140 



MB. T. T. WILKINSON ON 



and mostly treated in a style which for originality and ele- 

 gance would certainly have done no discredit to either the 

 "Author of the Elements" or the "Great Geometer of 

 Perga." Very few, however, of these interesting discussions 

 are presented in a connected form ; almost all the properties 

 belonging to any particular class are given as isolated ques- 

 tions, and often at considerable intervals, so that no small 

 amount of labour will be requisite whenever an attempt is 

 made to reduce any of these scattered theories to a regular 

 system. Isolation and promiscuous arrangement are, indeed, 

 among the characteristics of the Geometry of the Lancashire 

 School, and nowhere is this more distinctly exhibited than 

 in the Mathematical Companion; its members, as a body, 

 in common with many others, never appear to have become 

 sensible of the value of systematic research, nor did they 

 attribute much importance to arranged collections of the pro- 

 perties of geometrical figures. Hence, although the Symme- 

 trical properties of Plane Triangles, — the Harmonical division 

 of Lines, — the principal properties of Lines passing through 

 the same point, or of points ranging in the same Straight 

 Line, — the properties of Circles of Tangential Ratio, &c. &c., 

 may be found scattered throughout their writings, scarcely 

 an attempt was made to classify any of these results, before 

 the appearance of Swale's Geometrical Amusements and his 

 ApoUonius, beyond what we find eifected in the case of 

 "Lineal Sections" and the "Modern Geometry" of the 

 Strident. The former of these articles is well known to 

 relate to the determination of two lines, by having certain 

 relations given,* and is mostly due to Mr. Knowles, of 

 Liverpool; the latter article contains the additions made 



* To determine two lines, geometrically, when : — 



1. Their sum and difference ; — 



2. Their sum and rectangJe ;— 



3. Their sum and ratio j — 



4. Their sum and sum of their squares ; — 



6. Their sum and diflference of their squares ; — 



6. Their difference and rectangle ; — &c. &c. ; are giveOr 



