TELESCOPE. 



is called the lateral error, or t}ie aberration in latitude. It is 

 evident from the figure, tliat as the ray {an) approaches the 

 extreme ray A M, the point of interfeftion Q will approach 

 the axis ; and when a n coincides with A M, the point Q 

 will coincide with the point K in the axis ; and it is as ob- 

 vious that the point Q will coincide with F, when the ray {an), 

 approaching the axis a B, at laft becomes coincident with 

 it : therefore there is one pofition of the ray {an), in which 

 it will cut the ray N D in a point Q, which will make Q O 

 a maximum, or the greateft of all. If we take the arc 

 Bm = B «, and B M =;; B N, the rays incident on m and M 

 will intcrfeft in the point P on the other fide, and fo make 

 P Q = 2 Q O ; and it is alfo plain, that all tlie rays which 

 fall on the lens between N and M are refracted through tlie 

 fpace PQ. Now PQ is the diameter of tlie Icajl circular 

 fpace poffible, in which all the rays can be congregated, be- 

 caufe there will be fome ray {an) that will meet the extreme 

 ray N D, at the dillanoe Q O = i Q P from the axis. 

 Hence it follows, that the circular fpace is the focus, or 

 place of the image of an objeft, belonging to parallel rays 

 incident on the lens N M. Further, by reafon of fimiiar 

 triangles K O Q, K F D, and N G K, we have Q O : K O 

 :: D F : K F :: N G : G K. But it is demonftrable, (fee 

 Philof. Britannica, 3 edit. p. 58. art. 14.) that when Q O is 

 greateft, then K O = J: K F, and alfo that K F is always ■§. 

 of G B, the thicknefs of the lens ; fo then K O = 1 G B, 

 and confequently GK:GN::-g-GB:QO, whence 



qGBxGN _,^ , r.^ij- ri 



~ — 5-p7-i? ^ O O ; wlience P Q, the diameter of the 



circle of aberration, is known for any given lens. 



" It has been demonftrated, that the error P Q will always 



he proportioned to _^ ; fo that when the radius is given, 



the error will be .as the cube of the aiperture direftly : and 

 when the aperture is given, the faid error will be as the 

 fquare of the radius inverfely. It has alfo been demon- 

 ftrated, that v/hen the convex fide of the lens N B M is 

 turned towards parallel rays, the eiTor K F will be but ^ 

 of the thicknefs of the lens G B, and therefore near four 

 times lefs than in the other cafe ; for 4 G B : -J G B :: 54 

 , 14, which is almoft as 4 to i. 



" It has been further demonftrated, that the aberration P Q 

 is as the fquare of the fine of refraftion (the fine of inci- 

 dence being unity) in all media of different refraftive 

 powers : thus if a lens of the fame focal diftance and aper- 

 ture were made of glafs and water, and fuppofe thofe fines 

 in glafs to be as m : n, and in water as 'm '.'fi; then will P O 

 in the glafs lens be to the fame in the water lens as w' : '/?;% 

 or the area of the circles of aberration, and of courfe the 

 indiftinftnefs of the objeft will be as the refraftions m and 'm 

 of the media. 



" Whatever has been obferved with regard to convex and 

 plano-convex lenfes, will hold good in concave and plano- 

 concave ones. And in both forts, it is fuppoied that all of 

 them have the fame focal diftances, apertures, and thick- 

 nefles, while we are comparing their refpeftive aben-ations. 



" Hence it is very evident, that if rays proceed from any 

 point, as {a) at an infinite diftance to a lens N M, {Jig. 10.) 

 the im^age of that point will not be a point, but the area of 

 a circle, whofe diameter is P Q ; and, therefore, that point 

 cannot be diftinftly reprefented, but will be rendered in- 

 diftinft and confufed in proportion to the area of the faid 

 circle of aberration in the lens, as it is the image of this 

 circle (or dilated point) that is impreffed on the retina, and 

 excites the idea of the point in the mind. 



" Hence it appears alfo, that the points in the furfaccs or 



fubftances of bodiej cannot be pcrfeftly and dillinaiy feen, 

 as each of them will be dilated into a funfible area ; and 

 luch as are contiguous, as i, 2, 3, will have their confufed 

 images all blended together nearly in the fame fpace, viz. in 

 the circle of aberration, the diameter of which is P Q. 



" Therefore the ftars, which as to fcnfe are only lucid 

 points, \yill appear to have fome magnitude (and not as 

 points) in the focus of the beft fort of telelcopes, even 

 fuppofing there were no other caufe of confofion or in- 

 diftinft vifion, befides what refulted from the fpherical 

 figure of the lens. 



" Now, if tlie error from a fpherical furface, or, which is 

 the fame thing, the indijlinflncfs of vifion, depending on, and 

 comnienfiirate witii, tiie fpherical aberration of a lens, is as 

 the fquare of radius inverfely ; the tli/linclnr/s of vifion, on 

 the contrai7, will be as ihe fjuare of the radius direllly ; 

 and, therefore, if, by means of two glaftes, we can get the 

 view of an objeft, where the radii of the glaffes boar a greater 

 proportion to their refpeftive apertures, than the radius of 

 a fingle glafs of equal magnifying power does to its aper- 

 ture ; it is evident tlie diftinftnefs of that view will be 

 promoted in proportion to the fquare of that ratio. 



" For example, fuppofe (/^. 8. ) F — D = ji, or O F, to 

 be the focal diftance of the lens G H, fo that the focus of 

 each of the lenfes N M and G H falls on the fame point 

 F ; then, by tlic preper theorem, we have >: = i F, <5r 

 Q/= ^ C F : alfo, fince in this cafe we have F : ji :: * : /, 

 therefore/^ ijr, or O/ = i O F. Now, fince we liave the 

 fame optic angle G F O by both the glaffes, as by the 

 fingle one E E, the ratio of the radius O F to the aperture 

 G O, or of the radius C F to the aperture N C, is double the 

 ratio of 0/ to O G, or of the radius Q/to the aperture 

 E Q, and therefore the diJlinElnefs of vifion by both the lenfes 

 h, four times greater than that by ihefwgk lens E E. 



" The fame thing may be demonftrated from "the confidcr- 

 ation, that the aberration P Q is, in the fame glafs, glways 

 proportioned to the cube of the femi-aperlure E Q, or fine 

 ' of half the optic angle E/Q ; and that in fmall angles (as in 

 the glafles of telefcopes, &c. ) tlie fine E O is nearly as the 

 angle E/Q. The aberration, therefore, being as the cube 

 of that angle, it is plain, if we make the fame angle by two 

 refraftions inftead of one, the quantity of the aberration will 

 be greatly leffened, fince the fum of the cubes of the parts 

 will be much lefs than the cube of the whole ; and when the 

 parts are equal, the fum of the cubes of each will be but a 

 fourth part of the cube of the whole. Thus, if tlie whole 

 angle E/Q be as 1, the cube thereof is i ; but the half is 

 4, the cube of which is ^, and twice that, -J- = -i, which is 

 as the aberration arifing from the two halves, and is there- 

 fore but a. fourth part of the whole. 



" This is evidently the cafe when the optic angle GfO 

 ( = EyQ) is made by two refraftions, by the two lenfes 

 NM and G H, fo pofited, that the focus of eacli may fall 

 on the fame point F ; for then the angle G/O = L G/, 

 which is compofed of the two angles LGFr=TNF=r 

 GFO, (byreafonoftheparalleninesTN,LG, and FC,) 

 which is the part made by the lens N M. Alfo the angle 

 F Gyis the refraftion of the ray N G, or fecond refraftion 

 of the ray A N ; and fince, in the prefent cafe, Of = fY, 

 and Oy in fmall angles is equal to Gy nearly; therefore 

 the angle G F O is equal to the angle F Gy very nearly, 

 thofe angles being in the fame ratio with the cqu;il lines Gy 

 and fY, when they are not large ; and the optic angle 

 GyO = GF O -I- F Gf; confequently the aberration PQ 

 is but a fourth part fo great by the two lenfes N M and G H 

 together, as it is by lens E E alone. 



" But to render this theory general for any pofition or form 



of 



