28 



THE POPULAR EDUCATOR. 



I have it from the hatter. 19. For whom is it ? 20. It is for the 



tailor's son. 21. Have you gold, silver, or copper for the teacher ? 



22. I have silver for him. 23. Whom does the child love ? 24. It 

 loves the teacher's brother. 



EXERCISE 8 (VOL. I., page 62). 



1. Where is the mate's brother ? 2. He is with the captain in the 

 ship. 3. Is the nobleman's son with him also ? 4. No, he is in 

 Uermany. 5. Where is the father ? 6. He is with the captain in the 

 custom-house. 7: Does the captain praise the nobleman's son ? 

 8. Yes, and he praises the father also. 9. Does the nobleman love the 

 captain ? 10. Yes, he loves and praises him very much. 11. Is this 

 tuan the captain's son ? 12. No, he is the mate's son. 13. Is this 

 sailor rich? 14. No, he is poor and merry. 15. How old is this 

 man ? 16. He is not very old. 17. Is he sick ? 18. No, he is 

 hungry. 19. What does this girl give the child ? 20. She gives it 

 only sugar. 21. What do you give the servant ? 22. I give him 

 money. 23. What does the servant give the horse ? 24. He gives it 

 hay. 25. Does this child love the teacher ? 26. Yes, and the teacher 

 praises the child. 27. Is the hunter still in the forest ? 28. Yes, and 

 the nobleman's son is with him. 29. The huntsman goes to the 

 forest to the father, and I go to the brother. 



EXERCISE 9 (Vol. I., page 66). 



1. Has a man or a child this friend's stick ? 2. This man has an 

 enemy's sword, and this child has a friend's stick. 3. What has the 

 hunter ? 4. He has a dog and a gun. 5. Who has the peasant's 

 plough? 6. The father of this child has the plough. 7. Has this 

 blacksmith the merchant's money ? 8. No, he has only iron from a 

 merchant. 9. Have you the baker's wagon? 10. No, I have this 

 wagon from a carriage-maker. 11. Have you this girl's ribbon? 

 12. No, I have cloth from a draper. 13. Have you this friend's coat ? 

 14. No, I have this coat from a tailor. 15. Have you the teacher's 

 paper ? 16. No, I have this paper from a stationer, and a letter of 

 recommendation from the teacher. 17. Is the horse a draught- 

 animal ? 18. Yes, and it is also a beast of burden. 19. Is the camel 

 a draught-animal also ? 20. No, it is only a beast of burden. 21. 

 Whose law-book has the nobleman's son ? 22. He has the law-book of 

 the judge of the superior court. 



LESSONS IN GEOMETRY. XIV. 



THE CIRCLE AND ITS PROPERTIES. 



HITHERTO, in our lessons in Geometry, the attention of the 

 student has been directed to the construction of rectilineal 

 figures, or figures contained by right or straight lines : we shall 

 now enter on what may be termed the " geometry of the circle," 

 or the method of drawing circles and parts of circles under 

 various conditions; concluding our lessons on this subject 

 with instructions for drawing regular polygons by the aid 

 of the circle, protractor, and scale of chords, as well as the 

 ellipse and other figures bounded by curves or consisting of 

 curved lines. 



It may be useful to the student if we recapitulate briefly the 

 names of various parts of the circle, and mention its chief 

 properties as laid down in the Definitions (Vol. I., page 53), 

 before explaining one or two other points that will be necessary 

 for him to understand before he reads the problems that we are 

 about to bring under his notice. 



Firstly, let us ask, What is a circle ? It is a form that meets 

 the eye often enough as we go about our daily tasks. As we 

 pass through the streets of town or city, or along the highways 

 and byways of the country, it is brought before us in the wheels 

 of every vehicle we meet. It is exhibited in the form of the 

 majority of our cooking utensils. If we turn our eyes to the 

 face of the clock that stands on the mantel-shelf, or the watch 

 that is carried in the waistcoat-pocket, it is there. Nay, more, 

 it is found in every button that we wear on our attire, in the 

 cups and glasses out of which we drink, and in the plates off 

 which we eat our daily food. It is the most perfect, the most 

 elegant, the most useful of all forms. Under the figure of a 

 snake holding its tail in its mouth, the ancients adopted it as 

 the emblem of eternity, which had no beginning, and which has 

 no end. It is a figure which any one can describe by the aid of 

 a pair of compasses with the greatest ease, but one which it 

 would be most difficult to draw without the assistance of this 

 useful instrument. 



There was a man once, though, who could draw a perfect 

 circle with a simple sweep of his unerring arm and hand, and 

 mark its centre with tha same rapidity and precision. His 



name was Giotto or Angiolatto, an Italian painter, sculptor, 

 architect, and engineer, born at Vespignano in 1276. As a boy 

 he was employed as a shepherd, but Cimabue, who accidentally 

 discovered his innate talent for drawing, took him by the hand, 

 and made him a greater painter than himself. He had learnt 

 the rudiments of his art in the fields by sketching his sheep on 

 the earth with the end of his shepherd's crook, or with a nail 

 on any flat piece of stone that might come in his way. These 

 small beginnings had great results in Giotto's case, for he went 

 on step by step until he became the greatest Italian painter of 

 his time. When Benedict XI. was Pope of Rome, artists were 

 wanted to work at the decorations of the great cathedral dedi- 

 cated to St. Peter, and invitations were sent to the principal 

 painters of Italy to forward specimens of their skill for the 

 pope's inspection. Giotto contented himself with drawing a 

 circle on a piece of paper with a bit of charcoal, and handing it 

 to Pope Benedict's messenger. It was in vain that the mes- 

 senger urged that his master required some design as a speci- 

 men of Giotto's skill, for the painter refused to send anything 

 else. The circle so hastily drawn was found to be perfect when 

 tested with a pair of compasses, and so struck was the pope 

 and his advisers with this surprising proof of the artist's 

 capacity as a draughtsman, that he was immediately summoned 

 to Rome to carry out the work that Benedict wished to con- 

 tribute as his quota to the adornment of the finest cathedral 

 that has yet been built. 



But to return from this digression. The circle, in geometrical 

 terms, is a plane figure ; that is, a figure drawn on a plane or 

 level surface, and bounded by a curved line called the circum- 

 ference or periphery. Let us explain these terms ; for there is 

 nothing so well calculated to fix the meaning of a word and the 

 peculiar property of the figure that it is intended to describe as 

 to trace it to the primary source or root from which it is derived. 

 The word plane is derived from the Latin planus, flat, smooth, 

 level. It is merely another form of the word plain, which we 

 apply to a level tract of country because it is flat and devoid of 

 hills or any striking inequalities in its surface. A joiner or 

 cabinet-maker will now see at once the reason why the tool he 

 uses to give an even level surface to a piece of wood is called a- 

 plane. The word circumference, which is applied to the line 

 which is carried round about, or which bounds any figure, is 

 derived from the Latin words circum, round, and fero, I bear or 

 carry. The word periphery means precisely the same thing, 

 but it traces its source to the Greek instead of the Latin, being 

 derived from the Greek irepi (per'-ry), a/round, and <ptpw (fer'-ro), 

 I bear or carry. 



Look at the annexed figure. The whole of the superficies or 

 surface of the paper that lies within the curved line A c B E is 

 called a circle. The curved line A c B E 

 itself is called the circumference or periphery 

 of this circle. The point o is called its 

 centre, a word derived immediately from the 

 Latin centrum, and more remotely from the 

 Greek Ktvrpov (ken'-tron), a sharp point. 

 The position of this point has this peculiar 

 property : it is such that all straight lines 

 drawn from it to the circumference are 

 equal to one another. Thus the straight lines O A, o B, O c, 

 o E, drawn from the centre o to the points A, B, c, E, in the cir- 

 cumference, are equal to one another. These lines are called 

 radii of the eircle A c B E, from the Latin radius, a sunbeam or 

 ray of light, and hence applied to any line or any number of lines 

 that radiate in various directions from the same point, as rays 

 of light seem to proceed from the sun or any luminous centre, 

 as may be seen by looking at a candle or gas-light with half- 

 closed eyes, when the rays that seem to issue from it will become 

 distinctly visible. Any two radii that proceed from the centre 

 in opposite directions, and therefore lie in the same straight line, 

 form together a straight line called a diameter of the circle. In 

 the above figure (Fig. 49), A B is a diameter of the circle AC B E. 

 Its name, derived from the Greek Sia (di'-a), through, and 

 peTpfiv (met'-rine), to measure, implies that it is a line that 

 measures the circle across its superficies and through its centre. 

 Having arrived at the meaning of the word diameter, we arrive 

 at the full significance of the term " diametrically opposite." 

 Thus, when we say that the opinions entertained by any two 

 men are diametrically opposite, we mean that they are as con- 

 trary to each other as it is possible to bo as opposite, in fact, in 



Fig. 49. 



