348 



THE POPULAR EDUCATOR. 



of alum which possesses the property of absorbing the heat 

 rays, while it is transparent to light is interposed between 

 the lens, L, and the mirror, M. After reflection from the vibrating 

 mirror, the rays are brought parallel by means of another lens, L 2 , 

 and carried across space to the seleninm cell, p, contained 

 in its parabolic reflector, E. T is the telephone, and B 

 the battery. In the first experiment with this new form of 

 reflector sentences were clearly audible at a distance of over 

 800 feet. 



We have noted, in a former lesson, that Professor Page, in 

 America, had obtained from an electro-magnet an effect which 

 he called galvanic music, by causing an intermittent current 

 of electricity to traverse the coils. Now, Professor Bell, in 

 his photophone experiments, was led to assume that if an inter- 

 mittent beam of light were allowed to fall upon his selenium 

 cell, he would most surely hear a musical note in the attached 

 telephone, such note being in pitch according to the quickness 

 with which the light impressions followed one another. His 

 hopes were justified by the results attained. A beam of sun- 

 light was rendered intermittent by being interrupted in its 

 passage by a perforated wheel. As the wheel turned, the light 

 flashed through the perforations, and in this way the selenium 

 cell was subjected to pulsations of light, a distinct musical note 

 being heard when the telephone at the distant station was 

 applied to the ear. 



Now comes the most curious part of the experiment. It 

 became a matter of habit with the listener to occasionally cut 

 off the ray of light with his hand, so as to prevent it reaching 

 the selenium cell, with the result that any one would suppose 

 would happen the musical note ceased. It was afterwards 

 found convenient to substitute a sheet of hard rubber for the 

 hand, with the astonishing result that that opaque material 

 did not, as was fully expected, shut off all the sound. Some- 

 thing passed through the hard rubber, and continued to affect 

 the resistance of the selenium, but what that something was it 

 is at present impossible to say. Still more wonderful is the 

 fact that if the rubber is held on the other side of the revolving 

 disc, so that no light whatever can pass through its perfora- 

 tions, still the musical note is apparent. To prove that this 

 invisible force if we may call it so is not due to heat rays, 

 Professor Bell subsequently employed two sheets of rubber, 

 between which was a glass vessel holding a solution of alum. 

 The sounds were heard all the same. 



It next occurred to the experimenters to put aside the tele- 

 phone altogether, in order to ascertain whether the effect could 

 not be produced directly by holding the rubber to the ear while 

 an intermittent beam of light was allowed to play upon it. To 

 their surprise they heard clearly the mysterious note. By 

 forming the rubber into a thin diaphragm the effects were much 

 intensified. After this they were led to try experiments with 

 diaphragms of other materials, and were able to record that 

 this curious effect of light inducing sound can be obtained from 

 all the metals, glass, paper, indeed all kinds of substances, 

 provided they be in the form of thin sheets or diaphragms. 



After many experiments it occurred to Professor Bell that 

 the disturbance, whatever it was, occurred on the surface of 

 the material used, and that before a musical note could be 

 heard the vibrations must be carried through the substance 

 of the material to the ear ; hence the advantage of using thin 

 diaphragms, because the distance such vibrations had to 

 j traverse is thereby reduced to a minimum. At the same time, 

 granting that this supposition is correct, more favourable re- 

 sults still would be obtained by listening to the surface upon 

 , which the light pulsations actually fell. To do this conve- 

 . niently was not easy until the experimenters hit upon the 

 expedient of forming the material into a tube, and allowing 

 the light to impinge upon its interior, the other end of the tube 

 being applied to the ear. Using this method, they found that 

 india-rubber and all other substances responded to light im- 

 pressions in a very marked degree ; but they gave great varia- 

 tions with regard to intensity of sound, a circumstance which 

 in itself opens up a vast field for research. 



It is not even necessary that the material experimented upon 

 should be solid. It may be in the form of a liquid, or a gas 

 even, and exposed to the intermittent ray in a common glass 

 test-tube. Even tobacco-smoke, placed in the tube, will emit 

 its musical note ; indeed, it is difficult to point to anything 

 that refuses to take part in this strange orchestra. 



B D H 



Fig. 1. 



LESSONS IN ALGEBPvA. XLII. 



APPLICATION OF ALGEBRA TO GEOMETRY. 



IT is often expedient to make use of algebraical notation for 

 expressing the relations of geometrical quantities, and to throw 

 the several steps of a demonstration into the form of equations. 

 By this, the nature of the reasoning is not altered ; it is only 

 translated into a different language. Signs are substituted for 

 words, but they are intended to convey the same meaning. A 

 great part of the demonstrations in Geometry really consist of 

 a series of equations, though they may not be presented to us 

 under the algebraic forms. Thus the proposition, that the sum 

 of the three angles of a triangle is 

 equal to iivo right angles, may be 

 demonstrated, either in common lan- 

 guage, or by means of the signs used 

 in algebra. 



Let the side A B of the triangle 

 ABC (Fig. 1) be produced to D ; let 

 the line B E be drawn parallel to 

 A c ; and let G H i be a right angle. 



The demonstration in words is as 

 follows : 



1. The angle E B D is equal to the angle BAG (Euclid I. 29). 



2. The angle c B E is equal to the angle A c B. 



3. Therefore, the angle E B D added to c B E that is, the 



angle c B D is equal to B A c added to A c B. 



4. If to these equals wo add tho angle ABC, the angle 



CB D added to ABC is equal to BAC added to ACB 

 and ABC. 



5. But c B D added to A B c is' equal to twice G H I that is, 



to two right angles (Euclid I. 13). 



6. Therefore, the angles BAG and ACB and ABC are together 



equal to twice G H I, or two right angles. 



Now by substituting the sign + for the word added or and, 

 and the sign = for the word equal, we shall have the same 

 demonstration in the following form : 



1. By Euclid I. 29, E B D = B A c. 



2. And c B E = A c B (Euclid I. 29). 



3. Add the two equations EBD + CBE = B A c + A c B. 



4. Add ABC to both sides CBD + ABC = BAG + ACB +ABC. 



5. But by Euclid I. 13, c B D + A B G = 2 G H I. 



6. Therefore BAc + ACB + ABC = 2GHi. 



By comparing, one by one, the steps of these two demon- 

 strations, it will be seen that they are precisely the same, 

 except that they are differently expressed. 



It will be observed that the notation in the example just 

 given differs, in one respect, from that which is generally used 

 in algebra. Each quantity is represented, not by a single letter, 

 but by several. In common ftlgebra, when one letter stands 

 immediately before another, as ab, without any character 

 between them, they are to be considered as multiplied to- 

 gether. 



But in Geometry, AB is an expression for a single line, and 

 not for the product of A into B. Multiplication is denoted, 

 either by a poiat or by the sign X . The product of A B into 



C D is A B . C D, or A B X C D. 



There is no impropriety, however, in representing a geo- 

 metrical quantity by a single letter. We may make 6 stand 

 for a line or an angle, as well as for a number. 



If, in the example above, we put the angle 



E B D = a, 

 B A C = b, 

 C B E = C, 



A C B = d, 



C B D = g, 



ABC 

 OHI 



ft, 



E, 



the demonstration will stand thus : 



1. By Euclid L 29, 



2. And 



3. Adding the two equations, 



4. Adding h to both sides, 



5. By Euclid I. 13, 



6. Therefore, 



a = b, 



g + h = b + d + h. 

 g + h= 21. 



b -f d + h = 21. 



This notation is appcrently more simple than the other ; but 

 it deprives us of what is of great importance in geometrical 



