VOL. XXIV.] PHILOSOPHICAL TRANSACTIONS. 1§5 



I shall here attempt the same thing on other principles, as far as catoptrics are 

 concerned. 



Let DEF be the portion of a concave spherical speculum, fig. 3, pi. 7, its 

 centre being b, and radius be or bd : also let a be a radiating point placed in the 

 axis, from whence proceeds the ray ad, which at the point d is reflected into 

 DC. Now the distance of the focus c, from the vertex of the speculum e, is 

 to be investigated. 



It is to be. noted, that the point d is supposed to be very near to e. For the 

 remoter rays go beside the eye, placed in the axis ae ; and so contribute nothing 

 to vision. And because the arc de is indefinitely small, the angles dab, adb, 

 as also their sum dec, are very small, and therefore they will have to each other 

 the same ratio as their opposite sides. By reasoning on this principle it was 

 that Dr. Halley, professor of geometry at Oxford, arrived at his dioptrical 

 theorem. 



These being premised, put ab = Z», bd = be = r, bc = Zj ce = r — z 

 •=^f, suppose, for brevity sake. Then h and r are known quantities, because 

 the radius of the speculum and the distance of the lucid point from the vertex 

 are given ; but z and f are unknown and required quantities. Now in the 

 triangle dab, it will be, as the angle dab : the angle adb :: r\h\ and in the 

 triangle dbc, the angle bdc = the angle adb from the nature of reflection ; 

 also the angle dbc ^ dab -+- adb by Euclid's Elem. Therefore since the 

 angle dbc is as r ^ i, and the angle bdc as Z> ; it will be also, as the angle 

 DBC : bdc :: r + 6 : ^ ; and from the principle above-mentioned, as dc : bc : : 

 r -\- b \ h. But because the point d is very near the point e, dc may be ac- 

 counted equal to CE ; therefore it will be ce : bc :: r -|- ^ • ^> that is, y : z :: 

 r '\- h '.h, and hence, by compounding, / + z : y :: r -f- 2Z> : r + Z>; but/ + 



z = r, therefore r \ f v. r •\- lh : r -\- h, and hencey= ^^ — -£. a. e. i. 



If we put r -f" ^ = ae = rf, the theorem will contract to this form, f = 



rrf 



-J——. But in either way the theorem will serve for finding the focus, what-^ 



ever be the form of the speculum, or the condition of the rays. 



Carol, 1 . — It will be c?z = rf/* — r/i by substituting z for r — yi or ae X 

 bc = ab X CE, or, which is the same thing, the line ae is harmonically 

 divided in the points a, b, c, e : for that equality of rectangles is the property 

 of a line so divided. So that in every spherical speculum, the lines da, db, 

 DC, DE are harmonicals ; and the radiating point, the centre, the focus, and 

 the vertex, are points that constitute an harmonical division. 



VOL. ¥i> B B 



