Fig. 33.

B

D

Join BC. Because A B is parallel to CD, and BC meets them, the angle A ABC is equal (I. 29) to the alternate angle BCD. Because AB is equal to CD, and BC common to the two triangles ABC and DCB; the two sides A B and B c, are equal to the two DC and CB, each to each; and the angle ABC was proved to be equal to the angle BCD; therefore the base A c is equal (I. 4) to the base B D, and the triangle ABC to the triangle B CD. Also the remaining angles of the one are equal to the remaining angles of the other, each to each; viz., those to which the equal sides are opposite. Therefore the angle AC B is equal to the angle C B D. Again, because the straight line nc meets the two straight lines A C and B D, and makes the alternate angles AC B and C B D equal to one another, therefore a c is (I. 27) parallel to BD; and AC was proved to be equal to B D. Therefore, the straight lines which, etc. Q. E. D.

The enunciation of this proposition is more clearly expressed thus: "The straight lines which, without crossing each other, join the extremities of two equal and parallel straight lines, are themselves equal and parallel."

Corollary. A quadrilateral which has two of its opposite sides equal and parallel, is a parallelogram.-See the following definition:

DEFINITION XXXVI.

A parallelogram is a four-sided figure of which the opposite sides are parallel; and the diagonal is the straight line joining the vertices of two opposite angles.

N.B. In naming a parallelogram by letters, it is customary and it is sufficient, to take the two letters only which are at the vertices of any two of its opposite angles.

PROPOSITION XXXIV. THEOREM.

The opposite sides and angles of a parallelogram are equal to one another, and the diagonal bisects it, that is, divides it into two equal parts.

Let A D, fig. 34, be a parallelogram, of which D c is a diagonal. The opposite sides and angles of the parallelogram are equal to one another; and the diagonal B c bisects it.

A

B

D

EXERCISE I. TO PROPOSITION XXXIV.

If the opposite sides, or the opposite angles, of a quadrilateral figure be equal, it is a parallelogram.

In fig. 34, let A B D C be a quadrilateral figure; and first, let the opposite sides be equal. Then it is a parallelogram. Join CB. Because, in the two triangles ABC and BCD, the side A B is equal to the side c D, the side Bc is common to both, and the side a c is equal to BD; therefore (I. 8) the angle ABC is equal to the angle BCD; and they are alternate angles; wherefore A B is parallel to c D. In the same manner it may be shown that A c is parallel to BD; therefore (Def. 36) the quadrilateral figure A B DC is a parallelogram. Second, let the opposite angles be equal. Then the figure ABDC is a parallelogram.

Because all the interior angles of the figure ABD C are equal to four right angles (I. 32, Cor. 8), and that the two angles BAC and ACD are equal to the two angles A B D and BDC; therefore the two angles B A C and A CD are equal to two right angles, and (I. 28) A B is parallel to c D. In the same manner it may be shown that Ac is parallel to BD. Therefore the figure (Def. 36) ABDC is a parallelogram. Wherefore, if the opposite sides, or the opposite angles, etc. Q. E. D.

EXERCISE II. TO PROPOSITION XXXIV.

The diagonals of a parallelogram bisect each other; and if the diagonals of a quadrilateral bisect each other, it is a parallelogram.

In fig., let ABDC be a parallelogram, and 1ct AD and B c be its diagonals. Then, first, the diagonals A D and B c bisect each other.

From the 12th Axiom, it is plain that the diagonals AD and BC intersect each other. Let E be the point of their intersection. Because the straight line в c meets the parallels

Fig. 1.

Because A B is parallel to CD, and B c meets them, the angle ABC is equal (I. 29) to the alternate angle B C D. Because A C is parallel to B D, and BC meets them, Fig. 31. the angle ACB is equal (I. 29) to the alternate angle CB D. Because in the two triangles A B C and CBD, the two angies ABC and BCA, in the one, are equal to the two angles B CD and CBD in the other, each to each; and one side BC, adjacent to these equal angles, is common to the two triangles; therefore their other sides are equal, each to each, and the third angle of the one is equal to the third angle of the other (I. 26); viz., the side A B to the side CD, the side A c to the side BD, and the angle B A c to the angle BDC. Because the angle A B C is equal to the angle BCD, and the angle CBD to the angle ACB; therefore the whole angle A B D is equal (Ax. 2) to the whole angle ACD; and the angle BAC has been proved to be equal to the angle BDC;AD and BC are bisected in E. therefore the opposite sides and angles of a parallelogram are equal to one another.

Also the diagonal B c bisects the parallelogram A D. Because in the two triangles A B C and D C B, the side A B is equal to the side CD, and BC common, the two sides A B and B c are equal to the two sides DC and C B, each to each; and the angle ABC has been proved to be equal to the angle BCD; therefore the triangle ABC is equal (I. 4) to the triangle B CD. Wherefore the diagonal B C divides the parallelogram A D into two equal parts. Q. E. D.

Corollary 1.-If a parallelogram have one angle a right angle, all its angles are right angles.

Corollary 2.-Parallelograms having one angle equal in each, are equiangular.

Corollary 3.-Parallelograms which have one angle and two adjacent sides equal in each, are equal in all respects.

Corollary 4.-The adjacent angles of a parallelogram are supplements of each other,

E

Σ

AB and CD, the angle ABC is equal to the angle nCD (I. 29). For a similar reason, the angle BAD is equal to the angle ADC; therefore, in the two triangles B AE and CDE, the two angles ABE and BAE of the one, are equal to the two angles E CD and CDE of the other; but the side A B is equal to the side CD (I. 34); therefore the triangle ABE is equal to the triangle CRD (I. 26), and the remaining sides and angles of the one are equal to the remaining sides and angles of the other; wherefore A E is equal to ED, and B E to EC; and the diagonals

Second, let the diagonals AD and BC bisect each other in E; then the figure ABD C is a parallelogram.

Because in the two triangles BAE and CED, the two sides BE and EA are equal to the two sides c E and ED each to each (Hyp.), and the angle A E B is equal to the angle cED (I. 15); therefore the base A B is equal to CD, and the remaining angles of the one to the remaining angles of the other, viz. the angle ABC to the angle B C D, and the angle B A D to the angle ADC (I. 4). But the angles ABC and BCD are alternate angles; therefore AB is parallel to CD (I. 27); for a similar reason AC is parallel to BD. Wherefore, the diagonals of a parallelogram, Q. E. D.†

etc.

field; E. J. BREMNER (Carlisle); J. II. EASTWOOD (Middleton); and "This exercise was solved by T. Bocock (Great Warley); D. H. (Drif QUINTIN PRINGLE (Glasgow); who also solved the three latter exercises on the 32nd Proposition of the 1st Book.

This exercise was solved by those named in the preceding note, and by H. I. PUGH (Longsight); and T. WATKINS (Pembroke Dock).

EXERCISE III. TO PROPOSITION XXXIV.

The diagonals of rectangular parallelograms are equal; and in oblique angled parallelograms, those which join the vertices of the acute angles are greater than those which join the obtuse.

equal to the whole ACFE (Ar. 2). Wherefore the parallelo-
gram ABDC has been bisected by the straight line EF, which
Q. E. F.*
is drawn through the point B.

PROPOSITION XXXV. THEOREM.

In fig. m, let ABDC be a rectangular parallelogram; the Parallograms upon the same base, and between the same parallels, diagonal AD is equal to the diagonal B C.

Fig. m.

are equal to one another.

In fig. 35, Nos. 1, 2 and 3, let the parallelograms Ac and BF be upon the same base BC, and between the same parallels AF and BC; the parallelogram AC is equal to the parallelogram BF.

Because in the parallelogram A B D C, the side A c is equal to the side BD (I. 34), and the side c D is common to the two triangles ACD and BCD, and the angle ACD is equal to the angle BDC (Hyp.); therefore the base A D is equal to the base BC, and these are the diagonals of the parallelogram A B DC. In fig. 1, let ABD C be an oblique angled parallelogram; and let the angle ACD be acute, and the angle BDC obtuse; the Because in the diagonal B C is greater than the diagonal A D. parallelogram A B D c, the side Ac is equal to the side BD (I. 34), and the side cp common to the two triangles ACD and BCD, and the angle BDO greater than the angle A CD; therefore the base BC is greater than the base AD (I. 24); and the diagonal B C is that which joins the vertices of the acute angles Wherefore, the diagonals of rectangular parallelograms, etc. Q. E. D.

ABD and AC D.

EXERCISE IV. TO PROPOSITION XXXIV.

To divide a straight line into any number of equal parts. This problem may be solved in the same way as Exercise II. to Prop. xxxII. was done; and therefore it need not be repeated here.*

EXERCISE V. TO PROPOSITION XXXIV.

To bisect a parallelogram by a straight line drawn through any point in one of its sides.

In fig. 7, let A B D C be a parallelogram, and E a point in one of its sides AB; it is required to bisect the parallelogram ABDC by a straight line drawn through the point E.

and EBDF.

From CD, cut off or equal to B B (I. 3) and join EF; then EP divides the parallelogram A B D C into two equal parts A EFC Join AD. Because A B is equal to CD (Typ.), and BE to CF (Const.), therefore A E is equal to DF (4x. 3). Because in the two triangles AEG and GDF, the two angles EA G and AEG are equal to the two angles F D G and DFG, each to each (I. 29), and the side A E to the side Dr; therefore the triangle A EGI equal to the triangle DGF (I. 26). But the triangle ABD i equal to the triangle A CD (I. 34); therefore the trapezium E is equal to the trapezium AF (Ax. 3), and the whole EBD

(Middleton); E. J. BREMNER (Carlisle); QUINTIN PRINGLE (Glasgow); This and the preceding exercise were solved by J. H. EASTWOOD

and others.

First lct the sides A D and D F of the paallelograms a C and RF, opposite to the base B c, be terminated in the same point D, as in No. 1. Because each of the parallelograms A c and BF, is double (I. 34) of the triangle B DC, therefore the parallelogram A c is equal (Ax. 6) to the parallelogram B F. Next, let the sides A D and E F, opposite to the base BC, be not terminated in the same point, as in Nos. 2 and 3.

Because AC is a parallelogram, AD is equal (1. 34) to B C. For a similar reason, EF is equal to BC; therefore A D is equal (A. 1) to EF; and DE is common to both, wherefore the whole, or the remainder A E, is equal to the whole, or the remainder DF (4x. 2 or 3); and AB is equal (I. 34) to D C. But in the triangles EA B and FD C, the side FD is equal to the side E A, and the side DC to the side A B, and the exterior angle FDC is equal (I. 29) to the interior and opposite angle EA B; therefore the base Fc is equal (I. 4) to the base E B, and the triangle FDC to the triangle E A B. From the trapezium ABCF, take the triangle FDC, and the remainder is the parallelogram A c. From the same trapezium take the triangle EA B and the remainder is the parallelogram BF. But when equals are taken from equals, or from the same, the remainders (Ar. 3) are equal. Therefore the parallelogram A c is equal to the parallelogram BF. Therefore, parallelograms upon the same,

etc. Q. E. D.

C

In Dr. Thomson's edition of Euclid, this highly important proposition is simplified by the application of Prop. xxvi. of this book. It may be simplified still more in the following manner :-Because A B is equal to CD, and BD to CF (I. 34), therefore in the two triangles ABE and DCF, the two sides A B and B E, are equal to the two sides DC and CF. But the angle A B E is equal to the angle D CF (I. 29, Cor. 1). Therefore, the triangle AB E is equal (I. 4) to the triangle DCF. This equality being proved, the rest of the demonstration is the same as that in the text. That part of the demonstration indeed is often rendered obscure by reference to Axiom I., instead of a new one, tacitly assumed by Euclid; viz. that "if equals be taken from the same thing, the remainder are equal."

This proposition is the foundation of the mensuration of plane surfaces and hence of land-measuring. As the area of a rectangle is determined practically by multiplying its length by its breadth, or its base by its altitude, and as by this proposition, every parallelogram having the same base and altitude (that is, the same perpendicular breadth between the parallels) with a rectangle, is equal to that rectangle in area; therefore he area of every parallelogram is found by multiplying the length of its base by its altitude.

EXERCISE TO PROPOSITION XXXV.

Equal parallelograms upon the same base and on the same side of it, are between the same parallels.

In fig. o, let A B CD and EFGH be two equal parallelograms upon the same base BC; they are between the same parallels; that is, EF is in the same straight line with AD.

This exercise was solved by J H. EASTWOOD (Middleton); E. J. BREMNER (Carlisle); QUINTIN PRINGLE (Glasgow); and others."

For if not, let EG be in the same straight line with AD. Because EG is therefore parallel to BC, the figure EBCG is a parallelogram; and because the parallelograms ABCD and EBCG are upon the same base BC, and between the same parallels (I. 35), they are equal; but the parallelogram EBCF is equal to the parallelogram ABCD (Hyp.); therefore the parallelogram EBCO is equal to the parallelogram EBCF (Ax. 1) the less to the greater, which is impossible; therefore E G is not in the same straight line with AD. In the same manner, it may be proved that no other straight line but EF is in the same straight line with AD; therefore the parallelograms ABCD and E BCF are between the same parallels. Wherefore, equal parallelograms upon the same base, etc. Q. E. D.*

LESSONS IN READING AND ELOCUTION.
No. XII.

ANALYSIS OF THE VOICE.
EXERCISES ON INFLECTIONS.

Simple Concluding Series.

Exercise 1. "It is a subject interesting alike to the old, and to the young."

2. "Nature, by the very disposition of her elements, has commanded, as it were, and imposed upon men, at moderate intervals, a general intermission of their toils, their occupations, and their pursuits."

3. "The influence of true religion is mild, and soft, and noiseless, and constant, as the descent of the evening dew on the tender herbage, nourishing and refreshing all the amiable and social virtues; but enthusiasm is violent, sudden, rattling as a summer shower, rooting up the fairest flowers, and washing away the richest mould, in the pleasant garden of society."

Compound Concluding Series.

Exercise 1. "The winter of the good man's age is cheered with pleasing reflections on the past, and bright hopes of the fùture."

2. "It was a moment replete with joy, amazement, and anxiety."

3. "Nothing would tend more to remove apologies for inattention to religion than a fair, impartial, and full account of the education, the characters, the intellectual processes, and the dying moments of those who offer them."

4. "Then it would be seen that they had gained by their scepticism no new pleasures, no tranquillity of mind, no peace of conscience during life, and no consolation in the hour of death."

5. "Well-doing is the cause of a just sense of elevation of character; it clears and strengthens the spirits; it gives higher reaches of thought; it widens our benévolence, and makes the current of our peculiar affections swift and deep."

6. "A distant sail, gliding along the edge of the ocean, was sometimes a theme of speculation.-How interesting this fragment of a world, hastening to rejoin the great mass of existence! What a glorious monument of human invention, that has thus triumphed over wind and wave; has brought the

This exercise was solved by J. H. EASTWOOD (Middleton); J. JENKINS (Pembroke Dock); QUINTIN PRINGLE (Glasgow); and others.

ends of the earth in communion; has established an interchange of blessings, pouring into the steril regions of the north all the luxuries of the south; diffused the light of knowledge, and the charities of cultivated life; and has thus bound together those scattered portions of the human race, between which nature seemed to have thrown an insurmount able barrier!"

Exception 1.-'Disconnected Series.'-Exercise 1. "Youth, in the fulness of its spirits, defers religion to the sobriety of manhood; manhood, encumbered with cares, defers it to the leisure of old age? old age, weak and hesitating, is unable to enter on an untried mode of life."

2. "Let me prepare for the approach of eternity; let me give up my soul to meditation; let solitude and silence acquaint me with the mysteries of devotion; let me forget the world, and by the world be forgotten, till the moment arrives in which the veil of eternity shall fall, and I shall be found at the bar of the Almighty."

3. "Religion will grow up with you in youth, and grow old with you in age; it will attend you, with peculiar pleasure, to the hovels of the poor, or the chamber of the sick; it will retire with you to your closet, and watch by your bed, or walk with you, in gladsome union, to the house of God; it will fol low you beyond the confines of the world, and dwell with you for ever in heaven, as its native residence."

'Emphatic Series.'-Exercise 1. "Assemble in your parishes, villages, and hamlets. Resolve, petition, addrèss."

2. "This monument will speak of patriotism and courage; of civil and religious liberty; of free government; of the moral improvement and elevation of mankind; and of the immortal memory of those who, with heroic devotion, have sacrificed their lives for their country."

3. "I have roamed through the world, to find hearts nowhere warmer than those of New England, soldiers nowhere bràver, patriots nowhere pùrer, wives and mothers nowhere trùer, maidens nowhere lovelier, green valleys and bright rivers nowhere greener or brighter; and I will not be silent, when I hear her patriotism or her truth questioned with so much as a whisper of detraction."

4. "What is the most odious species of tyranny? That a handful of men, free themselves, should execute the most base and abominable despotism over' millions of their fellow-crea tures; that innocence should be the victim of oppression; that industry should toil for rapine; that the harmless labourer should sweat, not for his own benefit, but for the luxury and rapacity of tyrannic depredation :-in a word, that thirty mil lions of men, gifted by Providence with the ordinary endowments of humanity, should groan under a system of despotism, unmatched in all the histories of the world."

Exoeption 3.-Poetic Series.'

Ex. 1. "He looks in boundless majesty abroad,

2.

And sheds the shining day, that burnished plays
On rocks, and hills, and towers, and wandering streams,
High-gleaming from afar."
"Round thy beaming car,
High-seen, the Seasons lead, in sprightly dance
Harmonious knit, the rosy-fingered Hours,
The Zephyrs floating loose, the timely Rains,
Of bloom ethereal, the light-footed Déws,
And, softened into joy, the surly Storms."
3. "Hear him compare his happier lot, with his
Who bends his way across the wintery wolds,
A poor night-traveller, while the dismal snow
Beats in his face, and dubious of his paths,
He stops and thinks, in every length
He hears some village mastiff
And sees, far streaming,
Then, undecei»- ·
And cl

Si

4. "There was neither tree, nor shrub, nor field, nor house, nor living créatures, nor visible remnant of what human hands and reared."

5. "And I, creature of clay, like those here cast around, I travel through life, as I do on this road, with the remains of past generations strewed along my trembling path; and, whether my journey last a few hours more or less, must still, like those here deposited, shortly rejoin the silent tenants of some cluster of tómbs, and be stretched out by the side of some already sleeping corpse."

6. "I am charged with pride and ambition. The charge is true,
and I glory in its truth. Who ever achieved any thing great
in letters, arts, or arms, who was not ambitious? Cesar was

not more ambitious than Cicero. It was but in another way.
All greatness is born of ambition. Let the ambition be a noble
one, and who shall blame it? I confess I did once aspire to be
queen, not only of Palmyra, but of the East. That I am. I
now aspire to remain so. Is it not an honourable ambition?
Does it not become a descendant of the Ptolemies and of Cleo-
patra? I am applauded by you all for what I have already
done. You would not it should have been less.

"But why pause here? Is so much ambition praiseworthy,
and more criminal? Is it fixed in nature that the limits of
this empire should be Egypt on the one hand, the Hellespont
and the Euxine on the other? Were not Suez and Armenia
more natural limits? Or hath empire no natural limit, but is
broad as the genius that can devise, and the power that can win.
Rome has the West. Let Palmyra possess the East. Not
that nature prescribes this and no more. The gods prospering,
and I swear not that the Mediterranean shall hem me in upon
the west, or Persia on the east. Longinus is right,-I would
that the world were mine. I feel, within, the will and the power
to bless it, were it so.

Yet, though thou wear'st the glory of the sky,
Wilt thou not keep the same beloved name,
The same fair thoughtful brow, and gentle eye,
Lovelier in heaven's sweet climate, yet the same?
Shalt thou not teach me, in that calmer home,
The wisdom that I learned so ill in this,-
The wisdom which is love,-till I become
Thy fit companion in that land of bliss ?"

Both Inflections, in connexion.

RULE L-Exercise 1. "It is not a parchment of pédigree,it is not a name derived from the ashes of dead men, that make the only charter of a kíng. Englishmen were but slàves, if, in giving crown and sceptre to a mortal like ourselves, we ask not, in return, the kingly virtues."

2. "The true enjoyments of a reasonable being do not consist in unbounded indulgence, or luxurious éase, in the tumult of passions, the languor of indolence, or the flutter of light amusements. Yielding to immóral pleasures corrupts the mind; living to animal and trifling ones, debàses it: both, in their degree, disqualify it for genuine good, and consign it over to wretchedness."

3. "What constitutes a state?

Not high raised battlements, or laboured mound,
Thick wall, or moated gate;

Not cities proud, with spires and turrets crowned,
Not bays and broad-armed pórts,

Where, laughing at the storm, proud návies ride;
Not starred and spangled cóurts,-

Where low-browed baseness wafts perfume to príde!
No!-mèn,-high-minded ME'N,—

Men who their duties know,

But know their rights, and, knowing, dare maintain." Note. Concession and Unequal Antithesis.'

Ex. "The clouds of adversity may darken over the Christian's path. But he can look up with filial trust to the guardian care of a beneficent Father."

2. "I admit that the Greeks excelled in acuteness and ver

"Are not my people happy? I look upon the past and the
present, upon my nearer and remoter subjects, and ask nor
fear the answer. Whom have I wronged-what province
have I oppressed?-what city pillaged?-what region drained
with taxes?-whose life have I unjustly taken, or estates
coveted or robbed?-whose honour have I wantonly assailed?satility of mind. But, in the firm and manly traits of the
-whose rights, though of the weakest and poorest, have I Roman character, I see something more noble,-more worthy
trenched upon?-I dwell, where I would ever dwell, in the of admiration."
hearts of my people. It is written in your faces, that I reign
not more over you than within you. The foundation of my
throne is not more power than love."

7. "How shall I know thee in the sphere which keeps
The disembodied spirits of the dead,

When all of thee that time could wither, sleeps,
And perishes among the dust we tread?

For I shall feel the sting of ceaseless pain,

If there I meet thy gentle presence not;

Nor hear the voice I love, nor read again

In thy serenest eyes the tender thought.

Will not thy own meek heart demand me there?

That heart whose fondest throbs to me were given ?
My name on earth was ever in thy prayer,
Shall it be banished from thy tongue in heaven ?
In meadov.
thing wind

In the
And larg

Wilt

heaven's

helpless tools: we war against our oppressors,—not against 3. "We war against the leaders of evil,-not against the

our misguided brethren."

4.

"Still, still, for ever

Exercise 1.

'I'll keep them all;

He shall not have a Scòt of them;

No, if a Scot would save his sòul, he shall not."

2. "Do not descend to your graves with the disgraceful censure, that you suffered the liberties of your country to be taken away, and that you were mutes as well as cowards. Come forward, like men: protest against this atrocious attèmpt."

3. "I am not sounding the trumpet of war. There is no man who more sincerely deprecates its calamities, than I do." 4. "Rest assured that, in any case, we shall not be willing to rank last in this generous contest. You may depend on us for whatever heart or hand can dò, in so noble a cause." 5. "I will cheerfully concede every reasonable demand, for the sake of peace. But I will not submit to dictation."

RULE II. Question and Answer.-Exercise 1. "Do you think these yells of hostility will be forgotten?-Do you suppose their echo will not reach the plains of my injured and insulted

The penultimate inflection falls, when a sentence ends with the rising

country, that they will not be whispered in her green valleys, and heard from her lofty hills?-Oh! they will be heard there: yès, and they will not be forgotten."

2. "I will say, what have any classes of you, in Ireland, to hope from the French? Is it your property you wish to preserve?-Look to the example of Holland; and see how that nation has preserved its property by an alliance with the French! Is it independence you court?-Look to the example of unhappy Switzerland: sce to what a state of servile abasement that once manly territory has fallen, under France! Is it to the establishment of Catholicity that your hopes are directed?-The conduct of the First Consul, in subverting the power and authority of the Pope, and cultivating the friendship of the Mussulman in Egypt, under a boast of that subversion, proves the fallacy of such a reliance.-Is it civil liberty* you require?-Look to France itself, crouching under despotism, and groaning beneath a system of slavery, unparalleled by whatever has disgraced or insulted àny nation.' 3. "Shall I be left forgotten, in the dust,

When Fate, relenting, lets the flower revive?
Shall Nature's voice,-to man alone unjust,-

Bid him, though doomed to perish, hope to live?
Is it for this fair Virtue oft must strive
With disappointment, pénury, and pain?

No: Heaven's immortal spring shall yet arrive,
And man's majestic beauty bloom again,
Bright through the eternal year of Love's triumphant reign."
RULE III. Disjunctive "Or."'-Exercise 1. "Will you rise
like men and firmly assert your rights, or will you tamely
submit to be trampled on?"

2. "Did the Romans, in their boasted introduction of civilisation, act from a principle of humane interest in the welfare of the world? Or did they not rather proceed on the greedy and selfish policy of aggrandising their own nation, and extending its dominion ?"

3. "Do virtuous hábits, a high standard of morálity, proficiency in the arts and embellishments of life, depend upon physical formátion, or the latitude in which we are placed t Do they not depend upon the civil and religious institutions which distinguish the country?"

SIR,-Since the issue of your articles on Chemistry in the POPULAR EDUCATOR, I have been studying them practically, and with perfect success. I find, however, many drawbacks incident to the carrying on of some processes alone, owing to some of the experiments being inconvenient or dangerous without a room for the purpose; the want of apparatus that would prove too expensive; and, above all, inability to carry on any interchange of ideas. I believe that the consideration of these facts has deterred many from commencing the study of this useful branch of science, who otherwise would have entered upon it with spirit. As there may be many young students who like myself have prosecuted this study alone up to the present time, it has occurred to me to propose to those who may be residing in the vicinity of Camberwell, to unite and form a class for mutual improvement and advancement in various sciences; but more especially for the purpose of studying, in deep earnest, Chemistry practically and theoretically.

The expense being then divided amongst us, would be compara. tively trifling to each. The advantages of such a plan, when all are resolutely determined to persevere in their object, would be great. One or two have already joined in furtherance of these views. Your well-proved willingness to advance all efforts in the cause of progress, has induced me to request your insertion of this letter. T. G. LINSTEAD, 28, High Street, Newington Butts, London, 13th May, 1854.

Communications to be addressed to

ANSWERS TO CORRESPONDENTS.

T. BOCOCK (Great Warley): His algebraic solutions are correct.-W. BOOTH (Waterhead Mill): Very good, but not quite up to our mark.-A. WOOD: premature; we like fruits, poetical or otherwise, in their seasons.-W. C. His "contemp ation on the autumnal season" is as yet rather (Colchester): His solutions, amounting to 94 out of the 100 Problems, are all correct and very well done. We are much gratified with his account of his progress. To be able to solve so many questions since Christmas, when he only began to study Algebra in the P. E., and in Cassell's Algebra, is a very great achievement.

acquired; but that of Greek is the most valuable to the man who wishes to EXCELSIOR: The knowledge of French is, in our opinion, most easily read the New Testament in the original tongue; at the same time it is the most difficult of all.-AUDIENS (Portland-town): In Greek, the Gamma has always the hard sound, as in English the g has before a, o, and u. The Upsilon, whether spelt in English words with yor u, always sounds in Greek like the French u.-E. H. W.: The Latin is completed in the P. E., but there are other books in the Latin Language on Mr. Cassell's list; study

them.

R. F. T.: We are quite disposed to take the most favourable view of our correspondent's motives, and also of his endeavours to solve the problem relating to the four balls. We confess, however, our inability to comprehend his reasoning; but this may be as much, if not more, our fault than his; we are old, and accustomed to take Euclidean views of things; and as it seems that he cannot quite follow us in that direction, we must be content to remain as we are, ignorant of his peculiar method of explanation. But heaven forbid that we should harbour any "suspicion of evil intentions," we can't conceive how this entered his mind, for it never entered ours! "Honi soit que mal y pense."

A DEVERONSIDE PLOUGHMAN: His answer to the "Four Ball Query" is very nearly correct, and his solutions to the three hardest problems in the "Centenary of Problems" very good. Let him go on and prosper; perseverance overcomes all difficulties.-W. R. H. (Cowley): Thanks.-H. JONES (Islington): The very thing he wishes is preparing.-G. H. (W. (Hampton): The pronunciation which he has given of gibier and c'est are quite correct.-SIMPLICITAS (Wemyss): We shall keep his suggestion

in mind.

HIST: and THEO: NOVICE: Surely he knows that Hume, Gibbon, Hallam and Macaulay are celebrated as Historical Writers; and that Hall, Foster, Wilberforce and Smith are celebrated as Theological Writers; all for elegance and force; but he must beware of the infidel principles of Hume and Gibbon, and the flippancy of Macaulay; also of the heterodox opinions of Foster.-W. C. (Uxbridge) must really turn poet himself; our corres pondent quite coolly asks us to paraphrase a passage of Milton's "Faradise Lost" for him, and insert it in the P. E.! What next?