C 3% 3 
XVI. On the Roots of Equations. By James Wood, B.D. Fellow 
of St. Johns College , Cambridge. Communicated by the Rev. 
Nevil Maskelyne, D. D. F. R. S. and Astronomer Royal . 
Read May 17, 1798. 
pp 
1 he great Improvements in algebra, which modern writers 
have made, are chiefly to be ascribed to Vieta’s discovery, that 
“ every equation may have as many roots as it has dimensions/' 
This principle was at first considered as extending only to po- 
sitive roots; and even when it was found that the number might* 
in some cases, be made up by negative values of the unknown 
quantity, these were rejected as useless. It could not, however, 
long escape the penetration of the early writers on this subject, 
that in many equations, neither positive nor negative values 
could be discovered, which, when substituted for the unknown 
quantity, would cause the whole to vanish, or answer the con- 
dition of the question. In such cases, the roots were said to 
be impossible, without much attention to their nature, or inquiry 
whether they admit of any algebraical representation or not. 
As far as the actual solution of equations was carried, viz. in 
cubics and biquadratics, the imaginary roots were found to be 
of this form, <*+ V- F ; and subsequent writers, from this 
imperfect induction, concluded in general, that every equation 
has as many roots, of the form a =±= s/ ±6", as it has dimen- 
sions. In the present state of the science, this proposition is 
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