4a PHILOSOPHICAL TRANSACTIONS. [anNO 1684. 



A Cardanic equation, that is a cubic one wanting the second term, may be 

 multiplied or divided by a rank of continual proportionals, so as to render the 

 coefficient of the roots canonic, that is, to make it the same with the equations 

 of the table ; then find the sine, tangent, or secant of the third part of that 

 arch to which any sine, tangent, or secant is propounded, and so finding the 

 roots in the table, those sought are thence obtained by multiplication or divi- 

 sion. Also, the coefficient of the roots may in like manner be rendered an 

 unit, and then the resolvends sought in a table of the sums or differences of 

 the cubes of numbers and their roots, shall help to such roots, as multiplied 

 or divided as aforesaid shall be the true roots sought. 



Where equations have all their terms affected with the same sign, both + 

 or — , Mr. Newton, and Mr. Gregory deceased, have affirmed they are all 

 reduceable to some pure high power, which is of singular use in the infinite 

 series. And where this cannot be done, a learned German has asserted, that 

 they may be reduced to a higher power with a variable coefficient, which is the 

 root sought with a common addend or subducend. And even this would render 

 an easy tentative logarithmetical way for attaining the root. 



Lastly, as to constructions for equations, the following problem seems to be 

 universal. Any two analytic curves, viz. such as have the relation between the 

 base and ordinate expressed by an equation, being given in magnitude and po- 

 sition, and from the points of their intersection ordinates let fall to the axis of 

 either figure, or upon parallels to the said axis, the enquiry is of what equation 

 those ordinates are the roots ? Dr. Barrow liked the proposition as well grounded, 

 and left a discourse about doing it in the conic sections, in which there are 3 

 cases, either the axes are parallel ; or being produced concur, beyond the ver- 

 texes of the figures, without ; or otherwise intersect within the figures. Mr. 

 Gregory entered on the same contemplation, but death deprived us of the 

 benefit of his thoughts. 



Of analytic, or geometric curves, there are innumerable sorts, of which I 

 shall mention one or two kinds. Between an arithmetical progression, and 

 its squares, or between its squares and its cubes, or its cubes and biquadrats, 

 there may be interposed as many arithmetical or geometrical means as you 

 please : and thence loci or curves derived ; which some call paraboloids or para- 

 bolasters; see Gregory's Geometries pars universalis. 



On the Bridge of St. Esprit, in France. By Mr. Tancred Robinson. 



N" 160, p. 584. 



This famous Roman bridge is very crooked, bending in several places, and 

 forming many unequal angles, and most so in those places where the stream 



