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τέκες

τεκος, child

TER

TIRTW

The REKULT of action is denoted by

1. io, nominat. lov neuter, as 1, ματ, μα, neuter,

παιδ-ιο-ν, α little child stem maid 210m. παίς πραγ-μα(τ), a thing done from πραγ 2s in πρασσω

κηπ-to-y, a little garden

κηπο

κήπος ρή-μα(τ), 4 thing spoken ρε

ερω τμήμα(τ), a cut

Besides the form 10, there are these, namely, lilo (nom. τεμ Τεμνω.

ιδιον), αριo (non. αριον), υδριο (nom. υδριον), as οικιδιο-ν, ο With πραγμα compare το πεπραγμενον, that which has been | little house (οίκος) και παιδαριο-ν, α tittle child (παϊς) και μελυριο-ν, done (Latin, factum); with para compare ro elonjevov, that | a ditty (uelos, a song, our melody). which has been said (Latin, dictum); with tunua compare to TET Jnuevov, that which has been cut (Latin, sectum).

2. Μas. σκο, fem, ισκα, nom. ισκος, σκη, e. g. 2. ες nominative oς, neuter.

νεαν-ισκο-ς, α youth

stem νεανια nom. νεανίας Stem λαχες Nomin. λαχος, α ίot, from λαχ, as in λαγχανω παιδ-ισκη, α ίittle maiden παιο

παι-ς εθες εθος, custom εθ

ειωθα

Patronymics, or nouns denoting descent from a father (marup),

that is, an ancestor, are formed mostly by the suffix ia (nom. The same suffix in derived words denotes the peculiar in-s) for the masculine, and merely d (nom.s) for the feminine. quality.

This suffix is added immediately to the stem in a, as βαρος, weight, adj. sterm βαρυ, nominat. βαρυς

Mas.

Fem.
βαθος, depth

βαθυ
βαθυς

,

Nom,
μήκος, length

μακρο
μακρος

βορεα-δης βορεάς from βορέα βορεά-ς, the north tind The instrument or means of an action is denoted by mpo,

Αινεια-της

Αινεια Αινειά-ς, Εneas. nominative τρο-ν, neuter (the Latin tru-m).

To consonantal stems the suffix is appended by means of the

vowel 1; apo-tpo-v, a plough from apo, as in apow (Lat. ara-tru-m) λυ-τρο-ν, αγαnsom from λυ, as in λυω

Mas. Κεκροπ-ί-δη-s, fem. Κεκροπ-ι-ς, from Κεκροπ, norm. διδακ-τρο-ν, α teacher's Je, διδαχ, as in διδασκω.

Κεκροψ, Ccrops. Less definite is the meaning of the related feminine suffix, Stems in ev and o of the third declension also take the τρα ; as ξυ-σ-τρα (ξυω, I share), a curry- como ; ορχη-σ-τρα connecting vowel 1, before which the n or ευ disappears : (ορχεομαι, 1 dance), α dancing-place, our orchestra και παλαι-σ-τρα (Thatw I wrestle), a urestling-place.

Πηλε-ιδη-ς, from Πηλευ, nom. Πηλευ-ς. Place is signified by,

An Homeric equivalent is IInAniadns 1, τηριο, nominat. τηρι-ον, neuter (the Lat. toriu-m) ;

Λητο-ιδη-ς, from Λητο, nom. Λητω,
ακροα-τηριο-y, a place for hearing (Lat. auditorium),
from ακροα, as in ακροαομαι και δικαστηριο-ν, « judgment-

The o of the second declension is replaced by 1, as
hall, fronm δικα, as in δικαζω,

Mas.
Fem. .

Nom. 2. nominat. είο-ν, neuter.

Τανταλ-μη-ς Τανταλ-ι-ς from Τανταλο Τανταλο-ς λογ-είο-ν, α speaking-place, frστη λογο, as in λογος

Κρον-ιδης

Κρονο Κρονο-ς. κουρι- είο-ν, α barber's shop και κουρευ και κουρευς

Only to (nom. 10.s) is changed into ia, as
Μουσείο-ν, Museum

Μουσα Μουσα
Substantives denoting quality are derived from adjective

Mas.
Fem.

Nom. siems by means of the following suffixes.

θεστ-ια-η-ς θεστ-ια-ς from Θεστιο Θεστιο-ς

Μενοιτ-ια-δης 1. τητ, nom. της, fem. (Lat. tat, tut, nom. tas, tus).

» Μενοιτιο Μενοιτιο-ς Stem Nom.

Adj. St. Nom, A less frequent suffix for patronymics is cov, nominative ταχυτη ταχυτης, quickness

ταχυ ταχυς

των; as Κρογιων, Son of Kρονο-ς. νεοτης, youth

νέος

Gentilia, or nouns denoting the gens or race, the country o: ισοτης, equality

the tribe, whence a person is sprung, have the suflixes, 2. συνα, notminat, συνη, fem.

1, ευ, nom. ευ-ς. δικαιο-συνη, justice adj. st. deraio

norm, δικαιος σωφροσυνη, 8ense σωφρον σωφρων Μεγαρ-ευ-ς from the noun st. Μεγαρο, nom, τα Μεγαρα, δίcgara.

Ερετρι-ευ-ς
3. ια, nominat. ια, fem.

Ερετριο Ερετρια.
σοφ-ια, οίκdom
adj. st. σοφο nom. σοφος

Feminine gentilia end in 8 (nom. g), as
ευδαιμον-ια, happiness ευδαιμον ευδαιμων,

Μεγαριο 1om. Μεγαρ-ις, α woman of Megara. The suffix ia with the vowel of the adjective stem becomes Σικελιωτιδ Σικελιώτις

of Sicily. εια and oια:

2. τα, nom. της. αλεθεια (αλεθεια), truth adj. εt. αλεθε(c) nom, αληθης ευνο-ια (ευνοια), benevolence

Τεγεά-τη-ς (Τεγεα), Αιγινη-τη-ς (Αιγινη), Ηπειρω-τηευνους

(Ηπειρος), Σικελιω-τη-ς (Σικελια). Diminutives, or words denoting the quality in a less degree, are forned from nouns as stems by means of these suffixes :

The most important sufixes for the formation of adjectives are these :

ειο,

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νεο-τηισότητ

ισο

ισος

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S

Αλεξανδρος ο Φιλιππου Φωκιωνι μονω, φασι, το Αθηναιων
Suffixes forming Adjectives.

στρατηγό γραφων προςετιθει το χαιρειν. Αξιουντος δε του
1. ιο, nom, ιο-ς,

Αντιπατρου ποιήσαι τι των μη δικαιων αυτον, Ου δυνασαι, expresses in the most general way the idea involved in the ειπεν, Αντιπατρε, και φιλο Φωκιωνι χρήσθαι και κολακι. Μετά

δε την Αντιπατρου τελευτην δημοκρατιας Αθηναιοις γενομενης noun from which the adjective comes, as

κατεγνωσθη θανατος του Φωκιωνος εν εκκλησια και των φιλων ουραν-10-ς, heavenly, from the noun ουρανος, heaven

οι μεν ουν αλλοι κλαιοντες ηγoντo, το δε Φωκιωνι σιωπή βαδιεσπερ-ο-ς, δείonging to the evening, from the noun εσπερα (Lat. ζoντι των εχθρών τις ενεπτυσεν απαντησας εις το προσωπον. vesper), evening,

ο δε προς τους αρχοντας αποβλεψας, Ου παυσει τις, ειπε, With the Towel of a vowel-stem the 1sometimes grows into a τούτον ασχημονουντα ; Ηδη δε της κυλικος αυτω προςφερομενης, diphthong; as

ερωτηθεις, ει τι λεγει προς τον υίον, Εγω σοι, ειπεν, εντελλομαι αγορα, α market place αγορα-ιος (αγοραίος)

και παρακαλώ μηδεν Αθηναιοις μνησικακειν. Των δε μελλοναιδο (nom, αιδως), modesty αιδο-to-ς (αιδοίος).

των συναποθνησκειν ενος οδυρονενου και αγανακτούντος, θυκ

αγαπάς, ειπεν, ώ θουδιππε, μετα Φωκιωνος αποθανουμενος. A similiar result ensues after the dropping of the of the stem;

VOCABULARY AND REMARKS. stem θερες (θερος, Summer), θερε; θερε-10-ς (θερειος).

Εμπιπτω, I fall on ;

“ We have fallen on the enemy; why By appending to you also form adjectives from adjectives as may we not rather say that the enemy have fallen on us ?" stems;

Fallen on here denotes met with, implying a subject of con

gratulation. . ελευθερ-to-c, liberal from ελευθερ (ελευθερ-ο-ς, free);

Δωρεαν, ας α gift (from a noun not in use δωρεα, ας, ή, also gentile adjectives from names of places thus

a gift).

Καγαθον, that is, και αγαθον, honourable and good. from Mιλητ-ο-ς comes Μιλησ-to-ς (instead of Mιλητ-ιος).

Ουκουν, therefore.
Αθηναι
Αθηνα-to

Kai dovkalv kal, etc., both to appear and to be so, that is,
2. κo, nom. κο-ς.

honourable and good.

'O din nou, the noun vioc is understood, Alexander, the son which is generally appended to the stem by means of 1, and in of Philip. words derived from verbal stems, signifies fitness :

Κυλιξ, κυλικος, ή, a bout, the bowl of poison.

Το from apx (aoxw) comes

χαίρειν, the customary salutation, the complimentary words, apx-c-50-s, fit for governing. αισθητ (αισθανομαι) ασθητ-ι-κος, susceptible of feeling

this comes greeting, equivalent to our “Dear sir,” or “ Your

respectful servant."
πρακτ (πραττω)
πρακτ-ι-κος, practical.

Αυτον, is the accusative case before ποιήσαι, which depends
From nouns as stems are formed adjectives which denote

on αξιουντος. the peculiar quality of the noun, as

Kai oily, etc., you cannot use (have) Phocion as at once a

friend and a flatterer. βασιλι-κο-ς, kingly (βασιλευς, α king) ιατρ-ι-κος (ιατρος, ο Karayıyvoorw, I condemn; sentence of death was pronounced physician).

against (with the gen.) Phocion and his friends. θηλυ-κο-ς (θήλυ, female), womanly, feminine.

Eu TTUW, I spit on; one of his (Phocion's) enemies, meeting him,

spat in his face. 4. εo non. εο-ς, ους (Lat. eu-s)

Ασχημονείν, to be ill-mannerly, to behave ill, unbecomingly ;

will no one cause that person to cease being ill-mannerly? indicate the stuff or substance of which a thing is made, as Ovk ayarós, etc., art thou not content, secing that thou sufferest λιθ-ινο-ς, stony from λιθο-ς, α stone

death with Phocion ! ξυλ-ινο-ς, wooden

What is the derivation of συστρατηγος ? εξιοντος ? παραινείν! ξυλο-ν, wood

κατεγνωσθη? συναποθνησκειν μνησικακειν ? χρυσούς, χρυσο-ς, gold (Latin aur

ureus), Give the parts (also go through them) of the following

verbs :-ειπείν); ηρώτησε και εασατω; φασι; πεμψαντος : προς 5, εντ, nom. mas, ει-ς, fem. εσσα, neut. εν,

ετιθει; δυνασαι ; ηγοντο; ενεπτυσεν και εμπεπτωκαμεν.

Explain the augment in these forrns :-ενεπτυσε (ν); προς ετιθει και κατεγνωσθη.

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3. ινo nom, ινο-ς.

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Xpvo-to-s, golden

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denotes fulness, as

Xapr-El-s, full of grace or beauty, from xapı-s, grace, beauty υλη-ει-ς, full of wood

υλη, α wood or forest αμαθο-εις, Sandy

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αμαθος, sand.

LESSONS IN ITALIAN GRAMMAR.-No. XXXIII.

HISTORICAL ANECDOTES.

IRREGULAR VERES. Πελοπιδας, και συστρατηγος Επαμεινονδου, της γυναικος επι

The Irregular Verbs are those which deviate in some tenses μαχην εξιοντος αυτου, διομενης σωζειν εαυτον. Αλλοις, εφη, which is given for their model.

and persons from the regular verb of the same conjugation, ειν τουτο παραινείν, αρχοντι δε και στρατηγω σωζειν τους The first irregular conjugation contains only andare, dire, πολίτας. Ειποντος δε τινος των στρατιωτων, Εμπεπτωκαμεν γάνε, διανε, and their derivatives. εις τους πολεμιoυς, Τί μάλλον, ειπεν, η εις ημας εκεινοι και

IRREGULAR VERBS OF THE FIRST CONJUGATION, Αλεξανδρου του βασιλεως εκατον ταλαντα δωρεαν Φωκιωνι των Αθηναιω πεμψαντος, ηρώτησε τους κομιζοντας, τί δηποτε

[A3 all Italian verbs may be generally conjugated with or without per. πολλων οντων Αθηναίων, αυτω μονο ταύτα διδωσιν Αλεξανδρος | iPeg Perb, Teeling confident that to student is tutoroυς Ασημαία:

fonal pronouns, we now think proper omit them in the conjugation of the ποντων δε εκεινων ως μονον αυτον ηγείται καλον καγαθον ne

tenses, which the reader now will be easily able to form and conjugate for ειναι, Ουκούν, εφη, εασατω με και δοκείν και είναι τοιούτον. himself.]

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Indeterminate Preterite. dard, he will give 1

Predi or detti, I gave

darémo, we will give

daréte, you will give
disti, thou garest
sreds, dette or die, he gave

daranno, they will give
IN *

dem, we gave

Conditional Present.
dest, you gave

Darii, I should or would
dia , dicrono or dié. give
*
Peso hare

dun,
* they gave

darésti, thon wouldst give
Fature.

darebbe, he would give Hann var reci damely linkund: ese sentido ami r.

darémmo, we would give
24 Aug cene

Ders. I ses or will give dareste, you would give
arehe ni gire

darebbero, they would give
wat 'ondensapade

IVRATITI Moon. อด 8 8 คู่คองเชิงกางwข" 3 ,

1: F Person diamo, let us gire NAT

ore 23

date, give se or you) saved

kan tolke de 2h, let him give diano ox duens, let them give

SUBJUNCTIVE Moon.
Muro
Present.

Imperfect.

Che dia, that I may give
*****
model hall wr will go

Che dissi, that I might give
che dia or dii, that thou mayst | che dessi, that thou mightst
*** in de thuu wilt go

give
**** hu will go

gire
che dia, that he may give che desse, that be sight give
*** toets, we will go

che diumo, that we may give
you will go

che déssimo, that we might gire 1***

che diate, that you may give inuiranno, they will go

che diste, that you might give che diano, dieno, déano, that che dessera, that they might

Conditional Present. 13,4, he we going

they may give

give
Andréi, I should or would

So conjugate-
It's his in, they were going

andresti, thou wouldst go Ridare, to give again. Addáre or addársi, to apply one's self.
Turulmate liretorite,

andrebbe, he would go
Iubit, I went

andremmo, he would go
andréste, you would go

III.
andrebbero, they would go

Fáre, to make.
IMPERATIVE Mood.

INFINITIVE Mood.
Ivy wal Percon.)

andiamo, let us go
andate, go (ye or you)

Simple Tenses.

Compound Tenses.
vadano, let them go
Present: fáre, to make Past : avere fatto, to have

made
SUBJUNCTIVE Mood.

Present Gerund: facéndo, Past Gerund: avendo fútlo,
Present.

Imperfect.
making

having made
Chuda, that I may go Che andassi, that I might go Past Participle: fatto, made
st: ud or cudi, ihat thou che andussi, that thou mightst

во
1. 1. 1 1. may go
che andisse, that he might go

INDICATIVE Moop.
# 1 thai, tut we may go che andassimo, that we might go

Present.
in 1,4 km; 1t you may go
che anda ste, that you might go Fo or fáccio, I make

féce, fe, or féo, he made

facemmo, we made
11, tranki, u at they may go
che andassero, that they might fái, thou makest

facéste, you made
go
fa, he makes

fécero or fénno, they made
So conjugate-
facciamo, we make

Future.
fate, you make
Riandáre, to go again.

Farò, I shall or will make
fanno, they make

fardi, thou wilt make II,

Imperfect.

farà, he will make Faceva, facea, fè, I was making faréte

, you will make

farémo, we will make Dure, to give.

facévi, thou wast making INFINITIVE Mood.

faceva or facéa, he was making faranno, they will make facevamo, we were making

Conditional Present. simple Tenses,

Compound Tenses. facevate, you were making Faré, I should or would siebent: dire, to give Past: avere dáto, to have given facevano or facéano, they were make

making

farésti, thou wouldst make
Prezent Gerund: dindo, giv- Past Gerund: avendo dato,
ing

having given
Indeterminate Preterite.

farebbe, he would make

farémmo, we would make Put Participle : déto, given

Féci or féi, I made

faréste, you would make
facésti, thou madest

faribbero, they would make
INDICATIVE Mood.
Present.
Imperfect.

IMPERATIVE Mood.
Dáva, I was giving

(No First Person.]

facciamo, let us make
veet
dúvi, thou wast giving
Fa, make (thou)

fáte, make (ye or you)
dita, he was giving
fáccia, let him make

facciano, let them make
duodno, we were giving

daxote, you were giving
y one
divuno, they were giving

Diét, diéro, dénno, are used in poetry.

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SUBJUNCTIVE Mood,

IRREGULAR VERBS OF THE SECOND CONJUGATION.
Present.

Imperfect.

Verbs ending in ere are of two sorts. The first have their Che faccia, that I may make Che facessi, that I might make infinitives long, such as bére, cadére, etc.; the second short, che faccia or facci, that thou che facéssi, that thou mightst such as assórbere, conoscere, etc. make

1st. Irregular verbs ending in ére long. che faccia, that he may make che facesse, that he might make che facciamo, that we may che facéssimo, that we might

1. make

make che facciate, that you may make che facéste, that you might make

Bére, to drink. che facciano, that they may che facéssero, that they might

INFINITIVE Mood. make

make So conjugate

Simple Tenses.

Compound Tenses.

Present: bére, to drink Past : avere beúto, to have Assuefáre, to accustom Rifáre, to make up again

drunk Confáre, to become

Sfare, to undo
Contraffóre, to mimic

Sopraffäre, to ask too much Present Gerund : bevéndo, Past Gerund : avendo vento,
Disfáre, to undo
Soddisfare, to satisfy

drinking

having drunk
Liquefáre, to melt
Strafáre, to do too much

Past Participle: beuto, drunk
Misfáre, to do wrong
Stupefare, to stupefy

INDICATIVE Mood.
IV.

Present.

bévre, he drank Stáre, to stand.

Béo, I drink

beémmo, we drank INFINITIVE MOOD.

béi, thou drinkest

beéste, you drank bée, he drinks

bévvero, they drank Simple Tenses. Compound Tenses. beiamo, we drink

Future. Present: stáre, to stand Past : éssere stato, to have

beéte, you drink

berò, I shall or will drink
stood
bécno, they drink

berái, thou wilt drink Present Gerund : slándo,

berà, he will drink Past Gerund: essendo státo,

Imperfect.

berémo, we will drink standing

having stood
Beéva, I was drinking,

beréte, you will drink Past Participle : státo, stood

beévi, thou wast drinking

beranno, they will drink

beéva, he was drinking INDICATIVE Mood.

beeramo, we were drinking

Conditional Present.

becvate, you were drinking Beréi, I should or would drink
Present.
stétte, he stood

beévano, they were drinking berésti, thou wouldst drink
Sto, I stand
stémmo, we stood

berébbe, he would drink stái, thou standest stéste, you stood

Indeterminate Preterite.

berémmo, we would drink sta, he stands stéttero, they stood Bévvi, I drank

beréste, you would drink stiamo, we stand

Future.
beésti, thou drankest

berébbero, they would drink
státe, you stand
stánno, they stand
Starò, I shall or will stand

IMPERATIVE MOOD, starái, thou wilt stand Imperfect. stard, he will stand

[No First Person. beidmo, let us drink Stáva, I was standing starémo, we will stand Béi, drink (thou)

beéte, drink (ye or you) stávi, thou wast standing staréte, you will stand béa, let him drink

béano, let them drink stáva, he was standing,

staránno, they will stand starámo, we were standing Conditional Present.

SUBJUNCTIVE Mood. staváte, you were standing Staréi, I should or would stand

Present.

che beéssi, that thou mightst stávano, they were standing starésti, thou wouldst stand

Che béa, that I may drink

drink Indeterminate Preterite. starébbe, he would stand

che béa, that thou mayst drink i che beésse, that he might drink starémmo, we would stand Stétti, I stood

che béa, that he may drink

che beéssimo, that we might staréste, you would stand

drink stésti, thou stoodest

che beidmo, that we may drink
starébbero, they would stand
che beidte, that you may drink che beeste, that you might

drink
IMPERATIVE MOOD.

che béano, that they may drink

che beéssero, that they might (No First Person.]

Imperfect.

drink
stiámo, let us stand
státe, stand (ye or you)

Che beéssi, that I might drink stia or stie, let him stand stiano or stieno, let them stand

So conjugate-
SUBJUNCTIVE MOOD.

Imbére, to imbibe

| Ribére, to drink again

Strabére, to drink hard,
Present.
Imperfect.

The Italians prefer the regular verb bévere,
Che stia, that I may stand

Che stessi, that I might stand i che stia or stii, that thou mayst | che stéssi, that thou mightst; stand

II. stand che stia, that he may stand che stésse, that he might stand

Cadére, to fall. che stiámo, that we may stand che stéssimo, that we might che stiáte, that you may stand stand

INFINITIVE Mood. che stiano or stieno, that they che stéste, that you might stand may stand che stéssero, that they might

Simple Tenses.

Compound Tenses.
stand
Present: cadere, to fall Past: essere caduto, to have

fallen
So conjugate-

Present Gerund: cadéndo, fall. | Past Gerund: essendo caduto,
Distáre, to be distant
Ristáre, to stop

ing Instáre, to insist

having fallen Soprastúre, to differ

Past Participle: caduto, fallen

Sta, stand (thou)

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B

From the point a draw A E perpendicular to BC (I. 12). INDICATIVE Mood,

Then, because the angle A E B is a right angle (Cons.), therefore Present.

Future,

the square of A B is equal to the squares of B E and EA together Cádo or caggio, I fall Cadró or cadero, I shall or will

(I. 47).

For 'the same reason the square of A c is equal to the squares of cadi, thou tallest

A E and EC together. cade, he falls

cadrúi or caderái, thou wilt fall cadiamo or caggiumo, we fall

Therefore, adding equals to equals (I. Ax. 2), the squares of cadrà or cadera, he will fall

A B and Ac together are equal to the squares of B E and so cadéte, you fall

cadremo or caderémo, we will cudono or cággiono, they fall

together with twice the square of EA. fall

But because Bc is divided into two equal parts in D and two Imperfect. cadréte or caderéte, you will fall

unequal parts in E, therefore the squares of BE and BC are Cadeva or cadéa, I was falling

cadrunno or caderanno, they together equal to twice the square of B D together with twice the

will fall cadóvi, thou wast falling

square of d e (II. 9). cadéra or cadea, he was falling

Conditional Present. Therefore the squares of AB and Ac together are equal to cadevamo, we were falling Cadréi, or caderéi, caderia, ca

twice the squares of B D, D E, and E A together. cadevute, you were falling

dria, I should or would

But the squares of d e and Ea are together equal to the square cadévano or cadéano, they were full

of AD (I. 47), and the doubles of these equals are equal. Therefalling cadrésti or caderésti, thou

fore twice the squares of D E and E A are equal to twice the square Indeterminate Preterite,

wouldst fall

of Ad. Therefore the squares of A B and AC together are equal Caddi, cadetti, or cadei, I fell cadróbbe or caderebbe, caderia,

to twice the squares of BD and D A together, Q.E.D. cadésti, thou fellest

cadria, he would fall

Prop. B. Theorem. The squares of the two diagonals of a cadde, cadéo

, cadette, or cadè, cadremmo 'or caderémmo, we parallelogram are together equal to the squares of its four sides, he fell

would fall
cadémmo, we fell
cadréste or caderéste, you would

D
cadéste, you fell

fall
caddero, cadero, cader, cadettero, cadróbbero or caderebbero, cade-
cadérono, they fell

riano, they would fall
IMPERATIVE MOOD.
[No First Person.] cadiamo, let us fall
Cudi, iall (thou)

cadéte, sall (ye or you)
cada, let him fall
cadano, let them fall

Let ABCD be any parallelogram of which the diagonals are
SUBJUNCTIVE MOOD.

A C and B D cutting each other in E. Then the squares of a C and Present.

Imperfect.

BD together are equal to the squares of AB, BC, CD, and DA

together. By Exercise 2 to Proposition XXXIV. of the first book, Che cida or ciggia, that I may che cadossi, that I might fall fall che cadessi, that thou mightstof

which the side BD is bisected in e, therefore, by the last pro

the diagonals bisect each other. And because A B D is a triangle che cúda or ciggia, that thou fall che cadesse that he might fall position, the squares of A B and AD together are equal to twice

the squares of B E and A E together, che cada or cággia, that he may che cadessimo, that we might

For the same reason the squares of BC and cd are together fall

fall che cadiamo or caggiamo, that I che cadiste, that you might adding equals to equals, the squares of A B, BC, CD, and Da

equal to twice the squares of B E and E c together. Therefore, we may fall

fall
che cadicte or caggiátc, that you che cadeissero, that they might twice the squares of A z and E C (I. Ax. I. 2).

together are equal to four times the square of BE together with fall

But A e is equal to Eo, because the diagonals bisect each other, che cúdano or cággiano, that

and therefore the square of A E is equal to the square of so, they may fall

Therefore the squares of a B, BC, CD, and D A together are equal So conjugate

to four times the squares of B E and A 2 together.

But the square of any straight line is equal to four times the
Accadere, to happen

Discadere, to fall away &quare of half the line (II. 4, Cor. 2). Therefore the square of
Decadere, to decay
Ricadére, to fall again

BD is equal to four times the square of B E, and the square of BC
Scadere, to become due.

is equal to four times the square of A E.

Therefore the square of a B, BC, D, and D A are together equal to the squares of a C and B D together. Q.E.D.

Prop. c. Theorem. The squares of the four sides of a traSOLUTION OF EXERCISES TO THE SECOND pezium are together equal to the squares of its two diagonals and

four times the square of the straight line which joins the points BOOK OF EUCLID.

of the biscction of the diagonals.
Prop. A. Theorem. The squares of any two sides of a triangle
are together double of the squares of half the third side, and of
the straight line drawn from the opposite angle bisecting that
side.

с

mayst fall

may fall

А

D

А

F

E

C

B

B

D

E

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Let A B Co be any trapezium whose diagonals A c and B D are Let A B C be any triangle having one of its sides B o bisected bisected in the

points p and . Join er. Then the squares of a B, in D. Join A D. Then the squares of a s' and a c are together pc, cd, and D A are together equal to the squares of a and D

together with four times the square of E F. equal to twice the squares of B D and Ad together.

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