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Indeterminate Preterite. darà, he will give
Diédi or détti, I gave

darémo, we will give
Andáre, to go

désti, thou gavest

daréte, you will give diéde, détte or diè, he gave

dardnno, they will give INFINITIVE Mood.

démmo, we gave

Conditional Present.
Compound Tenses.

déste, you gave
Simple Tenses.

Daréi, I should or would diédero, déttero, diérono or diés

give Present: andare, to go Past : essere andato, to have

dono, * they gave

darésti, thou wouldst give or be gone


darébbe, he would give Present Gerund : andando, Past Gerund : essendo andato,

darémmo, we would give going

having, or being gone

Darò, I shall or will give daréste, you would give
darui, thou wilt give

darébbero, they would give
Past Participle : andato, an-
data, andati, andate, gone


[No First Person). diamo, let us give INDICATIVE Mood.

, give (thou)

dáte, give (ye or you)
andammo, we went

dia or déa, let him give diano or dieno, let them give Vado or vo, I go

anddste, you went tái, thou goest andárono, they went



Imperfect. andiamo, we go

Che dia, that I may give
Andrò, I shall or will go

Che déssi, that I might give
andáte, you go
andrai, thou wilt go

che dia or dii, that thou mayst | che déssi, that thou mightst oanno, they go andrà, he will go



che dia, that he may give
andrémo, we will go

che desse, that he might give
andrete, you will go

che diámo, that we may give Andava, I was going

che déssimo, that we might give

che dicite, that you may give andávi, thou wast going andranno, they will go

che déste, that you might give

che diano, dieno, déano, that che dessero, that they might andáva, he was going

Conditional Present.
they may give

give andavamo, we were going

Andrei, I should or would andavite, you were going

So conjugatego andrivano, they were going

andresti, thou wouldst go Ridáre, to give again. Addáre or addársi, to apply one's self. Indeterminate Preterite.

andrebbe, he would go Andai, I went andremmo, he would go

III. andásti, thou wentest

andréste, you would go
andò, he went
andrebbero, they would go

Fare, to make.

INFINITIVE MOOD. [No First Person.] andiamo, let us go

Simple Tenses.

Compound Tenses. Va, go (thou)

andate, go (ye or you) sáda, let him go vádano, let them go Present: fáre, to make Past : avére fatto, to have


Present Gerund : facéndo, Past Gerund : avendo fatto,


having made
Che cada, that I may go
Che andassi, that I might go

Past Participle : fátto, made che odda or vádi, that thou che andassi, that thou mightst mayst go go

INDICATIVE MOOD. che odda, that he may go

che andusse, that he might go che andiamo, that we may go che andassimo, that we mightgo


féce, , or féo, he made che andiate, that you may go che andúste, that you might go Fo or fáccio, I make

facemmo, we made che oddano, that they may go che andassero, that they might fái, thou makest

facéste, you made
fa, he makes

fécero or fénno, they made

facciamo, we make So conjugate

fáte, you make

Faro, I shall or will make
Riandáre, to go again.

fanno, they make

farái, thou wilt make Imperfect.

farà, he will make II.

Faceva, facéa, fè, I was making | faréte, you will make

farémo, we will make Dáre, to give.

facévi, thou wast making INFINITIVE MOOD.

faceva or facea

, he was making farinno, they will make
facevamo, we were making

Conditional Present.
Simple Tenses.
Compound Tenses. facevate, you were making Faréi, I should or would

make Present: dáre, to give Past: avere dito, to have given facevano or faceano, they were


farésti, thou wouldst make Present Gerund: dándo, giv- Past Gerund: avendo dato, ing

farebbe, he would make having given

Indeterminate Preterite.

farémmo, we would make Past Participle : dato, given

Fici or féi, I made

faréste, you would make facesti, thou madest

'farebbero, they would make


Do, I give
Dáva, I was giving,

[No First Person.) facciamo, let us make
dái, thou givest
davi, thou wast giving
Fa, make (thou)

fåte, make (ye or you) da, he gives

dura, he was giving
fáccia, let him make

facciano, let them make diumo, we give

daváno, we were giving dáte, you give

davate, you were giving alkinno, they give dávano, they were giving

Diér, diéro, dénno, are used in poetry,

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Verbs ending in ere are of two sorts. The first have their Che faccia, that I may

make Che facéssi, that I might make infinitives long, such as bére, cadere, etc.; the second short, che faccia or facci, that thou che facessi

, that thou mightst such as assórbere, conoscere, etc.

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 facessimo, that we might


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

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


make So conjugate

Simple Tenses.

Compound Tenses.

Present: bére, to drink
Assuefáre, to accustom Rifáre, to make up again

Past : avere beúto, to hare

Confáre, to become

Sfare, to undo
Contraffare, to mimic

Sopraffare, to ask too much Present Gerund : bevéndo, Past Gerund : avendo beute,
Disfare, to undo
Soddisfare, to satisfy


having drunk Liquefare, to melt

Strafore, to do too much

Past Participle: beúto, drunk
Misfare, to do wrong
Stupefáre, lo stupefy



bévre, he drank Stáre, to stand.

Béo, I drink

beémmo, we drank

beéste, you drank

béi, thou drinkest INFINITIVE Mood.

bée, he drinks

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

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

beéte, you drink

berò, I shall or will drink
béono, they drink

berdi, thou wilt drink
Present Gerund : stando, Past Gerund: essendo státo,


berà, he will drink
having stood

berémo, we will drink
Beéva, I was drinking

beréte, you will drink
Past Participle : státo, stood

beévi, thou wast drinking
beéva, he was drinking

berdnno, they will drink

beevamo, we were drinking

Conditional Present.

beeváte, 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

Indeterminate Preterite. stéste, you stood

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

beréste, you would drink
stiamo, we stand

beésti, thou drankest

berébbero, they would drink státe, you stand

Staro, I shall or will stand
stanno, they stand
arái, thou wilt stand

starà, he will stand

[No First Person. beiamo, 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 stavámo, we were standing Conditional Present.

SUBJUNCTIVE MOOD. staváte, you were standing Staré, I should or would stand

stávano, they were standing starésti, thou wouldst stand

Che béa, that I may drink
starébbe, he would stand
Indeterminate Preterite.

che béa, that thou mayst drink | che beesse, that he might drink starémmo, we would stand

che beéssimo, that we might

che béa, that he may drink
Stétti, I stood
staréste, you would stand

stésti, thou stoodest

che heidmo, that we may drink starebbero, they would stand che beiáte, that you may drink che beeste, that you might


che béano, that they may drink IMPERATIVE MOOD.

che beéssero, that they might Imperfect

[No First Person.] stiamo, let us stand
Sta, stand (thou)
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-

Imbére, to imbibe

| Ribére, to drink again

Strabére, to drink hard.

The Italians prefer the regular verb bévere.
Che stia, that I may stand Che stessi, that I might stand
che stia or stii, that thou mayst che stessi, that thou mightsti


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

che stiano or stieno, that they che stéste, that you might stand
che stéssero, that they might

Simple Tenses.

Compound Tenses. stand Present: cadére, to fall Past: éssere caduto, to have

So conjugate-

Present Gerund: cadendo, fall- Past Gerund: essendo caduto,
Dütare, to be distant
Ristáre, to stop


having fallen
Instáre, to insist
Soprastáre, to differ

Past Participle: cadúlo, fallen

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From the point a draw a B perpendicular to BC (I. 12). INDICATIVE MOOD.

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


the square of A B is equal to the squares of B E and zä together

(I. 47).
Cádo or caggio, I fall
Cadrò or caderò, I shall or will

For the same reason the square of a c is equal to the squares of cádi, thou fallest

fall code, he falls cadrdi or caderái, thou wilt fall 4 E and E c together. Therefore, adding equals to equals (I. Az. 2), the squares of

. , cadiómo or caggidmo, we fall cadrà or caderà, he will fall

A B and a c together are equal to the squares of BB and so cadéte, you fall

cadremo or caderémo, we will together with twice the square of Ba. cadono or cággiono, they 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 BB and Bo are Cadeva or cadéa, I was falling

cadranno or caderánno, they together equal to twice the square of BD together with twice the

will fall cadévi, thou wast falling

square of D E (II. 9). cadeva 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, cadería, ca- twice the squares of B D, D B, and B A together. cadeváte, you were falling

dría, I should or would

But the squares of D B and BA are together equal to the square cadévano or cadéano, they were fall

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 BA are equal to twice the square Indeterminate Preterite.

wouldst fall

of AD. Therefore the squares of AB and a c together are equal Caddi, cadetti, or cadéi, I fell cadrébbe or caderebbe, caderia,

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

cadria, he would fall

Prop. B. Theorem. The squares of the two diagonals of a cadde, cadeo, cadétte, or cadè, cadremmo 'or caderemmo, we parallelogram are together equal to the squares of its four sides. he fell

would fall cadémmo, we fell

cadréste or caderéste, you would
cadéste, you fell

caddero, cadero, cader, cadéttero, cadrébbero or caderébbero, cade-
cadérono, they fell

riano, they would fall
[No First Person.] cadimo, let us fall
Cádi, iall (thou)

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

Let ABCD be any parallelogram of which the diagonals are

AC and B D cutting each other in .. Then the equares of a c and Present.


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 cada or cággia, that I may che cadéssi, that I might fall fall che cadéssi, that thou mightst of which the side 8 D is bisected in , therefore, by the last pro

the diagonals bisect each other. And because ABD is a triangle che cada or caggia, that thou fall mayst fall

che cadesse that he might fall position, the squares of 13 and ap together are equal to twice che cada or caggia, that he may che cadessimo, that we might

the squares of B 2 and A B together.

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

fall che cadiamo or caggiamo, that che cadéste, 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 2 c together. Therefore, we may fall

che cadiáte or caggidte, that you che cadessero, that they might twice the squares of a 8 and B C (I. A. I. 2).

together are equal to four times the square of Be together with

fall che coidano or caggiano, that

But A e is equal to e c, because the diagonals bisect each other,

and therefore the square of AE is equal to the square of 2C they may fall

Therefore the squares of AB, BC, CD, and Da together are equal So conjugate

to four times the squares of B and as together,

But the square of any straight line is equal to four times the
Accadére, to happen
Discadere, to fall away

square 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 BB, and the square of A O

Scadere, to become due.

is equal to four times the square of A E.

Therefore the square of . B, BC, CD, 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 BOOK OF EUCLID.

four times the square of the straight line which joins the points

of the bisection 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


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Let A B C D be any trapezium whose diagonals a C and B D are Let A B C be any triangle having one of its sides B C bisected bisected in the points F and B. Join E f. Then the equares of A B, in D. Join a D. Then the squares of AB and Ac are together BC, CD, and D A are together equal to the squares of a cand BD equal to twice the squares of B D and A D together.

together with four times the square of E F.

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E C.


Join Ba and Ac. Then, because ABD is a triangle whose the rectangle contained by the whole line thus produced, and the base is bisected in E, and BA is drawn, therefore the squares of part produced, may be equal to a given square. A B and A D together are equal to twice the squares of BE and E A together (II. Prop. A).

For the same reason the squares of BC and CD are together equal to twice the squares of BE and EC together.

Therefore, adding equals to equals, the squares of A B, BD, CD, and D A, are together equal to four times the square of BE

C together with twice the squares of E A and E c together.

But the square of BD is equal to four times the square of BE (II. 4, Cor. 2).

Therefore the squares of A B, BC, CD, and D A are together equal to the square of B D together with twice the squares of Ea and Again, because EAC is a triangle, of which the base ao is

N bisected in e (Hyp.), therefore the squares of EA and ec are together equal to twice the squares of cr and EF together Let A B be the given straight line, and C D E F the given square. (II. Prop. A), And the doubles of these equals are equal; therefore the the whole line thus produced, and the part of it produced may be

It is required to produce AB so that the rectangle contained by doubles of the squares of EA and Ec are together equal to four equal to the given square CD e F. times the squares of CF and EP together.

From the point B in the straight line A B draw Bu at right Therefore the squares of a B, BC, CD, and D A are together equal angles to A B (I. 11)

and equal to CD (I. 3) one of the sides of the to the square of BD together with four times the squares of CP given square CD e F. and EF.

Bisect a B in the point G and join G H. But four times the square of cp is equal to the square of AC With centre G and distance Gh describe the circle umn.

ни (II. 4, Cor. 2).

Produce a B to meet the circumference in L. Then the rectTherefore the squares of A B, BC, CD, and D A are together angle A L, L B is equal to the square CDEF. equal to the squares of B D and á c together with four times the

Because Bu is equal to co (Cons.), therefore the square of square of Ep. Q.E.D.

Bh is equal to the square of CD, that is the square C D E F. Prop. D. Problem. To divide a given straight line into two Again, because G is the centre of the circle xy x, therefore G I parts, 80 that the rectangle contained by its segments shall be is equal to G L (I. Def. 15) and therefore the square of Gu is equal to a given square, not greater than the square of half the equal to the square of L. given straig bt line.

Again, because I B G is a right angled triangle having the right

angle u BG (Cons.), therefore the square of G u is equal to the H

squares of G B and B 1 together (I. 47).

But the square of G H has been above shown to be equal to the square of G i. Therefore (I. Ax. 1) the square of G L is equal to the squares of G B and BH together.

But the square of G L is equal to the square of G B together with the rectangle A L, L B (II. 6).

Therefore (I. Ax. 1) the squares of G B and rk are together equal to the square of G B together with the rectangle A L, L B.

From each of these equals take away the common square of & B, K

then (I. Ax. 3) the rectangle a L, LB is equal to the square of Let A B be the given straight line and CDEF the given square But the square of Ba is equal to the square CD E P(Cons.). Therenot greater than the square of half the given line AB. It is fore the rectangle A L, LB is equal to the square CDEP(I. As. 1). required to divide A B into two parts such that the rectangle Q.E.F.. contained by them may be equal to the given square C D E F.

Bisect a B in G (I. 10). From a draw gx at right angles to
AR (I. 11), and equal to DB (1. 3) a side of the given square CD ANSWERS TO CORRESPONDENTS.
Produce so to k making xx equal to GB (I. 3).

Joan BURROUGH: The POPULAR EDUCATOR has been published in the

United States. We cannot promise to act upon your suggestion. With centre 3 and distance is describe the circle x K L, GEORGE YOUNG will find our Lessons in Italian answer his purpose. cutting a B in 2. The straight line A B is divided in M, so that UN FRANÇAIS: It is doubtful whether we shall be able to find room for the rectangle Au, x B is equal to the square CD EF.

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G. LAXBMAN is referred to the two volumes of the "Historical Edu.

The most important parts of and two unequal parts in y, therefore the square of aB is equal Arithmetic have already been given in these pages. If we can find room for to the square of G x together with the rectangle ax, MB.

further lessons, we wil; but if not, we must reler to “Cassell's Arithmetic”

for any additional information that may be desired. But G B is equal to 11, by construction, and uk is equal to

BETH: Your questions are unsuitable for discussion in the " POPULAR XX (I. Def, 15).

EDOCATOR." We have no wish to engage in theological controversy. Therefore (I. Ax. 1) GB is equal to 1 m, and the square of G B DesUNT CBTERA: The promise will be fulfilled at the commencement of to the square of us.

SIMPLICITAS (Wemyos, Fireshire) has solved the first thirty-two of the Therefore the square of xx is equal to the square of GM Second Centenary of Algebraical Problems; D. HORSBY (Driffield) the first together with the rectangle a M, MB.

fifty-five, with the exception of Nos. 7, 12, 35, 37, 39, 45, and 54; GEORGE But the square of ax is equal to the squares of a G and GM And og. Fenchurch-street) the first thirty-two, except Nos. 16, 23, 20,

WILD (Dalton-on-Tees) the whole of the second portion from No.?3 to 70; together (I. 47).

28, 30, and 31, besides Nos. 46, 41, and 51 of the second portion. Therefore (I. Ax, 1) the squares of 1 G and ax are together D. BORNBY (Driffieid): The following is George Wild's solution of Proequal to the square of a'r together with the rectangle blem 49:A M, XB.

5 9 Days Days

: : 10 : 18 From each of these equals take the square of Gm, then the

9 rectangle a x, x B is equal to the square of 1 G (I. Ax. 3). But the number of days in which A can do the whole. Therefore he would the square of a G is equal to the square CDEP (Cons.).

Therefore the rectangle a M, M B is equal to the square CDEP reyuire 8 days to do the remaining (I. Az, 1). Q.E.F.

Prog. E. Problem. To produce a given straight line, so that The above five exercises were soived by J. H. EASTWOOD (Middleton).







next year.


of the work alone.

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Let I be the number of days in which B could do this part of the work

Now ready, price 93. strongly bound,

Whence x = 45.

In Two Parts :-1. German and English; 2. English and German. In

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18 54 54
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Cassell's Shilling Edition UP First LESSONS IN LATIN. By Pro- great value upon which we intend to give them instruction is,
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difficult to master, but which we hope to make no less popular CASSELL's Lessons in LATIN.-Price 28.6d. paper covers, or 38. neat than it is important. We have also prepared some excellent cloth.

A Key to CASSELL’s Lessons in Latin, Containivg Translations of lessons for the purpose of more fully illustrating the Pronun.
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Cassell's CLASSICAL LIBRARY.-The First Volume of this work, price ciation and Reading of the French Language than has yet
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Lessons in Latin."-Volume II. comprises LATIN EXERCISES, price 2s. neat on other subjects, will, we doubt not, secure a ready welcome
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