Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[ocr errors]

Most mute verbs, having a monosyllabic stem and for the stem-vowel, take in the Second Aorist active, middle, and passive, as well as in the Second Future passive, a as the converted rowel:

τρεπ-ω, I turn, aor. 2. act, ε-τρᾶπον; κλεπτω, I steal, aor. 2. pass. ε-κλάπην

τρεφω, I nourish, aor. 2. pass. ε-τράφ-ην ; πλεκω, I knit, aor. 2. pass. ε-πλάκ-ην.

The conversion does not take place in some verbs, as βλεπω, I behold, impf, ε-βλεπ-οv, aor. 2. pass. ε-βλεπ-ην. Some mute verbs with monosyllabic stems and for their stem-vowel take, in the Second Perfect and Pluperfect, the conversion o; and those which have a in those syllables take the conversion o; e. g.

τρέφω, I nourish, τέτροφα; λειπω, 1 leave, λελοιπα.

(a) pure characteristic: λER-w, I plait, ay-w, I drive, τεύχω, I frame.

(b.) impure characteristic: opto-w (Att. opirrw), I shudder (pure character K, pure stem PIK); raoc-w (Att. TATT-W), I set in order (y, TAT); ẞŋoo-w (Att. Bnrr-u), I cough (x, BHX).

3. Verbs whose characteristic is a t sound (r, d, 0, pure; , impure), as,

(a.) pure characteristic: avvr-w, I end, gô-w, I sing, wrɩd-w, I persuade.

(b.) impure characteristic: opal-w, I say (pure characteristic d, pure stem PAA).

Some verbs ending in rrw or cow have for their pure characteristic, not a k sound, but a t sound, as appоrтw, I adapt, put together, fut. οσω; ερεσσω, I steer, πασσω, I bestrew; πλάσσω, I form; Toow, I pound. The verb vasow, I press together, The same conversion is taken by the following verbs in the has both formations, as fut. važw, etc., perf. mid. or pass. First Perfect; namely, νενασμαι, verbal adj. ναστος,

[merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors]

Remarks on the Formation of the Second Tenses.

All the second tenses are distinguished from the first tenses partly in this, that they lack the tense characteristic, and consequently attach the person-endings, ον, ομην, ην, ήσομαι, α and v-immediately to the pure characteristic of the verb, as E-AT-ov; partly in this (yet with the exception of the Second Perfect), that they are formed from the pure unaltered verbal stem, as λειπ-ω, ε-λίπον; φευγω, aor. 2. εφύ-γον; and again, partly in this, that they take the conversion, as στρεφω, ε-στραφην, στραφ-ησομαι, but ε-στρεφθην.

The Second Perfect lengthens either the short stem-vowel, as à into ŋ (after p and vowels in a), or it retains the long

n

vowel of the Present; as,

[merged small][merged small][ocr errors][merged small][merged small][merged small]

Mute verbs, like the mute letters, are divided into three class s, according to their predominant letter: in each of these three classes are verbs with pure and verbs with impure characteristic in the Present and Imperfect.

1. Verbs whose characteristic is a p sound (π, ß, p, pure; πτ, impure).

(a.) pure characteristic: BEπ-w, I see, тpiß.w, I rub,
Ypap-w, I write.

(b.) impure characteristic: TUTT-W, I strike (pure charac-
teristic, pure stem Tro), Ban-w, I injure,
(B, BAAB), PITT-w, I cast (p, P1).

2. Verbs whose characteristic is a k sound (x, y, x, pure; TT or σo, impure).

The following verbs in Zw, which, for the most part, express a sound or call, have for their pure characteristic, not a t sound but a k sound, commonly ; namely, auaw, I lament (cry ai! ai!) aλaλazw, I shout the war-cry; koi¿w, I grunt (like a pig); paw, I croak (like a raven); paoricw, I whip; odalw, I bite; οιμώζω (f. ξομαι), I bewail (ery οι! οι !) ολολύζω, I howl; ῥυστάζω, I haul, σταζω and σταλάζω (hence stalagmite), I drop; oτevalo, I groan: orηpilw, I set fast; ori, I puncture, brand (our stick); σupiw (Att. ovpirTw), I whistle (f. ξομαι); σφάζω (Att. σφαττω), I hill, slay; σφύζω, I undulate; rpiw, I twitter; pλvZw, I bubble (like a fountain). The following in o have both formations: Baoral, I carry, f. άσω, etc., aor. pass, εβασταχθην ; νυστάζω, I nod, an drousy, fut. ασω and αξω; παίζω, 1 play, joke, fut. παίξουμαι aud παίξομαι, aor, έπαισα; perf. mid. or pass, πεπαίσμαι.

The following three in o have for their pure characteristic YY, namely, Kλaw, I sound, I clang, perf. 2. KEKλayya, f. κλαγξω, aor, εκλαγξα ή πλαζω, I mislead, lead astray, f. πλαγξω, etc.; oaλiw, I sound a trumpet, f. oaλñiɣžw, etc.

Formation of the Tenses in Mute Verbs.

The First Perfect and Pluperfect active have the aspirated terminations à and iv if the characteristic is a p sound or a k sound,

[ocr errors]

( sound : τριβω, τετριβά, which becomes τέτριφα k sound : πλεκ-ω, πε-πλεκ-ά, πεπλεγα) but the terminationз ka, Key, when the characteristic is a t sound; yet the t sound disappears before , as πε-πel-kα, from πείθω,

The vowels a, i, v, in verbs having a t sound as characteristic, are short before the terminations with the tense characteristics - and κ -κα, -κειν), as φράζω, φράσω, εφράσα, πεφράκα; πλάσσω, I mould, from επλάσα; νομίζω, I hold as a law, I am of opinion, evoμioa; kλvĽw, I bubble up, erλvoa, ete. : in the same way short vowels remain short, as appozw, I fit, ἡρμοκα.

When u precedes a p sound as the characteristic, as for tions beginning with in the Perfect middle or passive, as example in μлw, I send, μ is thrown out before the terminaπε-πεμ-μαι instead of (πε-πεμπομαι), πεπεμμ-μαι; καμπτω, I bend, κε-καμ-μαι instead of (κε-καμπ-μαι), κεκαμμ-μαι. when two gammas stand before μ, one of then vanishes, as σφιγγω, I lace, ε-σφιγμαι, instead of (ε-σφιγγ-μαι), εσφιγξαι, toyKral; inf. copyxoai, part. εopiyμevoc.

S

Verbs whose characteristic is a t sound do not, in ordinary speech, form the Second Aorist.

The terminations beginning with o after an immediately preceding mute, lose the o, whereon the mute assumes the aspirate form in consequence of the following 0, as keкpuplai, instead of KEKOVOαι (that is, KEROVπ-σÐαi),

The third persons plural Perfect and Pluperfect, middle or passive, which properly ends in vrat and vro (as we have seen not be so formed, on account of the coming together of so in the pure verbs), can, in the impure, both mute and liquid, many consonants. Consequently the person is commonly expressed with the aid of the plural of the participle Perfect middle or passive, and of the third person plural Present and

Imperfect of the verb ɛivai (εioi, are, and noav, were); sometimes, however, the v is thrown out, and an à put in its place, which, after a k sound and ap sound is aspirated, and remains un aspirated after a t sound: as

233

equal to DB (Ax. 1); but FC is also parallel to DB (Const.); wherefore D F is equal to BC, and is also parallel to it (I. 33). But it has been proved that D E is equal to EF; therefore, DE is half of DF; but BC is equal to DF; therefore, D E is equal to half of B C (Ax. 7); and it has been proved to be parallel to 3 plur. BC. Therefore, if two sides of a triangle, etc. Q. E. D.* τριβω τετριμ-μαι τετριφᾶται Corollary. The triangle A D E is one-fourth of the triangle πλέκω πε-πλεγμαι πεπλεχᾶται (πεπλεκνται) επεπλεχάτο parallelogram Da is equal to the parallelogram GF (I. 36), A B C. Through E, draw EG parallel to DB. Because the TUTT-W τέταγμαι τεταχᾶται (μεταγνται) ετεταχᾶτο therefore the parallelogram BF is double the parallelogram o F.

instead of plupf.
(πετρίβεται) ετετριφᾶτο

χωρίζω κε-χωρισομαι κεχωριδᾶται (κεχωριόνται) εκεχωριδάτο | But the parallelogram or is double the triangle EFc, or its φθείρω ε-φθαρ-μαι εφθαρᾶται (εφθαρνται) εφθαρᾶτο.

[blocks in formation]

For if A D be not parallel to в C, through B

C

the point A, draw A E parallel to B C (I. 31), meeting B D in E; and join E C.

The triangle ABC is equal to the triangle EBC (I. 37), because they are upon the same base B c, and between the same parallels B C and A E. But the triangle ABC is equal (Hyp.) to the triangle DBC; therefore the triangle DBC is equal to the triangle E B C, the greater to the less, which is impossible. Therefore a E is not parallel to B c. In the same manner it can be proved that no other straight line but A D is parallel to B C. Therefore A D is parallel to BC. Wherefore, equal triangles upon, etc. Q. E. D.

[blocks in formation]

and Ac bisected in the points D and E. The straight line DE which joins them, is parallel to the base B C and equal to half

of it.

is

duce DE till it meet or in the point F.
Through the point c, draw cr parallel to AB (I. 31). Pro-
Because, in the two triangles AED and FE C, the side a ¤
equal to the side Ec (Hyp.), and the angle AED to the angle
(I. 15), and the angle DAE to the angle FCE (1. 29);
unrefore, the triangle APD is equal to the triangle FE C (1.26);
and the side A D to FC, and the side D E to the side E F. Because
EC is equal to AD, and AD to DP (Hyp.); therefore, F C is

equal the triangle ADE (I. 34); therefore the parallelogram
Bris quadruple the triangle ADE; that is, the triangle ADE is one-
fourth of the parallelogram BF. But the parallelogram BF is
equal to the triangle ABC, because the trapezium DB CE is
common to both, and the triangle A DE is equal to the triangle
FEC; therefore the triangle A D E is equal to one-fourth of the
triangle ABC.

EXERCISE II. TO PROPOSITION XXXIX.

If the sides of any quadrilateral figure be bisected, and the points of
bisection be joined, the figure thus formed will be a parallelogram,
and equal to half the quadrilateral.

In fig. u, let A B C в be any quadrilateral figure; let its sides

[blocks in formation]

A B, BC, CD and DA, be bisected in the points E, F, G and н;
and let these points be successively joined by the straight lines
EF, FG, G H and HE. Then the figure E F G H is a parallelogram,
and it is equal to half the figure A B C D.

Join A c and B D. Because FE and G H are each parallel to
(I. 30); for the same reason, E H is parallel to FG; therefore,
BD, by the preceding exercise, therefore FE is parallel to GH
the figure E F G H is a parallelogram.

Because the triangle A EF is one-fourth of the triangle A B D,
and the triangle G C H one-fourth of the triangle BCD, by the
corollary to the preceding exercise; therefore the two triangles
AEF and GCH together are one-fourth of the triangles ABD
and BDC together, or of the figure A B C D which is equal to
them. In the same manner, it may be shown, that the two
triangles D E H and FBG are together one-fourth of the figure
ABCD; therefore the four triangles AEF, GCH, EDH and
FBG are together two-fourths or one-half of the figure ABCD;
wherefore the remainder, the parallelogram EFGH,
half of the figure A B C D. Therefore, if the sides of any
quadrilateral figure, etc. Q. E. D.†

PROPOSITION XL. THEOREM.

the other

Equal triangles upon equal bases in the same straight line, and on
the same side of it, are between the same parallels.

In fig. 40, let the equal triangles ABC and DEF be upon
equal bases B C and E F, in the same straight line BP, and
the same side of it; they are between the same parallels; that
is, if A D be joined, A D is parallel to B F.

been given by WARIN (East Dereham); E. J. BREMNER (Carlisle), and
QUINTIN PRINGLE (Glasgow); who ingeniously solved comes of und
previous exercises.

• This solution is by J. H. EASTWOOD (Middleton); solutions have alsd

(Glasgow); solutions were also given by J EASTWOOD (Middleton))
WARIN (East Dereham); and others,

This solution and the preceding corollary are by QUINTIN PRINGLE

[ocr errors][ocr errors]

For, if an be not parallel to BF, through a draw A G parallel the triangle BED half the parallelogram FD (I. 41); therefore the to BF (I. 31), meeting ED in G, and

join & F.

B

Fig 40.

D

A

The tringle ABC is equal to the triangle GEF (1. 38), because they are upon equal bases B C and E F, and between the same parallels BF and A G. But the triangle ABC is equal (Hyp.) to the triangle DEF. Therefore the triangle DEF is equal (Ax. 1) to the triangle GEF, the g eater equal to the less, which is impossible. Therefore AG is not parallel to B F. In the same manner it can be proved that no other straight line is parallel to BF, but AD. Therefore A D is parallel to B F. Wherefore, equal triangles upon, etc. Q. E. D.

Schol.-In this proposition and the preceding one, the demonstration would be conducted in the same manner, if the parallels AE and AG were to meet BD and ED produced. The argument would then rest on the absurdity of the less triangle being equal to the greater.

[blocks in formation]

two triangles AEC and BED, together, are half the two parallelograms FC and FD, that is, of the whole parallelogram A D. In the same manner may be shown that the two triangles ABB and CBD are together half the parallelogram AD. Therefore, if from any point within a parallelogram, etc. Q. E. D.*

EXERCISE II. TO PROPOSITION XLI.

In a trapezoid, if one of the sides which is not parallel be bisected, and straight lines be drawn from the point of bisection to the extremities of the other side which is not parallel, the triangle which they form with the latter side is equal to half the trapezoid.

In fig. w, let A B D C be a trapezoid having the side A B parallel to the side CD; let the side a c be bisected in the point E, and Fig. w.

[merged small][ocr errors][merged small][merged small]

let straight lines be drawn from E to the points B and D. Then the triangle BED is half the trapezoid A B D C.

Through the point E, draw FG parallel to BD, meeting CD in G, and A B produced in F (I. 31).

Because in the two triangles AFE and ECG the angle FA E is equal to the angle ECG (I. 29), the angle AEF to the angle CEG (I. 15), and the side A E to the side EC (Hyp.); therefore the triangle AFE is equal to the triangle CE G. To these equals add the figure AEG DB; and the parallelogram FGDB is equal to the trapezoid A B D C (Ax. 2). But the triangle BED is half the parallelogram FD (1.41); therefore the triangle BED is half the trapezoid ABDC (Ar. 7). Wherefore, in a trapezoid, if one of the sides, etc. Q. E. D.†

Schol. This proposition is the foundation of the mensuration of triangles, and consequently of all rectilineal figures, as they can easily be divided into triangles. Inasmuch as the area of a parallelogram is found by multiplying its base by its perpendicular altitude or breadth, so the area of a triangle is found by multiplying its base by its perpendicular altitude, and taking half the product. In a triangle the perpendicular altitude is the perpendicular drawn from the vertex of the angle opposite to the base, to the base itself or to the base To describe a parallelogram equal to a given triangle, and having produced.

Corollary.-A parallelogram is equal to a triangle on the double of its base, and between the same parallels.

EXERCISE I. TO PROPOSITION XLI.

If from any point within a parallelogram, straight lines be drawn to the extremities of two opposite sides, the two triangles upon these sides are together equal to half the parallelogram.

PROPOSITION XLII. PROBLEM.

one of its angles equal to a given rectilineal angle.

In fig. 42, Let ABC be the given triangle, and D the given rectilineal angle. It is required to describe a parallelogram equal to the given triangle ABC, and having one of its angles equal to D.

Bisect BC in E (I. 10), and join AE. At the point E in the straight line EC, make the angle CEF (I. 23) equal to the angle D. Through A, draw AFG (I. 31), parallel to BC, and through c, Then the

In fig. v, let A B D C be a parallelogram, and E any point draw ac parallel to EF.

[blocks in formation]

B

Fig. 42.

A F G

E

figure CEFG is a parallelogram (D ƒ. 36).
Because the two triangles A BE and AEC are on equal bases
BE and EC, and between the same parallels BC and AG; they
are equal (I. 38) to one another. Therefore the triangle ABC
is double of the triangle A E C. But the parallelogram FC is
double of the triangle A E c (I. 41), because they are upon the
same base EC, and between the same parallels EC and AG.
Therefore the parallelogram r c is equal (4x. 6) to the triangle
A B C, and it has one of its angles CEF equal to the given angle
D. Wherefore, a parallelogram PC has been described equal
to the given triangle A B C, and having one of its angles CEP
equal to the given angle D. Q. E. D.

Solved by EBENR JONES; J. JENKINS (Pembroke Dock); WARIN (East Dereham); Q PRINGLE (Glasgo); D. H., Driffield; J. H. EASTWOOD (Midleton; E. J BREMNER (Carlisle); and others.

Solved by Q. PRINGLE (Glasgow); H. J. WARIN (East Dereham); D. H. (Driffield); J. H. EASTWOOD (Middleton); E. J. BREMNER (Car

Through the point E, draw FG parallel to A C or BD (I. 31).
Because the triangle ▲ C E is half the parallelogram F C, and | liste); and others.

EXERCISE TO PROPOSITION XLII.

rectilineal angle. It is required to describe a triangle that shall be equal to the given parallelogram A D, and have one of

To describe a triangle equal to a given parallelogram, and having its angles equal to the given rectilinea! angle E. an angle equal to a given rectilineal angle.

In fig. x, let AD be the given parallelogram, and E the given given angle B (I. 23), and let the straight line CF meet A B

[blocks in formation]

Produce CD to G, making D G equal to DC; and at the point c, in the straight line co, make the angle GCF equal to the produced (if necessary) in F. Join F G. Then the triangle FCG is the triangle required.

Join F D. Because CD is equal to DG, therefore the triangle FCD is equal to the triangle FDG (I. 38), and the whole triangle FCG is double the triangle FCD. But the parallelogram AD is also double the triangle FCD (I 41); therefore the triangle FCG is equal to the parallelogram A D (Ax. 2); and it has one of its angles, F C G, equal to the given rectilineal angle E. Q. E. F.*

C

D

G

Solved by J. H. EASTWOOD (Middleton); Q. PRINGLE (Glasgow); H. J. WARIN (East Dereham); E. J. BREMNER (Carlisle); and others.

proceeds being on the Dr. side, and the sales on the Cr. side;

LESSONS IN BOOKKEEPING.-No. XXIII. sometimes, especially where the space admits of it, the charges,

(Continued from page 200.)

ACCOUNT SALES BOOK.

AN Account Sales is an account showing the amount of the sales of goods imported and sold for behoof of the merchant, or any of his correspondents, with the different charges attending the sales, and the net proceeds of the whole. The book in which such accounts are entered is called the Account Sales Book, or simply Sales Book. An account sales is frequently made up in the Dr. and Cr. form, the charges and the net

etc. are placed at the bottom of the account, so that the whole may be contained in one page, as exemplified in the two accounts in this book. The Account Sales Book is frequently made up from other books, where the particulars are entered as they can be obtained from time to time. As an account sales can rarely be made up at the period when the goods are sold, the copy in the Sal's Book must be marked with the date where it is entered in the Day Book, or the folio where it is entered in the Journal; as, like the Invoice Book, the entries may be made at once in the Journal, without passing through the Day Book.

(1)

ACCOUNT SALES BOOK.

(1)

ACCOUNT SALES of 7 Hhds. of Sugar (W. S. & Co.), received per the Ballarat, Captain Jones, from Barbadoes, and sold on account of Nathan Herschell of that place.

[graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][ocr errors][subsumed][subsumed][subsumed][merged small][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][merged small][subsumed][subsumed][subsumed][subsumed]

(2)

ACCOUNT SALES BOOK.

(2)

ACCOUNT SALES of 21 Tierces of Coffee (W. S. & Co.), received per the Wellington, Captain Browne, from Berbice, and sold on account of John Henderson of that place.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[ocr errors]

ANSWERS TO CORRESPONDENTS.

A PUPIL TEACHER (Ormskirk): The plan and rules of the association for mutual improvement are very good; but every such association must rely on its own resources, if it is to do good. Books and practice are the best things. Study well the P. E.-P. ALEXANDER (Glasgow): Right; go on and prosper.-W. ONN (Radford): However willing, we really cannot advise; but in all labour there is profit, profit sooner or later; therefore persevere, and seek direction from on high.-W. WATSON (Glasgow): Many persons in various parts of the empire, both at home and abroad, have taught themselves the different branches of learning, without the aid of a master, by means of the lessons in the P. E.; and we do not see why our orrespondent and his friends could not do the same; for, "what man has done, man can do." Let them TRY; where there's a WILL, there's a

WAY.

T. RAYNOR (St. John's-wood): His solutions of the quadratic equations are very ingenious, and coming from one self-taught are both curious and interesting; but as they are not quite up to the style of the present day, we forbear inserting them. Let him go on, however, and prosper.-A. Y.S.: Yes.-GEO. WRIGHT (Autodoctus ?): Pretty well, but not up to the P. E. mark.-J. W. (Fifeshire): Very well; go on and prosper in your Algebraic solutions.

PERU (Glasgow): We doubt whether the P. E. can be had in Peru; but nothing is impossible, for we have publishers at the Antipodes -J. JONES (Woolwich): His solution of the two-tower question is right.-T. Bocock Great Warley): His solution of the Geometrical Problems in this number, which came rather late to be noticed in order, are correct.

J. WORLEY (Reading) proposes the following varnish for maps in reply to some correspondents. First obtain some parchment shavings, and let them simmer down for some hours; then strain off the liquid, which if done enough will be a strong jelly when cool. Apply this to your paper or coloured print, twice, with a soft brush, letting the first coat well dry, before applying the second. Then, having ascertained that the size is well dried, and the map well covered in every part, varnish firmly with a soft brush, with the best white mastic varnish. In purchasing, reject it if it appear thick and gummy. The maps thus varnished will clean with sponge and water at any time.

LITERARY NOTICES.

Now Ready,

CASSELL'S FRENCH DICTIONARY:

In Two Parts:-1. French and English; 2. English and French. The French Department carefully edited by Professor DE LOLME, and the English Department by Professor WALLACE and H. BRIDGEMAN, Esq. In one large handsome Octavo Volume, price 98. 6d. strongly bound.

CA SELL'S LATIN DICTIONARY. In Two Parts:-1. Latin and English. 2. English and Latin. BY J. R. BEARD, D.D., and C. BEARD, B.A. In We ky Numbers, 3d. each, and Monthly Parts, ls. The First Four Monthly Parts are now ready, as also the First Sixteen Numbers. The Latin-English Division is now ready, price 4s. in paper covers, 58. in cloth.

Now Ready,

CASSELL'S GERMAN PRONOUNCING DICTIONARY: In Two Parts:-1. German and English; 2. English and German. In one large handsome Octavo Volume, price 9s. strongly bound.

CASSELL'S LESSONS IN FRENCH. Parts I. and II.-By Professor FASQUELLE. Price 2s. each in paper covers, or 23, 6d. bound in cloth. The Two Parts bound in One Volume, price 4s. 6d.

A KEY TO CASSELL'S LESSONS IN FRENCH, containing Translations of all the Exercises. Price 1s. paper covers, or 1s. 6d. cloth.

A COMPLETE MANUAL OF THE FRENCH LANGUAGE.-By Professor D LOLME. Price 3s. neatly bound.

A SERIES OF LESSONS IN FRENCH, on an entirely Novel and Simple Plan. Reprinted in a revised form ir m "The Working Man's Friend." Price 6d., by post 7d. Above 30,000 copies of this work have been sold.

« ΠροηγούμενηΣυνέχεια »