278 



THE POPULAR EDUCATOR. 



the magnitude and of the figure of the earth ; determination of 

 the distances of the moon, sun, and planets ; Kepler's laws of 

 planetary motion, and their consequences ; refraction and aber- 

 ration, as affecting astronomical observation. 



The subject of Astronomy will, in its mathematical treatment, 

 involve some acquaintance with the elements of spherical trigo- 

 nometry, and for this purpose the circles of the sphere, measures 

 of the sides and angles of spherical triangles, and the supple- 

 mental triangle, should be carefully read. The student cannot 

 be too familiar with the systems of great circles to which the 

 positions of the heavenly bodies are referred the ecliptic, the 

 equinoctial, and the celestial horizon. He should also be able 

 to describe and explain the principal phenomena depending on 

 the motion of the earth round the sun, and its rotatory motion 

 round its own axis ; to write a general description of the solar 

 system ; and to give a general explanation of solar and lunar 



LESSONS IN ALGEBRA. XXXI. 



DIVISION OF BADICAL QUANTITIES. 



THE division of radical quantities may be expressed by writing 

 the divisor under the dividend, in the form of a fraction. 



EXAMPLES. 

 Thus the quotient of 3 Va divided by */b is . 



i V t 



And 



divided by (b + jc) n is 



(6 + xft. 



The last branch of the examination is Mental and Moral 

 Science, which includes Psychology, Logic, and Ethics. In the 

 preparation of this branch it will be indispensable that the 

 student should master Whately's " Elements of Logic," which 

 will be found the best introduction to the subject ; that he 

 should be familiar with the leading principles of Mill's " System 

 of Logic ; " and that he should possess a general knowledge of 

 the philosophy of Sir William Hamilton, as disclosed in his 

 " Lectures on Metaphysics" and edition of Reid. We should 

 also advise strongly a careful study of Dr. Thompson's " Out- 

 lines of the Laws of Thought," and a personal analysis of Mill's 

 system. The requisite knowledge of "The Senses," which 

 must be regarded from a physiological no less than a meta- 

 physical point of view, may be derived from the work of 

 Professor Bain on "The Senses and the Intellect," from which 

 questions are almost invariably set, and from Sir William 

 Hamilton's " Lectures ; " and " The Intellect " will be found to 

 be dealt with in the same works. Bain on "The Will" must 

 also be carefully read and analysed, and the " Moral Philosophy" 

 of Paley and Dugald Stewart attentively studied. 



The names of successful candidates at the examination will 

 be published, arranged in two divisions, each in alphabetical 

 order. 



A Certificate under the Seal of the University, and signed by 

 the Chancellor, will be delivered at the Public Presentation for 

 Degrees to each Candidate who has passed. 



Candidates who have passed the second B.A. examination 

 may be examined for honours in (1) Mathematics ; (2) Classics ; 

 (3) Mental and Moral Science ; (4) French ; (5) English ; and 

 (6) German. Those who succeed are arranged in three classes 

 according to their respective degrees of proficiency, and if suf- 

 ficient merit be evinced, a scholarship or prize is awarded to 

 the most distinguished candidate in each branch. 



THE DEGREE OF MASTER OF ARTS. 



The examination for the M.A. degree takes place once in each 

 year, and commences on the first Monday in June. 



Candidates must be more than twenty years, and are not ad- 

 mitted to the examination until after the expiration of one aca- 

 demical year from the time of their obtaining the B.A. degree. 



The fee for this examination is <10. If, after payment of his 

 fee, a candidate withdraw, or fails to present himself at the ex- 

 amination, or fails to pass it, the fee will not be returned to him ; 

 but he will be allowed to enter for any subsequent M.A. exami- 

 nation upon payment, at every such entry, of .5. 



To obtain the degree of Master of Arts an examination must 

 be passed in one or more of the following : 



1. Classics, including the Greek and Latin classic authors ; 

 Prose Composition in Greek, Latin, and English ; and Ancient 

 History, and the History of Europe to the end of the eighteenth 

 century. 2. Mathematics and Natural Philosophy. 3. Mental 

 and Moral Science ; Political Philosophy ; the History' of Philo- 

 sophy ; and Political Economy. 4. Any two of the following 

 subjects: English language and literature (including Anglo- 

 Saxon language and literature) ; French language and litera- 

 ture ; German language and literature ; Italian language and 

 literature ; Hebrew language and literature ; Sanskrit language 

 and literature ; and Arabic language and literature. 



The examination is continued during four days, and a golc 

 medal of the value of .20 is awarded, if sufficient merit be 

 evinced, to the most distinguished candidate in each branch. 



In these instances, the radical sign or index is separately 

 applied to the numerator and denominator. But if the divisor 

 and dividend are reduced to the same index or radical sign, this 

 may be applied to the whole quotient. 



Thus n Va n Vb = " - = n d For the root of a frac- 



*Vb b 



tion is equal to the root of the numerator divided by the root of 

 the denominator. 



Again, *Vab -7- n -\/Z> = *Va. For the product of this 

 quotient into tho divisor is equal to the dividend ; that is, 

 *Va X n V6 n Vab. Hence 



Quantities under the same radical sign or index may be 

 divided like rational quantities, the quotient being placed under 

 the common radical sign or index. 



EXAMPLE. Divide (a; 3 !/ 2 )* by y*. 



These reduced to the same index are (a 3 ;/ 2 ) an< i ( ? /) 



131 



And the quotient is (a; 3 ) 5 = x* = x 2 . Ans. 



A root is divided by another root of the same letter or quantity, 

 by subtracting the index of the divisor from that of the dividend. 



1 1 11 31 8 1 



EXAMPLE. Thus a? -r a s = a*~ s = a*~ B = a? = a 3 . 



For a* = a* = a* X a> X a , and this divided by a j ia 

 at X a* X a* _ 



In the same manner, a 1 ^ -^- a = a a 



Powers and roots of the same letter may also be divided by 

 each other, according to the preceding article. 



Thus a 2 -T- a.3 = a 2 "' = a 3 . For a 3 X a 3 = a 5 = a 2 - 



When radical quantities which are reduced to the same index 

 have rational co-efficients, the rational parts may be divided 

 separately, and their quotient prefixed to the quotient of the 

 radical parts. 



Thus ac Vbd -r a */b c Vd. For this quotient multiplied 

 into the divisor is equal to the dividend. 



EXAMPLE. Divide ab(z 2 b)* by a ()'*. . . 



These reduced to the same index are ab(x^b)* and a(x^)a . 



Tho quotient then ia b(b)* = (b 5 )*. 



To save the trouble of reducing to a common index, the 

 division may be expressed in the form of a fraction. 



The quotient will then be i - 



a(rc) 2 

 Hence we deduce the following 



GENERAL RULE FOR DIVIDING RADICALS. 



If the radicals consist of the same letter or quantity, subtract 

 the index of the divisor from that of the dividend, and place the 

 remainder over the common radical part or root. 



If the radicals have co-efficients, the co-efficient of the dividend 

 must be divided by that of the divisor. 



If the quantities have the samo radical sign or index, divide 

 them as radical quantities, and place the quotient under the 

 common radical sign. f 



