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These prepositions generally correspond to the English prepositions between, betwixt, and among; e. g. tra il mú-ro ed i fiú-me, between the wall and the river; fra ti-mó-re e speran-za, between fear and hope; é-gli stá-va fra il si e il nô, he was undecided (i. e. he stood between yes and no); ré-gna fra és-si po-ca con-cór-dia, little concord subsists between them; ná-cque ú-na li-te in-tra le du-e dôn-ne, there arose a quarrel between the two women; di-stri-bui-te ciò fra i pô-ve-ri, "distribute that among the poor; tra gli dl-tri, among the rest; si the course of, in; e. g. tra pô-chi gior-ni, in a few days; ha prothe bad ones. They also very frequently signify within, in més-so di ri-tor-ná-re fra tre gior-ni, he promised to return within three days; in-fra un án-no tút-ti mo-ri-ro-no, all of them died in the course of one year.

Só-pra and só-vra commonly govern the accusative case, sometimes, however, the genitive and dative. Su, for the greatest part, has the accusative after it. E. g. só-pra un cár-ro é-gli se dé a, he was sitting on a cart. Só-pra la tá-vo-la, só-pra la têr-ra, on the table, on the earth or ground; por la má-no só-pra il pêt-to, to lay the hand on one's breast; és-se-re só-pra un u-fi-cio, to fill (i. e. to be appointed over or at the head of) an office or official situation; mon-tár só-pra ú-na bár-ca, to get into (i.e. to mount on) a boat; tené-re le má-ni só-pra il fuó-co, to hold the hands over the fire; vên-ga só-pra di me il giu-di-cio, non só-pra la non col-de-e an-no-ve-rá-ri tra i cat-ti-vi, he must be enumerated among pé-vo-le din-na, let the sentence be passed upon me, not upon the innocent woman; il sú-o sán-gue vên-ga sô-pra di noi e sópra i nô-stri fi-gliuô-li, may his blood come upon us and upon our children; sú-o pá-dre ha un gran po-té-re só-pra di lui, his father has a great power over him; co-min-ciò a pid-gne-re só-pra di lé-i, he began to shed tears at or over her; é-gli andá-va só-pra ad un al-be-ro, he climbed up a tree; lan-ciár-si só-pra ad ú-no, to rush upon one; sul mon-te, súl-la tá-vo-la, sull' ér-ba, on the hill, on the table, on the green turf; súl-la ci-ma di ú-na er-ta mon-tá-gna, on the top of a steep mountain; é-gli á-bi-ta sul sú-o po-dé-re, he lives on his estate; su quel tét-to vo-li-va un co-lôm-bo, a pigeon flew on that roof.

In su is frequently used in the place of su; e. g. sul tét-to, or in sult-to, on the roof; un cár-ro in sú dú-e rô-te, a cart on two wheels.

Before the personal pronouns me, se, etc., fra and tra have a
peculiar meaning corresponding to the English prepositions to,
with, and are used (as it were in the place of dén-tro me, dén-tro se,
within me, within himself; e. g. fra se me-dé-si-mo dis-se, he
said to himself; i-o di-cé-va fra it meo cuộn, per-che pa-tên-tỷ?
I said to my heart, why dost thou tremble? tra me so-ven-te
di-cên-do, frequently saying to myself; pen-sá-re fra or tra se
stés-so, to think with one's self.

In some phrases fra and tra merely signify in or at; e. g.
par-lár tra 'l són-no, to talk in one's sleep; so-vên-te fra 'l són-
no s' al-zá-va, he frequently rose in his sleep; tra più vol-te gli
instalments; tra ú-na vol-ta e l' al-tra, at different times.
pa-gò mil-le scu-di, he paid him a thousand crowns in several

Só-pra as well as su frequently denote nearness, and are used for vi-ci-no, ap-prês-so, near, close, hard by, on, etc.; e. g. ú-na cit-tà si-tuá-ta só-pra un fiú-me, sul Rê-no, súl-la ma-rí-na, a town situated on a river, on the Rhine, on the sea-coast; Tra (and sometimes also fra) is often used adverbially for ês-ser pó-sto só-pra il má-re, to lie on the sea; pas-sam-mo il pár-te, partly; e. g. tra per má-la con-dét-ta e per im-pen-sa-te Re-no só-pra Co-lô-nia, we crossed the Rhine near Cologne. scia-gu-re vên-ne a fal-li-re, he became bankrupt partly through They also frequently coincide with the use of the English bad conduct, partly through unforeseen misfortunes; Co-stanprepositions towards, about, at, etc., with regard to time; e. g. ti-o re-gnò più di trent' án ni, tra nell' im-pê-ro di Ró-ma e quél-lo só-pra sé-ra, towards evening; sul or in sul méz-zo dì, sul ve-spro, di Co-stan-ti-no-po-li, Constantine reigned more than thirty sul tra-mon-tár del só-le, towards, about noon, or twelve years, partly in the empire of Rome, partly in that of Constano'clock, towards evening, at sunset; súl-le diê-ci (ó-re), to- tinople; tra con pa-rô-le e con át-ti, partly with words, partly wards, about ten o'clock; per-chè ve-ni-te su qué-sta ó-ra? why with acts; é-gli pos-sie-de fra rô-ba e da-ná-ri 10,000 lí-re sterdo you come just now? at this hour? Sul co-min-ciá-re, in sulli-ne, he possesses 10,000 pounds sterling, partly in goods fi-ni-re, in the beginning, at the end or termination. They partly in money. also often denote on the point of, going to, etc.; e. g. stá-re sul Fra sometimes means together; e. g. fra uô-mi-ni e dôn-ne sópar-ti-re, to be on the point of departing; és-ser só-pra a far no 10,000, men and women together 10,000. chec-ches-si-a, to be on the point of doing something; és-ser só-pra a cá-de-re, to be on the point of falling, to be like to fall; i-gli e-ra sull' im-paz-zí-re, he was near becoming mad.

EXERCISES.-ITALIAN-ENGLISH.

Fran-co-for-te sull' O^-de-ra. Sul fát-to. Vi pro-mêt-to súl-
Ma-gón-za, cit-tà sul Rê-no. Fran-co-for-te, sul Mê no
la mi-a fé-de. Su qué-sta têr-ra. Su quál-che ta-vo-li-no.
Ric-car-do as-sí-so su d'un sás-so. Vô-glio suo-ná-re un á-ria
sul mí-o cla-vi-cém-ba-lo. Non sa-prê-i ri-spón-der-vi su tal
pún-to. I ba-ú-li só-no súl-la car-rôz-za. Ha pián-to súl-la
di lui dis-gra-zia. Ri-pó-ne súl-la sot-to-côp-pa il bic-chiê-re.
Non ha di-rít-to ve-ru-no súl-la mí-a ri-co-no-scên-za. Ri-

Besides these uses só-pra has other important meanings, as: in addition to; e. g. só-pra la fêb-bre mi è ve-nú-ta la po-da-gra, in addition to the fever I have got the gout; só-pra ciò, in addition to this, besides, moreover: against; e. g. an-dá-re só-pra i ne-mi-ci, to go against the enemy; u-sár ó-gni cru-del-tà sópra di ú-no, to commit all possible acts of cruelty against one: to; e. g. la nôt-te del ve-ner-di sán-to só-pra il sáb bat-to, the night from Good Friday to Saturday: on; e. g. pre-stá-re só-pra pé-gni, to lend on securities; man-gia-re sh-pra chec-ches-po-sa-te-vi súl-la mí-a pa-rô-la. Quel-lo che ha in cuô-re, lo si-a, to eat on credit; só-pra la mi-a fé-de, la mi-a pa-rô-la, stél-le non hán-no al-cún in-flús-so sú-gli uô-mi-ni. Sul far ha sêm-pre súl-le láb-bra. La cá-sa dà súl-la strá-da. Le upon my word; com-men-tá-re, scri-ve-re só-pra qual-che cô-sa, del giór-no (or in sul ná-sce-re del giór-no). Sul far dél-la sé-ra to comment, write on something: of; e. g. pen-sú-re, di-re, (or in súl-la sé-ra). Súl-la (or in sul-la) mêz-za nôt-te. In sul par-la-re, dis-cór-re-re só-pra qual-che cô-sa, to think, speak, món-te; in súl-la tá-vo-la. I'-o stô fra 'l ti-mó-re e la spetalk, discourse of something; és-ser só-pra al-cu-no, to fall foul rán-za. Fra a-mi-ci si può par-la-re li-be-ra-mén-te. Frál-lo of one or to take him to task (also to drub him, etc.): into scô-glio e 'l fiú-me. El più sfor-tu-ná-to fra' ge-ni-tó-ri. Fra e. g. qué-sta cá-sa ri-guár-da só-pra la piáz-za, this house looks out into the square: from; e. g. prên-de-re ú-na cit-tà só-pra il ne-mi-co, to take a town from the enemy: more than, above, beyond; e. g. l'a-má-va só-pra la ví-ta sú-a, he loved him more than his own life; só-pra ó-gni ál-tro, more than all the others, extremely, pre-eminently, especially, etc.; só-pra tút-to, above every thing; só-pra mô-do, beyond measure, exceedingly; só

*Star só-pra di se means: to stand absorbed in thought or to be in doubt, to waver, hesitate; fá-re or la vo-rá-re so-pra di se means: to work for one's self, i. e. without being a member of a tradesmen's company, etc.; só-pra di se generally means; at one's own expense, for one's self, etc.

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LESSONS IN ITALIAN.

gli uô-mi-ni. Fra gl' in-fe-li-ci. Dis-côr-dia fra ma-rí-to e
mó-glie. Ciò rê-sti fra di noi; sí-a dét-to fra noi. I'-o di-
cé-va fra me stés-so. E'-gli ver-rà fra diê-ci giór-ni. Fra qui
e Pá-squa; fra qui e ôt-to giór-ni. Non lo sô, ma lo sa-prô
bê-ne tra pô-co. Per-dêt-te-ro i ne-mí-ci tra môr-ti e pri-gio-
niê-ri nô-ve mi-la uô-mi-ni. Ve lo di-rò fra quát-tro ôc-chi.
Hán-no par-la-to
Fra noi pás-sa ú-na strét-ta a-mi-cí-zia.
sêm-pre sót-to vó-ce fra ló-ro dú-e.

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VOCABULARY,

Francoforte, Frankfort,
Meno, Maine (river).
Odera, the Oder (river).
Fatto, fact, deed, act (sul fatto,
in the act, in the very act).
Vi prometto, I promise you.
Fede, faith.

Qualche, some.

Tavolino, table.

Riccardo, Richard.

Assiso, seated, sitting.
Sasso, stone, rock.

Voglio suonare, I wish to play.
Aria, air, tune, song.
Clavicembalo, harpsicord, piano-
forte.

Non saprei rispondervi, I should
not be able to give you an

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knowledgment. Riposatevi, you may rely. Parola, word.

Quello che, what.

Cuore, heart.
Sempre, always.
Labbro, m., lip (pl. labbri, m.,
or, better, labbra, f.)
Da, gives (i. e. is situated to-

wards, faces or fronts). Strada, street.

Stella, star.

Non alcun, not any.
Influsso, influence.

Fare, to do, make, cause; na-
scere, to be born, come forth,
appear, arise (far del giorno,
nascere del giorno, dawn, day-
break; sul far or al far del
giorno, in sul nascere del gior-
no, at the break of day),
Sera, evening (far della sera,
evening, evening twilight,
dusk of evening; sul or in sul
far della sera, in sulla sera,
towards evening, when night
was coming on, when even-
ing drew near, etc.)
Mezza notte, midnight, twelve
o'clock at night (sulla or in
sulla mezza notte, a
notte, in the middle of the
night, at midnight).
Monte, mountain.

mezza

|

Tavola, table.

Io sto, I stand, I am.
Timore, fear.
Speranza, hope.

Si puo parlare liberamente, one
may speak unreservedly.
Scoglio, rock (in the sea, river,
etc.), cliff, crag.
Fiume, river.

COLLOQUIAL EXERCISES.-ITALIAN-ENGLISH.
I miê-i fra-têl-li, my brothers. |
I tub-i fra-tei-li, thy brothers.
brothers.
I suô-i fra-têl-li, his (her, its)

Il pó-mo, the apple.
La pé-ra, the pear.

La ci-riê-gia, the cherry.
Il ca-ne, the dog.
Il gát-to, the cat.
Fe-de-le, faithful, loyal, trusty,

true.

A-da-cquá-to, watered, washed,
bathed.

E-gli-no hán-no, they (m.) have.
E'l-le-no han-no, they (f.) have.
Il mí-o, mine.
Il tú-o, thine.

Il più sfortunato, the most un-Il sú-o, his, hers, its.
fortunate.

Genitore, father.

Infelice, unhappy, wretched,
unlucky.
Discordia, discord.
Marito, husband.
Moglie, wife.

Ciò resti, that must remain.
Sia detto, be it said (ciò resti fra
di noi, sia detto fra noi, we
must keep it a secret, or to
ourselves).

Il no-stro, ours.
Il vô-stro, yours.
Il ló-ro, theirs.

Il faz-zo-lét-to, the pocket-
handkerchief.

Il toc-ca-la-pis, the lead-pencil.
Tri-sto, sad, afflicted, melan-
choly.

sprightly, jolly, jovial.
Al-lé-gro, cheerful, gay, merry,
Tut-to,

every.

whole, entire, all,

Tut-to il món-do, the whole
world, all the world.
Tút-ta là cá-sa, the whole

house. Di-o, God.

La têr-ra, the earth, earth,
land, ground, soil; country,
province; estate, farm.
La not-te, the night,
Il prá-to, the meadow, pasture-
field.

Il món-do, the world.
Tut-ti i fió-ri, all flowers.
Tút-ti gli uố-mi-ni, all men,
every body.

Tút-te le cá-se, all houses.
Pián-to, wept, shed tears, be-
wailed.

Cred-to, created, produced,
caused.
Man-dd-to, sent.

A'-mo án-che i

A'-mo i miê-i fra-têl-li e le mi-e so-rêl-le. Ho miê-i cu-gí-ni e le mi-e cu-gi-ne. I tuô-i fió-ri só-no bêl-li; i miê-i só-no án-che bêl-li. Qué-sta dôn-na á-ma i suô-i fan. ciúl-li. Mí-o fra-têl-lo ha per-dú-to le sú-e pén-ne. ri-ce-vú-to dú-e pô-mi e quát-tro pé-re da qué-sto giar-diniê-re. Le mí-e ci-riê-ge só-no bel-lís-si-me. Quest' uô-mo ha per-dú-to i suô-i a-mí-ci, e qué-sta dôn-na ha per-dú-to le sú-e a-mi-che. Hô dá-to i miê-i pó-mi a tú-o cu-gi-no. Hai tu a-da-cquá-to i tuô-i fió-ri? Hỗ a-da-cquá-to i miei ed i tuô-i. Mi-a cu-gí-na ha an-che a-da-cquá-to i suô-i. I tuô-i fra-têl-li hán-no com-prá-to dú-e cá-ni che só-no mól-to fedé-li. E'-gli-no hán-no dá-to un cá-ne a mi-a cu-gi-na. Le mí-e so-rêl-le hán-no ri-ce-vú-to dú-e gát-ti da nô-stro zí-o, él-le-no só-no con-ten-tís-si-me. Luí-gi ha per-dú-to i suô-i lí-bri ed i miê-i. A-vé-te voi ve-dú-to i nô-stri fra-têl-li e le ere | nô-stre so-rêl-le ? Hô ve-dú-to i vô-stri cu-gi-ni e le vô-stre a|cu-gi-ne. D6-ve s6-no i nô-stri man-têl-li ed i vô-stri cap

Io diceva, I said.
Stesso, self (io diceva fra me
stesso, I said to myself).
Qui, here; now.
Pasqua, Easter.
Otto, eight (fra qui e otto giorni,
within a se'nnight).
Non lo so, ma lo saprò bene, I do
not know, but I shall be
sure to know it.
Poco, little (tra poco, fra poco,
in a little or short time,
long, soon, shortly, yet
little while).
Perdettero, lost.

Nemico, enemy.
Morto, m., dead, dead body.
Prigioniere, prisoner.
Nove mila, nine thousand.
Ve lo dirò, I shall tell
you.
Quattro, four.

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pêl-li? Ab-bia-mo per-du-to i nô-stri faz-zo-lét-ti, ed i và stri. I nô-stri cu-gí-ni só-no ar-ri-vá-ti. Le nô-stre so-rêl-le son par-ti-te qué-sta set-ti-má-na. Mi-o pá-dre e mí-a má-dre só-no trí-sti. Le mí-e cu-gi-ne só-no sêm-pre al-lé-gre. Abbia-mo a-da-cquá-to i nô-stri fió-ri. A-vé-te voi án-che a-dacquá-to i miê-i? Le vô-stre bot-ti-glie só-no píc-co-le, ma it le nô-stre só-no mól-to grán-di. I tuô-i fra-têl-li hán-no ri-ce-vú-to dú-e toc-ca-la-pis da mí-o cu-gi-no; é-gli-no só-no gli a-mí-ci di mí-o cu-gi-no. Hô com-prá-to tre toc-ca-lá-pis per i fan-ciúl-li di nô-stro zi-o. Dó-ve só-no le vô-stre so-relle? E'l-le-no só-no a Mi-lá-no. Ed i vô-stri fra-têl-li? E'gli-no só-no par-tí-ti per Pa-rí-gi. Nô-stra má-dre ha comprá-to sê-i bic-chiê-ri per le nô-stre cu-gi-ne. Tút-tii miên a-mí-ci só-no par-tí-ti. Qué-sta pô-ve-ra dôn-na ha per-dúto tút-ti i suô-i fan-ciúl-li. Mi-o zí-o ha ven-dú-to tút-te le sú-e cá-se. Tút-ti qué-sti tê-mi só-no fa-cil-lís-si-mi. Abgli uô-mi-ni. Luí-gia ha per-du-to tút-te le pén-ne. In biá-mo com-prá-to tút-te qué-ste bot-tí-glie. A'-mo tút-ti tút-te le stán-ze ci só-no quát-tro spêc-chj. Tút-to il tê-ma è fá-ci-le. La nô-stra cu-gi-na ha pián-to tút-ta la nôt-te. Dí-o ha creá-to tút-ta la têr-ra. A-vé-te ve-dú-to tút-ti quédi-niê-re ha man-dá-to tút-te qué-ste ci-riê-ge a mí-a so-rel-la. sti prá-ti? Hai tu scrít-to tút-te qué-ste lêt-te-re? Il giarA-vé-te voi a-da-cquá-to tút-ti qué-sti pic-co-li ál-be-ri e tút-ti Mí-o cu-gi-no è ar-ri-vá-to con tutt' i suô-i a-mí-ci. Mi-a qué-sti bê-i* fió-ri? Tút-te qué-ste cá-se só-no di mi-o zí-0. zí-a ha man-dá-to tút-te qué-ste pé-re e tút-ti qué-sti pó-mi a qué-sta pô-ve-ra dôn-na.

Passa, passes, passes on.
Stretto, m., stretta, f., narrow,
close, intimate.
Amicizia, friendship (fra noi
passa una stretta amicizia, we
are intimate friends).
Hanno parlato sempre, they have
always spoken.
Sotto, under, beneath, below.
Voce, voice (sotto voce, in a low
tone, low voice).
Loro, them.

Due, two (hanno parlato sempre
sotto voce fra loro due, both
of them always spoke with
in
low
each other
voice).

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I have watered thy flowers. Hast thou seen our glasses and our bottles? Our friends are always jolly. Where are your pocket-handkerchiefs and ours? My (female) cousin has lost our pens and hers. I have given (to) this poor child my pens and thine. My father has sold his dogs and mine. Have you also sold yours? Thy wife has bought ten glasses and four bottles for her daughter. I have given a lead-pencil to thy sister; she has lost hers. Our neighbour has sold all his houses. We have sold all our meadows. I have lost all my pocket-handkerchiefs. All these bottles belong to our uncle. The whole house is beautiful. I love all these beautiful flowers. I think every day (i. e. all days) of Henry and of Charles. Where have you bought these six glasses? I have seen the whole town. All your letters have (i. e. are) arrived. Louisa has (ie. is) departed with all her (female) friends. Have you found all these apples in the garden of your father? Our neighbour has been shedding tears (i. e. has shed tears) the whole week; she has lost all her children.

EXERCISE.-ENGLISH-ITALIAN.

The adversities are for the soul what a tempest is for the air. The count has bought the livery for the waiting-man and for the huntsman. The merchant was reputed to be (i. e. passed for) an honest man. The new singer made yesterday her appearance for the first time in the part of the prima donna. The indignant servant seized some by the hair (i e. hairs), others by the neck and by the arm. He went through the wood and not through the village. He put the dress on the chair, the watch, however, and the money on the table. God has punished him for his sins. The bird was now on the roof, now on the tree. Let us climb this hill together. We have taken to-day a long walk upon the bastion. The suspicion fell upon him and upon her. On similar matters books have been written by a great many excellent men. There is a great difference between him and his brother. Between longing and fear he knew not which resolution to take. excited mortal enmity between these (men) and those. VOCABULARY,

Adversity, av-ver-si-tà, f.
Soul, d-ni-ma, f.
What, ciò che

Tempest, tem-po-rá-le, m.
Air, á-ria, f.

Has bought, ha com-prá-to
Waiting-man, ca-me-riê-re, m.
Huntsman, cac-cia-tó-re, m.
Livery, livré a, f.
Merchant, mer-can-te, m.
Was reputed to be, pas-sti-va
Honest man, ga-lant-u-mo, m.
New singer, muô-va can ta-
tri ce, f.

ance,

Made yesterday her appear-
re-ci-to je-ri
First time, prí-ma vôl-ta, f.
Part, pair-te, f.

Servant, sér-ro, m.
Indignant, sde-gná-to
Seized some, pré-se al-cu-ni
Hair, ca pél-lo, m.

Other, al-tro, m.

Neck, col-lo, m.

Arm, brác-cio, m.

He went through, é gli è pas

sá-to per Wood, bô-sco, m. And not, e non

Village, vil-lg-gio, m.

He put, é-gli pó-se

Dress, d-bi-to, m.

On, su

Chair, sé-dia, f.

Watch, o-ri-ô lo, m.

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He

ó-ra Roof, tét- to, m. Tree, dl-be-ro, m. Let us climb together, sa-lidmo in-sie-mo (with su). This hill, qué-sta col-li-na, f. We have taken, ab-bid-mo fát-to

Long walk, lún-ga pas-seg-giá

ta, f.

Bastion, bastió-ne, m.
Suspicion, so-spêt-to, m.
Fell, & ca du-to
Him, lui
Her, le-i.

Similar, sí-mi-le
Matter, ma-te-ria, f.

Books have been written (i.e.
was written) fu scrit-to
A great many, mol-tis-si-mo
(i. e. very much).
Excellent man, và-lent-uô-mo

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However, all' in-cón tro

This, qué-sto, m.

That, quél-lo, m.

Money, da-na-ro, m.

Enmity, ni-mi-cí zia, f.

Has punished him for, lo ha Mortal, mor-tá-le.

Table, ta-vo-li-no, m.

pu-ni-to per

MATHEMATICAL ILLUSTRATIONS.-No. II.

(Continued from p. 150, Vol. IV.)

ARITHMETICAL LOGARITHMS.

1. The term Logarithms is derived from two Greek words (arithmoi and logōn), and literally signifies the numbers of ratios; but we ought to invert the words according to their present use and meaning (logoi and arithmōn), and then it will literally signify the ratios or indices of numbers. Hence logarithms are defined, a system of artificial numbers which indicate, or represent, the natural numbers.

2. By the use of logarithms instead of the natural numbers, the tedious and difficult calculations of arithmetic may be superseded; the irksome and laborious operations of multiplication and division may be effected by simple additions and subtractions; and the operose and intricate processes of involution and evolution, or the raising of powers and the extraction of roots, may be reduced to simple multiplications and divisions.

NATURE OF POWERS AND ROOTS.

3. If a number be multiplied by itself, the product is called the square or second power; and the number itself, the root or first power. If the second power be multiplied by the root, the product is called the cube or third power. If the third power be multiplied by the root, the product is called the fourth power; and so on. Thus, the multiplication of a number by itself, and by each successive product, produces the powers of that number. This process is called involution, or raising of

powers.

4. The numbers which indicate how often the root enters as a factor into each of the powers, are called the indices of those powers. Thus, the index of the first power is 1; of the second power, 2; of the third power 3; of the fourth power, 4; of the fifth power, 5; of the sixth power, 6: and so on.

5. If a number be considered as a power of another, the latter is called such a root of the former as is denoted by the name of that power. Thus, if one number be considered as the square of another, the latter is called the square root of the former. If one number be considered as the cube of another, the latter is called the cube root of the former. If one number be considered as the fourth power of another, the latter is called the fourth root of the former; if one number be considered as the fifth power of another, the latter is called the fifth root of the former; and so on. The process by which the roots are found is called evolution, or the extraction of roots, and is explained in the common treatises on arithmetic.

6. The reciprocal of a number is a fraction whose numerator the reciprocal of 2 is; of 3 is; of 4 is ; and so on. is unity, and whose denominator is the number itself. Thus, If the number be a fraction, its reciprocal is the fraction inverted. Thus, the reciprocal of is; of is; ofis or 4; and so

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8. If the number 2 be assumed as the root or first power, then the successive powers of 2 are calculated by simple multiplication in the following manner: Table of Products. 2=

2 X 2=

2 First Power 4 Second Power 8 Third Power

4 X 2:

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2, Fourth root of 16

2, Fifth root of 32

: 2, Sixth root of 61

- 2, Seventh root of 128

2, Eighth root of 256 =2, Ninth root of 512

2, Tenth root of 1024 =2, Eleventh root of 2048

=2, Twelfth root of 4096 etc.

10. The multiplication of the powers or roots of the same number, is indicated by the addition of the indices of the factors. Thus, if 16, the fourth power of 2, be multiplied by 8, its third power, the product 128 is its seventh power. This process is thus indicated: if 4 and 3, the indices of the factors, be added together, the sum 7 is the index of the product. Again, if 4, the cube root of 64, be multiplied by 2, its sixth root, the product 8 is its square root. This process is thus indicated if and, the indices of the factors, be added together, the sum is the index of the product. See Prob. VI. Addition of Fractions, p. 329, vol. II.

11. The division of powers or roots of the same number is indicated by the subtraction of the index of the divisor from that of the dividend. Thus, if 512, the ninth power of 2, be divided by 128, its seventh power, the quotient 4 is its square. This process is thus indicated: if 7, the index of the divisor, be subtracted from 9, the index of the dividend, the remainder 2 is the index of the quotient. Again, if 8, the square root of 64, be divided by 4, its cube root, the quotient 2, is its sixth root. This process is thus indicated: if, the index of the divisor, be subtracted from, the index of the dividend, the remainder is the index of the quotient. See Prob. VII. Subtraction of Fractions, p. 357, vol. II.

12. The involution of powers or roots of the same number is indicated by the multiplication of the indices of the given powers or roots of those of the required powers. Thus, if 16, the fourth power of 2, be squared, the square 256 is the eighth power of 2. This process is thus indicated: if 4, the index of the fourth power of 2, be multiplied by 2, the index of the square, the product 8 is the index of the required power of 2. Again, if 2, the sixth root of 64, be cubed, the cube 8 is the square root of 64. This process is thus indicated: if, the index of the sixth root, be multiplied by 3, the index of the cube, the product is the index of the required power of 64. See Prob. VIII. Multiplication of Fractions, p. 25, vol. III. 13. The evolution of powers or roots of the same number is indicated by the multiplication of the indices of the given powers or roots, by the indices of the required roots; or, by the division of those indices by the denominators of the indices of the required roots. Thus, if the cube root of 512, the ninth

power of 2, be extracted, the cube root 8 is the third power of 2. This process is thus indicated: if 9, the index of the ninth power, be divided by 3, the denominator of the index of the cubo root, the quotient 3 is the index of the required power of 2. Again, if the square root of 4, which is the same root of 16, be extracted, the square root 2 is the fourth root of 16. This process is thus indicated: if, the index of the given root, be multiplied by, the index of the required root of 4, the product is the required root of 16. See Prob. VIII. above cited.

14. If any power or root of a number be divided by itself, the quotient is unity. On the principle that the division of powers and roots of the same number is indicated by subtracting the index of the divisor from that of the dividend, the index of the quotient unity, in this case, is 0. Hence, that power of any number, whose index is 0, is unity. Consequently, in the expression 2° J, unity may be denominated the zero power of 2.

CORRESPONDENCE.

THE FOUR-BALL QUESTION, EUCLID, etc.

Sin, As I was informed by the last page of the last monthly part of the P. E. that the Ball Question remains unanswered, I was induced to give it a trial, with the humble assurance (though I have not studied Euclid) that the application of a little common sense would furnish a correct answer.*

I should like to know, by your favour, whether mathematics or algebra teaches usto find answers to similar questions, and with the same nicety, without the aid of paper and compasses.

As to the study of Euclid, which all the world esteems and recommends as an important item of education, excuse my (ignorance, perhaps,) want of veneration in saying, that I felt somewhat amused and deceived, on finding, after an hour's perusal of Euclid [your edition] for the first time, that it consisted in definitions that seemed to be a mere play upon words, and demonstrations shrouded in dark sentences, and mystified by letter repetitions to set forth so many apparently simple and self-evident facts. I therefore placed it aside for an undisturbed repose, till I could better understand the qualities which render it an essential portion of education, and the merits that have gained for it a world-wide fame. These, perhaps, you will direct me to discover. Pardon my having trespassed at this length; whilst I briefly add, that, as I have the happiness of being a young man in this golden age for the advantages of mental culture, and better opportunities for study than those who lived before me, I may have the pleasure of witnessing the consequent good effects on myself and society in general. To this end I hope that whilst the good aim is being made to bring knowledge within the reach of the many, that its genuineness will not lose in the attempt to make it cheap (rather let us have quality than quantity); that this may ever be the case with that boon to the world, the P. E., is the sincere R. F. T. desire of yours respectfully,

[We have inserted this let:er because it contains a strange mixture of good and evil-of the phrenological bumps marked Nos. 15 and 17, with those marked Nos. 5 and 10. We have so often recommended the study of Euclid and algebra, that our correspondent must excuse us this time; but if any of our readers will kindly take up the cudgels on our behalf, we shall be thankful; only let them remember the advice that David gave to his captains concerning Absalom-2 Samuel xviii. 5.]

MANUSCRIPT MAGAZINES.

SIR, Your correspondent W. B., in No. 100, gives a plan for This plan is the formation of "Mutual Instruction Societies." applicable where there are several young men in a town who pursue the same course of study; but there are, no doubt, many students of the P. E. who reside in small places, where it is impossible to form such classes, and many who cannot make it convenient to attend them when formed.

To such students the following plan may be of some use; it is one which I have tried, in conjunction with some friends, for the last eighteen months, and I have found it successful. The idea is not mine. Four or five young men agree to circulate a MS. Magazine on the subject which they wish to study. The first thing is to choose one of their number as conductor, who regulates the postage, the time of circulation, etc. He commences by writing a

Our correspondent's answer is 4 inches 3-16ths; and nearly the same answer has been sent by hundreds: but it is entirely wrong.

paper on the particular subject of study, and then forwards it by | join a Mutual Instruction or other Class on these subjects, it would be a post to the next member in rotation; this member contributes his quota to the work, asks any questions which he wishes to be answered, and passes it on to the next in rotation. Thus it circulates among all the members. When it comes into the conductor's hands the second time, he withdraws his first contribution and replaces it by a second one, prepared, for the most part, whilst No. 1 was in the possession of the other members. He again forwards the magazine, when each member does the same, circulating it as before.

know.

great help to him. He is right in the solution of the algebraic question.A. A. A. (Sherborne): Read "Cassell's Emigrants' Hand-book; or a Guide to the various Fields of Emigration in all parts of the Globe."-JOHN FORD to the most eminent physician he can find. Good men are generous in pro(Derby): Hutton's Mensuration.-ASPIRANT (Lower House) should apply portion to their greatness.-T. THOMAS (Derby): Still to be procured; we presume the Society of Arts has agents or connexions in all great towns.ELLEN SMITH (Westbury): We should be only too happy to answer her question to her satisfaction, but we cannot.-R. 8. (Bradford): We forget. -R. THOMPSON (Upleatham): Yes.-M. SCHOFIELD (Waterhead): The female elephants have tusks as well as the males, but they are not so large; It adds much to the value of such a plan, if the members mutu- and when the animals are young, the tusks are not visible.-JUVENIS (Carally agree to correct each other's spelling. lisle): His 45 solutions are received. The impropriety of his concluding Í should be glad to meet with four or five earnest young men remark is excused, because he is Juvenis.-T. WELLS (Linfield): We don't who desire to study physiology in this way; or I would be glad to unite with any of the would-be artists" of your "Correspon- 24 inches, it is very near the truth-very near, indeed, to be found by expeT. WATKINSON (Stanstead): If your answer to the Four-ball Question be dence" columns, in studying the theoretical part of their profes-riment alone. The word Excelsior means higher; it is from the same Latin sion. Address, "T. J., Post-office, Bedminster, Bristol." Yours root as the word excel.-W. CHEETHAM (Stockport): Study chemistry, and respectfully, T. J. you will see at once how the earth is neither lighter nor heavier since the creation, by the immense consumption of coal.-AN ASPIRING APPRENTICE is entitled to know why we wished him to call on us; we saw that he was enveloped in a mist of error, and we thought that a few words spoken to him in a kind and proper manner, in a free and easy conversation with him, would do him more good than a bushel of letters or arguments; with these useful: but we think his labour in learning French will be lost, if it be for we can't fill our pages. He may learn either Latin or French; both are the sole purpose of reading Voltaire, Volney, Mirabeau, Rousseau, or any of that motley infidel crew. His poetry is pretty fair, and indicates a better state of mind than his former letter. It is essential to blank verse that every line should contain ten syllables; and these syllables must, if possible, be made iambic, that is, consisting of a short and a long syllable alternately, as in Milton, thus:

the "

ALGEBRA, ETC.

SIR,-In compliance with your request, appended in a note to centenary of problems," which you gave in No. 101 of the P. E., I venture to send you two solutions. If I tell you that I never knew anything of algebra until last Christmas, when I was induced to commence the study from seeing the easy lessons in your truly excellent publication, I hope you will give me credit for my statement.

I am really a self-taught man; but as languages and other subjects have occupied so much of my time and attention, I was surprised at the improvement I made within the short space of a fortnight in the study of algebra. Although I began in real earnest, yet I am not insensible to the fact that my determination may be ascribed to the very simple and easy method laid down in the first lessons you published. For this acquisition, moderate as it is, I beg you will accept my most sincere thanks. It is, indeed, entirely owing to your useful and indefatigable exertions in the cause of self-improvement and of education and literature generally, that I have formed a slight acqua'atance with a science that always appeared to me to be involved in the most inexplicable mystery.

That you may go on and prosper, even beyond your most sanguine expectations, is the fervent wish of, Sir, your very grateful servant, ALEFH.

P. S. As your Lessons were not all published so as to enable me to prosecute my studies, I bought the shilling volume, "Cassell's Algebra," which I have found very acceptable. I should recommend all students in algebra to procure it.

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ANSWERS TO CORRESPONDENTS.

A. C. H. (Lambeth) is recommended to make his own solutions. The exact strength for most purposes of testing is of no consequence. The great consideration is to have them pure. When a solution of any particular strength is necessary, we shall take care to intimate this. The occasion will not occur so long as we are occupied with qualitative not quantitative chemistry.

J. G. CUNNINGHAM (Sunderland): Right, and thanks.—A. B. (Hoxton): The subject on which he makes inquiry will soon appear.-H. HOLBROOK (Ipswich): Cassell's Arithmetic, Algebra, and Geometry, price ls. each, are the best for him; and they are indispensable.-PETER (Ripponden) should apply to the Bishop's chaplain, as he says.-T. R. (Durham): We can't give the required information.-J. G. THORNLEY (Ballyclare): It is entirely out of our power to oblige him and his friends with the subject requested at present.-PEREGRINE PICKLE (Aberystwith): We would strongly advise him to persevere with the Lessons on Music and Drawing in the P. E.; he will be sure ultimately to succeed. If he could by any means

"Before the heav'ns thou wêrt, ănd at the võice
Ŏf God, ǎs with ǎ mäntlě didst Invest."

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