TELESCOPE. 



of the objeft which it repref«its, let DE (fig. 9.) be a 

 portion of a convex fpeculura, C its sentre, V the vertex, 

 O B an objeft, and I its image ; and let ft be required to 

 find tlie proportion that the objeft, or line B O, bears to 

 the image, or line 1 M. From the centre C, let the per- 

 pendicukr C A fall on the objeft or radiant ; and from its 

 extreme points O, B, draw O C and B C, to meet the fpe- 

 culum in the jtoints D and E ; and do the fame in Jii;. 10, 

 where the curve D E reprofents a concave fpeculum : then 

 the line A V will be the axis, in fome part of which the 

 rays proceedine from the points O and B will meet, and the 

 points of intericftion will be the foci refpeftively. From O, 

 let a ray O V pafs to the vertex of the fpeculum, fo as to 

 make the /_ F V A = /I O V A, then will V F be the rc- 

 flefted ray, which tending to the point I, in the axis CO, 

 (hall there form the image of the point O of the radiant. 

 In Lke nranncr, the ray B V will be reflefted in the direc- 

 tion V G, and interfefting the axis C B in M, will there 

 depift the point B of the faid radiant ; and thus all the in- 

 termediate points lying between O and B will be reprefejited 

 between I and M, and a complete image of O B will be 

 formed at I M. If we fuppofe the objeft at a great dif- 

 tance, and confequently fmall, the arc E D of the fpe- 

 culum will be ver)' minute, and not fenfibly different from a 

 right line, and confequently will be parallel to the radiant 

 B O, becaufe C A is perpendicular to both B O and E D. 

 Alfo, fmce the diftances O D, A V, and B E, are very 

 nearly equal, from their contiguity, it is plain that the focal 

 diftances D I, V a, and E M, will alfo be nearly equal ; 

 and, therefore, the image I M will be very nearly a right 

 line, and parallel to the radiant O B, as well as perpendi- 

 cular to C A. 



Now from the nature of refleftion, we have ^ B V A =; 

 Z.AVG= A«VM; therefore, ^OVA + BVA 

 = Z.aVl + aVM; namely, ^ O V B == I V M ; f o 

 that the radiant, or objeft B O, and its image I M, are feen 

 under the fame angle from the vertex of the fpeculum. But 

 Ihe triangles A V O and a V I are fimilar, for the zL O V A 

 ^= aV 1, and the angles at A and a are both right angles ; 

 therefore, VA:Va :: AO:aI. For the fame reafon, 

 VA:Va::AB:aM; andVA: Va::OA + AB: 

 la + aM :: OB : IM; or, in words, " the diftance of 

 the objeft is to the diftance of its image, from its vertex V, 

 as the length of the objeft is to the length of the image." 



From the analogy here deduced, it is eafy to form theo- 

 rems, that ftiall determine either d, f, or the proportion 

 O : I, when O is the length of the objeft and I of the 

 image, when the two others are given. For we have given 



O : I :: d : f, and confequently — = f; but our funda- 

 mental theorem was — = y, confequently -pr- = 



zd + r 



o 



2</ + 



-, and fo 2 rf I -f I r = /• O : and 2 dl — O r — 



r 



i r ; confequently for a convex fpeculum, the theorem will 



Or-Ir , 



"^ ~T = "; and for a concave, where r is negative, 



. ... , If -Or 



It will be = — = J, But if the focus be required 



2I 



to be negative in a convex, the theorem will be 



Or-li 



= d ; and in a concave, where r and / arc both negative, it 



•ill be 



Or-Hr 

 2I 



= .d. 



If r be required 



in a convex 



fpeculxun, when d and O 

 ild 



and 



in a concave 



I are given, the theorem will be 

 2ld 



1-6 



2ld 



= r ; but if the 



O + I 



=: r ; and the fecond 



O- 1 



focus is required to be negative, the firft will be 

 2ld 



Laftly, when J and r are given, to find O : I, we fhall 

 have this analogy for a convex Ipeculum O : I :: 2 d -{- r : r ; 

 and for a concave, O : I :: r — 2 d : r. But if the focus 

 be negative, for a convex, it will beO : I :: — 2d — r : r; 

 and for a concave, O : 1 :: 2 d — r : r ; {o that, as we have 

 faid, when anv two of the three terms are given, the other 

 may be determined by calculation. By way of exemplifica- 

 tion, let it be required to find the radius of a concave fpe- 

 culum, that ftiall n;ake the image of an objeft, placed at 

 100 feet, as I : 60, in front of the fpeculum. Now / being 

 in this cafe negative, we have the theorem, as before fpe- 



... 2IJ .. 2XIX 100 200 



ciiied, ^ ^ = r, or, in figures. 



O + I 



3.28 =:= r, nearly ; or if r be given. 



60+1 61 



and d required, the 



theorem 



Or + Ir 



2I 



= d will give 



60 X 3.28 + I X 3.28 



2x1 



200.08 



= 100.04 = ^; sf"^ 'f 3-28 had been the 



exaS radius, the diftance would have come out exaftly 100. 

 If the image and objeft had been given equal, they would 

 both have fallen exaftly at the centre of concavity of the 

 mirror ; which coincidence affords a ready method of deter- 

 mining the radius of concavity of ?ny fpeculum, by means 

 of a luminous point afed as an objeft, and brought fo that 

 its image will exaftly coincide with it. It is hardly necef- 

 fary to add here, that a concave fpeculum forms an inverted 

 and magnified image ; and that a convex one makes it eredt, 

 and at the fame time diminifhes it. 



We have before fhewn how the aberration of the rays of 

 light may be calculated, when reflefted to a focus by a fpe- 

 culura of a fpherical figure, when the rays are parallel be- 

 fore they fuffer refleftion ; and it has been demonftrated, that 

 for fuch rays a parabolic curve is the beft fuited for correft- 

 ing fuch aberration, particularly when the image is formed 

 by only one reflefting furface ; but when there is a fecond 

 or fmall fpeculum, either concave or convex, employed in 

 forming a fecondary image, or in affifting to form the pri- 

 mary one, a parabolic curve will not be the beft for correft- 

 ing the aberration of the rays ; becaufe each fpeculum will 

 have its own aberration ; and the praftical opticiaTa can em- 

 ploy his ikill in producing fuitable fpecula for counterafting 

 each other's errors, with refpeft to the united effeft of their 

 feparate aberrations, better than the calculating theorift can 

 pretend to direft ; for the moment he fcrews his eye-tube 

 alternately out and in, beyond and ftiort of diftinft vifion, 

 he knows the nature of the curves of his fpecula, and whe- 

 ther the indiftinftnefs jo-ifing from aberration is the confe- 

 quence of too much or too little curvature at the vertex of the 

 large fpecnlum, and can make the final alteration accordingly . 

 This praftical dexterity, arifing out of experience, fuper- 

 fedes the neceffity of tedious mathematical calculations, 

 where fome part of the data rauft neceflarily be affumed ; 

 and it is much to be wifned, that praftical men, v/ho have 



excelled 



