constituted a certain rule, from the supposition of which of the unknown quantities the analysis may best begin; nor out of the variety of equations, that at first appear, which we were best to choose; but the success will depend upon dexterity, upon formerly acquired science, and many times upon fortune. For no man can ever be a good analyst without being first a good geometrician; nor do the rules of analysis make a geometrician, as synthesis doth; which begins at the very elements, and proceeds by a logical use of the same. For the true teaching of geometry is by synthesis, according to Euclid's method; and he that hath Euclid for his master, may be a geometrician without Vieta, though Vieta was a most admirable geometrician; but he that has Vieta for his master, not so, without Euclid. And as for that part of analysis which works by the powers, though it be esteemed by some geometricians, not the chiefest, to be the best way of solving all problems, yet it is a thing of no great extent; it being all contained in the doctrine of rectangles, and rectangled solids. So that although they come to an equation which determines the quantity sought, yet they cannot sometimes by art exhibit that quantity in a plane, but in some conic section; that is, as geometricians say, not geometrically, but mechanically. Now such problems as these, they call _solid_; and when they cannot exhibit the quantity sought for with the help of a conic section, they call it a _lineary_ problem. And therefore in the quantities of angles, and of the arches of circles, there is no use at all of the analytics which proceed by the powers; so that the ancients pronounced it impossible to exhibit in a plane the division of angles, except bisection, and the bisection of the bisected parts, otherwise than mechanically. For Pappus, (before the 31st proposition of his fourth book) distinguishing and defining the several kinds of problems, says that "some are _plane_, others _solid_, and others _lineary_. Those, therefore, which may be solved by strait lines and the circumferences of circles, (that is, which may be described with the rule and compass, without any other instrument), are fitly called _plane_; for the lines, by which such problems are found out, have their generation in a plane. But those which are solved by the using of some one or more conic sections in their construction, are called _solid_, because their construction cannot be made without using the superficies of solid figures, namely, of cones. There remains the third kind, which is called _lineary_, because other lines besides those already mentioned are made use of in their construction, &c." And a little after he says, "of this kind are the _spiral_ lines, the _quadratrices_, the _conchoeides_, and the _cissoeides_, And geometricians think it no small fault, when for the finding out of a plane problem any man makes use of conics, or new lines." Now he ranks the trisection of an angle among