272 Mr. J. Walsh on Parallel Straight Lines. 



he imagine space less than which there is no space? If he 

 cannot, or more properly if there is not, then he cannot ob- 

 viate the difficulty of Euclid's postulate. I know not how 

 it happens that some illustrious geometers mistake the nature 

 of number, and the manner it presents itself in mathematical 

 investigations. I know not how it is, that the most glaring 

 absurdities are received by geometers, even as necessary 

 truths ; and that too in a science conversant only about truth. 

 It is acknowledged to be a fact antecedent to all hypotheses, 

 that the arc and tangent of a curve may become nearer to 

 each other than any given difference ; that is to say, there is 

 a given difference that is less than itself. It is said to be a 

 fact, in the theory of maxima and minima, that space can exist 

 and cannot exist at the same time. And these two proposi- 

 tions are the basis of the modern mathematics. Does this 

 arise from the nature of elementary education ; from spending 

 the important period of youth almost exclusively in the study of 

 the languages of Greece and Rome? This is a circumstance, 

 I suppose, which the well-being and advancement of society 

 render absolutely necessary. But are the powers of reasoning 

 by this means deranged, from having no fixed principles al- 

 ways to refer to ? On the subject of this paper, I find the 

 following observations in page 296, fourth edition of Leslie's 

 Geometry. It appears they were advanced by M. Legendre 

 in reply to the objections of Professor Leslie to his theory of 

 parallels. La loi de Vhomogcneite est nne lot generate qui n'est 

 jmnais en d'efaut. Every body is sensible of this. In the next 

 sentence he says : L'angle est une quantite que je mesure tou- 

 jour s par son rapport avec l'angle droit. Again he says : 

 U angle droit est l' unite naturelle des angles. That is to say, 

 particular sounds of the human voice find linear space are 

 homogeneous magnitudes. Locke falls into the same absurdity, 

 when he says, " Number measures all measurables." Geo- 

 meters are led away by the erroneous manner in which num- 

 ber presents itself in language. I define numbers to be arti- 

 culate sounds used as the signs of our ideas of certain relations 

 between homogeneous things. We measure magnitude by 

 magnitude homogeneous to it ; and number is the verbal ex- 

 pression of the relation found to exist by actual measurement. 

 And figures are the written characters which as it were re- 

 present numbers, and excite in us the ideas of certain rela- 

 tions. The term number is generally transferred from the 

 verbal expression to the written character. It is not until the 

 binomial calculus shall become a general object of study 

 among mathematicians, that reason, so long and so much 



distorted 



