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 (Ax. 2). Wherefore the parallelogram ABDC has been bisected by the straight line EF, which is drawn through the point E. Q. E. F.*

PROPOSITION XXXV. THEOREM.

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

A

Fig. m.

are equal to one another.

In fig. 35, Nos. 1, 2 and 3, let the parallelograms A c and DF be upon the same base BC, and between the same parallels AF and BC; the parallelogram Acis 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 A CD 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 D C. In fig. 7, let ABD c be an oblique angled parallelogram; and let the angle ACD be acute, and the angle BDC obtuse; the diagonal B C is greater than the diagonal AD. Because in the parallelogram ABDC, the side AC is equal to the side BD (I. 34), and the side cn common to the two triangles ACD and BCD, and the angle BDC 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 ABD and A CD. Wherefore, the diagonals of rectangular parallelograms, etc. Q. E. D.

EXERCISE IV. TO PROPOSITION XXXIV.

First let the sides AD and D F of the paallelograms A c and BF, 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 ac is equal (4x. 6) to the parallelogram B F.

Next, let the sides A D and EF, 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 BC. For a similar reason, EF is equal to BC; therefore A D is equal (4x. 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 E A B and FDC, the side FD is equal to the side E A, and the side DC to the side AB, and the exterior angle FDC is equal (I. 29) to the interior and opposite angle EAB; therefore the base Fc is equal (I. 4) to the base E B, and the triangle FDC to the triangle EA B. From the trapezium ABCF, take the triangle FD C, 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 (Ax. 3) are equal. Therefore the parallelogram Ac is equal to the To bisect a parallelogram by a straight line drawn through any parallelogram BF. Therefore, parallelograms upon the same,

To divide a straight line into any number of equal parts. This problem may be solved in the same way as Exercise 11. to Prop. XXXII. was done; and therefore it need not be repeated here.*

EXERCISE V. TO PROPOSITION XXXIV.

point in one of its sides.

etc. Q. E. D.

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 AB is equal to CD, and BD to CF (I. 34), therefore in the two triangles ABE and DC F, 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 DCF (I. 29, Cor. 1). Therefore, the triangle ABE 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 the area of every parallelogram is found by multiplying the length of its base by its altitude.

EXERCISE TO PROPOSITION XXXV.

From CD, cut off CF equal to BE (I. 3) and join EF; then EF divides the parallelogram A B D C into two equal parts A EFC and EBDF. Join A D. Because A B is equal to CD (Typ.), and BE to CF (Const.), therefore A E is equal to DF (Ax. 3). Because in the two triangles AEG and GDF, the two angles E AG and A EG are equal to the two angles FDG and DFG, each to each (I. 29), Equal parallelograms upon the same base and on the same side of it, and the side A E to the side DF; therefore the triangle A EGI equal to the triangle D G F (I. 26). But the triangle ABD i equal to the triangle A C D (I. 34); therefore the trapezium E is equal to the trapezium a F (Ax. 3), and the whole EBDF

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

are between the same parallels.

In fig. o, let A B C D and E F G H 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.

F

D

E

C

For if not, let EG be in the same straight line with AD. Because EG is therefore parallel to BC, the figure E BCG 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 E BCF is equal to the parallelogram ABCD (Hyp.); therefore the parallelogram EBCG is equal to the parallelogram E BCF (4x. 1) the less to the greater, which is impossible; therefore E G is not in the same straight line with a D. In the same manner, it may be proved that no other straight line but EF is in the same straight line with A D; therefore the parallelograms ABCD and EBCF 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 Conduding 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 occupátions, and their pursuits."

3. "The influence of true religion is mild, and soft, and nóiseless, and cònstant, 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 lífe, 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 insurmountable 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 devòtion; 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 truer, maidens nowhere lovelier, green valleys and bright ivers 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-crèatures; that innocence should be the victim of opprèssion; that industry should toil for ràpine; 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 millions 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.'

2.

66

Ex. 1. "He looks in boundless majesty abroad,
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 blooin 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 lengthening blast,
IIe hears some village mastiff's distant howl,
And sees, får streaming, some lone cottage light;
Then, undeceived, upturns his streaming eyes,
And clasps his shivering hands, or, overpowered.
Sinks on the frozen ground, weighed down with sleep
From which the hapless wretch shall never wàke."

• Accidental 'falling' inflection, for contrast.

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."

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."

Thomas Muffit,

K. Simpson,

Charles Curshaw,

do.

do.

do.

do.

do.

CHEMISTRY.

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 and, above all, inability to carry on any interchange of ideas. the purpose; the want of apparatus that would prove too expensive;

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: see 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?-Lock to France itself, crouching under despotism, and groaning beneath a system of slavery, unparal-may be many young students who like myself have prosecuted this leled 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 rèign."
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 nùtion, and extending its dominion ?"

3. 64

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

[The remaining rules on 'inflection,' as they are of less frequent application, are thought to be sufficiently illustrated by the examples appended to each rule. A repetition of these, however, may be useful as an exercise in review.]

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 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 comparatively 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. Communications to be addressed to T. G. LINSTEAD, 28, High Street, Newington Butts. London, 13th May, 1854.

ANSWERS TO CORRESPONDENTS. .

T. Bocock (Great Warley): His algebraic solutions are correct.-W. BOOTH (Waterhead Mill): Very good, but not quite no to our mark.—A. WOOD: His "contemp ation on the autumnal season" is as yet rather premature; we like fruits, poetical or otherwise, in their seasons.-W. C. (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.

EXCELSIOR: The knowledge of French is, in our opinion, most easily

The

acquired; but that of Greek is the most valuable to the man who wishes to 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. Upsilon, whether spelt in English words with y or 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 torce; 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 correspoudent quite coolly asks us to paraphrase a passage of Milton's "Paradise Lost" for him, and insert it in the P. E. ! What next?

ON PHYSICS, OR NATURAL PHILOSOPHY.

No. XXXVII.

(Continued from page 145.)

EVAPORATION AND EBULLITION. Acceleration of Evaporation.-It has been already observed in a former lesson, that evaporation is a slow production of vapour at the surface of a liquid. It is in consequense of spontaneous evaporation that wet clothes dry in the air, or that an open vessel full of water is completely emptied of it at the end of a certain time. It is evaporation which takes place at the surface of seas, lakes, rivers, and the ground, which produces the vapours that rise in the atmosphere, condense into clouds, and fall in the form of rain. Four causes have an influence on the rapidity of the evaporation of a liquid; 1st, the temperature; 2nd, the quantity of the vapour of the same liquid already dif fused in the surrounding atmosphere; 3rd, the renewal of the atmosphere; 4th, the extent of the surface evaporated. As to the first cause, the increase of temperature accelerates the evaporation by the excess of the elastic force which it produces in the vapours. In the case of the second, as the evaporation of a liquid will be stopped in a space saturated with the same liquid; and as it only attains its maximum in an air completely freed from that vapour; it is evident that between these two extremes, the rapidity of evaporation will vary according as the surrounding atmosphere is more or less already charged with the same vapour. As to the third cause, the renewal of the atmosphere, its effect may be explained on the same principle. For if the air or gas which envelopes a liquid be not renewed, it will be quickly saturated, and all evaporation will then cease. The influence of the fourth cause is self-evident.

The Laws of Ebullition.-To the rapid production of vapour, in bubbles of a larger or smaller size, in the same liquid, we give the name ebullition or boiling. When in a vessel, a liquid, such as water, is heated at its lower strata, the first bubbles which make their appearance are only those of the air held in solution in the water, which are disengaged on the application of heat. When these are dispelled, then small bubbles of vapour begin to rise from all the heated points of the sides of the vessel; these traversing the upper strata of the liquid, which have a lower temperature, are condensed before they reach the surface. The formation and condensation of these first bubbles of vapour are the cause of the singing noise which commonly precedes ebullition. Lastly, large bubbles begin to rise and burst at the surface of the liquid, and the continuance of this process constitutes the phenomenon of ebullition, see fig. 192,

Fig. 192.

All liquids which can undergo the process of ebullition exhibit phenomena according to the two following laws, which are proved by experiment:

VOL. V.

Several causes produce a variation in the boiling point of a liquid, viz. 1, the substances held in solution; 2, the nature of the vessel; and 3, the pressure.

1st. When a substance is dissolved in a liquid, and it is either not volatile or less than the liquid in quantity, the ebullition is retarded in proportion to the quantity of the substance held in solution.

Water which boils at 100° Centigrade, boils at the following temperatures, when saturated with the different salts :

purely in suspension in water, as earthy matters, wood shavings, Acid solutions present inilar results; but substances held etc., do not raise the temperature of ebullition. At this point, also, it is important to refer to the experiments of Rudberg, formerly cited under the subject of the Graduation of the Thermometer, experiments in which this philosopher shewed that when the temperature of the ebullition of water is above 100° Centigrade, in consequence of the substances held in solution, the temperature of the vapour which rises from it is still-1000 Centigrade, as in the case of pure water, provided the atmospheric pressure be at the standard point, viz. 29.922 inches.

2nd. Gay-Lussac observed that in a glass vessel, water boiled at a higher temperature than in a metallic vessel; a phenomenon which he attributed to the affinity of glass for water. For instance, he found that when distilled water boiled in a brass vessel at the standard temperature 'and pressure, the water did not enter into the state of ebullition in a glass vessel till the temperature was 101° instead of 100°, the pressure being the same; and when the glass vessel was rubbed with concentrated sulphuric acid, or potassa, the temperature of the water rose to 105° and 106° Centigrade before boiling. Yet a simple fragment of metal placed at the bottom of the vessel was sufficient to restore the temperature of ebullition to 100 Centigrade, and at the same time to dissipate the violent concussions which accompanied the ebullition of the saline or acid solutions in glass vessels. Moreover, in the case of substances held in solution, the temperature of the vapour is not influenced by that which the water assumes in glass vessels. At the standard pressure, the temperature of the vapour is still 100° Centigrade as in brass vessels.

3rd. According to the tables of the elastic force of the vapour of water and of steam given in our last lesson, it will be seen that at 100° Centigrade, the temperature at which water boils under the standard pressure, the vapour or steam has a tension precisely equal to this pressure. This principle is general, and may be thus stated: Every liquid enters into the state of ebullition only at the moment when the tension of its vapour is equal to the pressure which it supports. We infer, then, when this pressure increases or diminishes that the tension of the vapour, and consequently the temperature necessary to

115