TELESCOPE. 



d, and alfo the angle F A C as >■ + d; therefore we have as 

 / : z :: d : d + r ; and by compoiition of ratios,/ -f 2 : 

 /:: zd + r : d ; hut f + z = r -.■ r : f :: zd + r : d; 

 then by multiplying the extremes and means together, we 

 have the equation dr = 2 r//" -f fr, and dividing by zd + r, 



there refults the theorem — ; = f — EF. This may 



zd ■\- r ■" ' 



be confidered as the fundamental theorem in catoptrics, from 

 which the focus may be determined in any fpeculum, con- 

 cave, convex, or even plain, whether the rays fall on it 

 diverging, parallel, or converging ; and from a due variation 

 of the fymbols and figns, as the cafe may require, we have 

 all the variety of tlieorems for finding the focus contained 

 in the fubjoined table. 



Table for finding the Focus of Rays refledled by any 

 Speculum. « 



To illuftrate the utility of this little table, let it be re- 

 quired to determine the refpeftive foci of two fpecula, 

 both ground and polifhed, on tools of 30 inches radius, when 

 the radiant objeft is placed at 300 feet diftance, one fpe- 

 culum being convex, and the other concave ? In the firft 

 place, as the diftance is lefs than infinite, the rays will come 

 diverging from a luminous point ; and, therefore, with 

 refpeft to the convex fpeculum, we muft ufe the theorem 



— = /, which m figures will Hand thus, — — 



zd ■{■ r ■' ■ ^ 2X 300 -I- 2.5 



E= -— — = 1.245 f^^*' °'' '4'94 inches for the required 

 602.5 



foeus ; but for the concave fpeculum, the theorem — 



zd — r 



^ /"(or — f, becaufe the focus and centre of the curve are 

 on the fame fide of the fpeculum) will give us thefe numbers, 



400 X 2.5 750 r 



-^ i — = — i- — = 1.255 leet, or 15.00 



2 X 300 - 2.5 597.5 



inches for its focal diftance ; and in hke manner may the 

 proper focus be determined for any other radius and diftance, 

 however the rays may be circumftanced when they fall on 

 the fpeculum : whenever they come converging or diverging 

 from a firft to a fecond fpeculum, the focal point, real or 

 virtual, muft be confidered as the radiant, and its diftance 

 reckoued accordingly. 



Thefe theorems, however, imply that the fpeculum is 

 already made, whereas in many praftical cafes, the focus is 

 firft afllimed, and the proper radius of convexity or con- 

 cavity is required, or, which is the fame in efieft, the radius 

 «f the tool is required that (hall be proper for forming the 

 requifite curve. For inftance, let it be required to form a 

 tool of fuch a radius, that an image of any very remote 

 objeft may be formed, by a fpeculum ground and polilhed 



to its dimcnfions, at the didance from its rcflefting forface 

 of juft 18 inches ? In this cafe, the rays muft lie confidered 

 as parallel, becaufe the objeft is remote ; and, indeed, it i( 

 always for a remote objeft that the curve of a large fpe- 

 culum is formed ; confequently the theorems — = /, and 



— r 



—— = y, or — /, will be fuitablc for the required purpofe, 



in both which 2/ = /• ; therefore 18 x 2 = 36 will be 

 the proper radius of either the convexity or concavity of 

 the tools to bo ufed. If the ray had been diverging, and 

 the fucui ajirmallvc, or behind the fpeculum, for a convex 

 fpeculum the theorem arifiiig from transformation would 



have been . = r ; and for a concave — . 



d-f f-d 



in the cafe of a negative focus, or focus before the fpeculum, 



, = r : but 



the former would have been 

 zdf 



zdf 



'i+f 



= r, and the Utter 



-—, -J. = r. In like manner, the diftance may be deter- 

 mined from the radius and proper focus being given ; for, 

 fuppofing the focus affirmative, with a convex fpeculum, the 



transformed theorem for diverging rays will be — — =: d, 



r- 2/ 



and with a concave 



fr 



z/+ r 

 negative, the former will be 



= d: but when the focus i« 



i— . = d, and the latter 



r + 2/ 



2/- 



d. Hence, when any two of the terms/, r, and 



</are given, the third may be readily determined. 



Thus, in the cafe of a convex fpeculum with diverging 



/; when 



rays, 



d = 



if we pat d 

 4- r, then 



we ftiall have . — 

 3 



and when d = 



i r, thci) 



when d 



then — 

 5 



/; 



r 



/: from which refults we 



fee, that the points D and F both approach the fpeculum 

 in a regular manner, till at laft they will coincide at iti 

 vertex ; and the fame will be the cafe with a concave fpe- 

 culum, when the rays are converging, except that the focut 

 is negative, or on the fame fide with the centre of convexity. 

 Alfo, when a convex fpeculum is ufed with converging rays, 

 or a concave with diverging, when z d := r, f will be in- 

 finite ; or, which is the fame thing, the rays will be reflefted 

 parallel, as is the cafe in reflefting lamps ; and, generally 

 fpeaking, the focus of any concave or convex IpecuTum may 

 be made to fall in a given point, accordingly as the radiant 

 objeft is made to approach to or recede from its principal 

 focus with parallel rays ; and wlierever the focus is made to 

 fall, there an image is formed of the objeft by refleftion, ii^ 

 the fame wonderful manner as we have had occafion to men- 

 tion before, when fpeaking of its formation by refraflion. 

 Likewife the connexion between D, the radiant point, and 

 F, the focus, is fo intimate, that they may at any time 

 change places without error ; that is, when D is the radiant, 

 F will be the focus ; but if F is taken as the radiant, thea 

 D will be the proper focus in all cafes. 



In order to (hew what proportion the length of an image, 



formed in the focus of reflefted rays, will bear to tlie length 



LI 2 of 



