А, B K с нс may be H common piece B co k from tho trapezium A B C D and the ABCDEFG (Fig. 48), and proceed to construct a triangle equal *triangle B F C, we have the triangle A K B, the remainder of the to it in area. As the figura 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 i C. been drawn thicker than the lines which are necessary in working But these triangles are also parts of the triangles A D B, ont the process (as in Fig. 47), that the reader may the more B F D, which are equal in area, since they are on the same base, readily distinguish the relative areas of the figures in question. B D, 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 inte 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 A H, it the number of its sides being geven; a dodecagon, or twelve 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 AD H is also equal on the left side of the apex-by producing dc in the direction in superficial area to the irregular quadrilateral figure A B C D. of P, indefinitely; drawing B H parallel to Ac 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 sevenPROBLEM 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, ABCDE 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 F and G, and the original heptagon through B and d 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 AF L equal to the tripoints and K. Join CH, C K; the triangle cok is equal angle A F G, the highest triangle on the right side of the apex, wo 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 A HD E F G, and the heptagon C A E, and C E D, into which the pentagon was divided, the 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 C E D of the pentagon is content of each triangle would be found, and the five results equal to the triangle C E K 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 be sides, we will take the irregular seven-sided figure, or heptagon immediately opposite the base in each case as the apex Q E ABCDEFG. CHK. same F H CASSELL PETTER & GALPIN, BELLE SAUVAGE WORKS, LONDON, L.O. 277 INDEX TO CONTENTS. . PAGE Solium-Joint of same, 241 of Eunice-Proboscis of 2779 Glomeris Julus-Antenna nature of Rotary Illusion 313 pillar : Pupa, Imago - . 273, 305 . . . 337 PAGE PAGE Lower Oxides of Nitrogen Gas with Hydrogen 289 Hydro-carbons, Coal-gas, 362 Nitrogen and Sulphur- 121, 152 Tbe Halogens-Chlorine 399 153 CLASSES, " POPULAR the EDUCATOR" 411 177 COMPARATIVE ANATOMY : Introduction – Terms em- 17 Divisions of the Animal Kingdom - Vertebrata- Mollusca Molluscoida Annulosa --- Annuloida --Celenterata-Protozoa 81 218 Subdivisions of the Animal Kingdom-Table of Sub- 219 divisions of Classes-Pro- tozoa 133 273 Coelenterata-Hydrozor 145 Actinozoa (Rayed Animals) 183 Echinodermata (Hedgehog. skinned Animals) 215 Helminthozoa 2+1 314 Annelida: Ringed Worms. 279 Rotatoria-Myriapoda 311 Insecta 337 315 ILLUSTRATIONS : 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 Ameba-Shell of Polycys. ting-Sectional Diagram 377 showing circulation in a Sponge-Group of Vorti. 401 cellæ-Noctiluca Miliaris 113 Eadendrinn Ramosum 432 Hydrozoon encrusting a Shell-Rhopalonema Ve. 402 latum, the Veiled Club. Olive tentacled Medusa-Per- 402 pendicular Section of Sea Anemone Transverse Section of Sea Anemone - Pleurobrachia --Trans. verse Section of Pleuro- 145 Caryophyllia Smithii-Dry Coral of Caryophyllia Smithii – Diagrammatic Red Coral is secreted- Cestum Veneris--One of the Polypes of Alcyonaria 184 -Formation of Atoll 185 Plates and Holes on Echi- 100 nus Shell - Ambulacral 132 Plates--Echinus divided to show Alimentary Canal 168 -Spine--Jaws and Teeth 203, 235 --Side View of Single Jaw- Tooth -- Inside of 271 Purple-tipped Sco-Uxchin 217 7, 39 . . 199, ARITHMETIC, LESSONS IN: 7 37 101 with Compound Quan. 112 198 with Compound Quanti- 23+ tion 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 Apiacea - The Umbelli. ferous or Parsley Tribe. 21 25 sicaceae - The Cruciferous 56 DRAWING, LESSONS IN: Trees-Massing in the Foliage, etc. -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, 265 323 329, 392, 393 13 226 PAGE The Greek Element-Greek Stenis 202, 358, 391, 409 Convers itions on English Grainmar 134, 302, 331 . . . 9 . . 17–20. . : ESSAYS ON JIFE AND DUTY: 11 Patience 77 Unseliishness 131 Courage 193 Fidelity Perseverance 327 Economy. 398 • 259 UN . . 22, 58 FRENCH, LESSONS IN: XLVIII. Unipersonal Verbs 10 10 Irregular Verbs 76 LIV. The Past Anterior and the Pluperfect Tenses 106 LV. Idiomatic Constructions in Regimen . 103 LVI. Idiomatic Uses of Tens28 of Verbs 107 LVII. Liomatic Plırases 133 LVIII. Rules for the Plu ral of Compound Nouus 133 LIX. The Two Futures, 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. Idions relating to Avoir, etc. 237 LXVI. Idions 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 Infinitive Idioms 298 LXXI. The Subjunctive 330 LXXII., LXXIII. The Use of the Subjunctive 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. 406 of Map of Europe. 355, 388 Exs. 49 27 Exs. 27-33 222 The Key to the Exercises gisen 10 aby Lesson in Latin will be Longitudes of Places 11-16 95 38-41 . 233 found at the end of the text in Europe 389 119 42, 43 315 Lesson or the next Lesson but MAP3: 21-23 . 156 41-52 372 ope, 2120 180 53-59, 408 MECHANICS : The Pulley 12 Pacific Ocean 233 GREEK, LESSONS IN: Principle of Virtual Velo- 209 cities - The Three Sys- tems of Pulleys 300 Alphabet. Vowels-Consonants- Compound Pulleys 301 301 34 The Inclined Plane--The General Remarks on the Wedge-The Screw Noun, the Adjective, and Statical Forces--Friction the Prepositious The Illustrations of preceding Definite Article 66 Principles - Kite, Boat, GEOMETRICAL PERSPECCase-eudings of the Declen etc.-Elements of MeTIVE: sious 98 chinery Introduction - Definitions The First Declension. 98, 139 Priino Movers Avima! Steam 258, 291, 322, 351, 390 Dynamics Definitions The Three Laws tracted 390 Motion . tric Projection Pro Proof of Third Law of The Key to the Exercises given blems II.-VI. 235 Motion-Laws of Falling Problems VII.-XI. : in any Lesson in Greek will be 359 Bodies -- Atwood's Ms. found at the end of the next chine Iessou. Laws of Falling Bodies- Projectiles-Collision or Impact . Impact-Centrifugal Force -The Pendulum-Centre 124 of Oscillation. of Great Britain 85, 125 MUSIC, LESSONS IN: Meutal Effect of Notes 31 Regular Polygons 148, 191, 211 sessed of India 157 Character and Effect of Leading Notes 284, 307 Mental Effect Consonance of Notes, etc.. 308 Measurement of Intervals the Rebellion of 1745 253 --The Glass Harmonicon -German Concertina 317 Relation of Notes, etc. 3), 26 The Right Noble and Va- lorous Sir Walter Raleigb S41 OUR HOLIDAY : 27 Admiral Byng on the 14th Gymnastics. of March, 1757 373 The Hanging Rope The Giant's Stride. Vols. I. and II. 405 The Hanging Bar or Trequiring the Dative peze XLIX. Verbs requiring the HYDROSTATICS: The Hanging Stirraps The Hanging Rings Swimming Principle of Equality of Croquet 91 Laws of Croquet Press 366 from Official Handwriting .33, 6, 1. Centra of Pressure Business Handwriting governing the Accusative Levels--Springs and Ar- Legal Handwriting 118 396 Germau Handwriting. Greek Handwriting LATIN, LESSONS IN : READING AND ELOCUTIUS 18 quiring the Dative 155 Personal Pronouns 19 Analysis of the Voice : Possessive Exercises on Inflectious Adjective 155 54 Just Stress Demonstrative Pronouns 5+ Expressive Tones, Rules 156 83 Appropriate Modulatiou LVII. Examples illustra Indefinite Pronouns 8 Promiscuous Exercises ting the various uses of Correlative Pronouns 86 214, 250, 278, 26, 314,7 Prepositions 178 The Ninerals 122 RECREATIVE NATURAL HIS LVIII., LIX., LX. Pecu Prepositions 150 TORY: liar Idioms 18), 222, 216 The Latin Verb. 190 The Butterfy The Froz 216 -Coinpouads of Sum 210 12 English Snakes The Swalloir The Spider Conjugation WHITWORTH SCRO: SRthe Indicative 407 Ou Parsing 382 SHIPS, THE |