By Percival Frost Joseph Wolstenholme

The Authors of the next Treatise have endeavoured to give earlier than scholars as accomplished a view of the topic as yes boundaries have allowed them to do. the need of those barriers has built itself during getting ready the paintings in the course of a interval of 4 years. The research of innumerable papers, by means of the main celebrated mathematicians of all international locations, has confident the authors that the topic is sort of inexhaustible, and that, to have handled all elements of it with something forthcoming to the fulness with which they've got taken care of the 1st component, might have swelled their paintings in a frightened share to what it has already attained.

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5—9 of Archimedes’ treatise On Spirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K. It is then possible, says Archimedes, to draw a straight line KHF, meeting the circumference in H and the tangent in F, such that FH: HK < arc BH) : c, where c is the circumference of any circle. To prove this he assumes the following construction. ” Archimedes must of course have known how to effect this construction, which requires conies. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in H and KG in G, it is obvious that as G moves away from C, HG becomes greater and greater and may be made as great as we please.

G. ) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was “to prove that it is possible to find a point with such and such a property” or “a straight line on which lie all the points satisfying given conditions” etc. ” From the above it is easy to understand Pappus’ statement that loci constitute a large class of porisms. ” A difficult point, however, arises on the passage of Pappus, which says that a porism is “that which, in respect of its hypothesis, falls short of a locus-theorem” (τοπικο θεωρήματος).

Archimedes must of course have known how to effect this construction, which requires conies. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in H and KG in G, it is obvious that as G moves away from C, HG becomes greater and greater and may be made as great as we please. The “later writers” would no doubt have contented themselves with this consideration without actually constructing HG. 2 As Heiberg says, this translation is made certain by a preceding passage of Pappus (p.