a 
1819.], Haiiy..on the. Measuring of the Angles of Crystals. 416 
the accuracy of the results deduced from it; that the determina- 
tions of the primitive forms on which I have fixed, lead, with 
regard. to the secondary forms, to the true laws of decrement on 
which these forms depend ; and that the measurements made by 
the reflecting goniometer itself confirm the existence of those 
laws. 
I add-that the application of the theory to the mineralogical 
method has likewise all the accuracy necessary to make the forms 
of the molecules contribute to the distinction of the species. 
Finally, without excluding in certain particular cases, the use 
of measures obtained by the reflecting goniometer, 1 am con- 
vinced that those obtained by the common goniometer, which 
have the advantage of being at once direct and rapid, are suffi- 
cient, either to determine a new variety, or to ascertain to which 
of the varieties already classed in the method the crystal under 
examination belongs, though seen only for the first time. 
I shall give three examples in support of what Lhave just said. 
The first two, namely, quartz, and oxide of-tin, have been 
chosen from, those of which the determinations have been, pub- 
lished eithe’.in my Treatise, or my Tableau Comparatif. . With 
the last, sulphate of lead, I have occupied myself more recently. 
I shall compare the results obtained by the two goniometers ; 
and, I shall draw from the. comparison consequences which 
appear to me to guarantee the truth of all that | have advanced. 
Quartz. 
Though the crystals of quartz be subject to several anomalies 
which, occasion shght variations in the position of their faces, 
especially of those which are parallel to the axis ; yet it is not 
difficult to find, among the great variety deposited in collections, 
some possessed of all the requisite regularity for mechanical. 
measurement. Such in particular are those called hyacinths of 
compostella, many of which are isolated and complete, and all 
the faces. of which are smooth and perfectly level. 
With respect to these crystals then, I was in a favourable 
situation to bring the ratio of their dimensions to a simple limit, 
capable of leading to results sensibly the same with those of the 
erystallation. 
_ I took fora datum the inclination of one of the faces of the 
pyramid, such as P (fig, 1, Plate XCII.) on the adjacent face r. 
found that it was between 1414° and 142°. 1 supposed it, 
1413°. On this hypothesis, if from the centre c of the base of 
the pyramid, of which cs is the axis, we draw the perpendicular 
cr to one of the sides, and then the line 7 s, we shall have c rs 
= 51° 45’ and cr: cs :: sin. 38° 15’: sin. 51° 45’. To have the 
ratio ¢c,r:c s expressed in radical quantities, I take the loga- 
rithms of the squares of these two sines, and seeking in the 
table of natural numbers, those to which they correspond, 1 
obtaine r: cs :: ./ 3833 : / 6167, or nearly :: / 38: / 62 
