246 



THE POPULAR EDUCATOR. 



reader can now trace the original limits of the British American 

 colonies prior to 1776. The New England States so called 

 because they formed parts of New England, a tract settled by 

 the Plymouth Company are Maine, New Hampshire, Massa- 

 chusetts, Rhode Island, and Connecticut. 



Mexico, stretching southwards from the United States to the 

 Gulf of Tehuantepec, and including the peninsula of Yucatan, 

 was a dependency of Spain from its conquest by Hernando 

 Cortes in 1521 to 1821, when it regained its independence under 

 Iturbide, a native, who then became Emperor, but did not long 

 retain his crown. Since then it has been a prey to civil discord 

 and intestine strife. Mexico consists of twenty-six small states 

 in a federal union, with a constitution somewhat similar to 

 that of the United States. 



Central America, which includes the British territory of 

 Honduras, chief town Balize, under the jurisdiction of the 

 governor of Jamaica, contains the following republics, and the 

 Mosquito territory, which is under British protection : 



Guatemala, Capital New Guatemala. Nicaragua, 



San Salvador, 

 Honduras, 



Cojutejjeque. 

 Coniagua. 



Costa Eica, 



Capital Managua. 

 St. Jose. 



The West India Islands, which Ho like a breakwater at the 

 entrance to the Gulf of Mexico, and stretch southward between 

 the Caribbean Sea and the Atlantic Ocean to the north coast of 

 South America, are divided among various European powers. 

 The western part of Hayti, or St. Domingo, is the only 

 independent state. 



'British. The Bahamas, Jamaica, 

 Barbadoes, Antigua, St. Christo- 

 pher's or St. Kitts, St. Vincent, 

 Dominica, Grenada and the 

 Grenadines, Trinidad, Montser- 

 rat, Nevis, Anguilla, Barbuda, 

 the Virgin Isles, St. Lucia, and 



French. Martinique, Guadeloupe, 

 Grande Terre, Desinade, Mario 

 Galante, and St. Martin (north 

 side). 



Dani's7i. Santa Cruz, St. John, St. 

 Thomas. 



Swedish.. St. Bartholomew. 



Spanish. Cuba, Porto Eico, Isle 

 of Fines, and Dominica, on east 

 part of Hayti. 



Tobago. 

 Dutch. St. Eustatius, Saba, Cura- 



9oa, Buen Ayre, St. Martin 



(south side). 



The Bahamas, a cluster in the middle of the Atlantic Ocean, 

 belong to Great Britain. The chief town of the British West 

 Indies is Spanish Town, in Jamaica. The West India Islands, 

 it should be said, are divided into three groups 1, the Bahamas; 

 2, the Greater Antilles, including Cuba, Porto Eico, and Jamaica; 

 and 3, the Lesser Antilles, comprising the rest of the islands, 

 from the Virgin Islands in the north to Trinidad in the south. 

 A line drawn between Martinique and St. Lucia separates this 

 third group into the Windward and Leeward Islands, the latter 

 being the northern division, and the former the southern. 



A projection for a map of North America may be constructed 

 in the same way as projections for maps of Europe and Asia. 

 Those who wish to fill up a projection for themselves may obtain 

 the outline of the coast, and directions of rivers and mountain 

 ranges, from any good map, and the latitudes and longitudes 

 of the principal places, headlands, etc., from the index to any 

 atlas. 



LESSONS IN ALGEBRA. XX. 



INVOLUTION, OE EAISING OP POWEES. 



173. When a number is composed of the product of tlie same 

 factor any number of times, the result is called a power of the 

 factor. Powers are divided into different orders or degrees ; as 

 the first, second, third, fourth, fifth powers, etc., which cure also 

 called the root, square, cube, biquadrate, etc. 



The powers take their names from the number of times the 

 roof, or first power, is used as a factor in producing the given 

 power. 



The original quantity is called the first power, or root of all 

 the other powers, because they are all derived from it. 

 Thus, if 2 be the root or first power, then 



2x2 = 4, the square or second power of 2. 

 2X2 X 2 = 8, the cube or third power. 

 2 X 2 X 2 X 2 -- 16, the biquadrate or fourth power, etc. 

 And, if a be the root or first power, then 



a X a, = aa, the second power of a. 

 a X a X a = aaa, the third power. 

 aXXaX= aaaa, the fourth power, etc. 



174. The number of times a quantity is employed as & factor 

 to produce the given power is generally indicated by a figure or 

 letter placed above it on the right hand. This figure or letter is 

 called the index or exponent. Thus a X a = aa, is written a 2 

 instead of aa ; and a X a X a = aaa, is written a 3 . 



The index of the first power is 1 ; but this is commonly omitted, 

 that is, a 1 = a. 



An index is totally different from a co-efficient. The latter 

 shows how many times a quantity is taken as a part of a whole ; 

 the former how many times the quantity is taken as a factor. 

 Thus 4a = a + a,-}-a-j-a; but a 4 = a, X a X a, x a = aaaa. 

 If a = 4, then 4 .a = 16 ; and a 1 = 256. 



175. Powers are also divided into direct and reciprocal. 

 Direct Powers are those which have positive indices, as d 2 , d 5 , 



etc., and are produced by multiplying a quantity by itself, as 

 above described. Thus d X d = d* ; d X d X d = d 3 ; and 



The Reciprocal Power of a quantity is the quotient arising 

 from dividing a unit by the direct power of that quantity, as 



A reciprocal power is produced by dividing a direct power by 

 its root, till we come to the root itself ; and then continuing the 



division, we obtain the reciprocal powers. Thus = d 2 ; and 



d 2 d 1 1 1 d 1 



= d ; - = d == 1 ; and - -7- d = ; and - + d = -, etc. 

 d d d d 2 d 2 d 3 



176. For convenience of calculation and expression, reciprocal 

 powers are written like direct powers with the sign before the 



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



of d are d*, d 3 , d 2 , d 1 , d, d ~ \ d ~ 2 , d ~ 3 , d ~ 4 , etc., in which d 

 = 1. 



177. INVOLUTION is the process of finding any power of a 

 quantity, as explained in Art. 173. 



178. To involve a quantity to any required power. 



RULE. Multiply the quantity by itself, and by its successive 

 products, till it is taken as a factor as many times as there are 

 units in tlie index of tlie power to which the quantity is to be 

 raised. 



All powers of unity or 1 are the same, viz., 1. For IXlXl 

 X 1, etc. =1. 



179. A single letter is involved or raised to any power, by 

 giving it the index of the proposed power ; or by repeating it as 

 a factor as many times as there are units in that index. 



If the letter or quantity has a co-efficient, it must be raised to 

 the required power by actual multiplication. 



EXAMPLES. 



1. The 4th power of a is aaaa, or a 4 . 



2. The 6th power of y is yyyyyy, or y 6 . 



3. The nth power of x is xxx ... repeated n times, or w n . 



180. The method of involving a quantity which consists of 

 several factors, depends on the principle, that the power of the 

 product of several factors is equal to the product of their powers. 



EXAMPLE. What is the square of ay ? Here, (ay) 2 = a 2 !/ 2 . 

 For, by Art. 178, (ow/) 2 = ay X ay. 



But ay X ay = ayay = aayy = a 2 y 2 . Ans. 



In finding the power of a product, therefore, wo may either 

 involve the whole at once, or we may involve each of the factors 

 separately, and then multiply their several powers into each 

 other. 



181. When the root is positive, all its powers are positive also ; 

 but when the root is negative, the ODD powers are negative, while 

 the EVEN poivers are positive. 



Hence any odd power has the same sign as its root. But au 

 even power is positive, whether its root is positive or negative. 

 Thus (+ a) x (+a) = a 2 . And ( - a) X ( a) = a 2 . 



182. To involve a quantity which is already a power. 

 RULE. Multiply tlie index of the quantity by the index of the 



power to ivhich it is to be raised. 



EXAMPLE. Find the 3rd power of a 2 . Here, (a 2 ) 3 = a 6 . 

 For a 2 = aa : and the cube of aa is aa X aa X aa aaaaaa 

 a 6 ; which is the 6th power of a, but the third power of a 2 . 



