Review of T. P. Thompson's Geometry without ^a'ioms. 77 



case of the straight line." He forgets that we already know a straight line 

 of unlimited length to be determined by any two points in it Give five 

 points in a hyperbola, which is only just enough, and we shall find no 

 " new elements start up." But in fact his distinction between the data 

 W'hicli determine the value of quantities and the elements v\hence those 

 values are calculated, is utterly untenable and unj)hilosophical. Suppose 

 the central angle of an elliptical sector is given, and the axes of the ellipse: 

 if these are enough data to fix the length of the arc, then they are sufficient 

 elements for its calculation, however many be the intermediate shifts by 

 which the ultimate result is obtained. If Mr. T. can disprove this, he will 

 give a fatal blow to the whole of the modern analysis. This is in fact its 

 fundamental principle, wherever we reason concerning functions whose 

 form is as yet unknown. 



Before quitting this head, we will show how concisely the doctrine of 

 Parallels maybe proved from the principle that "lines cannot be calculated 

 (nor therefore be determined) by angles alone." Let A B, C D be each 

 perpendicular to A C ; and let A E divide the 

 angle CAB. If then A E, however far prolonged, 

 does not meet C D, shorten the distance A C, and 

 it is manifest that there is some least distance 

 A F, in which C D, coming into the position F G, 

 (still perpendicular to A C) is a true asymptote to 

 A E. Then the distance A F is determinate : yet there are no data to fix 

 it but the angles at A, which is absurd. 



To remark generally on Mr. T.'s philosophy ; while he aims at exceeding 

 precision, we often find him lax, or as we judge, erroneous. He, has re- 

 tained the old definition of a solid, " that which has length, breadth, and 

 thickness" — containing three words positively unintelligible till we con- 

 ceive of three rectangular axes. His definition of equality is exceedingly 

 questionable, as it involves us in the well-known paradox, that as a circular 

 area, however cut up, would never fit into a square, therefore no square 

 can be equal to a circle. Yet it is on this definition that he rests for 

 abolishing all the axioms that are not specially geometrical. He indeed 

 assumes that " all surfaces are of one kind," as regards magnitude; without 

 commenting on the difficulty. We have already complained of the laxity 

 of adove and below, and of the faces of a line that may be straight, or may 

 have double curvature, for aught we know. We disapprove also of his 

 defining an angle to be a plane surface, when he reprobates all inferences 

 drawn from it. " All references," says he, " to the equality of magnitude 

 of infinite areas, are intrinsically paralogisms." We are not at all con- 

 vinced of this. Yet if an angle be a surface, it is an infinite surface : and 

 when one angle is said to be double of another, it must mean that one 

 infinite surface is double of another infinite surface. It is surprising that 

 he should think any thing gained by the change. In his Appendix, how- 



