138 PHILOSOPHICAL TRANSACTIONS. [aNNO 1704. 



books De Locis Solidis^ which are these that Vincentio Viviani pretends to re- 

 store. Pappus also seems to say that he wrote a history of what had been done 

 in geometry until his time. And Campanus, in an annotation on Prop. 1 . 

 Lib. XIV. of the Elements, mentions a book of Aristaeus, entitled, Expositio 

 Scientiae quinque Corporum, from whence it may be inferred that he was of 

 the Platonic sect. 



Though the authour intended five books, as Aristaeus had written ; yet he 

 has published only three, and seems to despair of ever publishing the other 

 two. 



Liber I. In quo de Locis Ordinationum Conicarum Limitibus pertractatur. — 

 This book is divided into five parts. Part I, are 34 lemmatical propositions, 

 with considerable improvements in demonstrating the properties of the conic 

 sections from the regulatrix ; moduli ex semirecto, ex verticali, ex laterali ; and 

 in the hyperbola, from the modulus ex asymptoto triangulum circumactum a 

 symptotale, &c. all which he there defines. 



Part 2, prop. 35, shows that the altitudines normalium (or the subnormals) in 

 all the conic sections, erected from the points of the axis, where the ordinates 

 are erected, are ad locum planum ; and prop. p. 36 and 37, that the normals to 

 a right line and a circle erected as above (which is always understood) are ad 

 locum planum; but in the 38 to the 42, that the normals of the conic sections 

 are ad locos solidos, which he there determines. Part 3, prop. 43, shows that 

 in all the conic sections and the circle, the altitudines normalium super ramos 

 ex vertice are ad locum planum; but from thence to the 49, that the normales 

 super ramos ex vertice are ad locos solidos, which he there determines. 



Part 4, in the first three propositions, from the 50 to the 52 inclusive, he 

 determines the locus solidus of the rami from the vertex of a circle, or from an 

 origin between the vertex and the centre, or without the circle. Prop 53, he 

 shows that the rami from the focus of any conic section, erected to the axis, are 

 ad locum planum of a right line, there determined. In the following prop, 

 to the 58, he determines the loci solidi, made by the ordination of the rami 

 of a parabola, drawn from the principal vertex, and from an origin in the 

 axis between the vertex and the focus, and below the focus, and above or with- 

 out the vertex. In the next four, to the 62, he determines the loci solidi made 

 by the ordination of the rami drawn from the origin in the lesser axis of an 

 ellipsis; viz. either the vertex, the centre, between the vertex and centre, or 

 without the vertex. From the 63 to the 68, he determines the loci solidi made 

 by the like ordination of the rami on the greater axis of an ellipsis. From the 

 69 to the 77 the like is done in regard to the hyperbola, where there occurs a 

 greater variety, as it is here managed. The next two propositions are the like 



