EXERCISES IN EUCLID. 





OEOUP I. (FOUR OFFICES). 

 1. TREASURY, WHITEHALL. 



Open Competition. 



The business of the Treasury is to control the spending and 

 revenue departments of Government ; to decide upon all qnes- 

 tions of principle affecting the revenue of the country ; to 

 prepare the ways and means, subject to the revision of Parlia- 

 Mr carrying on the business of the country, and generally 

 t<> influence, to a very large extent, the whole machinery of the 

 administration. 



The establishment and salaries are as follow : 

 Four principal clerks, -1,000 to 1,200; 7 first-class, 700 

 to .900 ; 2 second-class, 250 to .600 ; 1 accountant, .000 

 to 800 ; 1 assistant accountant, .100 to .550. Supplemental 

 Establishment 6 clerks, .250 to .500 ; 3 ditto, 225 to 350; 

 2 second-class, 100 to 200 ; 10 clerks (lower division), 80 

 to 200. Solicitor's Office 3 clerks, 100 to 300: and 1 

 (shorthand), 100 to 150. 



2. HOME OFFICE, WHITEHALL. 



Open Competition. 



This office carries on the ministerial part of the Government 

 at home, is responsible for the preservation of the peace and 

 safety of the inhabitants of the country, and is the supreme 

 authority on all questions of internal policy. Age of admission 

 for established clerks, 18 to 24. 



The establishment and salaries are as follow : 



Three principal clerks, 900 to 1,000 ; 3 senior, 700 to 

 800 ; 7 junior, 200 to 600. Account Branch 1 clerk in 

 charge of accounts, 400 to 600 ; assistant ditto, 310 to 400 ; 

 1 supplementary clerk, second-class, 100 to 250 ; 3 clerks 

 (lower division), 95 to 250. /Statistical Branch -1 statis- 

 tical clerk, 350 to 500; 1 supplementary clerk, 100 to 

 250 ; 1 clerk (lower division), 90 to 250. Registry and 

 Copying Branch Superintendent, 350 to 500 ; superinten- 

 dent of copying branch, 250 to 350 ; 9 clerks (lower divi- 

 sion), 95 to 250. 



Mineral Statistic BrancJi^-l clerk, 300 ; 1 ditto, 250. 



Inspectors o/ Factories, Fisheries, Explosives, and Burials-^- 

 4 lower division clerks, 95 to 250 ; 1 ditto, 80 to 200. 



Messengers and office-keepers, for this as for all other de- 

 partments, do not come under the system of open competition, 

 but are appointed by nomination. They are required to be 

 British subjects under 40 years of age, and previous to en- 

 trance upon their duties are examined in 

 i. Beading, 

 ii. Writing from dictation. 



iii. Elementary Arithmetic (the first four rules). 



3. COLONIAL OFFICE, WHITEHALL. 

 Open Competition. 



The Colonial Office, governed by the Secretary of State for 

 the Colonies, carries out the colonial policy of the country, 

 instructs and supervises governors of British dependencies, 

 and watches generally over the interests of the colonies. Age 

 of admission for clerks, 18 to 24 ; for messengers and porters, 

 21 to 35. 



The establishment and salaries are as follow : 

 Four principal clerks, 900 to 1,000; 7 first-class clerks, 

 700 to 800 ; 11 second-class clerks, 250 to 600 ; and 

 several special and lower division clerks. 



In our next paper we shall describe the more elaborate 

 examinations required for the Foreign Office, a very important 

 nomination department. 



In succeeding papers we shall give similar information re- 

 specting other principal offices, subordinate situations in con- 

 nection with the Post Office and Telegraph Establishments, and 

 specimens of examination papers. 



It may save our readers some trouble if we inform them that 

 any application to the Commissioners, except in the manner we 

 have indicated would be fruitless, as they decline to advise 



candidates a* to the course of reading, particular tutor*, or 

 places of education, or to give information aa to m lariat, 

 duties, course of promotion, pensions, etc. 



EXERCISES IN EUCLID. I. 



THE question is frequently ankcd, What is the utility of the geo- 

 metrical element in mathematical studies ? Without in any 

 way attempting to discuss its merit*, we content ourselves with 

 giving the only satisfactory answer which can be given, that 

 such studies are pursued, not for their result*, but for the intel- 

 lectual habits which they generate. The power to apprehend 

 and the power to convince are both strengthened ; the habit of 

 clear and consecutive reasoning is developed by the successive 

 stages through which the mind is conducted in the course of a 

 geometrical investigation. It is our purpose, therefore, to aid 

 our readers by giving them a series of exercises upon the various 

 propositions of Euclid, consisting principally of " riders " which 

 may be deduced from them. Each article will take a certain 

 number of propositions, and the riders given will be deduced 

 from them without assuming any subsequent ones. At the con- 

 clusion of each article the ground to be covered by its successor 

 will be stated, and the enunciations given of those riders which 

 will bo proved, that the student may, if he so please, exercise 

 himself beforehand, by attempting their solution. It may be 

 mentioned here that the term " rider " is applied to a deduction 

 from any proposition of Euclid because the deduction is borne 

 up or supported by the mathematical reasoning worked out in 

 the proposition, as a horseman, or rider, is supported or carried 

 by the horse that he bestrides ; or, in other words, that the pro- 

 position carries the deduction on top of it, as it were, pretty 

 much as the horse carries its rider. This explanation of a term 

 which is familiar enough to any Cambridge man, may be neces- 

 sary for the information of many of our self-taught students who 

 now meet with the apparently singular but decidedly appropriate 

 expression for the first time. 



We assume a knowledge of definitions, axioms, and postulate*. 

 Definitions are the explanations of terms to be used in the 

 course of investigation ; axioms are statements of things obvious 

 to common sense ; postulates are statements of things requisite, 

 without which the investigation cannot be carried on. All those 

 requisite for the first book are given at the commencement of 

 any ordinary edition of Euclid's Elements, a book which, we 

 presume, is in the possession of all who intend to follow us in 

 these exercises. Those who have not yet provided themselves 

 with a copy may obtain a useful edition issued by the 

 publishers of the POPULAR EDUCATOR.* 



It will be seen that some of the propositions are headed 

 " Problem," some " Theorem." Strictly speaking, a " theorem " 

 is the proof of a geometrical fact ; a " problem " is the solution 

 of some geometrical difficulty, or a method of executing some 

 geometrical device which mayaid us in the solution of "theorems." 

 The riders which we shall give will be some " problems," some 

 " theorems." In either case the method of solution may be 

 thus indicated. We have certain given facts, and a certain end 

 to be deduced from them. Taking first the facts, endeavour to 

 argue from them towards the required end. Then, assuming 

 the required end to be accomplished, endeavour to argue back 

 from it to the original facts ; and if some common ground can 

 be found in which these processes meet in one, the problem is 

 solved or the theorem is proved. Thus, if we were endeavouring to 

 argue out for ourselves Prop. I., Book I., we have given a straight 

 line A B, and have to find a point c, such that c A, c B, and A B shall 

 be all equal. Assuming this accomplished, we see that A is the 

 centre of a circle passing through B and c ; and B is the centre of 

 a circle passing through A and c. Thence our method is obvious. 



Let us apply this method to the following proposition : 



PROPOSITION I. If in the fhure of Euclid I. 1 (Fig. 1), the 

 circles cut again in F, and the lino c F cut A B in o ; 



Then A o = o B ; (/3) 



Angle A c o = angle B c o ; (a) 



And angles A a c, B a c are right angles. (7) 



Join A F, B F ; then A c, A F are equal, being radii of the same 

 circle (Def. 15). And for the same reason B c. B F are equal. 



* Cassell's Euclid : beintf the First Six Books, with the Eleventh 

 and Twelfth of EUCLID, edited by Prof. Wallace, M. A. Cloth limp, Is. 

 CasselVs Sixpenny Euclid (Books I. to IV.), paper 6d. ; cloth, 9d. 



