72 Review of T. P. Thompsons Geometry lokhout A,vioms. 



the base of a tessera be acute, tlie angles opposite to the base are not 

 right." We should add, that by a tessera he understands, a trapezium 

 which has a diameter perpendicular to one of its sides. 



We have now to remark on the character of his reasoning, in these his 

 four main propositions. The first of them has four cases j that spheres 

 cannot touch externallj', neither in a surface, nor in a self-rejoining line, 

 nor in an open line, nor in isolated points. The proof of the first case is 

 virtually grounded on the principle, (we fear we ought to say a^iom .^ that 

 if any portion, however small, of a body's surface be immoveable, — as if 

 for instance it be glued down to another surface that is immoveable, — no 

 oscillating motion, however small, is conceivable in the former. We sus- 

 pect Ml-. T. would have bantered Le Gendre very cleverly for any such 

 assumption, telling him that for aught he knew, an axis of rotation might 

 have breadth. Did it ever occur to Mr. T. to inquire whether he could 

 prove that it cannot ? We do not say he cannot : but if he can, it should 

 be ])romiuent : while at present he labours hard in giving reasons where 

 we want none, and here he does not at all clearly tell his reader what he 

 is assuming, or why. We had written various objections to the proof of 

 the three remaining cases : wherein we exceedingly disapprove of tlie 

 vague and even unintelligible language of above and below, before we have 

 defined straight line ov plane, and when they cannot be changed into "this 

 and that side of such and such a surface :" and equally do we disapprove 

 of his arguing that " the line (/ D cannot turn its face (!) to all sides in 

 succession without change of place ;" more especially when he has not yet 

 proved that C D may not be such a line as we afterwards call straight. 

 His last case is proved only when there are two isolated points, and fails 

 when there are three. We had accumulated yet more objections, when we 

 met with the Quarterly Journal of Education, No. XIII., in which it is 

 remarked that the proof applies quite as well to intersecting spheres, as to 

 spheres in contact. This remark is obviously and instantly fatal to the 

 last three cases : but we think slight verbal changes would enable the first 

 to evade the charge. 



AVe are sorry to think the failure of this proof is so decisive : for the 

 proposition is to us the most interesting in the book. It seems an original 

 thought to prove directly from first principles, that the external contact of 

 spheres is but in one point ; and we are not at all in despair that it may 

 be done : but if done, it must be done by perspicuous and easy proof, or 

 we shall not value it. Out of this proposition instantly flows the inference, 

 (which we are surprised at Mr. T.'s not drawing) that a straight line is the 

 shortest between two points, and is therefore the measure of distance. For 

 it is manifest that no line connecting the centres can be the shortest, (or 

 as short as any) unless it pierce both surfaces in the point of contact. But 

 while we think that Mr. T. has decidedly failed of proving his proposition, 

 he has usefully set forth what is meant by three or more points lying evenly. 



