VOL. IV.] 



PHILOSOPHICAL TRANSACTIONS. 



33g 



progression, the resolvends as to their first, second or third, &c. differences 

 imitate the laws of the pure powers of an arithmetical progression, of the same 

 degree as the highest power or first term of the equation. Ex. Gra. In this 

 equation, aaa — 3 aa-\- A a =i N, 



N. 



If a be = 



10 -J Then iV,orthe 

 9 / absolute num- 

 8 > bers, or resol- 

 7 I vends, will be 

 6 J found to be 



3 dif. 



To wit, the 3d differences of those absolutes are equal, as in the cubes of an 

 arithmetical progression. 



To find what relation those differences have to the coefficients of the equa- 

 tion, it is best to begin from an unit. Now in any arithmetical progression if 

 you multiply numbers by pairs you shall create a rank of numbers whose 2d 

 differences are equal ; and if by ternaries, then the 3d differences of those pro- 

 ducts shall be equal. And how to find the greatest product of an arithmetical 

 progression of any number of terms, having any common difference assigned, 

 contained in any number proposed, is shown by Pascal in his Triangle Arithme- 

 tique, where he applies it to the extraction of the roots of simple powers. It is 

 manifest how this rank may be easily carried on by addition, till you have a 

 resolvend either equal or greater or less than that proposed. 



When you have a greater and less, you may interpolate as many more terms 

 in the arithmetical progression as you please, viz. Subdivide the common differ- 

 ence and render it less ; then renew and find the resolvends, which are easily 

 obtained by the powers and their coefficients, which are supposed known, and 

 may be readily raised from a table of squares and cubes, &c. By this means you 

 may obtain divers figures of the root ; and then the general method of Vieta 

 and Harriot proceeds more easily ; and after any figure is placed in the root by 



Mr. CoUins's birth and early prospects, like those of many other great men, were but low and 

 humble. He was born at Wood Eaton near Oxford in l624, and at l6 years of age was put ap- 

 prentice to a bookseller in this city j but appearing to have a remarkable turn for the mechanical 

 and mathematical sciences, he was taken under the protection of a Mr. Marr, a person who drew 

 several curious dials, which were placed in different positions in the king's gardens ; and under him 

 Mr. Collins made no small progress in matliematics. In the course of the civil wars he went to sea 

 for seven years, but still prosecuted his favourite study j and on his return he assumed the profession 

 of an accountant and civil engineer, giving his advice and directions in nice and critical cases, re- 

 lating to matters of commerce, of accounts, and of engineering, till the time of hi;, death, wbicb 

 happened in the year 1^83, in the 59th year of his age, 



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