direction, the first side of the one is equal to the first side of the other, the second side of the one, to the second side of the other, and so on. 22. Two polygons are equiangular, or mutually equi angular, when their angles, taken in the same order, are equal, each to each. 23. A DIAGONAL of a polygon is a line joining the ver tices of two angles, not consecutive. 24. A BASE of a polygon is any one of its sides on which the polygon is supposed to stand. 25. Triangles may be classified with reference either to their sides, or their angles. When classified with reference to their sides, there are two classes: scalene and isosceles. 1st. A SCALENE TRIANGLE is one which has no two of its sides equal. 2d. An ISOSCELES TRIANGLE is one which has two of its sides equal. When all of the sides are equal, the triangle is EQUILATERAL. When classified with reference to their angles, there are are two classes: right-angled and oblique-angled. 1st. A RIGHT-ANGLED TRIANGLE is one that has one right angle. The side opposite the right angle, is called the hypothe nuse. 2d. An OBLIQUE-ANGLED TRIANGLE is one whose angles are all oblique. If one angle of an oblique-angled triangle is obtuse, the triangle is said to be OBTUSE-ANGLED.. If all of the angles are acute, the triangle is said to be ACUTE-ANGLED. 26. Quadrilaterals are classified with reference to the relative directions of their sides. There are then two classes the first class embraces those which have no two sides par allel; the second class embraces those which have two sides parallel. Quadrilaterals of the first class, are called trapeziums. Quadrilaterals of the second class, are divided into two species: trapezoids and parallelograms. 27. A TRAPEZOID is a quadrilateral which has only two of its sides parallel. 28. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel, two and two. There are two varieties of parallelograms: rectangles and rhomboids. 1st. A RECTANGLE is a parallelogram whose angles are all right angles. A SQUARE is an equilateral rectangle. 2. A RHOMBOID is a parallelogram whose angles are all oblique. A RHOMBUS is an equilateral rhomboid. AXIOMS. 1. Things which are equal to the same thing, are equal to each other. 2. If equals be added to equals, the sums will be equal. 3 If equals be subtracted from equals, the remainders. will be equal. 4. If equals be added to unequals, the sums will be unequal. 5. If equals be subtracted from unequals, the remainders will be unequal. 6. If equals be multiplied by equals, the products will be equal. 7. If equals be divided by equals, the quotients will be equal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. 10. All right angles are equal. 11. Only one straight line can be drawn between two points. 12. The shortest distance between any two points is measured on the straight line which joins them. 13. Through the same point, only one line can be drawn parallel to a given line.. POSTULATES. 1. A straight line can be drawn between any two points. 2. A straight line may be prolonged to any length. 3. If two lines are unequal, the length of the less may be laid off on the greater. 4. A line may be bisected; that is, divided into two equal parts. 5. An angle may be bisected. A perpendicular may be drawn to a given line, either from a point without, or from a point on the line. 7. A line may be drawn, making with a given line an angle equal to a given angle. 8. A line may be drawn through a given point, parallel to a given line. NOTE. In making references, the following abbreviations are employed, viz. A. for Axiom; B. for Book; C. for Corollary; D. for Definition; 1. for Introduction ; P. for Proposition; Prob. for Problem; Post. for Postulate; and S. for Scholium. In referring to the same Book, the number of the Book is not given; in referring to any other Book, the number of the Book is given. PROPOSITION I THEOREM. If a straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles. Let DC meet AB at C: then will the sum of the angles DCA and DCB be equal to two right angles. E D ECA and ECB will both be right angles, and conse quently, their sum will be equal to two right angles. The angle DCA is equal to ECA and ECD (A. 9); hence, But, of the angles DCA+ DCB ECA + ECD + DCB; ECD + DCB is equal to ECB (A. 9); hence, DCA + DCB = ECA + ECB. The sum of the angles ECA and ECB, is equal to two right angles; consequently, its equal, that is, the sum of the angles DCA and DCB, must also be equal to two right angles; which was to be proved. Cor. 1. If one of the angles DCA, DCB, is a right angle, the other must also be a right angle. Cor. 2. The sum of the angles BAC, CAD, DAE, EAF, formed about a given point on the same side of a straight line BF, is equal to two right angles. For, their sum is equal to C E B -F A |