(iS Review ofT. P. Thompsons Geometry without Axioms. 



equal their fertility previous to the inundation, is about seven years ; their 

 average annual value before the inundation, fourteen shillings per acre ; 

 and from its commencement until all traces of it should have disappeared, 

 five shillings ; leaving a loss from the diminished produce of the lands of 

 about 3612,080, and a total direct loss from the inundation of sE 16,402 : 11*. 

 upon the above three Polders alone. 



REVIEW. 



Geometry without Aa'ioms. By T. Perronet Thompson, Queens College, 

 Camhridge. Fourth Edition, 1 833. 



The study of Geometry, rising, like most other sciences, out of utility, 

 was at a very early period fixed upon for its value as an exercise of the 

 reasoning powers: nor has all the brilliancy of modern analysis ever dis- 

 placed the earlier science from its claim to pre-eminence in this respect. 

 Not as though its reasonings were more rigorous than those of Algebraic 

 Mathematics, nor as though the latter did not vastly surpass the former in 

 the efforts of inventive genius and subtle discussion : but because the sub- 

 ject on which Geometry treats is far more popular in its character, than 

 any of the higher branches of Analysis can be made. But few can afford 

 the time and application needed to master these latter, nay, to gain an 

 apprehension even of the definitions, so far as to conceive the spirit of 

 the method : while all have a more or less accurate notion of spheres and 

 cubes, angles and distances. The subject matter is familiar, (if it be but 

 treated as such) and, like the first principles of Algebraic equations, affords 

 a method of calling the attention to accurate processes of reasoning ; with- 

 out so filling the young student's time, as though all were to be professed 

 mathematicians. 



When such is the object for which Geometry is pursued, — not for the 

 sake of its results, but for the sake of its processes, — we cannot easily be 

 hypercritical upon the soundness of the reasoning. That which would be 

 mere cavilling, in review of a practical treatise, may constitute valid objec- 

 tion against a treatise which is proposed as a model of perfect reasoning. 

 In such case, if the link of the argument be broken, the error or defect 

 should be avowed. To hush up the matter, and pawn off imperfectly con- 

 nected propositions as demonstratively proved, is like training a student in 

 the habit of smothering his doubts, and taking proof for granted. Perhaps 

 in this matter geometers have been far less honest than they should have 

 been. It is certainly strange, that when geometrical, as opposed to ana- 

 lytical, is popularly apprehended to mean rigorous as opposed to lax j 

 possibly no algebraic treatises have flaws so fundamental as those which 

 exist in the common elements of Geometry. And what is worst, geometers 



