_. are not in the same straight line. 
¥ 
GEOMETRY, 
PartI. OF LINES AND FIGURES UPON A PLANE. 
We have seen how the general ideas of a surface, a 
line, and a point, ma be acquired from the considera- 
tion of a solid. The nts of tpeartl admit of only 
two lines, the straight line, and the curve. ‘The straight 
line serves to determine the nature of the surface called 
a plane; and from both we get a correct notion of a 
-circle, But the nature of these, and the other thin 
to be discussed, will be particularly explained in the 
following Sections by precise definitions. , 
SECTION I. 
Tue Princreces or Geometry. 
Definitions. 
ar ht ee 
nitu 
2. A line is length without breadth. 
Cor. The extremities of a line are points, and the 
intersections of one Jine with anotlier are also points. 
8. A straight line is the shortest way from one point 
to another. 
ery line which is neither straight, nor com- 
posed of ight lines, is a curve line. 
Thus, in Fig. 1. Plate CCLXX. AB is a straight 
line, and ACB a curve line. : 2 
5A or superjicies, is that which has onl 
length Sinan 
mn. The extremities of a superficies are lines, and 
poe intersections of one superficies with another are also 
s. 
6. A plane superficies is that in which any two points 
‘being taken, the straight line between thiems lies wholly 
in that superficies. 
7. Every surface which is neither a plane, nor compo- 
sed of plane surfaces, is a curve surface. 
8. A plane rectilineal angle is the inclination of two 
straight lines to one another, which meet together, but 
The point in which 
- the lines meet one another is called the vertex of the 
‘angle; and the lines are called its sides. * 
__N.B. When several angles are at a point A (Fig. 6.) 
one of them is expressed by three letters, of which 
the letter that is at the vertex of the angle is put between 
the other two, and one of these is somewhere upon one 
of the straight lines, and the other upon the other line. 
Thus, the angle which is contained by AB and AC is 
named the angle CAB or BAC ; that which is contain- 
ed by AB and AD is named the angle DAB or BAD; 
‘and that which is contained by AC and AD is called 
the angle CAD or DAC. But if there be only one an- 
Fe 
is 
'  .o To.get-an.accurate notion of the nature of 
ig. 2.) is successively i 
A on D’, the line AB do not D‘E/, but has-another 
+ Dig 
_ Anangle ma; le up of several 
y ap angles 
gle at a point, as in Fig. 2. it may be expressed by a . 
an angle, we may suppose that the angle t : 
‘ly compared with the angles contained by the lines DE and DF (Fig. 3.); and D’E’ and D’/F’ (Fig. 4, and 5.) First, 
_ sa that the line AC (Fig, 2.) is placed on the line DF (Fig. 3.), so that the point A may fall on D; then, if AC coincide with DF, the 
contained by ABand AC te to the angle contained by DE and DF. But if, when AC is placed on D’F’ (Fig. 4 and Fig. 5.) 
a on position DG ; then the angle contai 
contained by D/E/ ‘and D/R!: Tt-is greater if D’Ky fall between DG and DF’, asin Wig. 45 but it is less.if DG fall between D’B’ 
letter placed at that point, as the angle at A, or the 
angle A. , 
199 
Principles. 
9. When a straight line standing on another rein on Fig. 7. 
an 
line makes the adjacent angles equal to one er, 
each of the angles is called a right angle; and the 
straight line which stands on the other is called a per- 
pendicular. (Fig. 7.) 
10. An obtuse angle is that which is greater than a right Figs. 8. and 
angle (Fig. 8.) ; and an acule angle is that which is 9 
less than a right angle (Fig. 9.) 
11. Parallel straight lines are such as are in the same 
Rate, and which being produced ever so far both ways, 
jo not meet, (Fig. 10. 
12. A plane figure is that which is enclosed by one 
or more lines on a plane. If the lines are straight, the 
space they enclose is called a rectilineal figure, and the 
lines themselves are called its perimeter, See Fig. 11, 
12, &c. to Fig. 22. 
13. A rectilineal figure having three sides is named 
a triangle; a figure of four sides is called a quadrilate- 
ral; that of five sides is a pentagon ; that of six sides 
is a hewagon, and so on. Figures of more than four 
sides are likewise called polygons. 
14. An equilateral titngle is that which has its three 
sides equal (Fig. 11.) An isosceles triangle is that 
which has only two equal sides (Fig. 12.), and a sca- 
lene triangle that which has all its sides unequal. (Fig. 
13. 
ty A ‘right angled triangle is that which has a right 
angle. ‘The side opposite to the right angle is called 
the hypothenuse (Fig. 14.) 
An obtuse angled triangle is that which has an obtuse 
angle (Fig. 15.) 
An acute angled triangle is that which has all its an- 
gles acute (Fig. 16.) 
16. Among four-sided figures the following are dis- 
tinguished by apa names : 
A square is that which has all its sides equal, and all 
its share right angles (Fig. 17. 
A rectangle is that which has its angles right angles, 
but its sides not equal (Fig. 18.) 
A rhombus is that which has all its sides equal, but 
its angles are not right angles (Fig. 19.) 
A parallelogram, or rhomboid, is that which has-its 
opposite sides parallel (Fig. 20.). 
A trapezium is that of which the opposite sides are 
not parallel (Fig. 21.) 
A trapezoid is that of which only-two of the opposite 
sides are parallel (Fig. 22.) ‘ 
17. The diagonal ofa figure is a straight line which 
joins the vertices of two angles which are not adjacent. 
Thus, in Fig. 48. AC, AD, AE, &c. are diagonals of 
the figure ABCDEFG. . 
18. An equilateral polygon’ is that which has all its 
contained by the straight lines AB and AC 
by AB and AC is not equal to the 
$ thus, in Fig. 6. the angle contained by the lines AB and AD is the sum of the two angles 
by AB and AC, _and by AC and AD. If these are equal, it is double any one of them. 
Hence it appears, that like other quantities, angles admit of addition, subtraction, multiplication, and divicion. . 
Fig. 10. 
Figs. 11, 
12, &e. to 
Fig. 22. 
Figs. 11, 
12, and 13. 
Fig. 14, 
Fig. 15. 
Fig. 16. 
Fig. 17. 
Fig. 18, 
Fig. 19. 
Fig. 20. 
Fig. 21. 
Fig. 22. 
Fig. 48. 
