SIMSON. 509 



terminate ; but these propositions may likewise arise 

 from a change in the conditions of determinate pro- 

 blems. Thus, if we seek for a number, such that its 

 multiple by the sum of two quantities shall be equal 

 to its multiple by the difference of these quantities, 

 together with twice its multiple by a third given 

 quantity, we have the equation ( + 6) x=(a &) x+ 

 2co? and 2b^=2ca: ; in which it is evident, that if 

 c=b t any number whatever will answer the conditions, 

 and thus we have this porism : Two numbers being 

 given a third may be found, such that the multiple 

 of any number whatever by the sum of the given 

 numbers, shall be equal to its multiple by their 

 differences, together with half its multiple by the num- 

 ber to be found. That number is in the ratio of 4 : 3 

 to the lesser given number. 



There are many porisms also in dynamics. One 

 relates to the centre of gravity which is the poris- 

 matic case of a problem. The porism may be thus 

 enunciated ; Any number of points being given, a 

 point may be found such, that if any straight line 

 whatever be drawn through it, the sum of the per- 

 pendiculars to it from the points on one side will 

 be equal to the sum of the perpendiculars from the 

 points on the other side. That point is consequently 

 the centre of gravity : for the system is in equilibrium 

 by the proposition. Another is famous in the his- 

 tory of the mixed mathematics. Sir Isaac Newton, 

 by a train of most profound and ingenious investi- 

 gation, reduced the problem of finding a comet's place 

 from three observations (a problem of such difficulty, 

 that he says of it, " hocce problema longe difficilimum 



