85 



CHAPTER OF VARIETIES. 



On the mathematical dances of gnats. — As an illustration 

 of the beautiful theorem in geometry (Eucl. i. 32,) which bears that 

 the " three interior angles of any triangle, are when taken together 

 equal to two right angles," I have just met with the following, given 

 with so much spirit too by the ingenious author, that perhaps he may 

 include himself among the " many persons " who " have burst into an 

 involuntary flood of tears upon first reading this proposition." 



" The maziest dance," says our author, " ever performed by three 

 gnats of a swarm in a summer's evening through the air, is yet subject 

 to this strict regulation : the flies can never separate from each other, 

 so that their angular distances, when added together, shall be less or 

 greater at one time than another, but will always be equal to the 

 stated quantity of two right angles, (unless when they happen to be 

 all in the same straight line, on which occasion their angular distances 

 from each other all vanish at once). But what shall we say to the 

 fact (which is, however, but a multiplication of this), that every trio of 

 gnats in the swarm, be it ever so large and so lively, every combina- 

 tion of three, in all the evolutions and involutions performed by the 

 whole crowd, keep always at the same collective angular distances 

 from each other; that is to say, the three angles formed by lines 

 joining every three flies in the swarm, when added, not only make up 

 the same quantity at all different times, but those formed by one trio 

 are exactly equal in a sum to those formed by any other ! The same 

 is true of all the bodies of the solar system ; planets, satellites, and 

 comets. Whatever changes or disturbances they may undergo, there 

 is one relation subsisting between them which no power can alter j 

 every three must, in every position, have the sum of their angular 

 distances what it was in every other position, namely, two right 

 angles (unless when they happen to be in the same right line.")— - 

 Darley's Geometrical Companion. 



b 



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