Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...respectively. Problems were plane if they could be solved by means of the straight line and circle only, solid if they could be solved by means of one or more conic sections, and linear if their solution required the use of other curves still more complicated and difficult to construct, such as spirals, quadratrice8, cochloids (conchoids) and cissoids, or again the various curves included in the class of ' loci on surfaces' (toitoi irpbs eirifaveiais), as they were called.1 There was a corresponding distinction between loci: plane loci are straight lines or circles; solid loci are, according to the most strict classification, conies only, which arise from the sections of certain solids, namely cones; while linear loci include all higher curves.1 Another classification of loci divides them into loci on lines (roiroi irpbs ypafifiaTs) and loci on surfaces (toitoi irpbs eirif)aveiais).2 The former term is found in Proclus, and seems to be used in the sense both of loci which are lines (including of course curves) and of loci which are spaces bounded by lines; e.g. Proclus speaks of 'the whole space between the parallels' in Eucl. I. 35 as being the locus of the (equal) parallelograms 'on the same base and in the same parallels'.3 Similarly loci on surfaces in Proclus may be loci which are surfaces; but Pappus, who gives lemmas to the two books of Euclid under that title, seems to imply that they were curves drawn on surfaces, e.g. the cylindrical helix.4 It is evident that the Greek geometers came very early to the conclusion that the three problems in question were not plane, but required for their solution either higher curves than circles or constructions more mechanical in character than the mere use of the ruler and compasses in the sense of Euclid's Post...
