ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 58 1 



cation is limited to finding length, althoagli it might be applied to 

 multiplying an angle if practically useful, both operations, however, not 

 being performed simultaneously. 



Representation of Areas and Volumes by Means of Segments. 



It may be assumed that any concrete quantity to be dealt with by a 

 graphic operation is given in terms of a numerical quantity, and may be 

 at once represented to some scale by a ' segment,' ' understanding by this 

 word, unless qualified (as, for instance, segment of a circle), some definite 

 distance between two points ; that is, the length of a straight line. 



A geometrical quantity, however, in the form of a curved line, or of 

 an area or volume, has frequently to be dealt with, and this must first be 

 represented by a segment. There are means of calculating such areas or 

 volumes, and in the second report a section was devoted to the mechanical 

 means of doing this. It is, however, necessary to have methods of doing 

 this graphically, and such methods are generally treated, not only in 

 books on graphic statics, but in treatises on descriptive geometry. 



Areas may be divided into polygons and figures bounded by curves. 

 The former may be always reduced to a single triangle by the well-known 

 method of drawing successive parallels. The area of any triangle may in 

 its turn be represented by a segment equal to the altitude of another 

 triangle, of the same area as the former, but with a base equal to twice 

 the unit of length. The polygon need not, however, be reduced to a 

 triangle, and there are various methods which avoid doing this, although 

 the principle of the operation is the same in all cases. 



If the boundary be curved the figure can be split into a polygon 

 bounded by the curved figures, which may be supposed to be segments 

 of parabolas. Now, the segment of a parabola is § the area of a triangle, 

 upon the same base and of the same altitude, and therefore by making 

 triangles upon each parabolic sector, having their altitudes respectively f 

 that of the parabola's segment, and adding them all to the original 

 polygon, the operation of reducing the area of the figure becomes merely 

 that of reducing the new formed polygon. 



A special case is that in which the bounding curve is an arc of a 

 circle, the area of which is occasionally required ; as, for instance, in the 

 case of an arch, the extrados and intrados of which may have different 

 centres. In this case the first step involves finding a segment equal in 

 length to an arc, or, as it is called, the ' rectification ' of the arc, which 

 is also occasionally required in other graphical problems. Rankine 

 suggests two methods, in one of which a tangent is drawn at one ex- 

 tremity to meet a radial line through a point on the arc a quarter of its 

 length from that extremity. The sum of the distances from the two 

 extremities to the above point differs from the length of the arc by a 

 distance (r being the radius). 



,_(arc)^ (arc )^ 



4320r* 3484648r6'^ ' ' ' 



This construction requires the centre of curvature of an arc. 



The other method does not require the centre of the curvature, but 



' Some American writers use the term ' sect,' and this term, though open to ob- 

 jection, is perhaps even less so than ' segment,' which has generally been used for a 

 ■different kind of geometrical quantity. 



