424. 
GEOMETRY, 
in the !>igh geometry could have been deduced from 
the dodlrine of infinites, it might have been expelled 
from this author ; but our ideas of infinites are too ob- 
fcure and inadequate to anfwer this end ; and there are 
many things advanced by all tliofe who have applied 
tliem with great freedom in geometry, that give ground 
to a remark like to Mr. de St. Evremond’s, w’hen he 
obferves, that “it is furprifing to find the ancient poets 
fo fcrupulous to preferve probability in adtions purely 
human, and fo ready to violate it in reprefenting the 
adtions ot the gods.” Some have not only admitted in¬ 
finites and infinitefimals of infinite orders, but have dif- 
tinguiflied even nothings into various kinds : and if fucit 
liberties continue, it is not eafy to forefee what abfur. 
dities may be advanced as difcoveries in w'hat is called 
the “ fublime geometry.” 
Of the elements of GEOMETRY. 
Definitions and Principles. 
1. Geometry is a fcience which treats of the defcrip- 
tions, properties, and relations, of magnitudes in gene¬ 
ral : tiiat is, of tilings in wliich length, or length and 
breadtli, or lengtli,breadth, and tliicknefs, are contidered. 
2. A point is that whicli has neither parts, nor dimcn- 
fions. It has been objected to this definition, that it 
contains only a negative, and tliat it is not convertible, 
as every good definition ought certainly to be. Tliat it 
is not convertible is evident, for thougdi every point is' 
unextended, or witliout magnitude, yet every thing un¬ 
extended, or without magnitude, is not a point. To 
fl'iis it is impodible to reply; and therefore it becomes 
necellary to change tlie definition altogether, which is 
accordingly done by Playfair ; a point being defined to 
he, “tliat which has pofition, but not magnitude.” 
Here the affirmative part includes all that is elfential to 
a point, and the negative part excludes every thing that 
is not elfential to it. 
3. A line is length without breadth: it is called a 
/w, when it lies in the fame direction, as A-B: 
or a curved line, w hen it does not lie in the fame direction, 
but as Q.^ -(A line is ufually denoted by two 
letters; viz. one at each end, as A B or CD).—After 
this definition, Euclid has introduced the following, 
“ the extremities of a line are points.” Nov/ this is 
certainly not a definition, but an inference, from the defi¬ 
nitions of a point and of a line. I'hat which terminates 
a line can have no breadth, as the line in which it is has 
none; and it can have no length, as it ivould not then 
be a termination, but a part of that which it is fuppol'ed 
to terminate. The termination of a line can therefore 
have no magnitude, and having necelfarily pofition, it 
is a point. Euclid has defined a ftraight line to be a 
line which “lies evenly between its extreme points.” 
This definition is obviouffy faulty, the word evenly Hand¬ 
ing as much in need of an explanation as the word 
fraight, which it is intended to define. In the original, 
Jiowever, it muff be confetfed, that this inaccuracy is at 
leaff lefs Itriking than in our tranflation ; for the word 
which we render evenly is equally, and is accordingly 
tranllated ex cequo, and equaliter, by Coinmandine and 
Gregory. The definition, therefore, is, that a firaight 
line is one which lies between its extreme points; 
and if by this we iinderlland a line that lies between its 
extreme points, fo as to be related exadfly alike to the 
Ipace on the one fide of it, and to the fpace on the otlier, 
we have a definition that is perhaps too metaphyfical, 
but which certainly contains in it the elfential character 
of a llraight line. 
4. A fuperficies, or furface, is that magnitude which 
has only length and breadth, and is bounded by lines. 
5. A /olid, is that magnitude which has length, 
breadth, and thicknefs. 
6. A figure^, is a bounded fpace, the limits or bounds 
of which may be either lines or furfaces. 
7. A plane, or a plane figure, Is a fuperficies which lies 
evenly, or perfectly flat, between its limits, and may be 
bounded by one curve line ; but not with lefs than 
three right lines, 
8- A circle is a plain figure, bounded A. 
by an uniformly curved line, called the 
circumference, as ABD, w'hich is every 
where equally diflant from one point, as 
C within the figure called the centre. 
9. A ratf/as is a right line drawn from the centre to 
the circumference, as CA, CD, or 
C E.—All the radii of the fame circle 
are equal. 
IQ. An arc is any part of the cir- 
m cumference.—As the arc AB, or tlie 
.j, arc A D. 
11. A chord is a right line joining 
the ends of an arc, as A B, and is faid 
to fubtend that arc; it divides the 
circle into two parts, called fegments. 
12. A diameter is a chord palling 
through the centre, as DE, and divides the circle into 
two equal parts, called femicircler.. 
13. The circumference of every circle is fuppofed to 
be divided into three hundred and lixty equal parts, 
called degrees-, each degree into fixty equal parts, called 
minutes-, each minute into fixty equal parts, called fe~ 
conds, &c. 
14. A plane angle, is the inclination of two lines on 
the fame plane meeting in a point, as A C B.—A right- 
lined angle is’formed by two right lines. 
The point where the lines meet is called 
the angular point, as C.—The lines which 
form the angle are called legs .—Thus 
C A and C B are the legs of the angle 
A C B.—(An angle is ufually marked by 
three letters, viz. one at the angular point, and one at 
the other end of each leg ;' but that at the angular 
point is always to be read the middle letter, as a'c B 
or B C A.) 
C 
; \B^ . The meafure of a right-lined angle 
-- is an arc, as BA, contained between the 
legs C B, CA, including the angle, the 
I angular point C being the centre of that 
i—^ arc. 
i6. A right angle is that, the meafure 
of which is a fourth part of the circum¬ 
ference of a circle, or ninety degrees. 
Thus the angle A C B is a right angle. 
17. A perpendicular is that right line which cuts anos 
ther at right angles ; or which makes 
H equal angles on bo.th fides. Thus 
D C is perpendicular to AB, when 
the angles D C A and D C B are equal, 
or are right angles. 
18. An acute angle, is that which is lefs tlian a right 
angle.' 
19. An obtufe angle, is that which is greater than a 
right angle.—Acute and obtufe angles are called obli.que 
angles. 
20. Parallel lines, are right lines in the fame plangj 
which do not incline to one another. 
21. A triangle is a plane figure bounded by three lines. 
22. An equilateral triangle, is that in which the three 
fides are equal 
23. An ifofceles triangle, is that which has two equal 
fides. 
24. A right-angled triangle, is that which has one right 
angle. 
25. An obtufe-angkd triangle, has one obtufe angle. 
26. An 
