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29. The word, therefore, or hence, frequently occurs. Το express either of these words, the sign .. is generally used.
30. If the quotients of two pairs of numbers, or quantities, are equal, the
quantities are said to be proportional: thus, if
; then, A is to B as C to D. And the abbreviations of the proportion is, A: B:: C: D; it is sometimes written A: B=C : D.
1. "A POINT is that which has position, but not magnitude*." (See Notes.)
2. A line is length without breadth.
"COROLLARY. The extremities of a line are points; and the intersections "of one line with another are also points."
3. "If two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line."
"COR. Hence two straight lines cannot inclose a space. Neither can two
straight lines have a common segment; that is, they cannot coincide "in part, without coinciding altogether."
4. A superficies is that which has only length and breadth.
"COR. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines."
5. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
6. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
N. B. 'When several angles are at one point B, any one of them is ex'pressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these 'two is somewhere upon one of those straight lines, and the other upon the 'other line: Thus the angle which is contained by the straight lines, AB, 'CB, is named the angle ABC, or CBA; that which is contained by AB,
*The definitions marked with inverted commas are different from those of Euclid,
BD, is named the angle ABD, or DBA; and that which is contained by BD, CB, is called the angle DBC, or CBD; but, if there be only one an'gle at a point, it may be expressed by a letter placed at that point; as the angle at E.'
7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other, is called a perpendicular to it.
8. An obtuse angle is that which is greater than a right angle.
9. An acute angle is that which is less than a right angle.
10. A figure is that which is enclosed by one or more boundaries.—The word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it.
11. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
12. And this point is called the centre of the circle.
13. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
14. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.
15. Two lines are said to be parallel, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced.
16. A plane figure, terminated on all sides by straight lines, is called a rectilineal figure, or polygon, and the lines themselves taken together form the contour, or perimeter of the polygon.
17. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; and so on.
18. Of three sided figures, an equilateral triangle is that which has three equal sides.
19. An isosceles triangle is that which has only two sides equal.
20. A scalene triangle is that which has three unequal sides.
21. A right angled triangle is that which has a right angle. The side. opposite the right angle is called the hypotenuse.
22. An obtuse angled triangle, is that which has an obtuse angle.
23. An acute angled triangle, is that which has three acute angles. 24. Of four sided figures, a square is that which has all its sides equal and all its angles right angles.
25. An oblong, or rectangle, is that which has all its angles right angles, but has not all its sides equal.
26. A lozenge, or rhombus, is that which has all its sides equal, but its angles are not right angles.
27. A parallelogram, or rhomboid, is that which has its opposite sides parallel, but all its sides are not equal, nor its angles right angles.
28. And, lastly, the trapezoid, only two of whose sides are parallel. 29. All other four sided figures are usually called trapeziums.
30. A diagonal is a line which joins the vertices of two angles not adjacent to each other. Thus, BC, in the diagram of Theor. 27. is a diagonal.
31. An equilateral polygon, is one which has all its sides equal; an equiangular polygon, one which has all its angles equal.
32. Two polygons are mutually equilateral, when they have their sides equal to each other, and placed in the same order; that is to say, when following their perimeters in the same direction, the first side of the one is equal to the first side of the other, the second of the one to the second of the other, the third to the third, and so on.
The phrase mutually equiangular has a corresponding signification. In both cases, the equal sides, or the equal angles, are named homologous sides or angles.
33. We shall give the name, equivalent figures, to such as have equal surfaces.
Two figures may be equivalent, though very dissimilar: a circle, for instance, may be equivalent to a square, a triangle to a rectangle.
34. The denomination, equal figures, we shall reserve for such as, when applied to each other, coincide in all their points of this kind are two circles, which have equal radii; two triangles, which have all their sides equal respectively, &c.
1. LET it be granted that a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line.
3. And that a circle may be described from any centre, at any distance from that centre.
1. THINGS Which are equal to the same thing, are equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be taken from equals, the remainders are equal.
4. If equals be added to unequals, the wholes are unequal.
5. If equals be taken from unequals, the remainders are unequal.
6. Things which are doubles of the same thing, are equal to one another. 7. Things which are halves of the same thing, are equal to one another. 8. Two magnitudes, lines, surfaces, or solids, are equal, if, when applied to each other, they coincide throughout their whole extent. They then fill the same space.
9. The whole is greater than any of its parts.
10. The whole is equal to the sum of all its parts.
11. All right angles are equal to one another.
12. "Two straight lines which intersect one another, cannot be both "rallel to the same straight line."