158 SIMSON. 



the centre of the given circle, and that therefore the 



Eoint is found. It is probable that the author began 

 is work with a simple case and gave it a peculiarly 

 rigorous investigation in order to explain, as he im- 

 mediately after does clearly in the scholium already 

 referred to, the nature of the porism, and to illustrate 

 the erroneous definitions of later times (veortpfKot) of 

 which Pappus complains as illogical. 



Of porisms, examples have been now given both in 

 plain geometry, in solid, and in the higher : that is, in 

 their connexion both with straight lines and circles, 

 with conic sections, and with curves of the third and 

 higher orders. Of an algebraical porism 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-f 6) x=(a b) x+2cx and 2bx=2cx ; in 

 which it is evident, that if c=>, 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 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 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 perpen- 

 diculars 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 



