40 Mr. C. HoltzapfFel on a Scale of Geometrical Equivalents 



ployed, will be true of any two, and the method is applicable to all solids, 

 of all forms and sections. It is desired to increase the contents of a pan 



or vessel 2.J times, the same as explained for the cu- ; 



bical vessel. Draw several ordinates across the figure. In 



1. The vessel being altered in height only, the 

 diameters remaining constant, increase the distance 

 between the ordinates 2^ times, whence the second 

 form immediately beneath would result. 



2. The figure being altered in diameter, the depth 

 remaining constant, the lengths of the ordinates must 



be increased v | times by the quadratic scales : the 



new figure would be of , — ■ . 



this form. -^-_ _ ; - -^ 



3. The three dimen- 

 sions being altered, the 

 space between the ordi- 

 nates, as well as their 



lengths, must be multiplied by Vf by means of the 

 cubic scales 5 and 2; this new figure would be produced, or a copy of the 

 original in the same proportions as to height and 

 diameter as the original : and further, should it 

 be desired to mix the two modes, that is to alter 

 both the general contents, and one of the mea- 

 sures in any defined ratio, for example, to make 

 a new vessel as before containing 2^ times as 

 much, but of only half the height, we must thus 

 learn the ratio. Half the height with the same diameters would give half 

 the contents ; the increase of the 

 diameters by the quadratic scales 

 must therefore be doubled by 

 employing the quadratic scale f , i X f being equal to 5 the ratio re- 

 quired for the new contents. 



We have therefore complete command over the capacity of 

 all vessels, under all proportions of general contents, and 

 specific variations of form. The truth of this method admits 

 of rigid demonstration, if we consider the ordinates to divide 

 the figure into so many zones, or as regards the section into 

 trapezoids and rectangles, and that the curved line running 

 through them is superseded by short right lines. Of course, 

 the more ordinates that are used the more nearly true will 

 be the result. The circle would by this treatment become 

 an ellipse, which may be taken as a further proof of the cor- 

 rectness of the result, as to find the area of an ellipse the two 

 diameters are multiplied into each other, and the product by 

 •7854- 



When vessels of complex form are constructed as retorts, 

 boilers, pans, stills, tanks, &c. it would be desirable to retain 

 drawings of them, with the several measures written upon 

 them, and also a memorandum of the cubic contents obtained 

 either by calculation or experiment, as by means of the scales 

 these data may be employed for obtaining the new contents, 



