CHAPTER III.

PRINCIPLES OF GEOMETRY.

Definitions.

1. GEOMETRY is a department of science, by means of which we demonstrate the properties, affections, and measures of all sorts of magnitude.

2. Magnitude is a continued quantity, or any thing that is extended; as a line, surface, or solid.

3. A point is that which has no parts: i. e. neither length, breadth, nor thickness.

4. A line is a length without breadth or thickness.

Cor. The extremes of a line are points.

5. A right line is that which lies evenly, or in the same direction, between two points. A curve line continually changes its direction.

Cor. Hence there can only be one species of right lines, but there is infinite variety in the species of curves.

6. An angle is the inclination of two lines to one another, meeting in a point, called the angular point. When it is formed by two right lines, it is a plane angle, as A; if by curve lines, it is a curvilinear angle.

7. A right angle is that which is made by one right line A в falling upon another c D, and making the angles on each side equal, A B C A B D; so that A B does not incline more to one side than another: A B is called a perpendicular. All other angles are called oblique angles.

8. An obtuse angle is greater than a right angle, as R.

A

с в

D

9. An acute angle is less than a right angle, as s. 10. Contiguous angles are those made by one line falling upon another, and joining to one another, as R, s.

11. Vertical or opposite angles, are those made on contrary sides of two lines intersecting one another, as A, B.

B

A

12. A surface has only length and breadth. The extremes or limits of a surface are lines.

drawn

13. A plane is that surface which lies perfectly even between its extremes; or in which, right lines any way coincide.

14. A solid is a magnitude extended has length, breadth, and depth.

every way, or which

The terms or extremes of a solid are surfaces.

15. The square of a right line is the space included by four right lines equal to it, set perpendicular to one another.

16. The rectangle of two lines is the space included by four lines equal to them, set perpendicular to one another, the opposite ones being equal.

SEC. I.-Of Angles and Right Lines, and their Rectangles.

Prop. 1. If to any point c in a right line a B, several other right lines D C, E c are drawn on the same side; all the angles formed at the point c, taken together, are equal to two right angles, ACD + DCE + E C B = two right angles.

A

C

B

Cor. 1. All the angles made about one point in a plane, being taken together, are equal to four right angles.

Cor. 2. If all the angles at c, on one side of the line A B, are found to be equal to two right angles: then A C B is a straight line.

2. If two right lines, A B, C D, cut one another, the opposite angles E and G will be equal.

3. A right line, в н, which is perpendicular to one of two parallels, is perpendicular to the other.

4. If a right line c G, intersects two parallels

A D, F H; the alternate angles, A B E, and B E H will be equal.

C D

A B

17°

E

F

G

Cor. 1. The external angle c B D, is equal to the internal angle on the same side B E H.

Cor. 2. The two internal angles on the same side are equal to two right angles D B E B E H = two right angles.

5. Right lines parallel to the same right line, are parallel to one another.

6. If a right line A c be divided into two parts, A B, B C the square of the whole line is equal to the squares of both the parts, and twice the rectangle of the parts, A c = A B+ В C3 + 2 АВ . В С.

7. The square of the difference of two lines A C, B c, is equal to the sum of their squares, wanting twice their rectangle, A B A C2 + B C2 - 2 A C. B C.

=

8. The rectangle of the sum and difference of two lines is equal to the difference of their squares.

9. The square of the sum, together with the square of the difference of two lines, is equal to twice the sum of their squares.

SECTION II.-Of Triangles.

Definitions.

1. A triangle is a plane figure bounded by three right lines, called the sides of the triangle.

2. An equilateral triangle is one which has three equal sides.

3. An equiangular triangle is one which has three equal angles; and two triangles are said to be equiangular, when the angles in the one are respectively equal to those in the other.

4. An isosceles triangle has two sides equal.

5. A right-angled triangle is that which has a right angle. The side opposite to the right angle is called the hypothenuse. 6. An oblique triangle is one having oblique angles. 7. An obtuse angled triangle has one obtuse angle. 8. An acute angled triangle has three acute angles.

9. A scalene triangle has three unequal sides.

10. Similar triangles are those whose angles are respectively equal, each to each. And homologous sides are those lying between equal angles.

1. The base of a triangle, is the side on which a perpen

A®

dicular is drawn from the opposite angle called the vertex; the two sides, proceeding from the vertex, are called the legs. Prop. 1. In any triangle A в C, if one side B c be produced or drawn out; the external angle A C D will be equal to the two internal opposite angles A B.

B

E

2. In any triangle, the sum of the three angles is equal to two right angles.

Cor. 1. If two angles in one triangle be equal to two angles in another the third will also be equal to the third.

Cor. 2. If one angle of a triangle be a right angle, the sum of the other two will be equal to a right angle.

3. The angles at the base of an isosceles triangle, are equal.

Cor. 1. An equilateral triangle is also equiangular; and the contrary.

Cor. 2. The line which is perpendicular to the base of an isosceles triangle, bisects it and the verticle angle.

4. In any triangle, the greatest side is opposite to the greatest angle, and the least to the least.

5. In any triangle A B C, the sum of any two sides B A, A c, is greater than the third в c, and their difference is less than the third side.

6. If two triangles A B C, a b c, have two sides, and the included angle equal in each; these triangles, and their corresponding parts, shall be equal.

B

B

D

7. If two triangles A B C and a b c, have two angles and an included side equal, each to each; the remaining parts shall be equal, and the whole triangles equal.

8. If two triangles have all their sides respectively equal: all the angles will be equal, and the wholes equal.

9. Triangles of equal bases and heights are equal.

10. Triangles of the same height, are in proportion to one another as their bases.

11. If a line D E be drawn parallel to one side B C, of a triangle; the segments of the other sides will be proportional; A D:DB:: AE: E C. B

D

C

Cor. 1. If the segments be proportional, A D : DB::AE: EC; then the line D E is parallel to the side B c.

Cor. 2. If several lines be drawn parallel to one side of a triangle, all the segments will be proportional.

Cor. 3. A line drawn parallel to any side of a triangle, cuts off a triangle similar to the whole.

12. In similar triangles the homologous sides are proportional;

A BACDED F.

e

B

E

C

13. Like triangles are in the duplicate ratio, or as the squares of their homologous side.

14. In a right angled triangle B A C, if a perpendicular be let fall from the right angle upon the hypothenuse, it will divide it into two triangles similar to one another and to the whole, A B D, A D C.

B

A

0 D

Cor. 1. The rectangle of the hypothenuse and either segment is equal to the square of the adjoining side.

15. The distance à o of the right angle, from the middle of the hypothenuse is equal to half the hypothenuse.

16. In a right-angled triangle, the square of the hypothenuse

is equal to the sum of the squares of the two sides.

17. If the square of one side of a triangle be equal to the sum of the squares of the other two sides; then the angle comprehended by them is a right angle.

18. If an angle A, of a triangle в A C, be bisected by a right line A D, which cuts the base ; the segments of the base will be proportional to the adjoining sides of the triangle;

BD:DC: ABA C.

B

D

E

C

19. If the sides be as the segments of the base, the line A D bisects the angle a.

20. Three lines drawn from the three angles of a triangle to the middle of the opposite sides, all meet in one point.

21. Three perpendicular lines erected on the middle of the three sides of any triangle, all meet in one point.

22. The point of intersection of the three perpendiculars, will be equally distant from the three angles: or, it will be the centre of the circumscribing circle.

23. Three perpendiculars drawn from the three angles of a triangle, upon the opposite sides, all meet in one point.

24. Three lines bisecting the three angles of a triangle, all meet in one point.

25. If D be any point in the base of a scalene triangle, A B C then is A B2. DC + A C2. B D = A D2.B C + BC. B D. DC.