• ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 583 



Hence it follows in practice that wj-otli of an inch (say ^]^ in.) will be 

 suflBciently accurate, and if this value be substituted for d we get 



where I is in inches; then if the ar-c to be measured is 4 in. long, a 

 must be taken -in. ; but if it be 25 in. long, then must 







r . 

 a=j-^xn. 



Culmann recommends that a should never be chosen smaller than 

 is required by these formulas ; for by plotting off the chord too many times 

 the accuracy due to the smaller difference between the arc and chord 

 is again lost. 



He remarks that this method is much more exact than attempting to 

 divide the arc into a certain number of equal parts. He also mentions 

 the method published by Herr Hanacek in the ' Zeitschrift des Oester- 

 reichischen Ingenieur und Architecten Vereins,' 1871, which is really the 

 construction of a length from the following formula : — 



when chord of ^ arc = \/l^+f^, and o7 is the segment of the base of a 



right-angled triangle whose height is / (i.e., the height of the arc), the 

 other segment being Bl, where 21 is the chord of the arc. 



Culmann remarks about the method : ' It is more practical and more 

 correct to take the length a, by means of which we measure an arc so 

 small that it should not require any correction, which is better than to 

 measure half the length of the arc, having to correct this measure before- 

 hand.' He also shows that the above formula when expanded gives 



'=2i(l+|''-^<H-i<'- • ■ •). 



where 



t-f- 

 whereas the correct value is really 



'=''{'^l''-r,''^l"- ■ ■ ■)• 



and by means of an example proves that for a semicircle the error is 

 inadmissible. 



Cremona, however, remarks about Culmann's method (' Graphical 

 Calculus,' English translation, page 114) : ' The method given by Cul- 

 mann for developing the circular arc A B along the tangent at one of its 

 points is much too long. The length of the circular arc may be found 

 graphically in a much simpler fashion by having recourse to auxiliary 

 curves, which drawn once for all can be employed in every example,' and 

 proceeds to give the methods suggested by Professor A. Sayno, of Milan, 



