" Geometrical Researches " in four papers, comprising numerous 

 new theorems and porisms, and complete solutions to celebrated 

 problems, by MARTIN GARDINER, C.E. 



DEFINITIONS. 



1. If A, B, 0, be any three points, then by " angle O AB " 

 we mean the angle formed by the revolution of a rigid straight 

 line round O as centre from a position coincident with A to a 

 position coincident with OB, the method of movement being such 

 as to sweep direct across the straight line AB from A to B. 

 And according as the revolution is right-handed or left-handed 

 we say the angle is of right formation or of left formation. 



2. The " rotative " of a straight line A B in respect to a 

 point O, is the method of rotatory movement of a rigid straight 

 line round as centre when the movement is such as to sweep 

 direct across the straight line from A to B. And according as the 

 revolution is right-handed or left-handed we say that A B is of 

 right rotative or left rotative in respect to O. 



3. By the rotative of a tangent drawn from a point to a curve, 

 we mean its rotative in respect to the centre of the osculating 

 circle at the point of contact of the tangent and curve. 



4. By the term n'gon, we mean a figure composed of n con- 

 nected portions of straight lines which we can conceive to be 

 formed by n successive movements of one point. 



5. The lines composing any n'gon are called sides and the 

 first point of the 1 st side, and the final point of the n th side are 

 called extremities. 



6. A closed n'gon has its extremities coincident ; and an open 

 n'gon has its extremities distinct. 



7. When any n'gon is represented by means of the letters 

 which indicate its first extremity and its various other angular 

 points and last extremity written in successive order, we say it is 

 of prescribed formation. 



