THE POPULAR EDUCATOR. 



KEY TO EXERCISES IN LESSONS IN ITALIAN. XXXII. 



EXEECISE 41. 



1. Mr. N. has invited me to dinner; I think you will find there a 

 large party. 2. Will you go out 011 horseback to-day ? 3. My sisters 

 will soon arrive. 4. Peter will return to you all that he has taken. 

 5. Why did you not return my salutation ? 6. Once we shall render 

 an account of our actions. 7. I will answer your letter on the ninth 

 of this month. 8. When will you leave off? 9. I should have 

 finished already if you had not hindered me. 10. Leave off, then. 

 11. I shall inform your father of your negligence. 12. I would eat a 

 fig if I did not fear the toothache. 13. I would not sell my meer- 

 schaum pipe if circumstances did not oblige me. 14. If you really 

 loved the Italian language, you would study it with more diligence. 

 15. I (should) wish that you would finish the work which you have 

 begun. 16. John brings plums, pears, and apples. 17. This watch 

 does not go well ; send it to the watchmaker that he may repair it. 

 18. Do not open the windows. 



EXERCISE 42. 



1. Returning to the house, I have found your brother. 2. Not 

 speaking Italian, you must feel yourself annoyed here. 3. Not 

 knowing where to find her, I have returned. 4. I am loved by my 

 school-fellows ; thou art praised by the master. 5. Frederick is 

 punished. 6. Good children are loved fey their parents. 7. The poor 

 man is forsaken by all the world. 8. Honour thy father and thy 

 mother, and thou shalt be honoured. 9. This book shall be bound 

 to-morrow. 10. Be virtuous, and you shall certainly be rewarded for 

 it. 11. The bad will one day be punished. 12. Harriet would be 

 praised by her masters if she were more diligent. 13. We were well 

 treated by our aunt. 14. John has been punished for not having 

 finished his exercise. 15. Speak loud, that you may be heard. 16. 

 It is sad to be hated by all. 17. He feels pleasure in being praised. 

 18. We have gathered many strawberries. 19. The strawberries which 

 we have gathered are delicious. 20. The figure which my brother 

 has drawn was very beautiful. 21. Have you sent my books to the 

 bookbinder ? 22. Yes ; I have sent them to him yesterday. 



EXERCISE 43. 



1. Our neighbour pretends to understand everything that we say. 



2. My uncle will arrive this evening ; we shall amuse ourselves well. 



3. Why do you grieve ? 4. I grieve for the death of my cousin. 5. 

 Rejoice, friends, in the little which you have. 6. Do not rely on him. 

 7. Remember your promise. 8. Wrap yourself with your cloak. 9. 

 T shall make use of your books. 10. They make use of mine. 11. 

 We often make use of this carriage. 12. I dress myself. 13. Dress your- 

 self also. 14. We shall dress ourselves by-and-by. 15. Francis, will 

 you not wash yourself yet ? 16. I will wash myself this instant. 17. 

 At what hour do you usually rise ? 18. I rise every morning at six, 

 and I go to bed at nine. 19. Charles will rise to-morrow at four ; 

 he will set out for Cronstadt. 20. We rise later than you. 21. 

 Formerly we did not rise so late. 22. Rest yourself a little. 23. I 

 will rest myself a moment ; I am very tired. 24. What is this 

 young man's name ? 25. I believe his name is William. 26. These 

 gentlemen are much amused at the ball. 27. They intend to go there 

 next week also. 



EXERCISE 44. 



1. They say that Mrs. Johnson will get married. 2. The bird is 

 known by its song. 3. One eats and drinks well in this hotel. 4. 

 People know their friends in misfortunes. 5. One most always seeks 

 a fortune where it is not. 6. They speak fifty-three languages in 

 Europe. 7. Have y ou heard what is reported of a boy in New York ? 

 3- It is no longer spoken of. 9. It was spoken of long since. 10. 

 What must be done to prevent such a misfortune ? 11. It is necessary 

 always to labour; it is not necessary to be idle. 12. It will be needful 

 to have patience. 13. What are you doing ? 14. I must write. 15. 

 It was necessary that I should write a letter. 16. Will you accom- 

 pany me ? 17. I am going. 18. Are you going already ? 19. It is 

 necessary for me to go. 20. Your mother is not going yet. 21. 

 Excuse me, my mother is already gone, and my brothers will go 

 directly. 22. Wait a moment longer ; we will go together. 23. Let 

 us go, gentlemen. 24. If I had come a littta later, I should have 

 come with your sisters. 25. Were yon in church ? 26. Yes ; I have 

 this moment come out of it. 



PLANE TRIGONOMETRY. I. 



INTRODUCTION - CIRCULAR MEASURE OF ANGLES FUNCTIONS 

 OF ANGLES RELATIONS OF TRIGONOMETRICAL KATIOS 

 TO ONE ANOTHER. 



TRIGONOMETRY is derived from two Greek words, rpiycavov (tri- 

 go'-non), a triangle, and fj.erpe<a (met'-re-o), I measure. Its mean- 

 ing would thus appear to be the science of computing triangles, 

 and its scope somewhat akin to Geometry. Geometry enables 

 us, certain sides and angles of a triangle being given, to construct 

 or draw the visible triangle to which they belong ; while Trigo- 

 nometry tells us how to calculate the parts or area of a triangle 

 when the numerical values of certain of its sides or angles, or 

 even the numerical value of the ratios they bear to one another, 

 are known to us. Trigonometry is used in the practical arts of 

 surveying and navigation ; and the power of computing tri- 

 angles and by that means many other figures, since all figures 

 bounded by straight lines may be split up into triangles -is 

 very useful. A moderate study of the science is enough for 

 these purposes that is to say, will establish a sufficient number 

 of formulae to enable us, with the aid of a book of tables, to 

 calculate the elements of any triangle when sufficient data are 

 given. It will also enable us to solve many mathematical 

 problems, for the formula} and equations of Trigonometry are 

 extensively used in calculations not relating to angles or tri- 

 angles at all. 



Trigonometry is divided into Plane and Spherical Trigono- 

 metry, the latter of which treats of triangles drawn upon 

 spherical surfaces, and is comparatively special in its applica- 

 tion. We are at present only concerned with Plane Trigo- 

 nometry. 



It is presumed that the learner is acquainted with the ordi- 

 nary or sexagesimal method of measuring angles, according to 

 which the circumference of every circle is considered as divided 

 into 360 equal parts, called degrees, each degree being divided 

 into 60 minutes, and each minute into 60 seconds, the signs for 

 which are respectively ' ". The fourth part of the circum- 

 ference, or 90, is called a quadrant, and subtends a right angle 

 at the centre. A right angle is thus described as 90, and every 

 angle is measured by the number of degrees, minutes, and 

 seconds in the arc or portion of the circumference which sub- 

 tends or lies opposite to it. 



I. Circular Measure of Angles. Trigonometry, it has been 

 before observed, is, in its primary signification, the science 

 which deals with the relations existing between the sides and 

 angles of triangles. But to enable us to deal freely with such 

 utterly dissimilar expressions as lines and angles in combination 

 with each other, it is necessary to bring them to speak figu- 

 ratively "to the same denomination;" and a system called 

 circular measure has been devised, by which any angle may be 

 described (or, in other words, its size expressed) by a statement 

 of the ratio existing between two lines, both of which are known, 

 and both of which may be obtained without difficulty for any 

 given angle. The unit by which all angles are measured on 

 this system is that angle whose subtending arc 

 is equal in length to the radius, aud is called 

 the circular unit, as the angle A c u in Fig. 1 , 

 where arc A u = radius A c. 



To express any other angle, A C B, in terms 

 of the circular unit : Let A be the value 

 sought, a the subtending arc, and r the radius. 

 By Euclid VI. 33 



A C B : A C U : : arc A B : arc A U ; 

 but A c u is the unit; or 1, and arc A u = radius. 

 Therefore A : 1 : : a -. r, 



Fig. 1. 



or 4.*. 



(1) 



That is to say, the size or value of an angle may be ex- 

 pressed in circular measure by the ratio subsisting between 

 the arc and the radius, or more specifically by dividing the 

 arc by the radius. We have thus found means to express the 

 size of an angle by the relation between the length of two 

 lines. 



By a calculation based upon the more abstruse results of the 

 science, it has been ascertained approximately that the circum- 

 ference of a circle = the diameter X 3 - 14159. This number 

 occurs so frequently, that it is the custom to represent it by a 



