GOA OPT 
five inches aperture, lately converted into a fuperb tranfit 
inftrument by Mr. Troughton, and placed in Greenwich 
Oblervatory. 
Soon after Peter Dollond’s telefcopes began to be in re¬ 
pute, namely, in the year 1759, Benjamin Martin, at the 
fame time a mathematician and a mechanic, who had long 
turned his attention to the conftruftion of telefcopes, and 
defcribed various conftruftions, publifhed his “New Ele¬ 
ments of Optics,” a book now, like Edwards’s Treatile, 
extremely fcarce, in which he has entered more minutely 
into the doftrine of both kinds of aberrations, as they 
relate to praftice, than any other author has done, either 
before or lince. He not only followed the fteps of J. Dol- 
lond, in determining by glafs wedges or prifms the relative 
refractive and difperfive powers of different fpecimens ot 
glafs, but ground tingle objeft-glaffes of feveral kinds of 
glafs, with tools of the fame radius, and then compared 
•the geometrical foci of each with the refrafted or real 
foci, by nice meafurements : by this means he afcertained 
the difference between the focus determined theoretically 
from the known radius, and the real or practical focus of 
the refrafted rays in each glafs by meafurement, coniider- 
ing at the fame time the diltance of the radiant point; 
and thus he gained the ratio between the fine of the angle 
of incidence and of the angle of refraction in each fepa- 
iv.te fpecimen, which ratio, in a ray paffing from air into 
glafs, had been affumed in all former optical theorems as 
3 to 2-in all kinds of glafs; and confequently the focus 
for parallel rays had been put equal to radius in double 
convex lenfes, and all'o equal to the diameter in fingle 
convex, without regard to the quality of the glafs, with 
relpeft to its refraftive power. The rectification of the 
old theorem, founded on the conftant ratio 3 to 2, formed 
the bafis of the “ New Elements of Optics,” in which one 
half of the difference between the old theoretic and the 
refrafted or practical foci was called a, and then — z=F 
with parallel rays became the bafis of the rectified theo¬ 
rems. According to thefe new elements, and from a mea- 
l'urement of the angles of difperfion, or of the coloured 
fpeCtra contained between the extreme rays, as given by 
a prifm of flint and another of crown-glafs refpeCtively, 
the ratio of which he determined to be as 5 to 3, he cal¬ 
culated that “ the radii of the lenfes muff have the fame 
proportion as the differences of the fines of incidence and 
refraCtion in red and violet rays-, in prifms of equal refraCt- 
ino-angles of white and crown glafs;” and that there¬ 
fore, “ the radii [or foci] of the lenfes muff have the fame 
proportion as the angles of diflipation in refraCtions by 
fuch prifms; and, of courfe, the fame proportion as the 
lengths of the coloured fpeCtra produced thereby.” From 
thefe confiderations the author-concludes, that “in all 
ca Jes of a compound lens for producing viiion without 
colours, the ratio of the radii, r and R, of the concave 
and convex lenfes (when two only are ufed.), muff: be that 
of 5 to 3 ; and that then the ratio of their focal diffances 
for parallel rays will be that of 3 to 2 nearly.” The ratio 
of the foci of two lenfes being thus determined that (hall 
make the colours vanifh, the longitudinal aberration arifing 
from the refpeftive curves was next conlidered ; and in 
doing this, care was taken that the comparative foci of 
the two lenfes was not to be altered by an alteration in the 
curves now to be rectified. By Huygens’s general theo¬ 
rem, the aberration arifing from the curves of any lens 
may be determined and compared ; and it being known 
from this theorem, that the longitudinal aberration is 
equal to fds of the thicknefs of a double convex lens of 
equal radii, a double concave was determined, from an 
equation of this aberration, fuch, that its contrary aber¬ 
ration might counteract the aberration of the affumed con¬ 
vex lens of equal radii ; and the numbers thus produced, 
for the radii of the double convex of crown-glafs and of 
the double concave of flint refpeCtively, were 8-36, 8-36, 10, 
and 23, inches, in which the focal diffances of the two 
kmfes are laid to be nearly as 2 to 3. In this combina- 
I c s. 
tion, the compound focus is dated to be 23'3 inches, and 
the radius r— 23 is contiguous to the convex-glafs. 
Other calculations were alfo made, where the radii of the 
convex lens were unequal, as well as thofe of the concave; 
but we do not learn that a good achromatic objeCt-glafs, 
put together agreeably to Martin’s calculations, was ever 
yet conftruCted. In the inftance before us, it is evident 
that the curve 8 36, coming in contaft with the concave 
23, muff touch it in the middle, and therefore the propor¬ 
tions are impracticable. 
But it remained for the ingenious optician of Iflington, 
C.Tulley, to calculate and manufacture, from any two 
given fpecimens of crown and flint glafs, a double objeCt- 
glals that fhall, generally fpeaking, be found both achro¬ 
matic, and alfo as free from the eftefts of fpherical aber¬ 
ration as art can make it. 
After this artiff had made himfelf mafter of Martin’s 
propofed plan of compounding an achromatic' objeft- 
glafs, he found that the curves calculated for this purpofe 
would not produce their defired effeft with any fpecimens 
of glafs that could be procured ; but ftill he thought that 
a careful repetition of Martin’s experiments might lead to 
refults favourable to his views, when fome modification 
was made in their application. He therefore, in the year 
1800, obtained fix forts of glafs, differing in fpecific gra¬ 
vity, and ground them all to the fame radius by a tool of 
fpeculum-metal, that did not much alter its figure by at¬ 
trition in grinding; and in giving a partial polilh : thefe 
lenfes were fitted lucceffively to one cell, that was received 
by a tube having an eye-niece at the oppolite end, in order 
that the l'olar focus of the refrafted rays might be the 
more accurately meafured with each glafs ufed as an ob- 
jeft-glafs of a telefcope ; and, though the polifn was im¬ 
perfect in thefe lenfes, ground and partially polilhed by 
the fame tool, yet the image of the fun was clearly de¬ 
fined by them. Thefe focal diffances, limited by the folar 
image, were in the next place meafured carefully by a 
nicely-divided ficale, and were found to differ from one 
another confiderably. The radius of curvature of the 
tool was alfo afcertained with equal care, and found to 
exceed in length the longeft of the focal lengths of the 
refrafted rays. The radius of the tool was then divided 
by each of the refrafted focal lengths, and the quotients 
were called fo many divifors or multipliers, accordingly as 
the geometrical was to be determined from the refrafted 
focus, or the contrary. Thefe quotients, therefore, bore 
the fame proportion to unity that the geometrical focus 
bore to the refrafted focus of each lens; and turned out 
to be very nearly the fame quantities that Martin had de¬ 
termined withglaffes of fimilar qualities, and that he de¬ 
noted by the expreflion za in his rectified theorems. In 
faft, they were the numbers from which the ratio of the 
fines of the angles of incidence to the fines of the angles 
of refraftion were accurately determined, as will be ex¬ 
plained hereafter. The fpecific gravities of the different 
lenfes were then taken with a good hydroftatic balance, 
and were found to increafe with their correfponding divi¬ 
fors, but not in a regular proportion. From thefe expe¬ 
riments a fet of tables was conftrufted, containing in pa¬ 
rallel columns, both for crown and flint glafs, the fpecific 
gravities, varying from 3-4.66 to 2^428, together with the 
correfponding ratios of the fines of the angles of incidence 
and of refraftion, and all'o the ratios of the two curves 
that fhall produce an afligned longitudinal fpherical aber¬ 
ration in any lens; all which calculations are extended 
from the ratios 1 : 1, 1 : roi, 1 : 1*02, &e. in fuccellion, 
up to 1 : 6, where the aberration is a minimum, as was 
long ago determined by Huygens; and, what is worthy of 
remark, the French plate-glals, which had the fpecific gra¬ 
vity loweff, and its diviloronly roo4, and which, confe¬ 
quently, had its refrafted focus nearly equal to its geome¬ 
trical focus, was, in all probability, fimilar to the glafs 
manufactured at the time when the experiments of fir ll'aac 
Newton were made, from which the original optical theo¬ 
rems were framed. 
From 
