THE POPULAR EDUCATOR 



Bouthern hemisphere, and thus have more power to produce heat 

 than if they fell obliquely, according to the illustration given 

 above. Now, as we in this country are inhabitants of the 

 northern hemisphere, and of that part which is within the circle 

 of illumination all the year round, we experience the vicissitudes 

 of the seasons just described as belonging 1 to it, and we are con- 

 sequently colder in winter than in summer, although the earth 

 be actually nearer the sun in winter than in summer. 



But we must explain more fully what we mean by the circle 

 f illumination. It is plain that the rays of light falling from 

 the sun upon the opaque or dark body of the earth in straight 

 lines, can never illuminate more than one-half of its surface at a 

 time ; as may be seen by the very simple experiment of making 

 the light of a candle fall upon a ball at a distance from it. 

 Now, as the earth revolves on its axis once every 24 hours, it is 

 evident that the illuminated half, and consequently the circle of 

 illumination which is the boundary of that half, is perpetually 

 changing, so that almost all parts of the globe receive light for 

 several hours in succession, and that they are also enveloped in 

 darkness for several hours in the same manner. If the axis of 

 the eartti, instead of being inclined at a certain angle to the 

 plane of its orbit, which we shall hereafter call the Ecliptic, 

 were at right angles to that plane, and preserved its parallelism, 

 then the circle of illumination would' continually extend from 

 pole to pole, and all places on the earth's surface would enjoy 

 light for 12 hours in succession, and would be enveloped in 

 darkness for exactly the same period the whole year round. 



On the other hand, if the axis of the earth were coincident 

 with the plane, and preserved its parallelism, this would happen 

 only twice a year; and each hemisphere would at opposite 

 periods be in total darkness for a whole day, while the 

 variations between these extremes would be both inconvenient 

 and injurious. In the former case the seasons would be all the 

 Bame, that is, there would bo perpetual sameness of season all 

 the year round ; in the latter case, the seasons, instead of being 

 four only, would be innumerable, that is, there would be per- 

 petual change. 



Here, then, creative wisdom shines unexpectedly forth. The 

 inclination of the earth's axis is such as to produce the four ', 

 seasons in a remarkable manner, and to permit sufficient time 

 for the earth to bring her fruits to perfection, as well to let her 

 lie fallow for a period that she may renew her fruitf ulness. 



In Pig. 1, when the earth is supposed to be at the point c, she 

 is at her mean distance from the sun at the vernal equinox, which 

 is the first time of the year when day and night are equal, which 

 happens on or about the 21st of March. Now, at this point the 

 inclination of the earth's axis to the minor axis of the ellipse 

 is a right angle, and as the focus F', in the case of the earth, 

 nearly coincides with the centre O, the rays of light proceeding 

 from the sun nearly in the straight line O c, fall upon that axis 

 nearly perpendicularly, and illuminate the globe from pole to 

 pole, so that the circle of illumination passes through the poles, 

 and the days and nights are equal all over the globe, each con- 

 sisting of 12 hours, while the earth is in this position. In the 

 opposite position at D, the earth is again at her mean distance 

 from the sun at the autumnal equinox, which is the second time 

 of the year when day and night are equal, which happens on or 

 about the 22nd of September. At this point the circumstances 

 of the globe and the circle of illumination are exactly the same 

 as we have just described. At these four points, A, c, B, and D, 

 in the orbit of the earth, are found the middle points of the four 

 seasons of the year, viz., at A, mid-winter ; at c, mid-spring ; at 

 B, mid-summer ; and at D, mid-autumn. At the point A, or mid- 

 winter, which is on or about the 21st of December, we have the 

 shortest day in the northern- hemisphere and the longest day in 

 the southern hemisphere ; and at the point B, or mid-summer, 

 which is on or about the 22nd of June, we have the longest day 

 in the northern hemisphere and the shortest in the southern 

 hemisphere. 



" Thus is primeval prophecy fulfilled : 



While earth continues, and the ground is tilled ; 



Spring time shall come, when seeds put in the soil 



Shall yield in harvest full reward for toil ; 



Heat follow cold, and fructify the ground, 



Winter and summer in alternate round ; 



And night and day iu close succession rise, 



While each is regulated by the skies. 



Supreme o'er all, at first, Jehovah stood, 



And, with creative voice, pronounced it good." 



LESSONS IN ARITHMETIC. XXIV 



1. FROM the tables given in Lessons XXI., XXII., XXIII. (Vol 

 I., pp. 366, 379, 394), it is evident that any compound quantity 

 could be expressed in a variety of ways, according as we use 

 one or other of the various units, or denominations, as they 

 are called, which are employed. Thus the compound quantity 

 2 3s. 6d. could be indicated as here written, or by 522 pence, 

 or again, by 43| shillings, etc. The process of expressing a 

 compound quantity given in any one denomination in another, 

 is called reducing the quantity to a given denomination. The 

 process is termed 



REDUCTION. 



2. EXAMPLE 1. Reduce 5 2s. 7fd. to farthings. 



Since there are 20 shillings in a pound, in 5 pounds there are 

 5 X 20, or 100 shillings ; and therefore, in ^85 2s., 100 + 2, or 

 102 shillings. Since there are 12 pence in a shilling, in 102 

 shillings there are 102 X 12, or 1224 pence ; and therefore, in 

 5 2s. 7d., 1224 -f 7, or 1231 pence. Since there are 4 

 farthings in a penny, in 1231 pence there are 1231 X 4, or 

 4924 farthings ; and therefore, in 5 2s. 7|d. there are 4924 

 + 3, or 4927 farthings. 



The process may be thus arranged : 



5 2s. 7|d. 

 20 



100 + 2 = 102s. 

 12 



1224 + 7 = 1231d. 



4 



4924 + 3 = 4927 farthings. 



EXAMPLE 2. In 4927 farthings how many pounds, shillings, 

 pence, and farthings are there ? 



4927 divided by 4 gives a quotient 1231, and a remainder 3; 

 hence 4927 farthings are 1231 pence and 3 farthings. 1231 

 divided by 12 gives a quotient 102, and a remainder 7 ; hence 

 1231 pence are 102 shillings and 7 pence. 102 divided by 20 

 gives a quotient of 5, and a remainder 2 ; hence 102 shillings 

 are 5 pounds and 2 shillings. Therefore 4927 farthings are 

 1231J pence, which is 102s. 7fd., which is .5 2s. 7|d. 



The operation may be thus arranged : 



4)4927 

 12 ) 1231 . . . 3f. 



20 ) 102 ... 7d. 



5 2s. 7^1. 



In dividing by 20, note the remark (Lesson VII., Art. 7). 



The same method would apply to compcind quantities of any 

 other kind. 



Hence we get the following 



Rule for the Reduction of Compound Quantities. 



(I.) To reduce quantities in given denominations to equivalent 

 quantities of lower denominations. 



Multiply the quantity of the highest denomination by that 

 number which it takes of the next lower denomination .to make 

 one of the higher ; and to the product add the number of quan- 

 tities of that lower denomination, if thf-re are any. Proceed in 

 like manner with the quantity thus obtained, and those of each 

 successive denomination, until the required denomination is 

 arrived at. 



(2.) To reduce quantities of given denominations to equiva- 

 lent quantities of higlier denominations. 



Divide the number of quantities of the given denomination by 

 that number which it takes of quantities of this denomination 

 to make one of the next higher. Proceed in the same manner 

 with this and each successive denomination, until the required 

 denomination is arrived at. The last quotient, with the several 

 remainders, will be the answer required. 



Obs. It is manifest uLa^ the correctness of an operation per- 

 formed in accordance with either of the foregoing rules may be 

 tested by reversing the operation that is, by reducing the 

 result to the original denomination. 



