Notices respecting New Books. 59 



and " breadth ") cannot be understood without a previous know- 

 ledge of the meaning of the word line. Compare the definition of 

 a line with that of a square — " a four-sided figure which has its 

 sides equal and its angles right angles ; " and the difference is at 

 once apparent : in the former the definition is performed by means 

 of words of an equal order of difficulty, in the latter by means of 

 an analysis of the name into others simpler than itself. Mr. Wil- 

 son is plainly not contented with Euclid's definition ; but he does 

 not mend matters by giving the definition of a hue thus : — "Aline 

 has position but neither breadth nor thickness." The fact is that 

 the difficulty cannot be got over. There are many words of which 

 we all know the meaning, but which we cannot define : such words 

 as red or blue are instances; and so are a point, aline, a superficies, 

 an angle. Mr ."Wilson's account of a straight line is quite a curious 

 instance of how nearly a writer may get to a distinction and yet 

 miss it. He tells us : — " Def . 5. A straight line is such that any one 

 part must, however placed, he wholly on any other part if its ex- 

 tremities are made to fall on that other part." This is plainly a 

 property of a straight line, or rather of two straight lines, not a 

 definition; it is in fact the second geometrical axiom with its 

 wording slightly altered. If a definition is to be attempted at all, 

 it would be hard to produce a better than the old one — " A straight 

 line is that which lies evenly between its extreme points ;" but, of 

 course, the word evenly as much requires definition as the word 

 straight. Mr. Wilson adds to his definition the remark, "A stretched 

 string suggests the notion of straightness, which is in fact incapable 

 of elucidation " (p. 6). If he had said that " a string stretched 

 between two points suggests the notion of a straight line, a term 

 which is in fact incapable of definition," we should have entirely 

 agreed with him ; but as to the notion of a straight line (or straight- 

 ness) being incapable of elucidation, that is quite another point. 

 The notion is elucidated in Whewell's ' Philosophy of the Inductive 

 Sciences,' vol. i. pp. 96, 97, as Mr. Wilson will find on reference. 



(3) We have hitherto dwelt only on axioms and definitions. We 

 will now advert to a fault which pervades the whole book, though 

 want of space compels us to consider it with regard to only a single 

 instance. The fault is impatience of the restrictions of Elementary 

 Geometry ; the instance is the treatment of angles. Euclid, as is 

 well known, does not recognize any angle which might not be an 

 angle of a triangle. There seems good reason for this : the notion 

 of an angle that the learner brings with him to the subject is that 

 of a knee or corner ; to make him thoroughly familiar with the no- 

 tion and able to reason upon it correctly is enough in the first 

 instance, and the more as the notion is fully sufficient for all the 

 requirements of the first four books. Mr. Wilson thinks other- 

 wise ; and so to angles right, acute, and obtuse he adds the " straight 

 angle" [one of two right angles] and the "reflex angle" [one 

 greater than two right angles]. Now, in the first place, he uses no 

 term to denote merely an angle less than a " straight angle ;" con- 

 sequently he always terms such an angle simply an angle ; in other 



