VOL. XXIV.] PHILOSOPHICAL TRANSACTIONS. ISQ 



in the opposite sections, where they superadd any thing to what was before said 

 of one hyperbola. And because a rectilinear angle may be considered as an in- 

 finitely narrow hyperbola, viz. whose transverse axis is a point, in the 80 and 

 last proposition of this book, he determines the locus solidus, made by the or- 

 dination of the rami to this angle, from an origin in its axis, either within or 

 without the angle. To this book he subjoins an epilogue, containing some 

 general corollaries useful, as he says, toward some things which he intended to 

 publish : as, that in a circle the loci solidi, made by the ordination of the rami 

 from an origin in the vertex, or within, are parabolas, which are all similar lines ; 

 and that the loci plani, arising by the ordination of the rami from the focus of a 

 conic section or circle, are straight lines, which are also similar lines; and so 

 in other cases, that like loci arise from such applications. 



Part 5, the first two propositions determine the loci solidi, arising when the 

 tangents of the parabola, intercepted between the section, and either the axis 

 or the tangent at the principal vertex, are made ordinates to the principal axis. 

 And the next two determine the loci solidi, arising when the normals, either to 

 the section, or to the rami, proceeding from the principal vertex, are made 

 ordinates to the tangent in the said vertex. 



Liber II. In quo Loci Ordinatarum Potentium Limites indicantur. — In this 

 he treats at large, in 71 propositions, of the loci, both plane and solid, arising 

 from ordinates on a straight line, whose squares are equal to the sums or differ- 

 ences of the rectangles and squares of a line, and its segments, and other 

 assumed lines, in all the variety and combinations of them. 



Liber III. In quo Loci Variarum Dispositionum Limites assignantur. — Here is 

 the determination of the loci plani and solidi that arise from several ways differ- 

 ent from the former. For example, if from two given points there be drawn 

 several pairs of straight lines, whose squares together are equal to a given 

 square, the concourse of each pair is in the locus planus of a circle there deter- 

 mined. And, the other conditions remaining, if of each pair of straight lines, 

 one be drawn from a given point, and the other be perpendicular to a given 

 straight line, the concourse is in the locus solidus of an ellipsis, there deter- 

 mined. Afterwards are several problems concerning arithmetical, geometrical, 

 and harmonical mean proportionals between two extremes, and divers methods 

 for describing the conic sections by points. There are also subjoined several 

 addenda to all the preceding three books. 



At the end of the books are prints of the orthography and gate of a stately 

 house built by the author Vincentio Viviani at Florence, with the inscriptions 

 on its front, in honour of the French King Louis XIV. from whom he had an 



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