£98 OPT 
nitely greater t'nan tang. R ; therefore qG vanilhes; or q 
bifeFts the chord of the arc cut oft’by the refracted ray. 
See fig. 14. 
Cor. 4. When tire rays are incident nearly perpendi¬ 
cularly upon the refracting furface, tang. I : tang. R :: 
ftn. I : fin. R ; alfo, D and G coincide with E; therefore 
qC : qE :: fin. I X QC : fin R X QE. See fig. 15. 
Cor 5. Similar conclufions may be drawn relpefting 
the refraCtion of a finall pencil of rays at any other fur- 
face, if E be the centre of curvature of the refraftor at the 
point ofincidence. 
On this fubjeCt, the reader may confult Hayes’s Flux¬ 
ions, Sett. ix, x. Smith’s Optics, book ii. ch. 9. Bou- 
guer’s Traite d’Optique; and Prieftley’s Hift. of Vifion, 
p. 233. 
CONSTRUCTION of OPTICAL INSTRUMENTS. 
Lenses. 
Aslenfes are the foundation of the mod important op¬ 
tical inftruments, as microfcopes, telefcopes, burning 
glaffes, it will be proper to fpeak of them in the firft place. 
Lenfes are diftinguifhed, with regard to the manner of 
their preparation, into blown and ground. 
The blown lenfes are merely little globules of glafs 
melted in the flame of a lamp or taper, and ufed for mi¬ 
crofcopes. 
The ground lenfes are moftly of a larger fize for tele¬ 
fcopes of different kinds; and the following is the method 
recommended by Fergufon, for grinding and polilhing 
them. 
Having fixed upon the proper aperture and focal dif- 
tance of the lens, take a piece of (heet-copper, and (Irike 
a fine arch upon its furface, with a radius equal to half 
that diftance, if it is to be plano-convex, and let the length 
of this arch be a little greater than the given aperture. 
Remove with a file that part of the copper which is with¬ 
out the circular arch, and a. convex gage will be formed. 
Strike another arch with the fame radius; and, having 
removed that part of the copper which is within it, a con¬ 
cave gage will be obtained. Prepare two circular plates 
of brals, about of an inch thick, and half an inch 
greater in aiameterthan the breadth of the lens, and fol¬ 
der them upon a cylinder of lead of the fame diameter, 
and about an inch high. Thefe tools are then to be fixed 
upon a turning-lathe, and one of them turned into a por¬ 
tion of a concave fphere, fo as to fuit the convex gage 5 
and the other into a portion of a convex fphere, fo as to 
anfwer the concave gage. After the furfaces of the brafs 
plates are turned as accurately as poffible, they muff: be 
ground upon one another, alternately, with flour-emery; 
and, when the two furfaces exattly coincide, the grinding- 
tools will be ready for ufe. 
Procure a^ece of glafs whofe difperfive power is as 
finall as pof^He, if the lens is not for achromatic inftru¬ 
ments, and whofe furfaces are parallel; and, by means of 
a pair of large fcilfars or pincers, cut it into a circular 
fhape, fo that its diameter may be a little greater than the 
required aperture of the lens. When the roughnefs is 
removed from its edges by a common grindftone, it is to 
be fixed with black pitch to a wooden handle of a fmaller 
diameter than the glafs, and about an inch high, fo that 
the centre of the handle may exattly coincide with the 
centre of the glafs. 
The glafs being thus prepared, it is then to be ground 
with fine emery, upon the concave tool if it is to be con¬ 
vex, and upon the convex tool if it is to be concave. To 
avoid circumlocution, we fhall fuppofe that the lens is to 
be convex. The concave tool, therefore, which is to be 
ufed, muff: be firmly fixed to a table or bench, and the 
glafs wrought upon it with circular ftrokes, fo that its cen¬ 
tre may never go beyond the edges of the tool. For every 
fix circular ftrokes, the glafs fliould receive two or three 
crofs ones along the diameter of the tool, and in different 
<$re£iions. When the glafs has received |ts proper fhape, 
I c s. 
and touches the tool in every point of its furface, which 
may be eaiily known by infpettion, the emery is to be 
waffled away, and finer kinds fucceflively fubftituted in 
its room, till, by the fame alternation of circular and tranf- 
verfe ftrokes, all the leratohes and afperities are removed 
from its furface. After the fineft emery has been ufed, 
the roughnefs which remains may be taken away, and a 
flight polifh fuperinduced, by grinding the glafs with 
pounded pumice-ftone, in the lame manner as before. 
While the operation of grinding is going on, the convex 
tool fhould, at the end of every five minutes, be wrought 
upon the concave one for a few feconds, in order to jfre- 
ferve the fame curvature to the tools and the glafs. 
When one fide is finilhed off with the pumice-ftone, the 
lens muff be feparated from its handle by infertin°- the 
point of a knife between it and the pitch, and giving it a 
gentle ftroke. The pitch which remains upon the'glafs 
may be removed by rubbing it with a little oil, or fplrits 
of wine ; and, after the ground fide of the glafs is fixed 
upon the handle, the other furface is to be wrought and 
finilhed in the very fame manner. 
When the glafs is thus brought into its proper form, 
the next and the moll difficult part of the operation is to* 
give it a fine polifli. The beft, though not the fimpleft, 
way of doing this, istocoverthe concave tool with alayer 
of pitch, hardened by the addition of a little rolin, to the 
thicknefs of one-fifteenth of an inch. Then, having taken 
a piece of thin writing-paper, prefs it upon the furface of 
the pitch with the convex tool, and pull the paper quickly 
from the pitch before it has adhered to it; and, if tne fur¬ 
face of the pitch is marked everv-where with the lines of 
the paper, it will be truly fpherical, having coincided ex¬ 
actly with the furface of the convex tool. If any paper 
remains on the furface of the pitch, it may be removed by 
foap and water; and, if the marks of the paper Ihouid not 
appear on every part of it, the operation mult be repeated 
till the polilher, or bed of pitch, is accurately fpherical. 
The glais is then to be wrought on the polilher by circular 
and crofs ftrokes, with the oxyd of tin, called the flowers 
of putty in the Ihops, or with the red oxyd of iron, other- 
wife called colcothar of vitriol, till it has received on both 
fidesa complete polilh. The polilhing will advance flowly 
at firft, but will proceed rapidly when the polilher becomes 
warm with the frittion. When it is nearly finilhed, no 
more putty or water Ihouid be put upon the polilher, which 
Ihouid be kept warm by breathing upon it; and, if the 
glafs moves with difficulty from its adhefion to the tool it 
fhould be quickly removed, left it fpoil the furface of the 
pitch. Alfo, if any particles of dull or pitch infinuate 
themfelves between the glafs and the polilher, which may 
be eafily known from the very unpleafant manner of work¬ 
ing, they Ihouid be carefully removed, by wafhin«- both 
the polilher and the glafs ; otherwife the lens will be 
fcratched, and the bed of pitch materially injured. 
The operation of polilhing may alfo be performed by 
covering the layer of pitch with a piece of cloth, and giv¬ 
ing it a fpherical form by prelfing it with the convex tool 
when the pitch is warm. The glafs is wrought as for¬ 
merly, upon the furface of the cloth, with putty or colco¬ 
thar of vitriol, tiil a fufficient polilh is induced. By this 
mode, the operation is flower, and the polilh lefs perfeff ; 
though it is beft fitted for thofe who have but little expe¬ 
rience, and would therefore be apt to injure the fio- ure of 
the lens by polilhing it on a bed of pitch. 
In this mariner, the fmall lenfes of Ample and compound 
microfcopes, the eye-glaffes and the objett-glaffes of tele¬ 
fcopes, are to be ground. In grinding concave lenfes, Mr. 
Imilori employs leaden wheels with the lame radius as the 
curvature of the lens, and with their circumferences of 
the fame convexity as the lens is to be concave. Thefe 
fpherical zones are fixed upon a turning-lathe ; and the 
lens, which is held fteadilyin the hand, is ground upon 
them with emery, while they are revolving on the fpindle 
of the lathe. In the fame way convex lenfes may be 
ground and polilhed, by fixing the concave tool upon 
