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 be perfectly colourless and transparent; not the slightest amount of nearly nine cwt., and this is the upward pressure which tends milkiness will be perceptible. I can fancy many a reader poring over his solution at this moment, and imagining the writer of these Fig. 93. 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 faot, 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 evideni 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, ve 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 experito raise the balloon. But in order to calculate the real force ment is best conducted in a Florence flask, which may be placed of the ascent, we must subtract from this pressure the weight in hot sand on a grate hob, in order that the injurious fumes of the hydrogen in the balloon, and of the globe of which it is which escape may be carried up the chimney. made, with its appendages. Now, the weight of hydrogen is When the operation of solution has been effected, remark well about part of the weight of air; whence, the weight of the the tint of the resulting fluid. The experimenter has employed gas in the balloon is about 9911-14371 lbs., nearly. Adding a silver coin, I have assumed, dissolved it in an acid, i. e. aqueous to this weight that of the globe and its appendages, formerly nitric acid or aquafortis. Having regard to the substances used, reckoned at about three cwt., we have upwards of 33 cwt., therefore, it would seem that a solution of nitrate of silver should say four cwt., for the weight to be subtracted from the nine result. Nevertheless the solution is no longer colourless but blue, ewt. just mentioned; this leaves a remainder of about five and if the student evaporates it, blue crystals will appear. It cwt. for the force of the ascent. But we have seen that it is follows, therefore, that if there be any truth in what I have sufficient for the force of ascent to be about 10 lbs.; whence, stated, the silver coin must have contained something in addition there is a little less than the weight of five cwt. remaining to silver. Now supposing the colouring agent to be metallic, and for the additional weight which à balloon may safely carry it must be so- by“construction," as geometers say-in other words, into the atmosphere. 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 LESSONS IN CHEMISTRY.-No. XIX. are capable of yielding such a blue solution. These metals are copper and nickel; and most people know, I presume, that copper The subject of our present lesson shall be the metal silver; is the metal used for alloying our silver coins. Pure silver not only so interesting for its commercial value, but as regards would be altogether too soft for the purpose, as the reader will its striking chemical qualities. not fail to see when he shall bave developed a little of that metal There are not many metals which admit of being traced through from its liquid combination. a long list of combinations, and again obtained in the metallic Put away this cupreous silver solution, duly labelled. To form, 80 easily as silver. Its chemical physiognomy is, in point of expatiate on it here would be so far out of order, that we are It is always well to begin the chemical examination of a substance, by less, come under our notice when we treat of the latter metal ; choosing the same in a pure condition, unmixed with any acces- indeed even before, for I shall put the student in possession of an sory that might veil its properties or obscure the result. I easy means by which all the silver may be separated, and the therefore recommend, as the source for obtaining a silver specimen, copper left behind. a few grains, say eighteen or twenty, of the salt called: nitrate of Returning now to our solution of nitrate of silver, let the silver. This substance occurs in commerce under two forms: student question it thus : cither as sticks something like slate-pencil, only whiter, or as (1) What is its nature: crystals. The latter will be somewhat the purer of the two; but To arrive at an answer to this question, drop a little of your the former, known popularly as " lunar caustic," will answer very strong solution, say twenty or thirty drops, into a wine-glass ; well. fill up the wine-glass with distillod water, and test with hydroLet the student then take about eighteen or twenty grains sulphuric acid solution. We get a well-pronounced black preciof lunar caustic, or rather more of crystallised nitrate of silver, pitate, on observing which we immediately deduce the following and effeot a solution of the substance in about half a pint of dis- truths. (1) The solution contains as its base, a metal. (2) A cal- . tilled water. The solution takes place with great facility, and cigenous metal (vide Lesson p. 39). (3) Neither zino, arsenic, 1 antimony, cadmium, nor tin, in the state of persalt; because the LE SAPEUR DE DIX ANS. SECTION III. witness. Let us now try another witness, namely,ferrocyanide of potassium; Cependant il entraita encore quelque hésitation dans la and once for all let the student remember that hydrosulphuric compagnie, et déjà deux fois le capitaine qui commandait acid, hydrosulphate of ammonia, and ferrocyanide of potassium, avait donné l'ordre au tambour-maître de prendre deux are the three witnesses always first cited in a court of chemical tambours, de se mettre en avant, et de battre la charge.2 inquiry, supposing the substance under question to be in Celui-ci restait appuyé sur sa grande canne,3 hochant la the state of liquidity and totally unknown. Whatever evidence tête et peu disposé à obéir. Pendant ce temps Bilboquet, à is to follow, theirs comes first; all three, if we want them, or chevalo sur son tambour 4 et les yeur levés sur son chef, two or one as the evidence may require. As regards the case sifflait un air de fifre et battait le pas aocéléré avec ses now under consideration, the reader will not fail to see that hydrosulphate of ammonia could only afford positive testimony, doigts. Enfin l'ordre venait d'être e donné une troisième fois au tambour-maître, et il ne paraissaitd pas disposé à given already negatively by hydrosulphuric acid. Now, in many chemical examinations, negative testimony is as valuable as posi obéir, lorsque tout à coup, Bilboquet se relève, accroche tive. It is so in the present instance. Let us now proceed to son tambour à son côté, prend ses baguettes, et passant use the third test, ferrocyanide of potassium ( yellow prussiate of sous le neze du tambour-maitre, il le toise avec orgueil , lui potash), in solution of course. For this purpose, add a few drops rend d'un seul mot toutes les injures qu'il avait sur le of the strong nitric acid solution to a little distilled water, and ceur, et luit dit:-Viens donc, grand poltron ! test with prussiate of potash. We now get a whitish sort of Le tambour-maître veuts lever sa canne, mais déjà Bilprecipitate. boquet était à la tête des deux compagnies, 10 battant la Omitting to repeat such of the evidence yielded by this test as charge comme un enragé. Les soldats, à cet aspect, we happen to know already, what novelty does it communicate ? s'avancent après lui et courent vers la terrible batterie.lí What has it to say of its own specific knowledge? Why it tells Elle décharge d'un seul coup ses six pièces de canon, et desi us that, in addition to all the metals amongst which ours is not, it furthermore is not rangs de nos braves voltigeurs s'abattent et ne se relèvent plus.!? La fumée, poussées par le vent, les enveloppe, le Copper fracas du canon les étourdit; mais la fumée passe, le bruit Uranium cesse: un instant, et ils voientk debout, à vingt pas devant Molybdenum eux, l'intrépide Bilboquet battant la charge, 13 et ils enTitanium ; tendent son bambour, li 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, 15 devant eux le anahogany 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 comat 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 listof 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 atrive at the conclusion that it must be partis, 17 et aussitôt, comme transporté d'une fureur de ven . appeal to the evidence of another test, either hydrochloric acid geance, il redouble de fracas : 18 on eût dito ringt tambours (spirit of salt), or else common salt dissolved in water; practi- | battant à la fois ; jamais le tambour-maitre 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 P aus Russes : be unknown) will throw down a dense white precipitate ; hence -Les morceaux en sont bons, les voici ; 20 attendez, we know at once that the metal we are hunting for is either silver attendez ! or mercury; no other metals being capable of producing a similar COLLOQUIAL EXERCISE. effect. Finally, the addition of a little hartshorn (liquor ammoniæ) 1. Que remarquait-on néan- | 11. Que firent les soldats en oauses the precipitate to dissolve and the whiteness totally to moins dans la compagnie ? voyant son intrépidité ? disappear; which characteristic result demonstrates the metal to 2. Quelordre le capitaine avait. 12. Quel effect produisit la débe silver, nothing but silver. il donné au tambour-maître ? charge des six pièces de canon? 3. Que fit celui-ci après avoir 13. Que virent les soldats quand reçu cet ordre? la fumée fut dissipée ? CURIOSITY. 4. Oà était Bilboquet pendant 14. Qu'entendaient-ils malgré ce temps là ? le bruit du canon ? Its aim oft idle, lovely in its end, 5. Que faisait-il ? 15. Que firent alors nos voltiWe turn to look, then linger to befriend; 6. Le tambour-maître paraisThe maid of Egypt thus was made to save sait-il disposé à obéir au troi- 16. Quel fut l'effet d'une seA nation's future leader from the wave; sième ordre ? conde décharge ? 7. Que fit alors le petit tam. 17. Combien d'hommes restait- 1 ? 8. Comment apostropha-t-il le 18. Que fit Bilboquet à la rue E'en in its slightest working, we may trace tambour-maître ? du carnage ? A deed that changed the fortunes of a race: 9. Que voulut faire le tambour- 19. Que firent alors les soldats? Bruce, banned and hunted on his native soil, maîtro? 20. Que cria alors le petit tamWith curious eye surveyed a spider's toil; 10. Où était alors notre héros ? NOTES AND REFERENCES.--a. Il entrait, there was; the verb " Once more,” he cried, “in thine, my doom I read, is unipersonal in French ; L. part ü., $ 43, R. (7).— à cheval, Once more I dare the fight, if thou succeed; seated across.—. venait d'être, had just been ; L. S. 25, R. 2.'Twas done : the insect's fate he made his own: d. from paraître ; I. part ü., p. 98.-e. sous le nez, close to the Once more the battle waged, and gained a throne. face; literally, under the nose--fi froni tenir ; L. part ii., p. geurs ? bour? i Yous assure. 80.--P. 108.-9. from vouloir ; L. part. i., p. 110.--h. enragé, madman. 13. Que dit le général, quand il 18. Les voltigeurs se moquaicuti -. S. 4, R. 1.--j. S. 98, R. 1.-k. from coir; I. part ü., p. 110. eut remarqué que Bilboquet ils de lui l. subjunctive of être.-m. from venir ; L. S. 25, R. 2.--. L. n'était qu'un enfant ? 19. Qu'allait-on faire en sa fa veur ? part ü., § 49, R. (4).--0. on eut dit, one would have thought 14. Que lui donna-t-il ? that; literally, said.--p. à tue-tête, with all his might. 15. Bilboquet prit-il la pièce ? 20. Que dit-il enfin au géné 16. Regardait-on le petit tam- ral. SECTION IV. bour? 21. Que fit-il après avoir mis 17. Que faisait-il al018? l'argent dans sa poche ? Pendant ce temps, Napoléon monté sur un tertre, regardait exécuter cette prise héroique. A chaque décharge, L. part i., p. 88.-c. se mirent, commenced ; L. S. 68, R. 3.-d. NOTES AND REFERENCES. d. L. S. 20, R. 2.-6. from dire ; il tressaillait sur son cheval isabelle ; puis, quand les soldats ils s'y connaissaient, they were good judges of such things ; L. S. entrèrent dans la batterie, il baissa sa lorgnette en disanto 86, R. 6.e. from courir ; L. part ü., p. 84.-f. from revenir tout bas : Braves gens ! 2 Et dix mille hommes de la garde, qui étaient derrière lui, mit, presented; from remettre; L. part ii., p. 102.i. fit en L. part ii., p. 104.g. from prendre; E. part ii., p. 100.-h. rese mirente à battre des mains et à applaudir: en criant: tendre, uttered; from faire; I. part ii., p. 92.j. accent, tone,-Bravo, les voltigeurs!! Et ils s'y connaissaient, d je K. L. p. ii., § 33, R. (8).–1. planté, standing ; literally, planted, posted.-m. j'en étais, I was one of them, of the number.- n. L. Aussitôt, sur l'ordre de Napoléon, un aide-de-camp cou- part ü., $ 33, R. (9).--. from battre; L. part ii., p. rute jusqu'à la batterie 4 et revint au galop. S. 80, R. 2. q. que veux-tu, how can I help it ; literally, what do Combien sont-ils arrivés ?5 dit l'Empereur. you wish.-r. L. S. 61, R. 5.-s. en attendant, meanwhile.-t. from -Quarante, répondit l'aide-de-camp. dire; L. part ii., p. 88.–4. il s'était fait, there was.--v. from Quaranté croix demain, dit l'Empereur en se retour- paraître ; L. part ii., p. 98.-20. L. S. 26, R. 2.--. toujours, nant vers son major-général. notwithstanding; literally, always. Véritablement, le lendemain, tout le régiment forma un SEOIION V. 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 remito à chacun petit Bilboquet, mais il n'en • devint pas pour cela plas 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 des dépenser son argent voix sortit du rang et fit entendre i ces mots,lo prononcés avec ses camarades, comme ceux-ci s'y attendaient, il le avec un singulier accenti de surprise : serra soigneusement.2 Et moi! moi!k je n'ai donc rien ? Quelque temps après, les troupes françaises entrèrent -Le général qui distribuait les croix, se retourna et vit à Smolensk.: victorieuses et pleines d'ardeur ; Bilboquet en plantél devant lui" notre camarade Bilboquet, les joues était, et le jur même de l'arrivée, il alla se promener dans rouges et l'ail presque en larmes.ll la ville,* paraissantb très content de presque tous les vi-Toi ? lui dit-il, que demandes-tu? sages qu'il rencontrait :5 il les considérait d'un air riants Mais, mon général, j'en étais » dit Bilboquet presque et semblait les examiner comme un amateur qui choisit en colère ; 12 c'est moi qui battais o la charge en avant, c'est des marchandises. Il fauti vous dire cependant, qu'il ne moi qui suis p entré le premier. Que veux-ta, 9 mon garçon ? on t'a oublié, répondit le regardait ainsi que les paysans qui portaient des grandes Elles étaient sans doute très longues et très général; d'ailleurs, ajouta-t-il en considérant que c'était un fournies, mais d'un roux si laid, qu'après un moment enfant, tu es encore bien jeune, on te la donnera quand tu d'examen Bilboquet tournait la tête et allait plus loin. auras? de la barbe au menton ;13 en attendant, voilà de Enfin, en allant ainsi , notre tambour arriva au quartier des quoi te consoler. Juifs. Les Juifs à Smùlensk, comme dans toute la Pologne En disantt ces paroles, le général tendit une pièce de et la Russie, vendent toutes sortes d'objets 9 et ont an penser à la pendre. 15 Il s'était fait un grand silence quartier particulier... Dès que Bilboquet yl fut entré, ce autour de lui, et chacun le considérait attentivement;26 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 doncm 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.16 Le marchand s'approche de notre ami et lui de -C'est bon, donnez toujours, ce sera pour une autre mande humblement en mauvais français : Que voulez-vous mon petit Monsieur ?16 Et sans plus de façons, il mit la pièce dans sa poche et -Je veuxta barbe répondit cavalièrement Bilboquet.17 s'en retourna dans son rang en sifflant d'un air délibéré et -Ma barbe ! dit le marchand stupéfait; vous voulezo satisfait.2 rire ? 18 COLLOQUIAL EXERCISE. -Je te dis, vaincu, que je veux ta barbe, reprend le vain queur superbe en posant la main sur son sabre ; mais ne 1. Que faisait Napoléon pen- 7. Que fit le régiment le len- crois pas que je veuillep te la voler:19 tiens, 9 voilà un nadant ce temps-là ? poléon, tu me rendras mon reste." 2. Que fit-il quand les soldats 8. Qu'appela-t-on successive entrèrent dans la batterie ? COLLOQUIAL EXERCISE. 3. Que firent les soldats de sa 9. Que donna-t-on à ces braves garde ? 1. A partir de ce jour, comment 5. De quoi paraissait-il con4. Quel ordre Napoléon donna- 10. Qu'arriva-t-il lorsque la cé- traita-t-on notre héros? t-il à un aide-de-camp? rémonie fut finie ? 2. Que fit-il de son argent ? 6. De quelle manière consi5. Que dit-il à l'aide-de-camp 11. Que vit alors le général ? 3. Que firent les troupes fran- dérait-il le visage des habià son retour? 12. Que répondit le petit tam- çaises quelque temps après ? 6. Quel ordre donna-t-il au bour à la question du géné- 4. Que fit le petit tambour le 7. Quelles personnes regardaitgénéral ? jour de son arrivée ? il particulièrement ? fois.20 demain ? ment? gens ? tent? tants? ral ? 8. Où arriva-t-il enfin ? 14. Où Bilboquet entra-t-il en- Let them'he produced and meet towards B and D, in the point G; 9. Que font les Juifs en Rus- fin? then Geris a triangle. sie? 15. Qui trouva-t-il dans la bou- Now, in the triangle G EF the exterior angle A BF is greater 10. Où demeurent-ils? tique ? (I. 16) than its interior and opposite angle E F G; but the angle 11. Quel sentiment éprouva 16. Que dit le marchand au A B F is equal (Hyp.) to the angle BFG; therefore the angle A EF Bilboquet, quand il fut entré petit tambour? is both greater than, and equal to, the angle BFG; which is imposdans ce quartier ? 17. Que lui demanda celui-ci ? sible. Wherefore the straight lines A B and C B, if produced, do 12. Pourquoi était-il si content? 18. Quelle fut la réponse du not meet towards B and D. In the same manner it may be proved, 13. Quelle remarque l'auteur marchand ? that they do not meet if produced towards A and c. But those fait-il à propos de la nation 19. Qu'ajouta Bilboquet en met- straight lines in the same plane, which do not meet when juive ? tant la main sur son sabre? produced ever so far either way, are parallel (Def. 33). There fore A B is parallel to CD. Wherefore, if a straight line falling NOTES AND REFERENCES.—. From se moquer; to laugh at. B. en, on that account.-c. from devenir; L. part ii., p. 88. upon two other straight lines, &c. Q. E. D. Scholium 1. The angles A E F and EFD are called alternate angles, d. L. §. 34, R. 4.e. ils s'y attendaient, they expected.f. L. or more properly interior alternate angles, because they are on oppopart ii., § 145.-9. L. S. 35, R.5.--h. from paraître; L. part ii., site sides of the straight line E F, and the one has its vertex at E p. 98.--. il faut, I must; from falloir ; L. S. 47; also L. part ii., the one extremity of the portion between the parallels, while the p., 92... portaient, wore.-K. fournies, thick.–1. I. part ii., s other has its vertex at the other extremity of the same. mm. voilà donc, behold then .. from vouloir ; L. Scholium 2. In the diagram the crooked lines B B G and FDA part ii., p. 110.--. vous voulez rire, you are joking, you are not must be considered straight lines, and the figure E F D G B a triangle, in earnest. P. from douloir.-9. tiens, here; literally, hold; for the sake of the argument. Otherwise, the figure might have from tenir, L. part ii., p. 108.--. reste, change. been constructed so that the straight lines AB and CD should actually converge and meet in a point. ر ر ر EXERCISE I. TO PROPOSITION XXVII. EXERCISE II. TO PROPOSITION XXVII. SKELETON MAPS.--No. V. If a straight line falling upon two other straight lines, make the exterior alternate angles equal to each other, these troo straight MAP OF SOUTH AMERICA. lines are parallel. Our Map of Russia in Europe (the approximate seat of war) not being ready, as intended, for this month, we insert in this In fig. 28, let the straight line EP, wbich falls upon the two Number a Skeleton or Outline Map of the Continent of South straight lines A B and CD, make the two exterior alternate angles America, including the continental part of the West Indies 4 GB and F. D equal to one another ; then A B is parallel to CD. called Guiana, and the small islands adjacent to the con Because (I. 13) the two angles A GE and A G H are equal to two tinent all around it. This Map will be useful to emigrants, right angles, and the two angles PhD and G A D are equal to two settlers, or colonists, who wish to transplant themselves to right angles; therefore (At. 1) the two angles A GE and A GK South America, where there is abundance of room for specu are equal to the two angles FHD and G HD. But (Hyp.) the angle lations of all kinds. If such persons have sufficient time and 4 GB is equal to the angle = ID; therefore (Ax. 3) the angle A & H skill to fill up this Map for themselves, the process of doing is equal to the angle GHD; and they are alternate angles, whereso will make them better acquainted with the country in fore (I. 27) the straight lines AB and C D are parallel. Q. E. D.* which they intend to settle, than meny Lessons in Geography, which consist of the mere descriptions of places, but give no If a straight line falling upon two other straight lines, make the two idea of their relative position in regard to one another. An extensive list of the latitudes and longitudes of the chief exterior anglos on the same side of it equal to trco right angles, these two straight lines are parallel. Ť or capital towns in the various countries and sub-divisions of the continent, and of the islands of South America, will be In fig. 28, let the straight line BF, which falls upon the straight found in Vol. iii., at page 250; and, as the continental part lines A B and CD, make the two exterior angles on the same side of of the West Indies is included in this Map, the latitudes and it, B G B and F D, equal to two right angles; then A B is parallel longitudes for the chief towns of this part will be found at to CD. page 118. On the marginal space of the Map, we have given Because (I. 13) the two angles EG B and EG A are equal to two the latitudes and longitudes of the principal islands, capes, right angles, and (Hyp.) the two angles E G B and FHD equal to bays, rivers, and ports along the eastern and western coasts of two right angles; therefore (Az. 1.) the two angles B G B and E GA the continent, from Cape Born to the Isthmus of Panama, in are equal to the two angles E G B and FHD; from these equals regular order, proceeding from south to north, and along the take away the common angle E G B, and (Ax. 3) the angle E GA is coast of America situated on the Caribbean Sea. These we equal to the angle FHD; but these are the two exterior alternate have added to the latitudes and longitudes of the places in the angles; wherefore, by the preceding exercise, the straight lines A B interior of the continent above-mentioned, so as to enable and c D are parallel. Q. E. D. our students to make their Map as complete as possible. PROPOSITION XXVIII.--THEOREM. If a straight line falling upon troo other straight lines, make the exterior angle equal to the interior and opposite angle upon the LESSONS IN GEOMETRY.-No. XXVI . same side of the straight line; or make the two interior anglos LECTURES ON EUCLID. upon the same side of it, together equal to two right angles; these two straight lines are parallel to one another. (Continued from page 256.) Let the straight line E F, falling upon the two straight lines A B PROPOSITION XXVII.--THEOREM. and CD, make the exterior angle E G B equal to the interior and If a straight line falling upon two other straight lines, make the opposite angle G Hd upon the same side of altorate 'angles equal to one another; these two straight linss EF; or make the two interior angles Fig. 28. are parallel. BGH and GHD on the same side of it, together equal to two right angles; then In fig. 27, let the straight line Fig. 27. A B is parallel to CD. EF which falls upon the two Because the angle E G B is equal (Hyp.) straight lines A B and CD, make A to the angle G HD, and the angle EGB is C the alternate angles a EF and EPD equal (I. 15) to the angle A GĦ; therefore equal to one another: then A B is the angle A G H is equal (Ax. 1) to the parallel to CD. For if AB be not parallel to cn, * Solved by Q. Pringle, Glasgow, J. H. Eastwood, Middleton; and these straight lines will E. J. Bremner, Carlisle. meet, if produced either towards A and C, or towards B and D. † See new edition of Cassell's Euclid, 1854. B F two &c. account. angle GHD; and they are alternate angles; wherefore A e is which parallel (straight] lines, though indefinitely produced, can parallel (I. 27) to cd. never intersect. This is, perhaps, the most ordinary idea of Again, because the two angles BGH and ghd are together parallelism. Almost every other property of parallels roquires equal (Hyp.) to two right angles ; and the two angles AG H and some consideration before an uninstructed mind assents to it; buu BG H are also together equal (I. 13) to two right angles ; there- the possibility of two such (straight] lines intersecting is repugfor the two angles A G H and B G H are equal (Ax. 1) to the two nant to every notion of parallelism. "When the possible existenco of the subject of a dofinition is not angles B G H and GID. Take away from these equals the com- self-evident, or presumed and declared to be so, it ought to be mon angle B G#, and the remaining angle a G H is equal (Ax. 3) proved so. This is the case with Euclid's definition of parallels. to the remaining angle GHD; but they are alternate angles; there- Row, it may be asked, does it appear that two right (straight] fore A B is parallel (I. 27) to cd. Wherefore, if a straight line, lines can be drawn upon the same plano so as never to intersect Q.E.D. though infinitely produoed & Euclid meets this objection in his Scholium 1. The twelfth axiom will now be admitted by the 27th proposition, where he shows that if two (straight] lines be student as a corollary to this proposition ; especially when Prop. inclined at equal alternate angles to a third, the supposed possiXVII. and the note added to the twelfth axiom are taken into bility of their intersection would lead to a manifest contradio tion. Thus it appears, that through a given point one right Scholium. 2: We think it right to introduce our students at this (straight] line at least may always be drawn parallel to a given point, to a discussion on the “Theory of parallel straight lines, right (straight] line. But it still remains to be shown, that no more which will be of immense advantage to them in their future studies. right (straight] line. And here the chain of proof is broken. Euclid than one parallel can be drawn through the same point to the same Our first extract shall be from the Gower-street edition of Euclid. was unable to demonstrate, that every other (straight] line except “The theory of parallel (straight] lines has always been con intersect the given right (straight] line if both be sufficiently that which makes the alternate angles equal will necessarily sidered as the reproach of Geometry. The beautiful chain of reasoning by which the truths of this science are connected here produced. He acoordingly found himself compelled to place the wants a link, and we are reluctantly compelled to assume as an deficient link among his axioms." acion what ought to be matter of demonstration. The most eminent We now add to this extract, notices of thirty different methods, geometers, ancient and modern, have attempted without success proposed at various epochs in the history of Geometry, for getting to remove this defect; and after the labours of the learned for over the difficulty of the Troelfth Aziom of Euclid's First Book. 2,000 years have failed to improve or supersede it, Euclid's theory This collection is taken from Col. P. Thompson's “Geometry of parallels maintains its superiority. We shall'here endeavour to explain the nature of the difficulty which attends this investi- without Axioms,” pp. 137-156. gation, and shall give some account of the theories which have • The uses of such a Collection are to throw light on the particubeen proposed as improvements on, or substitutes for, that of lars which experience has shown are not to be left unguarded in Euclid. any attempt at solution, and to prevent explorers from consuming “Of the properties by which two right[straight] lines described their time in exhausted tracts. To which may be added, that out upon the same plane are related, there are several which charac of so many efforts, some, either by improvement or by a fortunate terise two parallel (straight] lines and distinguish them from conjunction with others, may finally be found operative towards (straiglat] lines which intersect. If any one of such properties be the solution desired. assunied as the definition of parallel [straight] lines, all the others 1. The objection to Euclid's Axiom (independently of the objecshould flow demonstratively from it. As far, therefore, as thestrict tions common to all Axioms), is that there is no more reason why principles of logic are concerned, it is a matter of indifference it should be taken for granted without proof, than numerous other which of the properties be taken as the definition. In the choice propositions which are the subjects of formal demonstration, and of a definition, however, we should be directed also by other cir- the taking any one of which for granted would equally lead to the cumstances. That property is obviously to be preferred from which establishment of the matter in dispute, all the others follow with the greatest ease and clearness. It is also 2. Ptolemy the astronomer, who wrote a treatise on Parallel manifest that, cæteris paribus, that property should be selected Lines, of which extracts are preserved by Proclus, proposed to prove which is most conformable to the commonly received notion of that if a straight line cuts two parallel straight lines, the two the thing defined. These circunstances should be attended to interior angles on each side are together equal to two right angles, in every definition, and the exertion of considerable skill is by saying that if the interior angles on the one side are greater necessary almost in every case. But in the selection of a defini- than two right angles, then because the lines on one side are* no tion for parallel (straight] lines there is a difficulty of another more parallel than those on the other, the two interior angles on the kiun. It has been found, that whatever property of parallels be other side must likewise be together greater than two right angles, selected as the basis of their definition, the deduction of all the and the whole greater than four, which is impossible, and in the other properties from it is impracticable. Under these scircum- same way if they were supposed less. In which the palpable stances, the only expedient which presents itself, is to assume, weakness is, that there is no proof, evidence, or oause of probability besides the property selected for the definition, another property assigned, why parallelism should be connected with the angles on as an axiom. This is what every mathematician who has one side being together equal to those on the other; the very quesattempted to institute a theory of parallel (straight] lines has tion in debate being, whether they may not be a little more than done. Some, it is true, have professed to dispense with an axiom, two right angles on one side and a little less on the other, and still and to derive all the properties directly from their definition. the straight lines not meet. But these, with a single exception, which we shall mention here- 3. Proclus himself proposes" to take an Axiom of this sort, being after, have fallen into an illogicism inexcusable in geometers. the same that Aristotle employed to establish that the world is We find invariably a petitio principii, either incorporated in finite. If from the same point, two straight lines are drawn their definition, or lurking in some complicated demonstration. making an angle, the distance between them when they are proA rigorous dissection of the reasoning never fails to lay bare the longed to infinity will exceed any finite distance that inay be soplism. assigned. He then showed that if the straight lines prolonged Of the pretensions of those who avowedly assume an axiom it from this centre towards the circumference are of infinite length, is easy to judge. When Euclid's axiom is removed from the what is between them is also infinite; for if it was finite, to increase disadvantageous position which it has hitherto maintained, the distance would be impossible, and consequently the straight put in its natural place, and the terms in which it is expressed lines would not be infinite. The straight lines therefore on being somewhat changed, I think it will be acknowledged that no prolonged to infinity, will separate from each other by more than proposition whioh has ever yet been offered as a substitute any finite quantity assigned. But if this be previously admitted, for it, is so nearly self-evident. But it is not alone in the degree I affirm that if any straight line cuts one of two parallel straight of self-evidence of his axiom, if we be permitted the phrase, that lines, it will cut the other also. For let A B and cd be parallel, Euclid's theory of parallels is superior to those theories which and let E F cut A B in G; I say that it will cut c D also. For since are founded on other axiomis. The superior simplicity of the structure which he has raised upon it is still more conspicuous. When you have once admitted "Euclid's axion, all his theorems * ουδέν γαρ μάλλον αι αζ γη παράλληλοι ή αι ζο ηβ.-Procli Comment. in flow from that and his definition, as the most simple and obvious Prinun Euclidis Librum. Lib. 4. inferences. In other theories, after conceding an asiom much It is but right to notice, that Proclus calls this mapaloycouós and delFews further removed from self-eviđenoe than Euclid's , a labyrinth of kobévesa ; and Barocins the Venetian Translator in 1550, notes it iu the complicated and indirect demonstration remains to be threaded, margin as Flagiliosa Ptolemæi ratiocinatio. Professor Playlair says it is curious to observe in Proclus's account an requiring much subtlety and attention to be assured that error argument founded on the principle known to the moderns by the name of and fallacy do not lurk in its mazes. the sufficient reason (Elein. of Geom. p. 405). If the allusion is to this parta “Euclid selects for his definition that property in virtue of the "sufficient reason" of the moderns must be something very feeble. ܪ |