ANCIENT ANALYSIS. PORISMS. 83 



tions, and with curves of the third and higher orders. Of an 

 algebraical porisin it is easy to give examples from problems 

 becoming indeterminate ; but these propositions may likewise 

 arise from a change in the conditions of determinate problems. 

 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 

 (a+b*) x = (a V) x+2 c x and 26 x = 2 c x \ in which it is evident 

 that if c = b, 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 multi- 

 ple by the number 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 porismatic 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 at, the sum of the 

 perpendiculars 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 history of the mixed mathematics. Sir Isaac 

 Newton, by a train of most profound and ingenious investiga- 

 tion, 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 omnimodo aggres- 

 sus,"*) to the drawing a straight line through four lines 

 given by position, and which shall be cut by them in three 

 segments having given ratios to each other. Now his solution 

 of this problem, the corollary to the twenty-seventh lemma of 

 the first book, has a porismatic case, that is, a case in which 



* Principia, lib. iii. prop. xli. 



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