610 



Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



retracement, if y be negative) of y and the tangent : if <p y and - <p y y" differ, the contrary. 

 That is, y being measured positively upwards, the concavity is upwards or downwards, 

 according as y" is positive or negative. Again, a point on the tangent, just after contact, 

 abuts on convexity. Hence, a point close to the curve abuts on convexity or concavity 

 according as <p at the point and d) . y" at the contiguous points of the curve differ or agree in 

 sign. Hence another proof of the criterion of flexure : for if a point be a little nearer the 

 axis of x than (x, y), ((> and cf> y agree or differ, according as y is negative or positive ; if the 

 point abut on convexity, <p . (p y . y" is negative, so that y" agrees in sign with y. 



The point of contrary flexure, at which y" changes sign, stands apart from other points of 

 singular curvature, and seems to form a class by itself: but only because it belongs to a class 

 in which, though all the cases are singular, it is the only one remarkable to sight. Attempts 

 have been made to give name and character to the points at which y" , y' 1 ', &c. change sign, but, 

 though = may from the existence of such points be properly called a singular case of 

 <p = const., the points themselves are not remarkable points of <p = 0. It seems that our 

 senses are not capable of feeling anything beyond second differential coefficients. In geometry, 

 we recognise direction and flexure, and change of direction simultaneously with flexure : 

 in mechanics, velocity and pressure, and change of velocity simultaneously with pressure. But 

 change in the quantity of flexure, or of pressure, presents nothing which the common sense 

 of mankind embodies in a separate notion ; so that the third differential coefficient is not 

 presented to us in the form, as it were, of a separate* idea. 



Let two points of <b m be called similar, when a line drawn from one to the other 

 cuts (p = an even or an odd number of times, according as it begins and ends with 

 the same or different abutments : otherwise, let the points be called dissimilar. It follows 

 from the preceding connexion between <p y .y" at {x, y) and <p just after leaving (#, y), 

 that points are similar or dissimilar according as their values of rf> y .y" agree or differ 

 in sign. Also points are similar or dissimilar, according as, in passing from one to the 

 other, we pass over the curve (j> y *(p xx - 2 0^^^ + (pj*(f> yy - an even (0 included) 

 or an odd number of times. A change on the curve to points of a different character 

 takes place when there is a change of vertical flexure (flexure with respect to the axis of x, 

 or which depends on the sign of y") or passage over the vertical subordinate with change of 

 sign, but not both. And the equation <p y y" - (p?x" = connects the criteria with respect 

 to the two axes, and justifies one from the other. 



Hence no curve in which <p(x, y) is strictly real, uniyocal, and finite, can have an 

 abrupt termination : for in this case, the same two points would be pronounced similar or 

 dissimilar, according as the line of progress fell just on one side or the other of the 

 abrupt termination. 



* Voltaire's Satumian, who had only seventy-two senses, 

 lamented the poverty of his human nature to the Sirian, who 

 had a thousand : but the Sirian replied that the same complaint 

 was made in his own planet. Both had reason, for the train 

 of differential coefficients is interminable. Voltaire could not 

 make these worthies talk their own minds. If we had another 

 mode of perception, of which neither touch nor sight would 



give any notion, and which would associate with pressure the 

 idea of its cause, in the sense in which we say that pressure 

 causes acceleration, we should probably , as now, look upon the 

 cause of a cause, and that cause, in the light of cause and effect : 

 but we should, in associating the cause and effect of an inter- 

 mediate cause, pass over the third side of an intellectual trian- 

 gle of which at present we can only pass over the other two. 



