« AnteriorContinuar »
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
CASSELL PETTER & GALPIN, BELLE SAUVAGE WORKS, LONDON, L.O.
INDEX TO CONTENTS.
Solium-Joint of same,
of Eunice-Proboscis of
nature of Rotary Illusion 313
pillar : Pupa, Imago -
Lower Oxides of Nitrogen
Gas with Hydrogen 289
Nitrogen and Sulphur-
Tbe Halogens-Chlorine 399
CLASSES, " POPULAR
COMPARATIVE ANATOMY :
Introduction – Terms em-
Divisions of the Animal
Kingdom - Vertebrata-
Annulosa --- Annuloida
Subdivisions of the Animal
Kingdom-Table of Sub-
divisions of Classes-Pro-
Actinozoa (Rayed Animals) 183
Annelida: Ringed Worms. 279
375 Sketch of Haddock, show.
also the arrangement of
Transverse section of
Haddock Sketch of
Section of Lobster
Ameba-Shell of Polycys.
377 showing circulation in a
Sponge-Group of Vorti.
401 cellæ-Noctiluca Miliaris 113
432 Hydrozoon encrusting a
402 latum, the Veiled Club.
402 pendicular Section of Sea
Section of Sea Anemone
- Pleurobrachia --Trans.
verse Section of Pleuro-
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
--Side View of Single
Jaw- Tooth -- Inside of
271 Purple-tipped Sco-Uxchin 217
ARITHMETIC, LESSONS IN:
with Compound Quan.
with Compound Quanti-
tion with reference to
the Debtor and Creditor 154
Book, Cash Book, Bill
and Loss Accounts 348
Apiacea - The Umbelli.
ferous or Parsley Tribe. 21
sicaceae - The Cruciferous
DRAWING, LESSONS IN:
Trees-Massing in the
-Setting Drawings, etc. 72
263, 327, 392
Trees and Foliage, and
40, 41, 72, 73, 104
Vegetable Form to De.
Reflections in Water 136, 137
Angle in Men and Animals 201
329, 392, 393
PAGE The Greek Element-Greek
Stenis 202, 358, 391, 409 Convers itions on English Grainmar
134, 302, 331
ESSAYS ON JIFE AND DUTY:
193 Fidelity Perseverance
FRENCH, LESSONS IN:
XLVIII. Unipersonal Verbs 10
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
283, 299, 323 KEY TO EXERCISES IN LES. Regular Verbs—The Second Construction of Projection
SONS IN GERMAN:
406 of Map of Europe. 355, 388
Exs. 49 27 Exs. 27-33 222 The Key to the Exercises gisen
Lesson in Latin will be Longitudes of Places
11-16 95 38-41 . 233 found at the end of the text in Europe
119 42, 43
Lesson or the next Lesson but MAP3:
21-23 . 156 41-52 372
2120 180 53-59, 408 MECHANICS :
12 Pacific Ocean 233 GREEK, LESSONS IN:
Principle of Virtual Velo-
cities - The Three Sys-
tems of Pulleys 300
Vowels-Consonants- Compound Pulleys
34 The Inclined Plane--The
General Remarks on the
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:
98 chinery Introduction - Definitions The First Declension.
Priino Movers Avima!
258, 291, 322, 351, 390
The Three Laws
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
Laws of Falling Bodies-
of Great Britain
MUSIC, LESSONS IN:
Meutal Effect of Notes
31 Regular Polygons 148, 191, 211
sessed of India
Character and Effect of
Mental Effect Consonance
of Notes, etc..
Measurement of Intervals the Rebellion of 1745 253
--The Glass Harmonicon
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.
The Hanging Bar or Trequiring the Dative
peze XLIX. Verbs requiring the HYDROSTATICS:
The Hanging Stirraps
The Hanging Rings
Principle of Equality of
Laws of Croquet
Official Handwriting .33, 6, 1.
Centra of Pressure
Business Handwriting governing the Accusative
Levels--Springs and Ar- Legal Handwriting
LATIN, LESSONS IN :
READING AND ELOCUTIUS
18 quiring the Dative 155 Personal Pronouns
19 Analysis of the Voice :
Exercises on Inflectious
54 Just Stress
Demonstrative Pronouns 5+ Expressive Tones, Rules
83 Appropriate Modulatiou LVII. Examples illustra
Promiscuous Exercises ting the various uses of
214, 250, 278, 26, 314,7 Prepositions 178 The Ninerals
RECREATIVE NATURAL HIS LVIII., LIX., LX. Pecu
TORY: liar Idioms 18), 222, 216 The Latin Verb.
-Coinpouads of Sum 210
WHITWORTH SCRO: SRthe Indicative 407 Ou Parsing