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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
forrner is equal to the tri. heptagon A B C D E F G.
angle C HA of the latter, The learner will find the value of this geometrical process in
because they are on the same determining the areas of irregular polygons in mensuration. To
base, A C, and between the calculate the area of the heptagon A B C D E F G, it would be
same parallels ; and for the divided as in the above figure into five triangles, and by an

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

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Cyclopean or Pelasgic

Architecture Early
Monuments-First Re.
gular Constructions
Pyramids, etc.

15
Progress of Architecture in

Egypt and India-The
Column

65
Ancient Architecture of

Persia-Greek Architec-
ture-The Parthenon-
Doric Order of Architec-
ture

129
The Greek Orders of Ar-
chitecture

193
Roman Orders of Archi

tecture-Tuscan Order-
Composite Order

257
Explanation of some of the

Terms used in Architec-
ture

319
Architraves, Arches, and
Vaulted Roofs

383
ARITHMETIC, LESSONS IN :
Reduction

7
Compound Addition

37
Compound Subtraction 78
Compound Multiplication . 78
Compound Division

101
Fractions in connection

with Compound Quan.
tities

142
Practice

198
Decimals in connection

with Compound Quanti-
ties-Reduction of Deci.
mals

234
Short Methods of Redue-
tion

with reference to
Money

270
Rule of Three, Single and
Double

294, 326
Per-centage Profit and
Loss

362
Simple Interest

402
BOOKKEEPING, LESSONS IN:
Introduction

20
Principles of Double Entry 90
Principal Rules for finding

the Debtor and Creditor 154
Explanation of the Waste

Book, Cash Book, Bill
Book, Day Book, etc.-
Forms of Drafts, Promis-
sory Notes, and Foreign
Bills of Exchange

218
The Day Book, Journal, etc. 276
Property Accounts-Per-

sonal Accounts-Profit

and Loss Accounts 348
BOTANY, LESSONS IN:
XXV. Umbellifera, or

A piacex -- The Umbelli.

ferous or Parsley Tribe . 24
XXVI. Myrtaceae, or
Myrtle Tribe

25
XXVII. Cruciferæ, or Bras.

sicaces - The Cruciferous
(Cross-bearing) or Cab-
bage Tribe

PAGE

PAGE
XXVIII. Passifloraceæ, or Lower Oxides of Nitrogen

the Passion-flower Tribe 56 and Compounds of this
XXIX. Cucurbitacex, or

Gas with Hydrogen 289
the Cucumber Tribe 87 Carbon and its Oxides 335
XXX., XXXI. Solanacea, or Hydro-carbons, Coal-gas,
the Nightshade Tribe 88, 120 Flame, etc.

362
XXXII., XXXIII. Legu. Compounds of Carbon with
minose, or the Legumi.

Nitrogen and Sulphur-
2013 Tribe

121, 152

The Halogens-Chlorine 399
XXXIV. Boraginaceae, or
the Borage Tribe . 153

CLASSES, “ POPULAR
XXXV. Labiats, or the

EDUCATOR"
Lip-flower Tribe 177 COMPARATIVE ANATOMY :
XXXVI. Scrophulariacea,

Introduction - Terms em-
or the Foxglove Tribe 177

17

ployed in Classification
XXXVII., XXXVIII.,

Divisions of the Animal
XXXIX. Composite, or
Compound

Kingdom - Vertebrata-
flowered

Mollusca - Molluscoida
plants
178, 209, 217

Annulosa -- Annuloida
XL. Valerianacex, or Vale-

--Coelenterata-Protozoa 81
rian-worts

218

Subdivisions of the Animal
XLI. Caprifoliacere, or

Kingdom-Table of Sub-
Caprifoils

219

divisions of Classes-Pro.
XLII. Rubiacee, or Galia-

tozoa

113
cego

273
XLIII., xliv. Cinchona:

Cælenterata-Hydrozoa 145

Actinozon (Rayed Animals) 183
cece

273, 305

Echinodermata (Hedgehog.
XLV. Convolvulaced

317

skinned Animals). 215
XLVI. Polemoniaceæ, or

Helminthozoa

241
Phlox-worts

314

Annelida: Ringed Worms. 279
XLVII. Hydrophyllacea 3++

Rotatoria-Myriapoda

311
XLVIII. Gesneracere, or

Insecta
Gesner-worts.
XLIX. Bignonacere,

ILLUSTRATIONS :

or
Biguoniads

375 Sketch of Haddock, show.
L. Pedaliaceæ, or Pedaliads 375 ing its external form, and
LI. Acanthacese, or Acan-

also the arrangement of
thads

375
its internal

organs
LII. Selaginacem, or Sela-

Transverse section of
gids

375

Haddock Sketch of
LIII. Utriculariæ

376

Lobster Transverse
LIV. Plantaginaceæ, or

Section of Lobster

81
Ribworts

376 Amba-Shell of Polycys.
LV. Plumbaginaceæ, or

tina--Sectional Diagram
Leadworts

377 showing circulation in a
LVI. Primulacem,

Sponge-Group of Vorti.
Primeworts

401 cellæ-Noctiluca Miliaris 113
LVII. Ebenaceæ, or Ebe.

Eadendrinn Ramosum
nads

402 Hydrozoon encrusting a
LVIII. Aquifoliaceæ, or

Shell-Rhopalonema Ve-
Hollyworts

402 latum, the Veiled Club-
LIX. Oleaceæ, or Olive-

tentacled Medusa-Per-
worts

402 pendicular Section of Sea

Anemone Transverse
CHEMISTRY, LESSONS IN:

Section of Sea Anemone
Introduction - Attraction

-Pleurobrachia --Trans-
of Gravity-Force of Co.

verse Section of Pleuro-
hesion-Force of Affinity

1
brachia.

145
Elementary Substances-

Caryophyllia Smithii–Dry
Metalloids-Metals-Salt

Coral of Caryophyllia
--Acid - Alkali - Base-

Smithii – Diagrammatic
Chemical Nomenclature

Section, showing how
Measures of Length, Capa-

Red Coral is secreted-
city, and Weight-Ther-

Cestum Veneris-One of
mometry -To make a

the Polypes of Alcyonaria 184
Mercurial Thermometer 76 An Atoll-Fringing. Reefs
Chemical Apparatus

-Formation of Atoll 185
Tubing Blowpipes

Plates and Holes on Echi-
Flasks, etc.

100
nus Shell

Ambulacral
Oxygen

132 Plates--Echinus divided
Hydrogen
Diffusion of

to show Alimentary Canal
Gases, etc. etc.

168 -Spine--Jaws and Teeth
Wates

--Side View of Single
Peroxide of Hydrogen

235 Jaw- Tooth - Inside of
Nitrogen, Air, etc.

271 Purple-tipped Seo-Urchin 217

PAGE
Head and forepart of Tænia

Solium-Joint of same,
near tail ond-Head of
Larvå of Tenia Solium
-Distoma Conjunctum-
Liver Fluke Oxyuris
Vermicularis

241
Nereis-Transverse Section

of Eunice-Proboscis of
a nearly alliod species-
Leech Circulation in
Leech--Section of Earth-
worm - - Transverse Sec-
tion of Ditto - Circula-
tion in Lob-worm

279
Scolopendra Morsitans

Glomeris Julus-Antenna
and Eyes of Ditto-Under
side of double segment
of Ditto-Notomata Cen-
trura - Scheme showing

nature of Rotary Illusion 313
Priret Hawk Moth--Cater.

pillar : Pupa, Imago -
Common Wasp : Larva,
Pupa, Imago-Under side
of Head of Bed-bug -
Beetle with Dorsal In.
teguments removed to
show Viscera-Bee, Para-
sitical Insect, seen by
Transmitted Light, and
highly magnified to show
Tracheal System

337
DRAWING, LESSONS IN:
On Light and Shade in

Trees-Massing in the
Foliage, etc.

7, 39
Foregrounds-High Lights

--Setting Drawings, etc. 72
Treatment of Trees and
Foliage.

10+
Treatment of Reflections in
Water

135
The Human Figure 199,

263, 327, 392
ILLOSTRATIONS :
Figs. 100-110, Studies of

Trees and Foliage, and
Plants in Foreground 8, 9,

40, 41, 72, 73, 104
Fig. 111. Application of

Vegetable Form to De.
signing

105
Figs. 112-118. Treatment of

Reflections in Water 136, 137
Fig. 119. The Human Ske.
leton

200
Figs. 120-121. The Facial

Angle in Men and Animals 201
Figs. 122-127. Arms, Legs,
Hands, and Feet

264, 285
Fig. 128.

Head of Judas,
after Holbein.
Figs. 129-138. The Human
Face

329, 392, 393
ENGLISH, LESSONS IN:
Derivation: Prefixes .

13
Suffixes

38, 70, 110, 134, 165
Uncombined Suffixes. 206
Words that are both Pre-
fixes and Suffixes

226

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PAGE The Greek Element Greek

Stems 262, 358, 391, 409 Conversitions on English Grammar

134, 302, 334

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ESSAYS ON TIFE AND DUTY:
Charity

11 Patience

77 Unselfishness

131 Courage

199 Fidelity

259 Perseverance

327 Economy.

398

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22, 58

FRENCH, LESSONS IN :

XLVIII, Unipersonal Verbs 10 XLIX. Regimeu relating to some Verbs

10
L. The i ast Definite
LI. The Past Definite of

Irregular Verbs
LII., LIII. The Imperfect

Tense
LIV. The Past Anterior

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-
Bobully.

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
Europe

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
17-25 119

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 :
Atlantic Ocean

The Pulley
Pacific Ocean
233 GREEK, LESSONS IN:

Principle of Virtual Velo-
Europe

269
Introduction The Groek

cities - The Three SysNorway, Sweden, and Den

2 Alphabet.

tems of Palleys
mark

300
Vowels-Consonants-

Compound Pulleys
France

301
Punctuation, etc.

34

The Inclined Plane--The
Spain

301
General Remarks on the

Wedge-The Screw
Central Europe

. 324
Noun, the Adjective, and

Statical Forces--Friction 107
Projection of Map of Europe 356

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
--Proportional Scales · 16 The Second Declension 162, 195

Force, Water, Wind,
Methods of Construction-
The Third Declension 195, 229,

Steam
Station Point, etc.Pro-

258, 291, 322, 354, 390 Dynamics Definitions
blem I.
The Second Declension Con-

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
Problems VII.-XI.
359

Bodies

Atwoni's Ms. found at the end of the next

chine

Lessou.
GEOMETRY, LESSONS IN

Laws of Falling Bodies

Projectiles-Collision or
The Circle and its Proper- HISTORIC SKETCHES:

Impact .
ties
Problems in Practical Geo- How England and Scotland

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
Conic Sections-The Ellipse 251
The Dagger Scene in the

Leading Nutes
The Oval--Parabola . 284, 307

House of Commons 181

Meutal Effect-Consonance
The Hyperbola, Cycloid,
Origin of the United States 219

of Notes, etc..
Spiral, etc.

308
Charles Edward Stuart and

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
XLV. Peculiar Idioms 26 The Right Noble and Va-
XLVI. Verbs governing the

lorous Sir Walter Raleigh 341

OUR HOLIDAY :
Genitive

27 Admiral Byng on the 14th Gymnastics.
XLVII. Adjectives re-

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
LI. Verbs requiring the

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
Accusatives, and those

Pressure

Business Handwriting governing the Accusative

Levels--Springs and Ar

Legal Handwriting
with the Dative

118
tesian Wells.

396

Germau Handwriting. LIII. Prepositions

Greek Handwriting
quiring the Genitive 119

LATIN, LESSONS IN:
LIV.

READING AND ELOCUTIOS
Prepositions
Adverbs

18
quiring the Dative 155 Personal Pronouns

19 Analysis of the Voice : LV. Prepositions requiring Possessive or Adjective

Exercises on Inflections the Accusative

155
Pron.us

Just Stress
LVI. Prepositions
Demonstrative Pronouns 54 Expressive Tones, Rules

On
quiring the Dative

Relative and Interrogative
Accusative

156
Pronouns

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
LXI. Regimen of Nume.
Paradigm of the Verb Sum

The Froz.
rals, etc.
216 ---Compounds of Sum 210

Euglish Snakes,
LXII. - LXX. Various
The Latin Verb: its several

.

.

re.

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KEY TO EXERCISES IN LES

SONS IN FRENCH :
Exs. 1-3 111 Exs.29-35 . 233

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

.

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The Swallo:
Idioms 282, 283, 314, 315, 340, Terminations 250, 274

The Spider
371, 107 Regular Verbs --The First
LXXI. Passive Verbs in

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

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