A B к D common piece B C D K from the trapezium A B C D and the ABCDET G (Fig. 48), and proceed to construct a triangle equal "triangle B F C, wo have the triangle A K B, the remainder of the to it in area. As the fignra is complicated, the lines which concrapezium A B C D, and the triangle K F D, the remainder of the tain the heptagon and the triangle equivalent to it in area have triangle B 7 c. been drawn thicker than the lines which are necessary in working But these triangles are also parts of the triangles A D B, out the process (as in Fig. 47), that the reader may the more BF D, which are equal in area, since they are on the same base, readily distinguish the relative areas of the figures in question. BD, and between the same parallels A F, B D, and as the triangle The first step is to draw straight lines from A, the apex of KD B is common to both, the triangle A K B is equal to the the polygon, taking D E to represent its base, to the points triangle K F D. In the same manner, by drawing the diagonal C, D, E, F, or to each salient point of the polygon except the two A c of the tra- immediately on the right and left of the apex. The straight pezium A B C D, lines A C, A D, A E, A F divide the polygon A B C D E F G ints producing D c in five unequal triangles, A B C, A C D, A D E, A E F, and A FG. the direction of The reader will note that however many may be the sides of the G; drawing BH polygon, it is divided by this process into a number of triangles parallel to A C, always less by two than the number of its sides. Thus in the and meeting D G figure below the number of triangles into which it is divided by in H; and lastly, drawing straight lines from its apex to its salient points is five, joining An, it the number of its sides being seven; a dodecagon, or twelveнс may be shown sided figure, would be divided into ten triangles, and so on. Fig. 46. that the triangle Now—beginning with the triangle A B C, the highest triangle A D H is also equal | on the left side of the apex-by producing D c in the direction in superficial area to the ular quadrilateral figure A B C D. of P, indefinitely ; drawing B u parallel to A c to meet c D pro It will be useful for the student to repeat this construction as duced in h; and joining A H; we get a triangle, A H C, equal to an exercise, taking the sides C B, B A, and A d in succession as the triangle A B C, and by adding the polygon A C D E F G to the base of the trapezium A B C D, or the side on which it each of these triangles, we find that we have a hexagon or sixstands. sided figure, A H D E F G, equal in area to the original seven. PROBLEM XXXIV.—To draw a triangle that shall be equal in sided polygon A B C D E F G. By making the triangle A KD superficial area to any given multilateral figure or polygon. equal to the triangle First let us take a five-sided figure, as being next in order to A HD by the same a four-sided figure, as far as the number of its sides are con- construction, which we cerned, and let A B C D E (Fig. 47) represent the five-sided figure need not repeat, we or pentagon, to which it is required to draw a triangle equal in get a pentagon, or fivesuperficial area. From c, the apex of the pentagon, draw the sided figure, A K E F G, straight lines C A, C E, to the points A, E, the extremities of the equal in area to the base on which it stands. By doing this we divide the pentagon hexagon A H D E F G, A B C D E into three triangles A B C, C A E, and C ED. Produce the and consequently to base A E indefinitely both ways in the direction of r and g, and the original heptagon through B and draw the straight lines B H, D K, parallel to A B C D E F G. Con Fig. 48. C A, C E respectively, and meeting the base A E produced, in the tinuing the process with making the triangle A F L equal to the tripoints and K. Join C H, CK; the triangle cok is equal angle A F G, the highest triangle on the right side of the apex, we in superficial area to the pentagon A B C D E. That this is get an irregular quadrilateral figure, A K E L, equal to the pentrue may be seen as follows:-Of the three triangles A B C, tagon A K E F G, the hexagon AH D E F G, and the heptagon C A E, and C E D, into which the pentagon was divided, the A B C D E F G. Once more, by making by a similar construction triangle C A E is common to both the pentagon and the triangle the triangle A E M equal to the triangle A E L, we get at last a Of the remaining portions of the pentagon and triangle, triangle, A K M, equal in area to the quadrilateral figure A K EL, the triangle A B C of the and the above-named pentagon and hexagon and the original reason the triangle arithmetical process to be explained hereafter the superficial CED of the pentagon is content of each triangle would be found, and the five results Ć equal to the triangle CEK added together to obtain the area of the polygon. By reducing Fig. 47. of the triangle. the area of the polygon to a triangle, its area can be found by The learner will find it one calculation instead of five, and a sum in compound addition; useful to repeat this construction as an exercise, taking the sides or, to ensure accuracy, both processes may be gone through, each A B, BC, C D and D E in succession, as the base on which the proving a test whereby the correctness of the other may be pentagon is supposed to stand. ascertained. That the learner may thoroughly understand the process of As in the preceding propositions, let the learner repeat the drawing a triangle equal in superficial area to a polygon having above construction as an exercise, taking the sides E F, F G, GA, a great number of sides, and see that it is as easy as it is to A B, BC, and cd in succession, as the base on which the polygon draw a triangle equal in area to a pentagon, which has only five is supposed to stand, and the salient point which happens to bo sides, we will take the irregular seven-sided figure, or heptagon immediately opposite the base in each case as the apex QK E CHK. same F H CASSELL PETTER & GALPIN, BELLE SAUVAGE WORKS, LONDON, L.O. 277 . 315 • 337 . Cyclopean or Pelasgic Architecture Early 15 Egypt and India-The 65 Persia-Greek Architec- 129 193 tecture-Tuscan Order- 257 Terms used in Architec- 319 383 7 37 101 with Compound Quan. 142 198 with Compound Quanti- 234 with reference to 270 294, 326 362 402 20 the Debtor and Creditor 154 Book, Cash Book, Bill 218 sonal Accounts-Profit and Loss Accounts 348 A piacex -- The Umbelli. ferous or Parsley Tribe . 24 25 sicaces - The Cruciferous PAGE PAGE the Passion-flower Tribe 56 and Compounds of this Gas with Hydrogen 289 362 Nitrogen and Sulphur- 121, 152 The Halogens-Chlorine 399 CLASSES, “ POPULAR EDUCATOR" Introduction - Terms em- 17 ployed in Classification Divisions of the Animal Kingdom - Vertebrata- Mollusca - Molluscoida Annulosa -- Annuloida --Coelenterata-Protozoa 81 218 Subdivisions of the Animal Kingdom-Table of Sub- 219 divisions of Classes-Pro. tozoa 113 273 Cælenterata-Hydrozoa 145 Actinozon (Rayed Animals) 183 273, 305 Echinodermata (Hedgehog. 317 skinned Animals). 215 Helminthozoa 241 314 Annelida: Ringed Worms. 279 Rotatoria-Myriapoda 311 Insecta ILLUSTRATIONS : or 375 Sketch of Haddock, show. also the arrangement of 375 organs Transverse section of 375 Haddock Sketch of 376 Lobster Transverse Section of Lobster 81 376 Amba-Shell of Polycys. tina--Sectional Diagram 377 showing circulation in a Sponge-Group of Vorti. 401 cellæ-Noctiluca Miliaris 113 Eadendrinn Ramosum 402 Hydrozoon encrusting a Shell-Rhopalonema Ve- 402 latum, the Veiled Club- tentacled Medusa-Per- 402 pendicular Section of Sea Anemone Transverse Section of Sea Anemone -Pleurobrachia --Trans- verse Section of Pleuro- 1 145 Caryophyllia Smithii–Dry Coral of Caryophyllia Smithii – Diagrammatic Section, showing how Red Coral is secreted- Cestum Veneris-One of the Polypes of Alcyonaria 184 -Formation of Atoll 185 Plates and Holes on Echi- 100 Ambulacral 132 Plates--Echinus divided to show Alimentary Canal 168 -Spine--Jaws and Teeth --Side View of Single 235 Jaw- Tooth - Inside of 271 Purple-tipped Seo-Urchin 217 PAGE Solium-Joint of same, 241 of Eunice-Proboscis of 279 Glomeris Julus-Antenna nature of Rotary Illusion 313 pillar : Pupa, Imago - 337 Trees-Massing in the 7, 39 --Setting Drawings, etc. 72 10+ 135 263, 327, 392 Trees and Foliage, and 40, 41, 72, 73, 104 Vegetable Form to De. 105 Reflections in Water 136, 137 200 Angle in Men and Animals 201 264, 285 Head of Judas, 329, 392, 393 13 38, 70, 110, 134, 165 226 . ог . . PAGE The Greek Element Greek Stems 262, 358, 391, 409 Conversitions on English Grammar 134, 302, 334 9) 9 180 » ESSAYS ON TIFE AND DUTY: 11 Patience 77 Unselfishness 131 Courage 199 Fidelity 259 Perseverance 327 Economy. 398 . 98 . : 22, 58 FRENCH, LESSONS IN : XLVIII, Unipersonal Verbs 10 XLIX. Regimeu relating to some Verbs 10 Irregular Verbs Tense and the Pluperfect Tenses 106 LV. Idiomatic Constructions in Regimen 105 LVI. Idiomatic Uses of Tens28 of Verbs 107 LVII. Idiopatic Plırases 133 LVIII. Rules for the Plu ral of Compound Nouns 133 LIX. The Two Fatures, Simple and Anterior 172 LX. Irregularities of the Future 172 LXI. 'The Two Conditionals 173 LXII., LXIII. Idiomatic Phrases 174, 202 LXIV. Idioms : Faire used Reflectively and Uniper- 202 LXV. Idioms relating to Avoir, etc. 237 LXVI, Idioms relating to Avoir and Epouser 237 LXVII. Idioms relating to Dimension, Weight, etc. 266 LXVIII. Idioms relating to Mettre, etc. 266 LXIX. The Imperative 297 LXX The Imperative and the Intinitive Idioms 298 LXXI. The Subjunctive 330 LXXII., LXXIII. The Use of the Subjunctive 330, 361 LXXIV. The Imperfect and Pluperfect of the Subjunctive. 365 LXXV., LXXVI. Regimen, or Government of Verbs 386 . PAGE PAGE 283, 299, 323 KEY TO EXERCISES IN LES- Regular Verbs-The Second Construction of Projection SONS IN GERMAN: Conjugation. 405 of Map of Europe. 355, 388 Exs, 4-9 27 Exs. 27-33, 222 The Key to the Exercises given Table of Latitudes and 10 86-37 247 in avy Lesson in Latin will be Longitudes of Places 11-16 95 283 found at the end of the next in Europe 389 42, 43, 315 Lesson or the next Lesson but MAPS : 21-23 . 156 4152, 372 one. Map of the World 2+26. 53–59 408 MECHANICS : The Pulley Principle of Virtual Velo- 269 cities - The Three SysNorway, Sweden, and Den 2 Alphabet. tems of Palleys 300 Compound Pulleys 301 34 The Inclined Plane--The 301 Wedge-The Screw . 324 Statical Forces--Friction 107 the Prepositious The Illustrations of preceding Definite Article 66 Principles - Kite, Boat, GEOMETRICAL PERSPECCase-eudings of the Declen etc.-Elements of Ma. TIVE: sions chinery Introduction Definitions The First Declension. 98, 130 Prime Movers Animal Force, Water, Wind, Steam 258, 291, 322, 354, 390 Dynamics Definitions The Three Laws of Orthographic and Isome tracted . 390 Motion . tric Projection Pro Proof of Third Law of The Key to the Exercises given blems II.-VI. 295 Motion-Laws of Falling in any Lesson in Greek will be Bodies Atwoni's Ms. found at the end of the next chine Lessou. Laws of Falling Bodies Projectiles-Collision or Impact . Impact-Centrifugal Force -The Pendulum-Centre became one metry on the Circle 49, 92, 123 of Oscillation. Regular Polygons 124 How Ireland became part Problems in Practical Geo of Great Britain 85, 125 | MUSIC, LESSONS IN: metry on Constrnction of How England became pos Meutal Effect of Notes 31 Regular Polygons 148, 191, 211 sessed of India 157 Character and Effect of Leading Nutes House of Commons 181 Meutal Effect-Consonance of Notes, etc.. 308 Measurement of Intervals the Rebellion of 1745 --The Glass Harmonicon The Massacre of Glencoe 285 - German Concertina GERMAN, LESSONS IN : Wilkes and Liberty 317 Relation of Notes, etc. 386,5 lorous Sir Walter Raleigh 341 OUR HOLIDAY : 27 Admiral Byng on the 14th Gymnastics. of Murch, 1757 373 The Hanging Rope quiring the Genitive Summary of Sketches in The Giant's Stride. XLVIII. Adjectives re Vols. I. and II. 405 The Hanging Bar or Tra quiring the Dative peze XLIX. Verbs requiring the HYDROSTATICS : The Hanging Stirraps 411 Dative 92 The Hanging Rings L. Verbs requiring an ac- Objects of the Science Swimming cusative of a Person and Principle of Equality of Croquet a Genitive of a Thing 9 Pressure Hydrostatic Laws of Croquet Press 366 Dative or Accusative 117 Pressure of Liquids arising PENMANSHIP, LESSONS IN LII. Verbs requiring two from their weight Official Handwriting .33, 64, 13 Centra of Pressure Business Handwriting governing the Accusative Levels--Springs and Ar Legal Handwriting 118 396 Germau Handwriting. LIII. Prepositions Greek Handwriting LATIN, LESSONS IN: READING AND ELOCUTIOS 18 19 Analysis of the Voice : LV. Prepositions requiring Possessive or Adjective Exercises on Inflections the Accusative 155 Just Stress On Relative and Interrogative 156 83 Appropriate Modulation in LVII. Examples illustra Indefinite Pronouns 87 Promiscuous Exercises ting the various uses of Correlative Pronouns 86 214, 250, 278, 345, , Prepositions 178 The Nuwerals 122 RECREATIVE NATURAL HIS LVIII., LIX., LX. Pecu Prepositions 150 TORY : liar Idioms 18), 222, 246 The Latin Verb. 190 The Butterfey The Froz. Euglish Snakes, . . re. KEY TO EXERCISES IN LES SONS IN FRENCH : 36-41 .237 10 - 15 75 42-19.293 16, 17 1071 50-53 . 331 18-20 1391 5456.383 21-25 174 57-60.387 26-28, 2031 . . The Swallo: The Spider Conjugation 310, 350 WHITWORTH SCROLARthe Indicative 407 Ou Parsing 882 SHIPS, THE First Meridiau, etc. 1413 Construction of the Map of the World 166 Natural Divisions of the Earth's Surface 196, 231 |