_ the Injziiitefimal Calculus. C31. 



This refult is perfectly juft, becaufe it is conformable to 



that which was obtained bv the comparifon of the triangles 



MZ 

 CPM and MPT; and yet the equations TP = MP 7^ = 



_5' ^—~- and -— — = -2. — . are certainly both of them falfe. 



For the diftance of RS from 3/P, was not fuppofed nothing, 

 or even very fmall, but equal to a line arbitrarily afiumed. 

 It follows, therefore, of neceffity, that a mutual compenfation^ 

 of errors took place in the comparifon of the two erroneous 

 equations*. 



Why this Compen -fa Hon takes place. 

 10. Having fliown that there exift compensating errors, 

 whence they originated and how they are proved, I lha!l next 

 proceed to explain them, and to fearch for a mark by which 

 it mav be known that a compenfation takes place in calcula- 

 tions fimilar to the preceding, and the means of producing 

 fuch compenfation in particular cafes. 



' This " compenfation of errors'' the acute Dr. Berkeley, Biihop of 

 ., in The Analy/i, pubiilhed in 1734, laboured to magnify into a 

 ferious objection againft the differential calculus; not eenfideriug tb t 

 errors which we mar make as little as we pleafe, will affect the truth as 

 is we pleafe. Thus 1 '999999, Sec. is not, in rigid ftriftnefs, equal 

 to 1 ; but as, by annexing 9's, the error may foon be rendered - 



brnjibly minute, no one can doubt that its ultimate ratio to 2, is a ratio of 

 Hence the word falfr, applied by our ingenious author, in this 

 paragraph and clfe where, to differential equations, is abundantly too ftrong; 

 and, had he read The Analyll, he would certainly have.ufed a 1-jfs excep- 

 •: term, fuch as incorrect, or inaccurate, or imperfeil, a word which 

 in the lequcl he takes in a fimilar fenfe. But falft (faux, faitffe) fignifies 

 not minutenefs of error, but a total negation of truth. Jt is not always 

 . . deed, to find words exactly fuited to exprefs very abftracvt ideas. 

 The great Simpfon (J&eleQ Exercifes, p. 239.) admits that, had Newton 

 ufed the word limiting inftead of ultimate ratios, Dr. Berkeley might not 

 have conjured up his "Ghofts of departed Quantities.'' Yet Mr. Simp- 

 ion, and ail oitoer competent judges, allow that Newton, by repre (enting 

 fiuxions not as in the ;atio of the whole increments, but in the firft ratio 

 of thofc increments conlidered as nafcenc, or in their ultimate ratio con- 

 fidered as evanefcent, has effi 1 I every reasonable objec- 



tion; and this, no doubt, the learned and ingenious bifhop would have 

 cpnfcffed, had he given himfelf time todigeft this doctrine, which is partly 

 wofuldcd in the prcfent tuct. W. D. 



For 



