Again, because the angle DFB is bisected by FM, EM is to MD as EF to FD, that is in a given ratio; and there fore since Ell is given, EM, MD, arc also given, and like wise 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 Li\I, 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 Ell in L so that EL may be to LD in the given ratio of a to 13, and pro duce ED also to M,so that EM may be to Ml) in the same given ratio of a to [3. Bisect LM in N, and from the cen tre N, with the distance NL, describe the semicircle LFM, and the point F in which it intersects the circle ABC, is the point required, or that from which FE and PD are to be drawn.
It must, however, be remarked, that the construction fails when the cirle 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 impos sible. It is };lain also that in another case the construction will fail, rimelv, when it so happens that the circumference LFID wholly coincides with the circumference ABC. In this case it is farther evident, that every point in the cir cumference ABC will answer the conditions of the pro blem, which therefore admits of innumerable solutions.
The indefinite case of this proposition thus enumerated 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 circum ference of the circle, shall have to each other a given ratio, which ratio may be found.
From these examples, the definitions w hich have been given of the term porism will be better understood than by merely considering the words in which they are expressed. Dr. Simson has thus described them: " Porisma cst pro positio in qua proponitur demonstrare rem aliquam, vel plures Batas essc, cui, ref quibus, ut et cuilibet ex rebus innumeris, non quidem dabs, sect qua: ad ea qure data stint eundcm habent relationcm, convenire ostendendum est affectionetn quandam communem in propositions dcscrip tam."
The obscurity of this definition is such, that nothing but a comparison with an example can make it intelligible; that of Playfair is much happier, and is thus expressed : fiorism is a proposition, affirming the possibility of finding such conditions as will render a certain problem indetermi nate.
This latter has the advantage of indicating the course to be pursued in the discovery of porisms ; 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 example, the coinci dence of two circles, whose intersections should have deter mined the point required in the problem, rendered it inde terminate. This mode of analysis, for the discovery of porisms, has one disadvantage, that it supposes the solution of the problem to be first found ; that which was contrived by Simson is free from this objection, and when abridged by the considerations which Mayfair has introduced, is admirably adapted to its object.
It may be observed, that the points or magnitudes re quired 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 illustrated by Playfair in the various modes of enunciating the truth discovered in the second of the two propositions we have given.
Thus, when we say, if from two points, E and D (Plate CCCCLXVII. Fig. 3.) two lines, El', FD, are inflected to a third 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 position: we have a locus. But when conversely, it is said, if a circle ABC, of which the centre is 0, 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 .A0, 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 El: to DF will be a given ratio, and the same with that of EA to AI): we have a local theorem.