246 



THE POPULAR EDUCATOR. 



animal lives also influences the consumption ; it is considerably 

 greater on an animal than on a farinaceous diet. 



The increase of the carbonic a,cid is mainly dependent on the 

 absorption of oxygen, and this, therefore, is also affected by 

 like circumstances. In an ordinary way, it is calculated that a 

 man exhales 173 grains of carbon per hour, or rather more than 

 eight ounces in the twenty-four hours. Other authorities place 

 it as low as five, whilst Liebig estimates the amount from the skin 

 and lungs together at nearly fourteen ounces. Age and sex have 

 some influence in this matter : thus, the amount in males 

 regularly increases from eight to thirty years of age, and from 

 forty to extreme old age steadily diminishes. Temperature 

 also affects the result, the higher the temperature the less the 

 amount of carbonic acid exhaled. If the air is impure, as 

 where there is not sufficient ventilation, and in consequence the 

 same air is breathed more than once, the quantity of carbonic 

 acid is diminished, showing that in the absence of the proper 

 proportion of oxygen, the necessary purification of the blood does 

 not take place. By food the quantity is increased, by fasting 

 it is diminished ; physical exertion increases it ; rest, especially 

 sleep, diminishes it. The temperature of the expired air is in 

 almost all cases raised, its average heat being about 98 to 99 

 degrees. The moisture is also always increased, the increase 

 being greater the dryer the air is before it is inspired, the 

 expired air being always nearly saturated with moisture. The 

 quantity of water given off by the lungs during the twenty- 

 four hours is estimated to vary from six to twenty-seven 

 ounces. 



The changes produced in the blood during respiration are 

 manifested, first, by change of colour the dark venous blood 

 acquiring the bright arterial hue during its passage through the 

 lungs ; secondly, by the temperature of the blood being raised 

 by the same process. The way in which the oxygen inspired is 

 absorbed, and the carbonic acid expired is formed, has been 

 much disputed. It used to be formerly held that the oxygen 

 at once, at its entrance into the lungs, combined with the 

 carbon contained in the blood, and thus formed the carbonic 

 aoid ; but it has now been conclusively shown that though, no 

 doubt, some of the carbonic acid is produced in this way, yet the 

 greater part exists already in the blood by the time it reaches 

 the lungs. The origin of this, the larger part of the carbonic 

 acid, is thus explained : When the venous blood is passing 

 through the lungs, it gives up the carbonic acid with which it is 

 charged, and absorbs the oxygen, the red corpuscles being 

 credited with the greater part of this work ; the oxygen thus 

 held in solution, and not in combination, by the aerated blood, 

 is conveyed by the arteries to the capillary system, where it is 

 brought into intimate relation with the elementary tissues, and 

 the oxygen assists in the nutrition of the system, and, combin- 

 ing with the waste carbon of the worn-out structures, forms 

 carbonic acid and water, which are conveyed by the veins back 

 to the lungs, there to be removed from the body. In their 

 office of purification the lungs are powerfully assisted by the 

 skin. From the whole surface of the body there is constantly 

 going on an exudation of watery fluid containing many elements 

 derived from the wasted tissue and, notably, carbonic acid. It 

 is very difficult to estimate the amount of carbonic acid thus 

 excreted, but the importance of the proper performance of this 

 function of the skin is proved by the fact that animals whose 

 skin had been covered with an impermeable varnish, and thus 

 prevented from doing its duty, soon died with all the symptoms 

 of suffocation. This also shows how necessary for the preserva- 

 tion of health it is that the skin should be kept healthy and 

 active by the free use of baths, etc., to clear away the exuded 

 material from its surface. 



Closely connected with this function of respiration is the 

 question of animal heat, and the causes which produce and 

 maintain it ; the chief, if not the only one, is that combination 

 of oxygen with the various other elements of the body, which 

 is mainly brought about by respiration. The formation of car- 

 bonic acid, water, etc., in the body are all instances of chemical 

 action, and heat is necessarily produced ; and as these changes 

 are continually going on in all parts of the body, it follows that 

 a greater or less amount of increase of temperature is also being 

 constantly brought about. And it has been conclusively shown 

 by experiment that sufficient, or nearly sufficient, heat is pro- 

 duced during these processes to account for all the animal heat 

 of the body. 



LESSONS IX ALGEBRA. XL. 



CONTINUED GEOMETRICAL PROPORTION oil PROGRESSION. 

 WHEN all the ratios of a series of proportionals are equal, the 

 quantities are said to be in continued proportion or progression. 



As arithmetical proportion continued is arithmetical pro- 

 gression, so geometrical proportion continued is geometrical 

 progression. It is sometimes called progression by quotient. 



The numbers 64, 32, 16, 8, 4, are in continued geometrical 

 proportion. 



In this series, if each preceding term be divided by the 

 common ratio, the quotient will be the following term. Thus, 

 w = 32, and f = 16, and = 8, and a = 4. 



If the order of the series be inverted, the proportion will still 

 be preserved, and the common divisor will become a multiplier. 

 In the serios 4, 8, 16, 32, 64, etc., 



4x2=8, and 8x2 = 16, and 16X2 = 32, etc. 



Quantities then are in geometrical progression when they 

 increase by a, common multiplier, or decrease by a common 

 divisor. 



This common multiplier or divisor is called the ratio. For 

 most purposes, however, it will be more simple to consider the 

 ratio as always a multiplier, either integral or fractional. 



In the series 64, 32, 16, 8, 4, the ratio is either 2 considered 

 as a divisor, or considered as a multiplier. 



When several quantities are in continued proportion, the 

 number of couplets, and of course the number of ratios, is one 

 less than the number of quantities. Thus the five proportional 

 quantities, a, b, c, d, e, form four couplets containing four 

 ratios ; and the ratio of a : e is equal to the ratio of a 4 : b 4 , that 

 is, the ratio of the fourth power of the first quantity to the 

 fourth power of the second. Hence, 



If three quantities are proportional, the first is to the third as 

 the square of the first to the square of the second, or as the 

 square of the second to the square of the third. In other words, 

 the first has to the third a duplicate ratio of the first to the 

 second, And conversely, if the first of the three quantities is to 

 the third as the square of the first to the square of the second, 

 the three quantities are proportional. 



If a : b : : b : c, then a : c : : a* : 6 2 . And universally, 



If several quantities are in continued proportion, the ratio of 

 the first to the last is equal to one of the intervening ratios 

 raised to a power whose index is one less than the number of 

 quantities. 



If there are four proportionals, a, b, c, d, then a : d : : a 3 : b 3 . 



If there are five, a, b, c, d, e; a -. e -. : a* -. b 4 , etc. 



If several quantities are in continued proportion, they will 

 be proportional when the order of the whole is inverted. This 

 has already been proved with respect to four proportional 

 quantities. It may be extended to any number of quantities. 

 Between the numbers, 64, 32, 16, 8, 4, 



The ratios are, 2, 2, 2, 2. 



Between the same inverted, 4, 8, 16, 32, 64, 

 The ratios are, , , i, -}. 



So if the order of any proportional quantities be inverted, the 

 ratios in one series will be the reciprocals of those in the other. 

 For by the inversion each antecedent becomes a consequent, 

 and vice versa ; but the ratio of a consequent to its antecedent 

 is the reciprocal of the ratio of the antecedent to the consequent. 

 That the reciprocals of equal quantities are themselves equal is 

 evident from Ax. 4. 



To investigate the properties of geometrical progression, we 

 may take nearly the same course as in arithmetical progression, 

 observing to substitute continual multiplication and division, 

 instead of addition and subtraction. It is evident, in the first 

 place, that, 



In an ascending geometrical series, each succeeding term ia 

 found by multiplying the ratio into the preceding term. 



If the first term is a, and the ratio r, 



Then a X r = ar, the second term ; ar X f = ar 2 , the third ; 

 ar* X r = ar 3 , the fourth ; ar 3 X r = ar 4 , the fifth, etc. 



And the series is a, ar, ar 2 , ar 3 , ar*, ar 5 , etc. 



If the first term and the ratio are the same, the progression 

 is simply a series of powers. 



If the first term and ratio are each equal to r, 



Then r X r = r 2 , the second term ; r 9 X r r*, the third ; 

 r x r = r 4 , the fourth ; r 4 X r = r 5 , the fifth. 



