100 PHILOSOPHICAL TRANSACTIONS. [aNNO 1673. 



After some other communications with these two learned men, Huygens and 

 Slusius, in which the former acquiesces in the solution of the latter, but yet 

 thinks his own the more natural, Slusius sends another solution of the problem, 

 as follows : 



Let there be given a circle whose centre is A (fig. 7), and let the given points 

 be D and d. Suppose the thing done; and let DE be the incident, and Ed 

 the reflected ray; and from the point of reflexion E let fall on DA the perpen- 

 dicular EI, and from d on the same line the perpendicular dN, also let the 

 tangent EC and ray dE produced meet the same line in C and B. Now put 

 DA = z, AI = a, NA = n, EI = e, dN = Z?, BA = y, AE = q, CA = cc. 

 Then, since the angle DEC = CEB, and CEA a right angle by the hypo- 

 thesis, the three lines DA, CA, B A are harmonical proportionals, as is easily 

 shown. Therefore DA will be to B A, as DC to CB ; or, in algebraic terms, 



z'.y '.'.z — oc '. X — y \ hence Izy — xy ■=. zx, or ^ = x. And since the 

 rectangle CAI, or xa is equal to AE^ or qq, it will be a: = — and conse- 

 Guentlv — ^ = ^, or , ^'^^ = y. Again, it is, as f/N to EI, so is NB to 

 IB ; or as h \ e:'.y '— n '. y — a-, therefore ye — ne r=z. hy — ha, and y = 



ha we ^ ,1 2'70' ha — ne „ , ^ , , 



_- — - — . Consequently - — -- — = —, , or Izbaa — Iznae — qqba + 



b — e ^ 2za — qq b — e -' ^ ' 



qqjie = bzqq — "Z-qqe, which is the equation to an hyperbola about its asymp- 

 totes, the construction of which, with the given circle, answers the problem. 

 For since, from the nature of the circle, it is qq := aa -{• ee, if instead of 

 ibzaahe substituted its value 2bzqq — 2b zee, there will result another similar 

 equation to the hyperbola about its asymptotes, viz. bzqq — 2b zee — 2znae — 

 qqba-\- qqne z= — zqqe. And by this method, as also by that which is explain- 

 ed in my book on analysis, there will arise infinite equations to hyperbolas and 

 ellipses, which with the given circle will solve the problem ; only the efFections 

 will become so much more intricate, that it may not be worth while to try 

 them : but they may be constructed by that method used for the ellipsis at page 

 62 of the same book. 



The whole of the calculation, as you see, is referred to the line DA ; but you 

 will perceive that it might be as easily adapted to the line dA, which is likewise 

 given, viz. by drawing the dotted lines in the scheme. But there is no occasion 

 for the trouble of a new calculation. P'or if to the line d A, and its parts, there 

 be adapted the same symbols, viz. making d A = z, Dn = Z?, n A = tz, Ai = 

 a, i E = e, &c. ; there will then arise the same equation as before ; and you will 

 obtain infinite other hyperbolas and ellipses, which with the given circle will 

 satisfy the problem. It would be useless to prosecute the several cases, since 



