ANCIENT ANALYSIS. PORISMS. 79 



Dr. Sinison'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 datuni) 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 jis proposed to be 

 investigated. This is erroneous, and contrary to the rules of 



and dealing a good deal with Harmonical proportions. If from any point 

 whatever out of the ellipse there be drawn a straight line in any 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 harmonical pencil, and consequently cut in the Harmonical 

 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 circle. The ellipse, therefore, might be called the Har- 

 monical Curve, did not another of the 12th order rather merit tliat 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 



2dy + dx 2 = - : , which is reducible, and its integral is an equation 



of the 12th order. There is also another Harmonical Curve, a transcen- 

 dental one, in which chords vibrate isochronously. 



