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12,215 cubic feet; and such is the volume of displaced air at may be readily accomplished in a Florence flask,-all the more the first moment of its ascent. According to calculations | rapidly under the influence of a gentle heat. The solution will formerly shown, this quantity of air weighs about 991 lbs. or nearly nine cwt., and this is the upward pressure which tends Fig. 92.

to raise the balloon. But in order to calculate the real force of the ascent, we must subtract from this pressure the weight of the hydrogen in the balloon, and of the globe of which it is made, with its appendages. Now, the weight of hydrogen is about part of the weight of air; whence, the weight of the gas in the balloon is about 9911-14-71 lbs., nearly. Adding to this weight that of the globe and its appendages, formerly reckoned at about three cwt., we have upwards of 3 cwt., say four cwt., for the weight to be subtracted from the nine ewt. just mentioned; this leaves a remainder of about five cwt. for the force of the ascent. But we have seen that it is sufficient for the force of ascent to be about 10 lbs. ; whence, there is a little less than the weight of five cwt. remaining for the additional weight which a balloon may safely carry into the atmosphere.

LESSONS IN CHEMISTRY.-No. XIX. THE subject of our present lesson shall be the metal silver; not only so interesting for its commercial value, but as regards its striking chemical qualities.

I

There are not many metals which admit of being traced through a long list of combinations, and again obtained in the metallic form, so easily as silver. Its chemical physiognomy is, in point of fact, exceedingly well marked, as we shall presently see. It is always well to begin the chemical examination of a substance, by choosing the same in a pure condition, unmixed with any accessory that might veil its properties or obscure the result. therefore recommend, as the source for obtaining a silver specimen, a few grains, say eighteen or twenty, of the salt called nitrate of silver. This substance occurs in commerce under two forms: either as sticks something like slate-pencil, only whiter, or as crystals. The latter will be somewhat the purer of the two; but the former, known popularly as "lunar caustic," will answer very well.

Let the student then take about eighteen or twenty grains of lunar caustic, or rather more of crystallised nitrate of silver, and effect a solution of the substance in about half a pint of distilled water. The solution takes place with great facility, and

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 waterwill 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 Pure silver is the metal used for alloying our silver coins. 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. To

Put away this cupreous silver solution, duly labelled. 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:

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(1) What is its nature?

To arrive at an answer to this question, drop a little of your strong solution, say twenty or thirty drops, into a wine-glass; fill up the wine-glass with distilled water, and test with hydrosulphuric acid solution. We get a well-pronounced black precipitate, on observing which we immediately deduce the following truths. (1) The solution contains as its base, a metal. (2) A calcigenous metal (vide Lesson p. 39). (3) Neither zine, arsenic,

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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;

FRENCH READING S.--No. II.

LE SAPEUR DE DIX ANS.
SECTION III.

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 chevalt 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 done, grand poltron !s

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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.11 Elle décharge d'un seul coup ses six pièces de canon, et desi 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,13 et ils entendent son tambour, dont le bruit, tout faible qu'il soit, semble narguer tous ces gros canons qui viennent de tirer. because either of these, similarly treated, would have yielded a Les voltigeurs courent toujours, et toujours,13 devant eux le mahogany brown colour. This fact I have not brought before the tambour et son terrible rlan rlan les appelle;" enfin une student hitherto; let it therefore be committed to memory at once, second décharge de la batterie éclate et perce d'une and never forgotten. It follows, then, that our unknown metal is grèle de mitraille les débris acharnés des deux belles com at length hunted into an exceedingly narrow corner. If the student pagnies.16 A ce moment, Bilboquet se retourne et voit qu'il will only refer to a list of metals, and see the names of those of which reste à peine cinquante hommes des deux cents qui étaient the present is not, he will arrive at the conclusion that it must be partis, et aussitôt, comme transporté d'une fureur de venone of a very few. At this point I will assume the operator to appeal to the evidence of another test, either hydrochloric acid geance, il redouble de fracas: 15 on eût dit vingt tambours (spirit of salt), or else common salt dissolved in water; practi-battant à la fois ; jamais le tambour-maître n'avait si hardically, so far as relates to the present investigation, these tests are ment frappé une caisse. Les soldats s'élancent de nouveau the same, and the student may use whichsoever he pleases. et entrent dans la batterie,19 Bilboquet le premier, criant à Treated with either of these substances, our solution (assumed to tue-tête aux Russes: 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.

17

-Les morceaux en sont bons, les voici; 20 attendez, attendez !

COLLOQUIAL EXERCISE.

1. Que remarquait-on néan- | 11. Que firent les soldats en
moins dans la compagnie? voyant son intrépidité ?
12. Quel effect produisit la dé-

2. Quel ordre le capitaine avait-
il donné au tambour-maître ?
3. Que fit celui-ci après avoir
reçu cet ordre ?

4. Où était Bilboquet pendant
ce temps là ?

5. Que faisait-il?

6. Le tambour-maître parais-
sait-il disposé à obéir au troi-
sième ordre ?

7. Que fit alors le petit tam-
bour.

8. Comment apostropha-t-il le
tambour-maître ?

9. Que voulut faire le tambour-
maître ?

10. Où était alors notre héros?

|

charge des six pièces de canon? 13. Que virent les soldats quand la fumée fut dissipée ? 14. Qu'entendaient-ils malgré le bruit du canon?

15. Que firent alors nos voltigeurs ?

16. Quel fut l'effet d'une seconde décharge?

17. Combien d'hommes restaitil?

18. Que fit Bilboquet à la vue du carnage?

19. Que firent alors les soldats? 20. Que cria alors le petit tambour?

NOTES AND REFERENCES.-α. Il entrait, there was; the verb is unipersonal in French; L. part ii., § 43, R. (7).—6 à cheval, seated across.—ç. venait d'être, had just been; Ì. S. 25, R. 2. d. from paraître; L. part ii., p. 98.—e. sous le nez, close to the face; literally, under the nose.-f. from venir; L. part ii., p.

108.-9. from vouloir; L. part. ii., p. 110.—h. enragé, madman. | 13. Que dit le général, quand il
—¿. §. 4, R. 1.—j. S. 98, R. 1.—k. from voir; L. part ii., p. 110.
-7. 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 exécutera 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! 2

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

Yous assure.

Aussitôt, sur l'ordre de Napoléon, un aide-de-camp courut jusqu'à la batterie et revint au galop.

-Combien sont-ils arrivés ?5 dit l'Empereur. -Quarante, répondit l'aide-de-camp.

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

g

eut remarqué que Bilboquet
n'était qu'un enfant ?
14. Que lui donna-t-il ?
15. Bilboquet prit-il la pièce ?
16. Regardait-on le petit tam-
bour?

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19. Qu'allait-on faire en sa faveur ?

20. Que dit-il enfin au général.

21. Que fit-il après avoir mis l'argent dans sa poche?

17. Que faisait-il alors ?
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 ii., p. 84.-f. from revenir ;
mit, presented; from remettre; L. part ii., p. 102.-. fit en-
L. part ii., p. 104.-g. from prendre; L. part ii., p. 100.-h. re-
tendre, uttered; from faire; L. part ii., p. 92. j. accent, tone.-
z. L. p. ii., § 33, R. (8).—7. planté, standing; literally, planted,
posted.- -m. j'en étais, I was one of them, of the number.-n. L.
part ii., § 38, R. (9).—o. from battre, L. part ii., p. 80.-p. I.
S. 80, R. 2.-q. que veux-tu, how can I help it; literally, what do
you wish.―r. L. S. 61, R. 5.-s. en attendant, meanwhile.-t. from
dire; L. part iì., p. 88.-u. il s'était fait, there was.—v. from
paraître, L. part ii., p. 98.-w. L. S. 25, R. 2.—x. toujours,
notwithstanding; literally, always.

SECTION V.

b

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 a plus autant du braves qui avaient pris la batterie, et l'on remità à chacun petit Bilboquet,1 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 entendrei ces mots,10 prononcés avec ses camarades, comme ceux-ci s'y attendaient, il le avec un singulier accenti de surprise: serra soigneusement.2

-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."1

Toi? lui dit-il, que demandes-tu?

n

Mais, mon général, j'en étais dit Bilboquet presque en colère; 12 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.

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3

à Smolensk. victorieuses et pleines d'ardeur; Bilboquet en Quelque temps après, les troupes françaises entrèrent était, et le jour même de l'arrivée, il alla se promeners dans la ville, paraissant très content de presque tous les visages qu'il rencontrait: 5 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 portaienti des grandes barbes." 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. 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 un vingt francs 14 au pauvre Bilboquet, qui la regarda sans quartier particulier.10 Dès que Bilboquet y fut entré, ce penser à la pendre.15 Il s'était fait un grand silence fut pour lui un véritable ravissement:11 imaginez-vous les autour de lui, et chacun le considérait attentivement; 16 lui, plus belles barbes du monde, noires comme de l'ébène;12 demeurait immobile devant le général et de grosses larmes car la nation juive toute dispersée qu'elle est, parmi les roulaient dans ses yeux. 17 Ceux qui s'étaient le plus autres nations, a gardé la teinte brune de sa peau et le moqués de lui paraissaient attendris, 18 et peut-être allait-noir éclat de ses cheveux,13 Voilà donc notre Bilboquet on élever une réclamation 19 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.15 Le marchand s'approche de notre ami et lui de mande humblement en mauvais français:

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

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8. Où arriva-t-il enfin ?

9. Que font les Juifs en Russie ?

10. Où demeurent-ils ? 11. Quel sentiment éprouva Bilboquet, quand il fut entré dans ce quartier ? 12. Pourquoi était-il si content? 13. Quelle remarque l'auteur fait-il à propos de la nation juive ?

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14. Où Bilboquet entra-t-il en- Let them be produced and meet towards B and D, in the point &; fin? then GE is a triangle.

15. Qui trouva-t-il dans la boutique ?

16. Que dit le marchand au petit tambour? 17. Que lui demanda celui-ci ? 18. Quelle fut la réponse du marchand?

19. Qu'ajouta Bilboquet en mettant la main sur son sabre?

NOTES AND REFERENCES.-a. From se moquer; to laugh at. -b. en, on that account.-c. from devenir; L. part ii., p. 88.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.-i. il faut, I must; from falloir; L. S. 47; also L. part ii., p. 92. j. portaient, wore.-k. fournies, thick.-7. L. part ii., § 39, R (18).—m, 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.

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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 EPD equal to one another: then A B is parallel to c D.

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

Fig. 27.

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

Now, in the triangle GEF the exterior angle A EF is greater (I. 16) than its interior and opposite angle E F G ; but the angle ABF is equal (Hyp.) to the angle EFG; therefore the angle A È F is both greater than, and equal to, the angle E Fe; 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 AB 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 B B G and FDG must be considered straight lines, and the figure EFD G 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 AGB and F H D equal to one another; then A B is parallel to c D.

Because (I. 13) the two angles A G E and ▲ 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 G H D. But (Hyp.) the angle AGB is equal to the angle F HD; therefore (Ax. 3) the angle A & H fore (I. 27) the straight lines AB and C D are parallel. Q. E. D.* is equal to the angie G H D; and they are alternate angles; where

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.†

E

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

Because (I. 13) the two angles EG B and E GA are equal to two right angles, and (Hyp.) the two angles EGB and FHD equal to two right angles; therefore (Az. 1.) the two angles E GB and E G A are equal to the two angles EGB and FHD; from these equals take away the common angle E G 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 EF, falling upon the two straight lines A B and CD, make the exterior angle E G B equal to the interior and opposite angle G HD upon the same side of EF; or make the two interior angles 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 E G B is equal (Hyp.) to the angle GHD, and the angle EGB is C equal (I. Ï5) to the angle a GH; therefore the angle AGH is equal (Ar. 1) to the

Fig. 28.

E

G

B

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* Solved by Q. Pringle, Glasgow; J. H. Eastwood, Middleton; and

E. J. Bremner, Carlisle.

+ See new edition of Cassell's Euclid, 1854.

angle GHD; and they are alternate angles; wherefore A B is parallel (I. 27) to CD.

Again, because the two angles BGH and GHD are together equal (Hyp.) to two right angles; and the two angles A G H and BG H are also together equal (I. 13) to two right angles; therefore the two angles A GH and B G H are equal (Ax. 1) to the two angles B G H and G H D. Take away from these equals the common angle BGH, and the remaining angle A G H is equal (Ax. 3) to the remaining angle & HD; but they are alternate angles; therefore a B is parallel (I. 27) to c D. Wherefore, if a straight line, &c. Q. E. D. Scholium 1. The twelfth axiom will now be admitted by the student as a corollary to this proposition; especially when Prop. XVII. and the note added to the twelfth axiom are taken into

account.

Scholium. 2: We think it right to introduce our students at this point, to a discussion on the "Theory of parallel straight lines," which will be of immense advantage to them in their future studies. Our first extract shall be from the Gower-street edition of Euclid.

"The theory of parallel [straight] lines has always been considered as the reproach of Geometry. The beautiful chain of reasoning by which the truths of this science are connected here wants a link, and we are reluctantly compelled to assume as an axiom what ought to be matter of demonstration. The most eminent geometers, ancient and modern, have attempted without success to remove this defect; and after the labours of the learned for 2,000 years have failed to improve or supersede it, Euclid's theory of parallels maintains its superiority. We shall here endeavour to explain the nature of the difficulty which attends this investigation, and shall give some account of the theories which have been proposed as improvements on, or substitutes for, that of Euclid.

"Of the properties by which two right [straight] lines described upon the same plane are related, there are several which characterise two parallel [straight] lines and distinguish them from [straight] lines which intersect. If any one of such properties be assumed as the definition of parallel [straight] lines, all the others should flow demonstratively from it. As far, therefore, as the strict principles of logic are concerned, it is a matter of indifference which of the properties be taken as the definition. In the choice of a definition, however, we should be directed also by other circumstances. That property is obviously to be preferred from which all the others follow with the greatest ease and clearness. It is also manifest that, cæteris paribus, that property should be selected which is most conformable to the commonly received notion of the thing defined. These circumstances should be attended to in every definition, and the exertion of considerable skill is necessary almost in every case. But in the selection of a definition for parallel [straight] lines there is a difficulty of another kind. It has been found, that whatever property of parallels be selected as the basis of their definition, the deduction of all the other properties from it is impracticable. Under these circumstances, the only expedient which presents itself, is to assume, besides the property selected for the definition, another property as an axiom. This is what every mathematician who has attempted to institute a theory of parallel [straight] lines has done. Some, it is true, have professed to dispense with an axiom, and to derive all the properties directly from their definition. But these, with a single exception, which we shall mention hereafter, have fallen into an illogicism inexcusable in geometers. We find invariably a petitio principii, either incorporated in their definition, or lurking in some complicated demonstration. A rigorous dissection of the reasoning never fails to lay bare the sophism.

"Of the pretensions of those who avowedly assume an axiom it is easy to judge. When Euclid's axiom is removed from the disadvantageous position which it has hitherto maintained, put in its natural place, and the terms in which it is expressed somewhat changed, I think it will be acknowledged that no proposition which has ever yet been offered as a substitute for it, is so nearly self-evident. But it is not alone in the degree of self-evidence of his axiom, if we be permitted the phrase, that Euclid's theory of parallels is superior to those theories which are founded on other axioms. The superior simplicity of the structure which he has raised upon it is still more conspicuous. When you have once admitted Euclid's axiom, all his theorems flow from that and his definition, as the most simple and obvious inferences. In other theories, after conceding an axiom much further removed from self-evidence than Euclid's, a labyrinth of complicated and indirect demonstration remains to be threaded, requiring much subtlety and attention to be assured that error and fallacy do not lurk in its mazes.

"Euclid selects for his definition that property in virtue of

which parallel [straight] lines, though indefinitely produced, can
never intersect. This is, perhaps, the most ordinary idea of
parallelism. Almost every other property of parallels requires
some consideration before an uninstructed mind assents to it; but
the possibility of two such [straight] lines intersecting is repug-
nant to every notion of parallelism.
self-evident, or presumed and declared to be so, it ought to be
"When the possible existence of the subject of a definition is not
proved so. This is the case with Euclid's definition of parallels.
How, it may be asked, does it appear that two right [straight]
lines can be drawn upon the same plane so as never to intersect
though infinitely produced? Euclid meets this objection in his
27th proposition, where he shows that if two [straight] lines be
inclined at equal alternate angles to a third, the supposed possi-
bility of their intersection would lead to a manifest contradic-
tion. Thus it appears, that through a given point one right
[straight] line at least may always be drawn parallel to a given
than one parallel can be drawn through the same point to the same
right [straight] line. But it still remains to be shown, that no more
right [straight] line. And here the chain of proof is broken. Euclid
was unable to demonstrate, that every other [straight] line except
intersect the given right [straight] line if both be sufficiently
that which makes the alternate angles equal will necessarily
produced. He accordingly found himself compelled to place the
deficient link among his axioms."

We now add to this extract, notices of thirty different methods, proposed at various epochs in the history of Geometry, for getting over the difficulty of the Twelfth Axiom of Euclid's First Book. This collection is taken from Col. P. Thompson's "Geometry without Axioms," pp. 137-156.

"The uses of such a Collection are to throw light on the particulars which experience has shown are not to be left unguarded in any attempt at solution, and to prevent explorers from consuming their time in exhausted tracts. To which may be added, that out of so many efforts, some, either by improvement or by a fortunate conjunction with others, may finally be found operative towards the solution desired.

1. The objection to Euclid's Axiom (independently of the objections common to all Axioms), is that there is no more reason why it should be taken for granted without proof, than numerous other propositions which are the subjects of formal demonstration, and the taking any one of which for granted would equally lead to the establishment of the matter in dispute.

2. Ptolemy the astronomer, who wrote a treatise on Parallel Lines, of which extracts are preserved by Proclus, proposed to prove that if a straight line cuts two parallel straight lines, the two interior angles on each side are together equal to two right angles, by saying that if the interior angles on the one side are greater than two right angles, then because the lines on one side are no more parallel than those on the other, the two interior angles on the other side must likewise be together greater than two right angles, and the whole greater than four, which is impossible; and in the same way if they were supposed less. In which the palpable weakness is, that there is no proof, evidence, or cause of probability assigned, why parallelism should be connected with the angles on one side being together equal to those on the other; the very question in debate being, whether they may not be a little more than two right angles on one side and a little less on the other, and still the straight lines not meet.

3. Proclus himself proposes" to take an Axiom of this sort, being the same that Aristotle employed to establish that the world is finite. If from the same point, two straight lines are drawn making an angle, the distance between them when they are prolonged to infinity will exceed any finite distance that may be assigned. He then showed that if the straight lines prolonged from this centre towards the circumference are of infinite length, what is between them is also infinite; for if it was finite, to increase the distance would be impossible, and consequently the straight lines would not be infinite. The straight lines therefore on being prolonged to infinity, will separate from each other by more than any finite quantity assigned. But if this be previously admitted, I affirm that if any straight line cuts one of two parallel straight lines, it will cut the other also. For let A B and C D be parallel, and let E F cut A B in G; I say that it will cut C D also. For since

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* οὐδέν γαρ μᾶλλον αἱ αζ γη παράλληλοι ἢ αἱ ζδ ηβ.—Procli Comment. in Primum Euclidis Librum. Lib. 4. It is but right to notice, that Proclus calls this παραλογισμός and δείξεως a0évera; and Barocins the Venetian Translator in 1550, notes it in the margin as Flagitiosa Ptolemai ratiocinatio.

Professor Play air says it is curious to observe in Proclus's account an argument founded on the principle known to the moderns by the name of the sufficient reason (Elem. of Geom. p. 405). If the allusion is to this part, the "sufficient reason" of the moderns must be something very feeble.

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