Review of T. P. Thompson's Geometry without A^'ioms. 7 1 



The object proposed is to prefix or insert propositions to the first book 

 of Euclid, in such wise as to make the argument continuously logical, with- 

 out employing axioms. The defects are manifest. A straight line is defined 

 indeed by Euclid, but so defined as to be nowise clearer than before ; and 

 Euclid himself never appeals to the definition, but to his axioms. This is 

 a less guilty fallacy than that involved in his definition of the plane, in 

 which an assumption is smuggled j since he proposes more than enough 

 data for the generation of the plane surface. To remedy this, Mr. Thomp- 

 son has an " Intercalary Book," or more properly, an "^ Introductory 

 Book," ending in the establishment of that property of the plane, which 

 brings us from our airy flights on to the terra firma of plane Geometry. 

 He proceeds to dovetail the rotten parts of Euclid's first book, but of 

 course with the twelfth axiom chiefly in view. After various minor flou- 

 rishes, he concludes in an appendix by summing up and refuting all other 

 proofs that have been offered of the main principle of parallels. 



His mode of proceeding is as follows. — Equality of distance is deter- 

 mined by means of supraposition of points, even before the straight line is 

 defined : it is then easy to define a sphere. He proceeds to prove his first 

 important proposition, that spheres touching externalhj touch only in a 

 single point. It readily follows that this point lies evenly between the 

 centres : that is, turns about itself without change of place, if the two 

 spheres being united as one mass revolve together about the two centres. 

 Mr. T. does not use the word evenly ; but it is very appropriate ; and, so 

 explained, makes Euclid's definition of a straight line adequate. (Such is 

 nearly Mr. Leslie's view.) It follows, that by altering the size of the 

 spheres in contact, the point of contact is made to generate a path, con- 

 necting the centres, every point of which lies evenly betwixt the centres ; 

 and this is called a straight line. Out of this definition instantly flow all 

 the primary properties of the straight line. Mr. T, labours unnecessarily 

 at them. 



The second proposition of difficulty is, to show that intersecting spherical 

 surfaces coincide only in a circle ; which is virtually equivalent to Eucl. I. 8. 

 that the angles of a triangle are determined when the lengths of the sides 

 are given. 



His third difficult proposition is to generate a plane, and to show that 

 his plane has the property which Euclid makes its definition. And so ends 

 the Intercalary Book. 



There is moreover introduced into Euclid's First Book, to prove the 

 twelfth axiom, fifteen pages of close print, in this fourth edition. Diagrams 

 occupy a portion of the pages, and our author's style is that of over full 

 reasoning j leaving not even the very obvious steps and reasons to be 

 supplied. The quantity of matter is not therefore so formidable as might 

 appear : yet we must call it very hard and unreadable. The proposition 

 however which he is aiming to establish, is this: that "if the angles at 



