Gio-van-ni da Fit-so-le, Pie-tro da Cor-tó-na, Leo-nár-do da Vin-ci,

Guí-do da Siena, Po-li-dó-ro da Ca-ra-vág-gio, Ra-faél-lo da
Ur-bi-no, etc.

1. A gold snuff-box. 2. A silver vage. 3. A velvet dress. 4. Italian

wine. 5. A heart of stone. 6. The iron wire. 7. Fine leather gloves. Da, also, may denote any origin or commencement referring 8. A straw hat. 9. A gold, & silver mine. 10. English steel. 11. to time, and then it means since :

Iron of Styria. 12. Frankfort fair. 13. To-morrow's festival. 14,

To-day, now-a-days. 15. Yesterday's comedy. 16, The theatre of the Da che vi vi-di la pri-ma rólta, since (that day when) I saw you present day. 17. A four weeks' illuess. 18. The wine of eight, of the first time.

twenty years. 19. The Seven Years' War. 20. A fine prospect. 21, Dál-la mi-a gio-va-néz-za in si-no quo-slo tim-po, since my youth till The sound of the trumpet. 22. A person of fidelity. 23. He is this day.

obstinate. 24. A courtier, a fashionable man. 25. A learned man, Dau án-no pas-si-to in quà, since last year.

of agreeable temper, 26. A man of rank, a polite man. 27. An Da dú-e mé-si in qud, two months since.

ill-behared man. 28. A very clever man, of great fame. 29. A man Dál-la môr-te di mi-o pá-dre in qud, since the death of my father. of middle stature. 30. An ill-natured man, a warrior, a man of war.

31. A man of low birth. 32. A inan of indifferent health. 33. The The phrases da mat-ti-na, da sé-ra, da nôt-te, mean, in the matter is of great importance. 34. A celebrated physician. 35. The morning, in the evening, in the night (by night, at night) : art of the turner, of the dyer. 36. The types-foundry. 37. A camp Ô-pe-ra da far da mat-ti-na, work to be done in the morning.

of diversion. 38. A cotton-mill. 39. A bridge of boats, pontoon. 10. Non 6-sca da cc-sa che da sé-ra, he only goes from home in tho soldiers. 42. A law project. 43. The commission, the diploma. 44

An inspector of the cannou foundry. 41. The equipment of the evening Tá-li co-so non si fán-no da nót.to, such things are not done by The certificate of birth. 45. The United States of America. 46. The

Emperor of the Russians. 47. The frontiers of Saxony. 48. Within night.

the space of three months. 49. A prodigy of a man. 59. A man of Da also signifies about, nearly, close upon, not far off from, thirty. 51. A gentleman's model. 52. That blockhead of a servant of

yours. 53. This devil of a woman. .etc. :

54. That sly fox of a Wilham.

55. A little bit of a rogue ! 56. That poor brother of mine! 57. So Hỏ gua-da-gná-to da cin-que lire ster-li-ne, I have gained or won much wine and as much again of water. 58. It is such a bright moon. about five pounds sterling.

light. 59. One named Jonas. 60. Judas surnamed Thaddeus. 61. Ho per-dú-to da séi a 6t-to tál-le-ri, I have lost from about six to Three months' leave, discharge.

eight dollars. Da Ro-ma a Ná-po-li sa-rán-no da cên-to ses-sán-ta mi-glia, it is

EXERCISE 11. about a hundred and sixty miles from Rome to Naples. E-gli vi re-ste-rd da cin-que a si giór-ni, he will stay there from

1. This horse is beautiful. 2. This snuff-box is very small. 3. This about five to six days.

hotel is large. 4. This child is my brother. 5. This book is for my

father. Sti-má-va-si a-vé-re in Fi-rên-se da no-van-ta-mi-la bóc-che tra ub-mi- 8. Where have you found this ring! 9. Your little sister has a hand

6. This penknife is for my brother. 7. I have found a ring. ni e fém-mi-ne e fan-ciúl-li, about ninety thousand persons, men, women, and children, were estimated to be in Florence.

some book. 10. My mother has bought this hat. 11. Thy brother

has seen this fine rriage. 12. Your little brother is a good child. A logical contradiction and anomaly, though introduced and 13. Where hast thou bought this snuff-box? 14. This watch is very sanctionod by a universal usage, for the most part in the place good. 15. This beautiful ring is for this child. 16. My uncle has a of the preposition a, the constant employment of da in connec- lave received a present. 19. Have you written a letter ? 20.


son and a daughter. 17. I have seen thy brother and sister. 18. We tion with those verbs which, with some house, mansion, apart- sister has received a fine cap. 21. I have sold my carriage. 22. Hast ments, lodging, or any other place of continuance, denoto any thou also sold thy carriage ? 23. This present is for your aunt. 24. kind of motion to or towards, any kind of living or residing with, Your son is very little, but he is good. 25. My daughter is rery tall. and any kind of visit paid to, a person :

26. This father has a beautiful daughter. 27. This child is my son. An-dá-re dal mi-di-co, dal cal-co-la-jo, to go to the plysician, to the lost his hat and his umbrella. 30. Our uncle has sold his beautiful

28. The garden that I have seen is very large. 29. My father has shoemaker.

carriage. 31. My sister has found her ring. 32. This father has lost Do-má-ni ver-rò da voi, I shall come to you to-morrow.

his daughter. 33. This mother has lost her son, 34. My uncle has I'-o vi me-ne-rò da lui, I shall conduct you to him.

bought a hood for his little daughter. 35. This present is for my Ve-ní-te da me, dal mer-cán-te, come to me, to the merchant. So-no stá-to da lui, dal fra-tel-lo, I have been at his, at the brother's 37. Our aunt has bought a very fine cloak for her son. 38. Have you

sister. 36. This boy has written a very long letter to his mother. house (with him, with the brother).

found a ring? 39. My unclo has lost his cloak.
A-bi-ta, al-lóg-gia da sú-o zi-o, he lives or resides with his uncle.
Da is sometimes a substitute for di :--
Li bia-si-ma-va du-ra-mén-te, ó-ra da fol-lé-a, 6-ra da co-dar-di-a

KEY TO EXERCISES IN LESSONS IN ITALIAN.-IV. (instead of di follia, di codardia), he severely blamed them, now

EXERCISE 12. for their folly, now for their cowardice.

1. I tempi d' adesso non sono i migliori. 2. Egli si era nascosto Es-si han-no molti mó-di da al-log-gia-ne o da pas-sá-re quello (in- nella stanza di dietro. 3. Nostra città ha un ponte di pietra, la vostra

stead of di alleggiare, di passare), they havo many means to make ha solamente uno di legno. 4. Eduardo ha ricevuto da Londra un it easier or to pass over it.

oriuolo d' oro, una spada d' argento, e un pajo delle fibbie. 5. Una The particle da can never be really omitted, and the cases of volta

si portavono degli abiti di panno o de gile di veluto. 6. L'uso ellipsis that I mentioned only servo the purpose of grammatical trova da vendere della carne di manzo, di vitello, di castrone. Cho explanation.

significa questo suono di campane ? 9. Che dite del panno che ho

comprato ? 10. Esso è buono e fino. 11. E di colore ? 12. Esso è KEY TO EXERCISES IN LESSONS IN ITALIAN.--XIV.

bello. 13. Che pensate dell'uomo che vedete, del ragazzo ch'egli

mena seco, e del mendico che gli va dietro ? 14. Ecco dieci braccia EXERCISE 9.

di taffeta del quale volevate avere, e dodici braccia della tela batista 1. Il sartore domanda nove braccia di panno, due dozzine di bottoni, che arete domandata. 15. Mandatemi una dozzina di limone, e dua e mezza oncia di seta. 2. Mandato a prendere un pane di zucchero, é libbre di fichi che avete ricevuti da Smirna. 16. Cedetemi una fiusdue libbre di caffè. 3. Io ritornero in un quarto d'ora. 4. Finite di chetta dell' eau di Cologue che vi è stata mandata. bere questo bicchiere di vino e mangiato questo crostino di pane. 5. Prendete la carta geografica, e cercatemi la città di Parigi e la citta di Londra. 6. Io vengo per ordine del padrone a dirvi che si facciano le preparazioni per il giorno di domani. 7. Il mese d' Aprile è variabile ; LESSONS IN GEOGRAPHY.-XXXVIII. il mese di Maggio, incontro, è molto ameno. 8. I mesi di Decembre

CHIEF POLITICAL DIVISIONS OF NORTH AMERICA. e di Gennajo sono i piu rigidi dell' anno. 9. Che abito metterete per il ballo di domani ?' 10. Siete stati alla commedia diari? 11. Gli The following table exhibits the principal political divisions of furono assegnate le camere di sotto.

the continent of North America, showing the capital or most

important city in each division, and the river, etc., on which it * The English learner will, perhaps, best understand this use of da stands; the approximate area in English square miles; the by translating it with sprung from,

number of inhabitants, also stated approximately; and the + Since (denoting time, and not in the seuse of as or because) is nnmber of persons resident, on an average, on each square translated by fin da, da ... in p:i, or ds-, when it precodes a uoun,




THE CHIEF DIVISIONS OF NORTH AMERICA, THEIR CAPITALS, The most important colony on the west side of British North POPULATION, AREA, Etc.

America is Vancouver Island, which, from its rich coal and iron

mines, seems destined to become the seat of manufactures of Rivers, etc., Approxi

all kinds, and the focus in which will be centred a large trade Divisions. Capitals or Chief on which matearea

between our North American colonies and Eastern Asia and Cities. the Capitals in Square

Australia, when the Fertile Belt is traversed by a railway stand.

running from Canada to British Columbia. The last-named

colony, yet in its infancy, yields timber, coal, and gold; while of Brit. N. America Ottawa Ottawa 3,600,000 3,500,000 1

Stickeen, which lies to the north, and was constituted a British Greenland Julianshaab Sea Coast 380,000 10,000 1

colony in 1862, little is known at present, or will be known United States Washington Potomac 3,345,000 31,500,000

while more desirable localities for emigration are found in Mexico Mexico

L. Tezcuco. 857,500 8,000,000 Vancouver Island and British Columbia.
Central America New Guatemala. Montagua 192,000 2,515,000 13 Of the United States, which is fast approaching the comple.
W. I. Islands,etc. Havanna.

. C. of Cuba.
98,000 3,900,000| 40 tion of its first century of existence as an independent nation,

we can only give a list of the states of which it is composed, From the above table it will be seen that the great territorial distinguishing the states from the territories and territorial divisions are few in number ; but these are subdivided into a districts, and the free states from the former slave states. At number of lesser districts, as will be shown presently. Leaving the present time the republic of the United States, which Greenland out of the question, the reader will be struck at once stretches from the southern limits of British North America to. with the scanty population of British North America, when the head of the Gulf of California and the Rio Grande del viewed in comparison with its immense extent, which is larger Norte, the chief part of the boundary line that separates it. even than that of the United States. It speaks volumes about from Mexico, consists of one federal district, thirty-eight states, the defenceless state of by far the greater part of the British seven territories, and three territorial districts, including Alaska, territory, should the jealous and restless people of the United which was formerly called Russian America. The federal district States be inflamed by a furious desire for annexation; and it is Columbia, a small piece of land lying around the capital, tells a sad story of the negligence and indifference of the Washington, which stands on the Potomac river. Of the states, authorities at home, who, by the judicious use of emigration, territories, and territorial districts, there are four groups, supmight have settled the rich acres of the Fertile Belt-a track posing the whole country to be divided into two parts by the of rich woodland and meadows, stretching westward from Mississippi, and each of these parts to be again sub-divided Canada to the Rocky Mountains-with families who are wearily into two portiong, the eastern moiety into the north-eastern and dragging out a cheerless existence at home, thus stocking it south-eastern states by the rivers Ohio and Potomac, and the with men who would be ready and willing to defend their own, western half into the north-western and south-western states. and whose labour would soon render the country an abundant It will be as well to state at once that the dates preceding the source of food, both animal and vegetable, to those they have names of each state show the period at or about which it was left behind in the old mother-land in the distant east.

formed or settled, or admitted into the Union after the DeclaraGreenland, or Danish America, is a barren, snow-covered tion of Independence in 1776. The states in italics are the country, with a coast broken by deep inlets of the sea, on some former slave states, and those to which the letters c. s. are of which a few fishing towns and villages have been founded. added conjointly formed the federal union of the Confederate It is, in all probability, an island, though no man as yet has States from 1861 to 1865. The letters T. and T. D. affixed to been able to reach its northern limits, or even proceed to any the territories and territorial districts will serve to distinguish distance into the interior. Its inhabitants are chiefly Esquimaux, the former from the latter. who live on seals and whale-blubber. About 500 Danes are found in the fishing-towns of Julianshaab, Christianshaab,

1. NORTH-EASTERN STATES (16). Upernavik, Jacobshavn, and Frederickshaab, who work the 1820 Maine. Augusta. 1627 Delaware (7). Dover. whale fisheries and obtain copper on Discoe Islands, and an 1623 New Hampshire (1).. Concord. 1632 Maryland (8). Annapolis. inferior coal on some parts of the coast.

1791 Vermont. Montpelier, 1682 Pennsylvania (9). . Harrisburg.. British North America is divided from the United States by 1635 Rhode Island (3). Newport, etc. 1816 Indiana.

1620 Massachusetts (2).... Boston. 1802 Ohio.

Columbus. the parallel of 49° north latitude and the chain of great lakes in 1635 Connecticut(4). Hartford, etc. 1818 Illinois.


Springfield. the upper part of the river St. Lawrence. It is separated from 1614 New York (5). Albany. 1848 Wisconsiu. Madison. Alaska on the west by the meridian of 141° west longitude, from 1644 New Jersey (6). Trenton. 1837 Michigan. Lansing. the Arctic Ocean to Mount St. Elias, where the remainder of the line of demarcation from this mountain to the top of

2. SOUTH-EASTERN STATES (10). Observatory Inlet, one of the branches of the great strait that 1607 Virginia (10)c.s. Richmond. 1669 S. Carolina (12)cs.... Columbia. divides Queen Charlotte Islands from the mainland, is formed 1863 West Virginia, Wheeling. 1732 Georgia (13) c.S....Milledgville.. by the coast range of the Rocky Mountains. The following

1792 Kentucky. Frankfort. 1845 Florida, c.s. Tallahassee.

1796 Tennessee, C.S. Nashville. 1819 Alabama, c.s. Montgomery. are the principal divisions of British North America :

1669 North Carolina (11) c.9. Raleigh. 1817 Mississippi, C.s. Jackson.

Hudson Bay Ter. Fort York.

Brit. Columbia New Westminster. 1846 Iowa.

Des Moines. Montana (T.) Virginia.
Vancouver Is. Victoria.
1858 Minnesota. St. Paul.

Idaho (T.) Lewiston. } St. John's. and Labrador.

Dakotah (T.) Yankton. 1859 Oregon. Salem, 1867 Nebraska. Omaha.

Washington (T.) Olympia. Of these the Hudson Bay territory, lying round Hudson Bay

Wyoming (T.D.)

Alaska (T.D.) and stretching westward to the Rocky Mountains, a tract

Sitka, abounding in large lakes and rivers, is chiefly remarkable for

4. SOUTH-WESTERN STATES, Etc. (12). the Fertile Belt, which is intersected by Lakes Winnipeg and 1850 California. Sacramento. Indian Territory (T.D.) Winnipegoos, and traversed by the Saskatchewan river. This 1865 Nevada. Virginia City. 1861 Kansas. Topeka. belt is a fine wheat-producing country, and, abounding in wood,

Utah (T.) Filmore City, 1821 Missouri, Jefferson City. water, and fine pastures, offers considerable advantages_to

Arizona (T.) Prescott. 1836 Arkansas (c.s.) Littlerock. settlers proceeding thither from the United Kingdom. The

New Mexico (T.) Santa Fe. 1812 Louisiana (c.s.) Baton Rouge.

1867 Colorado. Dominion of Canada, which was formed in 1867, consists of the

Denver City. 1845 Texas (c.s.) Austin. provinces of Ontario and Quebec, formerly known as Upper and It should be stated that the territories send only one delegate Lower Canada, or Canada East and Canada West, New

Bruns- to Congress, while the districts are unrepresented. The states wick and Nova Scotia, to which last belongs Cape Breton Island. send members in proportion to their population. Several of the Ottawa, the capital, is in the

province of Ontario. Newfound present states were separated from the thirteen states that oriland is famous for its fisheries. The governor of this island ginally entered the Union in 1776: thus Maine was included in includes Labrador, with its bleak, iron-bound coast, within his Massachusetts, and Vermont in New York ; West Virginia and jurisdiction. In the Gulf of St. Lawrence is also found the Kentucky were portions of Virginia ; Tennessee belonged to little colony of Prince Edward Island.

South Carolina, and Alabama and Mississippi to Georgia. The




da = d; and

= 2°=1; and 2 + 2 =

reader can now trace the original limits of the British American 174. The number of times a quantity is employed as a factor colonies prior to 1776. The New England States—so called to produce the given power is generally indicated by a figure or because they formed parts of New England, a tract settled by letter placed above it on the right hand. This figure or letter is the Plymouth Company-aro Maine, New Hampshire, Massa- called the index or exponent. Thus a x a = aa, is written a* chusetts, Rhode Island, and Connecticut.

instead of aa; and a X a Xa= aaa, is written af. Mexico, stretching southwards from the United States to the The index of the first power is 1; but this is commonly omitted, Gulf of Tehuantepec, and including the peninsula of Yucatan, that is, al = a. was a dependency of Spain from its conquest by Hernando An index is totally different from a co-efficient. The latter Cortes in 1521 to 1821, when it regained its independence under shows how many times a quantity is taken as a part of a whole; Iturbide, & native, tho then became Emperor, but did not long the former how many times the quantity is taken as a factor, retain his crown. Since then it has been a prey to civil discord Thus 4a=a +a+a+a; but at= a xa xa xa = aaaa. and intestine strife. Mexico consists of twenty-six small states If a=4, then 4a = 16; and a' = 256. in a federal union, with a constitution somewhat similar to

175. Powers are also divided into direct and reciprocal. that of the United States. Central America, which includes the British territory of

Direct Powers are those which have positive indices, as d®, ds, Honduras, chief town Balize, under the jurisdiction of the etc., and are produced by multiplying a quantity by itself, as governor of Jamaica, contains the following republics, and the above described. Thus d x d = d; dxdxd= ds; and Mosquito territory, which is under British protection :

dxdxdxd= d. Guatemala, Capital New Guatemala. Nicaragua, Capital Managun. from dividing a unit by the direct power of that quantity, as

The Reciprocal Power of a quantity is the quotient arising San Salvador, » Cojutepeque. Costa Rica,

» St. Jose.

1 1 1 Honduras, Comagua.


da' da' ' The West India Islands, which lio like a breakwater at the entrance to the Gulf of Mexico, and stretch southward between

A reciprocal power is produced by dividing a direct power by the Caribbean Sea and the Atlantic Ocean to the

north coast of its root, till we come to the root itself ; and then continuing the South America, are divided among various European powers. division, we obtain the reciprocal powers. Thus The western part of Hayti, or St. Domingo, is the only d2

d d

1 1

dai and

index ; thus 2 = d–, etc. The direct and reciprocal powers

1 independent state.

= d;

d= etc. d

à British.-The Bahamas, Jamaica, French.-Martinique, Guadaloupe, 176. For convenience of calculation and expression, reciprocal

Barbadoes, Antigua, St. Christo. Grande Terre, Desinado, Marie pher's or St. Kitts, St. Vincent, Galante, and St. Martin (north powers are written like direct powers with the sign – before the

1 Dominica, Grenada and the side). Grenadines, Trinidad, Montser. Danish.-Santa Cruz, St. John, St. rat, Nevis, Avguilla, Barbuda, Thomas.

of d are d', d®, d, d', d', d-1, 2-3, 2-3, 2-, etc., in which ď!" the Virgin Isles, St. Lucia, and Swedish-St. Bartholomew.

=1. Tobago. Spanish.-Cuba, Porto Rico, Isle

177. INVOLUTION is the process of finding any power of a Dutch. St. Eastatius, Saba, Cura of Pines, and Dominica, on east çoa, Buen Ayre, St. Martin part of Hayti.

quantity, as explained in Art. 173. (south side).

178. To involve a quantity to any required power. The Bahamas, a cluster in the middle of the Atlantic Ocean, RULE.-Multiply the quantity by itself, and by its successive belong to Great Britain. The chief town of the British West products, till it is taken as a factor as many times as there are Indies is Spanish Town, in Jamaica. The West India Islands, units in the index of the power to which the quantity is to be it should be said, are divided into three groups-1, the Bahamas; raised. 2, the Greater Antilles, including Cuba, Porto Rico, and Jamaica;

All powers of unity or 1 are the same, viz., 1. For 1x1x1 and 3, tho Lesser Antilles, comprising the rest of the islands, from the Virgin Islands in tho north to Trinidad in the south.

x 1, etc. =1. A line drawn between Martinique and St. Lucia separates this

179. A single letter is involved or raised to any power, by third group into the Windward and Leeward Islands, the latter giving it the index of the proposed power; or by repeating it as being the northern division, and the former the southern. a factor as many times as there are units in that index. A projection for a map of North America may be constructed

If the letter or quantity has a co-efficient, it must be raised to in the same way as projections for maps of Europe and Asia. the required power by actual multiplication. Those who wish to fill up a projection for themselves may obtain

EXAMPLES. the outline of the coast, and directions of rivers and mountain 1. The 4th power of a is aaaa, or a'. ranges, from any good map, and the latitudes and longitudes 2. The 6th power of y is YYYYYY, or yo. of the principal places, headlands, etc., from the index to any

3. The nth power of æ is ezz ... repeated n times, or ä". atlas.

180. The method of involving a quantity which consists of

several factors, depends on the principle, that the power of the LESSONS IN ALGEBRA.-XX. product of several factors is equal to the product of their powers. INVOLUTION, OR RAISING OF POWERS.

EXAMPLE.—What is the square of ay? Here, (ay) = a*y*. 173. When a number is composed of the product of the same For, by Art. 178, (ay)*= ay X ay. factor any number of times, the result is called a power of the

But ay x ay=ayay = aayy=aRy? Ans. factor. Powers are divided into different orders or degrees; as

In finding the power of a product, therefore, we may either the first, second, third, fourth, fifth powers, etc., which are also involve the whole at once, or we may involve each of the factors called the root, square, cube, biquadrate, etc.

separately, and then multiply their several powers into each The powers take their names from the number of times the

other. root, or first power, is used as a factor in producing the given 181. When the root is positive, all its powers are positive also ; power,

but when the root is negative, the ODD powers are negative, while The original quantity is called the first power, or root of all the EVEN powers are positive. the other powers, because they are all derived from it.

Hence any odd power has the same sign as its root. But an Thus, if 2 be the root or first power, then

even power is positive, whether its root is positive or negative. 2 X 2 = 4, the square or second power of 2.

Thus (+ a) x (+a)=a. And (-a) X (-a)=a. 2 X 2 X 2=8, the cube or third power.

182. To involve a quantity which is already a power. 2 X 2 X 2 X 2 = 16, the biquadrate or fourth power, etc. RULE. --Multiply the index of the quantity by the index of the And, if a be the root or first power, then

power to which it is to be raised. a X a = aa, the second power of o.

EXAMPLE.—Find the 3rd power of a?. Here, (a )3 =a. axo Xa= ago, the third power,

For a”=aa: and the cube of aa is aa x aa X da=aaaaaa a xa xa X a = aaaa, the fourth power, etc.

=a®; which is the 6th power of a, but the third power of a'.


{(a +b)x(c + a) }or (a + b)2 x (c+d)?.



In like manner the

cube of a x (B+a) is {a x (b + a) }', or


EXERCISE 33. 1. Required the 3rd power of 3x. 11. Find the 3rd power of tasr. 1. Find the square of 2a + b. 4. Find the square of Gy + 3. 2. Required the 4th power of 4y. 12. Find the 4th power of 2a" x 2. Find the square of h + 1. 5. Find the square of 3d-h. 3. Required the 7th power of 2a.


3. Find the square of ab + cd. 6. Find the square of a -1. 4. What is the 3rd power of 13. Required the 5th power of 186. For many purposes it will be sufficient to express the bmx?

(a + b). 5. What is the nth power of 14. Required the 2nd power of powers of compound quantities by exponents without an actual

multiplication. ady? (a + b)".

EXAMPLES 6. What is the 4th power of 15. Required the nth power of dhy?


1. Find the square of a + b. Ans. (a + b). 7. What is the 3rd power of 4b ? | 16, Required the nth power of 2. Find the nth power of bc +8+ t. Ans. (bc + 8 + x)". 8. What is the nth power of (ic + y)"

In cases of this kind, all the terms of which the componnd bad?

17. Required the 2nd power of quantity consists must be included in the parenthesis. 9. What is the 3rd power of (a* * b*). 3m x 2y ? 18. Find the 3rd power

187. But if the root consists of several factors, the paren. 10. Find the 4th power of a'ba.


thesis used in expressing the power may either extend over the 183. A FRACTION is raised to a power by involving both the whole, or may be applied to each of the factors separately, as


may require. numerator and the denominator to the power required.

Thus the square of (a + b) x (c+d), is either EXAMPLE.—Find the square of i By the rule for the multiplication of fractions we have The first of these expressions is the square of the product of Ans.

the two factors, and the last is the product of their squares, bb

and these are equal to each other. 184. A compound quantity, consisting of terms connected by + and -, is involved by an actual multiplication of its several as x (6 + d)s. parts.

188. When a quantity, whose power has been expressed by a EXAMPLE.—Find the 2nd, 3rd, and 4th powers of a + b. parenthesis, with an index, is afterwards involved by an actual Here, (a + b)'=a + b,

... the frst power ; multiplication of the terms, it is said to be expanded. 0 + b

Thus (a + b), when expanded, becomes a + 2ab +62, and aa + ab

(a+b+ h)becomes aa + 2ab + 2ah + b2 + 2bh + h?.
+ ab + 8
(a + b)=a + 2ab + 69, the second power;


a3 + 2ab + aba

1. 9 and 19.
42. 30 and 19.

when m = n, and a 2. 17 and 31.

43. 16 and 21. + ab + 2ab + 13

greater or less than 3. 13. 44. 86 and 12.

nb, the values of a (a + b)3 =q3 + 3ab + 3ab2 + 63, . the third power ; 4. £189.

45. 42 and 26.

and y are infinite, a + b

5. 864.

46. 12, 4, and 18 miles. and the question
at + 3ab + 3a%b2
+ abs

6. 21 and 16.
47. 8, 12, and 16.

absurd. When xand
+ ab + 3aRb2 + 3abs +64
7. 9 and 3 years. 48. £740.

y are negative, the 8. 12 hours. 49. 24,009 men,

question will be (a+b)'=at + 4a3b + 6a*b+ 4ab3 + 64, the fourth

9, 6 and 9.
50. 53.

changed into one in power,

10. 156 days.
51. 23.

which the quantity EXERCISE 32.

11. £1,200.
52. s.

of rum is diminished
12. 8.
53. 20 and 13.

by a gallons, and 1. Find the 2nd, 3rd, and nth powers of a

13. 32} miles.
54. 233 and 142.

that of the water by 2xr

14. 480,
55. 24 feet.

b gallons.
2. Find the cube of

15. £1,125 and 1875. 56. 10 shillings.

78. .
16. £120.
57. 16 years.

79. 1.
3. Find the nth power of

17. A 200 miles, and B 58. 54 years.

80. £60 and £80. 160 miles. 59. 21 years.

81, 515 minutes before
4. Find the square of
-a** (d + m)

18. 48 years.
60. 61.

11 o'clock. (x + 1)*

19. 1,920.
61. 56 hours.

82, 24 and 20.
5. Find the square of a-b.

20. 35.
62. 20 days.

83. 18 and 60.
6. Find the cube of a + 1.

21. 100.
63. 197 sheep.

84. 679.
7. Find the square of a + b + h.

22. 800 trees.
64. 6 days.

85. Those of the first 8. Required the square of a + 2d + 3.

23. 47.
65. 30 days.

kind 21 shillings 9. Required the 4th power of b + 2.

24. £280.
66. 120 minutes.

each, and those 10. Required the 5th power of < + 1.

25. 84 years.

67. 36 farthings and 3 of the second 25 11. Required the 6th power of 1-b.

26. 84.


shillings. 185. The squares of binomial and residual quantities occur go

27. £450.

63. 5 shares and 120 86. At the end of 21 frequently in algebraic processes, that it is important to make | 29. 29 gals, brandy, 44 69. 300 leaps.

28. 1471bs.


days. them familiar. Thus,

87. 44 lines, each con

gals. wine, and 70. 60 days; and 1,800 taining 51 letters. If we multiply a + h into itself, and also a h into itself, we

73 gals. water.

and 1,200 miles 88. £1,200 and £200. ath a-h

30. A, 4317; B, £951; for A and B re- 89. In 48 minutes.

C, L1,268; and spectively. 90. £13. ath



[ocr errors]

D, £2,219. 71. £44 from B, and 91, £6,840.
a2 + ah

31. 17, 14, 27, 8, and 33. £88 from A. 92. £12,600.
+ ah + 12
- ah + ho

32. 153 shillings. 72. A £312, B 8412, 93. 200 cavalry, 600 aa + 2ah + ha az - 2ah + h?.

33. 147 sheep.

and C £476.

artillery, and 34. 20 days.

73. 12s., 23., and 16s. 1,800 foot. Here it will be seen, that in each case the first and last 35. A, 84; B, 42; C, 14. 74. 54 hours.

94, 28 and 18. terms are the squares of a and h; and that the middle term is 36. 20 yards and 26 75. 20s.

95. 189. twice the product of a by h. Hence the squares of binomial and yards.

76. 804 minutes. 96, 75 oz. gold, and 25 residual quantities, without multiplying each of the terms 37. £320.

nb -a 77. y =

and x

oz. silver, 38. 12.

97. £4 16s. separately, may be found by the following rule :(1.) The square of a BINOMIAL, the terms of which are both | 39. Chaise, £240;

98. 721bs.

m (nb - a) horse, £80; har

99. A, 1418; B, 1721; positive, is equal to the squares of the first and last terms, plus

ress, £40, twice the product of the two terms.

When m = "h, and a 100. 12 feet and 15 feet.

and C, 234t.

40. 126 gallons. (2.) The square of a RESIDUAL quantity is equal to the squares 41, 10, 14, 18, 22, 26,

= nb, the question of the first and last terms, minus twice the product of the two terms. and 30.

is indeterminate:

m - 1





whole of the visible sky, and from the remotest part of the THE REFLECTION OF LIGHT, AND DECEPTIONS WITH from the innumerable faces or facets which the roagh grinding

opposite wall, will be introduced into the apartment, reflected PLANE AND CONCAVE MIRRORS.-I.

of the glass has produced. The whole window will appear as A PENCIL or ray of light will continue in a straight path through if the sky were beyond it, and from every point of this luminous a medium of the same density, but may be acted upon in four surface light will radiate into all parts of the room.” “Further," ways if it passes into another portion of the same medium of a he says, " the opposite sides of the street or lane should be kept different density, or into any other gaseous, liquid, or solid body. whitewashed with lime ; and, for the same reason, the ceilings When light is affected in one or other of these ways, it is said and walls of the apartments should be as white as possible, and to be reflected,

all the furniture refracted, polar

of the lightest ised, or absorbed.

colours. Haring The reflection of

seen such effects light from the various objects pro

Fig. 2.

produced by im

perfect means, we duced by nature or art, enables us Fig. 1.

feel as if we had

introduced to see and enjoy

poor workmen and the works of God

noedle-women and man, which

from . dungeon must otherwise

into a have been invisiI

house, where the ble, or seen only

aged can read in outline, if they

their Bibles; did notone and all

where the inpossess the pro

mates can perty of throwing

each other, and off more or less of

carry on their the rays which

work in society fall upon them

and comfort." from luminous

The reflection of bodies. The in

light is governed fluence of the

by laws that adlight in promoting

mit of easy dethe well-being of

monstration. all is undoubted.

1st. The inciDr. Forbes Win

dent and reflected slow says, "Where light is not perzitted to permeate, there are rays are always found in the same plane or direction which is found, in the highest state of manifestation, bodily deformities, perpendicular to the surface reflecting the light. intellectual deterioration, crime, disease, early, and often sudden A pencil or ray of light can always be obtained in the death.” A material, as well as a moral and mental, etiolation daytime by making a small hole in the shutter of a darkened or blanching occurs when the vital stimulus of light is with room, and at night by putting a sheet-iron or copper chimney drawn. The reflection of

over a burning argand light into the dwellings of

oil or gas lamp. The the poor, where the direct

chimney must have a rays of the sun cannot

small hole punctured in reach them, has been in

it, or a series of holes of sisted on over and over

differont sizes may be again by philanthropists

bored, and each kept and philosophers. The

closed, with a circular late Sir D. Brewster, in

flap crossed at the upper one of his addresses to

edge by a wire properly the Royal Society of Edin

riveted; a glass chimney burgh, thought it a mat

may be coated with tinter of sufficient import

foil, in which holes of the ance to state how light

required size can be made might be reflected into

and pasted up with slips rooms situated in narrow

of foil when not required. lanes. He says: “ If, in

The experimental results a very narrow street or

are seen so much better lane, we look out of a

when the spectator conwindow with the eye in

fines his attention only to the same plane as the

the ray with which he is outer face of the wall in

operating which the window is

In Fig. 1 the rays of placed, we shall see the

light, r" B (made parallel whole of the sky by which

Fig. 3.

by a small double conver the apartment can be il.

lens), called the incident luminated. If we now withdraw the eye inwards, we shall gra- rays, and proceeding from the hole in the chimney, are falling on dually lose sight of the sky till it wholly disappears, which may the slanting mirror A B; and the rays R R", the reflected rays, are take place when the eye is only six or eight inches from its first proved to be in the same plane or direction, because they strike position. In such a case the apartment is illuminated only by or impinge on the chimney at R", and therefore in a perpendicular the light reflected from the opposite wall, or the sides of the stones plane corresponding with the upright chimney. which form the window; because if the glass of the window is 2nd. The angle of incidence is always equal to the angle of six or eight inches within the wall,

as it generally is, not a ray reflection ; thus, in Fig. 1, if a line perpendicular to the surface of light can fall opon it. If we now remove our window, and of the mirror, A B, be raised at R and drawn to P, then the angle substitute another, in which all the panes are roughly ground on pri" is equal to the angle Pre", or, in other words, the incident the outside, and flush with the outer wall, the light from the and reflected rays form equal angles. By moving the mirror

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