III'. 



831 



4. To jind tin- (?fnt .,n. Let A BCD K 



lines A c, A D to tho remote angles c ami i>. Tin- i*>lygon is 

 thns out up into Mir.- triangles. Lot a, n :m ! K ! tin; centres 

 of gravity of these latter figures ; there are thus thro.. 

 whose centres, a, u, aud K, are known, and whose masses are the 

 three areas of tho three triangles. 

 Suppose now that you had calcu- 

 late 1 these areas, and had thorn 

 written down in numbert. Then 

 join o with H and cut o n at x 

 'y as the numbers express- 

 ing tho areas of the triangles ABC, 

 ADC, Connect x now with K, 

 >o and cut K x at Y inversely, as tho 

 quadrilateral A D c D to the triangle 

 A E D ; tho point Y is the required 

 centre. If tho polygon had more 

 sides than are in Fig. 31, tho 

 process is the same, and must be 



continued until all tho triangles into which it is necessary to 

 divide tho polygon have been gone over. 



5. To find the Centre of Gravity of the Circumference of a 

 "Circle, Let tho circumference bo taken to be a curved lino of 

 atoms, as in a, in Fig. 32, to tho right ; and through tho centre 

 o of tho circle let any line, A G B, bo drawn passing through two ! 

 of thorn, one on either side. Since these two aro of equal ! 



31. 



tant from o, their oommon centre <*. 



i the middle of A B, that is, the point o. Bo, likewise, 



>und the figure, the oentre of gravity of every opposite 



Kiir of atoms is o, and therefore o is the oommon oentre of 



all, or of tho circumference. 



ntro of gravity of a ring I* thns seen to be the centre 

 of the circle in which it is formed, for the ring may be oon- 

 sidorod a handle of circles an atom thick, bound together, one 

 above and around the other, so as to have for oommon oentre of 

 gravity tho centre of 

 the central circle. 



The centre of gra- 

 vity of tho area of 

 a circle is also the 

 centro of figure of 

 tho circle, for tho 

 area may bo con- 

 sidered as made up 

 of a number of cir- 

 cles of atoms, lying 

 one inside tho other, and having tho same centre, o, which, by 

 the above, is therefore their common centre of gravity. 



The centro of gravity of a hollow sphere may, in like "*Ttfr 

 bo proved, by drawing lines through o to the atoms on its sur- 

 face, to bo the centro of figure of the sphere; and a solid 

 sphere may be considered as consisting of a number ut these 

 hollow ones inside one another. 



COPY-SLIP NO. 47. THE WORD tax. 



COPY-SLIP NO. 48. THE LETTER 6. 



COPY-SLIP NO. 49. THE WORD 0X6. 



LESSONS IN PENMANSHIP. XIV. 



IN Copy-slip No. 46 (page 196), an example was given of the 

 letter x. This letter is formed of the letter C twice repeated ; 

 the first, or the one to the left, being turned upside down, while 

 the second, or tho one to the right, is formed in the ordinary way. 

 The left half of the letter is commenced on tho line c c with a 

 hair-lino which is turned at tho top to the right, and brought 

 downwards without being thickened by pressure on the pen. 

 Tho hair-lino is turned to the left as it approaches tho line b b, 

 carried round, and terminated in a dot about midway between 

 tho lines bb,cc. The right half is then added. It is made in 

 precisely the same way as the letter c, the thick down-stroke 

 touching the thin down-stroko of tho turned C, and forming 

 the thickened centre of the letter. 



In Copy-slip No. 48 tho learner will find an example of the 

 letter e, which is commenced on tho central lino, c c, by a hair- 

 -ijoke carried up in a slanting direction to the right. This 

 hair-line is then turned at the top line, a a, and carried to the 

 left, and the letter ia finished in the same manner as tho letter 

 C-or tho right half of the letter X; but in making tho thick 

 down-stroke care must bo taken to let it pass over the point in 



the line c c, at which tho np -stroke forming the loop or bow of 

 the letter e was commenced. 



Copy-slips Nos. 47 and 49, comprising the words tax and 

 axe, are given to show the learner how the letters x and e are 

 connected with letters that precede or follow them. 



In the last lesson it was said that the letters c, X, and 6 are 

 modifications of tho letter o. Tho learner may prove this in a 

 practical manner for his own satisfaction, if ho will take the 

 trouble to make the letter o in pencil, on a piece of ruled paper, 

 and then trace tho letter C or e over it in ink ; or otherwise, by 

 making tho letters c and 6, and then adding to thorn the fine 

 hair-stroke on the right side that is required to form the com- 

 plete oval of the letter o. To show that X is a modification of O 

 it will be necessary to make tho letter o twice over, so that the 

 right side of tho first touches the right side of the second, and 

 then trace the letter x over tho donblo o thns formed ; or, as in 

 the case of c and e, the hair-stroke that is necessary to com- 

 plete the oval of o may be added on the right and left ef the 

 letter X. In the letters c, X, and e, tho bottom-turn is carried 

 to the right, beyond the limit of the bottom-turn of the 1< 

 in order to join them the more readily to any letter that may 

 follow them. 



