108 



THE POPULAR EDUCATOR. 



This is done upon the principle that the root of the product of 

 several factors is equal to the product of tlieir roots. 



Thus Vab = Va X vT> ; for each member of the equation, if 

 raised to any power, will give the same result. 



When, therefore, a quantity consists of several factors, we 

 may either extract the root of the whole together, or we may 

 find the root of the factors separately, and then multiply them 

 into each other. , . 



EXAMPLE. The cube root of xy is either (xy)*, or x j y*. 



The root of a fraction is equal to the root of the numerator 

 divided by the root of the denominator. 

 EXAMPLE. 



a J cfr a$ 



Thus the square root of T- = . For 



b* 



a 



'b' 



SIGNS. (1.) An odd root of any quantity has the same sign 

 as the quantity itself. 



(2.) An even root of a positive quantity is ambiguous. 



(3.) An even root of a negative quantity is impossible. 



But an even root of a positive quantity may be either positive 

 or negative. For tho quantity may be produced from the one, 

 as well as from the other. 



Thus the square root of a 2 is -f- a, or - a. 



An even root of a positive quantity is therefore said to be 

 ambiguous, and is marked with the sign +. Thus the square 



root of 3b is + -/3b. The 4th root of * is +*. 



The ambiguity does not exist, however, when, from the nature 

 of the case, or a previous multiplication, it is known whether 

 the power has actually been produced from a positive or from a 

 negative quantity. 



But no even root of a negative quantity can be found. 



The square root of - a'- is neither + a nor - a. 



For + a X + a = + r 5 and - a X - a = + a- also. 



An even root cf a negative quantity is therefore said to be 

 impossible or imaginary. 



The methods of extracting the roots of compound quantities 

 need not be considered here. But there is one class of them, 

 the squares of binomial and ramtfatal quantities, which it will be 

 proper to attend to in this place. The square of a -j- b, for 

 instance, is a~ + 2ab + b 2 , two terms of which, a 2 and b 2 , are 

 complete powers, and 2ab is twice the product of a into b, that 

 is, the root of or into the root of b 2 . 



Whenever, therefore, we meet with a quantity of this descrip- 

 tion, we may know that its square root is a binomial ; and this 

 may be found by taking the root of the two terms which are 

 complete powers, and connecting them by the sign +. The 

 other term disappears in the root. Thus, to find the square root 

 of ar + 2xy -f- y", take the root of x-, and the root of y", and 

 connect them by the sign +. The binomial root will then be 

 ,-fcy. 



In a residual quantity, the double product has the sign 

 prefixed, instead of + The square of a - b, for instance, is 

 a- 2ab -f- b 2 . And to obtain the root of a quantity of this 

 description, we have only to take the roots of the two complete 

 powers, and connect them by the sign - . Thus the square root 

 of & - 2xy + y 3 is x- y. Hence, to extract the square root of 

 a binomial or residual, 



Take the roots of the two terms which are complete powers, and 

 connect them by the sign which is prefaced to the other term. 



EXAMPLE. To find the root ot x-+2x+l. 



'The two terms which are complete powers are a; 2 and 1 . The 

 roots are x and 1. Then x -f- 1 = required root. 



EXERCISE 



1. Eequired the 5th root of ab. 11. 



?. Eequired the nth root of a 2 . 12. 



to 



3. Eequired the 7th root of I 



14. 



4. Eequired the 5th root of 



fa - *}3. 



5. Eequired the cube root of a 1 *. 



6. Eequired the 4th root of a- 1 . 



7. Eequirsd the cube root of a 1 . 



8. Eequired the nth root of j" 1 . 



9. Eequired the 3rd root of y 9 . 

 10. Eequired the 4th root of ii p . 



19. 



47. 



Eequired the 2nd root of x". 

 Eequired the 5th root of d 3 . 

 Eequired the 8th root of a*. 

 Eequired the 5th root of 3i/. 

 Eequired the 6th root of alh. 

 Eequired the cube root of 86. 

 Eequired the nth root of i"i/. 



Eequired the Jith root of ?, and 

 the cube root of . 



Eequired the square root of 



*-, and the 5th root of . 

 aj 0* 



20. Eequired the square root of 



x* - 2x + I. 



21. Ee'-v.ired the square root of 



22. Bequirei the square root of 



23. Eequired tho square root of 



a* + ab + b -. 



4 



2i. Eequired the square root of 



KEY TO EXEECISES IN LESSONS IN ALGEBEA. XXVI 

 EXERCISE 45. 



4. d. 



5. u + 7. 



6. if> or 1. 



7. v. 



8. fc. 



9. 2a. 



10. a"+ 

 11. 



1. a 3 denotes the 4th 

 power of the 3rd 

 root of a, or the 

 cube root of the 

 4th power of a. 



2. x 



denotes the 

 square root of 

 the cube of x. 



EXERCISE 46. 



3. y* denotes the 6th 



root of the 8th 

 power of y. 



4. b denotes the 8th 



root of the 7th 

 power of b. 



5. a*. 



6. A 



7. a<>- 25 . 



8. a-. 



9. a-. 



10. a>-8. 



11. a*-"'. 



12. a-'''. 



13. a 1 '"*. 



LESSONS IN CHEMISTRY. XXXIY. 



BASES AND ESSENTIAL OILS. 



THERE are certain compounds which are present in many 

 vegetables in combination with some vegetable acid, which are 

 called bases. The general plan adopted to isolate the base 

 is this : the plant is digested with water, acidulated with 

 sulphuric acid ; this being a stronger acid than the vegetable 

 acid, displaces the latter, and forms, with the base, a 

 sulphate. This solution being filtered is treated with potash, 

 which takes the sulphuric acid, and thus liberates the base. 

 Upon agitation with ether, the base is dissolved, and as the 

 ether is readily separated and evaporated, tae base is procured ; 

 generally, however, it is found necessary to again treat the 

 ether with acidulated water. The aqueous solution of the salt 

 thus obtained is acted upon by potash, and the liberated base 

 is removed by ether in a state of purity. We shall notice 

 some of the best known of these vegetable bases. 



Conia (C 8 H I5 N) is a volatile, colourless liquid, which is 

 procured by distilling hemlock seeds, digested in water which 

 contains a little potash. When exposed to the air it absorbs 

 oxygen, and becomes dark coloured and nearly solid. It is a 

 powerful base, and therefore salts of it are readily formed ; in 

 all its forms it is a deadly poison. Strong sulphuric acid turns 

 conia compounds, first a purple red, and then to an olive grcon ; 

 with nitric acid a blood-red colour is produced which fades into 

 orange. 



Nicotine (Cj H 14 N,,) is the active principle of tobacco, in 

 which plant it is in combination with malic and citric acids. 

 French tobacco contains 7 or 8 per cent. ; Virginia, G or 7 ; and 

 Maryland and Havannah, 2 per cent. When pure it is a colour- 

 less, oily liquid, very inflammable and volatile ; it is a powerful 

 poison. When exposed to the air it absorbs oxygen, and, like 

 conia, finally becomes a brown solid. Acetate of lead, corrosive 

 sublimate, and the tin chlorides, all give with it white pre- 

 cipitates. 



Cinchonia (C,, H 21 N 2 O) is the bitter principle in the bark of 

 different varieties of Cinchona. It is a solid, and appears 

 crystallised in quadrilateral prisms. The tree is found in South 

 America, in the forests of Bolivia and Peru, and several 

 varieties are furnished, probably produced by tho localities in 

 which they are found. They are divided into three classes 

 in the English market : Cinchona cordifolia (heart-leaved), the 

 bark is yellow ; Cinchona IsLncifolia (lance-leavod), pale bark ; 

 and Cinchona oblongifolia (oblong-leaved), red bark. 



A decoction of the bark is extensively prepared as a tonic ; 

 for this the pale variety is generally used. We give the London 

 Pharmacopoeia formulae for the "Decoction of Cinchona and the 

 Compound Tincture : ' ' 



" Take of lance-leaved cinchona, bruised, ten drachms ; dis- 

 tilled water, a pint ; boil for ten minutes in a lightly covered 

 vessel, and strain the liquor whilst hot." 



Compound Tincture of Cinchona. " Take of lance-leaved 



