any error involved in the latter theory, it must not only be infinitely small, but must remain infinitely small after all the magnifying processes to which it could possibly be subjected. But there is no error; for, if we suppose that there be an error which we may represent by A, since the aggregate of all the quantities neglected in arriving at the result is infinitely small, that is, as small as we choose, we may choose it to be smaller than A; and, therefore, the error A is greater than the greatest possible error which could be obtained, a manifest absurdity, but one which cannot be avoided as long as A is any thing. The term direction is introduced into this treatise without being defined; but it is regarded as a simple idea, and to be as incapable of definition as length, breadth, and thickness; and this innovation will probably be pardoned, when it is seen how much it contributes to the brevity and simplicity of demonstration, which I have everywhere studied. BENJAMIN PEIRCE. Complement and supplement of angle (22), Vertical and adjacent angles (23, 24), Sum of angles about a point (25, 26), Definition of parallel lines (27), Parallel lines cannot meet (28), . Angles which have their sides parallel (29), External-internal, alternate-internal angles (30), Interior angles on the same side (32), Cases of parallel lines (31, 33, 35, 36), PERPENDICULAR AND OBLIQUE LINES, Only one perpendicular from a point to a line (37), Oblique lines drawn from a point to a line (38, 39, 41), Shortest distance from a point to a line (39), Plane figure, polygon, and its perimeter (43); triangle, quadri- Equilateral and equiangular polygon (49); polygons equilateral and equiangular with respect to each other, and their homolo- Three cases of equal triangles (51, 53, 61), Greater side of a triangle opposite the greater angle (62), CONTENTS. Sum of the interior angles of a polygon (72-76), The diagonal of a parallelogram bisects it (77), Opposite sides and angles of a parallelogram (78), Parallel lines between two others (79), THE CIRCLE AND THE MEASURE OF ANGLES, Circumference and centre of circle (85); its radius and diame. Comparative magnitude of radii and diameters (87), Line inscribed in a circle (92); greatest inscribed line (93), Measure of angle formed by tangent and chord (121), Measure of angle formed by two secants, two tangents, or a tangent and secant (122); of angle formed by two chords (123), Arcs intercepted by parallels (124), Line joining the centres of circles which cut each other (126); CHAPTER IX. PROBLEMS RELATING TO THE FIRST EIGHT CHAPTERS, 37 Lines divided into equal parts (158), To divide a line into equal parts (159), A line parallel to one side of a triangle divides the other two To divide a line into parts proportional to given lines (164, 165), 47 To find a fourth proportional to three lines (165, 166), To divide one side of a triangle into two parts proportional to Similar polygons and their homologous sides; similar arcs (170), Intersections of lines drawn through the vertex of a triangle Division of the right triangle into similar triangles by a perpen- 53 |