342 PHILOSOPHICAL TRANSACTIONS. [aNNO 1707 , 



N, at a sufficient distance from the point of suspension f; or it maybe marked 

 by a small knot at n. Then taking at pleasure no for unity, by the middle 

 point A, in the said plane, draw the right line Aa parallel to the horizon, and 

 produced both ways as far as necessary. These general preparations being made, 

 then for the application to any particular case, make agl = r; the quantities 

 q, r, s, ty being first determined in the last equation above, either arithmetically 

 or geometrically, as the exigence of the proposed equation may require. Then 

 with a shar[)e style or bodkin, or with the fine point of a pair of compasses, let 

 the thread be inflected and moved from its place to such a point b as that the 

 point N may fall on the point a just fi)und. In sa take br = *, and at r raise 

 the perpendicular rb = q. But those lines aq, br, re, must be taken the con- 

 trary way when the values of q, r, s are negative. Lastly, fix one leg of the 

 compasses in the point e, and let the other leg, extended to the distance 

 Ez = t, be carried round with a circular motion, taking with it the thread 

 pzp. By this circulation of the thread the weight p will sometimes ascend 

 and sometimes descend with a reciprocal motion, and the knot n will be some- 

 times above and sometimes below the horizontal line aq. But whenever the 

 knot N shall be found in the line aq, as in the points d, d, A, «?, it will cut off 

 the right lines Da, dot, Aa, Ja, which will be all the real roots of the given 

 equation, viz. those on the right hand the affirmative roots, and those on the 

 left the negative ones. The demonstration of which is manifest from what is 

 before done, and by attending to the parabola passing through the points b, c, 

 c, k, K. For making f the focus of the parabola, whose distance from the 

 vertex is -Jon, it is known that all the lines, as fb -f- sa, fc -|- cd, &c. always 

 make up the same sum. 



And from the principles here laid down, it will not be difficult to construct a 

 sufficiently neat and accurate instrument, by means of which the roots of all 

 such equations may be easily found, and exhibited to the eye. 



^he Analytical Solution of any Equations, of the Third, Fifth, Seventh, Ninth, 



and the other higher uneven Powers, by Rules similar to those called Cardans. 



By Mr. Abr. Demoivre, F. R. S. N° 309, P- 2368. Translated from 



the Latin. 



If n denote any number whatever, y an unknown quantity, and a the abso- 

 lute known quantity, or what is called the homogeneum comparationis : also 

 let the relation between these be expressed by this equation. 



