APPENDIX. THIS Appendix contains a collection of miscellaneous propositions which will be found useful to those who wish to extend their studies beyond the range which is usually prescribed in a college course, and it includes brief notices of various topics which have not hitherto been generally admitted into the Elements of Geometry. Most of these topics are embraced under the comprehensive term of Modern Geometry, since they were either entirely unknown or had attracted but little attention until recent times. The treatise on Descriptive Geometry by Monge, published in 1794, gave a new impulse to the study of pure geometry. In 1803 Carnot published his Geometry of Position, and in 1806 an Essay on Transversals. In 1817 Brianchon extended this theory and that of harmonic pencils, applying them to the Conic Sections. In 1822 Poncelet published his treatise on the projective properties of figures, in which work he treated of Poles and Polars, Centres of Similitude, and, indeed, nearly every branch of Modern Geometry. Since that time the same subjects have been further developed by Steiner, Poinsot, Cayley, Chasles, Salmon, Cremona, and many other mathematicians. I take pleasure in acknowledging my obligations to Prof. H. A. Newton and Prof. J. E. Clark, to each of whom I am indebted for important assistance in preparing this Appendix. E. L. APPENDIX. MISCELLANEOUS PROPOSITIONS. PROPOSITION I. 1. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. Let ABCD be a quadrilateral whose diagonals AC, D BD bisect each other; then will ABCD be a parallelo A E B gram. Because AE-EC, and EB-ED, and the angles AEB, DEC, being opposite, are equal to each other, therefore the two triangles AEB and DEC are equal to each other (B. I., Pr. 6), and the angle EAB is equal to the angle ECD. Therefore AB is parallel to CD (B. I., Pr. 22). In like manner it may be proved that AD is parallel to BC, and therefore ABCD is a parallelogram. 2. Cor. If the diagonals of a quadrilateral bisect each other at right angles, the figure is a rhombus. PROPOSITION II. 3. The diagonals of a rectangle are equal. Let ABCD be a rectangle; then will its diagonals AC, p BD be equal to each other. Since the right angle ADC is equal to the right angle BCD, also the side AD is equal to BC, and CD is common A B to the two triangles ADC, BCD, therefore these two triangles are equal to each other, and hence AC=BD. 4. Cor. The diagonals of a square are equal; they bisect each other at right angles, and they also bisect the angles of the square. PROPOSITION III. 5. The straight line which joins the middle points of any two sides of a triangle is parallel to the third side, and is equal to one half of that side. |