to Mr. HoldreJ. 39 



tisfactory method, wliich only shows the truth of the rule as ap- 

 plied to the equation he has proposed. He has not given such 

 a demonstration as to furnish a general rule for all equations ; 

 but, reasoning by analogy, he has presumed the truth of the rule 

 without demonstration, by extending the idea to a general infer- 

 ence. This evident impropriety proceeds from his total ignorance 

 of the properties of figurate numbers, on which the whole de- 

 pends ; but on this head, he ungenerously observes, he did not 

 want any of my assistance. 



From what I have substantiated here, I shall be warranted in 

 restating what I have asserted in page 57 of the postscript, viz. 

 that Mr. Holdred's book does not contain a single idea but what 

 is found in my Essay on Involution &c., and which was given to 

 the world prior to the publication of his tract. 



In the postscript, page 64, which was written in conseqnence 

 of the misrepresentations to be foiuid in Mr. H.'s preface, I have 

 stated the circumstances which led me to the demonstration of 

 the non-flgurate method, and have there given a comparative view 

 of our methods by actual examples, from which the inquiring 

 reader may see, that if I was not the original inventor of the 

 principle, I have improved the demonstration and have simplified 

 the practice of both methods; but with respect to the non-figurate 

 operation, Mr. Holdred has no claim to style himself the original 

 inventor of it, as I was the first to give him any hint of it by sig- 

 nifying my intention to him (in confidence) of reducing it to a 

 formula, similar to that which I published in Essay 3, page 4, 

 of my Combinatorial Essays. However, I cannot help remark- 

 ing, that if what I have done is not purloined from him, it cer- 

 tainly has flowed from what he has done, as a consequence ; for 

 it was undoubtedly in consequence of the conduct of Mr. Holdred 

 that 1 was induced to publish, in my own defence, my own im- 

 provements in extracting the roots of equations ; for I was, prior 

 to our difference, resolutely bent on rehnquishinglhe study of the 

 mathematics, which interrupted the progress of mv professional 

 duties, and of other publications which I was then, and am still, 

 engaged in ; being fully aware, from dear-bought experience, that 

 analytical researches not only occupied too much of my time, 

 but obliged me to expend that money which I could have appro- 

 priated in a much more eligible and advantageous way. 



If, after all that has been not only said, but proved, Mr. Hol- 

 dred should still persist in asserting that my demonstrations were 

 extracted from his, I call upon him to show the reason why he 

 has occupied twelve (juarto pages in the demonstration of both 

 methods, when all that he has said might be comprised within 

 the compass of one-twelfth part of the space, which is what I 

 have done. It appears then, that either my demonstrations must 



be 



