle is perpendicular to that circle. FG is ± GH. For, by hypothesis, FG is a quadrant ; there- fore the great circle described with G as pole passes through F, and so the arc intercepted on it between GF and GH is, by 648, also a quadrant. But, by 651, the angle at G is to this quadrant as a straight angle is to a half-line. Inversely, any great-circle-arc perpendicular to a great cir- cle will pass through its pole. For if we use G as pole when the angle 'at (7 is a right angle, then FH, its corresponding arc, is a quadrant when GF is a quadrant ; therefore, by 649, F is the pole of GH. 254 THE ELEMENTS OF GEOMETRY. Theorem VIII. 653. The smallest line in a sphere^ between two points, is the great-circle-arc not greater than a semicircle^ which joins them. Hypothesis. AB is a greaf-c/rc/s-arc, not greater than a semicircle, joining any two points A and B on a sphere. First, let the points A and B be joined by the broken line A CB, which consists of the two great- circle -arcs A C and CB. Conclusion. AC + CB > AB. Proof. Join O, the center of the sphere, with A, By and C. ^ AOC -f ^ COB > ^ AOB. (603. If three lines not in the same plane meet at one point, any two of the angles formed are together greater than the third.) But the corresponding arcs are in the same ratio as these angles, .-. AC -\- CB>AB. Second, let F be any point whatever on the great-circle-arc AB. The smallest line on the sphere from A to B must pass through B. For by revolving the great-circle-arcs AB and BB about A and B as poles, describe circles. These circles touch at B, and lie wholly without each other ; for let F be any other point in the circle whose pole is B, and join FA, FB by great-circle-arcs, then, by our First, FA ^ FB> AB, .*. FA > FA, and F lies without the circle whose pole is A. TRI-DIMENSIONAL SPHERICS. 255 Now let ADEB be any line on the sphere from AXo B not passing through /*, and therefore cutting the two circles in different points, one in D, the other in E. A portion of the line ADEB, namely, DE, lies between the two circles. Hence if the portion AD be revolved about A until it takes the position AGP, and the portion BE be revolved about B into the position BHP, the line AGPHB will be less than ADEB. Hence the smallest line from A to B passes through P, that is, through any or every point in AB ; consequently it must be the arc AB itself. 654. Corollary. A sect is the smallest line in a plane between two points. BOOK IX. TWO-DIMENSIONAL SPHERICS. INTRODUCTION. . 655. Book IX. will develop the Geometry of the Sphere, from theorems and problems almost identical with those whose assumption gave us Plane Geometry. In Book VIII., these have been demonstrated by considering the sphere as contained in ordinary tri-dimensional space. But, if we really confine our- selves to the sphere itself, they do not admit of demonstration, except by making some more difficult assumption : and so they are the most fundamental properties of this surface and its characteristic line, the great circle ; just as the assumptions in our first book were the most fundamental properties of the plane and its characteristic line. So now we will call a great circle simply the spherical line ; and, whenever in this book the word line is used, it means spherical line. Sect now means a part of a line less than a half-line. 257 258 THE ELEMENTS OF GEOMETRY. Two-Dimensional Definition of the Sphere. 656. Suppose a closed line, such that any portion of it may be moved about through every portion of it without any other change. Suppose a portion of this line is such, that, when moved on the line until its first end point comes to the trace of its second end point, that second end point will have moved to the trace of its first end point. Call such a portion a half- line, and any lesser portion a sect. Suppose, that, while the extremities of a half-line are kept fixed, the whole line can be so moved that the slightest motion takes it completely out of its trace, except in the two fixed points. Such motion would generate a surface which we will call the Sphere. Fundamental Properties of the Sphere. ASSUMPTIONS. 657. A figure may be moved about in a sphere without any other change ; that is, figures are independent of their place on the sphere. 658. Through any two points in the sphere can be passed a line congruent with the generating line of the sphere. In Book IX., the word line will always mean such a line, and sect will mean a portion of it less than half. 659. Two sects cannot meet twice on the sphere ; that is, if two sects have two points in common, the two sects coincide between those two points. 660. If two lines have a common sect, they coincide throughout. Therefore through two points, not end points of a half-line, only one distinct line can pass. 661. A sect is the smallest path between its end points in the sphere. TWO-DIMENSIONAL SPHERICS. 259 662. A piece of the sphere from along one side of a line will fit either side of any other portion of the line. DEFINITIOXS. 663. If one end point of a sect is kept fixed, the other end point moving in the sphere describes what is called an arc, while the sect describes at the fixed point what is called a spherical angle. The angle and arc are greater as the amount of turning in the sphere is greater. 664. When a sect has turned sufficiently to fall again into the same line, but on the other side of the fixed point or ver- tex, the angle described is called a straight angle, and the arc described is called a semicircle. 665. Half a straight angle is called a right angle. 666. The whole angle about a point in a sphere is called a perigon : the whole arc is called a circle. The point is called the pole of the circle, and the equal sects are called its spherical radii. ASSUMPTIONS. 667. A circle can be described from any pole, with any sect as spherical radius. 668. All straight angles are equal. 26o THE ELEMENTS OF GEOMETRY. 669. Corollary I. All perigons are equal. 670. Corollary II. If one extremity of a sect is in a line, the two angles on the same side of the line as the sect are together a straight angle. 671. Corollary III. Defining adjacent angles as two angles having a common vertex, a common arm, and not over- lapping, it follows, that, if two adjacent angles together equal a straight angle, their two exterior arms fall into the same line. 672. Corollary IV. If two sects cut one another, the vertical angles are equal. 673. Any line turning in the sphere about one of its points, through a straight angle, comes to coincidence with its trace, and has described the sphere. 674. Corollary I. The sphere is a closed surface. TWO-DIMENSIONAL SPHERICS, 26 1 675. Corollary II. Every line bisects the sphere, — cuts it into hemispheres. Reduced Properties of the Sphere. Theorem I. 676. All lilies in a sphere ijttersect, and bisect each other at their points of intersection. Let BB' and CC be any t\vo lines. Since, by 675, each of them bisects the sphere, therefore the second cannot lie wholly in one of the hemispheres made by the first, therefore they intersect at two points ; let them intersect at A and A' . If BA C be revolved in the sphere about A until some point in the sect AB coincides with some point in the sect AB' y then ABA' will lie along AB'A'. (660. Through two points, not end points of a half-line, only one distinct line can pass.) Then, also, since the angles BA C and B^A C, being vertical, are, by 672, equal, AC A' will lie along AC A'. Therefore the second point of intersection of AB and A C must coincide with the second point of intersection of AB' and A C, and ABA' be a half-line, as also A CA' . 262 THE ELEMENTS OF GEOMETRY. 677. A spherical figure such as ABA'CA^ which is contained by two half -lines, is called a Lune. Theorem II. 678. The angle contained by the sides of a hme at one of their points of intersection equals the angle contained at the other For, if the two angles are not equal, one must be the greater. Sup- pose :^ A> :^ A', Move the lune in the sphere until point A' coincides with the trace of point A, and half-line A'BA coincides with the trace of half-line A CA' . Then, since we have supposed -4- A> ^ A' , the half-Hne A' CA would start from the trace of A between the trace of AC A' and the trace of ABA' . Starting between them, it could meet neither again TWO-DIMENSIONAL SPHERICS. 263 until it reached the trace of A' ; and so we would have the surface of the lune less than its trace, which is contrary to our first assumption (657). .*. the two angles cannot be unequal. DEFINITIONS. 679. The Supplement of a Sect is the sect bywhich it differs from a half-line. 680. One-quarter of a line is called a Quadrant. 618. A Spherical Polygoii is a closed figure in the sphere formed by sects. 682. A Spherical Triangle is a convex spherical polygon of three sides. 683. Symmetrical Spherical Polygons are those in which the sides and angles of the one are respectively equal to those of 264 THE ELEMENTS OF GEOMETRY, the other, but arranged in the reverse order. If one end of a sect were pivoted v^ithin one polygon, and one end of another sect pivoted within the symmetrical polygon, and the two sects revolved so as to pass over the equal parts at the same time, one sect would move clockwise, while the other moved counter- clockwise. 684. Two points are symmetrical with respect to a fixed line, called the axis of symmetry, when this axis bisects at right angles the sect joining the two points. Any two figures are symmetrical with respect to an axis when every point of one has its symmetrical point on the other. Theorem III. 685. The perimeter of a convex spherical polygon wholly con- tained within a second spherical polygon is less than the perimeter of the second. Let ABFGA be a convex spherical polygon wholly contained in ABCDEA, TWO-DIMENSIONAL SPHERICS. 26$ Produce the sides A G and GF to meet the containing perimeter at K and L respectively. By 66 1, a sect is the smallest path between its end points, .-. BF 4. EDF, Conclusion. BC > EF. Proof. When AB coincides with DE, if the points C and F are on the same side of the line AB, then, since ^ BAC > ^ EDF, the side DF will lie between BA and A C, and DF must stop either within the triangle ABC, on BC, or after cutting BC. When F is within the triangle, by 685, ^C + CA > EF + FD ; but^C= DF, .-. BC>EF, When F lies on BC, then EF is but a part of BC. In case DF cuts BC, call their intersection point G. Then BG + GF > EF, and GC -^ GD> AC, .-. BC ^ FD >AC ^ EF', hwi FD = AC, .'. BC>EF. If, when one pair of equal sides coincide, the other pair lie on oppo- site sides of the line of coincidence, the above proof will show the third side of a triangle symmetrical to the first to be greater than the third side of the second triangle, and therefore the third side of the first greater than the third side of the second. 268 THE ELEMENTS OF GEOMETRY. 690. From 6'^% and 689, by 33, Rule of Inversion, if two spherical triangles have two sides of the one equal respectively to two sides of the other, the included angle of the first is greater than, equal to, or less than, the included angle of the second, according as the third side of the first is greater than, equal to, or less than, the third side of the second. 691. Corollary. Therefore, by 688, two spherical triangles having three sides of the one equal respectively to three sides of the other are either congruent or symmetrical. Problem I. 692. To bisect a given spherical angle. Let BA C be the given 2^ . In its arms take equal sects AB and A C each less than a quadrant. Join BC. With B as pole, and any sect greater than half BC, but not greater than a quadrant, as a spherical radius, describe the circle EDF. With an equal spherical radius from C as pole describe an arc intersecting the circle EDF at D within the lune BAC. Join DB, DC, DA. i9^ shall bisect the angle ^^C. For the 'as ABD, A CD, having AD common and AB = AC, and BD = DC (spherical radii of equal circles), are, by 691, symmetrical, .'. ^ BAD = ^ CAD. TWO-DIMENSIONAL SPHERICS. 269 693. Corollary I. To bisect the re-entrant angle BAC, produce the bisector of the angle BAC. 694. Corollary II. To erect a perpendicular to a given line from a given point in the line, bisect the straight angle at that point. Problem II. 695. To bisect a given sect. With A and B, the extremities of the given sect, as poles, and equal spherical radii greater than half AB, but less than a quadrant, describe arcs intersecting at C. Join AC and BC; and, by 692, bisect the angle ACB, and produce the bisector to meet AB at Z>. B> is the mid point of the sect AB. For, by 688, a" A CD is symmetrical to 'aBCJD, 696. Corollary. The angles at the base of an isosceles spherical triangle are equal. 270 THE ELEMENTS OF GEOMETRY. Problem III. 697. To draw a perpendicular to a given line from a given point in the sphere not in the line. Given, the line AB, and point C. Take a point D on the other side of the line AB, and with C as pole, and CD as spherical radius, describe an arc cutting AB in F and G. Bisect the sect FG at B, and join CB. CB shall be ± AB. For, by 691, 'as GCB and FCB are symmetrical. Theorem VII. 698. If two lines be drawn in a sphere, at right angles to a given line, they will intersect at a point from which all sects drawn to the given line are equal. Let AB and CB, drawn at right angles to AC, intersect at B, and meet A C again at A' and C respectively. TWO-DIMENSIONAL SPHERICS. 27 1 Then 4. BA'C = ^ BAC, and 4. EC A' = 4 BCA' -, (678. The angles contained by the sides of a lune, at their two points of intersection, are equal.) moreover, AC = A'C\ for they have the common supplement A C . Hence, keeping A and C on the line AC, slide ABC until AC comes into coincidence with A' C . Then, the angles at A, C, A\ C, being all right, AB will lie along A'B, and CB along C'B, and hence the figures AB C and A'B C coincide. .-. each of the half-Unes ABA' and CB C is bisected at B. In like manner, any other line drawn at right angles to ^ C passes through B, the mid point of ABA' . Hence every sect from AC to B is a. quadrant. 699. Corollary I. A line is a circle whose spherical radius is a quadrant. 700. Corollary II. A point which is a quadrant from two points in a line, and not in the line, is its pole. 701. Corollary III. Any sect from the pole of a line to the line is perpendicular to it. 702. Corollary IV. Equal angles at the poles of lines intercept equal sects on those lines. 703. Corollary V. If K be the pole, and FG a sect of any other line, the angles and semilunes ABC and FKG are to one another as ^ 6^ to FG. 2/2 THE ELEMENTS OF GEOMETRY. 704. We see, from 698, that a spherical triangle may have two or even three right angles. If a spherical triangle ABC has two right angles, B and C^ it is called a bi-rectaiigular triangle. By 698, the vertex A is the pole of BC, and therefore AB and AC are quadrants. 705. The Polar of a given spherical triangle is a spherical triangle, the poles of whose sides are respectively the vertices of the given triangle, and its vertices each on the same side of a side of the given triangle as a given vertex. Theorem VIII. 706. If, of two spherical triangles, the first is the polar of the second, tJien the second is the polar of the first. Hypothesis. Lei A'B'C be the polar of ABC, Conclusion. Then ABC is the polar of A'B'C. TWO-DIMENSIONAL SPHERICS, 2/3 Proof. Join A'B and A' C, Since B is the pole of A' C , therefore BA' is a quadrant ; and since C is the pole of A'B\ therefore CA' is a quadrant ; .*. by 700, A' is the pole of BC. In like manner, B' is the pole of A C, and C of AB, Moreover, A and A' are on the same side oi B'C'y B and B' on the same side oi A'C, and C and C on the same side oi A'B\ .-. ^.5C is the polar of A'B'C, Theorem IX. 707. In a pair of polar triangles, any angle of either inter- ceptSy on the side of the other which lies opposite to it, a sect which is the supplement of that side. Let ABC and A'B^C be two polar triangles. Produce A'B' and B' C to meet BC at Z> and ^ respectively. Since B is the pole of B' C , therefore BE is a quadrant ; and since C is the pole of A'B' , therefore CD is a quadrant ; therefore BE -f CD = half-line, but BE -{- CD = BC -{- DE. Therefore DE, the sect of BC which A' intercepts, is the supplement of BC. Exercises. 109. Any lune is to a tri-rectangular triangle as its angle is to half a right angle. 274 THE ELEMENTS OF GEOMETRY. Theorem X. 708. If two angles of a spherical triangle be equals the sides which subtend them are equal. Hypothesis. In ^ ABC let 4. A = ^ C, Conclusion. BC = AB. Proof. For draw A'B'C, the polar of ABC. Now, on B' C and A'B' the equal 7^^ A and C intercept equal sects. Therefore B' C and A'B'y being, by 707, the supplements of these equal sects, are equal, .*. "4- A' — -^ C, (696. The angles at the base of an isosceles spherical triangle are equal.) .'. the supplements oi BC and AB are equal, /. BC ^ AB, Theorem XI. 709. If one angle of a spherical triangle be greater than a second^ the side opposite the first must be greater than the side opposite the second. In'^ABC, 4 C>4.A. Make ^ ACD = ^^; /. by 708, AD = CD, TWO-DIMENSIONAL SPHERICS. 2y$ But, by 66i, CD -^ DB> BC, .-. AD -^ DB>BC. 710. From 708 and 709, by 33, Rule of Inversion, If one side of a spherical triangle be greater than a second, the angle opposite the first must be greater than the angle opposite the second. Theorem XII. 711. Two spherical triangles having two angles and the included side of the one equal respectively to two angles and the included side of the other^ are either congrue7it or symmetricaL For the first triangle can be moved in the sphere into coincidence with the second, or with a triangle made symmetrical to the second. Theorem XIII. 712. Two spherical triangles having three angles of the one equal respectively to three angles of the other^ are either congruetit or symmetrical. Since the given triangles are respectively equiangular, their polars are respectively equilateral. (702. Equal angles at the poles of lines intercept equal sects on those lines; and, by 707, these equal sects are the supplements of corresponding sides.) 2/6 THE ELEMENTS OF GEOMETRY. Hence these polars, being, by 691, congruent or symmetrical, are respectively equiangular, and therefore the original spherical triangles are respectively equilateral. Theorem XIV. 713. An exterior angle of a spherical triangle is greater than^ eqnal to, or less tkaUy either of the interior opposite angles, accord- ing as the medial from the other interior opposite angle is less than, equal to, or greater than, a quadrant. Let ACD be an exterior angle of the a" ABC. By 695, bisect AC at E. Join BE, and produce to E, making EE = BE. Join EC» A ABE ^ A CEE, (688. Spherical triangles having two sides and the included angle equal are congru- ent or symmetrical.) .'. ^ BAE = ^ ECE. If, now, the medial BE be a quadrant, BEE is a half-line, and, by 676, E lies onBD; .'. 4.DCE coincides with ^ ECE, .\ ^ DCE = t BAE. If the medial BE be less than a quadrant, BEE is less than a half- line, and E lies between A C and CD ; .♦. ^ DCA > ^ ECE, .-. ^ DCA > ^ BAC. And if BE be greater than a quadrant, BEE is greater than a half- hne, and E lies between CD and A C produced ; .-. ^ DCA < -4. ECE, /. ^ DCA < 4 BAC. TWO-DIMENSIONAL SPHERICS. 2'J'J Thus, according as BE is greater than, equal to, or less than, a quadrant, the exterior ^ A CD is less than, equal to, or greater than, the interior opposite 2(1 BA C, 714. From 713, by 33, Rule of Inversion, According as the exterior angle ACD is greater than, equal to, or less than, the interior opposite angle BAC, the medial B£ is less than, equal to, or greater than, a quadrant. Theorem XV. 715. Two spherical triangles having the a^tgles at the base of the 07ie eqical to the a^tgles at the base of the other^ and the sides opposite one pair of equal angles eqical, are either coJigricent or symmetrical, provided that in neither triangle is the third angular point a quadrant from any point iii that half of its base not adjaceftt to 07ie of the sides equal by hypothesis. First, if the parts given equal lie clockwise in the two spherical triangles, as in ABC and DEF, where we suppose ^ B = }^ E, 4. C = 4. F,2ir\dAB == DE, make DE coincide with AB ; then EF will lie along BC, and DF must coincide with AC. For if it could take any other position, as AG, it would make 2^^ AGC with exterior ^ AGB = interior opposite ^ C, and therefore, by 714, with medial AH a quadrant, which is contrary to our hypothesis. Second, if the equal parts lie in one spherical triangle clockwise, in the other counter-clockwise, as in 'KA'B'C and ^ DEF, then A B>EF ^^ ABC, which is symmetrical to a A' B' C, 2/8 THE ELEMENTS OF GEOMETRY. Theorem XVI. 716. If a sect drawn in the sphere from a point perpendictilar to a line be less than a qnadranty it is the smallest which can be drawn in the sphere from the point to the line ; of others^ that which is nearer the perpendicular is less than that which is more remote ; also to every sect drawn on one side of the perpendicular there can be drawn one^ and only one^ sect equal on the other side. Let A be the point, BD the Hne, AB J. BD^ and AD nearer than AE to AB. Produce AB to C, making BC ^ AB. Join CD, CE, Then AD = CD. (688. Spherical triangles having two sides and the included angle equal are con- gruent or symmetrical.) But, by 661, AD -j- CD or iAD > AC or 2AB, .-. AD > AB. Also, by 685, AE + CE or 2AE > AD -f CD or 2 AD. Again, make BE — BD. Join AE. As before, AE = AD, and we have already shown that no two sects on the same side of the perpendicular can be equal. TWO-DIMENSIONAL SPHERICS. 279 717. Corollary. The greatest sect that can be drawn from A to BD is the supplement of AB. Theorem XVII. 718. If two spherical triangles have two sides of the one equal to two sides of the other^ and the angles opposite one pair of equal sides equals the angles opposite the other pair a7'e either equal or supplemental. First, given the equal parts AB = DE, AC = DF, and ^ B = ^ E arranged clockwise in the two spherical triangles. \i -2^ A =. }^ D, the spherical triangles are congruent. \i ^ A is not = ^ Z>, one must be the greater. Suppose -4. A> -i^ D. Make DE coincide with AB. Then, since ^B = :4E, side EF will he along BC; since ^ A> ^ D, side VF will lie between AB and A C, as at A G. Now, the two angles at G are supplemental ; but one is -^ F and the other = ^ C, because "a CFG is isosceles. Second, if the parts given equal lie clockwise in one spherical tri- angle and counter-clockwise in the other, as in "a A'B' C and 'a DEF, then, taking the "a ABC symmetrical to A'B' C, the above proof shows ^ F equal or supplemental to ^ C, that is, to ^ C\ 280 THE ELEMENTS OF GEOMETRY. Theorem XVIII. 719. Symmetrical isosceles spherical triangles are congruent. For, since four sides and four angles are equal, the distinction between clockwise and counter-clockwise is obliterated. Theorem XIX. 720. The locus of a point from which the two sects drawn to two given points are equal is the line bisecting at right angles the sect joining the two given points. For every point in this perpendicular bisector, and no point out of it, possesses the property. TWO-DIMENSIONAL SPHERICS. 281 Problem IV. 721. To pass a circle through auy three points , or to find the circumcenter of any spherical triangle. Find the intersection point of perpendiculars erected at the mid- points of two sides. Theorem XX. 722. Any angle made with a side of a spherical triangle by joining its extremity to the circwnc enter ^ equals half the angle- sum less the opposite aiigle of the triangle. FoT^A-}-^B-{-^C=2^ OCA + 2 ^ OCB ± 2 ^ OAB, ^A 4- ^B 4- ^ C t OCA = ^ \ U OCB ± ^ OAB) ^A -\- ^B + 4.C ^ ^B. 282 THE ELEMENTS OF GEOMETRY. 723. Corollary. Symmetrical spherical triangles are equiv- alent. For the three pairs of isosceles triangles formed by joining the vertices to the circumcenters having respectively a side and two adjacent angles equal, are congruent. Theorem XXI. 724. When three lines mutually intersecty the two triangles on opposite sides of any vertex are together equivalent to the lune with that vertical angle. "aABC-^-^ ADF = lune ABHCA. For DF — BC, having the common supplement CD ) and FA — CH, having the common supplement HF'^ and AD = BH^ having the common supplement HD ; .-. aADF^aBCFT, .-. A ABC + A ADF = A ABC -f a BCH = lune ABHCA. 725. The Spherical Excess of a spherical triangle is the excess of the sum of its angles over a straight angle. In gen- eral, the spherical excess of a spherical polygon is the excess of the sum of its angles over as many straight angles as it has sides, less two. TWO-DIMENSIONAL SPHERICS. 283 Theorem XXII. 726. A spherical triangle is equivalent to a lime whose angle is half the t7 iattgles spherical excess. Proof. Produce the sides of the "a ABC until they meet again, two and two, at Dy F^ and H. The a" ABC now forms part of three lunes, whose angles are A, Bj and C respectively. But, by 724, lune with 2^ ^ = a ABC + a ADF. Therefore the lunes whose angles are Ay By and C, are together equal to a hemisphere plus twice 'a ABC. But a hemisphere is a lune whose angle is a straight angle, .-. 2^ ABC = lune whose ^ is (^ + ^ + C - st. ^) = lune whose ^ is e, 727. Corollary I. The sum of the angles of a spherical triangle is greater than a straight angle and less than 3 straight angles. 728. Corollary II. Every angle of a spherical triangle is greater than ^e. T2.g, Corollary III. A spherical polygon is equivalent to a lune whose angle is half the polygon's spherical excess. 730. Corollary IV. Spherical polygons are to each other as their spherical excesses, since, by 703, lunes are as their angles. 284 THE ELEMENTS OF GEOMETRY. 731. Corollary V. To construct a lune equivalent to any spherical polygon, add its angles, subtract {71 — 2) straight angles, halve the remainder, and produce the arms of a half until they meet again. Theorem XXIII. 732. If the line be completed of which the base of a given spherical triangle is a sect^ and the other two sides of the triangle be produced to meet this line, and a circle be passed through these tzvo points of intersectio7i and the vertex of the triangle, the locns of the vertices of all triangles equivalent to the give?z triajtgle^ and on the same base with it, and on the same side of that base, is the arc of this circle terminated by the intersection points and cojttaining the vertex. C Let ABC be the given spherical triangle, AC its base. Produce AB and CB to meet AC produced in D and E respectively. By 719, pass a circle through B,D,E. Let F be any point in the arc EBD. Join AP and CP. AP produced passes through the opposite point D, and CP through E, forming with DE the a PDE. Join F, its circumcenter, with P, D, and E, Since, by 678, the two angles of a lune are equal, .*. 2^: PAC = St. ^ - :^ PDE, ^ PCA = St. ^ - 2< PED, 4.APC ^ 4.DPE', TWO-DIMENSIONAL SPHERICS. 285 /. 4.{PAC ■\- PCA 4- APC) = 2St. ^s - ^ {PDE + PED - DPE) = 2 St. ^ s -- 2 ^ EZ>E = a constant. (722. Any angle made with a side of a spherical triangle by joining its extremity to the circumcenter, equals half the angle-sum less the opposite angle of the spherical triangle.) 733. In a sphere a line is said to touch a circle when it meets the circle, but will not cut it. Theorem XXIV. 734. T/ie line drawn at right angles to the spherical radius of a circle at its extremity touches the circle. Let BD be perpendicular to the spherical radius AB, Join A with any point C in BD. By 7 1 6, ^ C > AB, therefore C is without the circle. And no other line through B, as BF, can be tangent. For draw AE _L BF. By 716, AB > AE, ,'. E is within the circle. 286 THE ELEMENTS OF GEOMETRY. Theorem XXV. 735. In a sphere the stun of one pair of opposite angles of a quadfilateral inscribed in a circle equals the sum of the other pair. Join E^ the circumcenter, with A, By C, D, the vertices of the inscribed quadrilateral. By 696, ^ ABC = 4. BAE + ^ BCE, and 4. ADC = 4. DAE + 4 DCE, /. 4. ABC + 4 ADC = 4. BAD + 4 BCD, Theorem XXVI. 736. In equal circles^ equal angles at the pole stand on equal arcs. For the figures may be slidden into coincidence. p TWO-DIMENSIONAL SPHERICS. 287 Theorem XXVII. 737. EquQ,l spherical chords cut equal circles into the same two arcs. Theorem XXVIII. 738. In equal circles^ angles at the corresponding poles have the same ratio as their arcs. Theorem XXIX. 739' Four pairs of equal circles ca7t be drawn to touch three non-concurrent lines in a sphere. BOOK X. POLYHEDRONS. 740. A Polyhedron is a solid bounded by planes. 741. The bounding planes, by their intersections, determine the Faces of the polyhedron, which are polygons. 742. The Edges of a polyhedron are the sects in which its faces meet. 290 THE ELEMENTS OF GEOMETRY. 743. The Summits of a polyhedron are the points in which its edges meet. 744. A Plane Section of a polyhedron is the polygon in which a plane passing through it cuts its faces. 745. A Pyramid is a polyhedron of which all the faces, except one, meet in a point. 746. The point of meeting is called the Apex^ and the face not passing through the apex is taken as the Base. 747. The faces and edges which meet at the apex are called Lateral Faces and Edges. 748. Two polygons are said to be parallel when each side of the one is parallel to a corresponding side of the other. 749. A Prism is a polyhedron two of whose faces are con- gruent parallel polygons, A»d the other faces are parallelo- grams. 750. The Bases of a prism are the congruent parallel poly- gons. 751. The Lateral Faces of a prism are all except its bases. 752. The Lateral Edges are the intersections of the lateral faces. 753. A Right Section of a prism is a section by a plane per- pendicular to its lateral edges. POL YHEDRONS. 291 754. The Altitude of a Prism is any sect perpendicular to both bases. 755. The Altitude of a Pyramid is the perpendicular from its vertex to the plane of its base. 756. A Right Prism is one whose lateral edges are perpen- dicular to its bases. \ 757. Prisms not right are oblique. 758. A Parallelopiped is a prism whose bases are parallelo- grams. 759. A Quader is a parallelopiped whose six faces are rect- angles. 760. A Citbe is a quader whose six faces are squares. 292 THE ELEMENTS OF GEOMETRY. Theorem I. 761. All the summits of any polyhedron may be joined by one closed line breaking only in themy and lying wholly on the surface. For, starting from one face, ABC . . . , each side belongs also to a neighboring polygon. Therefore, to join A and B, we may omit AB, and use the remain- der of the perimeter of the neighboring polygon a. In the same way, to join B and C, we may omit BC, and use the remainder of the perimeter of the neighboring polygon b, unless the polygons a and b have in common an edge from B. In such a case, draw from ^ in ^ the diagonal nearest the edge common to a and b ; take this diagonal and the perimeter of b beyond it around to C, as continuing the broken hne ; and proceed in the same way from C around the neighboring polygon c. When this procedure has taken in all summits in faces having an edge in common with ABC . . . , we may, by proceeding from the closed broken line so obtained, in the same way take in the summits on the next series of contiguous faces, etc. So continue until the single closed broken Hne goes once, and only once, through every summit. POL YHEDRONS. 593 Theorem II. 762. Cutting by diagonals the faces not triangles into tri- angleSy the whole surface of aiiy polyhedron contains four less triattgles than double its finmber of summits. For, joining all the summits by a single closed broken line, this cuts the surface into two bent polygons, each of which contains S — 2 tri- angles, where S is the number of summits. 763. Corollary. The sum of all the angles in the faces of any polyhedron is as many perigons as the polyhedron has summits, less two. 764. Remark. Theorem II. is called Descartes' Theorem, and is really the fundamental theorem on polyhedrons, though this place has long been held by Theorem III., called Euler's Theorem, which follows from it with remarkable elegance. 294 THE ELEMENTS OF GEOMETRY. Theorem III. 765. The number of faces and summits iri any polyhedron^ taken together, exceeds by two the member of its edges. Case I. If all the faces are triangles. Then, by 762, 7^ = 2(6'- 2). But also 2E = zF, for each edge belongs to two faces, and so we get a triangle for every time 3 is contained in 2E. By adding, we have 2E = 2F + 2{S — 2); that is, F -\- S = E -^ 2, Case II. If not all the faces are triangles. Since to any pyramidal summit go as many faces as edges, we may replace any polygonal face by a pyramidal summit without changing the POL YHEDRONS. 295 equality or inequality relation oi F -\- S \.o E -{■ 2 ; for such replace- ment only adds the same number to F as to E^ and changes one face to a summit. But, after all polygonal faces have been so replaced, F ■\- S =■ E -\- 2y \yy Case I. Therefore always the relation was equality. Theorem IV. 766. Qtiaders having Congruent bases are to each other as their altitudes. 1 jm~- - - — *' \ 1 y 7 at A / A 1 -•r hH ,>-.- 7 Hypothesis. Let a and c^ be the aliiiudes of two quaders, Q and Q, having congruent bases B. Conclusion. Q : Q w a \ a'. Proof. Of a take any multiple, ma ; then the quader on base B with altitude ma is mQ. In the same way, take equimultiples na' , nQ. According as m Q is greater than, equal to, or less than, n Q, we have ma greater than, equal to, or less than na' \ therefore, by defini- tion, Q \ Q \'. a \ a\ Exercises, no. In no polyhedron can triangles and three- faced summits both be absent ; together are present at least eight. Not all the faces nor all the summits have more than five sides. III. There is no seven-edged polyhedron. 296 THE ELEMENTS OF GEOMETRY. Theorem V. 767. Quaders having equal altitudes are to each other as their bases. "/HA A A A a 1 I 1 1 / f Jo- JP V A / i Hypothesis. Let the rectangles he and h', and DF \\ CE. Through CE and /?i^pass parallel planes. Now the solids DFB and CEA are congruent, having all their angles and edges respectively equal. Taking each in turn from the whole solid DFAj leaves parallelopiped AB = CD. In the line CD take GH = CD, and through G and H pass planes perpendicular to GH. The solids DFH and CEG are congruent, therefore parallelopiped CD = GH. Now prolong the four edges not parallel to GH, and take LM — HK, and through L and M pass planes perpendicular to LM. As before, parallelopiped GH = LM\ but LM is a quader of equivalent base and equal altitude to parallelopiped AB. POL YHEDRONS. 299 Theorem VIII. 770. A plane passed through two diagonally opposite edges of a parallelopiped divides it into two equivalent triangidar prisms. fa>\ V /!x' > B >. ./'.'T--^^^ <^/Jr^^ JT 5^s^-r n -k ^ . ' ^ ^ '• ^« ^*' 2 ^^ ^ • 7 /-"^ >^ K^ ^ If the lateral faces are all rectangles, the two prisms are congruent ; if not, the prisms are still equivalent. For draw planes perpendicular to AA' at the points A and A'. Then the prism ABC A'B'C is equivalent to the right prism AEM A'E'M', because the pyramid AEBCM'\% congruent to the pyr- amid A'E'B'C'M'. In the same way, ADC A'D'C is equivalent to ALM A'UM'. But AEM A'E'M' and ALM A'L'M' are congruent, /. ABC A'B'C and ADC A' U C are equivalent. 771. Corollary. Any triangular prism is half a parallelo- piped of twice its base but equal altitude. 300 THE ELEMENTS OF GEOMETRY. Theorem IX. 772. If a pyramid be ctit by a plane parallel to its base, the section is to the base as the square of tJie perpe7idicular on it, from the vertex, is to the square of the altitude of the pyramid. The section and base are similar, since corresponding diagonals cut them into triangles similar in pairs because having all their sides respectively proportional. For A'C : AC :: VC : VC : : VO" : VO, and B'C : BC :: VC : VC, .'. AC : AC :: B' C : BC; and in the same way, B'C : BC :: A'B' : AB, .'. t.A'B'C'^t.ABC,Q\c., section : base :-. A' C"" -. AC^ w VO^ : VC 773. Corollary. In pyramids of equivalent bases and equal altitudes, two sections having equal perpendiculars from the vertices are equivalent. POL YHEDRONS. 301 Theorem X. 774. Tetrahedra {triangular pyramids) having equivalent bases and equal altitudes are equivale7it. Divide the equal altitudes a into n equal parts, and through each point of division pass a plane parallel to the base. By 773, all sections in the first tetrahedron are triangles equivalent to the corresponding sections in the second. Beginning with the base of the first tetrahedron, construct on each a section, as lower base, a prism - high, with lateral edges parallel to one of the edges of the tetrahedron. In the second, similarly construct prisms on each section, as upper base. Since the first prism-sum is greater than the first tetrahedron, and the second prism-sum is less than the second tetrahedron, therefore the difference of the tetrahedra is less than the difference of the prism-sums. But, by 771, each prism in the second tetrahedron is equivalent to the prism next above it on the first tetrahedron. So the difference of the prism-sums is simply the lowest prism of the first series. As n increases, this decreases, and can be made as small 302 THE ELEMENTS OF GEOMETRY. as we please by taking n sufficiently great ; but it is always greater than the constant difference between the tetrahedra, and so that constant difference must be nought. Theorem XI. 775. A triangular pyramid is one-third of a triangular prison of the same base and altitude. Let E ABC be a triangular pyramid. Through one edge of the base, as A C, pass a plane parallel to the opposite lateral edge, EB, and through the vertex E pass a plane parallel to the base. The prism ABC DEF has the same base and altitude as the given pyramid. The plane AFE cuts the part added, into two triangular pyramids, each equivalent to the given pyramid; for E ABC and A DEF have the same altitude as the prism, and its bottom and top respectively as bases ; while E AFC and E AFD have the same altitude and equal bases. BOOK XL MENSURATION, OR METRICAL GEOMETRY. CHAPTER I. THE METRIC SYSTEM. — LENGTH, AREA. 776. In practical science, every quantity is expressed by a phrase consisting of two components, — one a number, the other the name of a thing of the same kind as the quantity to be expressed, but agreed on among men as a standard or Unit. 777. The Meastirement of a magnitude consists in finding this number. 778. Measurement, then, is the process of ascertaining approximately the ratio a magnitude bears to another chosen as the standard ; and the measure of a magnitude is this ratio expressed approximately in numbers. 779. For the continuous-quantity space, the fundamental unit actually adopted is the Meter, which is a bar of platinum preserved at Paris, the bar supposed to be taken at the tem- perature of melting ice. 780. This material meter is the ultimate standard univer- sally chosen, because of the advantages of the metric system 303 304 THE ELEMENTS OF GEOMETRY. of subsidiary units connected with it, which uses only decimal multiples and sub-multiples, being thus in harmony with the decimal nature of the notation of arithmetic. 781. The metric system designates multiples by prefixes derived from the Greek numerals, and sub-multiples by pi^e- fixes from the Latin numerals. Prefix. Meaning as used. Derivation. Abbreviation. myria- ten thousand fJiVpLOL kilo- one thousand XlXlol k- hecto- one hundred iKarov h- deka- ten SeKa da- deci- one-tenth decem d- centi- one-hundredth centum c- milli- one-thousandth mille m- The abbreviation for meter is m. ; hence km. for kilometer, mm. for millimeter. 782. The adoption of the meter gives the world one stand- ard sect as fundamental unit. 783. The Levgth of any sect is its ratio to the meter ex- pressed approximately in numbers. 10 mm I decimeter = 10 centimeters = 100 millimeters. Men of science often express their measurements in terms of a subsidiary unit, the Centimeter. The length of a sect referred to the centimeter as unit is one hundred times as great as referred to the meter. MENSURATION-, OR METRICAL GEOMETRY. 305 784. An accessible sect may be practically measured by the direct application of a known sect, such as the edge of a ruler suitably divided. But because of incommensurability, any description of a sect in terms of the standard sect must be usually imperfect and merely approximate. Moreover, in few physical measurements of any kird are more than six figures of such approximations accurate ; and that degree of exactness is very seldom obtainable, even by the most delicate instruments. 785. For the measurement of surfaces the standard is the square on the linear unit. 786. The Area of any surface is its ratio to this square. 787. If the unit for length be a meter, the unit for area, a square on the meter, is called a square meter (m.^^ ; better, ^■'^). In science the Square Centimeter (^'^•^) is adopted as the primary unit of surface. 788. To find the area of a rectangle. 3 f Rule. Multiply the base by the altitude. Formula. R = ab. Proof. — Special Case : When the base and altitude of the rectangle are commensurable. In this case, there is always a sect which will divide both base and altitude exactly. If this sect be assumed as linear unit, the lengths a and b are integral 306 THE ELEMENTS OF GEOMETRY. numbers. In the rectangle ABCD, divide AD into ^, and AB into b, equal parts. Through the points of division draw lines parallel to the sides of the rectangle. These lines divide the rectangle into a number of squares, each of which equals the assumed unit of surface. In the bottom row, there are b such squares ; and, since there are a rows, we have b squares re- peated a times, which gives, in all, ab squares. Note. The composition of ratios includes numerical multiplication as a particular case. But ordinary multiplication is also an independent growth from addition. In this latter point of view, the multiplier indicates the number of additions or repetitions, while the multiplicand indicates the thing added or repeated. This is not a mutual operation, and the product is always in terms of the unit of the multiplicand. The multiplicand may be any aggregate ; the multiplier is an aggregate of repetitions. To repeat a thing does not change it in kind, so the result is an aggre- gate of the same sort exactly as the multiplicand. But if the multiplicand itself is also an aggregate of repetitions, the two factors are the same in kind, and the multiplication is com- mutative. This is the only sort of multiplication needed in mensuration ; for all ratios are supposed to be expressed exactly or approximately in num- bers, and in our rules it is only of these numbers that we speak. Thus, when the rule says, " Multiply the base by the altitude," it means, Mul- tiply the number taken as the length of the base, by the number which is the measure of the altitude in terms of the same linear unit. The product is a tmmber, which we prove to be the area of the rectangle ; that is, its numerical measure in terms of the superficial unit. This is the meaning to be assigned whenever in mensuration we speak of the product of one sect by another. General Proof. If L represent the unit for length, and 5 the unit of surface, and ^ ab the rectangle, the length of MENSURATION, OR METRICAL GEOMETRY. 307 whose base is b and the length of whose altitude is a^ then, by 542, C7 cib aL bL But the first member of this equation is the area of the rect- angle, which number we may represent by R ; and the second member is equal to the product of the numbers a and b ; .'. R = ab. clZ Example i. Find the area of a ribbon I*"- long and i^i"- wdde. Answer, ydit'"** = loC^"^-^. Since a square is a rectangle having its length and breadth equal, therefore 789. To find the area of a square. Rule. Take the second power of the mimber denoting the length of its side. Note. This is why the product of a number into itself is called the square of that number. 790. Given, the area of a square, to find the length of a side. Rule. Extract the square root of the number deftoting the arecL. 308 THE ELEMENTS OF GEOMETRY, 791. Metric Units of Surface. I hectar (^^•) = i sq. hectometer = loooo sq. meters. I dekar = = 1000 sq. meters. I ar (a-) = I sq. dekameter = 100 sq. meters. I deciar = = 10 sq. meters. I centiar = i sq. meter = i sq. meter. I milliar = = .1 sq. meter. I sq. decimeter = .01 sq. meter. I sq. centimeter = .0001 sq. meter. I sq. millimeter = .000001 sq. meter. Example 2. How many square centimeters in 10 millimeters square? Answer, {iQ^'^-y = loo™"^-^ = i^m.^, 792. Remark. Distinguish carefully between square meters and meters square. We say 10 square kilometers (lo^"^-^), meaning a surface which would contain 10 others, each a square kilometer ; while the expression '* 5 kilometers square " (s''"^-)^ means a square whose sides are each 5 kilometers long, so that the figure con- tains 25^"^•^ Example 3. A square is looo'"-''. Find its side. Answer, yiooo"^- = 31.623™- 793. Because the sum of the squares on the two sides of a right-angled triangle is the square of the hypothenuse, therefore, also, > Given, ihe hypothenuse and one side, to find the other side. Rule. Multiply their sum by their difference^ and extract the square root. Formula, c^ — a^ =:i {c -\' a) {c — a) = ^^ MENSURATION, OR METRICAL GEOMETRY. 309 From this it follows, that, in an acute-angled triangle, if we are given two sides and the projection of one on the other, or two sides and an altitude, we can find the third side. Exercises. 112. What must be given in order to find the medials of a triangle } 113. If on the three sides of any triangle squares are described outward, the sects joining their outer corners are twice the medials of the triangle, and perpendicular to them. 310 THE ELEMENTS OF GEOMETRY. CHAPTER II. RATIO OF ANY CIRCLE TO ITS DIAMETER. Problem I. 794. Given^ the perimeters of a regular inscribed and a sim^ ilar circiLfHscribed polygon^ to compute the perimeters of the regular inscribed and circumscribed polygons of double the tium" ber of sides. a JO Take AB a side of the given inscribed polygon, and CD a side of the similar circumscribed polygon, tangent to the arc AB at its mid- point E. Join AE, and at A and B draw the tangents AF and BG; then AE is a side of the regular inscribed polygon of double the number of sides, and FG is a side of the circumscribed polygon of double the number of sides. Denote the perimeters of the given inscribed and circumscribed polygons by p and q respectively, and the required perimeters of the inscribed and circumscribed polygons of double the number of sides by/' and / respectively. MENSURATION, OR METRICAL GEOMETRY. 311 Since <9C is the radius of the circle circumscribed about the poly- gon whose perimeter is q, .-. q \ p w OC \ OE. (548. The perimeters of two similar regular polygons are as the radii of their cir- cumscribed circles.) But, since OF bisects the 4- COE, .'. OC : OE :: CE : FE, (523. The bisector of an angle of a triangle divides the opposite side in the ratio of the other two sides of the triangle.) /. q \ p \\ CF \ FE, whence, by composition, p ■\- q '. 2p \\ CF '\- FE \ 2FE ^ CE \ EG \\ q \ ^, since FG is a side of the polygon whose perimeter is /, and is con- tained as many times in / as CE is contained in q, .-. p -{- q : 2p '.'. q '. q'. If, now, the letters be taken to represent lengths in terms of the unit sect Z, this proportion is pL -f- qL : 2pL w qL \ q'L, which gives the number ^ = 7+i- <'> Again, right A AEH ~ EFN, since acute ^ EAH = FEN, ,\ AH '. AE :: EN : EF, .'. p :/ ::/ : /, since AH and AE are contained the same number of times in / and/' respectively, and EN and EF are contained twice that number of times in /' and / respectively. If, now, the letters be taken to repre- sent length in terms of the same unit sect Z, this proportion is pL : /L : : /Z ; /Z, which gives the number , , — -, Therefore from the given lengths p and q we compute q' by equa- tion (i), and then with / and / we compute /' by equation (2). 312 THE ELEMENTS OF GEOMETRY. Theorem I. 795* The length of a circle whose diameter is unity is 3.141592+. The length of the perimeter of the circumscribed square is 4. A side of the inscribed square is -|V2, therefore its perimeter is 2V 2. Now, putting p = 2V2 and ^ = 4 in 794, we find, for the perime- ters of the circumscribed and inscribed octagons, 2/^ / = = 3-3137085, / = V'//= 3.0614675. Then, taking these as given quantities, we put / = 3.0614675, and ^ = 3-3137085, and find by the same formulae, for the polygons of sixteen sides, / = 3.1825979, and / = 3.1214452. Continuing the process, the results will be found as in the following table : — Number of Sides. Perimeter of Circumscribed Polygon. Perimeter of Inscribed Polygon. 4 4.0000000 2.8284271 8 3-3137085 3.0614675 16 32 3.1825979 3-I517249 3.1214452 3-1365485 64 3.1441184 3.1403312 128 3.1422236 3-I412773 256 3.1417504 3-1415138 512 1024 3.I41632I 3.1416025 3-I415729 3-I415877 2048 3-I415951 3.1415914 4096 3-1415933 3-I415923 8192 3.1415928 3.1415926 MENSURATION, OR METRICAL GEOMETRY. 313 But since, by 654, a chord is shorter than its arc, therefore the circle is longer than any inscribed perimeter ; and, assuming that two tangents from an external point cannot be together shorter than the included arc, it follows that the circle is not longer than a circumscribed perimeter ; therefore the circle whose diameter is unity is longer than 3.1415926 and not longer than 3.1415928. 796. A Variable is a quantity which may have successively an indefinite number of different values. 797* If ^ variable which changes its value according to some law can be made to approach some fixed value as nearly as we please y but can never become equal to it, the constant is called the Limit of the variable. Example 4. The limit of the fraction -, as x increases indefitiitely, X is zero ; for, by taking x sufficiently great, we can make - less than any X assigned quantity, but we can never make it zero. PRINCIPLE OF LIMITS. 798. Ify while tending toward their respective limits, two variables are always in the same ratio, their limits will have that ratio. I 1 i hH-H M— H H^ ^ ' y C' x' C Z' y Z 2 Let the sects AC and AL represent the limits of any two variable magnitudes which are always in the same ratio, and let Ax, Ay, represent two corresponding values of the variables themselves ; then Ax -.Ay '.'.AC : AL. For if not, then Ax : Ay :: AC : to some sect > or < AL, 314 THE ELEMENTS OF GEOMETRY, Suppose, in the first place, that Ax \ Ay \\ AC \ AL', where AL' is less than AL. By hypothesis, the variable Ay continually approaches AL, and may be made to differ from it by less than any given quan- tity. Let Ax and Ay, then, continue to increase, always remaining in the same ratio, until Ay differs from AL by less than the quantity L'L ; or, in other words, until the point y passes the point Z', and reaches some point, as j/', between V and Z, and X reaches the corresponding point x' on the sect AC. Then, since the ratio of the two variables is always the same, Ax : Ay :: Ax' : A/, But, by hypothesis. Ax : Ay :: AC : AL\ .-. Ao^ : A/ :: AC : AL', But Ax' AL. Now, there is some sect, as AC, less than AC, which is to AL 2iS AC is to AL". Substituting this ratio for that oi AC to AL'\ we have Ax '.Ay :: AC : AL, MENSURATION, OR METRICAL GEOMETRY. 315 which, by a process of reasoning similar to the above, may be shown to be absurd. Hence, since the fourth term of the pro- portion can be neither greater nor less than ALy it must equal AL ; that is, Ax \ Ay w AC \ AL. 799. Corollary. If two variables are always equal, their limits are equal. Theorem II. 800. Ally two circles are to each other as their radii. Proof. By 548, the perimeters of any two regular polygons of the same number of sides have the same ratio as the radii of their circum- scribed circles. The inscribed regular polygons remaining similar to each other when the number of sides is doubled, their perimeters continue to have the same ratio. Assuming the circle to be the limit toward which the perimeter of the inscribed polygon increases, by 798, Principle of Limits, the circles have the same ratio as their radii. 801. Since c : c' 2r', c_ 2r 2r' that is, the ratio of any one circle to its diameter is the same as the ratio of any other circle to its diameter. 3l6 THE ELEMENTS OF GEOMETRY. This constant ratio is denoted by the Greek letter tt. But, by 795, this ratio for the circle with unit diameter, and there- fore for every circle, is TT = 3.I4I592-I-. 802. For any circle. Formula, c = 2r7r. CIRCULAR MEASURE OF AN ANGLE. 803. When its vertex is at the center of the circle, by 506, any ^ _ its intercepted arc _ arc St. ^ semicircle rTr' any ^ _ arc TT So, if we adopt as unit angle the radian^ or that part of a peri- st ^ gon denoted by -^^, that is, the angle subtended at the center Ih of every circle by an arc equal to its radius, and hence named a radian, then The number which expresses any angle in radianSy also expresses its intercepted arc in terms of the raditis. If ti denote the number of radians in any angle, and / the length of its intercepted arc, then / u — -, r MENSURATION, OR METRICAL GEOMETRY, 317 The fraction arc divided by radius^ or Uy is called the circu- lar measure of an angle. 804. Arcs are said to measure the angles at the center which include them, because these arcs contain their radius as often as the including angle contains the radian. In this sense an angle at the center is measured by the arc intercepted between its sides. 3i8 THE ELEMENTS OF GEOMETRY. CHAPTER III. MEASUREMENT OF SURFACES. 805. By 248, any parallelogram is equivalent to the rect- angle of its base and altitude ; therefore, ziy To find the area of any parallelogram. Rule. Multiply the base by the altitude. Formula, /u =. ab. 806. Corollary. The area of a parallelogram divided by the base gives the altitude. 807. By 252, any triangle is equivalent to one-half the rect- angle of its base and altitude ; therefore, Given, one side and the perpendicular upon it from the opposite vertex, to find the area of a triangle. Rule. Take half the product of the base into the altitude. Formula, a = '^ib. MENSURATION, OR METRICAL GEOMETRY. 319 808. Given, the three sides, to find the area of a triangle. Rule. From half the sum of the three sides subtract each side separately ; multiply together the half-sitm and the three remainders. The square root of this product is the area. Formula, t. :=i \s {s — a) {s — b) {s — c). Proof. Calling j the projection of c on b, by 306, a' =, b^ j^ c" — 2bj\ . . l^ -^ c^ - a"" 2b Calling the altitude //, this gives h^ =, c^ - j^ = c^ - {h" -\- c' - a^y 4b^ /. 4h^b^ = 4^V — (/r* + ^ - a^Yy 2/ib = V^4^V2 - (b^ ■\- c' - a'Y, 2hb 2hb^ sl{2bc -h b^ -\- c^ - - a'){2bc - b' — c^ + «^), = Vc^ -{-b -\- c){b + c - a){a -\- b — c){^a - b ■\- 0, ^ /(^ + ^ + (-^ + ^ -a)(a + b - c){a -b^ c) .,.. a -\- b -\- c ' b At c Writing s — \^ gives ^ .-. \hb = \ls{s - a){s - b){s - c). But, by 807, Ihb = A. 320 THE ELEMENTS OF GEOMETRY. 809. To find the area of a regular polygon. Rule. Take half the prodtict of its perimeter by the radius of the inscribed circle. rp Formula. TV = — . 2 Proof. Sects from the center to the vertices divide the polygon into congruent isosceles triangles whose altitude is the radius of the inscribed circle, and the sum of whose bases is the perimeter of the polygon. 810. To find ihe area of a circle. Rule. Multiply its squared radius by -k. Formula. O == rV. If a regular polygon be circumscribed about the circle, its area TV, by 809, is \rp. MENSURATION, OR METRICAL GEOMETRY, 32 1 If, now, the number of sides of the regular polygon be con- tinually doubled, the perimeter / decreases toward c as limit, and N toward circle. But the variables N and / are always in the constant ratio \r\ therefore, by 798, Principle of Limits, their limits are in the same ratio, .-. O = \rc. But, by 802, c = 2r7r, 811. To find the area of a sector. Rule. Multiply the length of the arc by half the radius. Formula. 5 = J/r = \ur''. Proof. By 506, 5 \ : / : c : \\ u \ 27r, s = 0/ c = t 27r s = c = r'^irU 27r s = ¥r = lur^. 812. An Amnihis is the figure included between two con- centric circles. Its height is the difference between the radii. 322 THE ELEMENTS OF GEOMETRY. 813. To find the area of a sector of an annu/us. Rule. Multiply the sum of the bounding arcs by half the difference of their radii. Formula. 5 . ^ . = J (^3 — r,) (/, + 4) = \h (/, + Q. Proof. The annular sector is the difference between the two sectors \rj^ and \rj^. But /i and 4 are arcs subtending the same angle ; therefore, by 804, ^ = 4 .-. -1^24 - -1^14 = Wi + i^2^2 - i^i4 — 1^,4 = K^. -^0(^ + 4). 814. To find the lateral area of a prism. ^ MENSURATION^ OR METRICAL GEOMETRY, 323 Rule. Multiply a lateral edge by the perimeter of a right section. Formula. P =z Ip. Proof. The lateral edges of a prism are all equal. The sides of a right section, being perpendicular to the lateral edges, are the altitudes of the parallelograms which form the lateral surface of the prism. 815. A Cylindric Surface is generated by a line so moving that every two of its positions are parallel. 816. The generatrix in any position is called an Elemejtt of the surface. 817. A Cylinder is a solid bounded by a cylindric surface and two parallel planes. 324 THE ELEMENTS OF GEOMETRY. 8i8. The Axis of a circular cylinder is the sect joining the centers of its bases. .--r-^ 819. A Truncated Cylinder is the portion between the base and a non-parallel section. 820. To find the lateral area of a right circular cylinder. Rule. Multiply its length by its circle. Formula. C = cl. MENSURATION, OR METRICAL GEOMETRY. 325 Proof. Imagine the curved surface slit along an element and then spread out flat. It thus becomes a rectangle, having for one side the circle, and for the adjacent side an element. 821. Corollary I. The curved surface of a truncated circular cylinder is the product of the circle of the cylinder by the intercepted axis. ^vK- For, by symmetry, substituting an oblique for the right sec- tion through the same point of the axis, alters neither the curved surface nor the volume, since the solid between the two sections will be the same above and below the right section. 822. Corollary II. The lateral area of any cylinder on any base equals the length of the cylinder multiplied by the perimeter of a right section. 823. The lateral surface of a prism or cylinder is called its Mantel. Exercises. 114. The mantel of a right circular cylinder is equivalent to a circle whose radius is a mean proportional between the altitude of the cylinder and the diameter of its base. 115. A plane through two elements of a right circular cylinder cuts its base in a chord which subtends at the center an angle whose circular measure is u \ find the ratio of the curved surfaces of the two parts of the cylinder. 326 THE ELEMENTS OF GEOMETRY. 824. A Conical Surface is generated by a straight line moving so as always to pass through a fixed point called the apex. 825. A Cone is a solid bounded by a conical surface and a plane. 826. A right circular cone may be generated by revolving a right triangle about one pependicular. All the elements are equal, and each is called the Slant Height of the cone. MENSURATION, OR METRICAL GEOMETRY. 327 827. The Frustum of a pyramid or cone is the portion included between the base and a parallel section. 828. To find ihe lateral area of a right circular cone. Rule. Multiply the circle of its base by half the sla7it height. Formula. K = \ch = -n-rh. Proof. If the curved surface be slit along a slant height, and spread out flat, it becomes a sector of a circle, with slant height as radius, and the circle of cone's base as arc ; therefore, by 811, its area is ^ch. 829. Corollary. Since, by 810, the base O = ^cr, there- fore the curved surface is to the base as the slant height is to the radius of base, as the length of circle with radius h is to circle with radius r, as the perigon is to the sector angle. 328 THE ELEMENTS OF GEOMETRY. 830. To find ihe lateral area of the frustum of a right circular cone. Rule. Midtiply the slajit height of the frustum by half the sum of the circles of its bases. Formula. F ^=^ \h{c^-\' c^ = irh {r^ + r^. Proof. Completing the cone, and slitting it along a slant height, the curved surface of the frustum develops into the difference of two similar sectors having a common angle, the arcs of the sectors being the circles of the bases of the frus- tum. By 813, the area of this annular sector, F = \h(^c, + c). 831. The axis of a right circular cone, or cone of revolution, is the line through its vertex and the center of its base. Exercises. 116. Given the two sides of a right-angled triangle. Find the area of the surface described when the triangle revolves about its hypothenuse. Hint. Calling a and b the given altitude and base, and x the per- pendicular from the right angle to the hypothenuse, by 828, the area of the surface described by a is itxa, and by b is irxb. Also a \ X w sla" -^ b-" \ b. 117. In the frustum of a right circular cone, on each base stands a cone with its apex in the center of the other base ; from the basal radii r^ and r^^ find the radius of the circle in which the two cones cut. MENSURATIONy OR METRICAL GEOMETRY. 329 Theorem III. 832. The lateral area of a friistmn of a cone of revolution is the product of the projection of the frnstiims slant height on the axis by twice tr times a perpendicular erected at the mid-point of this slant height^ and terminated by the axis. Proof. By 830, the lateral area of the frustum whose slant height is PR and axis MN is F = tt{PM -{- RN)FR, But if Q is the mid-point of PR, then PM + RN = 2QO, .'. P = 27r X PR X QO. But A PRL ~ A QCO, since the three sides of one are drawn per- pendicular to the sides of the other ; .-. PR X QO = PL X QC, /. F = 2it{LP X QC) = 27r{MN X CQ). Exercises. 118. Reckon the mantel from the two radii when the inclination of a slant height to one base is half a right angle. 119. If in the frustum of a right cone the diameter of the upper base equals the slant height, reckon the mantel from the altitude a and perimeter/ of an axial section. 330 THE ELEMENTS OF GEOMETRY. 833. To find the area of a sphere. Rule. Multiply four times its squared radius by tt. Formula. H = 4r*7r. Proof. In a circle inscribe a regular polygon of an even number of sides. Then a diameter through one vertex passes through the opposite vertex, halving the polygon symmetrically. Let PR be one of its sides ; draw PM, RNy perpendicular to the diameter BD. From the center C the perpendicular CQ bisects PR. Drop the perpendiculars PL^ QO. Now, if the whole figure revolve about BD as axis, the semicircle will generate a sphere, while each side of the in- scribed polygon, as PR, will generate the curved surface of the frustum of a cone. By 832, this F = 2ir{MN X CQ); and the sum of all the frustums, that is, the surface of the solid generated by the revolving semi-polygon, equals 2itCQ into the sum of the projections, BM, MN, NC, etc., which sum is BD, /. Sum of surfaces of frustums = ^-kCQ X BD. mensuration; or metrical geometry. 331 As we continually double the number of sides of the inscribed polygon, its semi-perimeter approaches the semicircle as limit, and its surface of revolution approaches the sphere as limit, while CQ, its apothem, approaches r, the radius of the sphere, as limit. Representing the sum of the surfaces of the frus- tums by 2/s and BD by 2r, we have CQ ^ That is, the variable sum is to the variable CQ in the constant ratio 4r7r; therefore, by 798, Principle of Limits, their limits have the same ratio, H .-. — = 4r7r, r ,'. H = 4r27r. 834. A Calot is a zone of only one base. 835. The last proof gives also the following rule : — To find the area of a zone. Rule. Multiply the altitude of the segment by twice ir times the radius of the sphere. Formula. Z = 2ar'K. 332 THE ELEMENTS OF GEOMETRY. CHAPTER IV. SPACE ANGLES. 836. A Plane Angle is the divergence of two straight lines which meet in a point. 837. A Space Angle is the spread of two or more planes which meet in a point. 838. Symmetrical Space Angles are those which cut out symmetrical spherical polygons on a sphere, when their vertices are placed at its center. 839. A Steregon is the whole amount of space angle round about a point in space. MENSURATION, OR METRICAL GEOMETRY. 333 840. A Steradian is the angle subtended at the center by that part of every sphere equal to the square of its radius. 841. The space angle made by only two planes corresponds to the lune intercepted on any sphere whose center is in the common section of the two planes. 842. A Spherical Pyramid is a portion of a globe bounded by a spherical polygon and the planes of the sides of the poly- gon. The center of the sphere is the apex of the pyramid ; the spherical polygon is its base. 843. Just as plane angles at the center of a circle are pro- portional to their intercepted arcs, and also sectors, so space angles at the center of a sphere are proportional to their inter- cepted spherical polygons y and also spherical pyramids. 334 "^^^ ELEMENTS OF GEOMETRY. Example. Find the ratio of the space angles of two right cones of altitude a^ and a^, but having the same slant height, h. These space angles are as the corresponding calots (or zones of one base) on the sphere of radius h. Therefore, by 835, the required ratio is 2Trh{h — a,) _ h — Uj^ 2i:h{h — a^ h — a^ the ratio of the calot altitudes. For the equilateral and right-angled cones this becomes 2 - V^ ^ 2 - V2* 844. To construct a space angle of two faces equivalent to any polyhedral angle, only involves constructing a lune equiva- lent to a spherical polygon, as in 731. 845. To find the area of a lune. Rule. Multiply its angle in radians by twice its squared radius. Formula. L = 2r^u. Proof. By 703, lunes are as their angles, /, a lune is to a hemisphere as its angle is to a straight angle, Z _ u 2r^ir TV .', L = 2r^u, MENSURATION, OR METRICAL GEOMETRY. 335 846. Corollary I. A lime measures twice as many stera- dians as its angle contains radians. 847. Corollary II. If two-faced space angles are equal, their lune angles are equal ; so a dihedral angle may be meas- ured by the plane angle between two perpendiculars, one in each face, from any point of its edge. 848. Suppose the vertex of a space angle is put at the cen- ter of a sphere, then the planes which form the space angle will cut the sphere in arcs of great circles, forming a spherical poly- gon, whose angles may be taken to measure the dihedral angles of the space angle, and whose sides measure its face angles. Hence from any property of spherical polygons we may infer an analogous property of space angles. For example, the following properties of trihedral angles have been proved in our treatment of spherical triangles : — I. Trihedral angles are either congruent or symmetrical which have the following parts equal : — 336 THE ELEMENTS OF GEOMETRY. (i) Two face angles and the included dihedral angle. (2) Two dihedral angles and the included face angle. (3) Three face angles. (4) Three dihedral angles. (5) Two dihedral angles and the face angle opposite one of them, provided the edge of the third dihedral angle of neither trihedral makes right angles with any line in the half of the opposite face not adjacent to one of the face angles equal by hypothesis. (6) Two face angles and the dihedral angle opposite one of them, provided the other pair of opposite dihedral angles are not supplemental. II. As one of the face angles of a trihedral angle is greater than, equal to, or less than, another, the dihedral angle which it subtends is greater than, equal to, or less than, the dihedral angle subtended by the other. III. Symmetrical trihedral angles are equivalent. IV. Space angles having the same number and sum of dihe- dral angles are equivalent. V. The face angles of a convex space angle are together less than a perigon. 849. To find ihe area of a spherical iriangle. Rule. Multiply its spheHcal excess in radians by its squared radius. Formula, a" = er^. Proof. By 711, a triangle is equal to a lune whose angle is \e ; therefore, by 845, 'a = r'^e. 850. Corollary I. A spherical triangle measures as many steradians as its e contains radians. 851. Corollary II. By 714, to find the area of a spherical polygon, multiply its spherical excess in radians by its squared radius. MENSURATION, OR METRICAL GEOMETRY. 337 CHAPTER V. THE MEASUREMENT OF VOLUMES. 852. The Volume of a solid is its ratio to an assumed unit. 853. The Unit for Measicremejit of Volume is a cube whose edge is the unit for length. Cubic Centimeter. 854. Metric Units for Volume. Solid. 1 kilostere I hectostere I dekastere I stere I decistere I centistere Liquid. Cubic. = I cubic dekameter = I kiloliter = i cubic meter = I hectoliter = I dekaliter = I liter = I cubic decimeter = I deciliter = I centiliter = I cubic centimeter = 1000 cubic meters. 100 cubic meters. 10 cubic meters. I cubic meter. .1 cubic meter. .01 cubic meter. .001 cubic meter. .0001 cubic meter. .ocxx)i cubic meter. .ocxxxDi cubic meter. I cubic millimeter = .000000001 cubic meter. The authorized abbreviation for cubic is the index ^ as in icm.3 foj. j cubic centi- meter ; that for stere is s., and for liter 1.. 338 THE ELEMENTS OF GEOMETRY, 855. To find the volume of a quader. Rule. Multiply together its lengthy breadth, and thickness. Proof. By "jG^, two quaclers have the ratio compounded of the ratios of their bases and altitudes. 856. Corollary. The volume of any cube is the third power of the length of an edge. This is why the third power of a number is called its cube. 857. To find the volume of any parallelepiped . Rule. Multiply its altitude by the area of its base. Proof. By 769, any parallelopiped is equivalent to a quader of equivalent base and equal altitude. 858. To find the volume of any prism. Rule. Multiply its altitude by its base. Formula. V . P = aB. Proof. By 771, any three-sided prism is half a parallelo- piped, with base twice the prism's base and the same altitude. Thus the rule is true for triangular prisms, and consequently for all prisms ; since, by passing planes through one lateral edge, and all the other lateral edges excepting the two adja- cent to this one, we can divide any prism into triangular prisms of the same altitude, whose triangular bases together make the polygonal base. MENSURATION, OR METRICAL GEOMETRY. 339 859. To find the volume of any cylinder. Rule. Multiply its altitude by its base. Proof. The cylinder is the limit of an inscribed prism when the number of sides of the prism is indefinitely increased, the base of the cylinder being the limit of the base of the prism. But always the variable prism is to its variable base in the constant ratio a ; therefore, by 798, Principle of Limits, their limits will be to one another in the same ratio. 860. To find the volume of any pyramid. Rule. Multiply one-third of its altitude by its base. Formula. Y = \aB. 340 THE ELEMENTS OF GEOMETRY. Proof. By 775, any triangular pyramid is one-third of a triangular prism of the same base and altitude. The rule thus proved for triangular pyramids is true for all pyramids ; since by passing planes through one lateral edge, and all the other lateral edges excepting the two adjacent to this one, we can divide any pyramid into triangular pyramids of the same altitude whose bases together make the polygonal base. 861. To find the volume of any cone. Rule. Multiply one-third its altitude by its base. Formula when Base is a Circle. V. K = ^ar^-n-. Proof. The base of a cone is the limit of the base of an inscribed pyramid, and the cone is the limit of the pyramid. But always the variable pyramid is to its variable base in the constant ratio \a. Therefore their limits are to one another in the same ratio, and V.K z=z ^aB. Scholium. This applies to all solids determined by an elastic string stretching from a fixed point to a point describing any closed plane figure. MENSURATION, OR METRICAL GEOMETRY. 341 862. Corollary. The volume of the solid generated by the revolution of any triangle about one of its sides as axis, is one-third the product of the triangle's area into the length of the circle described by its vertex. F= fTT^A. PRISMATOID. 863. A Prismatoid is a polyhedron whose bases are any two polygons in parallel planes, and whose lateral faces are triangles determined by so joining the vertices of these bases that each lateral edge, with the preceding, forms a triangle with one side of either base. 342 THE ELEMENTS OF GEOMETRY. 864. A number of different prismatoids thus pertain to the same two bases. 865. If two basal edges which form with the same lateral edge two sides of two adjoining faces are parallel, then these two triangular faces fall in the same plane, and together form a trapezoid. MENSURATION, OR METRICAL GEOMETRY. 343 866. A Prismoid is a prismatoid whose bases have the same number of sides, and every corresponding pair parallel. 867. A frustum of a pyramid is a prismoid whose two bases are similar. 868. Corollary. Every three-sided prismoid is the frus- tum of a pyramid. 869. If both bases of a prismatoid become sects, it is a tetrahedron. 870. A Wedge is a prismatoid whose lower base is a rect- angle, and upper base a sect parallel to a basal edge. 871. The altitude of a prismatoid is any sect perpendicular to both bases. 872. A Cross-Sectio7t of a prismatoid is a section made by a plane perpendicular to the altitude. 344 THE ELEMENTS OF GEOMETRY. 873. To find the volume of any prismaioid. Rule. Multiply one-fourth its altitude by the sum of one base and three times a cross-section at two-thirds the altitude from that base. Formula. D — -{B -{- z T). 4 Proof. Any prismatoid may be divided into tetrahedra, all of the same altitude as the prismatoid ; some, as C FGO, having their apex in the upper base of the prismatoid, and for base a portion of its lower base ; some, as O ABC, having base in the upper, and apex in the lower, base of the prismatoid ; and the others, as A COG, having for a pair of opposite edges a sect in the plane of each base of the prismatoid, as AC 3.nd OG. Therefore, if the formula holds good for tetrahedra in these three positions, it holds for the prismatoid, their sum. In (i), call 7"i the section at two-thirds the altitude from the base B^ ; then T^ is \a from the apex. Therefore, by 772, 4 4 3 3 which, by 860, equals Y, the volume of the tetrahedron. MENSURATION, OR METRICAL GEOMETRY. 345 In (2), B^ = o, being a point, and T^ is ^a from the apex ; /. T, : B, :: (|«)^ : a\ /. T, = |^„ ... A = ^(^, + 37;) = ^(o + ^^.) = -aB, = K 4 4 3 3 In (3), let KLMN be the section T^. Join CK, CN, OM, ON. Now A ANK : A ^(?6> : : AN^ : ^6^^ : : (^ay : a^ : : 1 : g. A GNM: A (9-^C : : GN^ : 6^^^ : : (|^)^ : tz^ : : 4 : 9. But the whole tetrahedron D^ and the pyramid C ANK may be considered as having their bases in the same plane, AGO, and the same altitude, a perpendicular from C; C ANK : Z>3 : : A ANK : A AGO : : i : 9, .-. C ANK = 4Z>3. 346 THE ELEMENTS OF GEOMETRY. In same way, O GNM \ D^ : : A GNM : A GAC : : 4 : 9, /. OGNM=%D^. C ANK + O GNM = %D^, .'. C KLMN + O KLMN = %D^, But, by 860, C KLMN -f O KLMN = \ . \aT, 4- i . 1^7; = \aT,, .-. 4_z)3 = i^r, /. i^3 = ^3r, = ^{B, 4- 37;), 9 3 4 4 since here B^ = o. 874. Corollary I. Since, in the frustum of a pyramid, B and T are similar. 4 \ ^i' / where w^ and ^e'^ are corresponding sides of B and T. 875. Corollary II. For the frustum of a cone of revolu- tion, where rj is the radius of T. V,F = -7rr,^ 4 MENSURATION, OR METRICAL GEOMETRY. 247 876. To find the volume of a globe. Rule. Multiply the cube of its radius by |-7r. Formula. G = ^-kt^. Proof. By 644, a tetrahedron on edge, and a globe with the tetrahedron's altitude for diameter, have all their corre- sponding cross-sections equivalent if any one pair are equiva- lent. Hence the volumes may be proved equivalent, as in 774, and G = \aT, But a = 2r, and, by 522, T = fr . f r . tt, .-. dP = I . 2r . |r . f r . TT = f 7rr3. 877. Corollary I. Globes are to each other as the cubes of their radii. 878. Corollary II. Similar solids are to one another as the cubes of any two corresponding edges or sects. DIRECTION. 879. If two points starting from a state of coincidence move along two equal sects which do not coincide, that quality of each movement which makes it differ from the other is its Direction. 880. If two equal sects are terminated at the same point, but do not coincide, that quality of each which makes it differ from the other is its direction. 881. The part of a line which could be generated by a tra- cing-point, starting from a given point on that line, and moving on the line without ever turning back, is called a Ray from that given point as origin. 348 THE ELEMENTS OF GEOMETRY. 882. Two rays from the same origin are said to have the same direction if they coincide ; otherwise, they are said to have different directions. 883. Two rays which have no point but the origin in com- mon, and fall into the same line, are said to have opposite direc- tions. 884. Two rays lying on parallel lines have parallel-same directions if they are on the same side of the line joining their origins. . 885. Two rays lying on parallel lines have parallel-opposite directions if they are on opposite sides of the line joining their origins. 886. A sect is said to have the same direction as the ray of which it is a part. 887. A sect is definitely fixed if we know its initial point, its parallel-same direction, and its length. 888. The operation by which a sect could be traced if we knew its initial point, that is, the operation of carrying a tracing-point in a certain parallel-same direction until it passes over a given number of units for length, is called a Vector. 889. The position of B relative to A is indicated by the length and parallel-same direction of the sect AB drawn from A to B. If you start from A, and travel, in the direction indi- cated by the ray from A through B^ and traverse the given number of units, you get to B. This parallel-same direction and length may be indicated equally well by any .other sect, such as A' B\ which is equal to AB and in parallel-same direction. 890. As indicating an operation, the vector ^^ is completely defined by the parallel-same direction and length of the trans- ferrence. All vectors which are of the same magnitude and parallel-same direction, and only those, are regarded as equal. Thus AB is not equal to BA. MENSURATION^ OR METRICAL GEOMETRY. 349 PRINCIPLE OF DUALITY. JOIN OF POINTS AND OF LINES. 891. The line joining two points is called the _/(?/;/ of the Two Points. The point common to two intersecting lines is called the Join of the Two Lines. 892. Pencil of Lines. A fixed point, A^ may be joined to all other points in space. We get thus all the lines which can be drawn through the point A. The aggregate of all these lines is called a Pencil of Lines. The fixed point is called the Base of the pencil. Any one of these lines is said to be a line in the pencil, and also to be a line in the fixed point. In this sense, we say not only that a point may lie in a line, but also that a line may lie in a point, meaning that the line passes through the point. 893. In most cases, we can, when one figure is given, con- struct another such that lines take the place of points in the first, and points the place of lines. Any theorem concerning the first thus gives rise to a corre- sponding theorem concerning the second figure. Figures or theorems related in this manner are called Reciprocal Figures or Recipi'ocal Theorems. 894. Small letters denote lines, and the join of two elements is denoted by writing the letters indicating the elements, to- gether. Thus the join of the points A and B is the line AB, while ab denotes the join, or point of intersection, of the lines a and b. ROW OF points, pencil OF LINES. 895. A line contains an infinite number of points, called a Row of Points, of which the line is the Base, A row is all points in a line. 350 THE ELEMENTS OF GEOMETRY. The reciprocal figure is all lines in a pointy or all lines pass- ing through the point. A Flat Pe7icil is the aggregate of all lines in a plane which pass through a given point. In plane geometry, by a pencil we mean a flat pencil. RECIPROCAL THEOREMS. 8961. A point moving along a line describes a row. 8971. A sect is a part of a row described by a point moving from one position, A^ to another posi- tion, B. 896'. A line turning about a point describes a pencil. 897'. An angle is a part of a pencil described by a line turning from one position, a, to another position, b. Thus to sect AB corresponds 4- ^^• LINKAGE. 898. The Peancellier Cell consists of a rhombus movably jointed, and two equal links movably pivoted at a fixed point, and at two opposite extremities of the rhombus. MENSURATION, OR METRICAL GEOMETRY. 351 TO DRAW A STRAIGHT LINE. 899. Take an extra link, and, while one extremity is on the fixed point of the cell, pivot the other extremity to a fixed point. Then pivot the first end to one of the free angles of the rhombus. The opposite vertex of the rhombus will now describe a straight line, however the linkage be pushed or moved. Proof. By the bar FD, the point D is constrained to move on the circle ADR ; therefore -^ ADR^ being the angle in a semicircle, is always right. If, now, E moves on EM ± AMy A ADR ~ A AME, .-. DA : AR : : AM : AE, /. DA , AE = RA , AM. Therefore, if AE . AD is constant, E moves on the straight line EAf. But because BDCE is a rhombus, and AB =: AC, .*. D and iV are always on the variable sect AE. 352 THE ELEMENTS OF GEOMETRY. Always AB^ = AN^ + NB% and ~BE' = EN^ + NB\ :. AB^ - ~BE' =■ AN' - JV£^ = {AN + iV^^) (^iV^ - NE) = ^^ . AD. 900. Corollary. The efficacy of our cell depends on its power to keep, however it be deformed, the product of two varying sects a constant. Therefore a Hart four-bar cell, constructed as in the accom- panying figure, may be substituted for the Peancellier six-bar cell, since AE . AD equals a constant. Also, for the Hart cell may be substituted the quadruplane, four pivoted planes. CROSS-RATIO. 901. If four points are collinear, two may be taken as the extremities of a sect, which each of the others divides inter- nally or externally in some ratio. The ratio of these two ratios is called the Cross-Ratio of the four points. The cross-ratio —— : -— — is written {ABCD). CB DB Distinguishing the "step" AB from BA, as of opposite ** sense," and taking the points in the two groups of two in a definite order, to write out a cross-ratio, make first the two MENSURATION, OR METRICAL GEOMETRY. 353 bars, and put crossing these the letters of the first group of A A two, thus — - : — - ; then fill up, crosswise, the first by the first B B letter of the second group of two, the second by the second. If we take the points in a different order, the value of the cross-ratio may change. We can do this in twenty-four different ways by forming all permutations of the letters. But of these twenty-four cross-ratios, groups of four are equal, so that there are really only six different ones. We have the following rules : — I. If in a cross-ratio the two groups be interchanged^ its value remains unaltered. {ABCD) = {CDAB). II. If in a cross-ratio the tzvo points belonging to one of the two groups be interchanged^ the cross-ratio cha?iges to its recip- rocal. I (ABCD) {ABDC) 1. and II. are proved by writing out their values. III. Fjvm II. it follozuSj that, if we inteixhange the elements in each pair^ the cross-ratio remains unaltered. {ABCD) = {BADC). IV. If in a cross-ratio the two middle letters be interchanged^ the cross-ratio changes into its complement. {ABCD) = I - {ACBD), This is proved by taking the step-equation for any four col- linear points, BC .AD ^ CA . BD + AB . CD = o, and dividing it by CB . AD. 354 THE ELEMENTS OF GEOMETRY. Theorem IV. go2. If any four concurrent lines are cut by a transversal, any cross-ratio of the four points of intersection is constant for all positions of the transversal. Hypothesis. Let A, B, C, D, be the intersection points for f, the transversa/ in one posit/on, and let A\ B', C , jy, be the cor- responding intersection points for t\ the transversal in another position. Conclusion. {ABCD) = {A'B'Cjy); that is, AC , AD ^ A[C_ . A'D' CB ' DB C'B' ' jyB'' Proof. Through the points A and B on t draw parallels to /', which cut the concurrent lines in C2, D2, B^, and A„ C^, D^. AACC^-^ £^BCC„ AC ^ AC^ CB C,B' and and A ADD:, ~ A BDD,, AD DB AD, D,B' where account is taken of " sense.' MENSURATION, OR METRICAL GEOMETRY. 355 Hence CB * Z>B C,B ' D,B AD^ ' Z>,b'' but ii^ = f^' and C^^C:^^ AD, A'ly D,B jyB" . AC , AD ^ A'C . CB' ^ y^^C^ . A'jy " CB' DB AD ' DB' CB' '' D'B'' EXERCISES. BOOK I. 1 20. Show how to make a rhombus having one of its diagonals equal to a given sect. 121. If two quadrilaterals have three consecutive sides, and the two contained angles in the one respectively equal to three consecutive sides and the two contained angles in the other, the quadrilaterals are congru- ent. 122. Two circles cannot cut one another at two points on the same side of the line joining their centers. 123. Prove, by an equilateral triangle, that, if a right-angled triangle have one of the acute angles double of the other, the hypothenuse is double of the side opposite the least angle. 1 24. Draw a perpendicular to a sect at one extremity. 125. Draw three figures to show that an exterior angle of a triangle may be greater than, equal to, or less than, the interior adjacent angle. 126. Any two exterior angles of a triangle are together greater than a straight angle. 127. The perpendicular from any vertex of an acute-angled triangle on the opposite side falls within the triangle. 128. The perpendicular from either of the acute angles of an obtuse- angled triangle on the opposite side falls outside the triangle. 129. The semi-perimeter of a triangle is greater than any one side^ and less than any two sides. 3S7 358 THE ELEMENTS OF GEOMETRY. 130. The perimeter of a quadrilateral is greater than the sum, and less than twice the sum, of the diagonals. 131. If a triangle and a quadrilateral stand on the same base, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter. 132. If one angle of a triangle be equal to the sum of the other two, the triangle can be divided into two isosceles triangles. 133. If any sect joining two parallels be bisected, this point will bisect any other sect drawn through it and terminated by the parallels. 134. Through a given point draw a line such that the part inter- cepted between two given parallels may equal a given sect. 135. The medial from vertex to base of a triangle bisects the inter- cept on every parallel to the base. 136. Show that the surface of a quadrilateral equals the surface of a triangle which has two of its sides equal to the diagonals of the quadri- lateral, and the included angle equal to either of the angles at which the diagonals intersect. 137. Describe a square, having given a diagonal. 138. ABC is a right-angled triangle; BCED is the square on the hypothenuse; ACKH and ABFG are the squares on the other sides. Find the center of the square ABFG (which may be done by drawing the two diagonals), and through it draw two lines, one parallel to BC, and the other perpendicular to BC. This divides the square ABFG into four congruent quadrilaterals. Through each mid-point of the sides of the square BCED draw a parallel to AB or A C. If each be extended until it meets the second of the other pair, they will cut the square BCED into a square and four quadrilaterals congruent to ACKH and the four quadrilaterals in ABFG. 139. The orthocenter, the centroid, and the circumcenter of a tri- angle are collinear, and the sect between the first two is double of the sect between the last two. 140. The perpendicular from the circumcenter to any side of a triangle is half the sect from the opposite vertex to the ortho- center. 141. Sects drawn from a given point to a given circle are bisected ; find the locus of their mid-points. EXERCISES. 359 142. The intersection of the Hnes joining the mid-points of opposite sides of a quadrilateral is the mid-point of the sect joining the mid-points of the diagonals. 143. A parallelogram has central symmetry. SYMMETRY. 144. No triangle can have a center of symmetry, and every axis of symmetry is a medial. 145. Of two sides of a triangle, that is the greater which is cut by the perpendicular bisector of the third side. 146. If a right-angled triangle is symmetrical, the axis bisects the right angle. 147. An angle in a triangle ^s-ill be acute, right, or obtuse, according as the medial through its vertex is greater than, equal to, or less than, half the opposite side. 148. If a quadrilateral has axial symmetry, the number of vertices not on the axis must be even ; if none, it is a symmetrical trapezoid ; if two, it is a kite. 149. A kite has the following seven properties ; from each prove all the others by proving that a quadrilateral possessing it is a kite. (i) One diagonal, the axis, is the perpendicular bisector of tlie other, which will be called the transverse axis. (2) The axis bisects the angles at the vertices which it joins. (3) The angles at the end-points of the transverse axis are equal, and equally divided by the latter. (4) Adjacent sides which meet on the axis are equal. (5) The axis divides the kite into two triangles which are congruent, with equal sides adjacent. (6) The transverse axis divides the kite into two triangles, each of which is symmetrical. (7) The Hnes joining the mid-points of opposite sides meet on the axis, and are equally inclined to it. 150. A symmetrical trapezoid has the following five properties ; from each prove all the others by proving that a quadrilateral possessing it is a symmetrical trapezoid. 360 THE ELEMENTS OF GEOMETRY. (i) Two opposite sides are parallel, and have a common perpendicu- lar bisector. (2) The other two opposite sides are equal, and equally inclined to either of the other sides. (3) Each angle is equal to one, and supplemental to the other, of its two adjacent angles. (4) The diagonals are equal, and divide each other equally. (5) One median line bisects the angle between those sides produced which it does not bisect, and likewise bisects the angle between the two diagonals. 151. Prove the properties of the parallelogram from its central sym- metry. 152. A kite with a center is a rhombus; a symmetrical trapezoid with a center is a rectangle ; if both a rhombus and a rectangle, it is a square. BOOK II. 153. The perpendicular from the centroid to a line outside the tri- angle equals one-third the sum of the perpendiculars to that line from the vertices. 154. If two sects be each divided internally into any number of parts, the rectangle contained by the two sects is equivalent to the sum of the rectangles contained by all the parts of the one, taken separately, with all the parts of the other. 155. The square on the sum of two sects is equivalent to the sum of the two rectangles contained by the sum and each of the sects. 156. The square on any sect is equivalent to four times the square on half the sect. 157. The rectangle contained by two internal segments of a sect grows less as the point of section moves from the mid-point. 158. The sum of the squares on the two segments of a sect is least when they are equal. 159. If the hypothenuse of an isosceles right-angled triangle be divided into internal or external segments, the sum of their squares is EXERCISES. 361 equivalent to twice the square on the sect joining the point of section to the right angle. 160. Describe a rectangle equivalent to a given square, and having one of its sides equal to a given sect. 161. Find the locus of the vertices of all triangles on the same base, having the sum of the squares of their sides constant. 162. The center of a fixed circle is the point of intersection of the diagonals of a parallelogram ; prove that the sum of the squares on the sects drawn from any point on the circle to the four vertices of the parallelogram is constant. 163. Thrice the sum of the squares on the sides of any pentagon is equivalent to the sum of the squares on the diagonals, together with four times the sum of the squares on the five sects joining, in order, the mid- points of those diagonals. 164. The sum of the squares on the sides of a triangle is less than twice the sum of the rectangles contained by every two of the sides. 165. If from the hypothenuse of a right-angled triangle sects be cut off equal to the adjacent sides, the square of the middle sect thus formed is equivalent to twice the rectangle contained by the extreme sects. BOOK III. 166. From a point, two equal sects are drawn to a circle. Prove that the bisector of their angle contains the center of the circle. 167. Describe a circle of given radius to pass through a given point and have its center in a given line. 168. Equal chords in a circle are all tangent to a concentric circle. 169. Two concentric circles intercept equal sects on any common secant. 1 70. If two equal chords intersect either within or without a circle, the segments of the one equal the segments of the other. 171. Divide a circle into two segments such that the angle in the one shall be seven times the angle in the other. 172. ABC and ABC zx^ two triangles such that AC = AC ', prove the circle through A, B, C, equal to that through A, B, C, 362 THE ELEMENTS OF GEOMETRY. 1 73. Pass a circle through four given points. When is the problem possible ? 174. \i AC and BD be two equal arcs in a circle ABCD, prove chord AB parallel to chord CD. 175. Circles described on any two sides of a triangle as diameters intersect on the third side, or the third side produced. 1 76. If one circle be described on the radius of another as diameter, any chord of the larger from the point of contact is bisected by the smaller. 177. If two circles cut, and from one of the points of intersection two diameters be drawn, their extremities and the other point of inter- section will be collinear. 178. Find a point inside a triangle at which the three sides shall subtend equal angles. 179. On the produced altitudes of a triangle the sects between the orthocenter and the circumscribed circle are bisected by the sides of the triangle. 180. If on the three sides of any triangle equilateral triangles be described outwardly, the sects joining their circumcenters form an equi- lateral triangle. 181. If two chords intersect at right angles, the sum of the squares on their four segments is equivalent to the square on the diameter. 182. The opposite sides of an inscribed quadrilateral are produced to meet ; prove that the bisectors of the two angles thus formed are at right angles. 183. The feet of the perpendiculars drawn from any point in its circumscribed circle to the sides of a triangle are collinear. 184. From a point P outside a circle, two secants, PAB, PDC, are drawn to the circle ABCD ; AC, BD, are joined, and intersect at O. Prove that O lies on the chord of contact of the tangents drawn from P to the circle. Hence devise a method of drawing tangents to a circle from an external point by means of a ruler only. 185. A variable chord of a given circle passes through a fixed point ; find the locus of its mid-point. 186. Find the locus of points from which tangents to a given circle contain a given angle. EXERCISES. 363 187. The hypothenuse of a right-angled triangle is given; find the loci of the comers of the squares described outwardly on the sides of the triangle. BOOK IV. 188. If a quadrilateral be circumscribed about a circle, the sum of two opposite sides is equal to the sum of the other two. 189. Find the center of a circle which shall cut off equal chords from the three sides of a triangle. 190. Draw a line which would bisect the angle between two hnes which are not parallel, but which cannot be produced to meet. 191. The angle between the altitude of a triangle and the line through vertex and circumcenter equals half the difference of the angles at the base. 192. The bisector of the angle at the vertex of a triangle also bisects the angle between the altitude and the line through vertex and circum- center. 193. If an angle bisector contains the circumcenter, the triangle is isosceles. 194. If a circle can be inscribed in a rectangle, it must be a square. 195. If a rhombus can be inscribed in a circle, it must be a square. 196. The square on the side of a regular pentagon is greater than the square on the side of the regular decagon inscribed in the same circle by the square on the radius. 197. The intersections of the diagonals of a regular pentagon are the vertices of another regular pentagon ; so also the intersections of the alternate sides. 198. The sum of the five angles formed by producing the alternate sides of a regular pentagon equals a straight angle. 199. The circle through the mid-points of the sides of a triangle, called the medioscribed circle, passes also through the feet of the alti- tudes, and bisects the sects between the orthocenter and vertices. Hint. Let ABC be the triangle ; H^ A', Z, the mid-points of its sides ; Xy V, Z, the feet of its altitudes ; O its orthocenter ; C/, F, JV, the mid-points of AO, BO, CO. LHWU is a rectangle ; so is HKUV\ and the angles at X^ Y, Z, are right. 364 THE ELEMENTS OF GEOMETRY. 200. The mediocenter is midway between the circumcenter and the orthocenter. 201. The diameter of the circle inscribed in aright-angled triangle, together with the hypothenuse, equals the sum of the other two sides. 202. An equilateral polygon circumscribed about a circle "is equi- angular if the number of sides be odd. 203. Given the vertical angle of a triangle and the sum of the sides containing it ; find the locus of the circumcenter. 204. Given the vertical angle and base of a triangle; find the locus of (i) The center of the inscribed circle. (2) The centers of the escribed circles. (3) The orthocenter. (4) The centroid. (5) The mediocenter. 205. Of all rectangles inscribable in a circle, show that a square is the greatest. BOOK VI. 206. The four extremities of two sects are concyclic if the sects cut so that the rectangle contained by the segments of one equals the rect- angle contained by the segments of the other. 207. If two circles intersect, and through any point in their common chord two other chords be drawn, one in each circle, their four extrem- ities are concyclic. 208. Does the magnitude of the third proportional to two given sects depend on the order in which the sects are taken ? 209. Through a given point inside a circle draw a chord so that it shall be divided at the point in a given ratio. 210. If two triangles have two angles supplemental and other two equal, the sides about their third angles are proportional. 211. Find two sects from any two of the six following data: their sum, their difference, the sum of their squares, the difference of their squares, their rectangle, their ratio. EXERCISES. 365 212. If two sides of a triangle be cut proportionally, the lines drawn from the points of section to the opposite vertices will intersect on the medial from the third vertex. 213. Given the base of a triangle and the ratio of the two sides; find the locus of the vertex. MISCELLANEOUS. 214. Find the locus of a point at which two adjacent sides of a rectangle subtend supplementary angles. 215. Draw through a given point a line making equal angles with two given Hnes. 216. From a given point place three given sects so that their extremities may be in the same line, and intercept equal sects on that line. 217. Divide a sect into two parts such that the square of one of the parts shall be half the square on the whole sect. 218. Find the locus of the point at which a given sect subtends a given angle. 219. Given a curve, to ascertain whether it is an arc of a circle or not. 220. If two opposite sides of a parallelogram be bisected, and lines be drawn from these two points of bisection to the opposite angles, these lines will be parallel, two and two, and will trisect both diagonals. 221. Through two given points on opposite sides of a line draw lines to meet in it silch that the angle they form is bisected. 222. The squares on the diagonals of a quadrilateral are double of the squares on the sides of the parallelogram formed by joining the mid-points of its sides. 223. If a quadrilateral be described about a circle, the angles sub- tended at the center by two opposite sides are supplemental. 224. Two circles touch at A and have a common tangent BC. Show that BA C is a right angle. 225. Find the locus of the point of intersection of bisectors of the angles at the base of triangles on the same base, and having a given vertical angle. 366 THE ELEMENTS OF GEOMETRY. 226. Find the locus of the centers of circles which touch a given circle at a given point. 227. Two equal given circles touch each other, and each touches one side of a right angle ; find the locus of their point of contact. 228. In a given Hne find a point at which a given sect subtends a given angle. 229. Describe a circle of given radius to touch a given line and have its center on another given line. 230. At any point in the circle circumscribing a square, show that one of the sides subtends an angle thrice the others. 231. Divide a given arc of a circle into two parts which have their chords in a given ratio. 232. The sect of a common tangent between its points of contact is a mean proportional between the diameters of two tangent circles. 233. Any regular polygon inscribed in a circle is a mean propor- tional between the inscribed and circumscribed regular polygons of half the number of sides. 234. The circumcenter, the centroid, the mediocenter, and the orthocenter form a harmonic range. EXERCISES IN GEOMETRY OF THREE DIMENSIONS AND IN MENSURATION. The most instructive problems in geometry of three dimensions are made by generalizing those first solved for plane geometry. This way of getting a theorem in solid geometry is often difficult, but a number of the exercises here given are specially adapted for it. In the author's "Mensuration" (published by Ginn & Co.) are given one hundred and six examples in metrical geometry worked out com- pletely, and five hundred and twenty-four exercises and problems, of which also more than twenty are solved completely, and many others have hints appended. si RETURN CIRCULATION DEPARTMENT TO^- r/lnip'i^^rnry. 198 Main Stocks LOAN PERIOD 1 HOME USE Ll I BOOKS MAY BE RECALLED AFTER 7 DAYS. J^nirond Recharges may ^ made 4 days prior ,o me due date. Books may be Renewed by calling 642-3405. DUE AS STAMPED BELOW FORM NO. DD6 ^^fJ^T^i^^^lSS^S U. C. BERKELEY LIBRARIES CO SfiS bEDl