840 
GEOMETRY. 
Practical geometry, is that which applies 
those speculations and theorems to particular 
uses in the solution of problems, and in the 
measurements in the ordinary concerns of life. 
Speculative geometry again may be di- 
vided into elementary and sublime. 
Elementary or common geometry, is that 
which is employed in the consideration of 
right lines and plane surfaces, with the solids 
generated from them . And the 
Higher or sublime geometry, is that which 
is employed in the consideration of curve lines, 
conic sections, and the bodies formed of 
them. This part lias been chiefly cultivated 
by the moderns, by help of the improved state 
of algebra, and the modern analysis or flux- 
ions. 
We shall now proceed to give the princi- 
ples of practical geometry, beginning with 
Definitions or explanation of terms. — 1 . 
A mathematical point has neither length, 
breadth, nor thickness. From this definition 
it may be easily understood that a mathema- 
tical point cannot be seen nor felt; it can only 
be imagined. What is commonly called a 
point, as a small dot made with a pencil or 
pen, or the point of a needle, is not in reality 
a mathematical point ; for however small 
such a dot may be, yet if it be examined 
with a magnifying glass, it will be found to 
be an irregular spot, of a very sensible length 
and breadth ; and our not being able to mea- 
sure its dimensions with the naked eye, arises 
only from its smallness. The same reason- 
ing may be applied to every thing that is 
i usually called a point ; even the point of the 
finest needle appears like that of a poker 
when examined with the microscope. 
2. A line is length without breadth or 
thickness. What was said above of a point, 
is also applicable to the definition of a line. 
What is drawn upon paper with a pencil or 
pen, is not in fact a line, but the representa- 
tion of a line. For however fine you may 
make these representations, they will still 
have some breadth. But by the definition, 
a line has no breadth whatever, yet it is im- 
possible to draw any thing so fine as to have 
• no breadth. A line therefore can only be 
imagined. The ends of a line are points. 
3. Parallel lines are such as always keep 
at the same distance from each other, and 
which, if prolonged ever so far, would never 
jneet. See Plate Geometry, fig. 1. 
4. A right line is what is commonly called 
a straight line, or one that tends every where 
the same way. 
5. A curve is a line which continually 
fchanges its direction between its extreme 
points. 
6. An angle is the inclination or opening of 
two lines meeting in a point, fig. 2. 
7. The lines AB, and B€, which form the 
angle, are called the legs or sides ; and the 
point B, where they meet, is called the ver- 
tex of the angle, or the angular point. An 
angle is sometimes expressed by a letter 
placed at the vertex, as the angle B, fig. 2 ; 
but most commonly bv three letters, ob- 
serving to place in the middle the letter at the 
vertex, and the other two are those at the 
end of each leg, as the angle ABC. 
8. When one line stands upon another, so 
as not to lean more to one side than to an- 
other, both the angles which it makes with 
the .other are called right angles, as the an- 
gles ABC and ABD, fig. 3; and all right an- 
gles are equal to each other, being ail equal 
to 90° ; and the line AB is said to be perpen- 
dicular to CD. 
Beginners are very apt to confound the 
terms perpendicular," and plumb or vertical 
line. A line is vertical when it is at right an- 
gles to the plane of the horizon, or level sur- 
face of the earth, or to the surface of water, 
which is always level. The sides of a house 
are vertical. But a Hue may be perpendicu- 
lar to another, whether it stands upright, or 
inclines to the ground, or even if it lies fiat 
upon it, provided only that it makes the two 
angles formed by meeting with the other line 
equal to each other; as for instance, if the 
angles ABC and ABD be equal, the line 
AB is perpendicular to CD, whatever maybe 
its position in other respects. 
9 ■ When one line BE (fig. 3), stands upon 
another, CD, so as to incline, the angle EBC, 
which is greater than a right angle, is called 
an obtuse angle ; and that which is less than 
a right angle is called an acute angle, as the 
angle EBD. 
10. Two angles which have one leg in 
common, as the angles ABC and ABE, are 
called contiguous angles, or adjoining angles ; 
those which are produced by the crossing of' 
two lines, as the angles EBD and CBF, 
formed by CD and EF, crossing each other, 
are called opposite or vertical angles. 
11. A figure is abounded space, and is ei- 
ther a surface or a solid. 
12. A superficies, or surface, has length and 
breadth only. The extremities of a superfi- 
cies are lines. 
13. A plane, or plane surface, is that which 
is every where perfectly flat and even, or 
which will touch every part of a straight line, 
in whatever direction it may be laid upon it. 
The top of a marble slab, for instance, is an 
example of this, which a straight edge will 
touch in eVery point, so that you cannot see 
light any where between. 
14. A curved surface is that which will 
not coincide with a straight line in any r 
part. Curved surfaces may be either con- 
vex or concave. 
15. A convex surface is when the surface 
rises up in the middle ; as, for instance, a part 
of the outside of a globe. 
16. A concave surface is when it sinks in 
the middle, or is hollow, and is the contrary 
to convex. 
A surface may be bounded either by straight 
lines, curved lines, or both these. 
17. Every surface bounded by straight 
lines only is called a polygon. If the sides 
are all equal, it is called a regular polygon. 
If they are unequal, it is called an irregular 
polygon. Every polygon, whether equal or 
unequal, has the same number of sides as 
angles, and they are denominated sometimes 
according to the number of sides, and some- 
times from the number of angles they con- 
tain. Thus a figure of three sides is called a 
triangle, and a figure of four sides a quadran- 
gle. 
A pentagon is a polygon of five sides ; a 
hexagon has six sides ; a heptagon seven 
sides; an octagon eight sides; a nonagon 
nine sides; a decagon ten sides; an undeca- 
gon eleven sides; a duodecagon twelve 
sides. See Pentagon, &c. 
When they have a greater number of sides 
it is usual to call them polygons of 13 sides, 
of 14 skies, and so on. 
Triangles are of different kinds, aec rdiag 
to the lengths ol their sides. 
18. An equilateral triangle lias all its side; 
equal, as ABC, fig. 4. 
19 An isosceles triangle has two equal 
sides, as DEF, fig. 5. 
':0. A scalene triangle has all its sides un- 
equal, as GI4I, fig. 6. 
1 rumples are also denominate 1 according 
to (he angles they contain. 
21 . A right angled triangle is one that has 
in it a right-angle, as ABC, fig. 7. 
22. A triangle cannot have more than one 
right angle. The side opposite to the right 
angle B, as AC, is called the hypothenuse, 
and is always the longest side. 
23. An obtuse-angled triangle has one ob- 
tuse angle, as fig. 8. 
24. An acute-angled triangle has all its an- 
gles acute, as fig. 4. 
25. An isosceles, or a scalene triangle, may 
be either right angled, obtuse, or acute. 
26. Any side of a triangle is said to subtend 
the angle opposite to it: thus. A B (lie. 7) 
subtends the angle ACB. 
27. It the side of a triangle be drawn out, 
beyond the figure, as AD (fig. 8), the angle 
A, or CAB, is called an int rnal angle, and 
the angle CAD, or that without the figure, 
an external angle. 
28. A quadrangle is also called a quadrila- 
teral figure. 'I hev are of various denomina- 
tions, as their sides are equal or unequal, or 
as all their angles are right-angles or not. 
29. Every four-sided figure whose opposite 
sides are parallel, is called a parallelogram. 
Provided that the sides opposite to each other 
be parallel, it is immaterial whether the an- 
gles are right or not. Figs. 9, 10, 11, and 
12, are ail parallelograms. 
30. \\ hen the angles of a parallelogram 
are all right angles, it is called a rectangular 
parallelogram, or a rectangle, as figs. 1 1 and 
31. A rectangle may have all its sides 
equal, or only the opposite sides equal. When 
ali its sides are equal, it is called a square, as 
fig. 12. 
32. \\ hen the opposite sides are parallel, 
and all the sides equal to each other, but the 
angles not right angles, the parallelogram is 
called a rhombus, as fig. 1 0. 
33. A parallelogram having all its angles 
oblique, and only its opposite equal, is called 
a rhomboid, as fig. 9. 
34. When a quadrilateral, or four-sided 
figure, has none of its sides parallel, it is 
called a trapezium, as fig. 13; consequently 
every quadrangle, or quadrilateral, which is 
not a parallelogram, is a trapezium. 
35. A trapezoid has only one pair of its 
sides parallel, as fig. 14. 
36. A diagonal is a right line drawn between 
any two angles that are opposite in a polygon, 
as IK, fig. 15. In parallelograms the diagonal 
js sometimes called the diameter, because 
it passes through the centre of the figure. 
37. Complements of a parallelogram. If 
any point, as E (fig. 15), be taken in the dia- 
gonal of a parallelogram, and through that 
point two lines are drawn parallel to the 
sides, as AB, CD, it will be divided into 
four parallelograms, D, D, L, F, G, G. The 
two divisions, L, F, through which (he dia- 
meter does not pass, are called the comple- 
ments. 
38. Base of a figure, is the side on which it 
