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The complements of the parallelograms, which are about the diagonal
of any parallelogram, are equal.

In fig. 43, let BD be a parallelogram, of which the diagonal is
AC; and let E H and or be the parallelograms about a c, that
is, through which AC passes. Then BK and K D are the other
parallelograms which make up the
Fig. 43.
whole figure B D, and are there-
fore called the complements.
complement B K is equal to the com-
plement K D.

The

E

A

H

K

13 G

D

Because BD is a parallelogram, and AC its diagonal, the triangle ABC is equal (I. 34) to the triangle AD C. Again, because в H is a parallelogram, and AK its diagonal, the triangle AEK is equal (I. 34) to the triangle AHK. For the same reason, the triangle K G C is equal to the triangle K F C. Therefore the two triangles A E K and K G C are equal (Ax. 2) to the two triangles A HK and K FC. But the whole triangle ABC is equal to the whole triangle ADC. Therefore the remaining complement BK is equal (4x. 3) to the remaining_complement KD. Wherefore the complements, etc. Q. E. D.

Corollary 1.-The parallelograms about the diagonal of a parallelogram, as also its complements, are equiangular to the whole parallelogram and to each other.

Corollary 2.-If through any point in the diagonal of a parallelogram, straight lines be drawn parallel to its sides, the parts into which each divides the parallelogram are equal, the greater to the greater, and the less to the less.

PROPOSITION XLIV. PROBLEM.

To a given straight line to apply a parallelogram, equal to a given triangle, and having one of its angles equal to a given rectilineal angle.

In fig. 44, let AB be the given straight line, c the given triangle, and D the given rectilineal angle. It is required to Fig. 44.

F

E

K

H A

M

B

F

Because FL is a parallelogram (Def. 36), of which the diagonal is H K ; AG and ME are the parallelograms about н K, and L B and B F are the complements. Therefore the complement LB is equal (I. 43) to the complement B F. But the complement BF is equal (Const.) to the triangle c. Therefore LB is equal (Ax. 1) to the triangle c. Because the angle G BE is equal (I. 15) to the angle A B M, and likewise (Const.) to the angle D. Therefore the angle ABM is equal (Az. 1) to the angle D. Wherefore to the straight line AB, is applied the parallelogram LB, equal to the triangle c, and having the angle A B M equal to the angle D. Q. E. F.

Corollary. From this proposition, it is manifest how to describe on a given straight line, a rectangle equal to a given triangle.

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to E.

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lelogram equal to the figure A B C D, and having an angle equal
Join D B. Describe the parallelogram FH equal to the
angle E.
triangle A D B, and having the angle FKH equal (I. 42) to the
To the straight line G H apply the parallelogram & M
equal to the triangle DB C, and having the angle GHM equal
(I. 44) to the angle E. Then the figure KL is the parallelo-
gram required.

Because the angle E is equal (Const.) to each of the angles
FK H and G H M, the angle FK H is equal (Ax. 1) to the angle
GH M. To each of these equals add the angle KH G. There-
fore the two angles FK H and K HG are equal to the two angles
K HG and G HM. But the two angles FK H and K HG are equal
(1.29) to two right angles. Therefore the two angles KHG
and G H M are also equal (Ax. 1) to two right angles. Because
at the point H, in the straight line GH, the two straight lines
KH and HM, upon opposite sides of it, make the adjacent
angles equal to two right angles, HK is in the same straight
line (I. 14) with H м. Again, because the straight line HG
meets the parallels K M and FG, therefore the angle мHG is
equal (I. 29) to the alternate angle HGF. To each of these
equals add the angle HGL; and the two angles MHG and
HGL are equal to the two angles HGF and HGL; but the two
angles MHG and HGL are equal (I. 29) to two right angles.
Therefore also the two angles HGF and HGL are equal (Ax. 1)
to two right angles. Wherefore, as before, FG (I. 14) is in
the same straight line with G L.
Because K F is parallel to HG,
and HG to ML; therefore KF is parallel (I. 30) to M L. And
Kм has been proved parallel to FL. Therefore, the figure
KL is a parallelogram (Def. 36). But the parallelogram F H is
equal (Const.) to the triangle ABD, and the parallelogram GM
Therefore the whole parallelogram K L
to the triangle BDC.
Wherefore the
is equal to the whole rectilineal figure ABCD.
lineal figure A B C D, and having the angle FKM equal to the
parallelogram KL has been described equal to the given recti-
Q. E. F.
given angle E.

apply to (that is, to describe upon) the straight line AB, a parallelogram equal to the triangle c, and having an angle equal to D. Make the parallelogram BF equal (I. 42) to the triangle c, and having the angle EBG equal to the angle D. Place the parallelogram so that its side BE shall be in the same straight line with AB. Produce FG to H. Through A, draw A H parallel to B G or E F (I. 31), and meeting FH in H. Join H B. Because the straight line HF falls upon the two parallels A and EF, the two angles A HF and HFE are together equal Corollary. From this it is manifest how, to a given straight (1.29) to two right angles. Therefore the two angles BHP line, to apply a parallelogram, which shall have an angle and HFE are less than two right angles. But those straight equal to a given rectilineal angle, and shall be equal to lines which, with another straight line, make the two interior a given rectilineal figure; viz., by applying to the given angles upon the same side of it less than two right angles, do straight line a parallelogram equal to the first triangle ABD meet (Ax. 12) if produced far enough. Therefore H B and F E (I. 44), and having an angle equal to the given angle, and co shall meet, if produced. Let them be produced and meet in K.structing the rest of the figure as in this proposition." Through K, draw KL parallel to EA or FH, and produce HA and GB, to meet in the points L and M.

By this proposition, also, upon a given straight line, a rec angle may be described, equal to a given rectilineal figure.

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FRENCH READING S.-No. XIX.

JOSEPHINE.

SECTION III.

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Les désastres
de Josephine. P

2

C

en Italie: et ià, comme selle se fit?

Inantes de son cœur

pas d'éclat spine conserva pople. A part Tabandonnerazne bien-aimée,3 A posterite lui confirmera.

Femme d'un des plus grands hommes des temps mo dernes, impératrice sacrée des Français, reine connaLLee quee d'Italie, aimée et vénérée de tout le monde, Josphine semblait étre arrivée au comble de la gloire et du bonheur, et n'avoir plus rien à désirer: mais son mariage avec cap mertel au cœur l'Empereur était stérile. Dans les premières années de vento as Decree $ Naposon regne, Napoléon parut peu sensible à ce malheur; il .5 parlager son malheur, considérait alors les fils de son frère Louis et d'Hortense, fille de Joséphine, comme ses héritiers naturels à l'empire: Apres catre des ennemis à Paris, les souverains étranet il avait désigné Eugène à la succession d'Italic. Maisen geis s'empresscent Culler porter a l'impératrice le tribut 1×0×, il céda' aux conseils pressants de ses sœurs, envieuses de leurs hommaren Joséphine, avez un instant par ce de Josi pine, et plus encore à la fausse vanités de s'allier à qu'elle crut être de l'interet de ses enfants, se laissa aller? l'un de ces grandes familles de l'europe qu'il avait succes, à recevoir les princes aules. Le dévouement maternel sivement écrasées de sa puissance populaire. excuse cet acte de faiblesse; et d'ailleurs la violence que

répétaltelli sans cuase.

11 we de vida donc à divorcer d'avec Joséphine, l'impéra- Joséphine dut faire à ses sentiments de femme et de tri fac raise, la compagne de sa destinée extraordinaire, Française, épuisa ses forces; son sang s'enflamma; elle se pour mettre a sa place une étrangère, qui en cette qualité trouva grievement indisposée. Le roi de Prusse étant pavait et ne devait jamais avoir aucune sympathie avec venu la voir, elle fut obligée de se lever. Un refroidisseFrance, Josephine consentit à ce sacrifice avec un coument garne's dans ses jardins aggrava son mal, une angine Tag heronque. Elle perdait par là l'Empereur, qu'elle se déclara, et trois jours après le vingt-neuf mai, elle ...t de toute la tendresse de son âme, et l'espérance de expira, chrétienne et résignée, dans les bras de ses enfants, Vour ses petits fils et son fils succéder à l'empire, et à la Ses dernières paroles furent: L'ile d'Elbe.... Napoléon! royauté d'Italie; et pourtant elle se résigna" sans faiblesse, Me voilà, me voilà.... Justement à la même époque,

$ou' sais une grande douleur.

Marie-Louise rentrait à Vienne avec son fils, abandonnant In presence des princes de la famille impériale et des volontairement et pour toujours, la France et son époux hauts dignitaires de l'empire, elle dut' lire elle-même l'acte malheureux. de renonciation à ce qu'elle avait de plus cher: telle était Le corps de Joséphine fut déposé dans 19 l'église de Ruei, la volonte de Napoleon; et elle l'accomplit. près de Paris. Sept ans plus tard ses enfants obtinrent la On lui offrit alors le gouvernement de Rome ou celui permission de lui élever un tombeau. de Bruxelles ; elles les refusa en disant que, celle qui avait été impératrice des Français ne pouvait ni descendre ni monter, Mais elle voulut rester en France, et elle y resta malgre les jalousies et les intrigues qui cherchaient à l'en éleigner.

La mémoire de Joséphine ne périra pas. Deux qualités précieuses lui assurent la perpétuité du souvenir populaire; elle fut bonne et Française. La postérité de l'impératrice se compose de deux petits-fils et de quatre petites-filles. LE BAS.

COLLOQUIAL EXERCISE.

NOTES AND REFERENCES.-a. L. S. 34, R. 2.-b. courtisans, courtiers.-c. lieu, reason.-d. from voir; L. part ii., p. 110.e. elle relates to majesté, which is feminine.-f. pauvre, beggar.11. A quels sentiments dut-elle. chapeau bas, with his hat off; literally, low-h. from suivre; L. part ii., p. 106.-i. refrain, reply; literally, chorus.

1. Que fit l'impératrice en mil | 10. Quel sentiment doit excuser
huit cent douze?
cette faiblesse ?

2. Comment fut-elle accueillie? 3. Qu'avaient donné à la couronne les qualités de son cœur?

4. Que conserva Joséphine après sa chute?

5. Quel surnom avait-on donné à Joséphine?

6. Quel fut l'effet des désastres de 1814?

7. Que répétait sans cesse Jo-
séphine?

8. Qu'arriva-t-il après l'entrée
des ennemis à Paris?
9. Que fit Joséphine?

faire violence?

12. Que s'ensuivit-il?

13. Comment se trouva-t-elle ?
14. Qu'arriva-t-il alors?

15. Par quoi son mal fut-il ag-
gravé?

16. Combien de jours vécut-
elle encore?

17. Quelles furent ses dernières
paroles?

18. Que se passait-il à la même
époque?

19. Où plaça-t-on le corps de
l'impératrice ?

NOTES AND REFERENCES.-a. from faire; L. part ii., p. 92. b. L. S. 41, R. 5.-c. à part quelques courtisans, if we except a few courtiers-d se laissa aller, consented.-e. dut, was compelled.f. angine, inflammation of the throat.-g. me voilà, here am.-h. from vivre; L. part ii., p. 110

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LE ROI ALPHONSE.
CERTAIN roi qui régnait sur les rives du Tage,1
Et que l'on surnomma le Sage,
Non parcequ'il était prudent,2
Mais parcequ'il était savant,

Alphonse fut, dit-on, un célèbre astronome.3
Il connaissait le ciel bien mieux que son royaume,
Et quittait souvent son conseil,
Pour la lune ou pour le soleil.
Un soir qu'il retournait à son observatoire,
Entouré de ses courtisans, b

Mes amis, disait-il, enfin' j'ai lieu de croire
Qu'avec mes nouveaux instruments

Je verrai1 cette nuit des hommes dans la lune.

Votre majesté les verra,

Répondait-on; la chose est même trop commune ;

Elle doit voir mieux que cela.

Pendant tous ces discours, un pauvre dans la rue,
S'approche en demandant humblement, chapeau bas,
Quelques maravédis. Le roi ne l'entend pas,10
Et sans le regarder, son chemin continue.
Le pauvre suit le roi, toujours tendant la main,
Toujours renouvelant sa prière importune:
Mais, les yeux vers le ciel, le roi pour tout refrain,
Répétait: Je verrai des hommes dans la lune.

11

Enfin le pauvre le saisit 13

Par son manteau royal, et gravement lui dit;
Ce n'est pas de là-haut, c'est des lieux où nous sommes
Que Dieu vous a fait souverain.

Regardez à vos pieds; là vous verrez des hommes,15
Et des hommes manquant de pain.

COLLOQUIAL EXERCISE.

FLORIAN.

1. Où régnait un certain roi? 9. Qu'arriva-t-il pendant ces 2. Pourquoi l'avait-on sur- discours?

nommé le sage?

sances?

10. Le roi fit-il attention au
pauvre ?

3. Que fut surtout Alphonse?
4. Quelles étaient ses connais- 11. Que fit alors le mendiant ?
12. Que répétait Alphonse les
yeux tournés vers le ciel ?
13. Que fit enfin le pauvre ?

5. Pourquoi quittait-il souvent

le conseil ?

6, Où allait-il un soir entouré 14. Que dit-il gravement au

de ses courtisans ?

7. Que leur disait-il ?

monarque?

15. Où lui dit-il de regarder?

8. Que lui répondirent les 16. Que devait-il voir à ses

courtisans ?

pieds ?

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DANS un temps qui n'est pas éloigné de nous, vivait en Allemagne un petit souverain1 qui gouvernait sa principauté en véritable père. C'était l'ancien landgrave de HesseHombourg. Un jour ce bon prince étant à table, s'entretenait avec sa femme et un de ses chambellans de la

position de ses sujets, qu'il connaissait en grande partie par leurs noms, car ils n'étaient qu'au nombre de quelques mille. Des flocons de neiges voltigeaient en dehors autour des fenêtres comme un léger duvet agité par le vent. Il faisait un bien grand froid. Avec quelle bonté,7 interrompit tout à coup la femme du landgrave, le Créateur n'a-t-il pas pris soin de nous ? Et pourtant nous nous montrons bien peu reconnaissants des bontés que nous envoie sa Providence. Sans peines comme sans souffrances," jouissant de tout ce que nous pouvons désirer, nous n'avons jusqu'ici vécu dans notre château héréditaire qu'au sein de la paix et du bonheur; et, tandis que le froid pénètre au fond des pauvres 10 cabanes et y fait entrer la misère, nous sommes ici, nous, dans un appartement bien chaud, nous savourons des mets délicats. Ah! remercions dans nos cœurs" le bon Dieu de toutes les faveurs dont il nous comble.

Le prince, secrètement ému 12 des paroles touchantes qu'il venait d'entendre, se retourna du côté de son chambellan, et lui dit: Quelles sont les familles les plus pauvres 13 et les plus honnêtes de ma principauté? Vous devez les connaître; nommez-les-moi pour que je leur distribue des

secours.

Le chambellan, fort honoré de cette marque de confiance, répondit:

-Je suis heureux, Monseigneur, que vous daigniez11 m'interroger sur ce point, car je puis vous satisfaire. Au village le plus prochain, dans la première cabane,15 végète dans la plus profonde misère, une famille intéressante; il y a là un bien digne homme, une brave femme et leurs deux petits enfants. Si vous voulez faire une bonne œuvre, elle sera bien placée.

17

Le prince répliqua: Votre avis me plait ; cependant, je voudrais être bien sûr 16 de l'honnêteté de cette famille, et particulièrement de celle du père.-Monseigneur, reprit le chambellan, je réponds des vertus de cet homme; il est si bon, en vérité, qu'il se dépouillerait pour un autre plus pauvre que lui.-Vraiment ? s'écria le prince.-Sur l'honneur! répondit le chambellan.-Eh bien? je veux, dit le prince, le soumettre 18 à une épreuve. Promettez-moi1 seulement de garder le secret. Le chambellan, après avoir fait 20 la promesse, salua et se retira.

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1. § 49, R. mpire; L.

ENCE.

SGS UNDER

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POPULAR EDUCATOR:
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I have got your "Euclid,"
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ya chaty of Problems given by you.
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1.

lit myself for a clerkship, as I think this is one which I should be able to manage.

me lame, I learnt a system of Phonography, so as to jolt a sermon delivered at the rate of 100 words and to read it afterwards. Since my lameness I have has to be able to read it. I have gone through it's Arithmetic," and have got as far as Equations ibuycne unknown quantity, "Cassell's Algebra;" so that, ith another, and books when I can get them, I contrive ml time #poke in the ribs, and thus get the better of

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a - Cult Birkby, near Huddersfield, 20th June, 1854.

1602

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ON PHYSICS, OR NATURAL PHILOSOPHY.
No. XLIV.

(Continued from page 258.)

WARMING AND VENTILATING.

Warming.-After the subject of Illumination, treated of in our last lesson, that of Warming and Ventilating naturally follows, as connected with the chemical sources of heat. The art of warming has for its object the proper employment of the sources of heat which nature presents to us, for the purposes of domestic economy and of manufacturing industry; in short, to warm our private dwellings and our public establishments so as to produce that degree of comfort necessary to the preservation of health. The chief sources of heat at present in use are the combustion of wood, charcoal, coal, coke, turf or peat, and anthracite. According to the apparatus employed in their combustion, we may arrange the subject under four distinct processes of heating:-1st, warming by the direct radiation of caloric, as in fire-places and stoves; 2nd, warming by hot air; 3rd, heating by steam; 4th, heating by the circulation of hot water. Let us consider these different processes of heating in their order.

injurious. This mass of cold air which flows towards the fire, from without, cools the apartment in such a manner that only a very small portion of the heat is rendered of any use for the purpose intended. Moreover, the velocity of the air in the chimney being small on account of its great size, the draught is easily influenced by the winds, and it often produces in the chimney two opposite currents, which occasion the discharge of smoke into the apartment. Large fire-places and wide chimneys have been long abandoned in cities and towns; but those which are still in use are not sufficiently have still some of the inconveniences of the old ones. contracted in their dimensions, and our ordinary fire-places

The proper theory of heating by means of common fires may be understood by reference to fig. 229, in our last lesson. If

Fig. 238.

[graphic]

Fire-places.-It is well known that modern fire-places are those in which open fires are placed against the walls of apartinents, and furnished with chimneys through which the products of combustion are permitted to escape. Combustion cannot be continued without a constant renewal of the air in the apartment where it takes place. This renewal is carried on by the process of heating itself, which occasions a continual inward flow or motion of the air from without. There is, therefore, an essential and natural connection between warming and ventilation. It is remarkable that the houses discovered in Herculaneum and in Pompeii had no fire-places. Yet the invention of fire-places appears to be dated from the we take the flame of the candle, enlarge it, and maintain it by first century of the Christian era. In the times more remote, any proper means; in a word, place it in a fire-place, and it the fire was placed in the middle of the apartment to be heated, becomes a fire. The air which keeps up this fire reaches it and the smoke escaped by an opening made in the roof of the from below, enters into the parts of the fuel, combines with it, house. Thus Vitruvius recommends that the winter apart-throws out its heat, and escapes in a heated state by the chimments should not be enriched with expensive works, because ney, which discharges it into the atmosphere: all this heat is they would be injured by the smoke and the soot. lost; the only heat which is felt in the apartment is the lateral The portable brasiers or chafing-dishes still in use in Spain, heat of radiation, which is at all times much less than the heat Italy, and France, indicate the manner in which the apart- which ascends. Hence it is not surprising that a fire-place ments of Pompeii and Herculaneum were heated; and the should give out to the room which it is intended to heat only fires still to be found in the huts of the savages show their two or three per cent. of the whole of the heat produced by origin. These warmed the apartments more by radiation than the fire which is made in it. It has been even observed by by direct combustion. The dwellings of the Romans, in the experiment, that in some extraordinary cases, only one-half per first period of the empire, appear to have been heated by ovens cent. has been given out, and there are very few which give or furnaces placed below the level of the ground, and the heat out more than six per cent. In our fire-places, therefore, we was distributed through the mass of the buildings; they were burn from twenty to thirty bundles of wood in order to obtain also heated by fixed fires open on all sides, set up in the the quantity of heat which in reality is given out by a single middle of the apartments, and their smoke escaped by an bundle. orifice in the ceiling. Both these modes of heating required an enormous quantity of fuel, because the greatest quantity of the heat ascended into the open air along with the products of combustion. In the time of Seneca, flues or passages were first made in the walls of houses in order to convey the heat into the upper stories; and it is probable that this is the origin of the chimneys now in use.

The first fire-places which were placed against the walls of houses were not surrounded with flues, but only surmounted with a sort of basket-funnel which gave a passage to the smoke. From an inscription found at Venice, we learn that in 1347, an earthquake threw down a great number of chimneys. This document, reckoned the most ancient in which mention is made of the existence of chimneys, coupled with the origin of chimney-sweepers in Piedmont, confirm the opinion that chimneys were invented in Italy.

The first work in which the true principles of the construction of fire-places were laid down, appeared in 1713, under the title of Mecanique du Feu, which was published anonymously by M. Gauger. This ingenious writer observed, first, that in order to reflect the heat the chimney, jambs should be of a parabolic form, and that the lower part of the mantelpiece should be horizontal. He next proposed to make under the hearth, behind the jambs, and close to the middle of the fire-place, a series of compartments through which the air might pass in succession, from a pipe or passage communicating with the exterior, into a mouth-piece which should throw it warm into the interior of the apartment. He discovered besides, that according to the testimony of the physician-architect Savot, in his L'Architecture Française des batimens particuliers, Paris, 1624, something analagous to this had already been done in the fire-place of the reading-room of the To the fire-places in which open fires were first made, as library of the Louvre. The only difference between the latter well as to the flues or chimneys for the smoke, a very dispro- and his own plan was, that the cavity or mouth-piece was portionate magnitude was given, and this disproportion still below the hearth and behind the middle of the fire-place, so exists in country houses, especially in the chimneys. Too great that the air rushed into the apartment with less velocity and an aperture for the fire-place and too great a width for the at a lower temperature. Besides, Gauger placed on the top chimney are very serious inconveniences. The ventilation in of the chimney an apparatus for effecting the escape of the such cases is enormous, and consequently the currents of cold smoke, whichever way the wind might blow. Notwithstandair which are admitted by the apertures around the doors and ing the excellence of these principles in heating of apartments, the windows, acquire too great a velocity, and become very they were not duly appreciated; and it was not till the begin

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