Mulhacen, the highest points of the Sierra Nevada, in Spain; and Etna, the Sicilian volcano, which are all nearly of the same elevation as the Pyrenean peaks.

The principal Rivers in Europe are the following:-The Volga, which rises in the Valdai Hills, in Russia, about 1,100 feet high, runs a course through that country of more than 2,000 miles, and falls into the Caspian Sea at Astrakan, where its delta (or mouths in the shape of the Greek letter of that name) is about fourteen miles wide. The Ural or Jaik, which rises in the Ural Mountains, runs a course of more than 1,000 miles, and falls into the Caspian Sea; the Don, which, after a course of more than 1,000 miles, falls into the Sea of Azov, at the town of Azov; the Dnieper, which, after a course of 1,200 miles, falls into the Black Sea at Kherson; the Dniester, which flows from the Carpathian Mountains, and, after a course of 700 miles, falls into the same sea at Akerman; the Danube, which has its source in the Black Forest (Schwarzwald), flows through Bavaria, Austria, and Turkey, and, after a course of about 1,630 miles, falls into the Black Sea at Ismail; the Rhine and the Rhone, which both spring from the Swiss Alps, take opposite courses: the former, running through the Lake of Constance, after a northern course of about 800 miles through Germany, France, and the Netherlands, falls into the German Ocean, after losing itself in the Waal, the Mæse or Meuse, the Leck, and the Old Rhine, and forming the great Delta of Holland between Rotterdam and Amsterdam; the latter, running through the Lake of Geneva, after a southern course of about 500 miles through France, falls into the Mediterranean at the Gulf of Lyons, where it also forms a delta; the Arno and the Tiber, on the west of Italy, fall into the Tuscan Sea; and the Po and the Adige, on the east of Italy, fall into the Adriatic; the Xucar and the Ebro, of Spain, fall into the Mediterranean; the Douro and the Tagus, of Portugal, with the Guadalquivir of Spain, fall into the Atlantic; the Dwina and the Onega, of Russia, fall into the White Sea; the Vistula and the Oder, of Germany, fall into the Black Sea; and the Elbe, of the same country, into the German Ocean, at Hamburgh. The chief Lakes of Europe are the following:-In the north, Ladoga and Onega, in Russia, the largest in Europe, the former being 130 miles long and 70 miles broad-the latter smaller; Peipus and Ilmen, in the same country, very considerably less; the latter is connected with Lake Ladoga and the River Volga by means of canals, and thus a line of communication is formed between the Baltic and the Caspian Seas; the Lakes Wener, Wetter, and Maeler, in Sweden, with the River Gotha, which form a line of communication between the middle of the Baltic Sea and its entrance; in the south of Europe, the Lakes of Constance, Lucerne, and Geneva, in Switzerland; and of Maggiore, Lugano, and Como, in Italy,-all remarkable for the beauty of their scenery. The following table contains some useful particulars relating to the lakes of Europe.

TABLE OF EUROPEAN LAKES.

Tuscany

Popedom

The Two Sicilies.
Modena.........

Parma

San Marino

TURKEY in Europe.
GREECE

Areas in
Square

Miles.

Elevations
in
Feet.

Greatest
Depths in
Feet.

SPAIN

PORTUGAL

ITALY

Ladoga

Russia

6,330

Onega

68,000 178 76,000 1,500 2,395,000 162 31,000 64,000 179,500 15,400,000 8410,800,000 119,000

35,260 3,413,000 93 2,246,000 28,000 104,576 18,745,000 179 12,442,000 227,400 29,000 4,650,000 160 4,800,000 146,000 8,553 1,700,000 198 $36,000 12,000 17,860 2,908,000 163 2,088,000 14.600 44,510 8,423,000 190 4,348,000 48,000 2,331 2,300

22

562,000 241 130,000 3,500 495,000 215 240,000 3,300 7,000 318 210,000 15,500,000 73 3,000,000 300,009 15,200 637,700 421 483,000 8.900 The numbers placed against GERMANY in this table form the aggregates of all the numbers placed against the respective countries and states of which it is composed, as given in the table; the same is to be understood in reference to the num bers placed against ITALY. As some of our readers may be curious to know some of the particulars relating to the British empire separately, we here give them on the same authorities:

BRITISH EMPIRE IN EUROPE.

very remote.

ON COMETS.

ON COMETS.-No. I.* COMETS are planets which describe ellipses of very great length round the sun, that is, the distance between the two foci of such ellipses is very great, and the sun occupies one focus; so that, at one time, comets are very near the sun, and at another they are These bodies are often accompanied by a vaporous and transparently luminous tail, through which the stars are visible. At first, when a comet becomes visible to us, there appears to be This only a nebulosity or cloudiness surrounding the comet. increases as it approaches the sun, and sometimes forms an immense train; the train in some comets reached to the length of 90°, or a The forms of fourth part of the whole circuit of the heavens. comets are also very variable; that which appeared at the birth of Mithridates, 130 years before the commencement of the Christian era, is reported to have presented to the spectators for the period of Ten years before the 80 days, a disc as large as that of the sun. last-mentioned date, a comet had appeared which seemed to extend over the whole face of the heavens. Justin, the historian, in Book 37, chap. 2nd, relates that it had a glare of light greater than the sun, and that it occupied a fourth part of the heavens. The comet which appeared in the year 117 A.D., presented the same phenomenon, and, according to Freret, that which appeared in 479, eclipsed the sun in a very extraordinary manner.

we

In the year 400 A. D., a comet was seen in the form of a sword, which extended from the zenith to the horizon. Another comet appeared in 531, which was denominated Lampadias; it was very large, and presented an alarming appearance to the beholder. The moon, if we believe history, compared with those which appeared in A.D. 1066, and 1505, presented a smaller disc than they did. The historians, however, of those early ages, unacquainted with the laws of nature, and surrounded by people as ignorant as themselves, would, no doubt, from fear and other causes, be led to exaggerate the accounts of such extraordinary sights as they had either heard of or seen in the heavens; and consequently, may conclude that there is much exaggeration mixed with truth in the records which they have transmitted to us of these singular bodies. Besides, either they or their informants may have mistaken occasional atmospheric phenomena, such as balls of fire, and the like, for astronomical phenomena; and thus sometimes they may have attributed to the heavens appearances which really belonged to the earth or its atmosphere. Even in the present day, the ignorant are often astonished by a very common appearance, viz., that of falling stars, which are nothing more than particles of ignited matter floating in the atmosphere; and, as soon as the combustion has nearly ceased, from the almost total consumption of the matter which supported it, the residue sinks in the air and presents the appearance to which the appellation of falling stars has been given. An ignis-fatuus, or what is vulgarly termed Will-o'-the-Wisp, may be traced to the same source. The most remarkable comets were those which appeared in the years A.D. 837, 1106, 1456, 1618, 1680, 1744, 1759, 1769, and 1811. The comet of 1744 had six tails, arranged in the form of a fan. There have been some, on the other hand, which had neither tail nor any sort of detached and diminished light, such as those which are popularly denominated beards, whiskers, or nebulosities. The phases which were observed in the comet of 1744, led to the opinion that these bodies are opaque, formed of a nucleus or inner orb, accompanied by, or surrounded with, a vaporous atmosphere or train of variable intensity and extent.

and

impulse of the particles of light thrown out by that body; and that
these vapours are condensed in the region to which they are after-
Moreover, these vapours, in starting from the
wards carried.
nucleus or body of the comet to which they belong, have less
velocity than it, and remain a little behind it; this makes them
take the form of a curve at the extremity of the tail. This expla-
nation, however, is not quite sufficient to account for the singular
forms which comets sometimes present, and particularly that of
1774, which had six tails. The proximity of the sun, and the con-
tinuity of the transmission of heat, must render its accumulation
enormous, and capable of melting and vapourising every substance
or matter in the comets similar to what is found in the earth;
it might even happen that the nucleus itself should lose its solidity.
The comet of 1680, when in perihelion, was one hundred and
sixty-six times nearer the sun than the earth is, or sixty-four times
nearer than the planet Mercury; in other words, if the distance
of the earth from the sun were divided into one thousand parts, the
perihelion distance of that comet was only six of those parts! The
heat which it then received from the sun, was twenty-eight thou-
sand times greater than that which the earth receives from it; the
temperature thus produced was about two thousand times greater
than that of melted iron, and two hundred and seventy-five thou-
sand times greater than that of summer heat. It would require fifty
thousand years to cool such a body, were it to be placed in the
same physical circumstances as the earth, and had it the same size
as our globe. With a telescope the tail of a comet is scarcely
visible: this vapour, light haze, or mist, can only be observed by
means of its density. The nucleus itself is sometimes a substance
amazingly rare or thin: as an instance, it was observed that when
the comet of 1811 passed over a star, a circumstance itself rather
uncommon, the star was seen even through the nucleus or body of
the comet.

Comets differ from planets not only in the phenomena which they
exhibit, but also in the diversity of their motions; these motions
orbit of small eccentricity, and little inclined to the ecliptic; yet
are neither always in the direction from east to west, nor in an
several comets agree in these circumstances with one another. In
comet is only visible to us when it approaches the perihelion, its
general, their orbits are ellipses very greatly elongated; and as a
orbit then, to all appearance, coincides with a parabola, which is a
of whose foci is supposed to be at an infinite distance from the
curve of the same origin as the ellipse, being indeed an ellipse, one
other. When a comet appears, astronomers make three observa-
tions on it, so as to ascertain its distance from two known stars, and
thus they ascertain its right ascension and declination at those
times; three points being sufficient to determine the form of the
visible curve which it describes is a parabola. In this manner,
parabolic curve, it is concluded from these observations that the
three observations, even very near each other, give the perihelion
distance, the position of the perihelion point, the instant of the
data are obtained the hour of its rising and setting for the whole
comet's passing this point, and the nodes of the orbit; from these
its course, &c.
period of its appearance to us,-the_stars that will be near it in

Nearly one hundred comets, which have been observed with care, have their orbits exactly represented by the parabolic curve, in the part of their orbits which is visible to us; consequently, this fact is established beyond a doubt. In the regions of space, where The opinion a comet escapes our observations, it completes its immense elliprespecting the opacity of comets is confirmed by the latest experi-tical orbit: it may even be the case that several of these bodies ments of M. Arago, on the polarisation of the light of the comet actually describe parabolas or hyperbolas, and consequently, they of 1819. A comet has never any tail when it is at a great distance will never return to our system again, but follow some other sun. from the sun; but as soon as it comes so near the sun as to be Burkhardt, the celebrated astronomer, believed from his observaonly at the distance of eighty millions of miles, the tail or vapour tions, that the comet of 1771 described an hyperbola. The sun begins to appear; it increases in extent, in proportion as the comet occupies the focus of all these orbits, whether elliptic, parabolic, or approaches the sun; and the tail acquires its greatest extent a little hyperbolic. Comets, like all the other heavenly bodies, partake of after passing the perihelion. After passing that point, in pro- the apparent diurnal motion, which is only the effect of the rotaportion as the comet goes to a greater distance from the sun, the tion of the earth on its axis in twenty-four hours; their motions tail diminishes in the same manner as it increased; it soon becomes are agreeable to the laws of Kepler, like all the bodies of our sysno more than a cloud; and at last, the comet itself even ceases to tem; but the calculation does not enable us to determine the return be perceptible. It is observed that the tail of a comet is always so correctly as the return of the planets in their revolutions; because thrown in the direction of the straight line which joins the sun and the arcs of their orbits which have hitherto been observed, are too the comet produced; if the comet be to the east of the sun, and small to lead to an exact determination of the whole of their orbits. set after it, the tail is thrown towards the east; if it be to the west The planets are almost always visible, and from numerous observaof the sun, and set before him, the tail is thrown towards the west. tions of them, we are enabled to correct the elements of their These phenomena induce the belief that the tail of a comet is but a motions. It is evident, therefore, that the same comet must be torrent of vapours raised by the heat of the sun, and driven by the observed in its return to the perihelion several times, before we

314

of the light of the moon. History affords us no account of its returns, till 1456, when it passed near the earth: its tail then occupied about 60° of the heavens, and had the form of a huge sabre. This appearance, which took place at the epoch when the Turks had made themselves masters of Constantinople, became a general subject of alarm, especially when the comet was seen

can determine with precision its periodic revolution; but we can judge only by the circumstance that the elements of the motions which are observed at one time, are the same as those observed at a former time, whether a comet which reappears be the same as one that has been formerly seen. This comparison is the sole method of resolving the problem. The causes, therefore, which have hitherto prevented the pre-approaching the moon at the full, at the very instant of an eclipse. diction of the return of comets, and the determination of their orbits, are the following:-1°. These bodies are only visible in a small part of their course, and that being very near the sun, is described with a prodigious velocity. But the more the ellipse is elongated, the more is the progress of the comet slackened from the action of the sun; so that it may happen towards the aphelion, that the comet may be almost immoveable, and cannot return to our system, till after the lapse of thousands of years. 29. Observations have only been made with care for about two hundred years past; and many comets must have escaped the observation of astronomers. This fact may be argued from the circumstance of the lateness of the discovery of the new planets revolving in the immense void which separates Mars and Jupiter. Even Uranus, of which the revolving period is but eighty-four years, had been previously observed by four astronomers, and yet it was not recognised as a planet till 1781; and it may be added that Mercury, whose period of revolution is only eighty-eight days, is so difficult to observe without optical assistance, that the celebrated Copernicus, who first promulgated the true theory of the present system of astronomy, died with this regret among others, that he never observed that planet.

In addition to all this, when a comet is above our horizon during the day, we cannot see it, though it be under our very eyes. That fine comet which astronomers observed only in July, 1818, after everybody else had seen it, was present in all its brilliancy (but in the daytime), a long time before its discovery. The comet which appeared sixty years before the Christian era, is related to have been visible only when a total eclipse of the sun took place, because that it was too near that body to be seen otherwise.

Halley predicted that this comet would return in 1757, but it was not observed that year; Clairault, the French astronomer, showed, in 1758, that it had been retarded by 618 days in consequence of the action of Jupiter and Saturn; so that, in fact, this comet only passed its perihelion about the 12th of March, 1759. This comet which had only been seen with dread and consternation in 1456, excited at that time a lively interest. The long train which it drew after it had formerly spread terror over Europe, which had been thrown into alarm by the rapid success of the Turks, who destroyed the Greek empire. The Pope Callixtus ordained on this occasion a prayer to be composed and read in the churches for allaying both the comet and the Turks. In those times of ignorance, men were far from thinking that the only means of arriving at a knowledge of nature, is to investigate her phenomena both by observation and calculation. According as those phenomena happened and followed each other with regularity or without apparent order, they made them depend on final causes or on chance; and when they presented anything extraordinary, and seemed to run contrary to the order of nature, they were regarded as so many signs of the But these imaginary causes have been succesanger of heaven. sively bounded by the limits of our knowledge, and should entirely disappear before sound philosophy, which perceives in them only the expression of our ignorance of the true causes. This comet was seen in the year 1835, first at Rome, on the 5th August, and then in Europe generally, on the 20th of the same month; but it may be asked, which of its former appearances did it then resemble? The other two comets whose returns are correctly predicted have short periods; for they return, to the perihelion, the one after 1,210 days, or about three years and four months, and the other after 2,465 days, or about six years and nine months. The first is called Encke's comet, and the other Biela's, because these astronomers calculated the elements of their orbits, and discovered the period of their revolution. The comet of Encke has been seen with and without a nucleus, with and without a tail, and under different appearances in 1786, 1795, 1801, and 1805; it ought to have returned in 1808, 1812, and 1815; but it escaped our observation in these three returns from its being so very small, in addition to the causes that have been already explained; but it re

3o. The appearances which comets present are very variable, because they depend on the place which the earth occupies in its orbit when they are so near the sun as to be visible; and this place must vary considerably in an orbit of one hundred and ninety millions of miles diameter. From this cause, a comet which may have at one time presented a very fine appearance, may, on its return, appear very small, or be even not at all visible, although near enough to the sun to be visible at another point of the earth's orbit, and its tail may have disappeared owing to a difference in its position. So many circumstances concur to pro-appeared in December 1818. duce changes in its aspect! The comet or 1811 was scarcely visible in April and May; it was afterwards immersed in the rays of the sun, and it only appeared at the end of August, beyond its perihelion point; but what a difference was there to us in its brilliancy, and how astonishingly magnificent was the appearance which it presented! Who could have recognised it as the same body? No one could have been certain on this point, except from the agreement of the elements of the orbit. But if the attraction of such masses as Jupiter and Saturn, and even Uranus, changes these elements, how shall the comet be recognised? This is a question that can only be solved by the application of the profoundest knowledge of the mathematical sciences.

4°. The greatest causes of error arise from the inaccuracy of observation. Small comets appear only as vaporous points; large ones seem to form a cloud, variable and ill-defined. The comet of 1729 was visible during six months; three astronomers calculated its orbit, and their results did not agree. What could be the reason but inaccuracy in the data obtained by observation? The comets of 1762, 1763, 1743, 1759, and 1766, afforded examples of the same description. How, then, is it possible to predict the returns of comets with such defective elements? It will not be considered surprising, therefore, if among one hundred and thirty comets, of which the course has been calculated, there are only three whose return can be predicted with certainty.

The first is that which bears the name of Dr. Halley, an English astronomer, who observed it in 1682: he found that the elements of its orbit were the same nearly as those of the comets observed in 1531 and 1670; he therefore concluded that it made its revolution in about seventy-six years, a shorter period than that of the planet Uranus. It had appeared in 1006, it then seemed four times greater than Venus, and threw out as much light as the fourth part

FRENCH EXTRACTS.

PENSEES MORALES ET MAXIMES

DOUTE.

Dans le doute, abstiens-toi.-Pythagore.

Il vaut mieux pencher vers le doute que vers l'assurance, aux choses de difficile preuve et dangereuse créance.-Saint Augustin,

DURETE.

L'insensibilité à la vue des misères, peut s'appeler dureté; s'il y entre du plaisir, c'est cruauté.-Vauvenargues.

Ne soyez pas trop rigoureux dans le châtiment. Il est rude, quelque léger qu'il soit. Ne vous en servez pas non plus trop fréquemment, vous pouvez arriver à votre but par d'autres voies que par celle-là.-Maximes des Orientaux,

ECONOMIE.

L'économie est fille de l'ordre et de l'assiduité.-Lévis.

Tout homme qui ne veut pas que sa fortune décroisse et qui veut rester constamment au niveau, doit se faire une loi de ne dépenser que la moitié de son revenu, et celui qui veut augmenter son bien ne doit dépenser que le tiers de sa rente.-Bacon.

EDUCATION.

Formez l'enfant à l'entrée de sa voie; car il ne s'en éloignera point, même dans sa vieillesse.-Salomon. L'avenir d'un enfant est toujours l'ouvrage de sa mère.-Napollos.

EGOISME.

L'égoïsme est une combinaison, un système, une volonté de tout ramener à son bonheur.-La Bruyère.

L'égoïste brûlerait votre maison pour se faire cuire un œuf.Chamfort

ENVIE.

Ceux qui font bien mériteraient seuls d'être enviés, s'il n'y avait encore un meilleur parti à prendre, qui est de faire mieux; c'est une douce vengeance contre ceux qui nous donnent cette jalousie. -La Bruyère.

L'envie qui parle et qui crie est toujours maladroite; c'est l'envie qui se tait qu'on doit craindre.-Rivarol.

L'envie est un sentiment de haine mêlé de désirs, sentiment que fait naître en nous le chagrin de voir posséder par un autre un bien que nous désirons. C'est la plus triste et la plus honteuse des passions. Elle devient le tourment de ceux qu'elle possède et de ceux qu'elle attaque. Elle est le fruit d'un amour-propre désor donné.-Locke.

ERREUR.

Une des erreurs les plus communes est de prendre la suite d'un événement pour sa conséquence.-Lévis.

ESPERANCE.

L'espérance est un emprunt fait au bonheur.-Rivarol.
Jouissez de ce que vous possédez; espérez ce qui vous manque.-
Lévis.

ESPRIT.

Le mot esprit est un mot générique qui a toujours besoin d'un autre mot qui le détermine....Ce qu'on appelle esprit est tantòt une comparaison nouvelle, tantôt une allusion fine; ici, l'abus d'un mot qu'on présente dans un sens et qu'on laisse entendre dans un autre; là, un rapport délicat entre deux idées peu communes ; c'est une métaphore singulière; c'est un recherche de ce qu'un objet ne présente pas d'abord, mais de ce qui est en effet dans lui; c'est l'art, ou de réunir deux choses éloignées, ou de diviser deux choses qui paraissent se joindre, ou de les opposer l'une à l'autre ; c'est celui de ne dire qu'à moitié sa pensée pour la laisser deviner.-Voltaire. On n'est pas un homme d'esprit pour avoir beaucoup d'idées, comme on n'est pas un bon général pour avoir beaucoup de soldats. --Chamfort.

Il est encore plus facile de juger de l'esprit d'un homme par ses questions que par ses réponses.-Lévis. L'esprit de parti abaisse les plus grands hommes jusqu'aux petitesses du peuple.-La Bruyère.

L'esprit de quelques personnes est comme une lanterne sourde, qui ne sert qu'à celui qui la porte, et qui n'éclaire que son chemin. -Pope.

L'une des marques de la médiocrité de l'esprit est de toujours

conter.-La Bruyère.

A mesure qu'on a plus d'esprit, on trouve qu'il y a plus d'hommes originaux. Les gens de commun ne trouvent pas de différence

entre les hommes.-Pascal.

ETUDE.

L'étude la plus utile est celle de soi-même les peines et les travaux des écoles ne servent à cette dernière étude que comme de degrés.-J. -J. Rousseau.

L'acier n'a l'avantage sur le fer que par le travail qui lui a fait acquérir plus de perfection.-Epictete. Etudiez, non pour savoir plus, mais pour savoir mieux que les autres.-Sénèque.

EXPERIENCE.

Il est sage d'interroger nos heures passées, leur réponse forme ce que l'on nomme l'expérience.-Young.

FAIBLESSE.

Les gens faibles sont les troupes légères de l'armée des méchants: ils font plus de mal que l'armée même; ils infestent et ils ravagent. -Chamfort.

FATUITE.

Un fat est celui que les sots croient un homme de mérite.-La Bruyère. Je n'aime pas un homme que je ne puis aborder le premier ni saluer avant qu'il me salue sans m'avilir à ses yeux et sans tremper dans la bonne opinion qu'il a de lui-même.-Idem.

SOLUTIONS OF PROBLEMS AND QUERIES.

NEW GEOMETRICAL DEMONSTRATION OF THE
PYTHAGOREAN THEOREM.

IN any right-angled triangle, the square which is described upon
the side subtending the right angle is equal to the squares described
upon the sides which contain the right angle

On B C describe the square B D E C (I. 46), and on BA, A C, the squares G B, H c; produce D B to meet F G in M (Post. 2); through it, draw A c, parallel to B c, meeting B M in N, and A P, parallel to MD (1.31); from P, draw P T and PQ, parallel to AB and A C respectively; from P, in PT, make P s equal to P Q (I. 3); and, from s and a, draw s L and a R, parallel to P Q and P s respectively (I. 31).

Then, because the angle B A C is a right angle (hyp.), and that the angle B A G is also a right angle (Def. 30), the two straight lines, ▲ C, ▲ G, upon the opposite sides of A B, make with it, at the AC is in the same straight line with A G (I. 14); for the same point A, the adjacent angles equal to two right angles; therefore, reason, B A and A H are in the same straight line. And because a parallelogram (Def. À.); whence N B is equal to A P (I. 34); for A N is parallel to B P (cons.), and A P to N B, therefore NA P B is a similar reason, B T is equal to A P; wherefore, N B is equal to And BT (AX. 1), and the triangle A N B is equal to the triangle PBT (I. 38), being between the same parallels, A P and N T. the side B C, and o B is equal to a c (I. 34); but N A is equal to because A OBC is a parallelogram (cons.), the side o A is equal to B P, therefore the remainder N o is equal to the remainder r c fore PQ, QC, are equal to C A, A P, respectively; but it has been (Ax. 3). And because A CQ P is a parallelogram (cons.), thereshown that CA, A P, are equal to o B, B N, respectively; wherefore, PQ, QC, are equal to o B, B N, each to each (Ax. 1), and the third side PC has been shown to be equal to the third side o N; thereI. 4). And because S R is parallel to P Q, and a R to P s (cons.), fore, the triangle P ca is equal to the triangle o N B (I. 8 and therefore P Q R s is a parallelogram (Def. A.); whence its opposite sides are equal (I. 34), and the angles which T P or Q P make with the angles B C, upon one side of it, are equal to two right angles (I. 13); but the angles BPT and cra, are equal to (I. 32) ABP and ACP of the triangle A B C, respectively; wherefore the remaining angle s Pa is equal to the remaining angle B A C (Ax. 3), that is, to a right angle; therefore, the remaining angles of P Q R S are right angles (Cor. I. 46), and the side PS is equal to the side P Q (cons.); whence, P Q R S is equilateral, and is therefore a square (Def. 30); and it is equal to the square H C, because a C is equal to P Q.

Again, because the angle C B M is a right angle, as also the angle A B F, take from these equals the common angle A B M, therefore, the remaining angles C B A and M B F are equal (Ax. 3); hence, in the two triangles A B C and F B M, the side A B is equal to the side FB, and the angles CB A, B A c, are equal to the angles M BF, BFM, respectively; therefore, the side F M is equal to the side A C, and B M is equal to B C (I. 26), or to B D, of which the part N B is equal to the part B T; therefore the remainder M N is equal to the remainder TD (AX. 3); now, r s is equal to P Q, or to a C, or to F M, and T P is equal to BA, or to F G; therefore, the remainder T s is equal to the remainder MG; and the angle S T D is equal to the angle G M N, because MD falls upon the parallels G M and 6 T

(I. 29); hence, the triangle s TD which would be formed by join- | ing S D is equal to the triangle G M N which would be formed by joining G N (I. 6); whence, the straight line joining S D is equal to the line joining GN, and the angles LDS and DSL are equal to the angles ANG and N GA, respectively, because their complements are equal in the former triangle; wherefore, the triangle S D L is equal to the triangle GNA (I. 26); therefore, the trapezium 8 TDL is equal to the trapezium G M NA (Ax. 2); in a similar manner, the trapezium Q ELR may be shown to be equal to the trapezium MNOF. And because the triangle A N B has been shown to be equal to the triangle P B T, and PCQ to o N B, as also the square HC to the square P Q R S, therefore the whole square B C E D is equal to the squares BG, CH (Ax. 2), which was to be demonstrated.

HORATIO ROBINSON, Sheerness. NOTE.-The trapezium 8 T D L might have been shown to be equal to the trapezium G M NA, as also QELR to MNO F, by proving the sides and angles equal, on the same principle which Euclid employs in the fourth proposition of the first book,-the well-known principle of supraposition, which results in coincidence and consequent equality.

The reader will understand this part better, by actually drawing the Raymond, and Finney, Judge Birnie, Horace Mann, Lewis Tappan, Horace lines DS and N G.-ED.