LESSONS IN f OKOGRAl'IIY. 



This is the required result, because, in multiplying any qnan- 

 u number, if we multiply separately the parta of which 

 tin- quantity ia composed, and then add the product* together, 

 the result in the same as would bo obtained by multiplying the 

 whole quantity by that number. The above operation would, in 

 practice, be thus arranged : 



5 2s. 7j.l. 

 1 



30 15*. lOAd. 



Hence wo BOO the truth of the following 



9. Rule for Compound Multiplication. 



Multiply each denomination separately, beginning with the 



and divide each product by that number which it takes 



of the denomination multiplied to make one of the next higher. 



Set down the remainder, and carry the quotient to the next 



product, as in addition of compound numbers. 



06s. Any multiplier ia of necessity an abstract number. 

 Two concrete quantities cannot bo multiplied together. Multi- 

 plication implies the repetition of some quantity a certain 

 number of times ; and we see, therefore, that to talk of multiply- 

 ing one concrete quantity by another is nonsense. 



In the case of geometrical magnitudes in finding the area of 

 a rectangle, for instance we do not multiply the feet in one side 

 by those in the other, but we multiply the number of feet in one 

 side by the number of feet in the other, and from geometrical 

 considerations we are able to show that this process will give us 

 the number of square feet which the rectangle contains. The 

 very idea of multiplication implies that the multiplier must be 

 an abstract number. It is of the nature of twice, thrice, etc. 

 (Vide Oba. of Art. 7, Lesson XXII., Vol. I., page 380.) 



10. ADDITIONAL EXAMPLE IN COMPOUND MULTIPLICATION. 

 Multiply 12 Ibs. 3 oz. 16 dwts. by 56. 



In a case like this, where the multiplier exceeds 12, it ia 

 often more convenient to separate it into factors, and to mul- 

 tiply the compound quantity successively by them (Lesson VI., 

 Art. 2, Vol. L, page 95). Now 56 = 7 X 8. 



Ibs. oz. dwts. 

 12 3 16 

 7 



86 2 



12 



8 



689 8 16 Answer. 



EXERCISE 45. 

 Work the "ollowing examples in compound multiplication : 



1. 35 6s. 7d. by 7. 



2. 1 6a. 8jd. by 18. 



3. 1 ton 270V Ibs. by 15. 



4. 16 tons 3 cwt. 10} Ibs. by 25 and 84. 



5. 17 dwts. 4J grs. by 96. 



6. 15 gals. 2 qts. 1 pt. by 63 and 126. 



7. 175 miles 7 fur. 18 rods by 84, 196, and 96. 



8. 40 leagues 2 m. 5 fur. 15 r. by 50, 200, and 385. 



9. 149 bush. 12 qts. by 60, 70, 80, and 90. 



10. 26 qrs. 7 bush. 3 pks. 5 qts. by 110 and 1008. 



11. 150 acres 65 rods by 52, 400, and 3000. 



12. 70 yrs. 6 mo. 3 wks. 5 d. by 17, 29, and 36. 



13. 265 cubic ft. 10 in. by 93, 496, and 5008. 



14. 75 40' 21" by 210, 300, and 528. 



15. 213 5s. 6jd. by 819 and by 918. 



16. 5 tons 15 cwt. 17 Ibs. 3 oz. by 7, by 637, and 763. 



17. 13 7s. 9|d. by 1086012 and by 1260108. 



(For the last three questions refer to Lesson VII., Arta. 15, 16, 

 Vol. I., page 111.) 



LESSONS IN GEOGRAPHY. XVI. 



HAVING explained, in a previous LCMOD (see VoL IL, page 4), 

 the nature of the seasons arising from the annual motion of the 

 earth in it* orbit or path round the ran, and the ptndtolirai of 

 ita axis, or tho invariable inclination of that axis to the plane of 

 ita orbit, we shall render this subject more strikingly evident by 

 means of the accompanying diagram of the HMHOTH. Hex* til* 

 nun JH considered to be fixed at the point r in Fig. 4 (page 60), 

 which ia considered to be the/ocu* of the elliptical or oral orbit 

 in which the earth moves, and which ia ao near to the centre of 

 tho curve that it may be, on this small scale of figure, reckoned 

 tho same with that centre ; and you know that the centre ia 

 tho point where the major axis, between summer and winter, 

 intersects or crosses the minor axis, between spring and autumn. 

 If you are curious enough to know how far the/oetu, r, ia from 

 tho real centre of the orbit, we shall tell you ; it ia about one- 

 sixtieth part of the half if the major ant, or of the mean Pit- 

 tance between the earth and the sun, from the real centre. Let 

 us sec if we can express this distance in some known measure. 

 Tho mean distance of the earth from tho sun, or the length of 

 the mean semi-diameter of the earth's orbit, ia about 23,109 

 times the length of the mean terrestrial radius, or of the mean 

 distance from the centre of the globe of the earth to its surface. 

 Tho earth's mean radius is 3,956} British miles, ita mean dia- 

 meter being 7,913 miles. Therefore multiplying 3,956} miles by 

 23,109, we have the mean distance of the earth from the sun, 

 that is, half the major axis of ita orbit, about 91,431,000 in 

 round numbers. This makes the mean diameter of the earth's 

 orbit about 182,862,000 miles, and its approximate circum- 

 ference about 574,709,000 miles. The linear eccentricity of the 

 earth's orbit being '0168, or about one-sixtieth of ita semi-axis 

 major, or mean distance of 91,431,000 miles, we have 1,523,850 

 miles for the distance between tho centre of the orbit and the 

 centre of the sun, or the focus of that orbit. Consequently, 

 the earth is about double this distance, or 3,047,700 miles nearer 

 to the sun in winter than in summer. 



In Fig. 4, tho earth is represented in four different positions 

 (momentary positions) in its orbit ; namely, at mid-summer, mid- 

 spring, mid-winter, and mid-autumn. In all these positions, 

 as well aa all round in its various positions in the orbit, the 

 parallelism of its axis, N s, is preserved. This axis is inclined 

 to the plane of the orbit, as we have before observed, at an angle 

 of 66 32' ; hence it makes an angle of 23 28' with the perpen- 

 dicular to the plane of its orbit ; for the perpendicular, repre- 

 sented by the dotted line passing through the centre, o, makes 

 an angle of 90 Q with the plane of the orbit ; and subtracting 

 66 32' from 90 gives the remainder 23 28', which is the angle 

 between the axis, N s, and the perpendicular, or dotted line. By 

 reason of this parallelism of the axis N s, it so happens that at 

 mid-sjyring, or March 20th, the half of the globe ia illuminated 

 from pole to pole, that is, from the northern extremity of the 

 axis N, to the southern extremity of the axis 8, and the day* 

 and nights are then exactly equal all over the earth ; that is, 

 there are twelve hours of light and twelve hoars of dark- 

 ness to every spot on the earth's surface for this day. Hence 

 this day is called the equinox (equal niyht) of spring, or the 

 vernal equinox. Again, at mid-summer, or June 21st, the half 

 of the globe is illuminated from the circumference of a small 

 circle of the globe at the distance of 23 28' from the north 

 pole, N, to the circumference of a small circle at the distance of 

 23 28' from the south pole, s ; and the day is twenty-four hours 

 long at all places of the earth contained in the apace between 

 tho small circle and the north pole ; that is, there are twenty- 

 four hours of light and no darkness at all to every spot within 

 this space on this day ; but the night ia twenty-four hours long 

 at all places of the earth contained in the space between the 

 small circle and the south pole, that ia, there are twenty-foot 

 hours of darkness and no light at all to every spot within this 

 space on this day. As at this point the earth begins to return 

 to a position similar to that at the vernal equinox, and the aun 

 seems to be stationary as to its appearance and effects on the 

 earth's surface for two or three days before and after this day, 

 it is called the summer solstice (sun-standing), or the tropic 

 (turning) of summer. Next, at mid-autumn, or Sept. 23rd, the 

 half of the globe is again illuminated from polo to pole, and the 

 same appearances take place aa at the equinox of the spring, 

 that ia, the day* and nights are then exactly equal all over the 



