42 Mr. C. Holtzapffel on a Scale of Geometrical Equivalents 



we obtain the proportion 10, 20, 30, 40 times, or J^, ^V, 

 ^1^, ^'^th part by the joint employment of the y'^th and the 

 1, 2, 3, 4 scales, &c. 



Again, the square root of 100 is 10, of 200, 14'142, of 

 300, 17*320, or ten times the square roots of 1, 2, 3; we may 

 .therefore call these scales 100, 200, 300, by simply multi- 

 plying the numerals by 10, so that to obtain ^/^^^ of any 

 quantity, seek it in 9, and the answer in 7, which, supposing 

 it to be 5*314, must be read 53*14; and lastly, y'^j being the 

 thousandth part of 100, by employing the Y\jth line on the 

 one hand, and the 1, 2, 3, 4, 5 lines, (multiplied by 10,) on 

 the other, we obtain the -^y^, ^-^fS>^ Woo' ^^^ ^° °"* these 

 matters are, however, of great simplicity to those at all ac-^ 

 customed to the employment of decimals. 



An example will better explain the utility of this arrange- 

 ment. 



A mine 650 feet deep has to be drained by means of a pump 17 inches 

 diameter : required first the total contents of the entire column of water in 

 cubic feet, gallons and tuns weight, and also the pressure on each square 

 inch of the column ; and the requisite diameter of a steam cylinder, work- 

 ing under the effective pressure of 12 pounds on the inch which shall ba- 

 lance the column in the mine. 



17 the diameter of the pump or column squared and multiplied by the 

 depth or 17*x650 = 187850, seek that number in G. (square inches) and 

 read in B. 1304 cubic feet, in I. 8140 gallons, and in L. 3231 tons avoir- 

 dupois, as the contents of the entire column. 



For the pressure on each square inch of the pump, seek the depth 650 

 feet in G, and against it 282*5 will be found in M. The pressure per inch 

 on the pump cylinder being 282-5 pounds, and the proposed effective press- 

 ure of the steam being 12 pounds per inch, the ratio in which the cyHnder 

 should be altered is f/y^^. 



The required result, and the others presented at the same view, are ex- 

 tracted in three columns : the first column shows the real titles of the 

 quadratic scales, — the second the assumed titles, or the former multiplied 

 by 3 (300 being nearest to 282-5,) ; and the third column the diameters of 

 the equivalent cylinders under each different pressure, the given diameter 

 of the pump having been sought in the 300 column. 



Scales 100, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 

 Pressures 300, 3, 6. 9, 12, 15, 18, 21, 24, 27, 30. 

 Diameters 17, 170, 121. 98, 85, 76, 70, 64, 60, 57, 54. 



The nearest whole numbers having been taken. The truth of this will be 

 apparent 17'x300 being 86700, and 852 x 12 being likewise 86700. 



But as a comprehensive example, suppose it is desired to 

 enlarge the drawings of a steam-engine, the new one to be as 

 5 to 4. 



The contents of all the vessels, as the cylinder, condenser, 

 air-pumj), &,c. will be increased by the cubic scales 5 and 4. 



The areas of the valves, passages, pipes, and the compli- 



