iSi Mr. P. Nicholson on derivative Analysis. 



integers not exceeding nine is the next figure of the root, that 

 is 80-=-52 gives 1 for the next figure of the root. 



We must now transform the function 3y^ + 5-2T)— '08 = 

 into 3(t;— 0-01)- + B,(t'— 0-01) — 0-08 by the same process; 

 therefore 



5-20 — -OSOO 



y x-01=-03 

 •03 



5-23 X -01 ='0523 | —0277 

 5-26 



Again : 2770-T-526 gives 5 for tlie next figure of the root, 

 therefore 



5-260 —-02 7700 



3x-005 = -015 I 5-275 X -00.5 = -026375 | —•001325 

 •015 I 5-290 

 Again: divide, 13250 by 5290 gives 2 for the next figure 

 of the root, therefore 



5-2900 — -001325000 



5^91^6 X -0002 = -00 1058 12 ] --00026688 

 5-2912 



3 X -0002 = -0006 

 -0006 

 and so on. 



The method which is here investigated and exemphfied in 

 various appUcations, is not only more general, more con- 

 venient, more obvious, but also less laborious, than any other 

 yet invented; it groups all the elementary branches of algebra 

 in one geueral formula. 



Schollnm. 



The world has been much indebted to Mr. Holdred, as be- 

 ino- not only the first who invented a general method of ex- 

 tractin^i- the roots of equations of all orders; but as being- also 

 the first to discover the best mode of abridging the labour, 

 when a certain or given number of figures was to be found in 

 the root. 



In confirmation of this assertion, I have to state, that so early 

 as the year 1810, which was nine years before the publication of 

 anyplan to accomplish the same object, Mr.Holdred submitted 

 his method to me ; but my engagements at that time prevented 

 me from entering into the subject. I have never made any 

 pretensions to the discovery of Mr. Holdred, or of any other 

 individual; but I solemnly affirm that upon my seeing his 

 fio-urate method, I discovered the non-figurate mode from a 

 consideration of my general method of transforming functions, 

 published in my Combinatorial Analysis in the year 1818, 

 whicli was just before Mr. Holdred had communicated to me 

 his method of extracting the roots of equations. 



Upon 



