504 SIMSON. 



Dr. Simson's definition is such that it connects 



itself with an indeterminate case of some problem 



solved, but it is defective, in appearance rather than 



in reality, from seeming to confine itself to one class 



of porisms. This appearance arises from using the 



word "given" (data or datum) in two different senses, 



both as describing the hypothesis and as affirming the 



possibility of finding the construction so as to answer 



the conditions. This double use of the word, indeed, 



runs through the book, and though purely classical, 



is yet very inconvenient ; for it would be much more 



distinct to make one class of things those which are 



assuredly data, and the other, things which may be 



found. Nevertheless, as his definition makes all the 



innumerable things not given have the same relation 



to those which are given, this should seem to be a 



limitation of the definition not necessary to the poristic 



nature. Pappus's definition, or rather that which he 



says the ancients gave, and which is not exposed to 



the objection taken by him to the modern one, is 



really no definition at all ; it is only that a porism is 



something between a theorem and a problem, and in 



which, instead of anything being proposed to be done, or 



to be proved, something is proposed to be investigated. 



might be called the Harmonical Curve, did not another of the 12th 

 order rather merit that name, which has its axis divided harmoni- 

 cally by the tangent, the normal, the ordinate, and a given point in 



the axis. Its differential equation is 2 d y*+d x*= - -' 



oc 



which is reducible, and its integral is an equation of the 12th order. 

 There is another Harmonical Curve, also, a transcendental one, in 

 which chords vibrate isochronously. 



