108.-g. from vouloir; L. part. ii., p. 110.—h. enragé, madman. -i. S. 4, R. 1.-j. S. 98, R. 1.—k. from voir; L. part ii., p. 110. -l. subjunctive of être.-m. from venir; L. S. 25, R. 2.—n. L. part ii., § 49, R. (4).-0. 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!"

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, je

vous assure.

Aussitôt, sur l'ordre de Napoléon, un aide-de-camp courute 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 nant vers son major-général.

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 braves qui avaient pris la batterie, et l'on remit à chacun d'eux la croix de la Légion-d'Honneur. La cérémonie était finie, et tout le monde allait se retirer, lorsqu'une voix sortit du rang et fit entendre ces mots,10 prononcés avec un singulier accent de surprise :

-Et moi! moi! je n'ai done 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; 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.

En disant ces paroles, le général tendit une pièce de vingt francs au pauvre Bilboquet, qui la regarda sans penser à la pendre.15 Il s'était fait un grand silence autour de lui, et chacun le considérait attentivement; lui, demeurait immobile devant le général et de grosses larmes roulaient dans ses yeux." Ceux qui s'étaient le plus moqués de lui paraissaient attendris, et peut-être allaiton élever une réclamationen sa faveur, lorsqu'il releva vivement la tête, comme s'il venait de prendre une grande résolution, et il dit au général :

W

C'est bon, donnez toujours, ce sera pour une autre

SECTION V.

a

A partir de ce jour, on ne se moqua plus autant du petit Bilboquet, mais il n'en devint pas pour cela plus communicatif; au contraire, il semblait rouler dans sa tête quelque fameux projet, et, au lieu de dépenser son argent avec ses camarades, comme ceux-ci s'y attendaient, il le serra soigneusement.2

barbes.7

Quelque temps après, les troupes françaises entrèrent à Smolensk, victorieuses et pleines d'ardeur; Bilboquet en était, et leur 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 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. Enfin, en allant ainsi, notre tambour arriva au quartier des Juifs. Les Juifs à Smolensk, comme dans toute la Pologne et la Russie, vendent toutes sortes d'objets et ont un quartier particulier. 10 Dès que Bilboquet y fut entré, ce fut pour lui un véritable ravissement: imaginez-vous les plus belles barbes du monde, noires comme de l'ébène;12 car la nation juive toute dispersée qu'elle est, parmi les autres nations, a gardé la teinte brune de sa peau et le noir éclat de ses cheveux.13 Voilà donc notre Bilboquet enchanté. Enfin il se décide, et entre dans une petite boutique où se trouvait un marchand magnifiquement barbu. Le marchand s'approche de notre ami et lui de mande humblement en mauvais français:

14

-Que voulez-vous mon petit Monsieur ?16

-Je veux" ta barbe répondit cavalièrement Bilboquet."7 -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, voilà un napoléon, tu me rendras mon reste."

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 ?

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.-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 MAPS.-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 A B 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 D, these two straight lines will

C

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 EFG; but the angle A EF is equal (Hyp.) to the angle EFG; therefore the angle A EF is both greater than, and equal to, the angle E F G; 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. 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.

But those

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 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 AGE and F H D equal to one another; then A B is parallel to c D.

Because (I. 13) the two angles A & 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 G H D. But (Hyp.) the angle AGE is equal to the angle HD; therefore (Ax. 3) the angle A G H fore (I. 27) the straight lines A B and C D are parallel. Q. E. D.* is equal to the angle G HD; 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.†

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

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

Fig. 28.

E

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 H D 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 EGB is equal (Hyp.) to the angle GHD, and the angle EGB is C equal (I. 15) to the angle AGH; therefore the angle AGH is equal (Ax. 1) to the

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 AG H and BG H are also together equal (I. 13) to two right angles; therefore the two angles A G H and BGH 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 GHD; 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 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.

account.

"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 charac terise 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 difliculty 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. "When the possible existence of the subject of a definition is not self-evident, or presumed and declared to be so, it ought to be 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 possibility of their intersection would lead to a manifest contradiction. Thus it appears, that through a given point one right [straight] line at least may always be drawn parallel to a given right [straight] line. But it still remains to be shown, that no more than one parallel can be drawn through the same point to the same 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 objec tions 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 ques tion 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

οὐδέν

Primum Euclidis Librum. Lib. 4. γὰρ μᾶλλον αἱ αζ γη παράλληλοι ἡ αἱ ζδ ηβ. — Procli Comment. in It is but right to notice, that Proclus calls this rapaλorioμos and deitens à@évela; and Barocius the Venetian Translator in 1560, notes it in the margin as Flagitiosa Ptolemæi ratiocinatio.

Professor Playtair 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" of the moderns must be something very feeble. the sufficient reason (Elem. of Geom. p. 405). If the allusion is to this part,

from the point G are drawn two straight lines G B, G F, and prolonged to an infinite length, the distance between them will become greater than any assigned magnitude, and consequently than that which may be the distance between the parallels; when, therefore. they are distant from each other by more than this, GF will cut CD."* Without disputing that the distance between the straight lines which make the angle will become greater than any assigned magnitude (though the reason given appears to be founded on ignorance of the fact that a magnitude may perpetually increase and still be always less than an assigned magnitude),-the defect is in begging the question, that the distance between the parallels is constant or at all events finite. For the very point in dispute is, whether the parallels (as for instance two perpendiculars to a common straight line, both of them prolonged both ways) may not open out or grow more distant as they are prolonged, and to do this so rapidly, that a straight line making some very small angle with one of them, shall never overtake the other, but chase it unsuccessfully through infinite space, after the manner of a line and its asymptote.

4. Clavius announces that "a line every point in which is equally distant from a straight line in the same plane, is a straight line;' upon taking which for granted, he finds himself able to infer the properties of Parallel Lines. And he supports it on the ground that because the acknowledged straight line is one which lies evenly [ex æquo] between its extreme points, the other line must do the same, or it would be impossible that it should be everywhere equidistant from the first.t Which is only settling one unknown by a reference to another unknown.

5 and 6. In a tract printed in 1604 by Dr. Thomas Oliver, of Bury, entitled, De rectarum linearum parallelismo et concursu doctrina Geometrica (Mus. Brit.), two demonstrations are proposed; both of them depending on taking for granted, that if a perpendicular of fixed length moves along a straight line, its extremity describes a straight line. Which is Clavius's axiom a little altered.

7. Wolfius, Boscovich, Thomas Simpson in the first edition of his" Elements," and Bonnycastle, alter the definitions of parallels, and substitute in substance, "that straight lines are parallel which preserve always the same distance from one another;" by distance being understood the length of the perpendicular drawn from a point in one of the straight lines to the other. Attempts to get rid of a difficulty by throwing it into the definition, are always to be suspected of introducing a theorem in disguise; and in the present instances, it is only the introduction of the proposition of Clavius. No proof is adduced that straight lines in any assignable position, will always preserve the same distance from one another; or that if a perpendicular of fixed length travels along a straight line keeping always at right angles to it, what mathematicians call the locus of the distant extremity is necessarily a straight line at

all.

8. D'Alembert proposed to define parallels as being straight lines "one of which has two of its points equally distant from the other line;" but acknowledged the difficulty of proving, that all the other points would be equally distant in consequence 1. 9. Thomas Simpson, in the second edition of his "Elements," proposed that the Axiom should be, that "If two points in a straight line are posited at unequal distances from another straight line in the same plane, those two lines being indefinitely produced on the side of the least distance will meet one another." 10. Robert Simpson proposes that the Axiom should be, "that a straight line cannot first come nearer to another straight line, and go further from it, before it cuts it." By coming nearer or

then

We omit the Greek.

"Nam si omnia puncta lineæ A B,æqualiter distant à rectâ D C, ex æquo sua interjacebit puncto, hoc est, nullum in eâ punctum intermedium ab extremis sursum, aut deorsum, vel huc, atque illuc deflectendo subsultabit, nihilque in eâ flexuosum reperietur, sed æquabiliter semper inter sua puncta extendetur, quemadmoduni recta DC. Alioquin non omnia ejus puncta æqualem à rectâ D D, distantiam haberent, quod est contra hypothesin. Neque verò cogitatione apprehendi potest aliam lineam præter rectam, posse habere omnia sua puncta à rectâ lineâ, quæ in eodem cum illâ plano existat, æqualiter distantia.'-Clavii Opera. In Euclidis Lib. I. p. 50. $ -"la vraie définition, ce me semble, et la plus nette qu'on puisse donner d'une parallèle, est de dire que c'est une ligne qui a deux de ses points également éloignés d'une autre ligne.-il faut ensuite démontrer (et c'est-là le plus difficile), que tous les autres points de cette seconde, seront également éloignées de la ligne droite donnée."-ENCLYCLOPEDIE. Parallèle.

Art.

This and most of what has preceded, is in the Arabic. In a manuscript copy of Euclid in Arabic but in a Persian hand, bought at Ahmedabad in 1817, the editor on the introduction of Euclid's Axiom comments as follows. "And this is what is said in the text. I maintain that the last proposition is not of the universally-acknowledged truths, nor of any thing that is demonstrated in any other part of the science of geometry. The best way therefore would be, that if it should be put among the questions instead of the principles; and I shall demonstrate it in a suitable place. And I lay down for this purpose another proposition, which is, that straight lines in the same

going from it, being understood the diminution or increase of the perpendicular from one to the other.

The objection to all these is, that no information has been given on the subject of the things termed straight lines, which points to any reason why the distance's growing smaller should be necessarily followed by the meeting of the lines. It may be true; but the reason why, is not upon the record. On the contrary, it is well known that there exist lines (as for instance the neighbouring sides of two conjugate hyperbolas) where the distance perpetually decreases and yet the lines never meet. It is open therefore to ask, what property of the lines called straight has been promulgated, which proves they may not do the like.

11. Varignon, Bezout, and others propose to define parallels to be "straight lines which are equally inclined to a third straight line," or in other words, make the exterior angle equal to the interior and opposite on the same side of the line. By which they either intend to take for granted the principal fact at issue, which is whether no straight lines but those that make such angles can fail to meet; or if their project is to admit none to be parallel lines of which it shall not be predicated that they make equal angles as above with some one straight line either expressed or understood, then they intend to take for granted that because they make equal angles with one straight line, they shall also do it with any other that shall in any way be drawn across them,-a thing utterly unestablished by any previous proof.

12. Professor Playfair proposes to employ as an Axiom, that "two straight lines, which cut one another, cannot be both parallel to the same straight line" in which he had been preceded by Ludlam and others, and which he says "is a proposition readily enough admitted as self-evident." The misfortune of which is, that instead of being self-evident, a man cannot see it if he tries. What he sees is, that he does not see it. He sees that a straight line's making certain angles with one of the parallels, causes it to meet the other; and he sees that by increasing the distance of the point of meeting, he can cause the angle with the first parallel to grow less and less. But if he feels a curiosity to know whether he can go on thus reducing the angle till he makes it less than any magnitude that shall have been assigned (or in other words whether there may not possibly be some angle so small that a straight line drawn to any point however remote in the other parallel shall fail to make so small a one), he discovers that this is the very thing nature has denied to his sight; an odd thing, certainly, to call self-evident.

13 The same objections appear to lie against Professor Leslie's proposed demonstration in p. 44 of his "Rudiments of Plane Geometry;" which consists in supposing a straight line of unlimited length both ways, to turn about a point situate in one of the parallels, which straight line, it is argued, will attain a certain position in which it does not meet the other straight line either way, while the slightest deviation from that precise direction would occasion a meeting.

14. Professor Playfair, in the Notes to his "Elements of Geometry," p. 409, has proposed another demonstration, founded on a remarkable non causa pro causa. It purports to collect the fact* that (on the sides being prolonged consecutively) the intercepted or exterior angles of a rectilinear triangle are together equal to four right angles, from the circumstance that a straight line carried round the perimeter of a triangle by being applied to all the sides in succession, is brought into its old situation again; the argument being, that because this line has made the sort of somerset it would do by being turned through four right angles about a fixed point, the exterior angles of the triangle have necessarily been equal to four right angles. The answer to which is, that there is no connexion between the things at all, and that the result will just as much take place where the exterior angles are avowedly not equal to four right angles, Take, for example, the plane triangle formed by three small arcs of the same or equal circles, as in the figure; and it is manifest that an arc of this circle may be carried round in the way described and return to its old situation, and yet there be no pretence for inferring that the exterior angles were equal to four right angles. And if it is urged that these are curved lines and the statement made was of straight; then the answer is by demanding to know, what property of straight lines has been laid down or established, which determines that what is not true in the case of other lines is true in theirs. It has been shown that, as

plane, if they are subject to an increase of distance on one side, will not be subject to a diminution of distance on that same side, and the contrary; but will cut one another. And in the demonstration of this I shall employ another proposition, which Euclid has employed in the Tenth Book and elsewhere, which is, that of any two finite magnitudes of the same kind, the smallest by being doubled over and over will become greater than the greatest. And it will further require to be laid down, that one straight line cannot be in the same straight line with straight lines more than one that do not coincide with one another; and that the angle which is equal to a right angle, is a right angle." We omit the Arabic.

I. 32. Cor. 2.

angles.

*

15. Franceschini, Professor of Mathematics in the University of Bologna, in an Essay entitled La Teoria delle parallele rigorosamente dimostrata, printed in his Opuscoli Matematichi at Bassano in 1787, offers a proof which may be reduced to the statement, that if two straight lines make with a third the interior angles on the same side one a right angle and the other an acute, perpendiculars drawn to the third line from points in the line which makes the acute angle, will cut off successively greater and greater portions of the line they fall on. From which it is inferred, that because the portions so cut off go on increasing, they must increase till they reach the other of the two first straight lines, which implies that these two straight lines will meet. Being a conclusion founded on neglect of the very early mathematical truth, that continually increasing is no evidence of ever arriving at a magnitude assigned.

The remainder in our next.

LESSONS IN ITALIAN GRAMMAR.-No. XIX.

By CHARLES TAUSENAU, M.D.,

Of the University of Pavia, and Professor of the German and Italian Languages at the Kensington Proprietary Grammar School.

In.

THE preposition in denotes being, continuance, or motion in the interior of a thing. It also denotes any kind of motion or penetration into it. The idea of existence in a time or in a certain condition, particularly in a certain state or disposition of the mind, likewise requires the use of in. The preposition a, on the contrary, merely expresses presence near or about a thing or motion, approach, and tendency to it; e. g. i-gli è nel giar-di-no, in quel-la cá-me-ra, in cit-tà, in piaz-za, he is in the garden, in that room, in the town, in the square; é-gli an-drà in In-ghil-tér-ra, in I-spú-gna, he will go to England, to Spain; nell' án-no mil-le set-te cen-to, in the year 1700; sog-yior-no al-quan-to in Ró-ma, he staid a while in Rome; Ge-su Cristo ná-cque in Be-te-lém-me, Jesus Christ was born in Bethlehem; é-gli mo-ri nel mil-le tre cên-to,

• See the Notes to Playfair's Elements of Geometry, p. 406; where there is a figure.

he died in 1300; im-mêr-ge-re ú-no nell' d-cqua, to plunge one in the water; é-gli é-ra qui in quest' i-stan-te, he was here (in) this moment; é-gli è in a-go-ni-a, he lies in the agonies of death; és-se-re in cól-le-ra, in gio-ja, in af-fli-zió-ne (i, e. nél·lo stá-to di col-le-ra, di giô-ja, di af-fl-zió-ne), to be angry, cheerful, sad (i. e. in a state of anger, joy, affliction); a-vér quál-che eôsa in boc-ca, in md-no, to have something in one's mouth, in one's hand; ês-se-re, stá-re in cam-pá-gna, to be, reside in the country; an-dd-re, en-trá-re in ú-na chiê-sa, to go into, enter a church; ca-scá-re in ú-na fôs-sa, to fall into a pit or hole; mét te-re le md-ni in tû-sca, to stick or thrust one's hands into one's pocket; me-ni-re il ca-vál-lo in i-stál-la, to lead a horse into the stable; sa-li-re in cá-me-ra, to go up into the room; vi-véva in un sê-co-lo di bar-bd-rie, he lived in an age of barbarity.

I have already remarked that the proper names of towns and similar localities are exceptions to the above-stated rule, for they have the preposition a as well as in placed before them, whenever a stay or arrival in them is expressed; e. g. é-gli stet-te per tre an-ni in (or a) Ró-ma, he lived for three years in Rome; la stu-te pas-sá-ta i-o stét-ti dú-e mé-si a (or in) Fi-rèn-ze, last summer I lived two months in Florence. There is, however, a shade of difference between the employment of a and in in such cases, which will be at once understood by the following examples; è in Lôn-dra, in the strictest sense of the word, means a person being or an occurrence taking place within the precincts properly called London; while è a Lón-dra, in the more enlarged or general meaning of the word, means a person not necessarily being in, or an occurrence not necessarily taking place within, those precincts, but perhaps in the neighbourhood of London; e. g. at Kensington.

The motion to or towards a town or village, conformably to the nature of the preposition, is always expressed by a Motion to or towards (and, naturally, being or staying in) parts of the world, countries, provinces, and islands, requires the preposition in. The reason of this appears to be, that in the latter instance, the idea of a penetration into the interior of these more extended localities prevails, though, strictly and logically speaking, the idea of going to or into Pie-tro-bur-go, let us go with him to St. Petersburgh; gli a town amounts to the same thing; e. g. an-diá-mo con lui a par-ti da Mô-na-co per re-car-si a Vi-ên-na, he departed from Munich to go to Vienna; -gli si por-tò a Cel-sé-a, he repaired to Chelsea; é-gli è an-dú-to a Pa-ri-gi e pói an-drà a Cel-te-nám, he is gone to Paris, and after that he will go to Cheltenham; quan-do an-dré-te in Fran-cia? when will you go to France? fa-ré-mo un viág-gio in Mo-scó-via, a Mo-scó-via, we shall go on a journey to Russia, to Moscow; i-o va-do in I-scó-zia, in I-svézid, I go to Scotland, to Sweden; il Ba-scia fu e-si-li-d-to nell' i-so-la di Ci-pri, the pasha was exiled to (the island of) Cyprus; è-gli è in Fran-cia, nél-la Chí-na, he is in France, in China; ná-cque nell' i-so-la di Lê-sbo, he was born in the island of Lesbos.

Usage allows the omission of the article after in before many nouns familiarly known and constantly recurring in conversation; e. g. é-gli va nél-la cá-me-ra, nél-la cit-tà, nel-la chiê-sa, nei-la can-ti-na, &c.; or, é-gli va in cá-me-ra, in cit-tà, in chiê-sa, in can-ti-na, &c., he goes to the room, to town, to church, to the cellar, &c.

Before the words day, week, month, year, morning, evening, when time is the subject, it is customary to omit the preposition in; e. g. l'an-no che mo-ri il Ga-li-lê-o, na-eque il Newton, in the year in which Galileo died, Newton was born; il me-se ven-tu-ro, (in the) next month; la set-ti-md-na scór-sa, (in the) last week; la not-te che vic-ne, (in the) next night, &c. ; instead of: nell' án-no, nel mé-se, &c.

have a proper or original and a figurative signification. In the The words cá-sa, cór-te, pa-láz-zo, teá-tro, lêt-to, and scuô-la, former case, they demand the preposition in; in the latter, the preposition a (without an article) before them; e. g.