76 Review o/T. P. Thompson's Geometry without A.rioms. 



facts alleged, and the consequences assigned to them." Would Mr. T. 

 excuse our attempt to clear the reasoning ? as we think the metaphysical 

 question worthy of attention, and moreover since it is very easy to deduce 

 from it all that we desire, without appealing to algebraic notions. We 

 conceive the argument to stand thus : — "Lines cannot be calculated from 

 angles alone : for when angles are given, no linear unit is hereby given. 

 It is OBJECTED that this proves too much ; for when lines alone are given, 

 no angular unit is hereby assigned ; yet angles may be hence calculated. 

 We REPLY, Not so ; of angles there is an extreme value, viz. the sum of 

 two right augles j which, for any thing which a priori we know to the 

 contrary, might virtually furnish an angular unit ; and this a posteriori we 

 positively know to be the case : but contrariwise, a straight line has no 

 maximum or minimum value. The cases then are not parallel, and the 

 objection falls to the ground." Whether the original argument be valid, is 

 to be considered again ; but assuredly the reply which Mr. T. thinks so 

 little to the purpose, is directed very accurately against his insuperable 

 objection. But he, as Mr. Leslie, seems to have much spite against this 

 principle. That lines cannot be calculated from angles, is a proposition 

 notoriously true ; and the truth of which we readily learn, without wading 

 through the properties of triangles. Why should he be incredulous as to 

 the possibility of giving a direct demonstration of it from first principles ? 



But he meets us on another ground : " The substantial inference," says 

 he, " is, that they have confounded the quantities which Euclid in his book 

 of Data would call given, with the quantities whicli must be employed as 

 elements in actual calculation." And here Mr. T. himself makes a great 

 blunder : and we are disposed to thiuk he has more acuteness in detail, 

 than sound philosophical views of the science. His very example might 

 have confuted him. If a, b, c, are the sides of a triangle, and A, B, C the 

 opposite angles ; we know c is determined when a, b, C are given. " Yet," 

 argues he, "c must be collected from the formula — 



Here start up among the practical elements of the calculation two straight 

 lines in the shape of the tangents of two arcs," &c. He would seem to 

 forget that c is at once determined by the formula — 



but to found an objection to Legendre's reasoning, on the idea that the 

 tangents and the cosines in the tables are straight lines, is truly extraor- 

 dinary — as though any linear unit were determined by them. With equal 

 want of consideration he compares a hyperbolic arc with a straight line, as 

 though "new elements, such as major and minor axes might start up in the 



