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THE POPULAR EDUCATOR. 



structor ' will take him by the hand, and lead him ' by the running 

 streams, and teach him all the principles of Science, as she comes 

 from her Maker ; as he would smell the fragrance ' of the rose, with- 

 out gathering it. 



This love of nature ; this adaptation of man ' to the place assigned 

 him ' by his heavenly Father ; this fulness ' of the mind || as it des- 

 cends into the works of God, is something, which has been felt ' by 

 every one, though to an imperfect degree, and therefore | needs no 

 explanation. It is the part of science, that this | be no longer a blind 

 affection ; but ' that the mind ' be opened | to a just perception ' of 

 what it is, which it loves. The affection, which the lover first feels ' 

 for his future wife, may be attended ' only by a general sense ' of her 

 external beauty ; but his mind ' gradually opens | to a perception ' of 

 the peculiar features of the soul, of which ' the external appearance | 

 is only an image. So it is ' with nature. Do we love to gaze on the 

 sun, the moon, the stars, and the planets ? This affection | contains 

 ' in its bosom | the whole science of astronomy, as the seed ' contains 

 the future tree. It is the office of the instructor ' to give it an 

 existence ' and a name, by making known the laws which govern the 

 motions of the heavenly bodies, the relation of these bodies to each 

 other, and their uses. 



Have we felt delight ' in beholding the animal creation, in watching 

 their pastimes ' and their labours ? It is the office of the instructor ' 

 to give birth to this affection, by describing the different classes of 

 animals, with their peculiar characteristics, which inhabit the earth, 

 the air, and the sea. Have we known the inexpressible pleasure | of 

 beholding the beauties ' of the vegetable world ? This affection | can 

 only expand ' in the science of botany. Thus it is, that the love of 

 nature ' in the mass || may become the love of all the sciences, and 

 the mind will grow and bring forth fruit [| from its own inherent 

 power of development. 



LESSONS IN GEOMETRY. X. 



IN our last lesson we considered the various series of data 

 necessary for the construction of an isosceles triangle : we will 

 now do the same for any kind of scalene triangle, or triangle of 

 which all three sides are unequal. 



A scalene triangle, as it has been stated, may be an acuto- 

 angled triangle, an obtuse-angled triangle, or a right-angled 

 triangle. To determine any scalene triangle, it is plain that wo 

 must have one of the following series of data. 



I. With regard to the sides without the angles : 



1. The length of each of the three unequal sides. 



2. The length of two sides and the altitude of the triangle. 



IE. With regard to the angles without the sides : 



3. Any two of the angles of the triangle. 



m. With regard to the sides and angles combined : 



4. The length of any two of the sides of the triangle and one of its 

 angles. 



5. The length of one side of the triangle and two of its angles. 



6. The length of one side of the triangle, its altitude, and one of its 

 angles adjacent to the given side. 



As in the construction of the isosceles triangle, the first case 

 ia met by Problem VIII. (page 191), but the second brings 

 us to 



PROBLEM XXIV. To draw a triangle of which tJie length of 

 jtco of its sides and the altitude are given. 



Let A and B (Fig. 32) represent the length of two of the sides 

 dL the triangle required, and c its altitude. In any straight line, 

 D E, of indefinite length, set off r a equal to B, and by Problem X. 

 (page 192), draw the indefinite straight line, H K, parallel to 

 D E, at a distance from it equal to c, the altitude of the required 

 triangle. Then from F as centre, with a radius equal to A, draw 

 an arc cutting H K in the point L. Join L F, L G ; the triangle 

 L F G is a triangle answering the requirements of the data, for 

 its sides, L F, F G, are equal to A and B respectively, and its 

 altitude shown by the dotted line L N is equal to the given 

 straight line c. The triangle M F G, drawn in the same way, is 

 also a triangle which meets the requirements of the data, for its 

 sides, M G, G F, are equal to A and B respectively, and its altitude, 

 shown by the dotted line M o, is equal to c. 



The triangles L F G, M F G, are equal to each other in every 

 respect, namely, the length of their sides, their altitude, and 

 their superficial area. They are upon the same base, F G, and 

 between the same parallels, D E, H K, and they are what we may 

 term symmetrical triangles. From this we learn that symme- 

 trical triangles on the same base and between the same parallels 

 are equal to one another ; and this ia true, not for symmetrical 



triangles only, but for any triang3es, whether symmetrical or 

 not, that are upon the same base and between the same 

 parallels. Thus, the triangles L F G, M F G are each of them 

 equal to the triangle P F G, which is on the same base, F G, and 

 between the same parallels, D E, H K, and each of them would 

 be equal to any triangle that may be formed by drawing lines 

 from the points F and G to any point in the straight line H K, 

 produced both ways indefinitely. 



Triangles also which stand upon equal bases and between the 

 same parallels are equal to one another. Thus, the triangles 



L N G, M o F, which 



B _ stand on equal 



c bases, N G, F O, and 



K between the same 

 parallels, D E, H K, 

 are equal to one 

 another, as are also 

 the triangles L N F, 

 M o G, which are 



Fig. 32. 



between the same 

 parallels and stand 

 on equal bases N P, 



G o. And this is also as true of unsymmetrical triangles as of 

 symmetrical triangles, for if we join the dotted line N P, the 

 triangles L N F, P N F, are equal to one another, because they 

 are on the same base, N F, and between the same parallels ; 

 and since the triangle M G o is equal to the triangle L N F, it 

 must also be equal to the triangle P N F. 



In Case 3, when two of the angles of the required triangle are 

 given, it is manifestly necessary only to make at two points in 

 the same straight line, and on the same side of it, two angles 

 equal to the given angles, each having its opening turned 

 towards the apex of the other, and then, if necessary in order to 

 complete the triangle, to produce the sides of the angles that are 

 inclined to the side that is common to both. The student must 

 notice that when two angles of a required triangle are given 

 without any special requirement as to their relative position, 

 an endless number of pairs of symmetrical triangles may be 

 drawn, similar in form but of different superficial areas, all 

 satisfying the general requirements set forth in the data. 



Thus, in Fig. 33, if A and B represent the given angles of the 

 triangle required, it is plain that to make a triangle having two 

 angles equal to the given angles A and B, we have only to make 

 at any point, C, in a straight line, x Y, of indefinite length, the 

 angle T C E equal to A, and at another point, r>, in the same 

 straight line, the angle x D E equal to B, each angle having its 

 opening opposite or turned towards the apex of the other, as, in 

 this figure, the opening of the angle at C is opposite the apex D 

 of the angle at r>, and vice versA ; and to complete the triangle 

 produce the sides, c E, D E, of the angles at c and D that are 

 inclined to the common side, c r>, until they meet. If we reverse 

 the position of the angles, making the angle at c equal to the 

 angle at B, and the angle at D 

 equal to the angle at A, the 

 triangle assumes the form shown 

 by the triangle F c D in the 

 figure. The triangles BCD, 

 F c D, are symmetrical and equal 

 in every respect. The triangles 

 K G H, L G H, shown by dotted 

 lines, are also equal and sym- 

 metrical in every respect, and 

 satisfy the general conditions of 

 the data, although their super- x 

 ficial area is greater than the 

 area of the triangles E c D, F c D, 

 because the points G and H, at which the angles necessary for 

 the construction of the triangle required are made equal to A 

 and B, are taken on the indefinite straight line, x Y, at a greater 

 distance apart than c and D. 



PROBLEM XXV. To draw a triangle of which two sides and 

 one of the angles are given. 



First, let the given angle be included between the given sides, 

 and let the straight lines B, C represent the length of the given 

 sides of the triangle required, and A the given angle included 

 between them (Fig. 34). Draw any straight line, x Y, of inde- 

 finite length, and at any point, D, in x Y, make the angle Y D E 

 equal to the given angle A. Along r> Y set off D F, equal to c, 



Fig. 



