GEOMETRY.

Various are the opinions upon the best modes of beginning to learn to draw: and it is by no means easy to decide upon this point, as so much must always depend upon the genius, turn of mind, and opportunities of the student. But, for general purposes, and when circumstances will admit of it, we have no hesitation in recommending to begin by the study of geometry and perspective.

The first forms the best introduction to a knowledge of form, by giving accurate ideas respecting the most simple forms, of which all the others may be considered as compounds: and the last is absolutely necessary, not only to enable us to draw the representations of regular objects, but even to see them correctly and it is certain that no one unacquainted with its rules can ever attain the power of drawing, without making the grossest mistakes.

Geometry is a branch of mathematics which treats of the description and properties of magnitudes in general.

Definitions or Explanations of Terms.

1. A point has neither length, breadth, nor thickness. From this definition it may easily be understood that a mathematical 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 measure its dimensions with the naked eye, arises only from its smallness. The same reasoning may be applied to every thing that is 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 representation 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 impossible 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 meet. Fig. 1.

Pl. 3.

4. A right line is what is commonly called a straight line, or that tends every where the same way.

5. A curve is a line which continually changes 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 A B, and B C, which form the angle, are called the legs or sides; and the point B where they meet, is called the vertex 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 by three letters, observing to place in the middle the

letter at the vertex, and the other two at the end of each leg, as the angle A B C.

8. When one line stands upon another, so as not to lean more to one side than to another, both the angles which it makes with the other are called right-angles, as the angles A B C and A B D, Fig. 3, and all right-angles are equal to each other, being all equal to 90°; and the line A B is said to be perpendicular 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-angles to the plane of the horizon, or level surface of the earth, or to the surface of water, which is always level. The sides of a house are vertical.. But a line may be perpendicular to another, whether it stands upright or inclines to the ground, or even if it lies flat 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 A B C, and ABD be equal, the line AB is perpendicular to CD, whatever may be its position in other respects.

9. When one line, BE (Fig. 3), stands upon another, CD,, so as to incline, the angle E BC, which is greater than a right-angle, is called an obtuse angle; and that which is less than a rightangle, is called an acute angle, as the angle E B D.

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 a bounded space, and is either a surface or a solid.

12. A superficies, or surface, has length and breadth only. The extremities of a superficies are lines.

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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.

A curved surface is that which will not coincide with a straight line in any part. Curved surfaces may be either convex or concave.

A convex surface is when the surface rises up in the middle, as, for instance, a part of the outside of a globe.

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.

13. Every surface bounded by straight lines only, is called a rectilinear figure.

14. Three is the fewest number of sides that a rectilinear figure can have; and it is then called a triangle.

15. Triangles are of different kinds, according to the lengths of their sides.

An equilateral triangle has all its sides equal, as A B C, Fig. 4.

An isosceles triangle has two equal sides, as DEF, Fig. 5.

A scalene triangle has all its sides unequal, as GHI, Fig. 6.

16. Triangles are also denominated according to the angles they contain.

A right-angled triangle is one that has in it a right-angle, as A B C, Fig. 7.

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.

An obtuse-angled triangle has one obtuse angle, as Fig. 8.

An acute-angled triangle has all its angles acute, as Fig. 4.

An isosceles, or a scalene triangle, may be either right-angled, obtuse, or acute.

Any side of a triangle is said to subtend the angle opposite to it: thus A B (Fig. 7), subtends the angle A CB.

If the side of a triangle be drawn out beyond the figure, as A D (Fig. 8), the angle A, or C A B, is called an internal angle, and the angle CAD, or that without the figure, an external angle.

17. A figure with four sides is called a quadrilateral figure. They are of various denominations, as their sides are equal or unequal, or as all their angles are right angles or not.

18. 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 angles are right or not. Fig. 9, 10, 11, and 12, are all parallelograms.

When the angles of a parallelogram are all right angles, it is called a rectangular parallelogram, or a rectangle, as Fig. 11 and 12.

19. A rectangle may have all its sides equal, or only the opposite sides equal. When all its sides are equal, it is called a square, as Fig. 12.

20. When the opposite sides are parallel, and all the sides equal to each other, but the angles not