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Join BA and Ac. Then, because ABD is a triangle whose the rectangle contained by the whole line thus produced, and the
base is bisected in E, and E A is drawn, therefore the squares of part produced, may be equal to a given square.
A B and a d together are equal to twice the squares of B E and E A
together (II. Prop. A).

For the same reason the squares of B C and cd are together
equal to twice the equares of B E and Eo together.

Therefore, adding equals to equals, the squares of A B, BD, CD,
and D A, are together equal to four times the square of BE
together with twice the squares of E A and E c together.

But the square of BD is equal to four times the square of B E
(II. 4, Cor. 2).

Therefore the squares of A B, BC, CD, and D A are together equal
to the square of BD together with twice the squares of Ea and

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Again, because eaC is a triangle, of which the base Ac is
bisected in E (Hyp.), therefore the squares of ea and ec are
together equal to twice the squares of cr and Ef together Let A B be the given straight line, and CDer the given square.
(II. Prop. A),

It is required to produce a B so that the rectangle contained by
And the doubles of these equals are equal; therefore the the whole line thus produced, and the part of it produced may be
doubles of the squares of Ea and E c are together equal to four equal to the given square CDEP:
times the squares of C P and Ep together.

From the point B in the straight line A B draw BH at right
Therefore the squares of a B, BC, CD, and D 4 are together equal angles to A B (I. 11) and equal to CD (I. 3) one of the sides of the
to the square of BD together with four times the squares of CF given square CD ef.

Bisect A B in the point G and join G 1.
But four times the square of cP is equal to the square of A O With centre G and distance GH describe the circle # MN.
(II. 4, Cor. 2).

Produce a B to meet the circumference in 1. Then the rect-
Therefore the squares of A B, BC, CD, and DA are together angle A L, L B is equal to the square C D E F.
equal to the squares of B D and á c together with four times the

Because Bu is equal to CD (Cons.), therefore the square of
square of E F. Q.E.D.

B H is equal to the square of on, that is the square CDEF.
Prop. D. Problem. To divide a given straight line into two Again, because G is the centre of the circle H MN, therefore G H
parts, so that the rectangle contained by its segments shall be is equal to GL (I. Def. 15) and therefore the square of gh is
equal a given square, not greater than the square of half the equal to the square of Gl.
given straight line.

Again, because a Bo is a right angled triangle having the right
angle u BQ (Cons.), therefore the square of an is equal to the
squares of G B and B i together (I. 47).

But the square of G H has been above shown to be equal to the square of G i. Therefore (I. Ax. 1) the square of a L is equal to the squares of G B and Bh together.

But the square of G L is equal to the square of a B together with the rectangle A L, L B (II. 6).

Therefore (I. Ax. 1) the squares of a B and RI are together
equal to the square of G B together with the rectangle A L, L B.

From each of these equals take away the common square of a B,
K

then (I. Ax. 3) the rectangle A L, L B is equal to the square of
Let A B be the given straight line and oder the given square But the square of Bu is equal to the square CD E F(Cons.). There-
not greater than the square of half the given line A B. It is fore the rectangle AL, L B is equal to the square C D E F (I. Ax. 1).
required to divide A B into two parts such that the rectangle Q.E.F.*
contained by them may be equal to the given square

Bisect AB in G (I. 10). From a draw ou at right angles to
AR (I. 11), and equal to D E (I. 3) a side of the given square CD

ANSWERS TO CORRESPONDENTS.

JOHN BURROUGH: The POPULAR EDUCATOR has been published in the
Produce id to k making # k equal to GB (1.3).

United States. We cannot promise to act upon your suggestion.
With centre I and distance * K describe the circle MK L, GEORGE YOUNG will find our Lessons in Italian answer his purpose.
cutting A B in m. The straight line A B is divided in m, so that UN FRANÇAIS: It is doubtful whether we shall be able to find room for
the rectangle A M, M B is equal to the square CD EF.

any lessons in Spanish, but we should be glad if it were possible.

G. LAKEMAN is referred to the two volumes of the “Historical Edu-
Join 1 m. Then because A B is divided into two equal parts in c, cator” for articles on ancient history.
and two unequal parts in M, therefore the square of a n is equal Arithmetic have

already been given in these pages. If we can find room for

further lessons, we will; but if not, we must refer to “Cassell's Arithmetic"
to the square of G M together with the rectangle A M, MB.

for any additional information that may be desired.
But G B is equal to x K, by construction, and n k is equal to BETH: Your questions are unsuitable for discussion in the " POPULAR

EDUCATOR.” We have no wish to engage in theological controversy.
Therefore (I. Ax. 1) a b is equal to 1 m, and the square of a B

DesunT CRTERA : The promise will be fulfilled at the commencement of
to the square of us.

SIMPLICITAS (Wemyss, Fireshire) has solved the first thirty-two of the Therefore the square of u m is equal to the square of GM second Centenary of Algebraical Problems; D. HORNBY (Drittiold) the first together with the rectangle A M, MB.

filty-five, with the exception of Nos. 7, 12, 35, 37, 39, 43, and 51; GEORGE

WILD (Dalton-on-Tees) the whole of the second portion from No. 23 to 70;
But the square of 1 M is equal to the squares of 10 and GM

and Q'Q. (Fenchurch-street) the first thirty-two, except Nos. 16, 23, 26,
together (I. 47).

28, 30, and 31, besides Nos, 40, 41, and 51 of the second portion.
Therefore (I. Ax. 1) the squares of ug and GM are together

D. HORNBY (Driffield): The following is George Wild's solution of Pro-
equal to the square of a m together with the rectangle
AM, MB.

5 9 Days Days

: : 10 : 18
From each of these equals take the square of GM, then the

9
rectangle A M, M B is equal to the square of 1 G (I. Ax. 3). But the number of days in which A can do the whole. Therefore he would
the

square of u g is equal to the square CD EF (Cons.).
Therefore the rectangle A m, n 1 is equal to the square oder require 8 days to do the remaining of the work alone,
(I. Ax. 1). Q.E.F.

Prop. E. Problem. To produce a given straight line, so that • The above five exercises were solved by J. H. EASTWOOD (Middleton).

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Let z be the number of days in which B could do this part of the work
alone.

Now ready, price 93. strongly bound,
1 1 1
Then

CASSELL'S GERMAN PRONOUNCING DICTIONARY. 8

In Two Parts:-1, German and English; 2. English and German. In Whence x = 4.

one large handsome Octavo Volume. The German-English Division, price 9 Days Dags

53. in paper covers, or 53. 64. neat cloth; the English-German Division, Again :: 44 : 105

3s. 6d. paper covers, or strongly bound in cloth, 4s. 9 9

CASSELL'S LESSONS IN GERMAX. Parts I. and I1.-Price 28. each in the time in which B could do the whole alone.

paper covers, or 2s.6d. in cloth. Two Parts bound together, price 4s. Gill
1 5 8
+

KT A Key to the above Lessons is now ready.
18 54 54
the fraction of the work done by A and B together in a single day.

54
Therefore

= 6 days 8

TO THE
is tbe time they would both require to fioish It.

Peacock's are the most profound works on Algebra, but the foorteenth
edition of Lund's Wood is perhaps the most useful for ordinary students. READERS OF THE POPULAR EDUCATOR
Goodwin's is an excellent work for those who have not the time or ability
requisite for going very deeply and extensively into the subjects. We con-
sider it better to go through this before entering upon the study of separate Our present number completes the Fifth Volume of the

POPULAR EDUCATOR. In the course of the five volumes, it has
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few, and to afford those who have been denied the advanNow ready, price 98. 6d. strongly bound.

tages of early training, every facility for supplying the defici

ency by the most valuable of all education-Self-instruction, CASSELL'S FRENCH AND ENGLISH DICTIONARY:

With this object in view, we have carried our readers over 8 Composed from the French Dictionaries of the French Academy, Bescherelle, wide range of important subjects, on all of which we have etc.; and from the Technological and Scientific Dictionaries of both Lan: given lessons of so popular and instructive a character, as to guuges. By Professor DE LOLME and HENRY BRIDGEMAN, Esq. require nothing more than industrious application on the part

The following are the distinctive features which render this work superior of persons of ordinary ability, to ensure the acquisition of
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Inost modern words and phrases-including those of science, art, manufaca
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are indispensable to a knowledge of language, but yet are rarely, if ever, to after week has it been our gratification to receive and acknow-
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CASSELL'S CLASSICAL LIBRARY.-The First Volume of this work, price ciation and Reading of the French Language than has yet
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from our numerous readers.

i

fessor, E. A. ANDREWS and 8. STODDARD.

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READING furnishes the mind only with materials of knowledge; it is thinking makes what we read ours. We are of the

ruminating kind, and it is not enough to cram ourselves with a great load of collections ; unless we chew them over
and over again, they will not give us strength and nourishment. There are, indeed, in some writers, visible instances
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if their readers would observe and imitate them : all the rest at best are but particulars fit to be turned into knowledge;
but that can only be done by our own meditation, and examining the reach, force, and coherence of what is said; and
then, as far as we apprehend and see the connection of ideas, so far is it ours; without that, it is but so much loose
matter floating in our brain.---Locke.

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JOHN CASSELL, LA BELLE SAUVAGE YARD, LUDGATE HILL.

MECOCLV.

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THE PRESENT Series of the Popular EDUCATOR is now completed. It was our intention to have closed it with a Supplement to Volume V., consisting of some 200 pages. But the difficulty of completing so many subjects in that limited space, together with urgent appeals from numerous Readers and Correspondents, induced us to postpone its termination till the completion of another Volume, uniformn with the five preceding. This will explain the continuity of the paging from the commencement of the Fifth Volume to the close of the work. The appearance of uniformity will be greatly preserved by having the entire Series bound in three double volumes.

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533

551

XXVIII. La vierge aus ruines. Sections I., II., III., IV.,

V., with exercises, etc.
XXIX. L'enfance du poète, with exercises, etc.

L'emploi du temps, with exercises, etc.
XXX. Les hirondelles, with exercises, etc. Le vieux

roi et la jeune fille ; Sections I., II., with ex

ercises, etc. XXXI. Sections III., IV., with exercises etc. La

charité; Section I., with exercises, etc. XXXII. Section II., with exercises. Eudoxie ; Sections

I. II., with exercises, etc.......
XXXIII. Section III., with exercises, etc.
XXXIV. Section IV., with exercises, etc. Le vaisseau

en péril, with exercises, etc.

533

599

615 632

647

587

LESSONS IN ALGEBRA.
XXI. Powers of Roots ; Evolution ; General Rule for

Evolution; Reduction of Radical Quantities 416
XXII. Addition of Radical Quantities; Subtraction of

Radical Quantities; Multiplication of Radical
Quantities...

454
XXIII. Division of Radical Quantities; Involution of

Radical Quantities; Evolution of Radical
Quantities....

453
XXIV. Reduction of Equations by Involution ; Reduce

tion of Equations by Evolution; Adfected
Quadratic Equations..

513
XXV. Method for Completing the Square; Second

Method for Completing the Square ; Demon-
stration; Other Methods of Completing the
Square, General Rule

527
XXVI. Problems in Adfected Quadratic Equations ;
Ratio and Proportion...

566
XXVII, Compound Ratio; Proportion; Arithmetical
Proportion and Progression..

591 XXVIII. Geometrical Proportion and Progression. 624 XXIX. Problems in Geometrical Progression; Evolu. tion of Compound Quantities

656
XXX. Application of Algebra to Geometry.

719
LESSONS IN ARITHMETIC.
XXX. Decimal Fractions; Decimal Table; Exercises;
Addition of Decimal Fractions

464
XXXI. Subtraction of Decimal Fractions : Multiplica-

tion of Decimals; Contractions in Multiplica-
tion of Decimals; Division of Decimal Frac-

tions; Contractions in Division of Decimals 486 XXXII. Periodical, or Circulating Decimal; Reduction

of Circulating Decimals; Addition of Circu-
lating Decimals, etc. etc.

510 XXXIII. Percentage; Commission; Brokerage; and Stocks

544 XXXIV. Interest; Compound Interest.

674 XXXV. Discount; Bank Discount; Insurance ;

606 XXXVI. Life Insurance ; Profit and Loss

640 XXXVII. Analysis

673
BIOGRAPHY.
XV. William Pitt

557
XVI. James Thomson

589
XVII. David Hume
XVIII. Samuel Johnson.

637
XIX. William Robertson

654 XX. Edward Gibbon

670 XXI. Oliver Goldsmith....

684 XXII. William Cowper

699 XXIII. Felicia Dorothea Hemans........

714 XXIV. Sarah Margaret Fuller

730 XXV. Laura Bridgman.......:

747 XXVI. Catharine Maria Sedgwick

762 XXVII. Mrs. Trollope XXVIII. Mary Russell Mitford

795 XXIX. Lydia Huntley Sigourney.

811
LESSONS IN FRENCH PRONUNCIATION.
I. Alphabet ; Accents ; Cedilla; Dieresis; Hy-
phen; Apostrophe; Euphonic t; Parenthesis;

412
II., III. Name and Sound of the Vowels
IV. Consonants ....

462
V. Compound Vowels

497 VI. Diphthongs; Nasal Vowel Sounds; Nasal Diphthongal Sounds

524 VII. Liquids; General Rule for Pronouncing and Reading French

559
FRENCH READINGS.
XXIV. La Marguerite et l'épi de blé. Section II.,

with exercises, etc. Le chien du Louvre,
Section I., with exercises ....

455 XXVII. Section II., with exercises

519

.............

623

LESSONS IN GEOLOGY.
LVII. Classification of Rocks; Permian Formation 555
LVIII.

» Lithological Character of the
Coal Rocks
LIX.
on the Coal Measures

617 LX.

651 LXI. Old Red Sandstone, or the Devonian Forma

tiun; Lithological Character of the Devonian;
Organic Remains of the Devonian; Some
Geological Phenomena of the Devonian
Period

716
LESSONS IN GREEK.
XLVII. Formation of Words ; Verbs; Compounds ;
Recapitulation...

421 XLVIII. Invariable Words

437 XLIX. Syntax; Preliminary Explanations; Subject; Predicate; Agreement; Government

450 L. The Parts of a Simple Sentence considered separately; their Agreement.

465 LI. Voices of the Verb...

479 LII. Tenses of the Verb..

499 LIII. Moods

515 LIV. Enlargement of Simple Sentences.

546 LV. Attributives; the Demonstrative Pronouns; the Article

576 LVI.. Attributive Words with Substantive Import;

Enlargement of the Predicate.......... 603 LVII.

the Single Accusative .. 641 LVIII. The Predicate with a Double Accusative. 674 LIX. Import and Use of the Dative .

701 LX. Import and Use of the Genitive....

736 LXI. Syntax of the Prepositions

76+ LXII. Interrogative and Imperative Sentences 1799

LESSONS IN ITALIAN.
XXXIV. Irregular Verbs ending in ére long (continued) 419
XXXV., XXXVI.

449, 467 XXXVII. Irregular Verbs ending in 'ere short;"Verbs ending in cere

431

501 XXXIX. Verbs ending in dere”.

530 XL. Verbs ending in urre, contracted from cere; Verbs ending in gere

561 XLI. Vebs ending in ggere ; Verbs ending in gliere ; Verbs ending in guere.......

593 XLII. Verbs ending in lore; Verbs ending in mere; Verbs ending in pere ...

627 XLIII. Verbs ending in orre; Verbs ending in rere;

Verbs ending in tere; Verbs ending in arre;
Verbs ending in vere....

659 XLIV. Table of Terminations of the Verbs in isco 676 XLV., XLVI.

688, 703 xLvii.' Impersonal Verbs; Participle : Adrerb

722 XLVIII. Preposition ; Conjunction; Interjection

739 XLIX. Syntax : Of Articles

762 L.

of Nouns, Adjectives, Comparatives and Superlatives, Numerals..

766 LI.

Cor Personal, Possessive, Demonstralire, Relative, and Interrogative Pronouns.. 786 LU.

Of Indefinite Pronouns; of Verbs 801 LIII. Qi Participles, Adverbs, and Conjunctions... 813

778 XXXVII.

............

Asterisks ...

432, 447

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