island of Corfu; la re-gi-na d' In-ghil-tér-ra, the Queen of England; ï Ré di Prús-sia, the king of Prussia; 'im-pe-ra-tó-re d'Au-stria, the emperor of Austria; l'as-sê-dio di Man-to-va, the siege of Mantua; lo stret-to di Gi-bil-têr-ra, the straits of Gibraltar; l'im-pê-ro di Rús-sia, the empire of Russia; le tragê-die di Al-fiê-ri, the tragedies of Alfieri; le com-mê-die di Goldó-ni, the comedies of Goldoni.

Adverbs of place or time before, nouns or even adiectives of this class, frequently, also, are translated by the genitive case; e. g. the back door or room, la pôr-ta la, stán-za di dié-tro (the door or room of behind); the hind-feet, i piê-di di diê-tro (the feet of behind); the following day, il giór-no di do-má-ni (the day of to-morrow); the present age, il món-do dog-gi-di (the world of now-a-days); after the present fashion or style, III. When words expressing quantity, weight, or any kind of al mó-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-tù di pé- last night, la not-ta-ta di jê-ri (the whole night of yesterday); co-re, a quantity of sheep; ú-na lib-bra di cár-ne, a pound of yesterday, il giór-no di jê-ri (the day 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-zi-na di cuc-chiá-ji, di guún-ti, d'uô-va, a dozen of spoons, fines another word, the preposition di must be placed before gloves, eg's; un brác-cio di pán-no, a yard of cloth; ú-na bot-it (just as the preposition of with the present participle of tí-glia di vi-no, a bottle of wine; ú-na ca-ráf-fa d'á-cquu, a English grammar in such cases); e. g. Ha ú-na gran vô-glia di decanter of water; un' in-cia di caf-fè (kahf-fe), an ounce of viag-gia-re, he has a great desire to travel or of travelling; è coffee; ví-no di diê-ci án-ni, wine of ten years. tém-po di an-dá-re, it is time to go or of going; ra-gió-ne di di ve-der-vi, the honour to see you or of seeing you'; li-cén-za. la-men-túr-si, right to complain or of complaining; l'o-nó-re di par-tir-si, permission to go away or of going away.

1

*

For the sake of elegance, the preposition di is, however, sometimes omitted after the words ca-sa, house; pa-luz-zo, palace; piáz-za, place, square; vil-la, villa; gal-le-ri-a, gallery; fa-mi-glia, family; pôr-ta, gate, entry, and some others: or long; al-quan-to, something, a little, some; tán-to, so much, Di is also placed after the words quin-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 cá-sa Al-tiê-ri, at little, few; mol-to, much, a great deal; nien-te, nothing; più, or great, or long; al-tret-tán-to, just as much, equal; pô-co, the Altieri-house; vi-cí-no al pa-láz-zo Bor-ghé-se, near the Borghese-palace; súl-la piáz-za Bar-be-ri-ni, on the Barberini- more; mé-no, less; tróp-po, too much, &c.; e. g. quán-to di square; per la vil-la Pan-fi-li, for the villa Panfili; nel-la gal-no-ja sa-réb-be per me, how a great a nuisance would it be for Is-ri-a Do-ria, in the Doria-gallery; dél-la fa-mi-glia Co-lin-na, me; dó-po al-quan-to di tem-po, after some time; tán-to di ví-no of the Colonna family; la pôr-ta di San Gio-van-ni, St. John-ed al-tret-tún-to d' á-cqua, so much wine and just as much gate or entry; a cá-sa'i zi-o (instead of a cá-sa del zi-0), at the water; pô-co di ú-ti-le ne ri-ca-ve-ré-te, you will derive from house of the uncle; ó-ra a cá-sa qué-sto, ó-ra a cá-sa quell' al- this little advantage; mól-to di má-le ne po-tréb-be se-guí-re, a tro (instead of di qué-sto, di quell' al-tro), now at the house of great deal of evil might be the consequence of it. this one, then at the house of the other.

English compound nouns, or combinations of nouns, for the greatest part must be decomposed by the genitive case with the case-sign di, especially when one of the nouns merely defines and qualifies the other, which is the principal word conveying the principal idea; e. g. garden-door, pôr-ta di giardi-no (door of the garden); stone-quarry, cá-va di piê-tra (quarry of stone); autumn fruits, frút-ti d'au-tún-no (fruits of autumn); a music amateur, un di-let-tán-te di mú-si-ca (an amateur of music); Leipzig fair, fiê-ra di Li-psia (fair of Leipzig); ox-tongue, lín-gua di bô-ve (tongue of an ox); horse'shead, tê-sta di ca-vál-lo (head of a horse); felt-hat, cap-pêl-lo di fél-tro (hat of felt); sugar-box, cús-sa di zuc-che-ro (box of sugar).

Whenever it is necessary with greater precision to define the noun in the genitive case so as to distinguish it from other objects of the same class, the article, according to its peculiar function of particularising that which is general, must be joined to the case-sign di.

In these two phrases, la Di-o mer-ce! thank God! and la Di-o gra-zia, the grace of God, the word di is understood, and in full they run thus: la di Dí-o mer-cè, and la di Dí-o grúia. When, however, Di-o is placed after the words mer-cè and gra-zia, the case-sign di cannot be omitted; e. g. la mer-cè di Di-o, and la grá-zia di Di-o.†

The word di is sometimes a mere expletive; e. g. é-gli di-ce di sì, ed i-o di-co di no, he says yes, and I say no; qué-sto diáro-lo di qué-sta fém-mi-na, that devil of a woman; quel po-ve-rino di mi-o fra-tél-lo, that poor brother of mine.

As a last remark on the use of the case-sign di for the of the Italian language, is of by far the most extensive use. present, I shall state that this word, among all the prepositions The reason of this is, that di, properly and philosophically speaking, merely cxpresses the mental separation of ideas or notions, while da indicates a real separation of objects, which distinction constitutes the principal and fundamental difference between these two important words di and da. The mere mental separation of ideas or notions may, however, serve any 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-ne dél-la cá-sa dó-ve a-bi-tiá-mo, the be surprised at meeting, in Italian books and conversations, master of the house where we live (il pa-dró-ne di cá-sa, is the with frequent substitutions of the case-sign di for many other master of the house in general); un boc-cá-le del vì-no che bev-prepositions; e. g. for a: I'-schia è ú-na í-so-la as-sú-i vi-cí-na vil' ál-tra sé-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-cá-le di vi-no, is a re del-la pri-gió-ne, to go out of or from prison; é-gli di pri-givmeasure of wine in general); il mer-cá-to dei ca-vál-li, the horse- ne il trás-se, he took him from prison; for con: di gran-dís-simarket (il mer-cá-to di ca-vál-li, is merely a place where horses ma för-za si com-bat-té-a da cia-scú-na pár-te, they fought with are sold); il mer-cá-to dél-la sel-vag-gi-na, the game-market; the greatest energy on each side; for in : í-o l' uc-cí-si di lea-le il ma-gaz-zí‐no dél-la pá-glia, the straw-magazine (ma-gaz-zí-no | bat-tá-glia, I killed him in fair fight; for per: é-gli pia-gné-a e di pá-glia, is merely a magazine full of straw); il ma-gaz-zi-no di gran pie-tà non po-té-a môt-to fa-re, he wept, and on account dél-le le-gna, the wood-magazine. of his great emotion he could not utter a word.

It is evident that the variable nature of di will admit of many translations into English; e. g. by with: só-no con-tén-to di te, I am satisfied with thee; by at: mi ri-do di lui, I laugh at him; by of: mo-rír di fá-me, to die of hunger; by as: scr vir di rê-go-la, to serve as a rule; by for: pre-gú-re ù-no di ú-no cô-sa, to request one for something; by than : più di du-e-mi-la scu-di, more than two thousand crowns.

English adjectives, indicating the material or stuff from which anything is manufactured, or denoting qualities attri- | buted or derived from proper names of countries, nations, or towns, for the greatest part will be best translated into Italian by means of nouns in the genitive case; e. g. a gold watch, un o-ro-lô-gio d' ô-ro (a watch of gold); a marble statue, ú-na stá-tua di már-mo (a statue of marble); a wooden table, ú-na tá-vo-la di lé-gno (à table of wood); an iron gate, ú-na pôr-ta di fer-ro (a gate of iron); a silver spoon, un cuc-chid-jo d'ar-gênto (a spoon of silver); a meritorious soldier, un sol-dá-to di mêri-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 di before the infinitive mood governed by them, silk, sé-ta d' I-tá-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-mô-rea, sol-dá-to me-ri-té-vo-le, gió-vane spi-ri-tó-so, cit-ta-di-ni Vien-né-si.

In some instances, the peculiarities in the use of di may, without difficulty or twisting, be explained by ellipsis, par

† Some other important omissions of the case-sign di will be explained hereafter.

ticularly when it denotes descent or children; e. g. Gian-nuôl đi Se-ve-ri-no, Céc-co di Mes-sé-re An-giu- liê-ri, in Boccaccio, where fi-gliuô-lo, child or son, is understood.

EXERCISES.-ENGLISH-ITALIAN.*

The rising of the sun. The dawn of the day. The return of spring. The warmth of the air. The beauty of the flower. The darkness of the night. The abyss of error. The fertility of the fields. The colours of the rainbow. The senses of man. The faults of young men. Money is the soul of commerce. Usage is the legislator of languages. The master of the garden is not here. The palace belongs to the prince. Here are the rooms of the uncle. The dresses belong to the cousin and not to the aunt. The brother tells the sister the will of the father. The children must always obey the parents. The physicians say the disorder shortens life. Exercise is useful to the body and to the mind. The countenance is the mirror of the soul. Tranquillity of mind is the highest degree of happiness. Temperance is the treasure of the wise man. The true ornament of the soldier is courage. The practice leads to perfection. Interest, pleasure, and glory, are the three motives of the actions and of the behaviour of men.

:

Rising, lá-var, m.

Sun, só-le, m.

Dawn, spun-tár, m.

Day, giór-no, m.

Return, ri-tór-no, nì,
Spring, pri-ma-vê-ra, f.
Warmth, ca-ló-re, m.
Air, á-ria, f.

Beauty, bel-léz-za (ts), f.
Darkness, o-scu-ri-tà, £.
Night, nut-te, f.

Abyss, a-bis-so, m.

Error, er-ró-re, m.

Fertility, fer-ti-li-tà, f.

Field, cám-p0, 2.

Colour, co-ló-re, m.

VOCABULARY.

Rainbow, ar-co-ba-lé-no, m.

Sense, sén-so, m.

Man, vô-mo, 12.

Fault, er-ró-re, m.

Young man, gió vane, m.

Money, da-ná-ro, m.
Is, è

Soul, d-ni-ma, f.

Commerce, com-mêr-cio, m.
Usage, ú-so, m.

Is, è

Legislator, le-gi-sla-tó-re, m.
Language, lin-gua, f.
Master, pa-dró-ne, m.
Garden, giar-di-no, m.
Is not here, non è qui
Palace, pa-láz-zo (ts), m.
Belongs, ap-par-tiê-ne
Prince, prin-ci-pe, m.
Here are, éc-co

Room, cú-me-ra, f.

Uncle, zí-o (ts), m.

Dress, d-bi-to, m.

Belong, ap-par-tên-go-no

Cousin, cu-gi-na, f.

And not, è non

Aunt, zí-a (ts), f.,
Brother, fra-têl-lo, m.
Tells, di-ce

Sister, so-rêl-la, f.

Will, vo-lon-tà, f.
Father, pá-dre, m.
Child, fan-ciúl-lo, m.

Must always obey, de-vo-no
sêm-pre ob-be-di-re
Parents, ge-ni-tó-ri, pl.
Physician, mê đi-co, mo
Says, di-ce

Disorder, dis-or'-di-ne, m.
Shortens, ac-cór-cia

Life, vi-ta, f.

Exercise, mô-to, m.

Is useful, gió và

Body, côr-po, m.

Mind, spi-ri-to, m.

Is, è

Mirror, spêc-chio, m.

EXERCISES.-ITALIAN-ENGLISH.

La me-mô-ria. Dél-la ciê-ra. Al-la col-lí-na. Dál-la spia-ná-ta. Le bec-che-rí-e. Dél-le lo-cán-de. Al-le pôrte. Dál-le strá-de. In fac-cia. Nél-la ví-gna. Nél-le fo-rêste. Con pá-glia. Cól-la ví-te. Cól-le pén-ne. Per disgrá-zia. Per la vál-le. Per le scioc-chéz-ze. Súl-la car-rôzza. Súl-le rú-pi. L'au-rô-ra. Súl-le rú-pi. L'au-rô-ra. Dell' al-le-gréz-za. All' opi-nió-ne. Dall' o-ste-rí-a. Le i-dê-e. Dell' êr-be. Al-le ar-ti. Dál-le cit-tà. In i-slit-te. Nell' im-ma-gi-na-zió-ne. Nél-le a-ni-me. Con á-cqua. Coll' ún-ghia. Cól-le in-ségne. Per a-mi-cí-zia. Per l' as-si-cu-ra-zió-ne. Per le azió-ni. Sull' in-sa-lá-ta. Súl-le in-fer-riá-te. Un fan-ciúllo. U-no stól-to. Un a-ni-má-le. U-na set-ti-má-na. D'un fiú-me. Ad ú-no schiop-pet-tiê-re. Da ú-na bal-le-rí-na. In ú-na chiê-sa. Con un ba-stó-ne. Per ú-no sco-lá-re. Su d' un sas-so, só-pra un sás-so. VOCABULARY.

Per, for, through, on account
of.

Countenance, fi-so-no-mí-a, f. Disgrazia, misfortune, dis

Soul, d-ni-ma, f.

Tranquillity, quiê-te, f.
Mind, á-ni-ma, f.

Highest degree, cól-mo, m.
Happiness, fe-li-ci-tà, f.
Temperance, tem-pe-rán-za, f.

Is, è

Treasure, te-sô-ro, m.
Wise man, sá-vio, m.
True, vé-ro

Ornament, or-na-mén-to, m.
Soldier, sol-dá-to, m.

Is, è

Courage, co-rág-gio, m.
Practice, es-er-cí-zio, m.
Leads, con-dú-ce

Perfection, per-fe-zió-ne, f..
Interest, in-ter-ês-se, m.
Pleasure, pia-cé-re, m.
And, e

Glory, glô-ria, f.

Are, só-no

grace (per dis-gra-zia, unfor-
tunately).

Valle, valley.
Sciocchezza (ts), folly.

Carrozza (ts), carriage, coach.
Rupe, rock.

Aurora, aurora, dawn.

Allegrezza (ts), joy.

Opinione, opinion.

Arte, art.

Città, town, city (no change in the pl.)

Slitta, sledge.

Immaginazione, imagination.

Anima; mind, soul.
Acqua, water.
Unghia, nail.

Insegna, sign, arms, colours.
Amicizia, friendship.

Assicurazione (ts), security, in

surance.

Azione, action.

Insalata, salad.

Inferriata, iron-grate.

Stolto, fool.

Fanciullo, child.

Animale, animal.

Settimana, week.

Fiume, river.

Schioppettiere, arquebusier, rifleman.

Ballerina (f.), dancer.

Chiesa, church.

Bastone, stick.

Scolare, pupil.

Su, sopra, upon.

LESSONS IN GEOMETRY.-No. XXIV.

LECTURES ON EUCLID.

(Contiuued from page 50).

BOOK I.-PROPOSITION XIX,-THEOREM.

The greater angle of every triangle is subtended by the greater side (that is, has the greater side opposite to it).

In fig. 19, let ABC be a triangle hav-
ing its angle ABC greater than its angle
BCA; then, the side Ac opposite the
angle ABC, is greater than the side AB

Three motives, tre mo-tz-vi, pl. opposite the angle B CA.
Action, a-zió-ne, f.

Fig. 19, A

Man, uô-mo, m.; pl. gli uô- AC is not equal to the side A B; for, if so, the angle A B C is equal to the angle A C B, by Prop. V., which is contrary to the hypothesis; therefore, the side a c is not equal to the side a B. Second, the side A c is not less than the side AB; for, if so, the angle ABC is less than the angle ACB, by Prop. XVIII., which is also contrary to the hypothesis; therefore, the side

* The pupil himself must examine whether he is to use before any noun or adjective the article or not, the prepositions di, a, and da, only being occasionally employed to denote the genitive, dative, and ablative. It is, moreover, to be noted, that the words are placed in the order in which they are to be translated into Italian. A c is not less than the side AB; and it has been proved that Wherefore, the greater angle I have thought it useful, in some cases, to denote the pronuncia- the side AC is not equal to the side A B. Therefore, the side tion of the zor zz. I have done so by placing after such words in AC is greater than the side A B. parenthesis ts, thus (ts), when the pronunciation of the z or zz is to of every triangle, &c. Q. E: D. be the sharp, hissing one; and ds, thus (ds), when the pronunciation of the z or zz is to be the soft one.

Corollary 1.-One side of a triangle is greater than, equal to, or less than another, according as the angle opposite to the

former is greater than, equal to, or less than the angle opposite
to the latter. This corollary was, by mistake, again appended
to Prop. XVIII. in our last lesson; whereas, the following
corollary should have been appended to it:-One angle of a
triangle is greater than, equal to, or less than another, accord-
ing as the side opposite to the former is greater than, equal to,
or less than the side opposite to the latter.
Corollary 2.-All the angles of a scalene triangle are un-
equal.

EXERCISE TO PROPOSITION XIX.

If from a point without a given straight line, any number of 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 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.

In fig. u, let a be the point, and BC the given straight line; also let any number of straight lines AD, A E, A F, and a G, be Fig. U.

A

drawn from the point a to meet the straight line BC, and let AD be perpendicular to в C, Prop. XII.; then, of all these straight lines A D is the least, and of the others, AE is less than A F, and a F than AG; also from the point a, only two equal straight lines can be drawn to the straight line в C, one upon

each side of a D.

sible, let the straight line A K be drawn meeting B c and equal to AE. Then, because ▲ н is equal to A E, as just proved, and AK is by hypothesis equal to a E, therefore A K is equal to AH, by Axiom I.; but it has been proved that ▲ H, a straight line nearer to the perpendicular AD is always less than AK, a straight line more remote from it; therefore, the straight line A K is both equal to AH and less than it, which is impossible; wherefore AK is not equal to AE; and in the same manner it can be proved that no other straight line than Aн can be equal Wherefore from the same point A, only two equal straight lines AH and AE can be drawn to the given straight Therefore, if from a point without a given straight line, &c. line BC, one upon each side of the least straight line a D. QE.D.

to AE.

BA and A c are together greater than в c. Produce в A to the point D, and make A D equal to a c by Prop. HI. Join D C.

Because, in the triangle A D C, the side DA is, by construction, equal to the side A c; therefore the angle ADC is equal to the angle AC D. But the angle B CD, by Axiom IX., is greater than the angle a CD; therefore, the angle B C D is also greater than the angle ADC, or BDC. Because the angle BCD, of the triangle B CD, is greater than its angle B D C, and the greater angle is subtended by the greater side, Prop. XIX., therefore the side B D is greater than the side B C. Again, in the triangle ADC, the side AD is equal to the side a c, by construction; to each of these equals, add the side BA; then, BD, the whole side of the triangle BCD, is equal to the two Because the straight line A D is by hypothesis perpendicular sides BA and AC of the triangle B AC. But the side B D of the to the straight line BC, therefore, by Def. 10, each of the triangle B CD, was proved to be greater than its side B ; thereangles A D E and ADH is a right angle; and they are, therefore, 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 B C. In the same manner, it may be angle ADH of the triangle ADE, is, by Prop. XVI., greater proved that the two sides A B and B c are together greater than than its interior and remote angle AED, therefore, also the Ac; and the two sides BC and CA are together greater than angle ADE is greater than the angle AED; wherefore, by AB. Therefore, any two sides of a triangle, &c. Q. E. D. Prop. XIX., the side AE is greater that the side AD. In the same manner, it may be shown that the straight lines AF and AG are also greater than the straight line AD. Wherefore of all the straight lines AD, A E, AF, and AG, the perpendicular AD is the least.

makes the following proper remarks:-"Proclus, in his comScholium.-Dr. Simson, in his notes to his edition of Euclid, mentary [on Euclid], relates, that the Epicureans derided Prop. XX. as being manifest even to asses, and needing no Next, because the exterior angle A D H 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 AFD, therefore also the angle A D F is greater than the angle AF D. 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 than the angle AFD or AFE, and that the exterior angle A EF of the triangle A D E is greater by Prop. XVI. than its interior not to be increased without necessity, as it must be, if these angle A D E, much more, therefore, is the angle AEF greater XX is merely a corollary to the definition of a straight line propositions be not demonstrated." It is true that this Prop. than the angle AFE; 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. manner, it tance between any two points;" for the distance between be proved that the straight line AG is greater the two points B and C, taken along the straight line, is than the straight line AF. Therefore, of the other straight lines, AE is less than AF, and A F than AG; that is, the straight line nearer the perpendicular is always less than the more

remote.

may

In the same

evidently less than the distance between these points taken along the crooked line BAC; and as even asses or drunken

men endeavour to take the shortest road to their desired Lastly, from the straight line DC, cut off the part DH equal object, there seems to be some foundation for the derision of to D E, by Prop III., and join a H. Because in the two triangles laugh and mock at everything that did not just exactly square the Epicureans; but these philosophers were accustomed to ADE and ADH, the two sides AD and DH are equal to the with their views; hence they said even of the great Apostle two sides AD and DE, and the angle ADH is equal to the angle Paul, when preaching Jesus and the resurrection at Athens: A DE, therefore, by Prop. IV., the base.A H is equal to the base AE; and no other straight line equal to AE, but AH, 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 A to the straight line BC. For, if pos-resurrection of the dead, they would not have believed him,

* By mistake printed greater, in the earlier editions of Cassell's

Euclid.

This exercise was solved by T. Bocock, Great Warley; Quintin Pringle, Glasgow; J. H. Eastwood, Middleton, and others.

but would have continued to mock on, like infidels in modern times. Now, they have both Moses and the prophets, and Christ and his apostles, and if they believe not them, neither would they believe if one rose from the dead.

Fig. V.

C

D

Fig. x.

B

Senotium 2.-This proposition may be demonstrated by | XV.; therefore, the angle A E G is equal to the angle FEG, by another method, as follows:-In fig. Axiom I. But the side A E is equal to the side Er, by construcv, let B A C be a triangle, and let it be required to prove that the two sides BA and A c are greater than the third side B C. Bisect the angle B A C by the straight line AD, meeting BC in D, by Prop. IX. Then, because the exterior angle BDA of the triangle D A C is greater than its interior and remote angle D A C, by Prop. XVI., and the exterior angle CDA of the triangle BDA 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 B DA is also greater than the angle DA B, and the angle CDA than the angle DAC; wherefore, by Prop. XIX., the side BA is greater than B D, and the side CA greater than CD; therefore the two sides BA and A c are greater than the whole side BC.

B

EXERCISE I. TO PROPOSITION XIX.

A

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
evident; for the difference between
Fig. w.
any two sides is nothing. If the
triangle be isosceles, the base or
third of the triangle is greater than
the difference between the other two
sides, which are equal, for the same

B

D

C

reason.

In any other case, let B A , fig. w, be a triangle, of which the side B c is greater than the side B A; then the remaining side a c is greater than the difference between the other two sides, BC and B A.

From BC, the greater side, cut off в D, a part equal to the less side B A, by Prop. III.

Because the two sides B A and A C are together greater than B C, by Prop. XX., and that BD is, by construction, equal to BA; therefore, taking these equals away, the remainder A c is greater than the remainder DC. Therefore, any side of a triangle, &c. Q. E. D.

EXERCISE II. TO PROPOSITION XX.

The three sides of a triangle taken together are greater than the double of any one side, but less than the double of any two sides.

Because any two sides of a triangle are greater than the third side, by Prop. XX.; therefore, if the third side be added to these unequals, the three sides taken together are greater than twice the third side. Again, because one side of a triangle is less than the other two sides, by Prop. XX., therefore, if the other two sides be added to these unequals, the three sides taken together are less than twice the other two sides. Therefore, the three sides, &c., Q. E. D,

EXERCISE III, TO PROPOSITION XX.

From two given points on the same side of a straight line given in position, to draw two straight lines which shall meet at a point in it, and which taken together shall be less than the sum of two straight lines drawn from the same points to any other point in the given straight line.

In fig. x, let A and B be the two given points, and cp the straight line given in position. From the points A and B it is required to draw two straight lines which shall meet at a point in the straight line c D, and which taken together shall be less than the sum of two straight lines drawn from the points A and B, to any other point in the straight line CD.

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F

tion, and the side EG is common to the two triangles A EG and FEG; therefore, the base GF is equal to the base A G. In the same manner, it may be shown that the straight line F H is equal to the straight line A н. Because the straight line A & is equal to the straight line G F, if to these equals we add the straight line GB, the two straight lines A G and G в are equal to the whole FB. But the two sides FH and HB, of the triangle FHB, are together greater than the side FB; and as A H is equal to FH, therefore AH and Hв are together greater than FB. But it has been shown that A G and G B are equal to FB; thefore AH and H в are together greater than A G and G B, that is, A G and GB are together less than the sum of AH and н г. And the same may be proved of the two straight lines drawn from the points A and B to any other point in the straight line CD. Therefore, from two given points A and B on the same side of the straight line CD, two straight lines have been drawn to a point & in it, which taken together are less than the sum of two straight lines drawn from the same points to any other point in CD. Q. E. F.

Scholium 2. In the preceding demonstrations, it is very properly remarked by T. Bocock, Great Warley, that this another, and if the same or equal magnitudes be added to each, axiom is taken for granted, viz. "If one magnitude be greater the same inequality will remain; that is, the sum of the greater magnitude and that which is added to it will be greater than the sum of the less magnitude and that which is added to it." Another axiom is also taken for granted, viz., "If one magnitude be greater than another, and if the same or equal magnitudes be taken from each, the same inequality will remain; that is, the difference between the greater magnitude and that which is taken from it, will be greater than the difference between the less magnitude and that which is taken from it."*

* The exercises on Problem XX., were solved as follows: I., II. and III. by J. H. Eastwood, Middleton; E. J. Bremner, Carlisle; T. Bocock, Great Warley; Quintin Pringle, Glasgow; C. L. Hadfield and J. Goodfellow, Bolton-le-Moors; I. and III. by E. L. Jones, Pembroke Dock; I. and II. by E. Russ, Pentonville; and I. by J. Jenkins, Pembroke Dock.

ANSWERS TO CORRESPONDENTS.

this journal; natural faith we dont understand, and the only book of E. WILKINSON (York): We eschew politics, and all mention of them in Christian faith is the BIBLE.-JOHN HEBN: Yes.-T. O. (Hainsworth): Very good.-A WELSHMAN (of Anglesea) was answered before. It is all From the point a, draw the straight line AE 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 EF equal to the part A E, by Prop. cus (Edinburgh), T. C. W. (London), and X. Y. Z. (Dublin): Yes.-T. III. Join FB, and let it cut CD in the point &. Join AG. SHEPHERD (Salford) and J. FARNDON (Birmingham): Thanks.-W. Then the straight lines A G and GB drawn from the points A and Thanks.-FAIR PLAY (Waterford) and LOUIS LE PLUS JEUNE: We don't WALKER, and ТLOH 32: Right.-AMATOR SCIENTIAE (Fenchurch-street): 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 н, and join А ¤ and в н. STUDETE (Hampstead-road), should call on Henry Moore, Esq., Secretary to Because the angle AEG 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 FG by Prop. BEST (Fordingbridge) Received.

LESSONS IN BOOKKEEPING.-No. XI.

THE JOURNAL.

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THE Journal, as we have before remarked, is no longer what 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 period into a focus, previous to their being entered made in Bookkeeping, in order to shorten 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 nor too 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 preceding ; may take place only once a month; and then with reference to for once, therefore, we leave this subject in their hands. All time, as we formerly observed, it might be called the Month- we shall say, is this: that gentlemen who have been in busiBook; 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 it is made up, and the connexion which it

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