154 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 de- 

 scribing the hypothesis and as affirming the possibility 

 of finding the construction so as to answer the con- 

 ditions. 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 dis- 

 tinct 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 investi- 

 gated. This is erroneous, and contrary to the rules 

 of logic from its generality ; it is, as the lawyers say, 



direction whatever cutting the ellipse, the line is cut harmonically by the 

 tangent, the ordinate, and the chords of the two arcs intercepted between 

 the point of contact of the tangent and the axis. The tangent, sine, and 

 chords are always an hannonical pencil, and consequently cut in the Har- 

 monica! ratio all lines drawn in all directions, from the given point. This 

 applies to all ellipses upon the same axis, (all having the same subtangent.) 

 and of course to the c?rcle. The ellipse, therefore, might be called the 

 Harmonica! Curve, did not another of the 12th order rather merit 

 that name, which has its axis divided harmonically by the tangent, 

 the normal, the ordinate, and a given point in the axis. Its differential 



equation is 2 dy* -\-d x* =- -' which is reducible, and its integral 



is an equation of the 12th order. There is also another Harmonical 

 Curve, a transcendental one, in which chords vibrate isochroin/usly. 



