306 OF THE POSITIONS OF EQUILIBRIUM 



or numerically, we shall obtain 



28 : 10 : : 1 : sin./>= 0.35714, 



and we have already found that 



cos.^ =: V (28 -f 10) (28 10) -f- 28 = 0.93406 ; 



but according to the arithmetic of sines, it is 



sin.20 2 sin.0 cos.0, 

 and by substituting the above numerical values, we get 



\ sin.20 = 0.35714 X 0.93406 = 0.33359. 



Then in the triangle ECH, there are given the two sides EC and HC, 

 respectively equal to 32.106 inches and 16.752 inches, together with 

 half the natural sine of the contained angle ; to find the area of the 

 triangle. 



Now, by the principles of mensuration, the area of any plane tri- 

 angle is expressed by half the product of any two of its sides, drawn 

 into the natural sine of the contained angle, hence we get 



32. 106X 16.752 X0.33359 = 179.417 square inches. 

 Again, in the isosceles triangle ABC, there are given the sides AC 

 and BC, respectively equal to 28 inches, and half the natural sine of 

 the contained angle ACB, equal to 0.33359 ; to find the area. 

 Here, by the principles of mensuration, we have 

 28X28X0.33359 zz 261.53456 square inches; 



then, by the property of floatation, it is 

 1000 : 686 : : 261.53456 : 179.413 square inches. 



388. Since this result agrees so very nearly with that derived from 

 a direct computation of the triangular area, we may reasonably con- 

 clude, that the positions exhibited in the diagram are those of equili- 

 brium ; it is however necessary to remark, that since the weight of the 

 body remains unaltered in what position soever it may be situated, it 

 does not readily appear in what manner the adequate quantity of fluid 

 is displaced, unless we conceive some physical plane, of sufficient 

 breadth and totally destitute of weight, to be fixed on that edge of the 

 solid which becomes immersed by reason of the change of position 

 that the body is supposed to undergo. 



This plane, during the oscillation of the prism, will dislodge the 

 fluid which occupies the space EAW or vbn, and the weight of this 

 quantity of fluid added to that which is displaced by the quadrilateral 

 figure cj mn or cbni, will be equal to the whole weight of the float- 

 ing body 



