P O R I S M S. 



109 



3. 



(ioth proposition in Dr. Simson'a restoration, slightly 

 iilt-.-n-.l in its statement. 



A another instance of a problem leading to a porism, 

 we will give one which :ip|>r.irs to have ltd to the in- 

 \.iition of the second porism in the treatise of Sim- 

 son. 



A i ircleABC,(Fig. 2.)andtwopointsI)andE,in a dia- 

 . meter of it being given, to find a point F in the circum- 

 ference of the given circle, from which, if straight lines 

 be drawn to the given points E and D, these straight 

 lines ahull have to one another the given ratio of 

 to/3. 



Suppose the problem resolved, and that F is found, 

 so that FE has to FD the given ratio of * to /3. Pro- 

 duce KF any how to B, bisect the angle EFD by the 

 line FL, and the angle DFB by the line FM. 



Then because the angle EFD is bisected by FL, EL 

 is to LD as EF is to FD, that is in a given ratio; nnd 

 as ED is given, each of the segments EL, LD is given, 

 and also the point L. 



Again, because the angle DFB is bisected by FM, 

 E.M is to MD as EF to FD, that is in a given ratio; 

 and therefore since ED is given, EM, MD, are also 

 given, and likewise the point M. 



But because the angle LFD is half of the angle EFD, 

 and the angle DFM half of the angle DFB, the two 

 angles LFD, DFM, are equal to the half of two right 

 angles, that is to a right angle. The angle LFM being 

 therefore a right angle, and the points L and M being 

 given, the point F is in the circumference of a circle 

 described on the diameter LM, and consequently given 

 in position. 



Now, the point F is also in the circumference of 

 the given circle ABC : it is therefore in the intersection 

 of two given circumferences, and therefore is found. 



Hence the following construction : Divide ED in L 

 so that EL may be to LD in the given ratio of 

 to A, and produce ED also to M, so that EM may be 

 to MD in the same given ratio of to j3. Bisect LM 

 in N, and from the centre N, with the distance NL, 

 describe the simicircle LFM, and the point F in which 

 it intersects the circle ABC, is the point required, or 

 that from which FE and FD are to be drawn. 



It must, however, be remarked, that the construc- 

 tion fails when the circle LFM falls either wholly 

 without or wholly within the circle ABC ; so that the 

 circumferences do not intersect ; and in these cases the 

 solution is impossible. It is plain also that in another 

 case the construction will fail, namely, when it so hap- 

 pens that the circumference LFM wholly coincides 

 with the circumference ABC. In this case it is farther 

 evident, that every point in the circumference ABC 

 will answer the conditions of the problem, which 

 therefore admits of innumerable solutions. 



The indefinite case of this proposition thus enume- 

 rated becomes a porism. 



A circle, ABC, (Fig. 3.) being given, and also a 

 point D, a point E may be found, such that two lines, 

 DF and EF, inflected from these points to any point 

 F in the circumference of the circle, shall have to each 

 other a given ratio, which ratio may be found. 



From these examples, the definitions which have 

 been given of the term porism will be better under- 

 stood than by merely considering the words in which 

 they are expressed. Dr. Simson has thus described 

 them: " Porisma est propositio in qua proponitur 

 demonstrare rem aliquam, vel plures datas esse, cui, 

 vel quibus, ut et cuilibet ex rebus innumeris, non 

 quideua datis, sed qua ad ea quit- data sunt eundem ha* 



bent relationem, convenire oUemh-ndum et affectionem l*ii. 

 quandnm com mu item in propositione descriptam." ^~~ ~*~ 



The obscurity of this definition in such, that nothing 

 but a companion with an example can make it int< 

 gible ; that of Playfair is much happier, and is thu 

 expressed : A p'.ritm it a proposition, affirming the pot. 

 tiltilittj of finding suck conditions at will render a certain 

 problem indeterminate. 



This latter has the advantage of indicating the course 

 to be pursued in the discovery of porism*; for in the 

 first case the problem was rendered indeterminate by 

 making two out of the three points, which determined 

 the position of a circle, coincide ; and in the last ex- 

 ample, the coincident e of two circles, whose intersec- 

 tions should have determined the point required in the 

 problem, rendered it indeterminate. This mode of 

 analysis,, for the discovery of poricm*, has one disad- 

 vantage, that it supposes the solution of the problem 

 to be first found ; that which was contrived by Simeon 

 is free from this objection, and when abridged by the 

 considerations which Playfair has introduced, is ad- 

 mirably adapted to its object. 



It may be observed, that the points or magnitude* 

 required may generally be discovered by considering 

 the extreme cases ; but that the relation between these 

 and the indefinite magnitudes cannot be arrived at by 

 such limited considerations. The difference between 

 a locus, a local theorem, and a porism, are well illus- 

 trated by Playfair in the various modes of enunciating 

 the truth discovered in the second of the two proposi- 

 tions we have given. 



Thus, when we say, if from two points, E and D PLATE 

 (Fig. 3.) two lines, EF, FD, are inflected to a third =CCLXVH. 

 point F, so as to be to one or other in a given ratio, the *' 

 point F is in the circumference of a circle given in po- 

 sition : we have a locus. But when conversely, it is said, 

 if a circle ABC, of which the centre is O, be given in 

 position, as also a point E, and if D be taken in the 

 line EO, so that the rectangle, EO, OD, be equal to 

 the square of AO, the semidiameter of the circle ; and if 

 from E and D, the lines EF and DF be inflected to 

 any point whatever in the circumference ABC; the 

 ratio of EF to DF will be a given ratio, and the same 

 with that of EA to AD : we have a local theorem. 



And, lastly, when it is said, if a circle ABC be given 

 in position, and also a point E, a point D may be found, 

 such that if two lines EF and FD be inflected from 

 E and D to any point whatever F, in the circum- 

 ference, these lines shall have a given ratio to one 

 another : the proposition becomes a porism, and is the 

 same we have just investigated. 



The algebraical method for the investigation of 

 porisms, may very readily be deduced from the consi- 

 deration of the definition which Playfair has given, 

 and the facilities which such a method presents in the 

 discovery of this class of truths, is another instance of 

 the advantages which result from a condensed method 

 of expressing the relations of quantity. It has been 

 stated, that a porism is a proposition affirming the pos- 

 sibility of finding the indeterminate case <>J a problem. 



If, therefore, any problem is proposed in which the 

 quantity sought is called x ; by means of the given 

 conditions some equation will be found between x and 

 known quantities, which may be reduced to the form 



A + Bj + C** -f ... +N*" = (a) 



A, B, . . . being known functions of the constant 

 quantities ; from this equation x may be determined, 

 or at least it cannot generally have more than a certain 



