140 



THE POPULAR EDUCATOR. 



the past of the infinitive, of pour which may be followed by 

 the present or by the past of the same mood, and of en which 

 requires the present participle, the prepositions, which admit of 

 being followed by a verb, require the present tense of the 

 infinitive : 



En arrivant, elle se mit a 

 pleurer. 



II riait tout en me parlant. 



Apres avoir parle, il sortit. 

 Elle sortit apres avoir dine". 

 Sans savoir ce qu'il faisait. . . 



Je 1'ai faifc pOUT vous plaire. 



On le chassa pour avoir 

 menti. 



Nous venions d'airiver. 



Us sont a travailler. 



On arriving, she began to weep. 



He was laughing while speaking 

 to me. 



After having spoken he went out. 



She went out after having dined. 



Without knowing what he was 

 doing. . . 



I have done it in order to please 



you- 



They expelled him for having told 

 a lie. 



We had just arrived. 



They are working. 



(2 ) In French a preposition must always precede its comple- 

 ment : What are you speaking of ? Whom is he speaking to ? 

 cannot be translated into French in this order ; the preposition 

 must be put in French before what and whom : 



De quo! parlez-vous ? Of what are you speaking ? 



A qui parlez-vous ? To whom are you speaking 1 



(3.) Prepositions are used between verbs having the same 

 subject ; conjunctions between verbs having different sub- 

 jects : 



Je 1'ai fait pour vous plaire. 

 Je 1'ai dit pour qu'il le sache. 



I have done it in order to please 

 you, i.e., in order that I might please 

 you. 



I have said it in order that he 

 should know it. 



(4.) When a conjunction is used between two verbs having 

 the same subject, the preposition de is added to it : 



Us s'avancerent afin de mieux 

 voir. 



They advanced in order to see 

 better. 



(5.) When a preposition is used between two verbs having 

 different subjects que is added to it : 



Je 1'ai fait avant qu'ils arri- I have done it before they arrived. 

 vassent. 



140. THE CONJUNCTION. GOVERNMENT OF CONJUNCTIONS. 



[See 123.] 



(1.) Conjunctions govern the verbs following them in the 

 indicative, in the conditional, or in the subjunctive mood : 



Ilest sur queje 1'ai dit, car 

 il m'a entendu. 

 II fut decide qu'il partirait. 

 Quoique vous le sachiez. 



He is sure I have said it, for he 

 has heard me. 



It was decided that he should start. 

 Although you know it. 



(2.) A conjunction cannot govern the infinitive; when, 

 therefore, a conjunction must be used between two verbs 

 having the same subject, de is added to it [ 139, (4.)] : 



II vint ici de peur d'etre vu. | He came here, lest he might be seen. 



(3.) The following conjunctions always require the subjunc- 

 tive after them in French, whatever mood they may take in 

 English. Those marked with an asterisk require ne before the 

 verb [ 134, (4.)] :- 



Afin que, in order that. 

 *A moms que, unless. 



Au cas que, if. 



Avant que, before that. 



Bien que, although. 

 *De crainte que, for fear. 



De peur que, lest. 



En cas que, in case. 



Encore que, although. 



Jusqu'a ce que, till, until that. 



Loin que, far from, not that. 



Quoique a peine a mes maux je 



puisse re'sister, 

 J'aitne mieux les souffrir, que de 



les me'riter. RACINE. 



Malgre' que.t although, in spite of. 

 Nonobstant que, notwithstanding. 

 Non que, not that. 

 Non pas que, not that. 

 Pos6 que, supposing that. 

 Pour que, that, in order that. 

 Pourvu que, provided that. 

 Quoique, although, though. 

 Sans que, without that. 

 Soit que, whether. 

 Suppose que, suppose that. 



Although I can scarcely bear my 

 misfortunes, I would rather suffer 

 under them than deserve them. 



f Only used with the verb avoir : malgxe qu'il en ait, in spite of 

 himself. 



En cas que vous persistiez, il 



faudra que j'allegue au prince et 

 au roi meme votre mauvaise saute 1 . 

 FENELON. 



In case you persist, I must men. 

 tion your bad health to the prince 

 and even to the king. 



PLANE TRIGONOMETRY. III. 



SOLUTION OF RIGHT-ANGLED TRIANGLES FUNDAMENTAL 

 PRINCIPLES, ETC. 



X. Solution of Right-angled Triangles. Every triangle con- 

 sists of six " elements," three sides and three angles. Any 

 three of these being given, including at least one side (this is 

 necessary, because triangles merely equiangular can be con- 

 structed in infinite number), Trigonometry enables us to calcu- 

 late the remaining elements. The formulae evolved as yet only 

 enable us to do this for right-angled triangles, and as these 

 involve one known quantity (the right angle), it is sufficient if 

 any two of the other elements (including one side) be given. We 

 may have (referring to Fig. 3), besides the right angle 

 (1.) Given two sides. 

 (2.) Given one side and one angle. 



Either of these cases may be solved by the ratios given in- 

 Section II., and by a table of natural sines and cosines, tangents 

 and cotangents, such as that given at the end of Galbraith and 

 Haughton's "Trigonometry." The following examples may all 

 be solved by the annexed table of ratios for a few angles only, 

 purposely restricted to three places of decimals : 



First, given two sides only, viz., a 15'58 ; b = 35. 

 A, B, and c. 



Find 



b~ 35 

 Referring to the table, we find '445 entered as tangent of 24. 



.-. A = 24, and B = 90 - A = 66. 



By Euclid I. 47, c 2 = a? + & 2 . 



.-. c = A/a 2 + 6 2 ; 

 which may readily be calculated, a and b being known. 



Again, given one side and hypothenuse, viz., 6 = 5; c = 10i 

 Find A, B, and a. 



c 10 



/. by the tables, A =60; 



= 30. 



a (from. Euclid I. 47, as before) = Vc 2 - b' 2 A/75. 

 Secondly, given one side and one angle, viz., a = 100; 

 36. Find A, 6, and c. 



A = 90 - B = 54. 



Since tan. B = ^ , b = a tan. B = 100 X '727 = 72'7 



= 1.-= 123-609. 



a 

 and since cos. B = 



cos. B -809 



Again, given hypothenuse and one angle, viz., c = 75 ; A 

 15. Find B, a and b. 



B = 90 - A=75. 



Since sin. A = a ~, a = c sin. A = 75 x '259 = 19'425 

 c 



and since cos. A = -, b = c cos. A = 75 X "966 = 72-45. 



c 

 These are merely specimens of the ways in which the four 





