Review of T. P. Thompson s Geometry without A.viom^. 8 1 



most of which we could advantageously part with : some only should be 

 turned into theorems, as in the fourth book. Farther, in the employment 

 of supraposition, he virtually uses another postulate, and when he ap- 

 proaches curvilinear areas, his system breaks down entirely. Thus, on 

 Euclid xii. 2. R. Simson remarks, that to complete the proof we must 

 insert : " For there is some square equal to the circle :" a thing true 

 enough ; but which rule and compass will never construct. 



To do him justice, we must remember that he himself must have thought 

 his fifth book unsatisfactory ; for he treats ratios over again, and that nu- 

 merically, in one of the books which we omit ; and in another, he collects 

 into a better order all the scattered properties of figures. 



But further ; in Euclid's day no doubt the second book was necessary to 

 Geometry. At present it is highly ungeometrical and useless ; because we 

 at once discern, that out of the infinite number of algebraic developments, 

 it selects a few to demonstrate geometrically. A sufficient substitute for 

 these, is, to show (what at any rate needs to be shown) how the areas of 

 rectangles are numerically represented, after making the square of the 

 linear unit our unit of surface. The twelfth and thirteenth propositions 

 of this book belong rather to Trigonometry or to Algebraic Geometry, than 

 to this department ; and the fourteenth is a problem misplaced. 



What then is needed to make Euclid's Elements a logical, well-ordered, 

 and perspicuous whole, confined to its proper limits, yet adapted to the 

 existing state of mathematical science? Strange liberties have already 

 been taken with it. The seventh, eighth, ninth, and tenth books, are 

 sunk in the darkness of allusion : the lengths and breadths and depths and 

 heights in the eleventh and twelfth are seen in shadowy distance ; but the 

 student is out of breath and comes to a halt, before he reaches the " method 

 of exhaustion :" the fifth book is very frequently given him in grace, and 

 an explanation of the fifth definition is thought enough. The second book 

 willvanish of itself by a diflerent arrangement, viz. by splitting the sixth 

 in twain, and kneading it up with the first and third. The fourth book 

 consists entirely of problems, most of which are to be wrenched out from 

 all the books ; twisting some into theorems, ramming down others into an 

 Appendix, or book of exercises. A better distribution of various subjects 

 is much needed. Similarity of shape should form a separate section, and 

 a single definition be given of the word similar, applicable to figures plane 

 or solid, curved or rectilinear. Areas equally need to be systematized. 

 Contact and curvature in the circle should be so treated as to prepare the 

 student for following out the subject with other curves ; in short, according 

 to the enlargement of view which the moderns have thereby gained. 

 Circular arcs and circular areas need to have something said about them, 

 and what is said, should be connected and compact. We have not noted 

 how large changes in the first book may be required for proving the twelfth 

 axiom : but, if attainable, we desire more method in exhibiting the proper- 

 No. I.— Vol. 1. M 



