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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
former is equal to the tri. heptagon A B C D E F G.
angle C H A 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 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

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CASSELL PETTER & GALPIN, BELLE SAUVAGE WORKS, LONDON, L.O.

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PAGE
Head and forepart of Tania

Solium-Joint of same,
near tail end-Head of
Larva of Tonia Solim
-Distoma Conjunctum-
Liver Fluke - Oxyuris
Vermicularis.

241
Nereis-Transverse Section

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

2779
Scolopendra Morsitans

Glomeris Julus-Antenna
and Eyes of Ditto-Under
side of double seginent
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

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273, 305

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337

PAGE

PAGE
XXVIII. Passifloraces, or

Lower Oxides of Nitrogen
the Passion-flower Tribe 56 and Compounds of this
XXIX. Cucurbitace, or

Gas with Hydrogen 289
the Cucumber Tribe 87 Carbon and its Oxides 335
XXX., XXXI. Solanaceæ, 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-
nou3 Tribe

121, 152

Tbe Halogens-Chlorine 399
XXXIV. Boraginaceæ, or
the Borage Tribe

153

CLASSES, " POPULAR

the
XXXV. Labiate, or

EDUCATOR"

411
Lip-flower Tribe

177

COMPARATIVE ANATOMY :
XXXVI. Scrophulariacem,

Introduction – Terms em-
or the Foxglove Tribe 177

17
ployed in Classification
XXXVIII.,
XXXVII.,

Divisions of the Animal
XXXIX. Composite, or

Kingdom - Vertebrata-
flowered
Compound

Mollusca Molluscoida
plants
178, 209, 217

Annulosa --- Annuloida
XL. Valerianaceae, or Vale-

--Celenterata-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

133
cego

273

Coelenterata-Hydrozor 145
XLIII., XLIV. Cinchona:

Actinozoa (Rayed Animals) 183
cewe

Echinodermata (Hedgehog.
XLV. Convolvulacew 311

skinned Animals)

215
XLVI. Polemouiaoeve, or

Helminthozoa

2+1
Phlox-worts .

314
XLVII. Hydrophyllaceae

Annelida: Ringed Worms. 279
316

Rotatoria-Myriapoda 311
XLVIII. Gesneracere, or

Insecta

337
Gesuer-worts.

315

ILLUSTRATIONS :
XLIX. Bignonaceæ, or
Bignoniads

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

also the arrangement of
thads

375
its internal

organs
LII. Selaginacex, or Sela-

Transverse section of
gids

375

Haddock Sketch of
LIIL, Utriculariæ

376

Lobster Transverse
LIV. Plantaginacea, or

Section of Lobster

81
Ribworts

376

Ameba-Shell of Polycys.
LV. Plumbaginaceæ, or

ting-Sectional Diagram
Leadworts

377 showing circulation in a
LVI. Primulacem, or

Sponge-Group of Vorti.
Primeworts

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

Eadendrinn Ramosum
nads

432 Hydrozoon encrusting a
LVIII. Aquifoliacex, 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 36 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

203, 235

--Side View of Single
Peroxide of Hydrogen

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

271 Purple-tipped Sco-Uxchin 217

7, 39

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ARITHMETIC, LESSONS IN:
Reduction

7
Compound Addition

37
Compound Subtraction 78
Compound Multiplication . 78
Compound Division

101
Fractions in connection

with Compound Quan.
tities

112
Practice

198
Decimals in connection

with Compound Quanti-
ties-Reduction of Deci-
mals

23+
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. Umbelliferre, or

Apiacea - The Umbelli.

ferous or Parsley Tribe. 21
XXVI. Myrtaceæ, or the
Myrtle Tribe

25
XXVII. Cruciferæ, or Bras-

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

56

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DRAWING, LESSONS IN:
On Light and Shade in

Trees-Massing in the

Foliage, etc.
Foregrounds-High Lights

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

10+
Treatment of Reflections in
Water

135
The Human Figure

263, 327, 392
ILLUSTRATIONS :
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, 265
Fig. 123. Head of Judas,
after Holbein.

323
Figs. 129-138. The Human
Faco

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

Stenis 202, 358, 391, 409 Convers itions on English Grainmar

134, 302, 331

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17–20.

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ESSAYS ON JIFE AND DUTY:
Charity

11 Patience

77 Unseliishness

131 Courage

193 Fidelity Perseverance

327 Economy.

398

• 259

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FRENCH, LESSONS IN:

XLVIII. Unipersonal Verbs 10
XLIX. Regimeu relating to
Bome Verbs

10
L. Tbe i ast Definito
LI. The Past Defipite of

Irregular Verbs
LII., LIII. The Imperfect
Tense

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-
Bomully.

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

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PAGE

PAGE
Europe

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
Table of Latitudes and

10
35-37 247 in

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,
Map of the World

2120 180 53-59, 408 MECHANICS :
Atlantic Ocean
232

The Pulley

12 Pacific Ocean 233 GREEK, LESSONS IN:

Principle of Virtual Velo-
Europe

209
Introduction The Groek

cities - The Three Sys-
Norway, Sweden, and Den-
mark

tems of Pulleys 300

Alphabet.

Vowels-Consonants- Compound Pulleys
France

301
Spain

301
Punctuation, etc.

34 The Inclined Plane--The

General Remarks on the
Ceutral Europe

Wedge-The Screw
324
Projection of Map of Europe 356

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!
-Proportional Scales 161 The Second Declension 162, 195 Force, Water, Wind,
Methods of Construction-
The Third Declension 195, 229,

Steam
Station Point, etc.-Pro-

258, 291, 322, 351, 390

Dynamics Definitions
blom I.
225 The Second Declension Con-

The Three Laws
Orthographic and Isome-

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.
GEOMETRY, LESSONS IN :

Laws of Falling Bodies-
The Circle and its Proper-

Projectiles-Collision or
HISTORIC SKETCHES:
ties

Impact .
Problems in Practical Geo- How England and Scotland

Impact-Centrifugal Force
metry on the Circle 49, 92, 123
became one

-The Pendulum-Centre
Regular Polygons

124
How Ireland became part

of Oscillation.
Problems in Practical Geo-

of Great Britain

85, 125

MUSIC, LESSONS IN:
metry on Construction 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 Notes
The Oval--Parabola.

284, 307
House of Commons 18:

Mental 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 253

--The Glass Harmonicon
The Massacre of Glencoe 285
GERMAN, LESSONS IN :

-German Concertina
Wilkes and Liberty

317
XLV. Peculiar Idioms

Relation of Notes, etc. 3), 26

The Right Noble and Va-
XLVI. Verbs governing the

lorous Sir Walter Raleigb S41

OUR HOLIDAY :
Genitive

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

of March, 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 Trequiring the Dative

peze XLIX. Verbs requiring the HYDROSTATICS:

The Hanging Stirraps
Dative
91

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

91
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, 6, 1.
Accusatives, and those

Centra of Pressure

Business Handwriting governing the Accusative

Levels--Springs and Ar- Legal Handwriting
with the Dative

118
tesian Wells.

396

Germau Handwriting.
LIII. Prepositions re.

Greek Handwriting
quiring the Genitive 119

LATIN, LESSONS IN :
LIV. Prepositions re.

READING AND ELOCUTIUS
Adverbs

18 quiring the Dative 155 Personal Pronouns

19 Analysis of the Voice :
LV. Prepositions requiring

Possessive
or

Exercises on Inflectious

Adjective
the Accusative

155
Pron - lins

54 Just Stress
LVI. Prepositions re-

Demonstrative Pronouns 5+ Expressive Tones, Rules
quiring the Dative or Relative and Interrogative
Accusative

156
Provouns

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
LXI. Regimen of Nume.
Paradigin of the Verb Sum

The Froz
rals, etc.

216

-Coinpouads of Sum 210
LXII.

12
LXX. Various
The Latin Verb: its several

English Snakes

The Swalloir
Idioms 282, 283, 314, 315, 310, Terminations 350, 27+

The Spider
371, 407 Regular Verbs --The First
LXXI. Passive Verbs in

Conjugation

WHITWORTH SCRO: SRthe Indicative 407 Ou Parsing

382

SHIPS, THE

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