Scientific Intelligence. — New Publications. ^9 



measure saved. In books of geometry, plates, especially folding 

 ones, are often fluttering in rags, whilst the work is otherwise 

 entire. Baron Dupin is entitled to great praise for the pains he 

 has taken to give a clear exposition of first principles ; and, in- 

 deed, the student who considers the first rudiments of any 

 science below his notice, is not likely to become a proficient. 

 We think, however, that some improvement might still be made 

 among the definitions. Thus, page 4, " A right line is the 

 shortest distance between any two points.'" This, to be sure, is 

 a characteristic feature of a straight line ; but unfortunately, it 

 is of no use at the outset of the elements of geometry. To sup- 

 ply the place of Euclid's tenth axiom, a second clause is added 

 defining a right line to be " that which we trace by always pro- 

 ceeding in the same direction."" Now, the term direction has 

 more need of definition than the other ; and we know of no 

 mode of defining direction, but by help of a previous knowledge 

 of a straight line. The tenth axiom of Euclid, or its converse, 

 forms the only definition of a straight line which has as yet 

 been found of any use in demonstrating the first propositions in 

 geometry. It has therefore been adopted in this form by some 

 authors of great note. In works exclusively devoted to elemen- 

 tary geometry, the demonstration of Euclid's twelfth axiom is 

 usually passed over as impossible ; and this makes it somewhat 

 curious, that, in page 18 of Dupin, a demonstration of that not- 

 able theorem, on which so many have foundered, should have 

 been attempted, as if it were a matter of no difficulty whatever. 

 The demonstration, however, is not new, but it is not exactly 

 given in its true colours ; for nothing is said of the infinite mag- 

 nitudes of the lines and areas on which the whole force of the 

 reasoning depends. A fairer representation of it may be seen 

 in Professor Duncan's " Supplement to Playfair's Geometry and 

 Wood's JZ^^^rft." This singular demonstration is somewhat allied 

 to the method of exhaustion, though not by infinitely small quan- 

 tities ; but areas infinitely great intercepted between lines of 

 infinite lengths ; and it is therefore doubtful if it be quite admis- 

 sible towards the beginning of the elements of geometry, or if 

 indeed it could be allowed in the higher branches of that science. 

 In short, it is such as neither Euclid nor Archimedes would have 

 tolerated ; and we are not sure if the modern supporters of the 



OCTOBEft — DECEMBER 18S6. O 



