844 
GEO 
G E R 
G E R 
-J- s.v; and from thence, by division, x — 
a J-J, -f ' 
a/ 3 " 
Pros. V. Having the area of a rectangle DEFG, in- 
scribed in a given triangle ABC ; to determine the 
sides of the rectangle. See Fig. 40, 
Pet Cl be perpendicular to AB, cutting DG 
in H ; and let Cl — a, AB = b, DG = x, and 
the given area = cc ; then it will be, as b ] x [ * 
ax 
— CH ; which, taken from Cl, leaves 
b 
ax 
T 
ax * 
IH ; and this, multiplied by x, gives 
ax ~r = cc = the area of the rectangle ; 
whence we have abx — ax 2 
bcc b 
a ' 2 
= ±s/t 
bcc 
r 2 — bx — 
, and x ” 
A + /** bcc 
2 ± v 
PROB. VI. Through a given point P, within a given 
circle , so to draw a right line , that the two parts 
thereof, PR, PQ, intercepted by that point and the 
circumference of the circle, may have a given differ- 
ence. See Fig. 41. 
Let the diameter APB be drawn ; and let AP 
and BP, the two parts thereof (which are sup- 
posed given) be denoted by a and b ; making 
PR — x, and PQ — 4 d (d being the given 
difference). Then, by the nature of the circle, 
PQ X PR being — PA X PB, we have x d 
X a- — ab, o r xx 4 dx = ab; whence x is found 
= \/ab 4 \dd — \d. 
Prob. VII. From a given point P, without a given 
circle, so to draw a right line PQ, that the part 
thereof RQ, intercepted by the circle, shall be to 
the external part PR, in a given ratio. See Fig. 42. 
Through the centre O, draw PAB ; put PA 
— a, PB — b, PR — x, and let the given ratio 
of PR to RQ be that of m to n ; then it will be, 
nx 
as m . n . . — = RQ ; therefore PQ — 
+ ~ ■ but PR X PQ — PA x PB, or x X 
. nx 
x rff— ab-, therefore mx 2 4 = ™ab, 
and 
Prob. IX. The segments of the base AD and BD, a,nd 
the ratio of the sides AC and BC of any plane tri- 
angle ABC, being given ; to fnd the sides. See 
Fig. 44. 
Put AD — a, BD = b, AC = x ; and let the 
given ratio of AC to BC, be as m to n, so shall 
BC = But AC- — AD 2 (— DC 2 ) — BC 2 
— • BD 2 , that is, in species, x 2 
— b 1 . Hence we have nfx 2 
n‘x 2 — m X 
bb, and x ~ m l / — — — . 
V mm — nn 
/ mab 
m 4 n 
+ 
Prob. VIII, The sum of the two sides of an isosceles 
triangle ABC, being equal to the sum of the base and 
perpendicular , and the area of the triangle bciner 
given ; to determine the sides. See Fig. 43. 
Put the semi-base AD = x, the perpendicular 
CD —y, and the given area ABC — a 1 ; so shall 
xy = a 2 , and 2^/xx 4 , y = 2x 4 y (by El. 47. 
1. and the conditions of the problem). Now, 
squaring both sides of the last equation, we have 
4xx 4 4yy == 4 xx 4 4 xy 4 ^ ; whence 3 yy — 
4 V 
4xy, and consequently y — — ; which value, 
s 
substituted i-n the former equation, gives 
~ a 2 ; from whence a- 
4x 
/3a 2 
V 7 = 
V 3 ; y 
*±X \ 
(= Y J — f'V’ 3 > and AC (= sfxx+yy = 
( ^ aa _L, ^ aa / v 
V 4 3 V ~12~ J ~ 3- = 
l-V 3 * 
Geometry, usefulness of. The useful- 
ness of this science extends to almost every 
art and science, it is by the help of it that 
astronomers turn their observations to ad- 
vantage, regulate the duration of times, sea- 
sons, years, cycles, and epochas ; and mea- 
sure the distance, motions, and magnitudes, 
of the heavenly bodies. It is by it that geo- 
graphers determine the figure and magni- 
tude of the whole earth, and delineate the 
extent and bearings of kingdoms, provinces, 
| harbours. See. it is from this science too 
| that architects derive their just measures, in 
i the construction of public edifices as well as 
of private houses. It is by the assistance of 
; geometry that engineers conduct ail their 
works, take the situation and plans of towns, 
the distances of places, and the measure ot 
! such things as are only accessible to the 
j sight. It is not only an introduction to forti- 
fication, but highly necessary to most me- 
chanics, especially carpenters, joiners, ma- 
thematical-instrument-makers, and all who 
profess designing. On geometry likewise 
depends the theory of music, optics, per- 
spective, drawing, mechanics, hydraulics, 
pneumatics, &c. 
GEORGE, or knights of St. George. See 
Garter. 
Religious of the order of St. George, 
form several congregations in Italy and other 
places. 
GEORG IC, a poetical composition upon 
the subject of husbandry. See Poetry. 
GERANIUM, cran-J s-bill, a genus of the 
decandria order, in the monadelphia class of 
plants, and in the natural method ranking un- 
der the 14th order, gruinales. Its characters 
are these : the flower has a permanent em- 
palement, composed of live small oval leaves, 
and five oval or heart-shaped petals, spread- 
ing open, which are in some species equal, 
and in others the upper two are much larger 
than the three lower. It has ten stamina, al- 
ternately longer than each other, but shorter 
than the petals, and terminated by oblong 
summits. In the bottom of the flower is 
situated a five-cornered germen, which is 
permanent. 1 he flower is succeeded by five 
seeds, each being wrapped up in the husk of 
the beak, where they are twisted together at 
the point, so as to form the resemblance of a 
stork s beak. r I here are 32 species. 
The common wild sorts of this plant, and 
those also which are brought to the curious 
from the colder climates, are hardy enough, 
and require little care ; but the southern spe- 
cies require care in their culture and propa- 
gation, and in general the protection of a 
greenhouse in the winter. The erodium, and 
the pelargorium, or African geranium, are 
vulgarly called by the name of geraniums, 
aud indeed resemble this genus in every- 
thing but the number of the stamina, 7Te 
erodium has live stamina, and the pelargo. 
rium seven. 
GERARDIA, in botany, a genus of the 
didynamia angiuspermia class ot plants, the 
corolla of which consists of a single ringent 
petal; the tube is roundish, and longer than 
the cup ; the upper lip is erect, obtuse, plain, 
and emarginated ; the lower lip is reflected, 
and divided into three segments ■. the fruit is 
an oval capsule, containing two cells, and 
consisting of two valves ; tne seeds are oval 
and single, 'i here are ten species, herba- 
ceous plants of America and the Cape. One 
species is much recommended in gouty dis- 
orders. 
GERMEN. See Botany. 
GERMINATION, in botany. Natural 
historians have proved, by a very complete 
induction or facts, that all plants arise from 
seeds. The pretended exceptions have dis- 
appeared, one after another, as our know- 
ledge of vegetables increased ; and now there 
remains scarcely a single objection intitlecl to 
the smallest regard. The late attempt of 
G Manner to revive the doctrine of equivocal 
generation deserves no attention whatever ; 
because his conclusions are absolutely incom- 
patible with the experiments of Mr. Senne- 
bier upon the very substance on which iris 
theory is founded. 
A seed consists of three parts, namely, the 
■cotyledons, the radicle, and the plumula, 
winch are usually inclosed in a cover. 
It we take a garden-bean, we may perceive 
each of these three parts with great ease ; 
for this seed is of so large a size, that all Its 
organs are exceedingly distinct. When we 
strip off the external coats ot the bean, which 
are two, and of deferent degrees of thickness 
in different parts, we iind^that it easily di- 
vides into two lobes, pretty nearly of the 
same size and figure. Each of these lobes is 
called a cotyledon. T he cotyledons of the 
bean, then, are two in number. 
Near that part of the lobes which is conti- 
guous to what is called the eye of the bean, 
there is a small round white body, which 
comes out between the two lobes. This body 
is called the radicle. 
Attached to the radicle there is another 
small round body, which lies between the co- 
tyledons, and wholly' within them, so that it 
cannot be seen fill they are separated from 
each other. 'I his body is called the plu- 
mula. 
The appearance and shape of these three 
parts differ very much in different seeds, but 
there is no seed which wants them. The 
figure and size of the seed depend chiefly 
upon the cotyledons. This is evidently the 
case with the bean, and it is so with all other 
seeds. The number of cotyledons is dif- 
ferent in different seeds. Some seeds have 
only one cotyledon, as the seeds .of wheat, 
oats, barley, and the whole tribe of grasses ; 
some have three ; others six, as the seeds of 
the garden-cress ; but most seeds, like the 
bean, have two cotyledons. 
W hen a seed is placed in a situation favour- 
able to vegetation, it very soon changes its 
appearance. 7 he radicle is converted into a 
root, and sinks into the earth ; the plumula, 
on the other hand, rises above the earth, and 
becomes the trunk or stem. When these 
changes take place the seed is said to gernii- 
