200 



THE POPULAR EDUCATOR. 



must rest very much, with the individual reader's conception of 

 the diabolical nature. 



The next important character to Faust himself and Mephis- 

 topheles is Margaret, or Gretchen, as she is sometimes called. 

 Of this peasant-girl, the late Mr. G. H. Lewes, the English 

 biographer of Goethe, has remarked that " Shakespeare himself 

 has drawn no such portrait ; no such peculiar union of passion, 

 simplicity, homeliness, and witchery." 



The moral character of Goethe's drama will always be a 

 subject of dispute. Some regard it as containing the noblest 

 lessons ; others bring against it the charges of irreverence, 

 scepticism, and laxity. What may most truly be affirmed is, 

 that it is a vast and varied picture of human life, reaching up- 

 wards into the heavens, and downwards into the infernal depths. 



MECHANICS. VI. 



FINDING CENTRES OF GRAVITY. 



IN the last lesson it was shown that every mass of matter has a 

 centre of gravity, but we did not inquire how such centres are 

 found in bodies of known shapes. To that part of our subject 

 we now proceed. 



As a general rule, the problem requires high mathematics for 

 its solution ; but there are some cases in which the centre can 

 be discovered without much difficulty. I take, first, the practical 

 method by suspension, which gives it exactly whenever the body 

 is of a uniform thickness, such as a deal board, or card, or piece 

 of paper. The two opposite faces should be equal and alike, 

 tho edges being either perpendicular or square to them, or run- 

 ning off at the same slope. In all such cases it is evident 

 that the centre of gravity is within the substance of the 

 board half-way across between the faces. If, therefore, we 

 can find the point on either face under which it lies, by 

 boring straight in half-way at that point, the required centre 

 is reached. 



But how find the outside point ? Let the board be of any 

 irregular shape, as at a (Fig. 27), and bore two holes through it 



perpendicularly at 

 any two points, near 

 its edge, o and Q. 

 Put a straight iron 

 rod now through o, 

 and on the rod, by 

 a small ring, hang 

 a plumb line, o A, 

 close to the board. 

 Put rod, line, and 

 board now across 

 two supports, so ar- 

 ra"nged that the rod 

 may be horizontal. 

 The board having 

 settled to rest, the 

 centre of gravity will, as I showed in the last lesson, be some- 

 where behind the plumb-line. Chalk now, or mark with a 

 pencil, the course, o A, of this line on the board. Perform the 

 same operation with the hole Q, pencilling in like manner the 

 line Q B. What now have we ? Two lines, behind both of 

 which the centre of gravity lies ; whence we infer that their 

 intersection, Q, is the point required. 



But the method in part applies to bodies which have not 

 parallel faces like boards, or are not cut perpendicularly, or at 

 the same slope across at their edges ; but in such cases the 

 sought centre is not midway across. All that is necessary is 

 that there should be one flat face on it, as in that represented at 

 b (Fig. 27). You can still determine the point a, behind which 

 the centre of gravity lies, by boring two passages at O and Q, 

 perpendicularly to the face, into its substance, suspending and 

 marking the lines o A, Q B, as before. The centre of gravity will 

 still be behind the point G ; but where, or how far in, is another 

 question, the answer to which depends on the shape of the body. 

 If the board which above first occupied our attention be sup- 

 posed to become very thin to be cardboard, or even paper 

 the body becomes almost all surface, and the point G and the 

 centre of gravity nearly coincide. Practically, they become 



identical ; and the operation is sometimes spoken of as " the 

 finding of the centre of gravity of an area or surface." In 

 strictness, a surface cannot have a centre of gravity, for (see 

 Lesson I. on Geometry) it has no thickness, and therefore can 

 have no weight, no force, no centre of force. But, for all that, 

 the inquiry is useful. We may agree, for mechanical purposes, 

 that a surface should have such a centre ; and the best course 

 for that purpose is to give it a thickness the smallest we can 

 conceive, namely, that of one particle or atom. Imagine, then, 

 a triangle, or polygon, or circle, one atom thick ; and let us 

 agree that, when we find its centre of gravity, we have the 

 centre of gravity of an " area " or " surface." Also let it be 

 understood that the centre of gravity of a line, straight or 

 curved, means that point for such a line of atoms. 



TO FIND CENTRES OF GRAVITY BY CONSTRUCTION. 



This is done by the rule for finding the centre of parallel 

 forces, given in Lesson IV. (page 125). We shall commence 

 with the most general case, namely : 



1. To find the common Centre of Gravity </f any number of 

 Bodies, the separate Centres and Weights of which are given. 

 The masses may be anyhow placed, but the operation is the 

 same whether they are all on the same plane, as in the case of 

 the balls on the ground, in Fig. 27, or whether some are in that 

 plane, some above, and some below. Let them be four in number 

 and on the same plane, their centres being A, B, c, D ; then four 

 parallel forces, the weights, act at these centres : what has to 

 be done ? Join first A with B, and cut the joining line at x in- 

 versely as the weights at these points. Next connect x and c, 

 and cut c x at Y inversely as the two first weights to that at o. 

 Lastly, Y being joined to D, divide D Y at z inversely, as the 

 weights of the three balls already used are to that of the 

 fourth, D. This last point, z, is the required common centre of 

 gravity. 



You observe that the joining and cutting of the lines is in no 

 way influenced by, or dependent on, the bodies being on the 

 same or in different planes, neither is it dependent upon their 

 number. How many soever they be, the operation is the same. 

 Note, also, that a common centre of gravity can be outside the 

 bodies of which it is the centre. 



2. To find the Centre of Gravity of a Eight Line. A mechanical 

 right line being, as we have agreed, a line of atoms of equal 

 size and weight, the case is that which we have considered in 

 Lesson IV., of a number of equal parallel forces acting at equal 

 distances from each other, along a right line. The resultant 

 passes through the middle point of that line ; hence the centre 

 of gravity of a right line is its middle point. 



This enables us to find the centre of gravity of a uniform rod. 

 By " uniform," I mean such that the cross sections are of the 

 same size 



andform - A, ^ b 



throughout 

 its length. 

 Such a body 

 may be con- 

 sidered a 

 collection of 

 equal mecha- 

 nical right 

 lines placed ^jjj^ 

 side by side, 

 their ends 

 being made 

 flat or level. 

 As the cen- 

 tre of each 

 line is in its 



middle, the centre of the whole bundle is in the cross section 

 at the rod's middle. And observe that this holds good of 

 all other bodies, besides mere rods, which can be considered 

 made up of equal parallel lines, such as of a cylinder or uniform 

 pillar, or of a beam of timber, or of a cubical block of stone ; 

 the centres of gravity will be in the cross sections at their 

 middle points. And it makes no difference whether the flat 

 ends of the cylinder, pillar, beam, or block are perpendicular to 

 the lines of which it is supposed to be composed, as in c and e 

 (Fig. 28), or oblique to them, as at d and / (Fig. 29) ; the 

 centre of gravity is still in the middle cross section parallel to 



Fig. 28. 



