Notices respecting New Books. 57 



An author who takes in hand to write a book on the Elements 

 of Geometry labours, however, under peculiar disadvantages. His 

 work will be compared with a high standard ; and mistakes which 

 might have passed without notice in other subjects will be chal- 

 lenged in this. Besides, as there are strong — we do not say con- 

 clusive — reasons against the substitution of a new work for the 

 recognized text-book, it will be expected that the author shall pro- 

 duce something substantially different and distinctly better. A 

 writer who attempts to produce a synthetical Compendium of the 

 Elements of Geometry substantially different from and distinctly 

 better than the first four books of Euclid, undertakes a task of 

 no ordinary difficulty ; and, if we may judge from the preface 

 to the third edition, Mr. Wilson is more aware of this than he 

 was a few years ago. To intersperse the text with remarks 

 and to add deductions are nothing to the point — notes and de- 

 ductions to Euclid are already in existence. Nor is it much 

 more to the point to make minor changes. Very many of Mr. 

 Wilson's changes fall under the latter head. Eor instance : — the 

 treatment of Parallels by Playf air's method ; the proofs of Euclid, 

 I. 8, I. 24, III. 26, 27, 28, 29 (Wilson, pp. 28, 29 ; 27 ; 112-120), 

 and others; the grouping of the propositions; the separation of 

 Problems from Theorems &c. It may be questioned whether in 

 this there is on the whole improvement; at all events it is im- 

 provement of no great importance. To set against this there is a 

 great deal which seems to us quite the reverse of improvement ; 

 and though we cannot exhaust the subject, we will go into details 

 on three points. 



(1) Mr. Wilson makes two classes of axioms — general and geo- 

 metrical. Now, of course, it is true that such propositions as " if 

 equals be added to equals the wholes are equal " are applicable to 

 other than geometrical magnitudes ; but then the application pre- 

 supposes notions which the student of geometry may or may not 

 possess. It may be true that when equal quantities of heat are 

 added to equal quantities of heat the wholes are equal ; but the stu- 

 dent of geometry has nothing to do with this ; he may not be able 

 to understand what is meant by equal quantities of heat ; and ac- 

 cordingly to him the axiom means that if equal angles or lines or 

 areas are added to equal angles or lines or areas, the wholes are 

 equal. The axiom, in fact, is to be construed with regard to geo- 

 metrical matter. The like is true of the other so-called general 

 axioms — at least as they stand in Euclid ; and consequently there 

 is nothing gained by the division. When we go further, however, 

 and see what Mr. Wilson includes in the list of General Axioms, 

 we cannot help suspecting the existence of some confusion of 

 thought. Here are two of the general axioms : — " (3) If equals be 

 added to equals the sums are equal." " (7) If it is known that < If 

 A is B then C is D,' it follows that 'If C is not D then A is not B.' " 

 Now the former of these, as above remarked, relates to the matter 

 of geometry'; and where necessary Euclid quotes it amongst the 

 premises of his reasoning, just as he would any other antecedent 



