jlli 5 a' rj X 1 12 cos. t*\ =s 0; 

 and the second, 



Z;> nt_r*l X jl 2 cos. o j =0; 



to the sums of the iquarex i>f the perpendiculars drawn 

 mes of the other, chord to the. given 



It is proper to observe, that in both these cases t be- 

 lower sign must be employed, otherwise the point 

 found will not be within the circle. In retranslating 

 algebra into geometry, it frequently happens that only 



ne one of the roots which satisfy the conditions of 

 the problem algebraically, will fulfil the geometrical 

 conditions. 



In both examples which have been given, the equa- 

 tion determining the value of the unknown quantity, 

 was multiplied by n factor independent of it ; thus the 

 first equation may be put under the form 



and as these equations contain all the conditions of the 

 respective problems, provided they are verified, solu- 

 tions will be found. If the relation amongst the 

 quantities a, v, r is such that the first factor vanishes, it 

 is evident that the equations are, in all cases, verified, 

 without assigning any particular values to cos. 6. 



This evanescence of a factor is not, however, the 

 only cause which produces porismatic cases, as will 

 appear in several of the subsequent examples. 



The following problem leads to a porism, which is 

 already well known by the writings of Simson and 

 Playfair. To show how it might have been discovered 

 by the algebraic method, will not therefore be without 

 interest. 



A circle and a straight line being given, and also a 

 point in that diameter of the circle which is perpen- 

 dicular to the given line, it is required to find a point 

 in the given line, such that if a line be drawn through 

 it, and the given point cutting the circle, then the rec- 

 tangle under the segments of that line contained be- 

 tween the points found and the circle, shall be a given 

 multiple of the square of the line joining the two 

 points. 



Employing the same notation and letters as in the 

 figure 4, let G be the point required ; then G will be 

 determined by the angle t, which the line CG makes 

 with the diameter ; and the values of the several lines 

 will be as follows : 



PG = U v cos. e 



COS. 6 



dbVT* 



sn. 



QG =r -f. v cos. / = v r 2 v sin. 



cos. t 



CG = 



a v 



CO.-. f 



Hence the equation expressing the condition is 



U cos. t ) 2 _,* + B . 8in .-,, =n MTLE.V 

 os. i ) \ C08> , ; 



Or, 



where n is the given multiple. From this equation the 

 angle t may easily be found, and consequently the po- 

 sition of the point G may be determined. But if = 1, 



Ill 



the angle altogether disappears from the formula, and ''^ 

 it can only be satisfied by supposing 



2a t>* r* = 



or, t; = a =r V'a* / ' 



1 1' this relation take place amongst the data, any va- 

 lue of t will fulfil the condition ; but if it does not, 

 value can satisfy it. 



Observing that the rectangle under the segments of 

 the line is equal to the square of the tangent to the 

 circle, we have the following porism. 



A circle and a right line GE (Fig. 5.) being given, a I'LATE 

 point C may be found tvilhiu the circle, such that, if from < -' i.xvn. 

 any point G in the given line a tangent GP be drann to v> f- 5 - 

 the circle, and also if that point G and the point found 

 C be joined, the line CG joining these points shall be 

 equal to the tangent to the circle GP. 



This porism is similar to propositions 63 and 66 oi* 

 Simson's restoration, and is one of the illustrations 

 made use of in the paper of Playfair. It would be easy 

 to investigate, by this method, many other of the por- 

 isms of Euclid ; but since it is proposed as a method 

 of invention, it will be more satisfactory to employ it 

 in the discovery of new ones. We shall therefore pro- 

 ceed to investigate a few others. 



A circle being given, and also a line, and a point si- 

 tuated in the line drawn from the centre, perpendicular 

 to the given line, and the quantities a, v, r, &c. remain- 

 ing the same as in the previous questions, if a chord 

 is drawn through the given point, and from its extre- 

 mities perpendiculars are drawn to the given line, we 

 shall have (in Fig. 4.) 



PL=a 



cos. 



=f=. cos. I Vr 1 v 1 cos. f 



QM=a v+v cos. f =J= cos. VV 1> 



cos. 



and the sum of these two perpendiculars, or 

 PL + QM= 2 (a v 4. v cos. **) 



which is independent of the value of r t or of the mag- 

 nitude of the radius of the circle. 



This circumstance allows us to enunciate this truth 

 as a very simple porism. 



Two lines making a given angle with each other GQ Fif . 6 

 and GM (Fig. 6.) being given, a point O may be found 

 in one of them, such that tf about that point as a centre, a 

 circle with any radius be described cutting one of the 

 given line? in two points P and Q, tlte sum of t lie perpen- 

 diculars drawn from these points to the other line shall be 

 equal to a given line. 



If FS perpendicular to GM be made equal to half 

 the line to which the sum of the two perpendiculars is 

 equal, and if SO be drawn perpendicular to GQ, any 

 point O may be taken as the centre of the circles. 



In Fig. ?. the same notation being preserved, the 

 sum of the perpendiculars is 



PL-f QM=2 (a v+v cos. *) 



Let some other point C, be taken, OC,=T/, and a 

 chord be drawn parallel to the former, then the sum of 

 the perpendiculars drawn from the extremities of thi* 



chord is 



P,L,+Q,M, = 2 (a v'+v' cos. /*) 



Let us now determine the value of cos. t t so that tht- 

 former sum shall differ from n times the latter by a con- 

 stant quantity 2 c, the resulting equation is 



