L'I-sti-tú-to po-li-tê-cni-co in Pa-rí-gi. La pô-sta pár-te ó-gni di per l' I-tá-lia, per Ve-nê-zia, per Ró-ma. E'-gli dê-ve recar-si a Mi-la-no. E re-sta-to tút-to il giór-no a cá-sa. E'-gli non va a pa-láz-zo, a cór-te. Di qui a Neo-ca-stêl-lo, a Jork.

Ha mandato, he has sent.
Lettera, letter.
Giovanni, John.

VOCABULARY.

Tirare, to draw, trail, drag; to shoot or fire, &c.

Uccello, bird.

Mercante, merchant.
Pensa, thinks.
Guadagno, profit.

Tocca, falls to the lot or share
(toc-cá-real-cú-na cô-sa ad
ú-no, to fall to the lot or
share of one).
Fiorino, florin.
Ognuno, every body.
Tira, draws, conveys.
Acqua, water.

Molino, mill (tirar l'acqua
al suo molino, to convey water
to one's mill; to look to the
main chance or to number
one).

Da, from.

Parola, word.

ENGLISH-ITALIAN.*

Thy mother has lost her umbrella. My sister has found a pen. Where have you bought this penknife? Hast thou seen our horse? We have seen a farge inn. Your little brother has a good watch. Our brother is tall,† but our sister is little. I have a hat which is very fine. The watch which you have bought is good. Our uncle has received a letter. This son has lost his mother. This daughter has lost her Nozze (ts), f. pl., wedding, mar- father. This present is for this child. ** riage feast.

Erano, were.

Caccia, chase.

Pranzo, dinner.

Cena, supper.

ENGLISH-ITALIAN.

Mr. Thomson has gone to the exchange. Let us go into the Festino (dancing, gaming, &c.), concert. The sisters have gone to-day to the evening enter

evening party.
Andrete, will you go.
Domani, to-morrow.
Ridotto (in some towns
Italy), public masquerade.
Concerto, concert.

of

Io andrò domani, I shall go to

Dogana, custom-house.
Soggiorna, he lives or resides.
Firenze, Florence.
Egli mori, he died.
Nottingamo, Nottingham.
Egli lo condurrà, he will bring
or conduct him.
Cestria, Chester.

Prossimo (m.), prossima (f.), Ella giunse, she arrived.

Presto, soon, quickly.

Lione, Lyons.

Egli è arrivato, he has ar-
rived.

Bristol (also Bri-stôl-le or Bri-
stó-lio), Bristol.

Egli è nato, he was born.
Plimutte, Plymouth.
Istituto, school or institute.
Politecnico, polytechnic.
Parigi, Paris.

Parte, starts.

Ogni, every.

Di, day.

tainment. He is at the ball, and the brother in the concert. We have paid a visit to the neighbour; he lives on the second floor, and the son on the ground floor. We are now sitting at table. Think of more serious things. The misers are like the horses that carry wine and drink water, and like the asses that bear gold and eat thistles. He lives at the Black Eagie, and not at the Golden Lion. I have spoken to him at the coffee-house. Shall we play a game at cards or at chess? VOCABULARY.

Mr. Thomson, il Si-gnór

↑ Thomson

Has gone, è an-dá-to
Exchange, bor-sa, f.
Let us go, an-did-mo
Concert, con-cêr-to, m.
Have gone, só-no an-dá-te
To-day, ông-gi
Evening entertainment, con-
ver-sa-zió-ne, f.
He is, é-gli è
Ball, bál-lo, m.
We have paid, ab-biá-mo fát-to
Visit, vi-si-ta, f.

Neighbour, vi-cí-no, m.
He lives, é-gli á-bi-ta (also al-
log-gia or sta, with a)
Second floor, se-con-do spid-no,

m.

Ground floor, pián ter-ré-no,m.
We are now sitting, noi se-did-
mo ó-raş

Table, tá-vo-la, f.
Think, pen-sd-te (i. e. direct
your thoughts to)
More serious, più sê-rio, m.,
più sê-ría, f.
Thing, co-sa, f.
Miser, d-vá-ro, m.

Da.

Are like, ras-so-mí-glia-no
Horse, ca vál-lo, m.
That, ehe

Carry, mé-na-no
Wine, vi-no, m.
Drink, bé-vo-no
Water, á-cqua, f.
Ass, d-si-no, m.
Bear, pôr-ta-nÓ›
Gold, 6-ro, m.
Eat, mán-gia-no
Thistle, car-do, m.

Black Eagle, d-cqui-la né-ra, f.
(with the preposition a) -
And not, e non
Golden Lion, león dó-ro, m.
(with the preposition a)
I have spoken to him, i-o gli
hộ par-lá-to
Coffee-house, caf-fè, m
Shall we play, vo-gliá-mo fá-
rell
Game, par-ti-ta, f.
Cards, car-te, f. pl.
Or, o

Chess, scác-chi, m. pl. (sing
scác-co, m., one of the sixty-
four houses or squares on a
chess-board)

I have already stated that the particle di denotes a mere mental separation of ideas or notions, while the particle da

After a careful study of the previous colloquial exercises, even Egli deve recarsi, he must de-ordinary pupils will be quite able to translate the following sen tences without the aid of a vocabulary.

In some of i's meanings, this word denotes discreditable places of resort, and. 10 avoid ambiguity, it should only be used with precaution in the above-stated signification.

+ In Italian, tall and great frequently are expressed by the same word.

When the word Si-gno-re is followed by a noun, the final e is dropped, except when the noun that follows begins with the s impure; e. g. il Signor An-to-nio, Mr. Anthony; il Signor Fran-cé-sco, Mr. Francis; il Signor cón-te, count; il Signor ba-ró-ne, baron; il Si-gnór dot-to-re, doctor; il Si-gnór con-si-glie-re, counsellor; il Signó-re Stê-fa-no, Mr. Stephen.

§ O-ra, now, at present; hour, time, and 6-ra (for aú-ra), light wind, breath of wind, gentle air, breeze.

The verbs giuo-cá-re, to play (at cards or at any other game), and fá-re ú-na par-ti-ta, to play a game or make up a match (at cards or any other game), invariably require the preposition a; e. g. giuo-cd-re a un giuô-co, ai dá-di (or a dd-di), ál-le cár-te, á gli scác-chi (or a scác-chi), a tre-sêt-te, all' óm-bre, ál-la pál-la, a pic-chét-to, &c., to play at a game, at dice, at cards, at chess, at tre-sept (an Italian game at cards), at omber, at tennis, at piquet, &c.; jac-cid-mc ú-na par-ti-ta al bi-gliár-do, al whist, al cribbage, &c., let us have a game or make up a match at billiards, at whist, at cribbage, &c.

expresses a real separation of objects. This is the fundamental | gi-re, to fly, escape, &c., admit of the preposition di before that signification of da, and, on this account, it must be pronounced place from which the going away or departure takes place, to be the very opposite or logical antagonist of the particle a. this apparent deviation from the general rule, without diffiThis latter word indicates any kind of tangible or mental and culty, will be explainca by ellipsis; i. e. by the omission of the imaginary approach or direction to or towards a person or thing, preposition da, with some other general noun; e. g. ve-ni-re, while da expresses any kind of tangible or mental and ima-par-ti-re di Ró-ma (i. e. dál-la cit-tà di Ró-ma), to arrive, to ginary, but clear and real separation, removal, distance, or depart from (the city of) Rome; é-gli è di A-ber-dô-nia (i. e, direction from a person or thing, and the ideas of direction to or dal-la cit-tà di A-ber-đô-nia), he is a native of (the town of) towards, and of a direction from a person or thing, are, to some Aberdeen; u-sci-re, sor-tí-re di cá-sa, di cór-te, di pa-láz-zo, di extent, the very poles or extremities of all relations in which tea-tro, di chie-sa, to go or come from home, from court, from words and things stand to each other; e. g. in this sentence, guild-hall, from theatre, from church. pár-lo di lui, I speak of him, it is evident that there is no direcThe particle da, also, is used, in order, by naming the birthtion whatever to or towards, but rather a direction from a place, to distinguish one person from others of the same appellaperson. This direction is, nevertheless, not sufficiently clear tion. The birth-place thus becomes, as it were, the surname of and real enough to justify the use of da; while, in the sentence the individual. vên-go da lui, I come from him, a real removal, distance, or conformable to its fundamental notion, for the birth-place This employment of da certainly is quite separation from the person, from which I come, is understood, is a part of the general idea of origin, descent, or extracwhich can only be expressed by the particle da. As a further tion; e. g. Gio-ván ni da Fiê-so-le, Pie-tro da Cor-tó-na, Leo-nárillustration, in the phrase un mer-can-te di Ve-ró-na, a merchant do da Vin-ci, Gui-do da Sie-na, Po-li-do-ro da Ca-ra-vag-gio, of Verona, the particle di is a mere sign or intimation to Ra-fael-lo da Ur-bi-no,* &e. distinguish the merchant from the town in which he lives, and not of his absence from it; while in the sentence é-gli viê-ne da Ve-ró-na, the particle da denotes an actual removal from that place. This fundamental explanation of the particle du, however, is not sufficient to convey a complete notion of all its uses; every language, generally speaking, being far too complex a vehicle of human thought anywhere to admit of such a summary discussion of its more important branches. Now, and hereafter, I shall be therefore obliged to explain the various modifications and exceptions of this general rule.

The ideas of removal, distance, separation, dependence, deduction, or derivation, and origin or descent, are, as it were, only collateral or subordinate branches of the fundamental notion of a direction from a person, or thing, and that word (person or thing), the removal, distance, deduction or derivation, origin or descent from which, and the dependence on which, is expressed, requires the particle da before it; e. g. scó-sta-ti da qué-sto luôgo, begone from this place; al-lon-ta-ná-re ú-no da un luô-go, to | remove one from a place; ca-vá-re á-cqua dal póz-zo, to draw water from the well; ve-ni-re da lon-tá-no, to come from afar; i-o vên-go dal giar-di-no, da cá-sa, I come from the garden, from home; l'uc-cêl-lo è u-sci-to dál-la gáb-bia, the bird | has flown out of the cage; ac-cat-tá re pá-ne da ú-no, to beg one's bread of one; ciò (pron. ciô) di-pên-de dál-la for-tú-na, da voi, that depends on good luck, on you; de-dúr-re ú-na ra-gióne da un prin-ci-pio fal-so, to deduce an argument, proof, or evidence from a false principle; dal-la qual cô-sa ná-cque-ro di-se pa- ú-re, from which arose various fears; de-ri. vá-re l'o-ri-gi-ne di ú-na cô-sa da un' al-tra, to deduce the origin of one thing from another; di-i-de-re ú-na cô-sa da un' ál-tra, to separate one thing from another.

vêr

Da, also, may denote any origin or commencement referring to time, and then it means since; e. g. da che vi ví-di la prí-ma vol-ta, since (that day when) I saw you the first time; dal-la mi-a gio-va-néz-za in si-no qué-sto têm-po, since my youth till this day; dall' in-no pas-sa-to in qua, since last year; da du-e mé-si in qua, two months since; dal-la môr-te di mi-o pá-dre in qua, since the death of my father.†

The phrases da mat-ti-na, da sé-ra, da nôt-te, mean: in the morning, in the evening, in the night (by night, at night); e. g. 6-pe-ra da far da mat-ti-na, work to be done in the morning; non e-sce da cai-sa che da sé-ra, he only goes from home in the evening; tá-li cô-se non si fán-no da not-te, such things are not done by night.

Da also signifies about, nearly, close upon, not far off from, &c., e. g. hổ gua-da-gná-to da cín-que li-re ster·li-ne, I have gained or won about five pounds sterling; ho per-dú-to da sêi a ôt-to tál-le-ri, I have lost from about six to eight dollars; da Ró-ma a Ná-po-li sa-rán-no da cên-to ses-sán-ta mí-glia, it is about a hundred and sixty miles from Rome to Naples; é-gli vi re-ste-rà da cín-que a séi giór-ni, he will stay there from about five to six days; sti-má-va-si a-vé-re in Fi-rên-ze da no-van-tami-la bóc-che tra uô-mi-ni e fém-mi-ne e fan-ciúl-li, about ninety thousand persons, men, women, and children, were estimated to be in Florence.

sanctioned by a universal usage, for the most part in the A logical contradiction and anomaly, though introduced and place of the preposition a, the constant employment of da in connexion with those verbs which, with some house, mansion, apartments, lodging, or any other place of continuance, denote any kind of motion to or towards, any kind of living or residing with, and any kind of visit It is obvious that the idea of origin, expressed by da, neces-paid to, a person; e. g. an-dá-re dal mê-di-co, dal cal-zo-bi-jo, sarily includes any action proceeding from a person or place. For to go to the physician, to the shoemaker; do-má-ni ver-rò this reason, on the one hand, the English preposition by, when- da voi, I shall come to you to-morrow; i-o vi me-ne-rò da ever in connexion with passive verbs it denotes cause, author- lui, I shall conduct you to him; ve-ni-te da me, dal mership, instrumentality, &c., must be translated by da; and, on cán-te, come to me, to the merchant; só-no stá-to da lui, dal the other hand, all verbs expressing a going away, or depar-fra-tel-lo, I have been at his, at the brother's house (with ture, generally demand this particle; e. g. Car-tá-gi-ne fu fab-him, with the brother); á-bi-ta, al-log-gia da sú-o zi-o, he lives bri-ca-ta da Di-dó-ne, Carthage was built by Dido; fu é-gli da or resides with his uncle.

al-cu-ni sub-t se-gré-ti ne-mí-ci ac-cu-sá-to, he was accused by

some of his secret enemies; a qué-sto giar-di-no l' á-cqua è ab- Da is sometimes a substitute for di; e. g. li bia-si-md-va dubon-de-vol-men-te som-mi-ni-strá-ta da u-na fre-schis-si-ma fon- follia, di codardia), he severely blamed them, now for their ra-men-te, ó-ra da fol-li-a, ó-ra da co-dar-dí-a (instead of di tá-na, the water for this garden is abundantly supplied by a very cool fountain; é-gli è par-ti-to da Lôn-dra, he has de-folly, now for their cowardice; és-si hán-no mól-ti mô-di da alparted from London; co-min-ciò a an-da-re da Na-za-rêt-te leg-giá-re o da pas-sá-re quel-lo (instead of di alleggiare, di a Ge-ru-sa-lém-me, be began to go from Nazareth to passare), they have many means to make it easier or to pass

Jerusalem.

Whenever the verbs u-sci-re or sor-ti-re, to go or come out or from; par-ti-re, to set off, depart; ve-ni-re, to come; fug

Da, as well as the English by, is in these cases the preposition, which must be placed before the nominative case of the original sentence of the active voice whenever the latter is to be changed into the passive; e. g. ú-na fre-schis-si-ma fon-tá-na som-mi-ní-stra ab-bon-de-vol-mén-te l' á-cqua a qué-sto giar-dí-no, a very cool fountain abundantly supplies the water for this garden.

over t.

of ellipsis that I mentioned only serve the purpose of The particle da can never be really omitted, and the cases grammatical explanation.

The English learner will, perhaps, best understand this use of da by translating it with sprung from.

↑ Since (denoting time, and not in the sense of as or because) is translated by fin da, da........in quà, or dó-po, when it precedes a

noun.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle is greater than the base of the other.

In fig. 24, let A B C and DEF be two triangles, which have the two sides AB and AC equal to the two sides DE and DF, each to each; viz., A B equal to D E, and AC to D F. But the angle B A C greater than the angle EDF. The base BC is greater than the base E F.

Of the two sides DE and DF, let DE be the side which is not A greater than the other. At the point D, in the straight line D E, make (I. 23) the angle EDG equal to the angle BAC. Make DG equal (I. 3) to AC or DF. And join E G and

GF.

Because DE is equal (Hyp.) to AB, and DG (Const:) to Ac, the

Fig. 24.
D

B

E

F

two sides, ED and DG, are equal to the two BA and AC, each to each; and the angle EDG is equal (Const.) to the angle B A C; therefore the base EG is equal (I. 4) to the base BC. Again, because DG is equal to DF, the angle DFG is equal (I. 5) to the angle DGF; but the angle DGF is greater (Ax. 9) than the angle EGF; therefore the angle DFG is also greater than EGF; much more then is the angle EFG greater than the angle EGF. Now, because the angle EFG of the triangle E F G is greater than its angle EGF, and the greater (I. 19) angle is subtended by the greater side; therefore the side EG is greater than the side E F. But G was proved to be equal to B c. Therefore B C is greater than E F. Therefore if two triangles, &c. Q. E. D.

ע

F

G

Next, if the point r falls within the triangle DEG, as in fig. 3., DF and F E, taken together, are less than DG and GE taken together (I. 24); but DG is equal to DF (Const.). Therefore EG is greater than EF (Axiom 5); but EG was shown to be equal to в C, therefore BC is greater than EF.

The case in which the point F falls without the triangle DEG, as in fig. 4, is demonstrated in the same manner as above, in the proposition itself, and therefore it need not be repeated here. Where

Scholium 1.-Dr. Simson, in the construction of this proposition, introduced these words: "of the two sides D E, DF, let DE before the exercise is demonstrated. Q. E. F. the side which is not greater than the other," in order to avoid three distinct cases of construction, which would arise by taking that side which is greater than the other.

Scholium II.-It has been remarked that Euclid's demonstration of this proposition appears to be defective, because of the omission of the words introduced by Dr. Simson, as stated in the preceding Scholium. But upon consideration of the three cases referred to in the following exercise, it would appear that Euclid had originally contemplated their insertion, inasmuch as the second case of it, as demonstrated below, requires only a simple and direct reference to Prop. XXI. Now Euclid is not guilty, in general, of bringing in propositions, in any book, which do not bear upon those that follow; but it has been universally admitted that Prop. XXI. was of this description; now if he considered the three cases of Prop. XXIV., this objection is at once removed. Why they are not found in the common Greek text, we cannot tell at present.

EXERCISE I. TO PROPOSITION XXIV.

Demonstrate this proposition, by making the construction on the greater of the two sides of the triangle DEF, and exhibit the three distinct cases above mentioned.

Let ABC, fig. 1, and DEF, figs. 2, 3, and 4, be two triangles which have two sides of the one equal to two sides of the other, each to each, viz., the side A B to the side D E, and the side AC to the side D F, but the angle BA C greater than the angle EDF. The base is greater than the base E F.

than the other. At the point D in the straight line DE make the Of the two sides DE and DF, let DE be that which is greater angle BDG equal to the angle BAC (I. 23). Make DG equal to AC or DF (I. 3) and join E G.

Because DE is equal to A B (Hyp.) and DG to A c (Const.), and the angle EDG to the angle BAC (Const.); therefore the base EG

EXERCISE II. TO PROPOSITION XXIV.
Demonstrate that, in Dr. Simson's construction, the straight line
EG cuts the straight line DF in some point between D and F.
point H; it is required to demonstrate
In fig. z. let DF meet EG in the
that the point lies between the D
points D and F.

the angle D GE is less (I. 18) than the
Because D E is less than D G (Hyp.),
greater (I. 16) than the angle DEG;
angle DEG. But the angle D H G is
much more, then, is the angle D H G
fore the side DG is greater than the
greater than the angle D G R. There-
side DH (I. 19). But DG is equal
to DF (Const.). Therefore D F is
also greater than D H. Therefore
EG cuts D P in the point н, between
the points D and F. Q. E. D.*

E

Fig. z.

PROPOSITION XXV.-THEOREM.

F

If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the two sides of that which has the greater base, is greater than the angle contained by the two sides sides A B and A C equal to the two sides D E and n F, each to each; equal to them of the other. In fig. 25, let A B C and D E F be two triangles which have the two viz., A B equal to D E, and AC to DF; but the base B c greater

* These exercises were solved by J. H. Eastwood, Middleton,

is equal to the base BC (I. 4). Now, if the point F falls upon EG, | C. L. Hadfield, Bolton-le-Moors; and Q. Pringle, Glasgow.

The angle BA c is greater than the is equal (Const.) to the side EF, and the side A в to (Hyp.)

Because, in the two triangles ACD and BCD, the two sides AC and C D are equal (Hyp.) to the two sides B C and CD, and the angle ACD is equal (Hyp.) to the angle BCD; therefore the base A D (I. 4) is equal to the base D B. Wherefore A B is bisected in D. Also, by I. 4, the remaining angles of the triangle A CD are equal to the remaining angles of the triangle BCD, each to each, viz., those to which the equal sides are opposite; therefore, the angle ADC is equal to the angle BDC; but these are adjacent angles; therefore (Def. X.) they are right Fangles.

In fig. b, let o be the given point, and A B and CD the two straight lines given in position. It is required to draw through o, straight line which shall make equal angles with A B and C D. Produce AB and CD till they meet in E; bisect the angle A EC (I. 9) by the straight line E F. From the point o (I. 12) draw OP at right angles to EF, and produce it to Q, so as to meet A B and CD in the points R and s. Then o q is the straight line required. Because in the two triangles REP and SEP, the angle REP is equal (Const.) to the angle s EP, and the angle RPE to the angle 8 PE, each of them being a right angle, for R P E is equal (I. 15) to SPF; and the side EP is common to both; therefore (I. 26) their other angles are equal, viz. the angle PRE to the angle P8 E. FWherefore a straight line o a has been drawn through the point o,

Let B c be the greater of the two. Make в н equal (I. 3) to E F, and join A H.

This exercise should have been been appended to Prop. VIII.; it was solved by E. L. JONES (Pembroke Dock); Q. PRINGLE (Glasgow); Because in the two triangles ABH and DE F, the side B HD. H. DRIFFIELD; E. JONES; E. J. BREMNER (Carlisle); and others.