GEO 
its sulphate predominates. The sulphate of 
lime, being verv sparingly soluble in water, 
is carried away by that liquid, and deposited 
to form gypsum ; while the >ther more solu- 
ble salts, remaining suspended, lonn vitriolic 
mineral waters. 
The pyritous schisti are frequently impreg- 
nated with bitumen, and. the proportions 
constitute tire various qualities' of pit-coal. 
It appears that we may lay it down as an 
incontestable principle, that the pyrites is 
abundant in proportion as the bituminous 
principle is more scarce. Hence it arises, 
t hat coals of a bad quality are the most sul- 
phureous, and destroy metallic vessels, by 
converting them into pyrites. r I lie foci of 
volcanos appear to be formed by a selristus ot 
this nature ; and in the analyses of the stony 
matters which are ejected, we find the same 
principles as those which constitute this 
schist us. We ought not therefore to be 
much surprised at finding schorls among vol- 
canic products ; and still less at observing 
that subterranean tires throw sulphuric salts, 
sulphur, and other analogous products, out ot 
the entrails ot the earth. 
3. The remains of terrestrial vegetables 
exhibit a mixture of primitive earths more 
or less coloured by iron: we may therefoie 
consider these as a 'matrix 'in which the seeds 
of all stony combinations are dispersed. The 
earthy principles assort themselves accoiding 
to the laws of their affinities; and form cry- 
stals of spar, of plaister, and even the rock 
crystals, according to all appearance : for we 
find ochreous earths in which these crystals 
are abundantly dispersed; we see them foi til- 
ed almost under our eyes. We have fre- 
quently observed indurated ochres full ot these 
crystals terminating in two pyramids. 
The ochreous earths appear to deserve the 
greatest attention of naturalists. Ihey con- 
stitute one of the most fertile means pi action 
which nature employs; and it is even in earths 
nearly similar to these that she elaborates the 
diamond, in the kingdoms ot Golcouda and 
Visapour. , . , .. .. 
The spoils of animals, which live on the 
surface of the globe, are entitled to some 
consideration among the number of causes 
which we assign to explain the various 
changes our planet is subjected to. W e find 
bones in a state of considerable preservation 
in certain places; we can even frequently 
enough distinguish the species of the animals 
to which they have belonged. 1* rom indica- 
tions of this sort it is that some writers have 
endeavoured to explain the disappearance 
of certain species ; and to draw conclu- 
sions thence, either that bur planet is per- 
ceptiblv cooled, or that a sensible change 
has taken place in the position oi the axis 
of the earth. The phosphoric salts and 
phosphorus which have been found, in our 
time, in combination with lead, iron, &c. 
prove that, in proportion as the principles 
are disengaged by animal decomposition, 
they combine with other bodies, and form 
the nitric acid, the alkalis, and in general all 
the numerous kinds of nitrous salts. See Mi- 
neralogy. . 
GEOMETRICAL lines, as observed by 
Newton, are distinguished into classes, 01 - 
ders, or genera, according to the number of 
the dimensions of the equation that expresses 
the relation between the ordinates and ab- 
scisses ; or, which comes to the same thing, 
GEO 
1 according to the number of points in which 
' they may be cut by a right line. 
Thus, a line ot the first order, is a right 
G E O 
line, since it can be only once cut by another 
right line, and is expressed by the simple 
equation y -}- ox b — 0 ; those of the 2d, 
or quadratic order, will be the circle, and 
the conic sections, since all of these may be 
cut in two points by a ri ght line , and express- 
ed by the equation y 2 -f- a v -j- b . y -j- ex' 1 -j~ dx 
_J_ e — 0: those of the 3d or cubic order will 
be such as may be cut in three points by a 
ri<dit line, whose most general equation is 
y ' 4- -f b ./ + w 4 - dx 4- • 3' 4 -f x ] +w 2 
_L /, v _j_ i — o ; as the cubical parabola, the 
nssoid, &c. And a line of an infinite order, 
is that which a right line may cut in infinite 
points; as the spiral, the cycloid, the. qua- 
dratrix, and every line that is generated by 
the infinite revolutions of a radius, or circle, 
or wheel, &x. 
In each of those equations, x is the absciss, 
y its corresponding ordinate, making any 
given angle with it; and a, o, c, txc. are 
given or constant quantities, affected with 
their signs -j- and — , of which one or more 
may vanish, be wanting or equal to nothing, 
provided that by such defect the line or equa- 
tion does not become one ot an inferior 
order. 
It is be observed that a curve of any kind 
is denominated by a number next less than 
the line of the same kind: thus, a curve ot 
the first order (because the right line cannot 
be reckoned among curves), is the same with 
a line of the second order; and a curve ot 
the second kind, the same with a line of the 
third order, &c. 
Jit is to be observed also, that it is not so 
much the equation, as the construction or 
description, that makes any curve, geometri- 
cal, or not. Thus, the circle is a geometri- 
cal line, not because it may be expressed by 
an equation, but because its description is 
a postulate; and it is not the simplicity of 
the equation, but the. easiness ot tne descrip- 
tion, that is to determine the choice of the 
lines for the construction of a problem. The 
equation that expresses a parabola, is more 
simple than that which expresses a circle ; 
and yet the circle, by reason of its more sim- 
ple construction, is admitted before it. Again, 
the circle and the conic sections, with re- 
spect to the dimensions of the equations, are 
of the same order; and yet the circle is not 
numbered with them in 'the construction of 
problems, but by reason of its simple descrip- 
tion is depressed to a lower order, viz. that 
of a right line ; so that it is not improper to 
express that by a circle, which may be ex- 
pressed by a right line ; but it is a fault to 
construct that by the conic sections, which 
may be constructed by a circle. 
Geometrical solution of a problem, 
is when the problem is directly resolved ac- 
cording to the strict rules and principles of 
geometry, and by lines that are truly geome- 
trical. This expression is used in contradis- 
tinction to an arithmetical, or a mechanical, 
or instrumental solution; the problem being 
resolved only by a ruler and compasses. 
The same term is likewise used in opposi- 
tion to all indirect and inadequate kinds of so- 
lutions, as by approximation, infinite series, 
&c. So, we have no geometrical way of 
finding the quadrature of the circle, the d y 
plicalure of the cube, or two mean propo* 
tionals; though there are mechanical ways, 
and others by infinite series, &c. 
Pappus informs us, that the antients en- 
deavoured in vain to trisect an angle and to 
find out two mean proportionals, by means 
of the right line and circle. Afterw ards they' 
began to consider the properties of several 
other lines ; as the conchoid, the c.ssoid, and 
the conic sections ; and by some ot these they 
endeavoured to resolve some of those pro- 
blems. At length, having more thoroughly 
examined the matter, and the conic sections 
being received into geometry, they distin- 
guished geometrical problems and solutions 
into three kinds ; \iz. 
1. Plane ones, which deriving their origin 
from lines on a plane, may be properly re- 
solved by a right line and a circle. 
2. Solid ones, which are resolved by lines 
deriving their original from the consideration 
of a solid ; that is, of a cone. 
3. Linear ones, to the solution of which, 
are required lines more compounded. 
According to this distinction, we are not 
to resolve solid problems by other lines than 
the conic sections, especially if no other lines 
beside the right line, circle, and the conic 
sections, must be received into geometry. 
But the moderns, advancing much farther, 
have received into geometry all lines that 
can be expressed by equations ; and have 
distinguished, according to the dimensions ot 
the equations, those lines into classes or or- 
ders ; and have laid it down as a law, not to 
construct a problem by a line of a higher 
order, that may be constructed by one of a 
lower. 
Geometrical progression, or propor- 
tion. See Algebra. 
GEOMETRY, the science and doctrine 
of local extension, as of lines, surfaces., and 
solids, with that of ratios, &c. 
The name geometry literally signifies mea- 
suring of the earth, as it was the necessity of 
measuring the land that first gave occasion 
to contemplate the principles and rules of 
this art, which lias since been extended to 
numberless other speculations; insomuch 
that, together with arithmetic, geometry 
forms now the chief foundation of all the ma- 
thematics. 
Herodotus and Proclus ascribe the inven- 
tion of geometry to the Egyptians, and assert 
that the annual inundations of the Nile gave 
occasion to it ; for those waters bearing away 
the bounds and landmarks of estates and 
farms, covering the face of the ground uni- 
formly with mud, the people, say they, were 
obliged every year to distinguish and lay- 
out their lands by tiie consideration of their 
figure and quantity ; and thus by experience 
and habit they formed a method or art, which 
was the origin of geometry.. A farther con- 
templation of the draughts of figures of fields, 
thus laid down and plotted in proportion, 
might naturally lead them to the discovery 
of some of their excellent and wonderful pro- 
perties ; which speculation continually impro- 
ving, the art continually gained ground, and 
made advances more and more towards per- 
tection. 
Geometry is distinguished into theoreti- 
cal or speculative, and practical. 
Theoretical or speculative geometry, treats 
of the various properties aud relations in 
magnitudes, demonstrating the theorems, &e\ 
And 
