angle B A D, the sum will likewise be 180°. Hence the angle EAC must be equal to the angle BAD. In the same manner, if the angle BAD be added either to DAC or BAE, it will give a sum of 180°, and, consequently, the angles DAC and BAE are equal. (21.) If from any proposed point P (fig. 12.), several straight lines be drawn to a given straight line A B, and if one, PM, of these straight lines be perpendicular to A B, it will be shorter than any of the others. Let PC be any one of the others, and suppose PM continued below A B until Mp shall be equal to MP, then let a straight line be drawn from C to p; now if we suppose the paper folded over so that the line Mp shall lie upon the line MP, the fold of the paper will correspond with the line A B, because the angle PMB is equal to the angle PMB; and since the line MP is equal to the line Mp, it is evident that the line Cp will precisely cover the line CP, and therefore must be equal to it. Now since a straight line is the shortest distance between two points, PMp will be less than PCp, and consequently PM which is half the former will be less than PC which is half the latter; and in like manner the line PM may be proved to be less than any other line which can be drawn from P to the line A B. fig. 12. P (22.) That only one perpendicular can be drawn from a given point P, to a straight line A B, is a proposition so nearly self evident that it admits of no other kind of proof but that which consists in showing that any thing contrary to it must be absurd. If it be admitted, for a moment, to be possible that a second perpendicular could be drawn, let the line PC, fig. 12., represent that per pendicular, and, as before, A C M B draw Cp; by the same process of folding back the figure, it may be shown that the angle pCM is equal to PCM, because the one exactly covers the other. But since PC is here supposed to be perpendicular to A B, the angle PCM is a right angle, therefore p CM must also be a right angle; and this being the case, PCp must be one continued straight line: but PMp is also one continued straight line. Thus there would be two different straight lines joining the same points, Pp, which is contrary to what has been already explained (4). Hence, the supposition of the possibility of drawing from a point to a straight line more than one perpendicular, involves an absurdity. (23.) From this reasoning it immediately follows, that if from any two points in a straight line two lines be drawn both perpendicular to that straight line, these lines can never meet, for, if they did, then they would, in fact, be two perpendiculars drawn from the point where they would meet to the same line, which is contrary to what has been just demonstrated. (24.) If several lines be drawn from the same point, P (fig. 13.), to the same straight line, A B, one of which is perpendicular to it, those lines will be equal which meet A B at points equally distant on different sides of the perpendicular, and the more distant from the perpendicular the points are at which such lines meet the line A B, the longer will such lines be. Let PM, as before, be the perpendicular, and take MC equal to M C', the lines PC and P C' will be equal; for if the paper be folded over along the line PM, the line M C' will fall upon the line MC, because the angle PMC' will be equal to the angle PMC, and the point C' will fall upon the point C, because the line MC' is equal to the line MC; since then the point C' falls upon the point C, the line P C must coincide with the line PC, and therefore they must be equal. Let D be a point on A B more distant from M than Cis. We are then to prove that the line PD must be greater than PC. Suppose a line CE, drawn from C at right angles to CP; since PC is perpendicular to CE it will be less than PE (21.), but PE is less than PD, therefore PC is less than PD, and in the same manner it may be proved that the more distant any line is from the perpendicular PM, the greater it is. (25.) The same process of investigation will easily show, that the lines drawn from P to points equally distant from the perpendicular are inclined at equal angles to the line A B, and that they are also inclined at equal angles to the perpendicular PM. It will also follow, that the more remote the lines are from the perpendicular, the less will be the angles at which they are inclined to A B, and the greater will be the angles at which they are inclined to the perpendicular. (26.) It is obvious that the lines more distant from the perpendicular will make greater angles with it; but it is not, at first view, so apparent that they will make less angles with the line A B. In fig. 14. let the lines CP and DP be continued beyond the point P, and let the angle MCP be imagined to be moved towards the point D, CM still remaining upon the line A B. It is evident that as the angle is thus moved, the point where its side crosses the side of the angle MDP, will move from its present position towards D, taking successively the positions P', P", &c.; the length of that portion DP', DP", &c. of the side DP, which is contained within the angle MDP, will gradually diminish, and when the angle C is moved to D its side will lie altogether above the side of the angle at D, and therefore the angle C. must necessarily be greater than the angle D. (27.) For the construction of a square, or the model of a right angle, it is necessary that we should be able to delineate an exact right angle by which such a square may be made. The preceding principles indicate a method of accomplishing this. Having drawn any straight line, such as A B (fig. 13.), take any point, M, upon it, and on each side of M take equal distances, MC, MC. Find a point, P, which shall be equally distant from C and C', and draw a straight line from this point P to M. That line PM will be perpendicular to A B. The greater the distances MC and MC' are taken, the more accurately will the position of the perpendicular be defined. CHAP. III. OF PARALLEL LINES. (28.) Ir was shown in the last Chapter (23.) that if two straight lines be drawn from any two points upon a given straight line, both perpendicular to it, they can never meet, to whatever distance they may be drawn. Two such lines are said to be parallel. The doctrine and properties of parallel lines have always held a conspicuous place in geometry, and have been the more remarkable, in that no geometrical skill has ever succeeded in reducing their investigation to the same simple and fundamental principles, which have always been considered as conferring the last degree of precision and clearness on the investigations of elementary geometry. Even the most remote and difficult propositions in other parts of the science, are deduced by rigorous demonstration from certain general axioms admitted to be so clear in their nature, that their demonstration, or their deduction from other more simple and evident truths, is equally unnecessary and impossible. But it has been the reproach of geometry, that the theory of parallel lines has never been established, without either introducing among the axioms some proposition whose truth is less evident than that of many other propositions of geometry already admitted to be capable of, and to require proof; or by introducing methods of investigation, deficient in the rigour and foreign to the spirit which characterises every other part of elemetary geometry. 1 Probably the origin of this difficulty may be traced to the very nature of parallels, and the hopelessness of |