Sidebilder
PDF
ePub

only necessary to place the balloon near a gasometer, and fill it by means of a connecting-pipe. In fig. 90 is represented the mode of filling a balloon with pure hydrogen. On the right of the figure is shown à series Fig. 91.

of casks, which contain iron filings, water, and sulphuric acid, substances necessary for the preparation of the hydrogen. From each cask, the gas is conveyed to a central cask, open at bottom, and immersed in a butt full of water. The gas, after passing through this water, is conveyed into the balloon by a long canvas pipe, fixed at one end to the central cask, and at the other to the bottom.

In order to facilitate the filling of the balloon, two masts are erected, having at their top pulleys traversed by a rope, which passes through a ring fixed at the top of the valve. By this means, the balloon being at first raised about a yard above the ground, the gas is admitted; then, in proportion as the balloon is filled, it is raised a little higher, and it is allowed to expand more and more, until it frees itself from this apparatus, It is now necessary to oppose the force with which it begins to ascend. For this purpose, a number of men are employed to hold it down by means of cords fixed to the netting. When the balloon is completely filled, it is then necessary to remove the pipe which conveyed the gas, and to attach the car to the net-work. These different preparatory operations require at least two hours. The aeronaut is then seated in the car, and at a given signal, the cords are loosed, and the balloon ascends with a velocity in proportion to its lightness as compared with the air which it displaces.

It is important to observe that a balloon should not be completely inflated; for the atmospheric pressure diminishing in proportion to the height of the ascent, the interior gas expands in consequence of its elastic force, and tends to make the balloon burst. It is sufficient that the force of ascent; that is,

the excess of the weight of the air displaced above the whole weight of the apparatus, be about ten pounds. It is to be observed that this force remains constant so long as the balloon is not completely inflated by the expansion of the interior gas. For, if the atmospheric pressure be reduced to one-half, the gas in the balloon, according to Mariotte's law, is increased to double its volume. Whence it follows, that the volume of air displaced is itself doubled, and its density is reduced to one-half; therefore its weight, and consequently its upward pressure or buoyancy are still the same. But as soon as the balloon is completely inflated, if it continue to rise, the force of ascent diminishes; for the volume of air displaced remaining the same, the density diminishes. Accordingly, the balloon will ere long reach a point where the upward pressure is zero. Consequently, the balloon can only take then a horizontal direction, being carried by the currents of air which exist in the atmosphere.

The indications of the barometer are the most certain means by which the aeronaut knows when he is ascending and when he is descending. In the former case, the column of mercury falls; in the latter, it rises. By the assistance of the same instrument, he is enabled to ascertain the height which he has reached. A long streamer fixed to the car, fig. 91, also indicates, by the position which it takes above or below the car, whether he is ascending or descending. When the aeronaut wishes to descend, he draws the cord which opens the valve placed at the upper part of the balloon; the hydrogen mixes with the exterior air, and the balloon descends. On the contrary, in order to slacken his descent when it is too rapid, or to re-ascend if placed in a perilous situation, the aeronaut empties bags full of sand, a sufficient quantity of which had been placed in the car for this purpose. Thus lightened, the balloon rises again, in order to descend in a more suitable place. The descent is facilitated by suspending an anchor to the car by means of a long cord. When this anchor has taken hold of a proper obstacle on the ground, the ear and balloon are lowered by gently drawing the cord.

Balloons have not as yet received any important applications. At the battle of Fleurus, in 1794, a balloon, retained by a cord, was employed to discover the movements of the enemy, which were made known to the army by signals made by an observer seated in the car. Several ascents have also been undertaken with the view of making meteorological observations in the higher regions of the atmosphere. But balloons will only be come of real utility when the power of directing them has been attained. The trials hitherto made for this purpose have completely failed. At present, we can only rise in the atmo sphere until we meet a current of air which will carry us in the direction answerable to the end we have in view.

The Parachute.-The object of the parachute (from the French, a guard from falling) is to enable the aeronaut to leave his balloon, by giving him the means of slackening the velocity of his descent. This apparatus is composed of a large circular sail, fig. 92, of about five or six yards in diameter, which, by the effect of the resistance of the air, expands and forms a huge umbrella which slowly descends to the ground. On its edges are fastened cords, which support a car, in which the aeronaut is seated. In the centre of the parachute, there is an opening for the escape of the air which is compressed by the effect of the descent; without this, the air would produce oscillations on the parachute, which would be communicated to the car and render the position of the aeronaut perilous. In fig. 91 is shown, on the side of the balloon, a parachute folded and attached to the netting, by means of a cord passing over a pulley and fixed to the car. By loosening this cord, the parachute is placed in the power of the aeronaut. M. J. Garneri was the first who descended in a parachute; but M. Blanchard appears to have been the inventor.

Weight required to raise a Balloon.-In order to calculate the weight required to raise a balloon of given dimensions, when it is supposed to be perfectly spherical, the following formula is employed: v=D3, which represents geometrically the volume of a sphere, whose diameter is D, being the ratio of the circumference to the diameter, or 31416 nearly. Thus, if a balloon of thirty-six feet in diameter were completely filled with hydrogen, its volume would be about 24,430 cubic feet. But in general, the balloon, when it begins to ascend, is only about half filled, whence its volume may be assumed at

[graphic]
[graphic]

may be readily accomplished in a Florence flask,-all the more rapidly under the influence of a gentle heat. The solution will be perfectly colourless and transparent; not the slightest amount of milkiness will be perceptible. I can fancy many a reader poring over his solution at this moment, and imagining the writer of these lessons to have erred. Some, in looking at a milky opalescent solution, will be ready to think that the assurance of "perfect clearness" is altogether untrue. If the water be quite pure, the solution will be absolutely transparent; but inasmuch as nitrate of silver is a most delicate test for certain classes of impurities, it is more than probable that many students may get a turbid solution.

Should this be the case in the present instance, heed it not. The occurrence will serve to mark a fact, without interfering with the current of our experiments. You have only to wait awhile, and the turbidity will settle, leaving a clear solution above, well adapted for our purposes. Having followed out the preceding directions, it is evident that a solution of nitrate of silver in water will have been obtained. We will proceed to investigate its chemical characters presently; meantime, let it be well impressed upon the mind that the solution is colourless: hence it follows that any solution which is not colourless, must contain some other substance besides nitrate of silver. We may generalise still more, and say that all silver solutions are colourless, Strictly true this assertion is not, I am aware; but it is, nevertheless, so nearly true, as to warrant its being considered by the student as a universal fact. Accepting the proposition as absolute, we may then make the further assertion, that, though a metallic coloured solution may contain silver, it must contain some metal in addition to silver.

The appreciation of these broad qualities-these general characteristics, are of the highest importance in chemistry: several metals being recognisable at once, by noticing the colour of their solution. That the reader may at once see the force of this remark, let him dissolve a small silver coin in some pure aquafortis, diluted with about an equal volume of water, for the purpose of moderating the violence of the action which ensues. The experiment is best conducted in a Florence flask, which may be placed in hot sand on a grate hob, in order that the injurious fumes which escape may be carried up the chimney.

When the operation of solution has been effected, remark well the tint of the resulting fluid. The experimenter has employed a silver coin, I have assumed, dissolved it in an acid, i. e. aqueous nitric acid or aquafortis. Having regard to the substances used, therefore, it would seem that a solution of nitrate of silver should result. Nevertheless the solution is no longer colourless but blue, and if the student evaporates it, blue crystals will appear. It follows, therefore, that if there be any truth in what I have stated, the silver coin must have contained something in addition to silver. Now supposing the colouring agent to be metallic, and it must be so-by "construction," as geometers say-in other words, it must be so, because we have only used a metallic coin, then it follows, firstly, that the coin was not of pure silver, but an alloy. Secondly, that the alloying substance was a metal yielding a blue nitric acid solution. Now I am only aware of two metals which are capable of yielding such a blue solution. These metals are copper and nickel; and most people know, I presume, that copper is the metal used for alloying our silver coins. Pure silver would be altogether too soft for the purpose, as the reader will not fail to see when he shall have developed a little of that metal from its liquid combination.

Put away this cupreous silver solution, duly labelled. To expatiate on it here would be so far out of order, that we are discussing the properties of silver, not copper. It will, nevertheless, come under our notice when we treat of the latter metal; indeed even before, for I shall put the student in possession of an easy means by which all the silver may be separated, and the copper left behind.

Returning now to our solution of nitrate of silver, let the student question it thus:

antimony, cadmium, nor tin, in the state of persalt; because the precipitate would either have been white or yellow. (4) Nor iron, manganese, nickel, cobalt, or uranium, because hydrosulphuric acid without ammonia does not precipitate them. Consider, then, the nature of these deductions, and see into what a corner we are driving metal, even by the evidence of one single witness.

Let us now try another witness, namely, ferrocyanide of potassium; and once for all let the student remember that hydrosulphuric acid, hydrosulphate of ammonia, and ferrocyanide of potassium, are the three witnesses always first cited in a court of chemical inquiry, supposing the substance under question to be in the state of liquidity and totally unknown. Whatever evidence is to follow, theirs comes first; all three, if we want them, or two or one as the evidence may require. As regards the case now under consideration, the reader will not fail to see that hydrosulphate of ammonia could only afford positive testimony, given already negatively by hydrosulphuric acid. Now, in many chemical examinations, negative testimony is as valuable as positive. It is so in the present instance. Let us now proceed to use the third test, ferrocyanide of potassium (yellow prussiate of potash), in solution of course. For this purpose, add a few drops of the strong nitric acid solution to a little distilled water, and test with prussiate of potash. We now get a whitish sort of precipitate.

Omitting to repeat such of the evidence yielded by this test as we happen to know already, what novelty does it communicate? What has it to say of its own specific knowledge? Why it tells us that, in addition to all the metals amongst which 'ours is not, it furthermore is not

Copper Uranium Molybdenum

Titanium;

because either of these, similarly treated, would have yielded a mahogany brown colour. This fact I have not brought before the student hitherto; let it therefore be committed to memory at once, and never forgotten. It follows, then, that our unknown metal is at length hunted into an exceedingly narrow corner. If the student will only refer to a list of metals, and see the names of those of which the present is not, he will arrive at the conclusion that it must be one of a very few. At this point I will assume the operator to appeal to the evidence of another test, either hydrochloric acid (spirit of salt), or else common salt dissolved in water; practically, so far as relates to the present investigation, these tests are the same, and the student may use whichsoever he pleases.

Treated with either of these substances, our solution (assumed to be unknown) will throw down a dense white precipitate; hence we know at once that the metal we are hunting for is either silver or mercury; no other metals being capable of producing a similar effect.

Finally, the addition of a little hartshorn (liquor ammonia) causes the precipitate to dissolve and the whiteness totally to disappear; which characteristic result demonstrates the metal to be silver, nothing but silver.

CURIOSITY.

Its aim oft idle, lovely in its end,

We turn to look, then linger to befriend;
The maid of Egypt thus was made to save
A nation's future leader from the wave;
New things to hear, when erst the Gentiles ran,
Truth closed what Curiosity began.
How many a noble art, now widely known,
Owes its young impulse to this power alone;
E'en in its slightest working, we may trace
A deed that changed the fortunes of a race:
Bruce, banned and hunted on his native soil,
With curious eye surveyed a spider's toil;
Six times the little spider strove and failed;
Six times the chief before his foes had quailed;
"Once more," he cried, "in thine, my doom I read,
Once more I dare the fight, if thou succeed;
"Twas done: the insect's fate he made his own:
Once more the battle waged, and gained a throne.

[blocks in formation]

Cependant il entrait encore quelque hésitation dans la compagnie,' et déjà deux fois le capitaine qui commandait avait donné l'ordre au tambour-maître de prendre deux tambours, de se mettre en avant, et de battre la charge.2 Celui-ci restait appuyé sur sa grande canne,3 hochant la tête et peu disposé à obéir. Pendant ce temps Bilboquet, à cheval sur son tambour et les yeux levés sur son chef, sifflait un air de fifre et battait le pas accéléré avec ses doigts. Enfin l'ordre venait d'être donné une troisième fois au tambour-maître, et il ne paraissait pas disposé à obéir, lorsque tout à coup, Bilboquet se relève, accroche son tambour à son côté, prend ses baguettes, et passant sous le nez du tambour-maître, il le toise avec orgueil, lui rend d'un seul mot toutes les injures qu'il avait sur le cœur, et luit dit :-Viens donc, grand poltron!

Le tambour-maître veuts lever sa canne, mais déjà Bilboquet était à la tête des deux compagnies,10 battant la charge comme un enragé. Les soldats, à cet aspect, s'avancent après lui et courent vers la terrible batterie." Elle décharge d'un seul coup ses six pièces de canon, et des rangs de nos braves voltigeurs s'abattent et ne se relèvent plus.12 La fumée, poussée par le vent, les enveloppe, le fracas du canon les étourdit; mais la fumée passe, le bruit cesse, un instant, et ils voient debout, à vingt pas devant eux, l'intrépide Bilboquet battant la charge, et ils entendent son tambour,14 dont le bruit, tout faible qu'il soit,' semble narguer tous ces gros canons qui viennent de tirer. Les voltigeurs courent toujours, et toujours, 15 devant eux le tambour et son terrible rlan rlan les appelle;" enfin une second décharge de la batterie éclate et perce d'une grèle de mitraille les débris acharnés des deux belles compagnies.16 A ce moment, Bilboquet se retourne et voit qu'il reste à peine cinquante hommes des deux cents qui étaient partis, et aussitôt, comme transporté d'une fureur de vengeance, il redouble de fracas: 13 on eût dit vingt tambours battant à la fois ; jamais le tambour-maître n'avait si hardiment frappé une caisse. Les soldats s'élancent de nouveau et entrent dans la batterie,19 Bilboquet le premier, criant à tue-tête aux Russes:

17

[blocks in formation]

108.-g. from vouloir; L. part. ii., p. 110.—h. enragé, madman.
i. 8. 4, R. 1.-J. S. 98, R. 1.-k. from voir; L. part ii., p. 110.
-. subjunctive of être.-m. from venir; L. S. 25, R. 2.-n. L.
part ii., § 49, R. (4).-
.-o. on eut dit, one would have thought
that; literally, said.-p. à tue-tête, with all his might.

SECTION IV.

Pendant ce temps, Napoléon monté sur un tertre, regardait exécuter cette prise héroique. A chaque décharge, il tressaillait sur son cheval isabelle; puis, quand les soldats entrèrent dans la batterie, il baissa sa lorgnette en disant tout bas: Braves gens!"

Et dix mille hommes de la garde, qui étaient derrière lui, se mirent à battre des mains et à applaudir3 en criant: -Bravo, les voltigeurs!! Et ils s'y connaissaient,

vous assure

d

[blocks in formation]

NOTES AND REFERENCES.-a. L. S. 20, R. 2.-b. from dire, part ii., p. 88.-c. se mirent, commenced; L. S. 68, R. 3.—d. ils s'y connaissaient, they were good judges of such things; L. S. 86, R. 6.-e. from courir; L. part ., p. 84-f. from revenir; mit, presented; from remettre; L. part ii., p. 102.-i. fit enL. part ii., p. 104.-g. from prendre; L. part ii., p. 100.-k. re tendre, uttered; from faire; L. part ii., p. 92. j. accent, tone.jek. L. p. ii., § 38, R. (8).-. planté, standing; literally, planted, posted.- -m. j'en étais, I was one of them, of the number.—n. L. part ii., § 33, R. (9).- -o. from battre; L. part ii., p. 80.-p. L. S. 80, R. 2.-q. que veux-tu, how can I help it; literally, what do you wish.-r. L. §. 61, R. 5.—s. en attendant, meanwhile.—t. from dire; L. part ii., p. 88.-u. il s'était fait, there was.-v. from retour-paraître; L. part ii., p. 98.-w. L. S. 26, R. 2.—. toujours, notwithstanding; literally, always.

Aussitôt, sur l'ordre de Napoléon, un aide-de-camp cou-
rut jusqu'à la batterie et revint' au galop.
-Combien sont-ils arrivés ? dit l'Empereur.
-Quarante, répondit l'aide-de-camp.

Quarante croix demain, dit l'Empereur en se nant vers son major-général.

SECTION V.

a

Véritablement, le lendemain, tout le régiment forma un grand cercle autour des restes des deux compagnies de voltigeurs, et on appela successivement le nom des quarante A partir de ce jour, on ne se moqua plus autant du braves qui avaient pris la batterie, et l'on remit à chacun petit Bilboquet,' mais il n'en devint pas pour cela plus d'eux la croix de la Légion-d'Honneur. La cérémonie communicatif; au contraire, il semblait rouler dans sa tête était finie, et tout le monde allait se retirer, lorsqu'une quelque fameux projet, et, au lieu de dépenser son argent voix sortit du rang et fit entendre ces mots, prononcés avec ses camarades, comme ceux-ci s'y attendaient, il le avec un singulier accent de surprise: serra soigneusement."

10

-Et moi! moi! je n'ai donc rien? -Le général qui distribuait les croix, se retourna et vit planté devant lui notre camarade Bilboquet, les joues rouges et l'œil presque en larmes."

-Toi? lui dit-il, que demandes-tu? -Mais, mon général, j'en étais dit Bilboquet presque en colère; c'est moi qui battais la charge en avant, c'est moi qui suis entré le premier.

-Que veux-tu, mon garçon ? on t'a oublié, répondit le général; d'ailleurs, ajouta-t-il en considérant que c'était un enfant, tu es encore bien jeune, on te la donnera quand tu auras de la barbe au menton; 13 en attendant, voilà de quoi te consoler.

k

Quelque temps après, les troupes françaises entrèrent à Smolensk, victorieuses et pleines d'ardeur; Bilboquet en était, et le jour même de l'arrivée, il alla se promener dans la ville, paraissant très content de presque tous les visages qu'il rencontrait: il les considérait d'un air riant et semblait les examiner comme un amateur qui choisit des marchandises. Il faut vous dire cependant, qu'il ne regardait ainsi que les paysans qui portaient des grandes barbes.7 Elles étaient sans doute très longues et très fournies, mais d'un roux si laid, qu'après un moment d'examen Bilboquet tournait la tête et allait plus loin. Eufin, en allant ainsi, notre tambour arriva au quartier des Juifs.8 Les Juifs à Smolensk, comme dans toute la Pologne En disant ces paroles, le général tendit une pièce de et la Russie, vendent toutes sortes d'objets et ont an vingt francs au pauvre Bilboquet, qui la regarda sans quartier particulier.10 Dès que Bilboquet y fut entré, ce penser à la pendre. Il s'était fait un grand silence fut pour lui un véritable ravissement: imaginez-vous les autour de lui, et chacun le considérait attentivement; 19 lui, plus belles barbes du monde, noires comme de l'ébène;" demeurait immobile devant le général et de grosses larmes car la nation juive toute dispersée qu'elle est, parmi les roulaient dans ses yeux.1 Ceux qui s'étaient le plus autres nations, a gardé la teinte brune de sa peau et le moqués de lui paraissaient attendris,1s et peut-être allait-noir éclat de ses cheveux.13 Voilà donc notre Bilboquet on élever une réclamation" en sa faveur, lorsqu'il releva enchanté. Enfin il se décide, et entre dans une petite vivement la tête, comme s'il venait de prendre une grande boutique 14 où se trouvait un marchand magnifiquement résolution, et il dit au général : barbu. Le marchand s'approche de notre ami et lui de mande humblement en mauvais français:

17

W

-C'est bon, donnez toujours, ce sera pour une autre fois.20

Et sans plus de façons, il mit la pièce dans sa poche et s'en retourna dans son rang en sifflant d'un air délibéré et satisfait.21

[blocks in formation]

-Que voulez-vous mon petit Monsieur ?16

-Je veux ta barbe répondit cavalièrement Bilboquet.17 -Ma barbe! dit le marchand stupéfait; vous voulez rire ? 18

-Je te dis, vaincu, que je veux ta barbe, reprend le vainqueur superbe en posant la main sur son sabre; mais ne crois pas que je veuille te la voler:19 tiens, voila un napoléon, tu me rendras mon reste.

[blocks in formation]
[blocks in formation]

p.

88.

NOTES AND REFERENCES.-a. From se moquer; to laugh at. -b. en, on that account.-c. from devenir; L. part ii., d. L. S. 34, R. 4.-e. ils s'y attendaient, they expected. f. L. part ii., § 145.-g. L. S. 35, R. 5.-h. from paraître; L. part ii., p. 98.. il faut, I must; from falloir; L. S. 47; also L. part ii., p. 92.j. portaient, wore.-k fournies, thick.-l. L. part ii., § 39, R (18).-m. voilà donc, behold then.-n. from vouloir; L. part ii., p. 110.-o. vous voulez rire, you are joking, you are not in earnest.-p. from vouloir.-q. tiens, here; literally, hold; from tenir, L. part ii., p. 108.-r. reste, change.

SKELETON MAP S.-No. V.

MAP OF SOUTH AMERICA.

OUR Map of Russia in Europe (the approximate seat of war) not being ready, as intended, for this month, we insert in this Number a Skeleton or Outline Map of the Continent of South America, including the continental part of the West Indies called Guiana, and the small islands adjacent to the continent all around it. This Map will be useful to emigrants, settlers, or colonists, who wish to transplant themselves to South America, where there is abundance of room for speculations of all kinds. If such persons have sufficient time and skill to fill up this Map for themselves, the process of doing so will make them better acquainted with the country in which they intend to settle, than many Lessons in Geography, which consist of the mere descriptions of places, but give no idea of their relative position in regard to one another.

An extensive list of the latitudes and longitudes of the chief or capital towns in the various countries and sub-divisions of the continent, and of the islands of South America, will be found in Vol. iii., at page 250; and, as the continental part of the West Indies is included in this Map, the latitudes and longitudes for the chief towns of this part will be found at page 118. On the marginal space of the Map, we have given the latitudes and longitudes of the principal islands, capes, bays, rivers, and ports along the eastern and western coasts of the continent, from Cape Horn to the Isthmus of Panama, in regular order, proceeding from south to north, and along the coast of America situated on the Caribbean Sea. These we have added to the latitudes and longitudes of the places in the interior of the continent above-mentioned, so as to enable our students to make their Map as complete as possible.

LESSONS IN GEOMETRY.-No. XXVI.
LECTURES ON EUCLID.
(Continued from page 256.)
PROPOSITION XXVII.-THEOREM.

If a straight line falling upon two other straight lines, make the
alternate 'angles equal to one another; these two straight lines
are parallel.

In fig. 27, let the straight line EF which falls upon the two straight lines AB and CD, make the alternate angles A EF and EFD equal to one another: then a B is parallel to c D.

For if AB be not parallel to c D2 these two straight lines will

с

Fig. 27.

meet, if produced either towards A and c, or towards B and D.

Let them be produced and meet towards B and D, in the point G; then GE F is a triangle.

Now, in the triangle GEF the exterior angle ABF is greater (I. 16) than its interior and opposite angle EFG; but the angle AEF is equal (Hyp.) to the angle EFG; therefore the angle A EF is both greater than, and equal to, the angle EFG; which is impossible. Wherefore the straight lines A B and C B, if produced, do not meet towards B and D. In the same manner it may be proved, that they do not meet if produced towards A and c. But those straight lines in the same plane, which do not meet when produced ever so far either way, are parallel (Def. 33). Therefore A B is parallel to CD. Wherefore, if a straight line falling upon two other straight lines, &c. Q. E. D.

Scholium 1. The angles A E F and EFD are called alternate angles, or more properly interior alternate angles, because they are on opposite sides of the straight line EF, and the one has its vertex at E the one extremity of the portion between the parallels, while the other has its vertex at F the other extremity of the same.

Scholium 2. In the diagram the crooked lines E B G and FDG must be considered straight lines, and the figure EFDG B a triangle, for the sake of the argument. Otherwise, the figure might have been constructed so that the straight lines AB and CD should actually converge and meet in a point.

EXERCISE I. TO PROPOSITION XXVII.

If a straight line falling upon two other straight lines, make the exterior alternate angles equal to each other, these two straight lines are parallel.

In fig. 28, let the straight line EF, which falls upon the two straight lines A B and CD, make the two exterior alternate angles AGE and F HD equal to one another; then A B is parallel to c n. Because (I. 13) the two angles A G E and A G H are equal to two right angles, and the two angles FHD and GHD are equal to two right angles; therefore (Ax. 1) the two angles AGE and AGH are equal to the two angles FHD and GHD. But (Hyp.) the angle AGE is equal to the angle FHD; therefore (Ax. 3) the angle & G H is equal to the angle G H D; and they are alternate angles; wherefore (1. 27) the straight lines A B and C D are parallel. Q. E. D.

EXERCISE II. TO PROPOSITION XXVII.

If a straight line falling upon two other straight lines, make the two exterior angles on the same side of it equal to two right angles, these two straight lines are parallel. †

In fig. 28, let the straight line EF, which falls upon the straight lines ▲ B and C D, make the two exterior angles on the same side of it, E G B and FH D, equal to two right angles; then A B is parallel

to C D.

Because (I. 13) the two angles EG B and EG A are equal to two right angles, and (Hyp.) the two angles EGB and FHD equal to two right angles; therefore (A. 1.) the two angles E GB and E GA are equal to the two angles BG B and FHD; from these equals take away the common angle EG B, and (Ax. 3) the angle E GA is equal to the angle FHD; but these are the two exterior alternate angles; wherefore, by the preceding exercise, the straight lines a B and CD are parallel. Q. E. D.

PROPOSITION XXVIII-THEOREM.

If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite angle upon the same side of the straight line; or make the two interior angles upon the same side of it, together equal to two right angles; these two straight lines are parallel to one another.

Let the straight line Er, falling upon the two straight lines A B and CD, make the exterior angle E G B equal to the interior and opposite angle & HD upon the same side of or make the two interior angles Fi BGH and GHD on the same side of it, together equal to two right angles; then A B is parallel to CD.

Because the angle EGR is equal (Hyp.)

to the angle G HD, and the angle EGB is C
equal (I. 15) to the angle AG H; therefore

the angle AG is equal (Ax. 1) to the

Fig. 28.

E

G

[blocks in formation]
« ForrigeFortsett »