ly the forms which bodies present, and the spaces which they occupy. Geometry is thus founded likewise on external observation; but such observation is so familiar and obvious, that the primary notions which it furnishes might seem intuitive, and have often been regarded as innate. This science, proceeding from a basis of extreme simplicity, is therefore supereminently distinguished, by the luminous evidence which constantly attends every step of its progress. PRINCIPLES. In contemplating an external object, we can, by successive acts of abstraction, reduce the complex idea which arises in the mind into others that are successively simpler. Body, divested of all its essential characters, presents the mere idea of surface; a surface, considered apart from its peculiar qualities, exhibits only linear boundaries; and a line, abstracting its continuity, leaves nothing in the imagination, but the points which form its extremities. A solid is bounded by surfaces; a surface is circumscribed by lines; and a line is terminated by points. A point marks position; a line measures distance; and a surface presents extension. A line has only length; a surface has both length and breadth; and a solid combines all the three dimensions of length, breadth, and thickness. 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. [5] 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 con B tained 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. 4. A right angle is the fourth part of an entire circuit or revolution of a straight line. 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 C 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, a bout the point B, a complete revolu E tion; or the line BA in forming them, 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 C F -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. |