Euclid and his modern rivals . d, A, B, are so related, that whena is greater than b, A is greater than B ; and when a isequal to h, A is equal to B: then, conversely^ when A isgreater than B, a is greater than h; and, when A is equalto B, a is equal to b. The truth of this principle, which extends to everykind of magnitude, is thus made evident:—If, when Ais greater than B, a is not greater than b, it must be eitherless than or equal to b. But it cannot be less ; for, if itwere, A should, by the antecedent part of the proposition,be less than B, w^hich is contrary to the supposition
Euclid and his modern rivals . d, A, B, are so related, that whena is greater than b, A is greater than B ; and when a isequal to h, A is equal to B: then, conversely^ when A isgreater than B, a is greater than h; and, when A is equalto B, a is equal to b. The truth of this principle, which extends to everykind of magnitude, is thus made evident:—If, when Ais greater than B, a is not greater than b, it must be eitherless than or equal to b. But it cannot be less ; for, if itwere, A should, by the antecedent part of the proposition,be less than B, w^hich is contrary to the supposition can it be equal to b ; for, in that case, A should beequal to B^ also contrary to supposition. Since, therefore,a is neither less than nor equal to ^, it remains that it mustbe greater than b: Now let a and A be variables and represent the ordinates 156 WILLOCK. [Act II. to two curves, mnr and MNB^ for the same abscissa ; andlet h and B be constants and represent their intercepts onthe Z-axis; let On — b, and ON = Does not this diagram fairly represent the data of theproposition ? You see, when we take a negative abscissa,so as to make a greater than ^, we are on the left-handbranch of the curve, and A is also greater than B; andagain, when a is equal to h, we are crossing the Z-axis,where A is also equal to B. Nie. It seems fair enough. Mln. But the conclusion does not follow ? With a posi-tive abscissa, A is greater than B, but a less than b. Nie. We cannot deny it. 3Iin. What then do you suppose would be the effecton a simple-minded student who should wrestle with thisterrible theorem, firm in the conviction that, being in aprinted book, it must someJwiv be true ? Nie. [gravely) Insomnia, certainly; followed by acuteCephalalgia; and, in all probability, Epistaxis. Min. Ah, those terrible names! AVho would suppose Sc. VI. § 3.] INCOMMEXSURABLES. 157 that a man could have all those three maladies, and sur-vive ? And yet the thing is possible ! Let me now read you
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectgeometry, bookyear188
