EXPLANATIONS. THE arguments employed to demonstrate any proposition in Geometry must be founded more or less on the antecedent Propositions, or on the Definitions, Postulates, Axioms, Hypothesis, or Construction. These are respectively referred to, in the following pages, under the abridged forms Def. Post. Ax. Hyp. and Const. The Hypothesis is the condition assumed or taken for granted. Thus, when it is affirmed that in an isosceles triangle, the angles at the base are equal, the Hypothesis of the proposition is, that the triangle is isosceles, or that its legs are equal. The Construction is the addition made to, or change made in the original figure, by dividing or drawing lines, &c., in order to adapt it to the argument of the demonstration or the solution of the problem. The conditions under which these changes are made, are as indisputable as those contained in the hypothesis. Thus, if we draw a line and make it equal to a given line, these two lines are said to be equal by Construction. A+ B, or A plus B, A — B, or A minus B, The signs and are to be read plus and minus. means A with B added to it, or the sum of A and B. means the difference of A and B, or A with B taken from it. The sign of multiplication is X; thus AX B signifies A multiplied by B. In speaking of rectangles, the adjacent sides are written with a point between them; thus ABCD expresses the rectangle contained by the sides AB and CD. The square of AB, or AB'AB, may be also written AB2. Equality is expressed by the sign, which may be read equal to, or is equal to, or are equal to. Thus, AB signifies that A is equal to B; A+B C, expresses that A and B together are equal to C. The sign is the distinction of an angle. Thus CAB means the angle CAB. N.B.-The portions between brackets occurring in the following pages, are editorial glosses, and not in the original text. EUCLID'S ELEMENTS OF PLANE GEOMETRY. BOOK I. DEFINITIONS. 1. A POINT is that which has position, but not magnitude. 2. A LINE is length without breadth. COROLLARY.-The extremities of a line are points; and the intersection of one line with another is also a point. 3. A right or straight line is that of which the successive points lie in the same direction. COR.-Hence, two straight lines cannot enclose a space; for if they meet in two points, since they both lie in the same direction with those points, they must coincide between them, and they must form throughout one continued straight line. 4. SURFACE is that which has length and breadth, without thickness. COR.-The extremities of a surface are lines. 5. A PLANE, or plane surface, is a surface in which any two points being taken, the straight line joining them, lies wholly in that surface. 6. A plane rectilinear ANGLE is the inclination of two straight lines to one another in the same plane; which B lines meet together, but do not lie in continuation of each other. [The two straight lines which, meeting together, make an angle, are called the LEGS of that angle; and it must be observed, that the magnitude of an angle does not depend on the length of its legs, but solely on the degree of their inclination to each other. The point at which the legs meet is called the VERTEX of the angle. A F G E B An angle may be designated by a single letter when its legs are the only lines which meet together at its vertex thus if GC and EC alone met at C, the angle made by them might be called the angle C. But when more than two lines meet at the same point, it is necessary, in order to avoid confusion, to employ three letters to designate an angle about that point, the letter which marks the vertex of the angle being always placed in the middle: thus the lines GC and EC meeting together at C make the angle GCE or ECG: the lines GC and EC are the legs of the angle; the point C is its vertex. In like manner may be designated the angle GCA; or the angle ECA, which is the sum of GCE and GCA; and so of the other angles ECB, BCD, ECD, &c., round the same point. When the legs of an angle are produced (or continued) beyond its vertex, the angles made by them on both sides of the vertex are said to be vertically opposite to each other thus, since GC is continued to D, and EC to F, the angles GCE and DCF are vertically opposite to each other. In like manner, GCF and DCE, FCA and ECB, ACG and BCD, ACE and BCF, GCB and DCA, are pairs of vertically opposite angles.] 7. When a straight line, standing on another straight line, makes with it the adjacent angles equal to each other, each of these angles is called a RIGHT at right angles with, or to be perpendicular, or a 8. An obtuse angle is an angle greater than a right angle. 9. An acute angle is an angle less than a right angle. 10. Straight lines are said to be parallel to one are incapable of meeting in a single point, 11. A FIGURE is that which is enclosed by one or more boundaries. [The space enclosed within a figure is called its AREA.] 12. A CIRCLE is a plane figure bounded by one line called the CIRCUMFERENCE or periphery; to which all straight lines drawn from a certain point within the figure, are equal. 13. That point is called the CENTRE of the circle. Ө 14. A DIAMETER of a circle is a straight line drawn through the centre, and terminating on both sides in the circumference. 15. A SEMICIRCLE is the figure contained by the diameter and the part of the circle cut off by the diameter. 16. Rectilinear figures are those contained by right or straight lines. 17. Of rectilinear figures, a TRIANGLE is that which has three sides. 18. A quadrilateral figure is that which has four sides. 19. A POLYGON is a rectilinear figure having more than four sides. 20. Of triangles, that which has all its sides equal, is said to be equilateral. 21. That which has two sides equal is called an isosceles triangle. A 22. A scalene triangle is that which has no two sides equal. 23. A right-angled triangle is that which has a right angle. 24. An obtuse-angled triangle is that which has an obtuse angle. 25. An acute-angled triangle is that which has three acute angles. [One side of a triangle, when considered apart from the other two sides, may be called the BASE of the triangle; or, considered in reference to the angle opposite to it, and made by the other two sides, it may be said to subtend that angle, or to be its SUBTENSE. In a rightangled triangle, the side opposite to the right angle is distinguished as the subtending side or HYPOTENUSE. When a side of a triangle is produced, the angle made by the produced part, with the other leg of the angle, from the vertex of which it was produced, is called an external angle; and the angle of the triangle having the same vertex with it, is the internal adjacent angle. The other two angles of the triangle are together called the internal remote angles; and of these, that of which the produced side is a leg, is the internal opposite; the other is the internal alternate angle.] 26. Of quadrilateral figures, a PARALLELOGRAM is that of which the opposite sides are parallel. 27. Of parallelograms, that is a SQUARE which has all its sides and angles equal. 28. An OBLONG is that which has equal angles, but not equal sides. 29. A RHOMB is that which has equal sides, but not equal angles. |