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Fut. ολ-ώ, -είς

2 Aor, ωλ-ομην

2. ζε-ννυμι, Ι οοίί (transitive), f, ζεσω, a. εζεσα, pf. m. or p. Middle.

εζεσμαι, a. p. εζεσθην (ζεω is commonly intransitive). Pres. σεεδα ννϋ μαι

3. σβε ννυμι, Ιαείinguish, f. σβεσω; aor. εσβεσα ; 2 Α, εσβην, κορέ-ννύμαι στρω-ννι-μαι

I went out, I was extinguished; pf. counka, I have been Imp. ε-σκεδα-γνύ-μην ε-κορε-γνύ-μην έ-στρω-γνύ μην put out : mid. σβεννυμαι, I go out ; pt. m. or p. εσβεσμαι; Pert. ε-σκεδα-σ-μαι κε-κορε-σ-μαι ε στρω μαι

3. p. εσβεσθην. There is no other vero in ννυμι, except Plpf. ε-σκεδα-σ-μην έ-κε κορε-σ-μην έ-στρω μην

this, with a Second Aorist,
Put,
,

4, στορε-γνύ-μι, 1 spread over, f. στορεσω, Attic, στορώ; 8.
κορέ-σομαι
Aor.

εστορεσα.
έ-κορέσαμην
3 Fut.
κε-κόρε - σομαι,

The stem ends in o, lengthened into wa
Passite.

1. ζω-ννυ μι, I gird, f. ξωσω; a, εζωσα ; mid. I gird myself, a.

εζωσαμην; pf. m. or p. εζωσμαι. Αστ. ε-σκεδα-σ-θην ε-κορε-σ-θην ε-στρω-θην

2. ρω ννυμι, I strengthen; f. ρωσω ; 8. ερρωσα ; pf. m, or p. Fut. σεεδα-σθησομαι

κορε-σ-θησομαι στρω-θησομαι ερρωμαι, (εόρωσο, vale, farewell); it. ερρωσθαι; a. p. Verb, adj. σκεδαστος, σκεδαστεως και κορεστος, κορεστεος και στρω

ερρωσθην. τος, στρωτέος.

3. στρω-ννυμι, Ιεpread out ; f. στρωσω; a. εστρωσα, etc., see

στορέννυμι. Another form of the Present and Imperfect is, σκεδα-ννυω. 4. χρω-ννυμι, I colour: f. χρωσω ; 3. έχρωσα ; pf. m. Or P. ε-σκεδα-ννυον: κορε-ννυω, ε-κορε-ννυον: στρωγνύω, ε-στρωvyvov: the v being always short.

κεχρωσμαι; Imp. εχρωσθην, 6. Verbs ucith a stem terminating in a consonant, as od-pl

ii. Verbs whose stem ends in a consonant and take vy. (ΟΛ.), I destroy, and νυ-μι (OM-), Iswear.

1. αγ-νύμι, 1 break; f. αξω: a. εαξα; inf. αξαι ; 2 pf. εάγα, I

λαve been broken; a. p. εάγην.
Pres. oλλυ μι (Lat. perdo) | ομ-νύμι: ομ-νύ-μαι
ολ-λύμαι (poreo)

2. είργ:νυμι («r είργω), I restrain, enclose; f. ειρξω; a. είρξα ;
Imp. ωλ-λυ-ν
ωμ-νυν: ωμ-νύ μην

8. μ. είχθην; μι. ιn, or p. είργμαι (but ειργω, ειρξω, ειρξα, 1Perf. ολ-ωλε-κα

I erclude)
(ΟΛΕΩ) ομ-ωμο-κα:

ομ-ωμο-σ-μαι,
perdidi
(ΟΜΟΩ)

3. ζευγ-νύμι, I joke. bind; f. ζευξω; a. εζευξα; mid. I oud 2 Perf, oλ -ωλα, per

Jor myself; a. έζευξαμην; pf. m. or p. εζευγμαι; a. p. 1 Plpf. ωλ ωλε-κειν, perdid- | ομ-ωμο- κέιν: ομωμο σ-μην

εζευχθην ; and iOre cornonly εζύγην.

4. μιγ-νύμι, I mis; f. μιξω ; Α, εμιξα, μιξαι; pf. μεμίχα; pf. 2 Plpf. A.W.-Elv perieram

11. Or μ. μεμιγμαι; 8. μ. εμιχθην, εμιγην; f. p. μιχθησομαι,

μιγησομαι, 3. 1. μεμιζομαι.
ολ-ούμαι, και
ομ-ούμαι, ή

5. οιγ-νυ μι, I open, com τηonly as a compound, ανοιγνυμι (for 1 Aor. ωλε-σα

which, however, ανοιγω is more usual in the present, and ωμο-σα : ωμο-σαμην

in the imperfect ανεωγον is always used); f. ανοιξω; a. 1 Α.Ρ. ωμο-σ-θην

ανεφξα, ανοίξαι ; 1 μι. ανεφχα, Ιhave opened; 2 pf. ανεφγα,

1 F. P. ομο-σθησομαι. I stand open (for which the Artic ανευγμαι); impf. ανεμOf the Present and Imperfect there is another form with the

γομην; 4. p. ανεφχθην, ανοιχθήναι. υ short, namely, oλλυ ω, ωλλυ-ον, ομνύω, ωμνυ- ον.

6, ομοργ-νυ-μι, I wipe off, f. oμoρξω; 8. ωμορζα; mid. I wipe

off for myself, f. oμoρξομαι και 8. ωμορξαμην ; 8. μ. present, the fir-c perfect, and the first pluperfect have a transitive signification ; this, pres. I destroy, i perf. I have

ωμορχθην. destroyed, 1 pipf. I had destroyed; and the 2 perf. and 2 plupt. 7. πηγνύ μι, Ifasten; f. πηξω, 8. επηξα; 1 pf. πεποχα, I have have an intransitive meaning, as 2 perf. I have perished (I am fastened; 2. pf. πεπηγα, I am fast; mid. πηγνύμαι, I relost

, it is all over wiih me), 2 plpf. I perished. i'ne middle of main Just; pt. πεπηγμαι, I am fast; a. p. επάγην; f. p. ολλυμι, nam» 1y, oλλυμαι, είgnities Ian perishing.

παγησομαι.
Ολ-λυ-μι has arisen by the assimilation of ν το λ, from 8. ρηγνυμι. Ιδreak, 1 tear, f. ρηξω; a. ερρηξα : 2 p. ερρωγα,
ολ-νυ-μι. Τhe perfect participle middle or passive of oμνυμι I am orchen; 8. η. ερρηξαμην, a. μ. ερράγην; f. p.

ομωμοσμενος. The other forms of the perfect and pluperfect ράγησομαι.
are in the Attic writers generally without the o, as Olla jotai,

VOCABULARY.
In particular instances belonging to this class of verðs tho stem ends, Avačevyvūpei

, I yoke. Λυχνος, ον, o, a torch, a light.

Διαρρηγνύμι, I break in | Αντισθενης, ους, ό. Αntisthenes i, in a vowel, and takes yvū.

pieces. .

Ξενοφανης, ους, ό,Χenophanes.

Συμπηγνύμι, I fasten together. | Φυσημα, ατος, τo, breath. The stem ends in a

Αποσβεννύμι, I put out, ex. Φως, φωτος, τo, light.

tinguish (the prepos. gives | Μαλθακος, η, ον, weak, soft, 1. κερα-ννύμι, I mis, f, κεράσω, Attie κερω ; 8. εκεράσα ; pf. force).

del cate, deceitful. κεκράκα; mid. I mia for myself, a. εκερασαμην; pf. ηuid. | Εξολλύμι, I destroy utterly. Ολυμπιος, α, ον, Olympic, or pass, κεκράμαι ; 8. p. εκράθην (by metathesis), also

Ανακαιω, I set on fire.

subst. an Olympic combaεκερασθην.

Eγκαλλωπιζομαι, I adorn my2, κρεμα ννύμι. I hang, f. κρεμάσω, Att, κρεμώ; a. εκρεμάσα ; deli, I make a display. Ψευδορκος, ον, perjured, το nd, or pass. Kpepavvuar, I hang myself or am hanged, bul Kwocłw, I talk, chutter.

ψευδορκον, perjury. κρεμάμαι, I hang; f. μ. κρεμασθήσομαι, a. εκρεμασθην, 1 Στυγεω, I hate.

Αηδης, ες, unpleasant, diswas hanged or I hung (intrans.)

Βδελυγμια, ή, disgust.

pleasing, repulsive.
3. πετα-ννύμι, 1 spread out, I open , f. πετάσω, Att, πετώ;

EXERCISES. -GRIEK- ENGLISH,
pf. m, or p. πεπταμαι (oy syncope), a. p. επετασθην.
The stem ends in

Τών βρωματων τα ήδιστα, εαν τις προσφερη πριν επιθυμειν,

o-
1. εννώ με, I clothe, in prose αμφιέννυμι; impf. αμφιένυον,

αηδη φαινεται, κεκορεσμενοις δε βδελυγμιαν παρεχει. Τω αυτό Thout augment ; . αμφιεσώ Αεο αμφιώ, Αμφιτα; | φυσηματι το μεν πόρ ανακαυσειας αν, το δε του λυχνου φώς E. A wanting ; pt. m. ο. ημφιεσμαι, ήμφιεσαι, ημφέσται, αποσβεσειας. Η ύβρις πολλα ηδη των ανθρωπων απωλεσεν etc., 1af. ήμφιεσθαι; f. m. αμφιεσομαι, Αττίς αμφιούμαι.

εργά. Ει μη φυλαξεις μικρή, απολείς τα μειζόνα. Οι πολεμιοι

The

IIT

ομωμοτο.

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ωμοσαν τας συνθηκας φυλάξαι. Ξενοφανης ελεγε την γήν εξ append the person-endings immediately to the stem. A11

other parts of these verbs, however, follow the formations
αερος και πυρος συμπαγήναι. Σωκρατης, ιδων Αντισθενη το
διερόωγος ιματιου μερος αει ποιουντα φανερον: Ου παυση, εφη.

The formation of this second aorist active corresponds in all
εγκαλλωπιζομενος ημιν ; Ψευδορκον στυγεί θεος όςτις ομείται. | the moods and the participle, το that of the second aorist
Ζευς ανέρ' εξολεσει Ολυμπιος, ός τον έταιρον μαλθακα κωτιλλων | active of the verbs in μι. Τhe characteristic vowel is length.
εξαπατάν εθελει.

ened throughout, as in xorny, ă and E being changed into n, o

into w, and ŭ into ū; and so remains, as in cotny, through all ENGLISH-GREEK,

the indicative, imperative, and infinitive. The imperative ter

mination, nou, in verbs with a for the characteristic vowel He mises. We mix. They mix. It was mixed. It has is abridged into å, as apopå instead of apoßnoi. been mixed. Tnry hang. They will hang. They have hung.

Char, a

Char, ε Char. o
I spread out. Thou wilt spread out. He has clad. They

Char, και
have clad. I shall clothe inyself. We shall clothe ourselves.

ΒΑ-Ω

ΣΒΕ-Ω ΓΝΟ-Ω
He boils. They will boil. He extinguishes. I am extin.

βαινω
σβεννυμι γιγνώ

δυ-ω
guished. Thou wilt gird. I shall strengthen. They will

I step, I go.

I put out.
I learn.

I cover.
strengthen. I have been strengthened. They have coloured.
He will colour. He is broken. They will break. They will

Indicative. .
yoke. He has yoked. I have been voked. He opens. They
open. Thou hast opened. He will wipe away. He has

I went. I went out. . I learnt.

I went into. wiped away. Thou wilt fasten. He will fasten. I am broken.

εσβην εγνων

εδυν O boy, boil the water. The boy cannot boil the water. The S. 1.8.Bn-v

2. ε-βης
boss boiled the water. The garment is torn. The doors stand

εσβης
έγνως

εούς
(are) open.
The wine will be mixed with water. The 3. <-311

εσβη
εγνω

εού
cup is broken. The light is put out. They put out the light. D. 2. ε-βη-τον εσβητον εγνωτον

εδυτον I swear not (subj. aor.) rashly. Pride will destroy you all

3. ε-βη-την

εσβητην

εγνωτην εούτην
Put out (aor ) the light. The women in grief (grieving) tore

Ρ.1.ε-βη-μεν
εσβημεν έγνωμεν

εδυμεν
(aor, mid.) their garments.

2. ε-βη-τε εσβητε εγνωτε

EO UTE 3. ε-βησαν εσβησαν

έγνωσαν Inflexions of the two Present perfect forins, krīļai, I lie, and

εδυσαν
ήμαι, Isit.

Subjunctive.
Ktipal, the perf. mid, or pass. means properly I have laid
myself, and so, as the result of so doing, I lie. Keijat,

S. βω, βης, β) σβω, ης, ή γνώ, γνως, γνω δυω, ης, η then, is perfect in form and present in signification. D. βήτον

σβήτον γνώτον δυητον Perf. Ιnd. κείμαι, κείσαι, κείται, κείμεθα, κείσθε, κείνται; subj. P. βωμεν, ήτε, ώσι σβωμεν, etc. γνώμεν, etc.

δυωμεν, etc. κείωμαι, κεφ, κεηται, etc.; impr. κείσο, κεισθω, etc.; inf. κείσθαι, part, κειμενος.

Optatire.

. Plpf. Ιnd. εκειμην, εκεΐσο, εκείτο, 3 pl. εκειντο; opt. κεoιμην,

S. 1. βαιην

σβεην

γνοιαν κέoιο, κεοιτο, etc.

2. βαιης

σβειης γνοιας Fut. κεισομαι

3. βαιη

σβειη

γνοια Compounds. ανακειμαι, κατακειμαι, κατακεισαι, etc.,

inf. κατακείσθαι; impr. κατακεισο, εγκεισο.

D. 2. βαιητον σβειητον γνοιητον

and αίτoν and είτον and oίτον Ημαι, I sit.

3. βαιητην σβειητην γνοιητην

and oιτην
Huai, a perfect passive form, signifying I have seated myself,

or I have been sealed, and so coming to signify I sit. P. 1. Bainjuev σβειημεν γνοιημεν
The stem is ΗΔ (compare ήσ-ται instead of ήδ-ται, and

and αίμεν

and είμεν

and oίμεν
the Latin sed-to, where the 8 represents the aspirate in 2. βαιητε σβειητε γνοιητε
'HA; compare also the Greek jopa, a srat)

and aite

and cire

and oire
Perf. Ιnd. ήμαι. ήσαι, ήσται, ήμεθα, ήσθε, ήνται; impr. ήσο, 3. βαιεν

σβειεν γνοιεν
ήσθω, etc.; inf. ήσθαι; part. ήμενος.
Pipf. ήμην, ήσο, ήτο, ήμεθα, ήσθε, ήντο.

Imperatiro. .
S. βήθι, βητω

σβήθι, ητω
both verbs the pluperfect is equivalent to the

EUTE imperfect.

P. 2. Bijte

σβητε Instead of the simple ήμαι in prose, we find καθημαι,

3. βητωσαν

σβητωσαν

γνωτωσαν the inflexions of which vary in this, that in the

and βαντων

and σβεντων and γνoντων and δυντων 3ril person singular of the perfect, it never takes o, and in the pluperfect only then when it has

Infinitire. the temporal argument.

δύναι βήναι

σβήναι
Perf. καθημαι, καθησαι, καθηται : sub, καθώμαι, καθή, καθήται;

Participle.
in pr. καθησο; Inf. καθήσθαι; p. καθημενος.
Plpf. εκαθημην and καθημην, εκανησο And καθησο, επαθητo and

βας, ασα, αν σβεις, είσα, εν γνους, ούσα, ον δυς, ύσα, ύν καθηστο; opt. καθoιμην, καθοιο, καθοτο.

G. βαντος

G. σβεντος

G.

γνοντος The forms wanting in ήμαι are made up by έζεσθαι or

The subjunctive Bw. Bys, Bö, etc., appears in the compounds, ίζεσθαι (prose καθεζεσθαι, καθιζεσθαι).

as, αναβώ, αναβής αναβ, etc. και αποσβώ, διαγνώ, αναδυω.

The imperative Bnei, ßntw, etc, appears in the compounds Verbs in w which in the Second Aorist Active and Middle follow

ανάβηθι αναβά, αναβητε; αποσβηθι; διαγνωθι; αναδύθι. the Analogy of the Verbs in fllo

In ordinary style the second aorist middle is found in rery

few verbs, as πετομαι, I ft); πριασθαι, I purchase. Several verbs, having the characteristics a, e, 0, v, form a

Besides the verbs just adduced, some others follow this active after the analogy of the forma- formation, ,

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and αιτην

and ειτην

γνώθι, γνωτω δύθι, δύτω

As the perfect form has a present meaning, so in Β. Βήτον βηνων σβήτον, ητων γνωτον, ωτων δυτον, ύτων

γνώτε

δυτωσαν

γνωναι

G. δυντος

tions in op, since those tense of want the need towel, anal 1. Biopeix», commonly arodvopaoxw (with acc.), I run eving

τον.

from ; α. (ΔΡΑ) απεδράν, ας, α, αμεν, ατε, άσαν; subj. | Εμπιπτω (with dat. or with | Πτερυξ, υγος, ή, a wing. αποδρώ. δράς, δρά. δρώμεν, δράτε, δρώσι(ν): opt δραιών; els and accus.), I fall into. Συλλας, α, ο, S, lla,

imp. αποδραθι, ατω : inf. αποδράναι, part. δράς, ασα, αν. Tηκω, Ι ιnelt (transitive); mid. | Νεκρος, α, ον, dead ; ο νεκρος, 2. πετομαι, I ty; a. (ΠΤΑ) επτην; imp. πτηναι ; p. πτας, a.

aor. and 2 iut. pass., I melt a corpse, m, επταμην, inf. πτασθαι (by syncope).

(intransitive), I waste awav. Πτερον, ου, τo, a wing. 3. σκελλω or σκελεω, I dry, I dry up (hence our oλoleton). | Εμπιπλημι (τι τινος), I full | Πελαγος, ου, τo, the sea. 2. (ΣΚΛΑ) εσκλην, I am dried up; inf. σκλήναι; opt. | Δαιδαλος, ου, ο, Daedalus.

Προσθετος, η, ον, added (arti

ficielly), adventitious, arti. σκλαιην.

Ικαρος, ου, ο, Icarus.

ficial. 4, φθα-νω (with acc.), I get before, I anticipate ; 8. εφθην, φθω.

Κηρος, ου, o, wax.

Αχρηστος, ον, useless. φθαιην, φθήναι, φθας.

Αίμα, ατος, τo, blood.

οξυς. εια, υ, quick, feet, ex: 5. καιω,

Iourn (transit.); a. (ΚΑΕ) εκαην, 1 burnt (intrans.), Κεραμεικος, ό, the pottery quisite. but I aorist ekavoa (transit.), I set on fire.

inarket in Athens; here, a Aκολαστως, unrestrainedly. 6. ρεω, I low ; a. (PYE), ερρυην, I flowed; f. ρυησομαι; pf. place out of the city where | 'QOTE, so that (with infin. and ερρυηκα.

were buried those who fell indicat.)

in war. 7. χαιρω, Irojoice; a. (ΧΑΡΕ) εχαρην. 8. αλισκομαι, I am taken, caught; a. (ΑΛΟ) ηλων and εάλων.

EXERCISES. -GREEK-ENGLISH, 9. βιοω, I live; a. εβιων; subj. βιώ, ώς, ω, etc. ; opt. βιωην (not βιoιην, as γνoιην, το distinguish this part from the

Οι ανθρωποι την αληθειαν γνώναι σπευδουσιν. Γνώθι σεαυoptative impf. βιoιην); inf. βιώναι; part. βιους, ούσα (the

Γνώναι χαλεπον (sc. εστι) μετρον. Η πολις εκινδυνευσεν neuter does not occur); the cases, however, are supplied by the 1 aor. βιωσας (so, ανεβιων, I lived again, from υπο των πολεμιων αλώναι. Φεύγε τους ακολαστως βιωσαντας. αναβιωσκομαι). The present and innperfect are little used Συγγνωθι μοι, ω πατερ. Εις τι καταδυς του κοσμου μερος και by the Attics, instead of which they employ ζω (ζαω), αποκρυψας σεαυτον πιστευεις, οτι τον θεον αποπεφευγας; Λιμος which on the other hand borrows the remaining tenses | μεγιστον αλγος ανθρωποις εφυ. Οξεία ήδονη παραπτάσα φθανει. from βιοω, thus pres. ζω, impf. εζων, fut. βιωσομαι, a. | δουλος ελαθεν αποδρας τον δεσποτην. Οι στρατηγοι εγνωσαν εβιων, pf. βεβιωκα, pf. p. βεβιωται, part. βεβιωμενος.

τους πολιταις βοηθειν. Μηποτε σεαυτον απογνώς. Δαιδαλος 10. qvw, I bring forth; 2 aor. egūr, I arose, came into being;

ρύναι, φυς, subj. φυω (no op. in Attic), 1 aor. εφύσα, 1 ποιησας πτερυγας προςθετας εξεπτη μετα του Ικαρου. Συλλας brought forth; t. φυσω, I shall bring forth. The perfect | ενεπλησε την πολιν φονου και νεκρών, ώστε τον Κεραμεικον αιματι πεφύκα, I have come into being, I have beco:ne, is also ρυήναι. Ικαρος, ο Λαιδαλου υίος, τακεντος του κηρού και των intransitive. The mid. pres φυομαι, f. φυσομαι.

πτερών περιρρυεντων, εις το πελαγυς ενεπεσεν. Οι πολεμιοι Particular attention must be paid to a verb of frequent την γήν τεμοντες απεβησαν. Αχρηστον (sc. εστι) προειδεναι occurrence, namely, olda (stem EIA, vid-co in Latin), I know. τα μελλοντα. Οι αγαθοι παντων μετρον ισασιν εχειν. Πολλοι

ανθρωποι ουτε δικας δεσαν, ουτε νομους.
Perfect.
Ιnd. S. 1, οιδα
Subj. ειδώ Impr. . Infnit.

ENGLISH - GREEκ. 2. φισθα

ειδής
ισθι ειδεναι

They know. They knew. He may know. Thou knewest. ειδη

He became. The woman will bring forth a child. They may ειδήτον

Particip. . live. I have lived. He is dried up. He went. They knew ειδητον

ELCws, via, os (resolved). He sits. They sit. Thou liest. They lie. He will ειδωμεν

lie. He runs away. Go away, 0, boy. The whole city flowed ειδήτε

with blood. The bird flew away. The general determined

to succour the city. The father pardoned the son. Do not ειδωσι

live without restraint! The city was taken by the enemy. Pluperfect.

Let us not despair. The slave ran away from his master. Dual Plural φδειμεν

ιστω

ιστον

ιστων

3. οιδε(ν) D. 2. ιστον

3. ιστον Ρ,1. ισμεν

2. ιστε 3. ισάσι(ν)

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Ιnd. 8. 1. φδειν,

Attic pón

2. μδεις and φδειτον δειτε

LESSONS IN ALGEBRA.-No. XX. ηδεισθα or

(Continued from page 290.) ηδησθα 3. δει οι ηδη φδειτην μεσαν

ADDITION OF POWERS. Opt. 8. ειδειην, ης, η ειδειητον, η την ειειημεν, ητε,

It is obvious that powers may be added, like other quanti. ειδειεν

ties, by writing them one after another, with their signs. Fut. εισομαι, I shall know or experience. Of oda there is

1. Thus the sum of a3 and b, is a3 + b?. this compound, συνοιδα. I am conscious, inf. συνει- 2. And the sum of a? bn and hó - dt, is a? - buto 13 d. δεναι, imp. συνισθι, subj. συνειδώ, etc.

The same powers of the same letters are like quantities, hence The

present flow (not in use) has the force of I see, so ciòov, their co-efficients may be added or subtracted. I saw; and as the perfect denotes the result of the present act,

3. Thus the sum of 2a2 and 3a", is ba?. and as the result of seeing is knowledge, on the perfect oioa

4
6

6 comes to signify I know; the pluperfect accordingly means I

7

8 To -36y 33m

-y

-5a350 3(a+)η Add -22"ys 65m -7α'ya

6a3/06 4(a+) VOCABULARY.

But powers of different letters, and different powers of the 1

Παραπετομαι, I fy away. same letter, are unlike quantities, hence they can be added only Απογιγνωσκω, Irefuse consent | Περιρρεω, I low round. by writing them down with their signs. 1o, I reject ; εμαυτον, I con. Προσιδα, I foreknow.

9. The sum of a’ and a3, is a + a3. aign or resign mrself, and A ROKOUT TW, I conceal.

It is evident that the square cfa, and the cube of a, are give myself up to; I doubt. Bondex, I hasten to help, I neither iwice the square of a, por ewice the cube of a. Συγγιγνωσω, I pardon, Εκπιτομαι, I Hy out, I ly succour (compare the French

10. The sum of us and βαθύ6, is αδη + 3αθή8. aux secours).

From the preceding principles We deduce the following

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15. Mult. y.n into y-into y-*. GENERAL RULE FOR ADDING POWERS,

16. Mult, a.? into G} into a s.

17. Mult, a-2 into 2-3 inta aus. If the porers are like quantities, add their co-effictents, and to

18. Mult, an into aminti ala. the sum annex the common letter or letters with their given indices.

19. Mult. y.? into y2 into — yo* y*. If the powers are unlike quantities, they must be added by writing 20. If a + b be multiplied into a b, the product will be them, one after another, without altering their signs.

am b", that is, 11. Add 5x(a - b)3 + xfa--6)3 in 2x(a - b)3 + 10x(1 - 0)3. The product of the sum and difference of two quantities, is equal 12. Add 3(x + y) + 5a: - 4(x + y)* to 10a + 6(x + y)! to the difference of their squares. 13. Add a b2 + xy + a band -x*yt ta'bs.

This is an instance of the facility with which general truths 14. Add 5a-bc, 3a-bel, abe and 2a2bc*.

are demonstrated in algebra. 15. Add 3a3 + bc? + 5a' t- 2be and a3 + 5bc2 to 6a3 + 2bc. If the sum and difference of the squares be multiplied, the 16. Add }(zy - cm), 3(xy - em), * (xy - em)and product will be equal to the difference of the fourth powers, $(xy - cm).

etc.

21. Mult. (a −y) into (a +y). SUBTRACTION OF POWERS,

22. Muli, (22 - y2) into (a1 + y). Roll.-- Subtraction of powers is performed in the same manner

23. Mult. (at -- y) into (ar + v). as addition, except that the signs of the subtrahend must be changed

24. Mult, at art ac into a? - 1, 48 in simple subtraction.

25. Mult. 3a(x2 – ) into 2a(x?

26. Mult. (az + 63)3 into da? + 63) 1, From 2at take -6a, Ans, 8a",

27. Mult, a 62 into a3 + b?.
2. From -3) 3. 37.66 4. aiba 6. 6(@ - A) 28. Mult. *3 + xy + xy? t gå into a ty
Take 4加

4208
abo 24 --- h) 29. Mult, at

2a3b + 4a26 8ab3 + '166+ into a it 25. 6. From 69(a + b)* take a(a + b).

30. Mult, a? + činio a 7. From 17a_x3 + 5xy take 12a2} 4zy. 8. From 323(67 8): take a3 (62 – 8).

DIVISION OF POWERS. 9. From n°63 + x3yd take asbe -- x'y? 10. From 6213 y)- 3(a? — b)s take 348 - 63)3 to 4 Powers may be divided, like other quantities, by rejecting (x3 + y)

from the dividend a factor equal to the divisor; or by placing 11. From 2xín -b): + 3(2-=)) take 3(a --b) + 3a =0)3. the divisor under the dividend, in the forni of a fraction. 12. From $(x + y)2 + 3(a + b)3 take ili++ ila +593 Thus the quotient of cuba divided by b2, is a'.

1.
2,

3.
MULTIPLICATION OF POWERS.

Div. Da’y! 12631 a) + 3ary dx (

at Powers may be multiplied, like other quantities, by writing By --32 263

(a+y) the factors one after another, either with, or without, the sign

as of multiplication between them.

5. The quotient of as divided by as, is But this is equal

as 1. The product of as into b? is a3b2; and x3 into am is amg3. 2. Mult. Men 3. Babya 4, dhe 5. aby?

to a?. For in the series Into

2x
4 by
a3b-y

a', as, as, a', a, a-, a-, a-, a-, etc., If the quantities to be multiplied are powers of the same root, if any term be divided by another, the index of the quotient instead or writing the factors one after another, as in the last will be equal to the difference between the index of the diviarticle, we may add their exponents, and he sum placed at the end and that of the cirvisur. right hand of the root will be the product required. The reason of this operation may be illustra ed thus:

Thus as a3 = Sa?. And am a ? x 43 is a?a3. But a? = aa, and a3 = aan,

And aa x aga

Hence we deduce the following = aaaaa = a'. The sum of the exponents 2 + 3, is also b. So dn X du = dmta.

GENERAL RULE FOR DIVIDING POWERS. N.B. The same principles hold true in all other powers of

A power may be divided by another power of the same root boy the same root.

subtracting the index nf the divisor from that of the diridend. Hence we deduce the following

If the divisor and dividend have co-efficients, the co-efficient of the

dividend must be divided by that of the divisor. GENERAL RULE FOR MULTIPLYING POWERS. if the divisor and dividend are both compound quantities, the

terms must be arranged, and the operation conducted in the same Powers of the same root may be multiplied by adding their expo- munnes, as in simple division of compound quantities, nents.

If the powers have co-efficients, these must be multiplied together, and their product prefired to the common letter or letters.

Thus y + y =y=y! That is,

уу Powers of different roots are multiplied by uriting them one after

7. Divide at å by a.

8. Divide to by ** another, either with, or without, the sign of multiplication between

9.
10.
11.
12.

13. them.

Divide wit

84***

12(6 + y)" Thus a? X = a +6 = a. And 23 X 3 X 4 = 34241

By
23

810 +

(The above rule is equally applicable to reciprocal powers.] 6. 7. 8,

9.

10. Mult, 8.04 by: а?ьту? (6th-y)

ROOTS Into 2a 233 by a'b'y ( (+h - y)

If we resolve 6s, or bbb, into equal factors, viz. b, 6 and by 11. Mult. + y + xy + y into 4 - y.

each of these equal factors is said to be a root of W. So if we 12. Muit. 4x2y + 3xy Immo 2x

resolve 27 into is three equal factors, as 3 x 3X3 13. Mult. Il + - 5 into 2.1? + I +1.

each of these equal factors is said to be a root of 27. And The rule is equally applicable to powers whose exponents when any quantity is resolved into any number of equal are negative, i. e. lo reciprocul powers,

factors, each of those factors is said to be a root of utan 1 1 1

quanti'x. 14. Thus ami x pula. That is,

х

A root of a quantity, then, is a factor, which multipliends like ats aussi itself a certain newiber of times, will produce that quantily

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The number of times the root must be taken as a factor to D'abord, je pourrais vous répondre qu'un massif de blé produce the given quantity, is denoted by the name of the n'a rien de laid. Au printemps, c'est de l'émeraude, en étés

c'est de l'or. Un de mes voisins, qui s'y connaît,' a dans son Thus 2 is the fourth ront of 16; because 2 X 2 X 2 X 2 beau parc un champ d'avoine encadré de géraniums, et qui = 16, where two is taken four times as a fa tor to produce 16. forme un tableau splendide et varié. Mais, chez moi, le fro

So as is the square root of a® ; for as X ad ab.

Powers and roots are correlative terms. If one quantity is ment est séparé" des fleurs et dérobé à l'oeil par une haie
a power of another, the latter is a root of the former. Aš 63 vive. Il ne saurait- donc défigurer mon jardin. 17
is the cube of l, 80 b is the cube root of 63,

Je prétends au contraire, qu'il lui fait honneur,12 et je
Tnere are two methods in use, for expressing the roots of fonde ma prétention sur un souvenir d'enfance, qui m'est
quantities; one by means of the radical sign V, and the other sacré.
by a fractional index. The latter is generally to be preferred ; J'étais enfant, et je me promenais dans le jardin de mon
but the former has its uses on particular occasions.

père.13 le même que je cultive après lui. L'hérédité est When a root is expressed by the radical sign, the sign is bonnels aux jardins comme à tant d'autres choses. A la placed before the given quantity, in this manner, Va, même place qu'aujourd'hui, il y avait, non pas un champ, Thus ? Va is the 2nd, or square root of a.

mais un simple épi de blé,16 pauvre enfant du hasard, sva is the 3rd, or cube root.

qui avait jeté là un grain, à côté d'une plate-bande de The figure placed over the radical sign, denotes the number marguerites. of factors, into which the given quantity is resolved; i. e. Je trouvai que le voisinage de l'épi déshonorait l'éclat du the number of times the root must be taken as a factor to produce parterrels et j'allais l'arracher avec la tige, lorsque mon père the given quantity.

m'arrêta la main. Thus ?V/a? shows that az is to be resolved into two factors,

- Il faut y regarder à deux fois,17 me dit-il avant de déand V as, into three factors; and "Vao, into n factors.

The figure for the square root is commonly omitted, and the truire une æuvre de Dieu, toute petite et toute modeste radical sign is simply written before the quantity, thus V az qu'elle soit. Qui sait ce que deviendra' celle-ci ? Laissons

la vivre auprès des marguerites. Nous verrons et nous When a figure or letter is prefixed to the radical sign with comparerons leurs destins. out any character between them, the two quantities are to be Comme mon père achevait ces mots, deux enfants pasconsidered as multiplied together.

sèrent, derrière la haie.18 C'étaients les deux filles d'un ferThus 2 Va, is 2 x Va, that is, 2 multiplied into the root of mier voisin ; l'une vive, alerte, brune, aux yeux noirs et &, or which is the same thing, twice the root of a.

pétillants ;19 l'autre blonde, påle, aux yeux bleus, à l'air doux And & v'b, is x XV b, or <iimes the root of b.

et réfléchi.20 J'ai retenu leurs noms. La première s'appeWhen no co-efficient is pretixed to the radical sign, 1 is lait Marie, et la seconde Louise.—Marie s'écria : always understood; Va being the same as I v a, thai is, once

Les belles marguerites !21 voulez-vous m'en donner une, the root of a.

Monsieur ?
The cube root of as is . For a? x a X a? = al.
Here the index is divided into three equal parts, and the

Mon père me fit un signe. Je choisis la fleur la plus quantity itself resolved into three equal factors,

grosse,22 la plus variée de couleure, la mieux disposée en The square root of a’ is al or a. For a Xa= a.

couronne, et j'en fis cadeau à la petite fille, qui la mit" aussiBy extending the same plan of notation, fractional indices tôt sur ses cheveux. Mon père alors observant sa sæur, lui are obtained.

demanda si elle ne voulait pas une fleur aussi 223— Pour toute Thus, in taking the square root of al or a, the index 1 is réponse, Louise regarda, en rougissant, l'épi de blé, qui se divided into two equal parts, $ and t; and the root is a

at dressait avec ses pointes et était déjà gonflé de quatre rangs

de grairs jaunes.- Je vous remercie, Monsieur, dit-elle entin, On the same principle, the cube root of a, is a” = 3V a je vous demanderai ce bel épi,ut quand il sera' tout à fait mûr. The nth root, is an = "va, etc.

Ce sera ma première moisson. - Frappé de ces mots, mon
Every root, as well as every power of 1, is 1. For a root is père répliqua: Très bien, ma petite,-5 tu peux compter sur
a factor, which multiplied into itself wiil produce the given ton épi.
quantity. But no factor except I can pruduce 1, by being
multiplied into itself.

COLLOQUIAL EXERCISE.
So ih-t 19, 1, N 1,7 1, etc., are all equal,

1. Comment l'auteur parle-t-il | 12. Que prétend-il au Negative indices are used in the notation of roots, as well as des fleurs ?

traire ?
of
Гро веrѕ, ,

2. Quelles autres avantages les 13. Que faisait-il étant en

gazons offrent-ils, outre les fant ? Thus

promenades ?

14. Que dit-il de l'hérédité ? 3. Que donnent les arbres tour 15. Qu'y avait-il à la même à tour?

place? 4. Queréserve l'auteur, tous les 16. Que trouva l'auteur en

voyant l'épi de blé?
FRENCH READINGS.-No. XXV.

5. Quelle exclamation met-il 17. Que lui dit son père ?
dans la bouche de son lec. 18. Qu'arriva-t-il comme le père
teur ?

achevait ces mots?
LA MARGUERITE ET L'EʻPI DE BLE'.

6. Quelle réponse fait-il à l'ex. | 19. Pourez-vous décrire la preSECTION I.

clamation supposée ?

mière ?

7. Que pourrait-il répondre 20. Et l'autre petite fille ? J'AIME bien les fleurs, ces sourires de la nature ;' mais je ne d'abord ?

21. Que dit Marie? leur livre pas mon jardin tout entier.

8. Que présente le blé au prin- 22. Que fit l'auteur, sur un signe Outre les gazons qui, après avoir offerta leurs tapis à mes

temps et en éié?

de son père ? promenades, vont garvir de foin les ràteliers de mon écurie ; 9. Qua le voisin qui s'y con- 23. Que denjanda le père de ontre les arbres qui donnent tour à tour l'ombre de leur feuil

nait?

l'auteur à la

de

Marie?
lage à ma têtes et le suc de leurs fruits à mon palais, je ré - 10. En est-il ainsi chez l'au-
serve, tous les ans, un petit coin de mon enclos pour en

teur?

24. Que réponait-elle enfin ? faire un champ de blé.

11. Quelle conclusion tire-t-il de 23. Que réplique le père de Quelle barbarie!: m'allez-vous! dire, vous défigurez votre

cela?

l'auteur ?

NOTES AND REFERENCES.- .- a. from offrir; L. part ii., p. 96 ; Ecoutez-moi, s'il vous plaît,e avant de me juger,

also L. S. 41, R. 6. 8.-6. allez-vous, will you ; literally, are you

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