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XXVIII. Notices respecting New Books. 



An Introduction to the Elements of Euclid, being a familiar explana- 

 tion of the first twelve propositions of the first booh. By the Rev. 

 Stephen Hawtrey, A.M., late Assistant Master at Eton* London: 

 Longmans, Green, and Co. 1874. (Pp. 105.) 



ONE'S first impression on looking at this book might be to ask, 

 Is it possible that the first twelve propositions of Euclid's 

 first book require 105 pages of explanation? But if the querist has 

 ever taught the elements of geometry to a child and remembers in 

 what his verbal instruction consisted, he will probably find on read- 

 ing the book that his explanations were at least as voluminous as 

 those here printed ; and doubtless he will be both interested and in- 

 structed in this record of the method of oral teaching which was 

 used for many years by Mr. Hawtrey " with some success." 



We will notice very briefly the leading points of the book. In 

 the first place the author insists on the learner being provided with 

 " a pair of compasses having a pen- and pencil-leg, a small flat ruler, 

 and a hard pencil;" besides these, he adds, it is well to have "a 

 little box-wood triangle, called . . . . ' a set square.' " He also 

 insists on having the data of each problem drawn in ink before the 

 construction is completed in pencil. In the next place he treats 

 the definitions very fully, those at least which the learner needs to 

 begin upon ; the postulates and axioms are noticed, with a suffi- 

 ciency of comment, the twelfth axiom being left out as not needed 

 at present. The propositions are then taken in order, directions are 

 given for drawing the figure line by line, and each step of the de- 

 monstration commented on. At the end of each exposition the 

 proposition is given as it stands hi Euclid; — general enunciation, 

 particular enunciation, construction, demonstration ; and the learner 

 is directed to write it out. 



The fourth proposition is treated at great length, and its import- 

 ance in regard to what follows insisted on more forcibly (not than 

 it deserves to be, but) than we have ever seen it insisted on before 

 "It is a great and most important proposition, and has been called 

 the key to Euclid." The exposition of the proposition is followed 

 by seven exercises, all fully worked out ; and as they cannot be 

 done without assuming something more than the first four propo- 

 sitions, the assumptions are carefully explained; as a deduction 

 from two of the exercises the fifth proposition is proved in fact 

 though not in name. After this the learner is ready to encounter 

 the fifth proposition in its proper form, which is, as Mr. Hawtrey 

 says, nothing but an exercise on the fourth proposition, though too 

 hard to be taken immediately after it. 



"We fear that this account of the contents is somewhat meagre ; 

 but there is, in fact, not much to say about the book in the way of 

 description. Its merit consists in the admirable clearness with 

 which all the points of a well-trodden road are brought under notice 

 and much insisted on, which, in these days of quick travelling, is 



