GEOMETRICAL PEBBFBOTIVE. 



G. The last question might also have been wired thus : 



35 men eat the quantity in 20 days ; 

 Therefore 1 man eats 1*1 of the quantity in 20 days ; 

 Therefore 1 man eata of the quantity in 1 day; 



x> X JO 



Therefor* 50 men i 



.it ., - of the quantity in 1 day; 

 *u x x> | 



Ami therefore 50 men would occupy" 50 , or 



eating tlio bread. 



20 x 35 



20 x 35 

 50 



lilVH ill 



-To got tho timo occupied in doing a certain work, 

 \\hcii tho amount done per unit of time (say per day) is given, 

 wo must evidently divide the whole quantity of work by the 

 amount dono in a day. In tho caso given above, the bread 



50 

 being considered the unit, ^ 5^ of the bread is eaten in 1 day, 



and therefore 50 , which is tho whole amount eaten divided 



20x 35 

 by tho amount eaten in one day, will be the whole time occupied. 



7. Hence we get the following statement of 



Simple or Single Rule of Three. 



Writo down the ratio of tho two quantities which are of the 

 sui n ! kind, putting the greater in the first place. Then ob- 

 serving from the nature of the question whether the fourth 

 quantity required will be greater or less than the third one 

 which is given, place the greater of tho two in the third place 

 of the proportion, and multiply the extremes and means 

 together. 



EXERCISE 51. EXAMPLES IN SINGLE RULE OF THREE. 



1. If 16 barrels of flour cost 28, what will 129 cost? 



2. If 641 sheep cost 485 15s., what will 75 cost ? 



3. If 11 5s. buy 63 pounds of tea, how many con be bought for 



385? 



4. A bankrupt pays 6s. 4d. in the pound : what will be received on 



a debt of 2,563 10s. F 



5. What is 1,460 worth in dollars, allowing 4 dollars 84 cents to 



a pound, and 100 cents to a dollar ? 



6. If } Ib. of snuff cost t \, what will 150 Ibs. cost ? 



7. A man bought { of a vessel, and sold { of what he bought 



for 8,240, which was just the cost of it: what was the 

 whole vessel worth ? 



8. If f of a yard cost | of a crowa, what will 3| yards cost ? 



9. If 10 men build a wall in 7 days, how long would it take 24 



men to build it ? 



10. If 6 men build a wall in 15 days, how many men would it take 



just to finish it in 22A days ? 



11. If f of a ton costs 9s. 8jd., what would 4? of a cwt. cost? 



12. If a twopenny loaf weighs 1 Ib. 2 oz. when wheat is 50s. a 



quarter, what should it weigh when wheat sells for 60s. ? 



13. If the weight of a cubic inch of distilled water be 253J& groins, 



and a cubic foot of water weighs 1000 oz. avoirdupois, find the 

 number of grains in a pound avoirdupois. 



14. If 1 Ib. avoirdupois weighs 7,000 grains, and 1 Ib. troy weighs 



5,760 grains, find how many pounds avoirdupois are equal to 

 175 Ibs. troy. 



15. Find the rent of 27a. 3r. 15p. at 1 3s. 6d. per acre. 



16. The price of standard silver being 5s. 6d. per ounce, how many 



shillings are coined out of a pound troy ? 



17. A bankrupt's assets are 1,500 10s., and he pays 9s. 3Jd. in the 



pound : what are his debts ? 



18. If standard gold is worth li'id. per groin, how many sovereigns 



would be coined out of a pound troy of gold ? 



19. What is the income of a man who pays 53s. lOd. tax when it 



is 7d. in the pound ? 



20. Raising the income-tax Id. in tho pound increases my amount 



of tax by 2 3s. 4d., and the tax I actually pay is 15 3s. 4d. : 

 what is the rate of the income-tax ? 



21. A barrel of beer lasts a man and his wife 3 weeks, she drinking 



half the amount he does : how long would it last 5 such 

 couples ? 



KEY TO EXEKCISE 50, LESSON YYTT. (Vol. II., page 270). 



GEOMETRICAL PERSPECTIVE. 1 1 1 



BEFOEB proceeding farther and deeper into our subject, we wiak 

 to draw tho pupil's attention to an explanation of projection, a 

 term applied not only to perspective but also to other sjstOM of 

 ;:, namely, orthographic and itometnc. Our reason 

 for Jut u..-.i.-ii,g this now, is in order to make it clearly under- 

 stood aow tho plan of an object is to be treated when we an 

 about to make a perspective drawing of that object, as we rery 

 fr> Min-ntly meet with cases when the plan of the object to be 

 represented most be drawn according to the position which that 

 object presents, whether horizontal or inclined. The plan, an 

 wo said in Lesson I., is produced by perpendicular linen drawn 

 from every part of an object upon a horizontal plane. Now, 

 there can be no difficulty in drawing a plan when the subject. 

 represented by it ia parallel with the ground or horizontal. 

 plane ; but it occurs sometimes that it is placed at an angle 

 with both planes, that is, with the picture-plane and ground 

 piano : therefore in cases of this kind it is necessary to under- 

 stand tho first principles of orthographic projection, namely, projec- 

 tion by straight lines upon vertical and horizontal plaaex. We 

 have mentioned above another method of projection, isometric ,- 

 as the term has been introduced, we will explain its meaning 

 and then pass it by, as it does not, like orthographic, form any 

 auxiliary to perspective. The term isometric signifies likt- 

 measurement, that is, all the parts of the drawing, both near 

 and distant, are drawn to one and the same scale, also the plan 

 and elevation are combined in one drawing. It is a method 

 much used by architects and engineers when they wish to give 

 what is generally called a bird's-eye view of a building, etc., 

 without diminishing the distant parts, as shown in perspective 

 projection. A drawing made isometrically will enable a stranger 

 to understand the proportions, position, and general character 

 of a subject probably better than any other system ; hence the 

 reason of its frequent use. 



The extent to which we intend to proceed with orthographic 

 projection must be limited to that which relates to, and can 

 assist us in, our present subject, by which we hope to make it 

 a valuable auxiliary in our efforts to render the science of per. 

 spective easy and intelligible. 



The difference between tho results of perspective and orthiv 

 graphic projection is caused by the altered position of the eye 

 when viewing tho object. In perspective the eye is in one plaet 

 only, and from that place is included all that can be seen within 

 the angle of sight. In orthographic projection the eye is tup- 

 posed to be opposite every part at the same time, above the object 

 when the plan is represented, and before it when the elei-ation is 

 represented ; consequently, in perspective, all the visual rays 

 proceeding from tho object to the eye converge to one point; 

 but in orthographic projection these rays are drawn parallel 

 with each other, and perpendicularly to the plane of projection, 

 whether the plane is horizontal or vertical. To make this clear, 

 we request the pupil to compare Figs. 5 and 6 of the last 

 Lesson with Fig. 8, when he will notice that the characteristic 

 difference between the two systems rests entirely upon the 

 different treatment of the lines of projection, which, as we have 

 said, converge in one case, and are parallel in the other. Fig. 8 

 is to ' hew how a cube is projected orthographically upon vertical 

 and horizontal planes of projection. A is the vertical, and B the 

 horizontal, c is tho cube in space, that is, at a distance from 

 both planes of projection. If straight lines are drawn from the 

 angles of the cube perpendicularly to and meeting the plane B, 

 and then lines (a, b, e, d) be drawn to unite them, we shall have 

 a plan of tho cube ; and as the edges in this case are placed 

 perpendicularly with the ground, the plan will be a square. 

 Again, if horizontal and parallel lines are drawn from the 

 angles of the cube until they meet the vertical plane A, and aro 

 then joined by the lines e, f, </, A, we shall produce the elevation ; 

 and because the horizontal edges of the cube are perpendicular 

 to the vertical plane of projection, the drawing in this case alao 

 will be a square. Consequently, it will be seen that the drawing 

 of the plan or the elevation is the same size as the object on the 

 respective plane to which the object is parallel, according to the 

 given scale of that object, as in Figs. 10 and 11. This remit 

 makes orthographic projection of much importance for practical 

 purposes. The working drawings for the guidance of builders 

 and mechanists are made by this method. Horizontal lengths 

 and breadths are shown both in the plan and elevation, but 



