The uniform tracing of a line which through its whole extent stretches in the same direction, gives the idea of a straight line. No more than one straight line can therefore join two points; and if a straight line be conceived to turn as an axis about both extremities, none of its intermediate points will change their position. From our idea of the straight line is derived that of a plane surface, which, though more complex, has a like uniformity of character. A straight line connecting any two points situate in a plane, lies wholly on the surface; and consequently planes must admit, in every way, a mutual and perfect application. Two points ascertain the position of a straight line; for the line may continue to turn about one of the points till it falls upon the other. But to determine the position of a plane, it requires three points; because a plane touching the straight line which joins two of the points, may be made to revolve, till it meets the third point. The separation or opening of two straight lines at their point of intersection, constitutes an angle. If we obtain the idea of distance, or linear extent, from contemplating progressive motion, we derive that of divergence, or angular magnitude, from the consideration of revolving motion. GEOMETRY is divided into Plane and Solid; the former confining its views to the properties of space figured on the same plane; the latter embracing the relations of different planes or surfaces, and of the solids which these describe or terminate. In the following definitions, therefore, the points and lines are all considered as existing in the same plane. BOOK I. DEFINITIONS. 1. A crooked line is that which consists of straight lines not continued in the same direction. 2. A curved line is that of which no portion is a straight line. 3. The straight lines which contain an angle are termed its sides, and their point of origin or intersection, its vertex. To abridge the reference, it is usual to denote an angle by tracing over its sides; the letter at the vertex, which is common to them both, being placed in the middle. Thus, the angle contained by the straight lines AB and BC, or the opening formed by turning BA about the point B into the position BC, is named ABC or CBA. A 4. A right angle is the fourth part of an entire circuit or revolution of a straight line. B It is manifest that all right angles, being derived from the same measure, must be equal to each other. If a straight line CB stand at equal angles CBA and CBD on another straight line AD, and if the surface ACD be conceived laid over towards the opposite side, the point B and the line AD remaining in the same place; CB will, in this new position EB, make angles EBA and EBD equal to the former, and therefore all of them equal to each other. But the four angles ABC, CBD, DBE, and EBA constitute, about the point B, a complete revolution; or the line BA in forming them, -D B by its successive openings, would return into its original place, -and consequently each of those angles is a right angle. The angle contained by the opposite portions DA and DB of a straight line is hence equal to two right angles; and, for the same reason, all the angles ADC, CDE, EDF and FDB, formed at the point D and on the same side of the straight line AB, are together equal to two right angles. E -B D 5. The sides of a right angle are said to be perpendicu lar to each other. 6. An acute angle is less than a right angle. 7. An obtuse angle is greater than a right angle. 8. One side of an angle forms with the other produced a supplemental or exterior angle. 9. A vertical angle is formed by the production of both its sides. 10. The inverted divergence of the two sides of an angle, or the defect of the angle from four right angles, is named the reverse angle. The angle DBE is vertical to ABC, ABD is the supplemental or exterior angle, and the angle made up of ABD, DBE, and EBC, or the -D B E It is apparent that vertical angles, or those formed by the same lines in opposite directions, must be equal; for the angles CBA and ABD which stand on the straight line CD, being equal to two right angles, are equal to ABD and DBE, and, omitting the common angle ABD, there remains CBA equal to DBE. 11. Two straight lines are said to be inclined to each other, if they meet when produced; and the angle so formed is called their inclination. 12. Straight lines which have no inclination, are termed parallel. 13. A figure is a plane surface included by a linear boundary called its perimeter. 14. Of rectilineal figures, the triangle is contained by three straight lines. |