KKUM NATI-UK. 



'-'I* I 



inultiplior iindor the corresponding terms of the multiplicand. 

 .Multiply eoh term of tin- multiplicand by each term of tho 

 multiplier xtpanktely, beginning with tho lowest denomination in 

 :m multiplic.iiKl iui'1 tho highest in tho multiplier, and plin-in^ 

 the first iiguru of eaah line one pluoo to the right of that of tin- 

 preceding lino, under ita corresponding denomination. J''in:illy, 

 uld tlio Hcvi-nil lines together, carrying 1 for each 12 both in 

 multiplying and adding. The Hum will be tlioanswor required. 

 I tho area of a board 9 ft. 7' 2" long, and 

 3 ft. 4' 7" wide. 



9 ft. T 2* 



8 ft. 47* 



SW sq.ft. 9' 6" 

 8 sq.ft. 2* 4" 8" 



5* 7' 2-* 2" 



32 sq.ft. 5' 5* 10" 2" Antvier. 



Wo now subjoin some other examples connected with square 

 and cubic measure, such as are of frequent occurrence, not con- 

 fining ourselves to the method of duodecimals. 



EXAMPLE. (3.) Find the oost of carpeting a room 24 ft. 6 in. 

 long by 18 ft. 4 in. wide, with carpet which is 2 of a yard broad, 

 and costs 3s. ltd. a yard. 



24; x I8i = number of square feet in the floor, 

 and tho breadth of the carpet is * of a foot. 



Therefore the length of carpet in feet x ft. = area of floor in sq. 

 feet = 24} x 181. 



241 x 181 

 Therefore the number of feet of carpet required = . 



And each foot of carpet costs Is. 3d., or 1,8. 



24 1 - x 18 l - 5 

 Therefore, required cost = x - shillings 



49x55x4x5_ 13475 



2x3x9x4 



-2-T =-^ Bhillings = 12 9s. 6Jd. 



5t 



EXAMPLE. (4.) What is the height of a room which contains 

 223 cub. yds. 7 cub. ft. 624 cub. in. of air, the area of the 

 floor being 41 sq. yds. 12 sq. in. ? 



Since the cubical contents are obtained by multiplying the 

 three dimensions of length, breadth, and height together, and 

 the area of the floor is obtained by multiplying the length and 

 breadth together, we shall evidently get the height of tho room 

 by dividing the cubical contents by the area of the floor. 



223 cub. yds. 7 cub. ft. 624 cub. in. = 6028^ cub. ft. = 6028-J-J- cub. ft. 

 41 sq. yds. 12 sq. in. = 369^ sq. ft. = 369^ sq. ft. 



Hence required _ 60281* _ 217021 12 _ 217021 

 height in feet ~ 369A 36 X 4429 ~ 3 



4429 ) 217021 ( 49 

 17716 



39861 

 39861 



4429. 



Hence required height = V feet = 16 feet 4 inches. 



EXEBCISE 63. EXAMPLES IN SCALES OP NOTATION, DUO- 

 DECIMALS, CROSS MULTIPLICATION, ETC. 



1. Transform 12345678 from the decimal to the duodecimal scale, and 

 also to the scale of 7. 



2. Transform 58367 from the decimal to the duodecimal scale. 



3. Transform 57(39 from the duodecimal to the decimal scale. 



4. Transform 67535 from the scale of 8 to the scale of 6. 



5. Transform 79658 from the duodecimal scale to the scale of 8. 



6. Transform tetet from the duodecimal to the decimal scale. 



7. How many square feet are there in a beard 15 ft. 7 in. long, and 



1 it. 10 in. wide ? 



8. How many cubic feet are there in a block 18 ft. 5 in. long, 4 ft. 



2 in. wide, and 3 ft. 6 in. thick? 



9. Find the area of a space 16 ft. Sf 4" by 6 ft. 5' 8" 10"*. 



10. Find the area of a space 18 ft. 0' 5" 10'" by 4 ft. 8' 7" 9"'. 



11. What will it cost to plaster a room 20 ft. 6 in. long, 18 ft. wide, 

 and 10 ft. high, at 6jd. a square yard ? 



12. How many brick* 8 in. long, 4 in. wide, and 2 in. thick, will make 

 a wall 50 ft. long, 10 ft. high, and L> ft. <> in. thick ? 



13. Find the cost of carpeting the following rooms : 



(a) 18 ft. 4 in. long, 13 ft. 6 in. broad, with carpet { of a yard 

 wide, at 2s. 9d. a yard. 



(b) 11 ft. 6 in. long, 10 ft. 4 in. broad, at 10s. 6d. a square yard. 



(c) 16 ft. 11 in. long, 13 ft. 3 in. wide, with carpet 2 ft. 3 in. 



wide, at 4s. 7d. a yard. 



14. A Turkey carpet 11 ft. 6 in. long by 9 ft. 8 in. wide is laid down 

 In a room 14 ft. long by 12 ft. 6 in. wide; find the amount of oil-clot Ji 



necessary to complete the covering of tho floor, and its cost at 5s. par 

 square yard. 



15. Tho length of a room is twice ita breadth, and ita ana is 1152 

 square foet ; what Is ita length ? 



10. Find the cost of carpeting a room 17 ft. in. loom; by 13 ft. 9 in. 

 wide, with carpet } yard wide, at 4s. 84. a yard. 



17. If a cubic foot of water weighs 1000 oonoes, what most b* Uta 

 depth of a rectangular tank, which is 35 ft. Ions; and 10 ft. broad, that 

 it limy JUKI <.,,,! uii 1000 tons of water / 



' roller being ft. in. in circumference and 2 ft. 3 in. wide, 

 makes 12 revolution* as it moves from one end of a grass-plot to th 

 < .! !i< T, and passes 10 times from one end to the other ; find tho area, of 

 the grass-plot. 



19. What length must be cut off a plank 1{ ft. broad and 9 in. deep, 

 in order that it may contain 11-J cubic feet? 



20. A block of wood in the form of a rectangular parallelepiped* 

 measures along ita edges 18} ft., 5i ft., and 3 ft. respectively ; And ita 

 value if a cubical block of the same wood, whose edges are all 11 in. 

 long, is worth 3s. 6d. 



21. On laying down a bowling-green with soda 2 ft. 6 in. long by 9 

 in. wide, it is found that it requires 75 sods to form one strip the whole 

 length of the green, and that a man can lay down one strip and a 

 quarter each day ; find the area laid down in 8 days. 



22. If a cubic foot of water weighs 1000 oz., find to what depth a 

 ton of water will cover an acre. 



23. A square foot of paper weighed 104*68 grains, and when 320 

 figures had been written on it, weighed 105*155 grains. If a strip of 

 this paper 5j inches wide be taken to have written on it the circulating 

 period of Too'nn (which contains 100102 figures) in two lines at the 

 rate of 5 figures in an inch, find the weight of the whole, and the length 

 of the paper, and express this act in terms of the height of Salisbury 

 spire, which is 400 feet. 



Errata.. In page 142 of this vol., col. 2, line 40, for "2j" read "\ ;" 

 line 41, for "1 : 1 : j," read "1:1: J;" and in line 42, for "7 Ibe. of the 

 third," read "5 Ibs." 



SKETCHING FROM NATURE. IV. 



THEORY OF SKETCHING. 



THE previous lessons upon Sketching from Nature have been 

 almost entirely devoted to the practice ; we must now say some- 

 thing upon the theory, and offer our pupils some advice upon the 

 course they must pursue amongst the difficulties they will find in 

 the principles and application of the art. During their progress 

 they will meet with many perplexities, for which technical rules 

 can afford but little assistance, because the theory of art depends 

 upon laws of quite a different nature. The rules we have given 

 will help them over grammatical difficulties and assist them in 

 the work of construction, and for these reasons they cannot be 

 dispensed with ; but they are incapable of giving those charms 

 to a picture which it is the province of theory to impart, founded 

 upon a right feeling for the beauties and effects of nature. Our 

 pupils have now at their command a sufficient supply of geome- 

 trical information, as well as directions where to find it in these 

 pages, and of which we hope and trust they will make good use: 

 it will prove to be the best and most solid foundation whereupon 

 to build other principles to be derived from a close observation 

 of nature, and from a careful study of the numerous works of 

 our most eminent artists. The few remarks we made in our 

 first lesson upon the choice of a subject were offered with a view 

 of cautioning our pupils not to overburden themselves with too 

 many details at first, but to make their early essays from the 

 most simple subjects they could find. It is remarkable, but very 

 little experience will make it evident, that many subjects which 

 at first sight appear to be easy, from the fact that they are com- 

 posed of few prominent objects, will upon close examination, and 

 especially during the process of drawing, seem to expand into a 

 quantity of detail beyond all previous anticipation. Very fre- 

 quently a feeling of discouragement is the consequence. We 

 make this last observation, knowing from experience that in the 

 majority of cases the cause of the discouragement did not arise 

 from the unexpected amount of details, but because too little 

 value was placed upon them. The results were failures ; indeed, 

 how could they be otherwise ? As the pupil progresses his con- 

 fidence will increase, and he will thus decide for himself t.ie kind 

 of subject, and its extent, that he may feel capable of undertak- 

 ing. Whilst allowing this, a few words of advice may be useful. 

 There are various reasons why, in our individual estimation, one 



* A parallelepiped is a six-sided rectangular figure whose opposite 

 sides are of equal dimensions. 



