ioo Mr. Woodhouse on the Independence of the 
< A , a fortiori, ^ +1 is < A n+l , and so on; the terms after the 
n — ith term constantly diminishing.* 
The above method is purely analytical : it has no tacit 
reference to other methods ; it does not virtually suppose the 
existence either of an hyperbola or circle. If practical commodi- 
ousness, however, be aimed at, it is convenient to give a different 
expression to the values of the roots, or to translate them into 
geometrical language : and this, because tables have been calcu- 
lated, exhibiting the numerical values of the cosines, &c. of 
circular arcs. Now, since it has already appeared that the cosine 
of an arc z=2~ 1 i -j- the 3 roots of the equation 
x 3 — • qx = r may be said to equal 
XII, In the fifth volume of his Opuscules,' f D’Alembert 
* In the Phil. Trans, for xSoi. p. 116, I mentioned M. Nicole as the first ma- 
thematician who shewed the expression of the root in the irreducible case, when 
expanded, to be reah But the subjoined passage, in Leibnitz’s Letter to Wallis, 
causes me to retract my assertion. “ Diu est quod ipse quoque judicavi — i 
f( ^a-yb\J \ ~ z esse quantitatem realem, etsi speciem habeat imaginarise ; 
“ ob virtualem nimirum imaginariee destructionem, perinde ac in destruction actuaii 
V_i qa-JV'-i -2a. Hinc, si ex vA<± bv ' — 1 extrahamus radicem 
*« ope seriei infinitae, ad inveniendum valorem ipsius z serie tali expressum, efficere 
possumus, ut reapse evanescat imaginaria quantitas. Atque ita etiam, in casu ima- 
ginario, regulis Cardanicis cum fructu utimur,” &c. Vol. III. p. 126. See also p. 54. 
f “ Elle etoit neanmoins d’autant plus essentielle, que l’expression de l’arc par 3 e 
dx , 
" sinus, fondee sur la serie connue, qui est l’integrale de — — , ne peut etre regardee 
V 1- — x z 
if comme exacte, c’est a dire, comme representant a la tois tous les arcs qui ont le 
it meme sinus ; puisque cette serie ne represente evidemment qu’un seul des arcs qui 
repondent au sinus dont il s’agit, savoir, le plus petit de ces arcs, celui qui est infe- 
“ rieur, ou tout au plus egal, a 90 degres. Cependant, c’est d’un autre cote une sorte 
«< de paradoxe remarquable, que 1’expression de l’arc par le sinus ne representant qu’un 
