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Lagiar-di-niê-ra, f., the female Ad En-ri-co, to Henry.

gardener, the gardener's Da En-ri-co, from or by wife.

I uô-mo, m., the male person,| man, husband,

La dôn-na, f., the woman, wife,

lady, mistress.

Il sol dá-to, m., the soldier.
Il ser-vo, m., the servant.
Lo sco-lá-re, m., the pupil,
learner, scholar.

Lo scul-tó-re, m., the sculptor,
statuary.

*

Lo spêc-chio, m., the lookingglass, mirror.

Lo scri-gno, m., the coffer, casket, safe, iron safe, strongbox, small money-box, drawer, portable desk.

Lo scán-no, m., the long stool,
form, bench.

Lo scrit-to, m., the writing.
Car-lo, Charles.

Di Cár-lo, of Charles.

A Cár-lo, to Charles.

Da Cár-lo, from or by Charles.
En-ri-co, Henry.

D' En-rí-co, of Henry.

Henry.
Mi-lá-no, Milan.

Di Mi-lá-no, of Milan.

ENGLISH-ITALIAN.

Our gardener is a good man. Your gardener's wife is a good woman. My friend is the uncle of this young man. I have bought this tree from your gardener. Our (female) neighbour

A Mi-lá-no, to, in or at Milan. has a very good son and a very good daughter. Hast thou seen this poor man's child? My (male) cousin's looking-glass is very large. Thy (male) neighbour is the pupil of my father. My book is on the form. I have given my hat to this poor child. The book which I have received from a friend is lost. Louisa has lost her bonnet. Have you (sing.) found Charles's ring? Henry's father (i. e. the father of Henry) is very rich. John's garden is very small. William's friend has departed. My cousin has (i. e. is) arrived. We have received a letter from Louis; he is at Milan. Have you seen Francis and Ferdinand? Rodolph has departed for Venice. We have written a letter to Stephen in Paris. Have you (sing.) seen thich of Louis? Has (i. e. is) your (sing.) uncle departed for Paris Caroline's aunt is in London. Our (male) neighbour has a son, who is called Adolphus, and a daughter who is called Louisa.

Da Mi-lá-no, from Milan.
Gio-ván-ni, John.
Lu-i-gi, Lewis.
Fran-cé-sco, Francis.
Gu-gli-él-mo, William.
A-dôl-fo, Adolphus.
Ki-dôl-fo, Rodolph, Ralph.
An-tô-nio, Anthony.
Stê-fa-no, Stephen.
Fer-di-nán-do, Ferdinand.
Car-li-na, Caroline.
Lu-í-gia, Louisa.
Vi-ên-na, Vienna,
Ve-né-zia, Venice.
Pa-rí-gi, Paris.
Lôn-dra, London.
Ar-ri-vá-to, arrived.
Par-li-to per-departed for—
Si chia-ma, is called (i. e. one
calls or names, we, they,
people call or name).
Edi-belongs to—(i.e. is of
—(i.e. is of—)

EXERCISES.-ITALIAN-ENGLISH.

|

|

ENGLISH-ITALIAN.

The nephew has gone with the general's son and daughter into the park to dine there. Next week they will go together into the country. A courier has arrived with the news of the conclusion of peace. The cousin came here with the express order to buy a horse and a coach. I have never offended him with one single word. In time, and with patience, one learns everything. Man ought to spend the first part of his life with the dead, the second with the living, and the third with himself. The world is filled with ungrateful persons: we live with the ungrateful, we work for the ungrateful, and we always have to do with the ungrateful.

Nephew, ni-pó-te, m.

ra, f.

re

VOCABULARY.

Together, tút-ti iu-si-me
Country, cam-pu-ra, f.
(There) has arrived, è giún-to
Courier, cor-riê-re, m.
News, nuô-va, f.

One single word, ú-ra só-la pa-
rò-la, f.

Time, têm-po, m.
Patience, po-zi-èn-zo (ts—ts), f.
One learns everything, s' im-
pá-ra tút-to

Ought to spend, de-ve pas-sá

re

Hô ve-dú-to l'om-brêl-la di vô-stro pá-dre. L'a-mí-co di
mí-o zí-o è ric-co. Quest' uô-mo è l' a-mí-co i mi-o pá-dre. I
Il fan-ciúl-lo di quest' uô-mo è am-ma-lá-to. Qué-sto fan-
ciúl-lo è an-có-ra gió-va-ne. A-vé-te voi ve-dứ-to l’ al-be-ro
che mí-o pá-dre ha com-prá-to? L'uô-mo, che a-vé-te ve-dú-
to, è mól-to pô-ve-ro. Sú-o fí-glio è am-ma-lá-to. Ho dá-to Has gone to dine there, è on-
dá-to a pran-za-re
la pén-na a qué-sto pô-ve-ro fan-ciúl-lo. A-vé-te voi ve-dú-
to 1 o-ro-lô-gio che mí-o zí-o hạ ri-ce vú-to? E-gli ha ven- Park, bo-schét-to, m.
ha
dú-to qué-sto o-ro-lô-gio a mí-o pá-dre. La zí-a di qué-sto General, ge-ne-ru-le, m.
gió-va-ne è an-có-ra am-ma-lá-ta. Qué-sto pô-ve-ro fan- Week, set-ti-má-na, f.
ciúl-lo ha per-dú-to sú-a má-dre. Il 1í-o a-mí-co è un uô-mo| Next, ven-tú-ro, m., ven-tú-
mol-to ríc co Quest’uô-mo è il nô-stro giar-di-niê-re. Qué
sta dôn-na è la nô-stra giar-di-niê-ra. Il nô-stro vi-cí-no è They will go, rô-glio-no an-dá-
ric-chís-si-mo. La vô-stra vi-cí-na è ú-na buô-na dồn ng.
A-vé-te voi ve-dú-to mí-o cu-gi-no. Hô ve-dú-to vô-stro cu-|
gi-no e vô-stra eu-gi-na. Vô-stro cu-gi-no è l’a-mi-co di
mí-o fra-têl-lo. Mi-a so-rêl-la è l'a-mí-ca di vô-stra cu-gi-na.
La buô-na giar-di-niê-ra ha per-dú-to sú-o fí-glio; sú-a fi-glia
è an-có-ra am-ma-lá-ta. Là vì-c1-na di mí-0 z1-0 ha un gran
zí‐o
dís-si-mo fi-glio. Il nô-stro giar-di-niê-re è il pá-dre di qué-
sto fan-ciúl-lo. La fi-glia di qué-sta pô-ve-ra dôn-na è am-ma-
lá tạ. Hô ri-ce-vú-to un re-ga-lo da tứ-o cu-gi-no. Mi-a so-
rel-la ha serít-to ú-na lột-te-ra a vô-stro cu-gi-no. Il sol-dá-to |
che a-vé-te ve-dú-to é mí-o cu-gi-no. Lo sco-lá-re di mi-o
zí-o ha per-dú-to lo spêc-chio dì sú-a má-dre. Dov'è lo scrít-
to di mí-a so-rêl-la? E's-so è nél-lo scri-guo. Lo spêc-chio è
súl-lo scán-no. Il sêr-vo ha ri-ce-vú-to qué-sta ta-bac-chiê-ra
da ú-no scul-tó-re. Mi-o cu-gi-no è un buô-no sco-lá-re. A-
vé-te ri-ce-vú-to ú-no spêc-chio da mí-a má-dre. Ca-nô-va è
un grán-de scul-tó-re. Il fi-glio di mí-o zí-o si chiá-ma Cár lo
e sú-a fi-glia si chia-ma Lu-í-gia. Il fan-ciúl-lo di qué-sto
scul-tó-re si chia-ma Gu-gli-êl-mc. La zi-a di Fer-di-nán-do
è ar-ri-vá-ta; ma sú-o pá-dre è par-tí-to per Lôn-dra. La so-
rêl-la di Lu-i-gi è gran-dís-si-ma. Pên-so ad En-rí-co ed a
Siê-fa-no. La žî-a di Lu-í-gia ha scrít-to ú-na grán-de lêt-te-
ra ad A-dôl-fo. Fran-cé-sco ha ri-ce-vu-to qué-sta pén-na da
un gió-va-ne, che si chia-ma Ri-dôl-fo. Il cu-gi-no di Gio-
ván-ni è par-tí-to per Pa-ri-gi. L'om-brêl-la di Car-li-na è
pic-co-lis-si-ma. Ab-bá-mo da-to la nô strà ta-bac-chiê-ra a
Gu-gli-êl-mo. Qué-sto cap-pêl-lo è di Gio-ván-ni, e qué-sto
man-têl-lo è d' A-dôl-fo. No-stra zi-a è a Mi-lá-no. Lo scul-
tó-re è a Vi-ên-na. Qué-sta dôn-na è di Ve-nê-zia. Il nô- Person.
stro a-mí-co è di Pa-rí-gi. Il no-stro sêr-vo è ar-ri-vá-to da First.
Lôn-dra. Mi-a so-rêl-la pên-sa a Car-li-na. Ste-fa-no ha
per-dú-to il tem-pe-ri-no ch' é-gli ha ri-ce-vú-to da A-dôl-fo.
Lu-i-gia è la so-rêl-la di Car-lí-ña, ed An-tô-nio è il fra-têl-lo Person.
di Gio-ván ni

First part, pri-ma pár-te, f.
His life, la sú-a ví-ta
Dead, môr-to, m.

Second, se-con-do, m., se-cón-
da, f.

Living, ví-ro, m.

Last, úl-ti-mo, m., úl-ti-ma, t.

Peace, pá-ce, f. (i.e. news of Himself, sè stés-so
the peace)
World, món-do, m.
Cousin, ca-gi-no, m.
Came here, ar-ri-ao qui
Express order, ór-di-ne es-pris-

so, m.

To buy, di com-prá-re
Horse, ca-vál-lo, m.
Coach, car-rô-sa, f.

I have never offended him, í-o
non l' of-fé-si má-i

Is filled with, è piê-no di
Ungrateful (person), in-grá

to, m.
We live, si ví-ve

We work, si la-vó-ra
For, per

And we always have to do, e si
há da far sêm-pre.

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From this instance you learn that the Greek verb varies, or | is modified in person, in number, in tense, in mood, and in voice. Accordingly, it is the business of the learner to become familiar with the verb in all these its modifications, so as to at once recognise every form he may meet with in reading, and be ready at first sight to assign its meaning. The task is not an easy one, but will yield to persevering application. The task, being difficult, must be undertaken in detail.

Before we proceed to the general conjugation of the Greek verbs, we must present a peculiar form, namely, that of the

substantive verb, or verb of existence, ειναι, to be.

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G.

εστε εστωσαν.
εστον εστων
εστω

ειναι

Μ. ων

ουτος

Η ουσα

ουσης

Ν. ον

οντος.

Infinitive.

Participle.

S. D. P.-That is, the singular number, the dual, the
plural.
N. denotes the nominative case.
G. denotes the genitive case.
M. denotes the masculine gender.
F. denotes the feminine gender.
N. denotes the neuter gender.

The English is only given in part, it being presumed that the learner can easily supply the rest; thus, when he knows that eu means I am, he can hardly fail to know that the plural runs we are, you are, they are.

Ειμι

Let it be premised that the significations given in the paradigms, or examples of conjugation, are sometimes only approximately correct; for the exact meaning the student must wait until he is familiar with the details of Syntax and other details, which will follow.

The verb whose forms are given above, belong, it will be seen, to the class of the verbs in μι. There is another forma, distinguished in part by accents, namely, είμι, I go, (εἰμί, I am); the conjugation of which will be given in its place under the verbs in

μι.

The second person of the present, et, is more used than εις. In the imperfect, the second person, ης, often becomes ησθα, by the addition of a euphonic suffix; the third person is nv, more frequently than η.

ην,

Instances are found, particularly in the first person singular and the third person plural, of another imperfect, which resembles the imperfect of the middle voice.

S. ημην ησο ητο. Ρ. ημεθα ησθε ηντο.

A middle imperative form is also found in the second person singular, namely, εσο, be thou.

The entire present subjunctive, namely, ω ης η, &c., supplies terminations to all the verbs in w. The second and third person singular have the iota subscript, as seen above. The optative forms, ετην ειης ειη, lend their terminations την, &c., to the optative of the verbs in μι. For the form είημεν, ειμεν is used; and for ειησαν, ειεν is much more common; ειεν is also found in the sense of well! very well! be it so!

The future, in all its moods, is a middle form ; its termination, σομαι, is that of all the middle verbs in the future. The original forms were

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In soɛoaι the second σ was elided, and the word became The ɛa was contracted into n, the was written under, and thus toy arose.

εσεαι.

This observation extends to all the second persons in 7, of the middle and passive verbs. Also, in the optative, εσοιο stands for εσοισο εσται, a contracted form of εσεται, is more commor than εσεται.

The participle εσομενος (the Latin futurus) is declined like αγαθος, αγαθή, αγαθόν.

The substantive verb lacks the perfect, the pluperfect, and the aorist; these tenses are supplied from γιγνομαι, I become.

The stem of the verb is εs, as found in εσμεν, εσομαι, &c.

The present participle is declined thus-
Singular.

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Before I make any remarks on this verb, I will explain the contractions, the rather because they will recur again and again. LIST OF CONTRACTIONS, WITH EXPLANATIONS.

So decline the participles in wv, of all the verbs.

By the aid of prepositions various compounds of u are Ειμι formed, and these compounds are conjugated like their primitive; as παρ-ειμι (adsum), I am present ; απ-ε:με (absum), I Ι am absent; μετ-ειμι (intersum), I am among; συν-ειμι (una sum), I am with; προς- ειμι (insum, accedo), I am near, I approach; περι-ειμι (supersum, superior sum), I survive, I am 1. 2. 3.-That is, the first person, the second person, the superior; and others. The preposition remains invariable; third person.

only the verb undergoes the conjugational changes.

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3. ντι By studying these terminations now, and by reverting to them afterwards, the student will be materially assisted.

One thing let me enforce on the student, he must make himself thoroughly master of all the paradigms before he attempts to set a step in advance. The remarks made thereon are, in the commencement, of less consequence.

VOCABULARY,

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apa, as, ǹ, an hour (Latin,
hora), time.

AρiσTOV, OU, TO, breakfast;
ἡμιν αριστον εστι, we have
breakfast; avaι, with a date
of the person, has the force ui.
to have; the pronoun must
be put in the dative, the
person being preserved, thus
εOTI μoι is I have; or σo,
EOTL μOL
thou hast, &c.
Φυτεία, ας, ή, planting, care.
Γεωργικος, η, ον, agricultural,

t

The subject is either a noun or that which is the representative or equivalent of a noun; the predicate is either a verb alone, or a verb in conjunction with some other part or parts of speech. All other words entering into a sentence are to be regarded as mere adjuncts. The following sentences exhibit the subject and the predicate under several varieties of form:

Subject.
God

Man

To be,

Throwing the stone

Predicate.
exists.

is mortal.

contents his natural desire. was his crime.

In the sentence, God exists, the verb exists is the predicate: affirming, as it does, existence of the Almighty. But in the sentence, man is mortal, mortality is what is affirmed of man; and the verb (is) is the mere link that connects the subject and the predicate together. It is thence called the copula. § 158.

Sentences are either simple, that is, contain a single assertion or proposition; or compound, that is, contain two or more assertions or propositions. Of the various parts of a sentence, detail; so as to show the relation, agreement, government, and whether principal or adjunct, we come now to speak more in rangement of words in construction.

§ 120. THE ARTICLES.
RULE.

The article in German, whether definite or indefinite, is generally employed wherever the corresponding article would be used in English.

OBSERVATION.

This rule is of course founded upon the presumption that the student is familiar with the usage of the English in respect to the article. In the specification that follows, therefore, he is to look only for the points in which the German differs from the usage of our own language.

hence the name Georgies, given to Virgil's Didactic Poem on agriculture. Ayopa, aç, ǹ, a market. Aropos, a, ov, impassable; ra aroрa, straits, extremities; observe that ɛval, with the (a) Before words of abstract or universal signification; as, preposition επ, signifies "tober Mensch ist sterblich), man (i. e. every man) is mortal; das Gold be in the power of." ist dehnbar, gold is ductible; das Leben ist kurz, life is short; die Tugend führt zum Glücke, virtue leads to happiness :

EXERCISES.-GREEK-ENGLISH.

‘H ražis nv έkatov avdoes. Hv тs was μikov Tρо duvrоç Η ταξις ην ήλιου. Οἱ νομοι ζημιαι εισι των αμαρτωλων. Τούτοις θανατος εστιν ἡ ζημια. Ὁ σιτος επέλιπε, και πρίασθαι ουκ ην. Εστι ὁρᾶν το ορος. Η Αγησιλαου αρετη παραδειγμα ην. 'Hur αριστον ουκ εστιν. Εγω εσομαι ὁ συγκαλῶν. ὁ συγκαλῶν. Οὗτος εστι ὁ νικῶν. Εγω μια τούτων ειμι. Βασιλευς νομίζει ύμας αυτοῦ EOTIV OUR TYS Yεwpyikns texvns ǹ twv devdowv putsa, Έστιν αυτοις αγορα. Εν τοις απόροις ημεν. Ὁ Κύρος εν Ον μικρον αγαθον τῳ ἁρμοττειν προςεστιν. Τη βις προςεισιν εχθραι και κινδυνοι. Τη επιμελεια περιείναι των φιλών θελω. Παρην Αγεσίλαος δωρα φερων. Κυρῳ παρησαν εκ Πελοποννησου νηες.

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(1) The Germans insert the definite article:

(b) before the names of certain divisions or periods of time: as, der Sonntag, Sunday; der Montag, Monday; der Dezember, De. cember; der August, August; der Sommer, Summer:

surfet, Turkey ; bie Gywei, Switzerland, vie Sombarbei, Lombardy: (c) before certain names (feminines) of countries; as, dic

(d) before the names of authors, when used to denote their works; as, ich lese den Lessing, I am reading Lessing:

(e) before the proper names or titles of persons, when used in a way denoting familiarity or inferiority; as, grüße bie Marie, greet (or remember to me) Mary; sage dem Luther, daß ich ihn zu sehen wünsche, tell Luther that I wish to see him: also, when connected with attributive adjectives: as, die kleine Sophie, little Sophia:

(f) before words (especiairy proper names of persons) whose cases are not made known either by a change of termination, or by the presence of a preposition; as, das Leben der Fürften, the

life of princes; die Frau des Sokrates, the wife of Socrates; der] Cicero was called the father of his country; Er ist Alexander geLag der Rache, the day of (the) vengeance: tauft worden, he has been christened Alexander. From this re

(g) before the names of ranks, bodies, or systems of doc-mark, however, must be excepted the verb lehren, since it has trine: as, das Parlament, Parliament; die Regierung, government no passive. die Monarchie, monarchy; das Chrißtenthum, Christianity: also in such phrases as, in der Stadt, in town; in der Kirche, at church; die meisten Menschen, most men :

(h) before the words (signifying) half and both: as, vie halbe (not halbe die) Zahl, half the number; die beiden (not beiden die) Brüder, both the brothers:

(i) before words denoting the limit within which certain specified numbers or amounts are confined, wherein in English, the indefinite article would be used: as, zweimal die Woche, twice a week:

(2) Note, further, that the German differs from the English in omitting the definite article,—

(a) before certain law appellatives: as, Beklagter, (the) defendant; Kläger, (the) plaintiff; Apvellant, (the) appellant; Supplicant, (the) petitioner:

(b) before certain common expressions; such as, in bester Ord. nung, in (the) best order; Ueberbringer dieses. (the) bearer of this; and certain adjectives and participles treated as nouns; as, erfterer, (the) former; letterer, (the) latter; bejagter, (the) before- | said (person):

(c) before certain proper names and places: as, Ditindien, (the) | East Indies: Westindien, (the) West Indies; and before the names of the cardinal points: as, Often, (the) East; Westen, (the) West; Süden, (the) South; Norten, (the) North :

(d) before a past participle joined with a noun, which, in English, precedes the participle: as, das verlorene Paravies, (literally, the lost Paradise) Paradise Lost.

(3) Note, again, that the Germans, in using certain collective terms preceded by adjectives, employ the indefinite article where the English would use the definite: as, ein hochweiser Rath, the (lit. a) most learned Senate; cine löbliche Universität, the (a) honourable University.

§ 123. RULE.

fying a different thing, is put in the genitive; as,
A noun used to limit the application of another noun signi-

Der Lauf der Sonne, the course of the sun.
Der Sohn meines Freundes, the son of my friend.
Die Erziehung der Kinder, the education of the children.
Die Wahl eines Freuntes, the choice of a friend.

How this limitation is made, is easily seen thus, der Lauf der
Sonne, the course of the sun. Here we speak not of any course
indefinitely, but of the sun's course definitely: the word der
Sonne, is the genitive, limiting der Lauf, which is the governing
word.
OBSERVATIONS.

(1) If, however, the limiting noun (unless restricted itself by an adjective or some other qualifying word) signify measure, number, weight, or quantity, it is then put in the same case with that which it limits; as, zwei Olas Wein, (not Weines), two glasses (of) wine; sechs Bfund Thee (not Thees), six pound (of) tea: but (with a restrictive term), sechs Pfund dieses Thees; zwei, Glas dieses Weines.

(2) It should be observed that the two nouns under this rule must be of different signification; for two nouns standing for the same thing would be in the same case, forming an instance of apposition. See § 133. (1).

(3) The noun in the genitive, that is, the limiting noun, is commonly said to be governed by the other one. This genitive is either subjective or objective; subjective, when it denotes that which does something or has something: objective, when it denotes that which suffers something, or which is the object of what is expressed by the governing word. To illustrate this, we have only to take have only to take the examples given above: der Lauf der Sonne, the course of the sun; die Erziehung der tater, the education of the children; where, in the first ex

(4) In German, also, the indefinite article stands before (not after, as in English) the words, such, half: thus, ein solcher Manu, (not solcher ein Mann), such a man; ein halbes Jahr, (not halbes ein Jahr), half a year. In questions, direct or indirect, like the following: Gunen wie langen Spazierritt hat er gemacht, how longariple, the sun is represented as performing or having a course, ride has he taken; it must be noticed that the article stan before wic: thus, cinen mie langen (a how long) and not, as in English, how long a.

(5) The German differs again from the English in not using an article at all in the phrases answering to the English; a few; a thousand; a hundred.

§ 121. THE NOUN,
RULE.

and is consequently subjective; and, in the second example, the ldren are represented as being the objects of education, and the word is consequently objective. This objective genitive, it should be added, occurs only after verbal nouns, and chiefly those ending in the suffixes cr, which marks the doer, and ung, which marks the doing of an action.

(4) It seems hardly necessary to observe that under this rule come all words which perform the office of nouns; as, pronouns, adjectives used substantively, &c.; thus, die Gnade der

A noun or pronoun which is the subject of a sentence must Großen, the favour of the great. be in the nominative case: as,

Der Mensch denkt, Gott lenkt, man devises, God disposes.
Die Berge donnern, the mountains thunder.

OBSERVATIONS.

(1) The subject or nominative in German is seldom omitted, except in the case of the pronouns agreeing with verbs in the second person (singular and plural) of the imperative: as, Lese (bu), read! Gehet und saget (Ihr) ihn, go and tell him. See, however, § 136. 2.

§ 122. RULE.

(5) We say often in English, He is a friend to, or an enemy to, or a nephew to any one; where, were these phrases put into German, we might expect the dative to be used. But, in such cases, the Germans always employ the genitive: thus, er ist ein Feind seines Vaterlandes, he is an enemy of his native country.

(6) We say in English, the month of August, the city of London, and the like: where the common and the proper name of the same thing are connected by the preposition of. The Germans put the two nouns in apposition. See § 133. (2).

(7) So, too, in English we say, the fifth of August; but in German, the numeral is put in direct agreement with the name of the month: as, ter fünfte August, the fifth (of) August, or

A noun or pronoun which is the predicate of a sentence, must August fifth. be in the nominative case: as,

Er war ein großer König, he was a great king.

}

Dieser Knabe ist Kaufmann geworten, this boy is become a merchant.
Merander hieß der Große, Alexander was called the Great.

(8) In place of the genitive, the preposition von, followed by the dative, is, in the following instances, generally used: a. When succeeded by nouns signifying quality, rank, measure, weight, age, distance, and the like; as, cin Mann von hohem Stante, a man of high standing; ein Schiff von zwei hundert Tonnen, a ship of two hundred tons; ein Gewicht von fünf Pfund, a weight (1) This rule applies, where the subject and the predicate are of five pounds; ein Mann von achtzig Jahren, a man of eighty connected, as above, by such verbs as sein, to be; werden, to be-years; eine Reise von drei Meilen, a journey of three miles; cin come; heißen, to be called; bleiben, to remain, &c. Engländer von Geburt, an Englishman by birth, &c.

OBSERVATIONS.

(2) So, also, the rule becomes applicable when any of those b. When followed by nouns denoting the material or subverbs which in the active govern two accusatives (§ 132. 2.), are stance of which any thing is made: as, ein Becher von Silber, a employed passively: as, Cicero wurte der Vater des Vaterlances genannt, | cup of silver, i. e. a silver cup; cine Uhr von Golte, a gold watch

c. When followed by nouns whose cases are not indicated by the terminations of declension nor by the presence of the article: as, der Schein von Redlichkeit, the appearance of honesty; ein Vater von sechs Kindern, a father of six children; die Königin von England, the queen of England; die Grenzen von Frankreich, the boundaries of France; der Bischof von Konstanz, the bishop of Constance.

d. When followed by a word indicating the whole, of which the word preceding expresses but a part: as, einer von meinen Bekannten, one of my acquaintances; welcher von beiden? which of the two?

LESSONS IN GEOMETRY.-No. XXVII.
LECTURES ON EUCLID.
PROPOSITION XXVIII.—THEOREM.

(Continued from page 298.)

16. A fallacy somewhat more subtle than Franceschini's, though akin to it, may be framed on the consideration of the CD'be the angle of intersection. Let A B, CD be two straight lines in the same plane making with a third straight line AC the angles C a 3, ACD, of which ACD is a right angle and CA B less than a right angle. And to improve the appearance though this is not indispensable) draw a straight line A E on the other side of A C and in the same plane, making an angle C A B equal to CAB. And

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the asymptotes, be placed so that the perpendiculars shall coin-
cide and the asymptotes in consequence be in one straight line,
as s T in the figure below. Upon which it is clear, that how-
smaller than A H G or BHX between WA and A B, W X cannot
ever it may be pleaded that there may always be an angle
be carried beyond the line of the asymptotes S T without ceasing
to meet A B ; and consequently cannot be carried till it meets CD,
if CD lies on the other side of s T as represented in the figure.
It follows therefore, that to say there will always be the possi-
bility of a further diminution of the angle, is not enough. It may
be (as it is in the case of the hyperbolic angle above) the sophism
of Achilles and the tortoise; which argues, that because after
running a mile, half a mile, a quarter of a mile, &c., Achilles
would always be behind by the last-mentioned fraction of a mile,
he would never overtake or pass the tortoise. The solution
resolving itself into the fact, that these quantities, though endless
in number, are limited and surpassable în amount.

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To establish the union of the lines to any particular extent that may be desired, it is consequently necessary to prove, not only that the angle at the intersection is capable of diminution, but that the, angle on each side of the travelling line (that is to say, both the angle AH G and the angle GP) will never be reduced to less than some given angle. Which is what has been attended to in Proposition XXVIII. C of the "Geometry without Axioms. accordingly, in the case of the intersections with a travelling line 17. Another course taken has been to define a straight line to be one "of which every successive portion has the same direction," and parallel straight lines to be "straight lines having the same direction with each other." From which it is purported to be collected, that a straight line cutting two parallel straight lines makes the interior angle equal to the exterior and opposite on the same side of the line. For, it is argued, the direction of the cutting line is at the two points of intersection the same; and the directions of the two parallel straight lines cut, are at those points the same with each other's; whence the differences of direction, which are the angles, will be equal. To which reply may be made by asking, what definite idea is attached to two lines having the same direction. It does not mean that they tend to the same point, for they do not. It means, then, that they never run against each other; or are parallel., And a line every successive portion of which has the same direction, means 2 ne of which the parts all lie in the straight line leading to a partiçalar point or object. The argument therefore resolves itseu into the proposition, that if a straight line falls upon two parallel straight lines, it makes the exterior angle equal to the interior and opposite on the same side of the line; propounded in other terms, without the intervention of any new or explanatory idea. 18. In a traet entitled "The Theory of Parallel Lines perfected; or the Twelfth Axiom of Euclid's Elements demonstrated. By Thomas Exley, A. M.-London. Hatchard 1818" the proof rests on taking for granted (in the Second Proposition) that if four equal straight lines in the same plane, making right angles with one another successively towards the same hand, do not meet and enclose a space, a fifth if prolonged both be resolved into taking for granted that the three angles of a rectilinear triangle are greater than a right angle and a half; for if they were equal to this, the angles of an equilateral and equiangular octagon would be right angles, and the fifth straight line in the series proposed would never meet the first; still more if they were less. And in the same manner. if it was urged that a sixth, seventh, &c. perpendicular must meet the first straight line, it would only resolve itself into a demand for admitting, without proof, that the three angles of a triangle are greater than some other amount capable of being specified. There is no obscurity about the fact that four such straight lines, and still more five, are found on experiment to meet; but the object was to discover why they necessarily meet. And between the observed fact and the explained fact there is a difference of the same kind as between Kepler's observation of the proportion between the periodic times and distances of the planets, and Newton's explanation of the cause.

from a let a straight line of unlimited length as w x travel along the straight lines A B and A E, cutting ac always at right angles in some point & between a and c. This line will represent Franceschini's succession of perpendiculars. But instead of arguing from its continually cutting off greater and greater portions AG, let it be argued that because it at any time makes with A B an angle A H G or B HX, it may always be removed to a position further from A without ceasing to cut A B and AE. From which it at first sight might appear to be a reasonable conclusion, that the straight line w x may be carried forward without the possibility of failing to cut A B and A E, till it arrives at c. And the fallacy will perhaps he still more taking, if a B and A E are made to begin by being placed at c, and so are moved from cways must inevitably accomplish it. A conclusion which may towards A, as represented by a b and a e; under which circumstances the allegation that there must always be an angle of some kind at h, has a very inviting appearance as a reason why ab and c D, being continually prolonged, cannot quit one another or fail to meet and make an angle of some magnitude or other, the consequence of which would be that ab and ae might be moved till a coincides with A and ab with AB, without the possibility of parting company with C D by the way.

The answer to this is by inquiring, whether there are no lines in which the same facts may be determinable on the subject of the angle, but where it is certain that a straight line as wx cannot be carried on to an unlimited extent as proposed. And here it is easy to show that there may. Take, for example, any hyperbola; and from the vertex draw a perpendicular to each of the asymptotes; and let the two halves of the linear hyperbola,

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19. In "A Treatise on Geometry, by D. Cresswell, M. A. Cambridge, 1819," the principle proposed to be taken for granted is, that through any point within an angle less than the sum of two right angles, "a straight line may be supposed to pass, which shall cut the two straight lines that contain the angle." The grounds for such admission being stated to be, that "if it be granted that among the infinite number of points outside the angle, there are two, on contrary sides of the angle, which are in the same straight line with the point within; or if it be granted, that, inasmuch as the sides that make the angle are unlimited in length in the directions removed from the angular point, two points in those sides may be taken, one in each, so that every point in the straight line which joins them shall be farther

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