Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Perceived visual angle wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
Contour line wikipedia , lookup
THE THEORY OF PARALLEL 72 LINES. THEORY OF PARALLEL LINES, BEING AN ATTEMPT TO DEMONSTRATE THE TWELFTH AXIOM OF EUCLID. BY THE REV. D. WILKIE. ART. Ill— THE READ BEFORE THE SOCIETY ON THE 21 ST the truth of the In the following paper, axioms, and of the first — I. eleven first twenty-six propositions of the hook of Euclid, and of these DEFINITIONS. JAN., 1SS2. only, When is first to be admitted. a line cuts two other lines, the angles on the opposite sides of the cutting line, and at its opposite extremities, are called II. When a line cuts two other nate angles equal, these two Alternate angles. making the lines, lines are said to alter- be Parallel. H C D B \ E I Thus, gles if ACF the line and CFD. Theorem 1. CFE HI cut the two lines are called alternate If a line cut two other ; AB, DE, and also lines, the an- BCF and making one -pair of the alternate angles equal, the remaining pair shall also be equal. Thus, if the angle pair of alternate equal. ACF angles, CFE, and CFD, be equal to the remaining BCF shall also be THE THEORY OF PARALLEL LINES. 73 ACF, BCF, being equal to two right angles (XIII. 1. Eucl.), and CFD, CFE, being also equal to two right angles the two ACF, BCF, are equal to CFD, CFE (Ax. 1.). And since ACF is equal to CFE, For, the angles ; the remainder Theorem BCF is equal to the remainder CFD. — Q. E. D. If a 2. line cut two parallel secting the line which cuts them lines, ivill have any line bi- its alternate angles equal. M A O Let AB any N B AM be parallel NB, to NM, rallel lines, shall For, the angle AB bisecting have and likewise the consequence of the its alternate angles equal in C, and meeting the pa- MAC being equal to CBN by supposition, side AC BCN also equal (XV.l.Eucl.) equal to CB by construction, the (XXVI. 1. Eucl.) the angle CNB, and triangles are entirely equal the angle AMC alternate angles. NMS. is equal to Hence, Wherefore, any parallel lines, line alternate angles also equal. its and the opposite angles ACM, two in cutting them, and having line, as „ makes its . also, the angle MNO is Hence, they are equal to AB, and meeting the angles equal, as was to line bisecting alternate be shown. Cor. —Hence, a line passing through the point C, and perpendicular to the one * This was first line, is perpendicular to both*. proved by Dr. R. Simson ; but his proposition is limited to and falling on the parallels at right angles, as i«J lines bisecting the cutting line, the Cor. K THE THEORY OF PARALLEL LINES. 74 Theorem lines, 3. — The perpendiculars drawn from points in distances from the line to one of two parallel the other, ivhich are at equal which perpendicular to both, is are equal to one another. Let AC and DE A B C D E F be parallel, and let (by Cor. Prop. 2.) points A BE be perpendicular to both is the line which ; and the let and C, be at equal distances from B, the perpen- diculars let fall from these points upon the other parallel, are equal to one another. AE and EC be drawn, the triangles ABE, CBE having AB equal to CB by supposition, BE common, and the angle ABE equal CBE being both right angles; are entirely equal. (Eucl. Consequently the line AE 4.) equal to CE and the angle BEA equal to BEC. Also, let AD be perpendicular to DF, and having made For, if I. EF equal to ED, join BED, and being equal to the whole BEA, AED. the part der the remainder Hence, the triangles first.) equal to the angle at D, which equal to is BEC BEF equal to equal to the remain- having the two sides in the one, and the angle between them equal, (by the IV. of the is CEF the part AED, CEF and the angle between them sides Then, the whole angle CF. equal to the two in the other, are entirely F CF Therefore the angle at is a right angle, and AD. Wherefore AD and CF, which are perpendicular and drawn from the points A to DF x and C at equal distances from the point B, are equal to one another. — Q. E. D. THE THEORY OF PARALLEL Theorem making If a line cut two lines, 4. 75 LINES. angles unequal, every line bisecting that the alternate and meet- line, ing the other two, shall make unequal angles with these two. A B \F E Thus, if AF the line AFH, FAI, the alternate angles ing cut the two lines AI, unequal, every line bisect- HF AF, and meeting AI, and HF, making have shall alternate its angles unequal. For, since the angle FAI is AFE, let AB equal to AF two that be bisected is in C, : sequently greater than In the lines CEF. — Hence it is A I. That If a line cut CBA, That — Q. let draw ECB, cutting the ABC (Eucl. is, is equal to CDA is 16.) I. CEF greater and con- the alternate angles HF may be AI and line bisecting shown to to AF have its E. D. manifest that the line bisecting and drawn perpendicular with it, same manner, any alternate angles unequal. Cor. let AFE, But the angle than the interior opposite one and meeting the be parallel to HF, then, the angle (by the 2d of this paper.) are unequal. be and through B and E parallels in FAB unequal to HF, will make unequal AF angles is, two lines, qual, the line which to one of the two making the bisects that line lines, alternate angles une- and is perpendicular makes oblique angles with the other. Theorem o. — If a line cut two other lines, alternate angles unequal, the lines drawn from making the the one line THE THEORY OF PARALLEL 76 LINES. perpendicular to the other, will continually decrease, the nearer to the acute angle being always greater than the more remote on : the other hand, these perpendiculars will continually increase on the side of the obtuse angle, the nearer being always less than the one more remote. U j) Thus, if BF V M 1 qual angles with them, so that while FBC is AC, DE, and make une- cut the two lines BFD may be a right angle, acute (by the 4th of this paper,) then, from if L and K in the line AC, perpendiculars DE at M and I, LM shall be less than BF, any other points be let fall and KI less upon less LM. than the triangle FK to FK FBK, (Eucl. Much greater than right-angled at FK, drawing LM is less than GH, and GH it. K, the side BF For the same reason, 19.) I. more, then, Further, let I, BF the side FK is greater is in the tri- greater than which was proved be to greater than KI. L be any point between B and K, then, Upon perpendicular to have the angle 16.) is be perpendicular to AC, then, in angle FKI, right-angled at KI. BF than any perpendicular beyond For, supposing than Also, BLF DE, the triangle Therefore, in the triangle BLF, to the acute angle is FBL son, in the triangle must obtuse because greater (by Eucl. than the inward opposite angle the obtuse angle BFL BFL, FKL which FLM, I. is right. the side BF, opposite to greater than the side (Eucl. I. 19.) the side FL FL opposite For the same rea- opposite the right- THE THEORY OF PARALLEL angle FML, is LM, opposite greater than 77 LINES. to the acute angle LFM, which is acute because only a part of the right-angle BFM. Much more must the line BF, which was proved to be greater than FL, be greater than LM. For the like LM reasons, is Much more, KI. Thus, it MK, greater than then, is AC, may from points DE, drawn from continually decrease as these points recede from the acute angle at B. it greater than LM greater than KI. appears, that perpendiculars to points in the line ner, MK and And in the be proved that perpendiculars to in the AC, same man- DE, drawn continually increase as these points recede from the obtuse angle at B, as asserted.* Theorem hvo parallel Let the Every 6. lines, is lines AC, line which is perpendicular to one of ABC perpendicular to the other. F H D E DF be parallel, having BE ular to both (by the Cor. our 2d to ;) perpendic- any other line perpendicular to the one shall be perpendicular to the other. For, if BA be taken equal to BC, and drawn perpendicular to DF, But if CF had its alternate angles unequal, no other perpendicular as to DF, could be equal equal to CF, the angle — be both these perpendiculars shall be equal (by the 3d of this paper. ) AC AD, CF to BCF CF. is AD, drawn from Therefore since equal to AD is CFH, which is a *Note. This is the most important part of the theory, now attempted, as all It implies, that, if the rest of the demonstrations succeeding depend upon it. two lines are not parallel, according to the definition given above, no two per~ pendiculars drawn from the one upon the other, can be found equal. THE THEORY OF PARALLEL 78 Hence, right angle. FC is LINES. same manner, any other perpendicular shown to be so to AC. the Theorem AC. perpendicular to to And, DF may be Parallel lines are every where at the 7. in same perpendicular distance. Let LM and OQ be formed by some L M N O P Q parallel, or have the alternate angles line cutting these equal, any perpendiculars NQ shall be equal. For, bisect LN in M (Eucl. 10.) and join LP and PN. Also, let PM be perpendicular to LM, and will be so to OQ, Then, the triangles LMP, MNP are entirely equal, (6.) (Eucl. 4.) and have the side LP equal to NP and the to them, as LO, I. it I. angles Now PLM, LPM, since the the part to the MLP whole equal to MLO is MNP, equal to MNP, MPN. MNQ the remainder and (6.) OLP is equal QNP. In like manner the angle LPO is angle NPQ. Wherefore the triangles OLP remainder equal to the and respectively equal to QNP, having two angles equal, and the side PN £ucl. LO is lequal to 26.) are entirely equal. LP equal Hence the to I. lars cutting the NQ, and thus all the perpendicutwo lines, may be shown to be equal. Theorem 8. side the side If a line cut two other alternate angles unequal, the two lines, lines will duced, on the side of the acute angle. making the meet y if pro- THE THEORY OF PARALLEL LINES. 79 THE THEORY OF PARALLEL 80 A, F, N, decrease by the equal points, EG. LINES. same manner, In the if more points AE, be taken on AB, differences, AF, the perpendiculars let fall from these points upon CD, will decrease by differences each equal to AE. By taking. AE, therefore, a sufficient number of times, it must at last, either end at C, or beyond But, by taking AF the same number of times it at X. at distances each equal to — upon AB, it must at last end in the line CD, or on the other has been just shown, that every line joining side, since it AC, and AB, is perpendicular If, for example, four parallel to CD. the corresponding points in AC, and to times AE therefore should be found equal to must end precisely in 11. Cor.) ; but if AC some point of the Hence is AB CD four times line CD, AF (Eucl. I. should be found to be more than four times, but less than five times would not meet AC, AE, then, four times AF produced, but five times would pass and CD, being both produced, must meet, and it. it on the side of the acute angle, that they meet. Quebec, December 27th, 1831. In the preceding paper, defect under I have endeavored to remove the which the science of Geometry has hitherto laboured from the imperfect elucidation of parallel To this attempt Length," inserted I alluded in in the my lines. paper on " Space and second volume of the transactions of this society. I have not had the pleasure of seeing the methods posed by Legendre to remedy this defect. Professor Playfair, one of his methods is pro- According to founded on the abstract doctrine of Functions, and therefore unsuitable to THE THEORY OF PARALLEL LINES. 81 be proposed to such as are entering on the study of Geometry. The other method is also stated by the same Professor and intricate a nature to to be of too abstruse, extensive An form part of an elementary system of that science. ingenious demonstration of this property of Parallel lines, was published Mr. Ivory in the Philosophical but ; it is Magazine As also very complicated. can judge, the preceding Theory is for 1822, by far as I sufficiently simple to be put into the hands of the youngest students of Geometry. The more 5th proposition and the last might be rendered easy of comprehension by dividing each of them into two, a distribution of which they will readily admit. In communicating this it Theory I beg further to state, that my forms part of a system of Geometry which occupied now nearly comfragment now presented, may leisure hours for several years, pleted. Of that system, the be considered as a necessary to omit fair and which is specimen, except as far as it was reference to the larger work, in an all extract requiring to be read independently by the lovers of demonstrative science. altered as to render it The extract has only been so far susceptible of separate perusal. This system of Geometry, on which have bestowed a I good deal of attention, possesses the following which, of course, 1. The I consider as so many peculiarities, advantages. usual treatises on this subject, presuppose the use of the ruler and compass. That which presupposes only the use of the ruler. use of the compass are in The derived from it I speak of, properties and first principles, by the same synthetical reasoning, upon which the whole science depends. 2. The science is definitions, on which it is entirely built, are all strictly L known that this logical. The terms well THE THEORY OF PARALLEL 82 which cannot be logically defined, are explained, principles they involve, 3. No involves 4. assumed on definition is attempted, is shown LINES. their proper grounds. till to be possible, or and the the property which assumed All the propositions are simple. it to be so. It is at no time at- tempted to prove two or more properties of a geometrical figure in the same demonstration. But where a series of truths are intimately connected so as to require a connected view to be given of them, such a scholia or notes appended view is given of them in to the propositions. There are other minor points of difference which need not be here enumerated. The the strictest geometrical reasoning with the most lucid illustration of object has been, to combine which the subject admits. Particular care has been taken to render the introductory demonstrations as simple and easy as possible. trical When a habit of geome- reasoning has once been formed, greater conciseness of expression, and be safely admitted. more difficult forms of ratiocination, may