Royal Astronomical Society. 513 



supposing him to employ the gift in the payment of his debts, then 

 concludes that he had less than nothing, because, being now richer 

 than before, he has only nothing. Others admitted the negative 

 and impossible quantities as mysteries, and, reversing Mr. Frend's 

 process, brought them forward as auxiliaries to the mysteries of the 

 orthodox forms of Christanity ; a practice not extinct in our own 

 day, even after all that was inexplicable about impossible quantities 

 has disappeared. At the time when Mr. Frend first thought on 

 the subject, the assertion of mystery was the escape from the con- 

 fession of incompleteness ; the great mass of readers followed with 

 implicit confidence, while, of those who thought for themselves, an 

 enormous majority was too sensible of the value of the results of 

 algebra to abandon it on account of difficulties. Some few rejected 

 the peculiar doctrines of algebra altogether ; among whom those of 

 most note were, in succession, Robert Simson, Baron Maseres, and 

 Mr. Frend. Most of those who were independent of authority united 

 in blaming the method of the elementary writings, and were content 

 to hope that a palpable guide to truth would not always be without 

 rational connexion with undeniable axioms. Woodhouse, the re- 

 storer of thought on first principles at Cambridge, in a letter to Baron 

 Maseres, preserved among Mr. Frend's papers, and dated November 

 16, 1801, distinctly lays it down that, in these matters, it is not the 

 principles which prove the conclusions, but the truth of the conclu- 

 sions which proves that there must, somewhere or other, be prin- 

 ciples. " Whether or not," says he, " I have found a logic, by the 

 rules of which operations with imaginary quantities are conducted, is 

 not now the question : but surely this is evident, that, since they 

 lead to right conclusions, they must have a logic." And he goes on 

 thus : " Till the doctrines of negative and imaginary quantities are 

 better taught than they are at present taught in the University of 

 Cambridge, I agree with you that they had better not be taught ; 

 and the plan of our system of mathematical education, much as it is 

 praised, needs, in my opinion, considerable alteration and reform ; 

 and perhaps you think that our late mathematical publications will 

 not much increase the love or improve the taste for luminous and 

 strict deduction." As concerns the mystics, then, there is no need 

 to object to Mr. Frend's entire abandonment of their principles, but 

 the reverse ; for it may be asserted that most of those who thought 

 about first principles did the same. Those who imposed on 

 matter, in the name of Newton, a primary power of attracting other 

 matter, with those who could, on their own definitions, be made to 

 say that a command to subtract 2, repeated as many times as there are 

 units in a command to subtract 3, gives a command to add 6, ought to 

 have been surprised that they found. so little opposition. 



But the circumstance relative to Mr. Frend's ultimate views which 

 is peculiar to himself and which cannot be remembered without sur- 

 prise, is, that in clearing the trammels of mystery he had to force so 

 thick an enclosure, that he left behind him not only the mysterious 

 explanation, but the very facts which were professed to be explained, 

 and which, it may be thought, could have admitted of no doubt. It 

 Phil. Mag. S. 3. No. 141. Suppl. Vol. 21. 2 M 



