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proportional, which is the 4th proposition of the 6th book of Euclid ; and thence be derives at once a proof of the 12th axiom. This method, however ingenious, is much too abstract and difficult for an elementary treatise. The principle of homogeneity on which it depends, is in itself sufficiently obvious ; but the mode in which it is here applied, would be intelligible to very few learners, when commencing the study of Geometry.
4. In Leslie's Elements of Geometry, a work which contains much curious and interesting matter, there is a singular failure in the attempt to prove that a straight line GE, falling on two parallels, AB, CD, makes the exterior angle EFC equal to the interior and remote angle EGA. The following is a correct abstract of his reasoning. Suppose a revolving straight line, extended in both directions, to pass always through F, and first let it take the position IFH. Then (I. 16.) the exterior angle EFH is greater than EGH, in the triangle HGF. Again, let it take the position LFK. Then (I. 16.) in the triangle FKG, the angle KFG, or its equal EFL, is less than EGK. When the incident line, therefore, meets AB above G, it makes an angle EFH greater than EGH, and when it meets AB below that point, it makes an angle EFL, which is less than the same angle. There is consequently a certain intermediate position, CD, in which the revolving line, not meeting AB on either side, and being therefore parallel to it, makes the exterior angle EFC equal to the interior EGA.
Now it is plain from the slightest consideration, that what is here demonstrated, is not what is intended, but simply that the straight line which makes the angle EFC equal to EGA, does not cut AB, but is parallel to it; which is the first part of Euclid's 28th proposition.
5. A method of proving the 12th axiom, has been proposed by M. Bertrand of Geneva, which is in substance the same as the following. Let AB, CD make with AC, the angles BAC, ACD, less than two right angles : AB, CD will meet, if produced. Draw AE, so that the angles EAC, ACD may be equal to two right angles : make also CF equal to AC, and the angle CFG equal to ACD. Then it would be shown by superposition, that the bands between AE, CD, and CD, FG, are equal, if taken of the same length. Now it is plain, that by the continued repetition of the angle EAB, as is shown by the dotted lines, the indefinite space contained between AE, AF, will be at length exhausted ; but that same space can be exhausted by no number of repetitions of the indefinite band, or space contained between AE, CD, as would appear by constructing more such bands, on bases taken in AF produced,
each equal to AC, or CF. Hence, therefore, the space contained between AE, AB, extended without limit, must be greater than the space contained between AE, CD, extended without limit; which, since AE is a common boundary of these spaces, could not take place, if AB lay wholly between AE and CD; and therefore AB must cut CD.
This method is very ingenious, and, on full consideration, will perhaps be regarded as satisfactory. The comparison, however, of the unlimited bands, and angular spaces is scarcely conformable to the strict precision which we expect and admire in geometry. A resemblance will be readily traced between this method and that of Pappus, illustrated hereafter. *
6. In Playfair's Elements of Geometry, the theory of parallels is founded on the following axiom, suggested by Ludlam in his Rudiments of Mathematics : “ Two straight lines which intersect one another, cannot be both parallel to the same straight line:” and from this he derives, indirectly, in a very simple manner,
the proof of the first part of the 29th proposition: and the proofs of the second and third parts are then given as in Euclid. It escaped the notice of these writers, however, that this axiom is virtually the same as Euclid's 12th axiom. To perceive this, it is only necessary to consider, that by the latter part of the 28th proposition, AGB is parallel to CHD, if the angles BGH, GHD be equal to two right angles ; while it is assumed in the 12th axiom, that
any other straight line drawn through G, will cut CHD, so that through G, only one straight line can be drawn parallel to CRD.
7. The theory of parallels has been founded by Thomas Simpson, in the second edition of his Elements of Geometry, on the following axiom : “ If two given points in a straight line are posited at unequal distances from another straight line, in the same plane, those two lines, being indefinitely produced on the side of the least distance, will meet one another."
From this it is easy to prove, that a straight line falling on two parallels, makes the alternate angles equal. For by drawing lines through the points in which it intersects the parallels, perpendicular to one of them, and which therefore (1. 28.) are parallel to one another, these perpendiculars are equal, as otherwise the given lines, by the axiom, would meet.
* In addition to the attempts to establish the theory of parallels without the 12th axiom, or the assumption of any new principle, may be mentioned a very acute and ingenious one contained in an edition of the first book of Euclid, published in 1830, by Mr. T. P. Thompson of Queen's College, Cambridge. As it would be difficult to give, in small compass, an adequate and intelligible idea of the method followed in that publication, the reader is referred to the work itself. It may be farther remarked, that the author adopts only the one axiom, “ that things which are equal to the same, are equal to one another ;" deriving from this, as corollaries, such of the other axioms, as he requires in his subsequent reasonings.
For the same reason, the parts of the given lines between these perpendiculars are equal; and therefore (1. 8.) the alternate angles are equal.
This method, if the axiom be admitted, is very simple and easy. When it is known, however, that the distance between two curves, or between a straight line and a curve, may continually decrease, and yet never vanish, it cannot be regarded as a perfect axiom, that two straight lines must meet, if their distances at different points be unequal.
8. Dr. Simson, in the Notes to his edition of Euclid, admits that the 11th axiom is not self-evident; and gives a proof of it founded on the following axiom : “A straight line cannot first come nearer to another straight line, and then go farther from it, before cuts it; and, in like manner, a straight line cannot go farther from another straight line, and then come nearer to it; nor can a straight line keep the same distance from another straight line, and then come nearer to it, or go farther from it; for a straight line keeps always the same direction." This axiom is preceded by two definitions,* explaining the terms employed in it; and it is followed by five propositions, the last of which is the 12th axiom, and which therefore prepares the reader for the 29th proposition. Such a lengthened and prolix proof, as has been well remarked by Playfair respecting the method of Clavius, “ leaves a strong suspicion, that the road pursued is by no means the shortest possible.'
9. Clavius, after offering objections to the method of Proclus next to be explained, assumes as an axiom, that “a line, all the points of which are equally distant from a straight line in the same plane with it, is itself a straight line.” Of this he endeavours to show the reasonableness ; and though he does not demonstrate its truth, he concludes, that it will gain assent much more readily than the 12th axiom of Euclid. He then applies the assumed principle in demonstrating the 12th axiom, in doing which he employs a very tedious and operose process. On the whole, therefore, his method has little to recommend it.
10. Proclus, in his Commentaries, adopts the principle previously assumed by Aristotle, that if two straight lines forming an angle, be extended infinitely from their point of intersection, their distance asunder will also increase, so as to exceed any given magnitude what
On this assumption, he proves that a straight line which intersects one of two parallels, intersects the other also;t as the two
The latter of these concludes thus: “ Two straight lines are said to keep the same distance from one another, when the distance of the points of one of them from the other is always the same.” It will be seen that this is liable to objection on the grounds stated in the next paragrapb of the text.
+ This principle is assumed without proof in Bonnycastle's Elements, as the basis of the theory of parallels ; and instead of being called an axiom, it is im.. properly classed among the postulates.
lines which by hypothesis, intersect one another, may, by the assumed principle, be extended so far, that their distance asunder will exceed the distance between the parallels, so that the line which cuts the one parallel, must also cut the other. He then demonstrates the 12th axiom by showing that, of the two lines on which the third falls, one cuts a line drawn parallel to the other ; and that consequently these lines must cut one another.
This method, were it developed with more care and rigour than have been employed by Proclus, would be preferable, on account of its conciseness, to several others; and the principle assumed, though not self-evident, will be admitted as readily as most others that have been proposed for the same purpose.
11. Many writers employ a principle similar to that of Clavius ; but instead of calling it an axiom, they assume it as the definition of parallel lines. Thus, in Elrington's Euclid, the method of Euclid is retained in the text; but in the Notes another method is given, as the foundation of which, parallel lines are defined to be those which are always equally distant however far they are produced. This definition, which is also adopted by Boscovich, Wolfius, Pardies, Fernandez, Dechales, and Emerson, and by Thomas Simpson in the first edition of his Elements, is liable to objection on nearly the same ground as the axiom of Clavius. If through two given points in a straight line, two equal perpendiculars be drawn on the same side, one straight line (I. post. 1.) and (I. def. 3. cor.) only one can be drawn through their other extremities : and a definition in which it would be assumed, that this line is parallel to the proposed line, would be perfectly legitimate. Without proof, however, we are not entitled to infer, that other perpendiculars from the first of these lines to the second are each equal to one of the first two perpendiculars, or that a perpendicular to one of the lines, is also perpendicular to the other, or finally, that the two lines will never meet, however far they may be extended. These propositions, though all true, require proof; and if we employ them without it, we should at least be aware, that we do what is inconsistent with strict reasoning.
12. A method different from all the foregoing is given by Dr. Cresswell of Cambridge, in his “Treatise of Geometry," published in 1819. He assumes the principle, that through any point within an angle a straight line may be supposed to pass, which shall cut the two straight lines that contain the angle. * Admitting this axiom, he demonstrates that if ABD be a right angle, and BAC an acute one, BD and AC will meet, if produced through C and D. To prove this he makes the angle BAE equal to BAC. Then, by the axiom, some line EBF may be drawn cutting AC, AE. If
This principle, which is tacitly assumed by Euclid, in the 20th proposition of the eleventh book, is easily derived from Euc. 1. ax. 12, and I. 31, when the theory of parallel lines is established by other means.
this coincide with BD, the proposition is plain; as it is also, if BF fall below BD. But if BF fall within the angle ABD, DB produced to G will fall within the angle ABE, and will therefore cut AE, since BE cuts AE. The proof is then completed by applying the angle BAE to BAC, so that AE may fall on AC, and BG will fall on BD, whence the truth of the proposition is evident.
From this proposition, the theory of parallel lines is easily derived ; and the assumption, or axiom, will perhaps be as readily admitted, as the others that have been assumed for the same purpose.
The following method of establishing the theory of parallel lines, which occurred to me several years ago, I shall venture to insert here. I am aware that it is liable to some of the objections that have been brought against other methods. At the same time, the axiom on which it is founded, will be more readily admitted than some others which have been proposed for the same purpose ; and it gives short and simple proofs of the 32d proposition, and the 12th axiom. The 29th proposition may then be demonstrated in the manner given by Euclid.
AXIOM. If a triangle be moved along a plane, so that its base may always be on the same straight line, its vertex describes a straight line equal to that which is described by either extremity of the base.*
BOOK I. PROP. XXXII. THEOR. If a side of any triangle be produced, the exterior angle is equal to the two interior and remote angles; and the three interior angles of every triangle are together equal to two right angles.
Let the side BC of the triangle ABC be produced to D: the exterior angle ACD is equal to the two interior and remote angles CAB, ABC; and the three interior angles of the triangle are together equal to two right angles.
Let the triangle ABC move along the plane in which it lies, so that the base BC may always be on the straight line BD, till the point B shall fall on C, and the triangle ABC take the posi
This axiom might be generalized ; since it is evident, that the same will hold respecting any figure whatever, having a straight line for one of its sides, and respecting any point whatever in that figure. As given above, however, it is sufficient for the intended purpose.