61 



inches fall of rain into it, shall fill the cone ; and a scale so gra- 

 duated, as that when it is put down to the bottom of the cone, the 

 water mark left on it shall show the quantity of rain fallen. 



The first thing to be attended to in the construction of this gage, 

 is the scale. And as the heights of similar cones are as the cube 

 roots of their contents, the contents of the cone being given and 

 divided into any number of equal parts, numbers corresponding to 

 the cube roots of those parts, will give the proportions of the divi- 



ln the example now given, the cone is intended to contain three 

 inches or thirty tenths of an inch fall of rain. In order then to 

 make a scale to measure these thirty tenths of an inch, the cube 

 root of thirty < mmencing with unity, must 



be found ^ the proportions of the scale in tenths of an inch. These 

 cube roots are found by dividing the logarithm of each number 

 by three and rinding the numbers corresponding to the quotients. 

 Hut the cube root of* the highest number 30, being only 3,107, 

 this must be multiplied by such a number as will give the desired 

 length of the scale, or height of the cone. 



For the construction of the contemplated gage, I have assumed 

 6 for the multiplier, which will give 3,107 x6=18,642 inches for 

 the length of the scale or heighth of the cone. All the other cube 

 roots, found as before mentioned, being also multiplied by 6, will 

 give the several divisions of the scale measuring tenths of an inch 

 fall of rain. If the three first tenths of an inch be graduated ac- 

 cording to the proportions found for the graduation of three inches, 

 the result will be hundredths of an inch ; and this may be easily 

 done, as shown by the diagram, where A B is made equal to 0.3 

 and parallel to it, and has transferred to it a ^ ^hen 1 token * fof tbe 

 hundred the scale. The 



tenths above the third, may be graduated sufficiently accurate for 

 hundredths, by dividing them into equal parts. 



The diameter of the base of the cone is assumed at pleasure ; I 

 have taken 6 inches, and its heighth as before stated is 18,642 in- 



C With these data the problem is then presented. What will be 

 the diameter of the opening of the funnel, or rather what will be the 

 diameter of a cylinder 3 inches high, the contents of which shall 

 be equal to the contents of a cone 18,642 inches high with a base 

 of 6 inches diameter ? 



As the capacity of a hollow cone is equal to one third of it ^cir- 

 cumscribing cylinder, let D equal the diameter of the base of the 

 cone = 6 inches, H its height =18,642, and 3 = the height of the 

 cylinder; required d, the diameter of the cylinder, the contents ot 

 which shall be equal to that of the cone. 



Cones and cylinders being as the squares of the diameters of 

 their bases multiplied by their heights, the proposition, stated 

 algebraically, will be thus : 



D' XH=d * XSy or D * XH= d 2 or y'D * XH-d. 



