to prove the Properties qf parallel Lines. 167 



The modern analysis, from the universality of its application, 

 has the supremacy of the mathematical sciences. It is the 

 most powerful instrument of investigation that has yet been 

 invented. But this potent engine cannot put forth its strength, 

 until a proper fulcrum has been prepared for it. 



Professor Leslie has himself attempted to improve the theory 

 of parallel lines, and we shall add a few observations on his 

 manner of treating this difficult subject. His definition is 

 most faulty, involving the idea of infinity. For what notion 

 can we have of two lines with no mutual inclination, except 

 we conceive that they make at first a small angle which de- 

 creases to zero. But the definition is set down only pro 

 forma, and may be blotted out without being missed, since it 

 is doubtful whether it be once referred to in the whole treatise. 

 In reality we are left to gather the sense we must affix to pa- 

 rallel lines from the descriptive and illustrative writing in 

 Prop, 22. book 1st; for demonstration it cannot be called. 

 All this may be very elegant, but it is not in the spirit of the 

 Greek geometry. It is directly opposed to the example of 

 Euclid, who has taken so much pains to avoid whatever is 

 vague, and to exhibit every point to the understanding of his 

 readers in a definite form. But the learned Professor has 

 not been successful in removing any difficulty, or in throwing 

 light upon any obscure subject, in elementary science. If new 

 proofs of this were wanting, we might refer to the manner in 

 which he has treated the composition of forces; the perma- 

 nent axes of rotation ; and other matters, in his late volume 

 on Natural Philosophy. 



Legendre complains that a part of a private letter of his to 

 the learned Professor was published without leave asked and 

 obtained. But the anonymous correspondent who has been 

 d ragged before the public in so notable a maimer in this dis- 

 pute, has a much more grievous cause of complaint. His 

 opinion with respect to these demonstrations, communicated 

 in a private letter, was partially published without his know- 

 ledge: the part of his letter which seemed to corroborate the 

 opinion of the Professor being laid before the public, while 

 no notice was taken of another part which coincided with the 

 opinion of Legendre and was opposed to that of the Pro- 

 fessor. 



March 3, 1824. DlS-IOTA. 



XXIX. Ri 



