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John.

island of Corfu; la re-gi-na d' In-ghil-tér-ra, the Queen of Eng- Adverbs of place or time before, nouns or even adiectives land ; il Re di Prús-sia, the king of Prussia ; l' im-pe-ra--re of this class, frequently, also, are translated by the genitive d' -stria, the emperor of Austria ; l' as--dio di Mán-to-va, case ; e. g. the back door or room, la pôr-ta la, stán-za di dié-tro the siege of Mantua; lo strét-to di Gi-bil-têr-ra, the straits of|(the door or room of behind); the hind-feet, i piê-di di diê-tro Gibraltar ; l' im--ro di Rús-sia, the empire of Russia ; tra- (the feet of behind); the following day, il giór-no di do-inc-ni -die di Al-fiê-ri, the tragedies of Alfieri; le coin--die di Gol(the day of to-morrow); the present age, it món-do ďog-gi-di -ni, the comedies of Goldoni.

(the world of now-a-days); after the present fashion or style, III. When words expressing quantity, weight, or any kind of al mo-do d' og-gi-di (after the fashion of now-a-days); the whole measure are joined to other nouns ; e. g. ú-na quan-ti-di pea last night, la not-ta-ta di -ri (the whole night of yesterday); meat ; un cen-ti-na-jo di fie-no, a hundred-weight of hay; ú-na! Whenever the infinitive mood of any verb explains and dedoz-si-na di, cuc-chiá-ji, di guin-ti, d'uô-va, a dozen of spoons, fines another word, the preposition di must be placed before gloves, egi's ; un brúc-cio di pán-no, a yard of cloth ; ú-na bot- it (just as the preposition of with the present participle of ti-glia di 'virno, a bottle of wine ; 'ú-na ca-rif-fa á-cquu, a English grammar in such cases); e. g. Ha ú-na gran -glia di decanter of water ; un ón-cia di caf-fè (kahf-fe), an ounce or viag-gisi-re, he has a great desire to travel or of travelling; è coffee; vi-ro di diê-ci in-ni, wine of ten years.

téin-po di an--re, it is time to go or of going; ra-gió-ne di For the sake of elegance, the preposition di is, however, di re-dér-vi, the honour to see you or of seeing you; li-cén-os.

lo-inen-tur-si, right to complain or of complaining; ľ o-no-re sometimes omitted after the words ca-sa, house; pa-láz-zo, di par-tir-si, permission to go away or of going away.* palace; piás-zd, place, square; vil-la, villa; gal-le-ri-d, gallery; fa-mi-glia, family; pôr-ta, gate, entry, and some others: or long; al-quun-to, something, a little, some ; tán-to, so much,

Di is also placed after the words quir-to, how much, or great, when they are followed by the name of the owner or of the person after which they are called ; e. g. in cúzsa Al-tie-ri, at little

, few ; mol-to, much, a great deal; nién-te

, nothing; più,

or great, or long; al-tret-tán-to, just as much, equal; -co, the · Altieri-house; viací-no al pa-láz.co Bor-ghé-se, near the Borghese-palace; súl-la piáz-za Bar-be-ri-ni, on the Barberini- more; inerno, less; tróp-po, too much, &c.; e. g. quán-di square; per la vil-la Pan-fi-li, for the villa Panfili; nél-la gal- -ja sa-réb-be per me, how a great a nuisance would it be for ls-ri-a Do-ria, in the Doria-gallery; dél-la fa-mi-glia Co-lin-na, me; -po al-quan-to di tém-po, after some time; tán-to di vi-no

'--, just of the Colonna family; la pôr-ta di San Gio-ván-ni, St. gate or entry; a ca-sa’i zi-o instead of a ca-sa del zi.o), at the water; po-co di ú-ti-le ne ri-ca-ve--te, you will derive from

this little advantage; inol-to di mu-le ne po-tréb-be se-gui-re, a house of the uncle ; 6-ra a -sa qué-sto, óra a -sa quell' altro (instead of di qué-sto, di quell' ál-tro), now at the house of great deal of evil might be the consequence of it.

In these two phrases, la Di-o mer-ce! thank God! and la this one, then at the house of the other.

Di-o grui-cia, the grace of God, the word di is understood, English compound nouns, or combinations of nouns, for the and in full they run thus : la di Di-o mer-cè, and la di Di-o grugreatest part must be decomposed by the genitive case with zia. Then, however, Di-o is placed after the words iner-cè the case-sign di, especially when one of the nouns merely de- and gra-zia, the case-sign di cannot be omitted; e. g. la mer-fines and qualifies the other, which is the principal word di Di-o, and la grá-zia di Di-out conveying the principal idea; e. g. garden-door, pôr-ta di giar

The word di is sometimes a mere expletive; e. g. é-gli di-ce di-no (door of the garden); stone-quarry, cu-va di pre-tra di si, ed i-o di-co di no, he says yes, and I say no; qué-sto dia(quarry of stone); autumn fruits, frút-ti d'au-tun-no (fruits of ro-lo di qué-sta fém-mi-na, that devil of a woman; quel po-ve-riautumn); a music amateur, un di-let-tun-te di mu-si-cm (an no di mi-o fra-tél-lo, that poor brother of mine. amateur of music) ; Leipzig fair, fiê-ra di Li-psia (fair of Leip

As a last remark on the use of the case-sign di for the zig); ox-tongue, lin-gua di -ve (tongue of an ox); horse's present, I shall state that this word, among all the prepositions head, tê-sta di ca-vál-lo (head of a horse); felt-hat, cap-pêl-lo di of the Italian language, is of by far the most extensive use. fél-tro (hat of felt); sugar-box, cus-sa di zúc-che-ro (box of The reason of this is, that di, properly and philosophically sugar).

speaking, merely copresses the mental separation of ideas or Whenever it is necessary with greater precision to define notions, while da indicates a real separation of objects, which the noun in the genitive case so as to distinguish it from other distinction constitutes the principal and fundamental differe objects of the same class, the article, according to its peculiarence between these two important words di and da. The mere function of particularising that which is general, must be mental separation of ideas or notions may, however, serve any joined to the case-sign di.

connexion and relation between words, ever so loose and geneThe disregard of this rule will not unfrequently cause ral, and no reader, bearing this truth in mind, henceforth need ambiguity; e. g. il pa-dró-re dél-la cưi-sa -ve a-bi-tia-mo, the be surprised at meeting, in Italian books and conversations, master of the house where we live (il pa-dró-ne di ca-sa, is the with frequent substitutions of the case-sign di for many other master of the house in general); un boc-ca-le del vi-no che bév- prepositions; e. g. for a: I'-schia è ú-na i-80-la as-sc-i vi-ci-na vi l dl-tra -ra, a measure = about two pints) of the wine di Na-po-li, Ischia is an island close to Naples; for da : u-sciwhich I drank the other evening (un boc-ca-le di vi-no, is a re del-la pri-gió-ne, to go out of or from prison ; égli di pri-giúmeasure of wine in general); il mer-co-to dei ca-vál-li, the horse- ne il tras-se, he took him from prison; for cour: di gran-dis-simarket (il mer-ce-to di ca-vul-li, is merely a place where horses ma för-za si com-bat--a da cin-scu-na pár-te, they fought with are sold); il mer-ca-to dél-la sel-vag-gi-na, the game-market; the greatest energy on each side; for in :ico l' uc-ci-si di lvi-lo il ma-gaz-zi-no dél-la pu-glia, the straw-magazine (ma-gas-zi-no bat-ta-glia, I killed him in fair fight; for per : é gli pia-gne-a e di -glia, is merely a magazine full of straw); il ma-gaz-zi-no di gran pie-non po--a một-to fi-re, he wept, and on account dél-le -gna, the wood-magazine.

of his great emotion he could not utter a word. English adjectives, indicating the material or stuff from

It is evident that the variable nature of di will admit of which anything is manufactured, or denoting qualities attri- many translations into English; e. g. by with: -no con-tén-ta buted or derived from proper names of countries, nations, or di te, I am satisfied with thee; by at : mi ri-do di lui, I laugh at towns, for the greatest part will be best translated into Italian him; by of: mo-rir di -me, to die of hunger; by as : scr «vir by means of nouns in the genitive case ; e. g. a gold watch, di -go-la, to serve as a rule; by for : pre-ju-re i-no di á-hi un 0-90--gio -ro (a watch of gold); a marble statue, ú-na cô.sa, to request one for something; by than : più di du-e-mi-la stá-tua di már-mo (a statue of marble) a wooden table, á-na scu-di, more than two thousand crowns. -vo-la di -gno (à table of wood); an iron gate, ú-na pôr-ta di In some instances, the peculiarities in the use of di may, fêr-ro (a gate of iron); a silver spoon, un cuc-chia-jo d' ar-gên- without difficulty or twisting, be explained by ellipsis, para to (a spoon of silver); a meritorious soldier, un sol-dd-to di mepi-to (a soldier of merit); a spirited or talented youth, un gió- * Which special class of verbs, nouns, and adjectives, requires va-ne di spi-rito, di ta-lên-to (a youth of spirit, of talent); Italian the preposition đi before the infinitive mood governed by them, silk, -ta d' I--lia (silk of Italy); Viennese citizens, cit-ta-di- will be explained hereafter. For the present, the above-stated ni di Viên-na (citizens of Vienna).' It is, however, quite allow- merely general rule will be, I think, sufficient. able to say: stá-tua mar--rea, sol-da-to me-ri--eo-le, gió-va- + Some other important omissions of the case-sign di will be He spi-ri--so, cit-ta-di-ni Vien-ne-si.

explained hereafter.

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surance,

ticularly when it denotes descent or children; e. g. Gian-nuól

EXERCISES.ITALIAN-ENGLISH, di Se-te-ri-no, Cêc-co di Mes--re An-giu-liê-ri, in Boccaccio, where fi-gliuô-lo, child or son, is understood.

La me-mô-ria. Del-la Giê-ra. Al-la col-lí-na. Dal-la ESERCISES- ENGLISH-ITALIAN."

spia-na-ta. Le bec-che-rí-e. Dél-le lo-can-de. Al-le pôr

te. Dál le strade. In fác-cia. Nél-la ví-gna. Nél-le fo-rêThe rising of the sun. The dawn of the day. The return ste. Con pá-glia. Cól-la ví-te. Cól-le pén-ne. Per dis of spring. The warmth of the air. The beauty of the flower. grá-zia. Per la val-le. Per le scioc-chez-ze. Súl-la car-rôzThe darkness of the night. The abyss of error. The fertility za. Sul-le rú-pi. L'au-rô-ra. Dell'al-le-gréz-za. All' oof the fields. The colours of the rainbow. The senses of man. pi-nió-ne. Dall'o-ste-ri-a.

Le i-de-e. Dell' êr-be. Al-le The faults of young men. Money is the soul of commerce. ár-ti. Dal-le cit-tà. In i-slít-te. Nell' im-ma-gi-na-zió-ne. Usage is the legislator of languages. The master of the garden Nél-le a-ni-me. Con a-cqua. Coll' un-ghia. ČOl-le in-séis not here. The palace belongs to the prince. Here are the one. Per a-mi-cí-zia. Per l' as-si-cu-ra-zió-ne. Per le arooms of the uncle. The dresses belong to the cousin and not zió-ni. Sull' in-sa-la-ta. Súl-le in-fer-ria-te. Un fan-ciúl. to the aunt. The brother tells the sister the will of the lo. U-no stól-to. Un a-ni-má-le. U-na set-ti-ma-na. D'un father. The children must always obey the parents. The fiú-me. Ad û-no schiop-pet-tiê-re. Da u-na bal-le-ri-na. physicians say: the disorder shortens life. Exercise is useful In ú-na chiê-sa. Con un ba-stó-ne. Per ú-no sco-la-re. Su to the body and to the mind. The countenance is the mirror d'un sas-so, só-pra un sás-80. of the soul. Tranquillity of mind is the highest degree of

VOCABULARY. happiness. Temperance is the treasure of the

wise man. The true ornament of the soldier is courage. The practice leads Ciera, mien, look, air of the Idea, idea, notion.

Memoria, memory.

Osteria, public-house, tavern. to perfection.

Interest, pleasure, and glory, are the three motives of the actions and of the behaviour of men.

face,

Erba, herb.
VOCABULARY.

Collina, hill.

Arte, art.

Spianata, plain, esplanade. Città, town, city (no change Rising, Da-da, m. Will, vo-lon-, f. Beccheria, slaughter-house,

in the pl.) Sun, -le, m. Father, pa-dre, m.

shambles.

Slitta, sledge.
Dawn, spun-túr, m.
Child, fan-ciúl-lo, m.

Locanda, inn, hotel.
Must always obey, de-vo-no Porta, gate, door.

Immaginazione, imagination. Day, giór-no, m.

Anima; mind, soul.
Return, ri-tór-no, ni,

se-pe ob-be-di-xe
Strada, street, road.

Acqua, water.
Spring, pri-ma--ra, f. Parents, ge-ni--ri, pl.

Faccia, face (di-re in fác-cia, Unghia, nail.
Varmth, ca-lóvre, m.
Physician, mê- da-co, m.

to tell one to one's face). Insegna, sign, arms, colours.
Air, á-ria, f.
Says, di-ce
Vigna, vineyard.

Amicizia, friendship: Beauty, bel-léz-za (ts), f. Disorder, dis-or-di-ne, m.

Foresta, forest.

Assicurazione (ts), security, inDarkness, 0-SCU-ri-tà, f. Shortens, accor-cia

Con, with.
Night, nit-te, f.
Life, vi-ta, f.
Paglia, straw.

Azione, action.
Abyss, a-bis-so, m.
Exercise, -to, m.
Vite, vine.

Insalata, salad.
Error, er-96-te, m.
Is useful, gió-da
Penna, pen.

Inferriata, iron-grate.
Fertility, fer-ti-li-, f.

Body, côr-10, m.
Field, cont-po, m.
Mind, spí-ri-to, m.

Per, for, through, on account Fanciullo, child.
of.

Stolto, fool,
Colour, co--re, m.
Counteñance, fi-80-no-mi-lg f. Disgrazia, misfortune, dis-

Animale, animal.
Rainbow, ar-co-ba--no, m.
Is, è

grace (per dis-gra-zia, unfor. Settimana, week.
Sense, sen-so, m.
Mirror, spéc-chio, m.

tunately).

Fiume, river.
Man, 2.5-ao, m.
Soul, á-ni-ma, f.
Valle, valley.

Schioppettiere, arquebusier,
Fault, er--re, m.
Tranquillity, quié-te, f.
Sciocchezza (ts), folly.

rifleman. Young man, gió-00-70e, m. Mind, á-hi-ma, f.

Carrozza (ts), carriage, coach. Ballerina (f.), dancer.
Money, da-ni-ro, m,
Highest degree, cól-mo, m.
Rupe, rock,

Chiesa, church.
Is, è
Happiness, fe-li-ce-, f.

Aurora, aurora, dawn.
Temperance, tem-pe-rán-la, f. Allegrezza (ts), joy.

Bastone, stick,
Soul, d-ni-ma, f.

Scolare, pupil. Commerce, com-mêr-cio, m. Is, é

Opinione, opinion.

Su, sopro, upon. Usage, ú-so, m.

Treasure, te--r0, m. Is, è

Wise man, -po, m. Legislator, le-gi-sla-túre, m. True, -ro

LESSONS IN GEOMETRY.No. XXIV.
Language, lin-gua, f.

Ornament, orana-men-to, m.
Master, pa-dró-ne, m.
Soldier, sol-dd-to, m.

LECTURES ON EUCLID.
Garden, giar-di-no, m.

Is, è
Is not here, non è qui
Courage, co-rág-gio, m.

(Contiuued from page 50). Palace, pa-láz-zo (ts), m. Practice, es-er-ci-zio, m.

BOOK I.PROPOSITION XIX.-THEOREM.
Belongs, p-pa--rue

Leads, con -ce
Prince, prin-ci-pe, m.
Perfection, per-fe-zió-ne, f.

The greater angle of every triangle is subtended by the greater sido

(that is, has the greater side opposite to it). Here are, be-co Interest, in-ter-es-se, m.

Fig. 19,
Room, cu-me-la, f.
Pleasure, pia--re, m.

In fig. 19, let ABC be a triangle hav-
Uncle, zi-o (ts), m.
And, e
ing its angle A BC greater than its angle

A
Dress, d-bi-to, m.
Glory, glória, f.

BCA; then, the side ac opposite the
Belong, ap-par-tën-go-no
Are, -no

angle A B C, is greater than the side AB Cousin, cu-gi-na, f.

Three motives, tre mo-ti-vi, pl. opposite the angle BCA.
And not, è non
Action, a-zió-ne, f.

For if the side A C be not greater than
Aunt, zi-a (ts), f.,
And, e
the side A B, it must be either equal to

B
Brother, fra-têl-lo, m.
Behaviour, con-dót-ta, f.

or less than the side AB. First, the side Tells, di-ce

Man, uô-mo, m.; pl. gli uôn ac is not equal to the side A B; for, if so, the angle A B C is Sister, so-rêl-10, f.

mi-ni.

equal to the angle A CB, by Prop. V., which is contrary to the

hypothesis ; therefore, the side A c is not equal to the side A B. * The pupil himself must examine whether he is to use before second, the side A c is not less than the side A B; for, if so, any noun or adjective the article or not, the prepositions di, a, and the angle A B C is less than the angle ACB, by Prop. XVIII., da, only being occasionally employed to denote the genitive, dative, which is also contrary to the hypothesis ; therefore, the side placed in the order in which they are to be translated into Italian. 14 c is not less than the side A B; and it has been proved that placed in the order in which they are to be translated into Italian. the side A o is not equal to the side A B. Therefore, the side I have thought it useful, in some cases, to denote the

Wherefore, the greater angle tion of the % or 27. I have done so by placing after such words in a c is greater than the side A B. parenthesis ts, thus (ts), when the pronunciation of the zor zz is to of every triangle, &c. Q. E. D. be the charp, hissing one; and ds, thus (ds), when the pronuncia- Corollary 1.-One side of a triangle is greater than, equal ciod of the zor zz is to be the soft one.

to, or less than another, according as the angle opposite to the

, .

EXERCISE TO PROPOSITION XIX.

to A E.

PROPOSITION XX.-THEOREM.

D

A

А.

B

former is greater than, equal to, or less than the angle opposite sible, let the straight line A k be drawn meeting B C and equal to the latter. This corollary was, by mistake, again appended to AE. Then, because A 1 is equal to A B, as just proved, and to Prop. XVIII. in our last lesson; whereas, the following ak is by hypothesis equal to a 7, therefore a k is equal to A H, corollary should have been appended to it:-One angle of a by Axiom 1.; but it has been proved that A y, a straight line triangle is greater than, equal to, or less than another, accord-bearer to the perpendicular AD is always less than ak, a ing as the side opposite to the former is greater than, equal to, straight line more remote from it; therefore, the straight line or less than the side opposite to the latter.

A K is both equal to AH and less than it, which is impossible; Corollary 2.--All the angles of a scalene triangle are un- wherefore A. K is not equal to AB; and in the same manner it equal.

can be proved that no other straight line than A H can be equal

Wherefore from the same point A, only two equal If from a point without a given straight line, any number of line Bc, one upon each side of the least straight line A D.

straight lines A H and As can be drawn to the given straight straight lines be drawn to meet it; of all these straight lines, that which is p rpendicular to the given straight line is the least; and of Therefore, if from a point without a given straight line, &c.

Qu.E.D. others, that which is nearer to the perpendicular is always less* than the more remote; also from the same point only two equal straight lines can be drawn to the given straight line, one upon each side of the least straight line.

Any two sides of a triangle are together greater than the third In fig. v, let A be the point, and Bc the given straight line; side. also let any number of straight lines AD, A E, A F, and A G, be

In fig. 20, let ABC be a triangle;

Fig. 20,
Fig. u.

any two of its sides are together
greater than the third side; viz., the two
sides BA and AB are together greater
than BC; the two sides A B and BC
are together greater than Ac; and the
two sides-B C and ca are together
greater than a B.

First, to prove that the two sides
B A and A c are together greater than Bc. Produce B A to the
point D, and make A D equal to A c by Prop. H. Join D c.

Because, in the triangle ADC, the side DA is, by construcB

tion, equal to the side A C; therefore the angle ADC is equal to the angle a CD. But the angle BCD, by Axiom IX., is greater than the angle A CD; therefore, the angle BCD is also

greater than the angle ADC, or BDC. Because the angle drawn from the point A to meet the straight line B C, and let B CD, of the triangle BCD, is greater than its angle B D C, and AD be perpendicular to BC, Prop. XII. ; then, of all these the greater angle is subtended by the greater side, Prop. XIX., straight lines a d is the least, and of the others, A E is less than therefore the side B D is greater than the side BC. Again, in A F, and AF than AG; also from the point A, only two equal the triangle ADC, the side ad is equal to the side a c, by constraight lines can be drawn to the straight line B C, one uponstruction; to each of these equals, add the side Ba; then, each side of AD.

BD, the whole side of the triangle BCD, is equal to the two

But the side BD of the to the straight line BC, therefore, by Def. 10, each of the triangle BCD, was proved to be greater than its side B C; there

gles A D E and ADH is a right angle; and they are, there fore, the two sides BA and a c of the triangle B A C are together fore, by Axiom XI., equal to one another; but the exterior greater than its third side BC. In the same manner, it may be angle Ad u of the triangle ADE, is, by Prop. XVI., greater proved that the two sides A B and Bc are together greater than than its interior and remote angle A ED, therefore, also the i c; and the two sides Bc and ca are together greater than angle ADE is greater than the angle A ED; wherefore, by AB. Therefore, any two sides of a triangle, &c. Q. E. D. Prop. XIX., the side as is greater that the side AD. In the

Scholium.-Dr. Simson, in his notes to his edition of Euclid, same manner, it may be shown that the straight lines A AG line

makes the following proper remarks :- “Proclus, in his comWherefore of all the straight lines A D, A E, A F, and AG, the mentary [on Euclid], relates, that the Epicureans derided

Prop. XX, as being manifest even to asses, and needing no perpendicular AD is the least. Next, because the exterior angle A D A of the triangle AFD it be manifest to our senses, yet it is science which must give

demonstration ; and his answer is, that though the truth of is by Prop. XVI. greater than its interior and remote angle the reason why two sides of a triangle are greater than the A FD, therefore also the angle A DF is greater than the angle AFD. third; but the right answer to this objection against this and Again, because the angle ADF has been shown to be greater some other propositions is, that the number of axioms ought

the of the triangle A D B is greater by Prop. XVI. than its interior not to be increased without necessity, as it must be, if these angle A DE, much more, therefore, is the angle A EF greater &xis merely a corollary to the definition of a straight line

propositions be not demonstrated." It is true that this Prop. than the angle A FE; wherefore, in the triangle A Ef, by Prop. given by Archimedes, namely, that “it is the shortest disXVIII., the side A F is greater than the side A E. In the same manner, it may be proved that the straight line 4 4 is greater the two points B and c, taken along the straight line, is

tance between any two points ; for the distance between than the straight line A F. Therefore, of the other straight evidently less than the distance between these points taken lines, A is less than a F, and a F than Ag; that is, the straight along the crooked line BAC; and as even asses or drunken line nearer the perpendicular is always less than the more

men endeavour to take the shortest road to their desired to D Е, by Prop III., and join a H. Because in the two triangles laugh and mock at everything that did not just exactly.square Lastly, from the straight line D e, cut off the part de equal object, there seems to be some foundation for the derision of

the Epicureans; but these philosophers were accustomed to ADE and AD #; the two sides A D and d are equal to the with their views; hence they said even of the great Apostle two sides A D and D E, and the angle Ad = is equal to the angle Paul

, when preaching Jesus and the resurrection at Athens: AD E, therefore, by Prop. IV., the base.A H is equal to the base A E; and no other straight line equal to a e, but As, can be Paul' had given them a mathematical demonstration of the

“What will this babbler say?" Hence, it is evident, that if drawn from the point À to the straight line BC. For, if pose resurrection of the dead, they would not have believed him, * By mistake printed greater, in the earlier editions of Cassell's but would Hare continued to mock on, like infidels in modern

times. Now, they have both Moses and the prophets, and This exercise was solved by T. Bocock, Great Warley; Quintin Christ and his apostles, and if they believe not them, neither Pringie, Glasgow; J. H. Eastwood, Middleton and others.

would they believe if one rose from the dead.

[graphic]

remote.

Euclid.

с

B

D

Scrotium 2. This proposition may be demonstrated by | XV.; therefore, the angle A E G is equal to the angle F E q, by another method, as follows:- In fig.

Axiom I. But the sides E is equal to the side E F, by construcV, let B A C be a triangle, and let it be

Fig. V.

Fig. x. required to prove that the two sides BA and a c are greater than the third side BC. Bisect the angle B AC by the straight line A D, meeting Bc in D, by Prop. IX. Then, because the exterior angle BDA of the triangle DAC is greater than its interior and remote angle D A C, by Prop. XVI., and the exterior angle CDA of the triangle BD A is greater than its interior and remote angle DAB; and that the angles D A c and D A B are equal; therefore, the angle BDA is also greater than the angle DAB, and the angle CDA than the angle DAC; wherefore, by Prop. XIX., the side BA is greater than BD, and the side c A greater than CD; therefore the two sides BA and A c are greater than the whole side BC.

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EXERCISE I. TO PROPOSITION XIX.

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EXERCISE II. TO PROPOSITION XX.

Any side of a triangle is greater than the difference between the other two sides. If the triangle be equilateral, the truth of the proposition is tion, and the side EG is common to the two triangles A EG

evident; for the difference between and FEG; therefore, the base of is equal to the base A G. In Fig. w.

any two sides is nothing. If the the same manner, it may be shown that the straight line ry is triangle be isosceles, the base or equal to the straight line A #. Because the straight line A G is third of the triangle is greater than equal to the straight line GF, if to these equals we add the the difference between the other two straight line GB, the two straight lines AG and G B are equal to sides, which are equal, for the same the whole FB. But the two sides FI and HB, of the triangle

FHB, are together greater than the side FB; and as a H is In any other case, let B A ., fig. w, equal to Fu, therefore AH and u B are together greater than be a triangle, of which the side b c is greater than the side BA;

But it has been shown that A G and G B are equal to FB; then the remaining side A C is greater than the difference be-thefore a I and H B are together greater than A G and a B, that twpe: the other two sides, Bc and BA.

is, A G and GB are together less than the sum of A H and H 1. From BC, the greater side, cut of BD, a part equal to the And the same may be proved of the two straight lines drawn less side BA, by Prop. III.

from the points A and B to any other point in the straight line Because the two sides e A and a C are together greater than CD. Therefore, from two given points A and B on the same BC by Prop. XX., and that BD is, by construction, equal to side of the straight line CD, two straight lines have been drawn BA; therefore, taking these equals away, the remainder A c is to a point G in it, which taken together are less than the sum greater than the remainder Dc. Therefore, any side of a trio of two straight lines drawn from the same points to any other angle, &c. Q. E. D.

point in cd. Q. E. F.

Scholium 2. In the preceding demonstrations, it is very

properly remarked by T. Bocock, Great Warley, that this The three sides of a triangle taken together are greater than the another, and if the same or equal magnitudes be added to each,

axium is taken for granted, viz; “If one magnitude be greater double of any one side, but less than the double of any two sides.

the same inequality will remain ; that is, the sum of the Because any two sides of a triangle are greater than the greater magnitude and that which is added to it will be third side, by Prop. XX.; therefore, if the third side be added to greater than the sum of the less magnitude and that which is these unequals, the three sides taken together are greater than added to it.” Another axiom is also taken for granted, viz., twice the third side. Again, because one side of a triangle is "If one magnitude be greater than another, and if the same less than the other two sides, by Prop. XX., therefore, if the or equal magnitudes be taken from each, the same inequality other two sides be added to these unequals, the three sides will remain ; that is, the difference between the greater magtaken together are less than twice the other two sides. There- nitude and that which is taken from it, will be greater than fore, the three sides, &c., Q. E, D,

the difference between the less magnitude and that which is

taken from it." EXERCISE III, TO PROPOSITION XX.

* The exercises on Problem XX., were solved as follows: I., II. and From two given points on the same side of a straight line given III. by J. H. Eastwood, Middleton; E. J. Bremner, Carlisle ; T. in position, to draw two straight lines which shall meet at a point Bocock, Great Warley; Quintin Pringle, Glasgow; C. L. Hadfield in it, and which taken together shall be less than the sum of two and J. Goodfellow, Bolton-le-Moors; I. and III. by E. L. Jones, Pemstraight lines drawn from the same points to any other point in the broke Dock; I. and II. by E. Russ, Pentonville; and I. by J. Jenkins, given straight line.

Pembroke Dock, In fig. x, let A and B be the two given points, and op the straight line given in position. From the points A and B it

ANSWERS TO CORRESPONDENTS. is required to draw two straight lines which shall meet at a point in the straight line c D, and which taken together shall this journal; natural faith we dont understand, and the only book of

E. WILKINSON (York): We eschew politics, and all mention of them in be less than the sum of two straight lines drawn from the Christian faiin is the BIBLE.-JOHN HEBN : Yes.-T. 0. (Hainsworth); points A and B, to any other point in the straight line CD. Very good. A WELSHMAN (of Anglesea) was answered before. It is all From the point A, draw the straight line a E at right angles horse power he must just eat what is good for him, and this he can only

stuff about physiology and food; man is not a steam-engine of a certain to the straight line cd, by Prop. XI., and produce it to the find out by experience.—YOUNG NATURALIST: We don't know.-GERMANIpoint F, making the part er equal to the part A E, by Prop. cus (Edinburgh), T. C. W. (London), and X. Y. Z. (Dublin): Yes.-T.

Join FB, and let it cut cd in the point G. Join AG, WALKER, and Tlou 32: Right.-AMATOR SCIENTIAL (Fenchurch-street): Then the straight lines A G and QB drawn from the points A and Thanks.—-FAIR PLAY (Waterford) and LOUIS LE PLUS JEUNE: We don't B and meeting cd in G, are the two straight lines required. know.-E. MORRIS: Write to Mr. Bell.-A SUBSCRIBER (Colne): RightIn CD. take away any other point w, and join A I and BH.

STUDETE (Hampstead-road), should call on Henry Moore, Esq., Secretary to Because the angle A E G is equal to the angle A EO by con- | Greek Scriptures, apply to Messrs. Bagster, Paternoster-row.S. F. HEN

the University, Somerset House, for a solution of all his queries. For the struction, and the angle A E C equal to the angle Foo by Prop. I BEST (Fordingbridge)? Received.

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III.

LESSONS IN BOOKKEEPING.-No. X I.

THE JOURNAL.

(Continued from page 177).

Tre Journal, as we have before remarked, is no longer what I has with the Ledger, we mean the GENERAL POSTING Book, its name denotes, a Day Book ; but is now used, in Double | Some of our students who are, no doubt, keen business-men, Entry, as a book for collecting all the transactions of business and are on the alert to discover any improvements that can be for a give i period into a focus, previous to their being entered made in Bookkeeping, in order to shorter their labour, and in the Ledger. In an ordinary business, where the transactions produce more accurate results, or, rather, to effect less frequent are neither too numerous por tou complicated, the formation liability to error, will

, if they have gone with us thus far, proof this book from the various subsidiary books of the concern, pose some shorter or more pointed name than the precedings may take place only once a month; and then with reference to for once, therefore, we leave this subject in their hands. Ali time, as we fornierly observed, it might be called the Month- we shall say, is this : that gentlemen who have been in busi. Book ; and in the same way, according to the regular intervals ness for twenty, thirty, aye, and forty years, have thanked us when this collective book is made up, it might be called Week- personally many times for the lessons on this subject which Book, or even Day-Book. The best name, however, which they have received from us, and particularly in reference to could be given to it, would be one indicative of its actual use, our method of striking a General Balance, exemplified at the without reference to time; we have already suggested the end of this Journal, but which cannot be fully explained in this name Sub-Ledger, and. we may now propose a name which lesson, as the Trial Balance and Ledger have not yet been subwould, perhaps, be more accurate and distinct, as regards the | mitted to the student. This will be done in our next lesson. method in which is is inade up, and the connexion which it

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