378 



THE POPULAR EDUCATOR 



|, ' 1 5, and j, and called respectively a major tone, a minor tone, 

 and a major semitone. The latter is the interval between E 

 and F, and between B and C, and is .not divided, as the other 

 intervals usually are, by tho insertion of intermediate notes, 

 known as flats and sharps. 



When, instead of sounding two musical pipes which are in 

 unison, wo take two which differ very slightly in their number 

 of vibrations as, for example, one making 256 vibrations, while 

 the other makes 258 or 260 wo shall cease to have a uniform 

 flow of sound. The sound-waves produced by the one alter- 

 nately increase and diminish those of the other, and thus we 

 have a series of beats which become less frequent as the notes 

 approach more nearly to unison. These are called Tartini's 

 beats, and may easily be noticed by sounding two adjacent 

 notes in the bass of an organ. 



By sounding different notes simultaneously, we find that some 

 combinations produce a much more pleasing effect than others. 

 The most pleasing result is attained when one is just an octavo 

 above tho other, and consequently produces twice the number of 

 pulsations. In this case, every other pulsation of the higher 

 note corresponds with one of the lower, somewhat as shown at 

 Fig. 24. This interval is called an octave, because in the 

 gamut any note is the eighth above the previous one of the 

 same name. 



Next to the octavo, the most pleasing chord is produced when 



three pulsations of the one note correspond to two of the other. 



This may be produced by sounding together C and G, and is 



known as a, fifth. If we sound G and the C above it we shall 



obtain the combination known as a fourth, in which four 



vibrations of one correspond to 



rrxvE. 2:1 -^ three of the other. Both these 



* 7 are represented in Fig. 24. 



*** Tho other concords are known 

 FIFTH. 3 : 2. as the waj'or third, in which the 



1 * ratio is 5 : 4, and the minor third, 

 I in which it is 6 : 5. These may re- 

 spectively be produced by striking 



FOURTH. 4 : 3. together C and E, and E and G. 



When the numbers representing 



* the ratio are high as, for in- 



Fig. 24. stance, 13:14 we get unpleasant 



jarring sounds or discords. 



A perfect chord is produced when three notes are sounded 

 together whose vibrations bear to one another the ratio 4 : 5 : 6. 

 Illustrations of this may be obtained by sounding C, E, and G 

 or G, B, and D. 



Before concluding these lessons we must just briefly notice 

 the construction of that most wonderful and important of all 

 acoustic instruments the human ear. 



The external ear is so shaped as to convey the pulsations of 

 the air to a circular membrane, known as the tympanum. 

 Behind this is a small cavity known as the drum of the ear, 

 across which there stretches a chain of four small bones. At 

 the further side of the drum are two apertures, also closed by 

 membranes ; through these the vibrations are conveyed to a 

 remarkable cavity hollowed out of the bone. This cavity, 

 which is _ known as the labyrinth, is filled with water, and the 

 ramifications of the auditory nerve are distributed over its 

 surface. In certain parts of it minute bristles project, and in 

 another part we have minute crystalline particles called otolithes, 

 all of which seem specially fitted to receive the faintest vibra- 

 tions. A remarkable organ has further been discovered by 

 Corti ; this consists of a vast number of vibrating cords, each 

 of which is apparently tuned to receive and render audible some 

 special _ vibrations. The intimate structure of these delicate 

 organs is, however, as yet but imperfectly understood ; further 

 investigation will doubtless serve to throw much fresh light on 

 the whole subject ; but the way in which external sensations of 

 any kind are communicated to the brain is at present veiled in 

 mystery. 



| We have purposely made the foregoing remarks on the con- 

 struction of the human ear extremely short a mere outline, in 

 fact, of the form and disposition of the component parts of this 

 wonderful organ as v- full description of its structure, with 

 numerous illustrations, has already been brought under the 

 notice of our readers in our lessons on " Animal Physiology" 

 in Vol. I. of the POPULAR EDUCATOR. 



Fig. 12. 



Fig. 13. 



LESSONS IN ALGEBRA. XLIII. 



WE now offer to our readers, as a useful and almost indis- 

 pensable supplement to our " Lessons in Algebra," an exercise 

 (Exercise 75) on the application of Algebra k> Geometry, and 

 two more (Exercises 76 and 77, of which the latter will be 

 found in our next and concluding lesson) of Miscellaneous 

 Examples for Practice. 



EXERCISE 75. 



1. If the hypothenuse of a right-angled triangle ABC (Fig 1 . 10, page 

 350) is Jifeet, and the difference of the oth^r two sides d feet, what 

 are the lengths of A B and B c ? Apply this when cl = 2, and Jv = 10. 



2. If the hypothenuse (Fig. 10, page 350) is 20 rods, and the base ia 

 to the perpendicular as 3 to 4, find their lengths. 



3. Having the perimeter of a parallelogram A B c D = 2p, and the 

 diagonal = d; to find the length (i), 



and the breadth (b). Apply this when 

 d = 15, and 2p = 42 (Fig. 11, page 350). 



4. The area of a right-angled tri- 

 angle ABC (Fig. 12) is d square feet, 

 and the sides, 1>E, DF, of the inscribed 

 parallelogram are respectively b and a 

 feet. Find B c. 



5. The perimeter of a right-angled 

 triangle is s feet, and its area is a 

 square feet ; find the hypothenuse. 



6. Havingtheareaof aparallelograin 

 D E PG (Fig. 14) inscribed in a given tri- 

 angle A B c = c square feet, find D G. 



Draw c i perpendicular to A B ; by supposition i> a is parallel to A B. Let 



C i = a, A B = b, and D a = x ; then c H = , i H = a - x -, etc. 



I) O 



7. The three sides of a right-angled triangle ABC (Fig. 13) are as 



follows : A c = 10, B C = 6, aud A B = 8 ; 

 find the segments c D and D A made by 

 a perpendicular from B oil A c. 



8. Through a given point (P) in the 

 diameter A B of the circle A Q B R (Fig. 

 15) to draw a right line so that its 

 two parts,' P R and P Q, shall have a 

 given difference (d), if AP = a, and B P 

 = b. , 



9. The height of an arch is 6 feet, an<J 

 its span 20 feet; what radius was it 

 struck with ? 



10. Find the side of that square whoso 

 area is 2\ times its perimeter. 



11. Find the side of that cube whose solid content and surface aro 

 expressed by the same number. 



12. The area of a right-angled triangle is 54 square feet, and its sides 

 are in arithmetical progression; find 



their lengths. Also give a general solu- 

 tion when the area = a. 



13. A rectangle contains 98 square feet, 

 and the difference between the adjacent 

 sides is 7 feet ; find the sides. 



14. The perimeter of a right-angled 

 isosceles triangle is m feet ; find tho 

 sides. 



15. The difference between the peri- 

 meter and perpendicular of an equilateral 

 triangle is m feet; find the length of one 

 of its sides. 



16. One side of a right-angled triangle is 



15, and the excess of twice the other side above the hypothenuse is 33 ; 

 find the side and hypothenuse. 



17. The sides of a right-angled triangle are in geometrical progression 



its area is m square feet ; find the 

 side which is a mean between the other 

 two. 



18. Given the difference between the 

 diagonal and side of a square = d feet to 

 find its area a ; apply this when d = 

 9-9411255. 



19. If a right-angled triangle has it? 

 sides in arithmetical progression, they 

 may be found by multiplying the square 

 root of ^ of the area by 3, 4, and 5 

 respectively; required the demonstra- 

 tion. 



20. Given the area (a) and base (b) of a 

 triangle A B c to divide it into two equal 



parts by a line (F G) drawn parallel to the base (A B) ; find the length oi 

 FG, and its perpendicular, c E. 



EXERCISE 76. MISCELLANEOUS EXAMPLES FOR PRACTICE. 



1. If a = 8, b = 7, c = 6, d = 5, and 6 = 1, prove that (ab + ce bd) 



+ C?._ b + ?5r:_ 2c ) = 3a - 2b + 4c - c. 

 \c e a d ' 



2. Also prove that V(a-3d) x V(b 3 -c 3 -2c) =(d*+ (a-c)- 3c 2 + d) 

 -3Va(ab + d). 



Fig. 14. 



E 



I F 



3. Prove that 



-(- &) =!_/!_(!_ 1^)1 . 

 2 C ' 



4. Prove that ^ + + <* ~ *> = - [- | - ( - a) }} 



