CHAPTER VIDefinition of Parallel Lines
As a matter of fact the question of definition becomes now of less importance than before, because of the method of our proof. As long as some form of definition was to be postulated as representing a thing really existing, it was all important that the definition should be impeccable, since it was to be employed in proof. But as we have completed our demonstration without recurring to such a proceeding, it becomes now a mere matter of method and proper logical definition. Both propositions, the one containing the definition of parallels, and that known as the fifth postulate, have now become legitimate and valid, for the reality of both constructions has been shown. Concerning the exact manner of the definition of Euclid and others there may be more reason to dispute, and it is with this that we shall concern ourselves at present. First, let us consider Euclid’s own definition, or at least that which is found in his Elements, whether his or accepted by him from his predecessors. The latter is nearer the truth, for we find the same idea in Aristotle. Euclid thus defines parallel lines: Parallel lines are straight lines which lie in the same plane and which being produced in both directions indefinitely, do not meet. This definition has been considered faulty.1 Many others, as we have seen, adopted other definitions as being more capable of substantiation, and more useful for demonstrating subsequent theorems, particularly those that depend on the parallel postulate. The contention is, and it is this contention that is also at the base of non-Euclidean Geometry, that of several straight lines in a plane it is impossible to know if there are some that will not meet. A second objection is that this definition does not give us the means of recognizing parallel lines, since the criterion is negative. Finally, it does not give us the means of constructing parallel lines, which every geometric definition should do. These are the chief difficulties urged against Euclid’s definition. It may be somewhat true that, according to the strict rules of definition, Euclid’s definition is somewhat faulty. We are not defending his actual definition as such, and the truth or falsehood of the geometry that goes by his name does not depend on the formal way he defines parallel lines. It certainly is faulty in this sense, that it is negative, in that it tells us what parallel lines will not do. Definitions wherever possible should be positive. In the second place, the introduction of the idea of infinity, even though it be the idea of negative infinity is not a proper element in the specific difference that determines the nature of a line or lines. It is an idea that is also negative, and we can grasp it only by denying the opposite positive element. The very name tells this; infinite means not finite. But we must distinguish between faulty and incorrect definition; between the actual definition or explanation of a thing as it compares with the rules of logic, and the actual content of the definition as representing reality and truth. It may be that a certain definition is faulty in logic, but the conception back of it which the author attempts to put in words, may be correct. So long as the latter is true, and the truth is kept in sight, there is no danger to the science. The defect in the definition can be easily corrected. As an actual fact, Euclid’s definition has a correct idea back of it, and the actual wording need not worry us very much. The correct form can easily be given. He accepted this form doubtless to tally with the assumption in the fifth postulate. But the definition expresses a real scientific characteristic property of parallel lines, and it could very easily be drawn as a conclusion or a corollary from any real definition. Let us then glance at the objections made to this definition, and see how far they hold. The first one is, that of several straight lines in a plane infinitely or rather indefinitely produced, it is impossible to know whether any certain ones meet or not. The objection as given here in a general form takes two different meanings, according to the theories with regard to parallel lines that sprang up as a result of rejecting the fifth postulate. In the theory of Lobatschewsky it is assumed, as a result of denying the truth of the fifth postulate, that there are an infinite number of straight lines passing through a certain point that will not meet a given straight line. In the theory of Riemann, which goes to the opposite extreme, there are none at all that do not meet; for, according to his Geometry, all meet. In other words, he refuses the definition of parallel lines altogether as representing anything real, and makes all lines follow the fifth postulate. For him, therefore, the parallelogram is a real impossibility. The best we can arrive at are near-parallelograms. Lobatschewsky and Bolyai, accept the definition of parallel lines, but deny the fifth postulate. They are, therefore, at opposite extremes, and the geometry of Euclid holds the golden mean. First, let us examine the general objection that is at the back of these absurd theories. Is it really impossible to know whether or not there are such things as parallel lines according to this definition; in other words, to know that there are some that will not meet, and be able to tell precisely which lines they are? It certainly is not. A definition of itself tells us nothing as to the existence or non-existence of a thing, that is, a definition such as Euclid’s definition of parallel lines is placed as a preliminary to all scientific demonstration. Such a definition is simply a conceptual form for our notion of a thing, but says nothing about whether or not it actually exists. This is a matter for actual demonstration, or to be taken from experience as a postulate. Of course, it was not the hypothesis of the indemonstrability of the fifth postulate that these objections were made, and the new Geometries constructed. If it is possible to demonstrate it, then we can tell whether or not straight lines will or will not meet, and we can tell precisely those which meet and those which will not meet. It is true that the parallel lines we demonstrated to exist are not directly representative of the definition of Euclid, or of lines that will not meet no matter how far they are produced, but as definitely finite lines that are equidistant everywhere. Yet the other definition can easily be deduced by using the second postulate of Euclid, that straight lines can be produced continuously in a straight line, and as by definition and construction they remain equidistant, they will never meet. The lines that will never meet, or that remain equidistant, are quite definite. They are those that are on opposite sides of a quadrilateral figure formed by two equal and corresponding triangles erected on opposite sides of the same base; they are lines where the alternate-interior angles, and the exterior and opposite interior angles, are equal, and where the sum of the angles on the same side of the transversal is equal to two right angles. But the definition of Euclid can be proved directly from the given proof that the sum of the angles of a triangle is equal to two right angles. For if the sum of all the angles is equal to two right angles, then any two angles taken together are less than two right angles, a theorem which Euclid proves in Proposition 17, but which also becomes useless since it is a direct corollary of the proof that the sum of the angles actually equals two right angles, and therefore does not need proof as a theorem. The two base angles in any triangle are then necessarily less than two right angles, and when the base angles are less, the lines meet and form a triangle. The contradictory is therefore also true. If the base angles equal two right angles, the lines cannot meet, for if they did, they would form a triangle the sum of whose angles would be greater than two right angles. The parallel lines of Euclid can, therefore, be perfectly established, as well as any other construction in geometry. But what is to be said of the second objection that the definition does not give us the means of recognizing parallel lines, and likewise does not give us the means of constructing them, which every geometric definition should do? A definition ought always be of such a kind that we can distinguish the figure defined, and construct it. Euclid is supposed to have sinned against this rule. The idea is that infinity is beyond our ken and the scope of our experience, and thus we cannot tell whether lines infinitely produced will or will not meet. This objection might hold, if in constructing or recognizing parallel lines we had to produce them actually to infinity to find out whether or not they meet. This of course would be impossible, and therefore could not be a criterion. But we can find the other actual conditions of lines that will never meet, and apply them to constructing and recognizing parallel lines. Since the conditions enumerated above are essentially connected with lines not meeting, and since they are easily applicable, there is no difficulty either in recognizing or constructing parallel lines. The worst that can be said of the definition is that it defines by one essential property, but that actual recognition and construction must come through another essential property that is also necessarily connected with the first. Euclid’s definition is then not really wrong. Another definition introduced into modern projective geometry, and essentially connected with this definition of Euclid, is that wherein parallel lines are defined as meeting at infinity, or having a common point at infinity. The idea of having a common point at infinity was first mentioned by Kepler2 in 1604 as a “façon de parler,“ then used again in 1639 by Desargues3 and then adopted as one of the definitions of Projective Geometry. This definition is objected to by some mathematicians for a reason such as Gauss gave in his letter to Schumacher,4 “There is nothing contradictory in that finite man lacks the power to consider the infinite as something granted and to be grasped by the ordinary means of observation.” The difficulty is ordinarily interpreted in the sense that we know nothing of the behavior of lines at infinity; and although we may know that they do not meet within a finite distance, how can we assume that they will meet at infinity. But this is not the difficulty at all. Infinite in this sense simply means nothing, or zero. The definition is then only an obscure way of saying that lines do not meet. When we say they meet at infinity, we mean they never meet. In other words, there is no such thing as an actually infinite line, as Aristotle proved long ago. A line is essentially a distance between definite points, or according to Euclid’s postulate is always extended from one point to another. It must, therefore, always be finite because its extremities are necessarily determined, definite points, which are the limits of the line. The line is then essentially limited or finite. Infinite extent actually existing is simply a non-sense; infinite distance or an infinite line is simply a possibly infinite line, that to which we cannot put a term; it by no means entitles us to assume any line as positively infinite. Gauss spoke correctly in the same letter when he copied the words of Kepler, “The infinite is only a façon de parler.”5 Parallel lines meeting at infinity mean nothing more than that they never meet, which is the definition of Euclid. Another definition in the same line of thought as that of Euclid is one held by geometers such as Ingrami6 and Enriques.7 They define parallel lines as “Those straight lines in a plane which have not any point common.” Hilbert takes the same view in his axiom of parallels.8 It is the same in sense as that of Euclid, for meeting or intersecting simply means that lines have a point in common. If they have no point in common, they do not intersect. Not having a point in common is therefore subject to the same objections as the idea of not meeting of Euclid, and also must be demonstrated. Another method of defining parallels is by what is called the direction theory. According to Killing,9 this method of defining parallel lines originated with Leibnitz. Heath seems to think that there are traces of this theory also among the Greek writers. There is a passage from Philoponus, one of the Commentators on Aristotle, that explains the petitio principii alluded to by the philosopher, about which we have already spoken, which Heath explains in this sense.10 Parallels in this theory are straight lines that lie in the same direction. Now, while direction itself is a primary notion, two parallel directions are not, but are the result of a comparison, and for a comparison we must have a common term. There is no identity of direction, and similarity of direction cannot be assumed, but must be proved. The definition is therefore no improvement on Euclid, since geometry does not deal directly with direction, but rather with relations of directions. One relation of directions in geometry is the parallel relation. Positive direction belongs rather to physical science. The parallel relation must be demonstrated. The last theory is the equidistance theory, about which we have spoken in the historical sketch. This also was found among the ancient Greeks who sought in this way to fill the gap in Euclid’s method. The following is attributed by Proclus to Posidonius: “Parallel lines are lines which being in one plane, neither converge nor diverge, but have all the perpendiculars equal which are drawn from the points of one line to the other, while such as make the perpendiculars less and less, continually converge to one another; for the perpendicular is enough to define the heights of areas and the distance between lines. For this reason, when the perpendiculars are equal, the distances between the straight lines are equal, but when they become greater and less, the interval is lessened, and the straight lines converge to one another in the direction in which the less perpendiculars are.”11 This definition will recall Nasiraddin’s attempt to prove the fifth postulate; it was along the lines of this explanation. This definition is practically the same as that quoted by Simplicius from a certain Aganis and used in an attempt to prove the fifth postulate. This idea Simplicius himself adopts. The same idea is found in a quotation by Proclus12 from Geminus. The quotation is as follows: “Of lines which do not meet, some are in one plane with one another, others not. Of those which do not meet and are in one plane, some are always the same distance from one another, others lessen the distance continually, as the hyperbola to the straight line and the conchoid to the straight line. For these, while the distance is being continually lessened, are continually in the position of not meeting, though they converge towards one another; they never converge entirely, and this is the most paradoxical theorem in geometry, since it shows that the convergence of some lines is non-convergent. But of lines which are always an equal distance apart, those which are straight and never make the distance between them smaller, and which are in one plane, are parallel.” Geminus here distinguishes between asymptotes and parallel lines and defines the latter as straight lines that are an equal distance apart, and always keep this same distance, whereas asymptotic lines, of which he instances the hyperbola and the conchoid of Nicomedes, never meet, and yet they are continually converging. His reason apparently for not taking the definition of Euclid is that it does not cover the situation, since there are lines which do not meet, but are still not parallel. The genus is therefore lines that do not meet, and the specific difference of parallel lines from the others consists precisely in this, not that they do not meet, but that they always keep the same distance between them. And here we may say a word about the objection that has been so often urged against the connection of parallel lines with lines that do not meet, from ancient times till our own day. We find the objection that non-parallel lines are not necessarily lines that do not meet, because in the case of the asymptote and its curve we have nonparallel lines that do not meet, as already mentioned by Proclus. The same objection was urged later by Clavius and Saccheri, who adduced both the parabola and the conchoid of Nicomedes as furnishing such an example. But the objection is not a valid one at all, and would never furnish a difficulty to the theory. The asymptotic relation is not a relation between straight lines at all, but one between a curve and a straight line, and hence cannot logically be adduced as an objection to the behavior of two straight lines. There is no parity at all. To make a valid objection, the instance would have to be one in the same order of facts as that to which it is objected. A difficulty from one order of facts can never be urged to render illegitimate or invalid a general proposition in another order. The difficulty in the parallel theory was not that an instance had been found in which the general assertion did not hold, but it was in the theory itself; it was not found in a different order of facts, but in the actual order itself. It was simply this, that the general principle was not logically established till it was proved, since it was not of a kind that could legitimately be made a postulate. The middle term had not been found. The bringing forward of the asymptotic relation was therefore a mere bit of dust-throwing that was later adopted and made use of by the non-Euclideans to bolster up their theory of a new Geometry that would be independent of the parallel theory. The argument—the only one they ever adduced—s a mere sophism, and not a real difficulty at all. The parallel theory was not established or proved by Euclid or his successors, and that was all. And that was the only valid objection that could be urged against it. Here as everywhere metageometry has only sophistry for its basis, and a sophism that needs no great logical penetration to set aside. It is solely a question of confusing the straight with the curved line. In more modern times this equidistance definition again makes its appearance in an attempt to solve the difficulty in the fifth postulate. Clavius, a well known editor of Euclid in the sixteenth century, uses it, although he does not actually seem to have given the precise definition. Borelli in his Euclides restitutus criticises Euclid’s parallel theory for bringing in the idea of infinity into the definition, and himself sets out from the definition that parallels are straight lines that have a common perpendicular. But as he soon perceived that this was not sufficient to prove the fifth postulate, he uses as an axiom, a proposition already employed by Clavius, that a straight line lying perpendicular on another straight line will, if moved forward in the same position with one end on the straight line, and the angle remaining a right angle, describe a parallel line with the other end.13 From this time on to the beginning of the eighteenth century most of the textbooks of elementary geometry defined parallels as lines that are equidistant. Giordano da Bitonto seems to have been the first that really understood that this “axiom” required proof, and he himself tried to supply it. He erects two perpendiculars at the ends of a straight line, and makes these two perpendiculars of equal length; then he tries to prove that the line joining the free ends of these perpendiculars is everywhere equally distant from the groundline. His proof is rather involved, and it centers around the theorem that if the line that joins the two is in any way curved the perpendiculars meeting it cannot be of equal length. In this proof Giordano makes use of a figure that we already find in Clavius, and meet again in Saccheri.14 Saccheri takes up the same question and discusses it, and objects to what he calls the new definition. In the Introduction to his Euclides ab omni naevo vindicatus he says: “But since the investigations of the ancients do not seem to lead to the goal, it came about that many eminent geometers of later times undertook the task, and found necessary a new definition of parallel lines. While Euclid defined parallels as lines, which, lying in the same plane, and produced infinitely will never meet, the others instead of the last phrase of the foregoing definition put these words, ‘which are always equally distant from one another,’ so that all perpendiculars, drawn from any point on one line to the other, will always be equal. “But here arose another split. Some indeed, and those the keenest, endeavored to prove the actuality of such parallel lines, and set out from that ground to prove the famous axiom, which is, as Euclid announces it, debatable; for on it rests the twenty-ninth proposition of the first book, and with few exceptions, the whole of geometry. “Others, however (not without a great blunder in strict logic), assumed beforehand such parallel lines as actual, viz., lines equidistant from one another, and made this their starting point in the proof of the other propositions.”15 A variation of this equidistance definition is found in Veronese.16 He defines parallel lines as follows: “Two straight lines are called parallels, if one of them contains two points opposite to two points of the other with respect to the middle point of a common transversal.” This is scarcely clearer than the ordinary equidistance definitions, and, although it is physically correct, it is not a proper definition, since it is rather the corollary of a proposition, where proof precedes. The true definition precedes its proof. This is really an equidistance definition, for it depends on the equidistance of the opposite points. Instead, however, of taking the equidistance as between corresponding opposite points, the equidistance is taken from the alternately opposite points to the middle point of the transversals formed by joining the two pairs of alternate opposite points. The idea back of all the equidistance definitions of parallel lines that have been brought forward consists in assuming that the perpendicular is the distance, or shortest distance between them. The idea in itself is quite legitimate, and is sufficiently clear to be used as the basis of a definition. The fundamental notion at the base of geometry and all measurement is distance. No geometry is possible, unless we admit the possibility of being able to measure distance between two points, and therefore between two points on different lines. The most that can be objected to it is that it is abstract theory, which is perfect, but that we do not know which two points to choose as representing the shortest distance. The objection is one of practical application to construction, and in this sense the objection is correctly made. A definition by itself is not a geometric actuality, or a construction; it becomes such only when we are able to demonstrate that the figure that represents the definition is of possible and demonstrated construction. In the above definition this rests on several assumptions. First, we must be able to construct a perpendicular, and prove that it realizes the definition. This is possible and proved in the eleventh proposition of Euclid, and could therefore be legitimately used in establishing the parallel postulate which is only applied in the twenty-ninth. The next assumption is that a perpendicular is the shortest distance from a point to the line on which it is erected as perpendicular. This assumption is also legitimatized by the twentieth proposition which proves that in any triangle two sides in any manner taken together are greater than the third side. From this proposition can be easily deduced the conclusion that the perpendicular is the shortest distance from a point to a straight line. The third assumption involved is that we are able to choose the proper point on the second line from which to measure the perpendicular distance to the first line, or that we are able to realize the construction required or from points not on a line to drop perpendiculars to the line. It is true that this is possible by proposition twelve. But we have no means of proving that a series of these perpendiculars will be equal. To prove this requires a further assumption that the perpendiculars so drawn are also perpendiculars to the second line. Then the perpendicular to both lines would be the shortest distance from line to line, since the distance from a point on one line by the shortest distance to the other line coincides each way, and hence, also, is the shortest distance from line to line. Further than this we need to know that, when we have one transversal perpendicular to two straight lines, all other perpendiculars between the same lines will be equal to this transversal. For this last we require to know that such lines are equidistant. We need, therefore, to know that parallel lines exist having the property of equidistance to satisfy the second of these conditions, and further to satisfy the other condition that the shortest distance between such parallel lines is the common perpendicular, or that the angles formed by the line of shortest distance are right angles, that is to say, that the sum of the interior angles formed by the transversal with the two parallel straight lines is two right angles. Now the first of these is supposed to be realized in the construction of proposition twenty-seven, although in Euclid equidistance is not shown; it cannot then be validly used. The second assumption is shown only in the twenty-ninth, and hence cannot be employed. For since the twenty-ninth proposition makes use of the supposed reality of parallel lines to establish the conclusion that the sum of the angles made by a transversal with them is equal to two right angles, this cannot be validly brought into a proof of parallel lines; for such a proof would be circular, or a petitio principii. It is easy to see, then, why it was always found impossible to establish the construction of parallel lines on the equidistance theory. Perpendicular equidistance is a conclusion to be deduced from the properties of parallel lines, rather than an essential element in the definition. While theoretically correct, it requires conditions that already demand parallel lines to exist having the interior angles formed with them by the common perpendicular equal to two right angles, and furthermore that these lines be continuously parallel in the sense of equidistance. In Euclid’s order of propositions this is impossible. For before Euclid establishes this, he already requires the existence of parallel lines as a necessary element in his demonstration. The perpendicular equidistance theory cannot then be established under the geometric order of Euclid’s Elements, and hence all attempts at proof were failures. The mistake made by all who employ the equidistance theory of parallel lines, has been to connect the idea of distance with perpendiculars and parallels. This they were never able to do without assuming implicitly the fifth postulate, as we have shown, or implicitly or explicitly some postulate equivalent to the fifth postulate, according as those who used the definition fall into one or other of the categories mentioned by Saccheri as constituting the split in the method of proving the fifth postulate of Euclid. That they are actually connected there can be no doubt, but that is quite a different matter from demonstrating the fact scientifically. If it is assumed, the new assumption does not better the case in the slightest. While we have adopted the equidistance theory of parallels as giving the better logical definition, we do not hold that Euclid’s definition is in any sense false. But again our definition is quite different from that of the others, and is not subject to the same objections. Let us then begin by defining parallel lines. Parallel lines are straight lines that are equidistant at equidistant points. For instance, the lines x and y are parallel if, taking any two points on one line, as a and c, and two points, such as b and d, on the other line, so that
the distance between b and d is equal to the distance between a and c; the distance between a and b, and c and d are also equal. Or ac = bd, This definition is different from the others for the reason that our choice of points is in no wise determined. We may choose any points we will on one line, and also any points we will on the other, with the sole proviso that they be the same distance apart on each of the two lines. By equidistant therefore we do not necessarily mean the shortest distance, or the perpendicular distance, but any distance that will be the direct distance between corresponding points, and by corresponding points are meant points that are themselves situated at equal distances from one another. Whatever way the distances are taken, perpendicular or otherwise, provided the distances are taken between corresponding points, they are always equal if the lines are parallel. The points may be fixed on one line, and any corresponding points may be taken on the other, and all the points on the other line may be made corresponding points, and vice versa. For example, in the figure above, instead of b and d, we can take the corresponding points at b’ and d’ , or at b “and d”, and the distances will always be equal. Our definition then is absolutely universal, and is not restricted to the special distance that is measured by the perpendicular between the lines. In universality it corresponds exactly to proposition twenty-nine, where the sum of the interior angles on the same side of a transversal in the case of parallel lines is always equal to two right angles. It does not make any difference whether the angles are actually right angles or not, if the sum is equal to two right angles. The equidistance definition which is built on the perpendicular, holds of parallels only in the case where the two interior angles are really right angles. It is not, therefore, a general case. Our definition is therefore much more general than the other. Just as the twenty-ninth proposition would not be a general proposition, if it only proved the existence of parallel lines when the interior angles on the same side of the transversal would actually be two right angles; so the perpendicular distance between parallel lines is not a general case. There is then a further difference between the two equidistance definitions. Our definition is absolutely universal; the definition using the perpendicular distance is not universal but only a special case of distance, i.e., the shortest. The construction corresponding to our definition is the parallelogram rather than the rectangle which corresponds to the other equidistance definitions; it is more general, just as the parallelogram is more general than the rectangle or square. For this reason it has nothing to do with shortest distance, or perpendicular distance; it is not at all necessary that our distances be perpendicular distances; and even in this case we do not need the perpendicular theory to prove it. All we require is a four-sided figure, whose opposite angles are equal and opposite sides equal and parallel, and this is proved with no further construction than that of equal triangles correspondingly disposed on opposite sides of the same base. With this definition and the construction we have used, both the construction and recognition of parallels become a matter of the greatest facility. Our definition is then not only a valid definition, which we also concede to that of Euclid, but it is logically legitimate as well. It is according to some such definition or property as described in this definition that parallels in the sense of the Euclidean definition would have to be recognized and constructed. There could be no question of drawing them to infinity, to see whether or not they would meet. Our definition need only apply Euclid’s postulates for the straight line, for the points can be taken anywhere on an indefinite line, even if two points on one line are arbitrarily fixed. We have claimed for this definition of parallel lines absolute validity as well as logical legitimacy. What then becomes of the objection by Saccheri against the proofs of the perpendicular equidistance theory as brought forward by Clavius and particularly by Giordano Vitale? Giordano constructs two equal perpendiculars at the extremities of a straight line, and then joins the free ends of the perpendiculars by another straight line, which he claims is parallel to the original straight line or groundline. He attempts to prove it by showing that if this line is not a straight line, but some form of curve, the perpendiculars cannot all be equal. Saccheri objects to) this that it has to be proved that “the geometrical locus of points equidistant from a straight line is a straight line.” Does not this apply to our construction, which likewise only takes the two end points of the sides of a quadrilateral for constructing parallel lines, or rather opposite sides formed by two corresponding and equal triangles? As usual Saccheri and his predecessors Clavius, Borelli, and Giordano have succeeded in misplacing the question. They take two equidistant pairs of points corresponding perpendicularly and try to prove that the lines joining them are parallel, if one of the lines is given as a straight line. We have constructed our straight lines and proved them equidistant, and therefore parallel. There is then no problem for us requiring us to prove what Saccheri demands, that the locus of points equidistant from corresponding points on a straight line, is a straight line. Our lines are straight by construction, and all we have to do is to prove corresponding points equidistant. It is true Giordano also makes the lines joining the extremities of his perpendiculars straight lines, but since his proof is an attempted reductio ad absurdum, he immediately makes the other hypothesis, that it is not a straight line, and then tries to prove it so. It was this evidently that threw Saccheri off the track in his criticism of the proof. The objection Saccheri should have made was not that the line joining the free ends of the perpendiculars was not proved a straight line, but that we had no means of knowing whether the perpendiculars joining its points to the base line were the shortest distances from line to line. That they were the shortest distances from the points chosen to the base line is clear. His perpendiculars were the shortest distances from the ends of the line to the groundline, but one cannot conclude thence that the same perpendiculars were the shortest distances between the two lines on which they fall, for this is what requires proof. All that we have in Giordano’s proof is that the shortest distances from two certain points to a groundline are two perpendiculars. We know further that the line joining the free ends of these perpendiculars is a straight line. But what we do not know, and what we require to know for the proof, is that any lines thus drawn perpendicular to one line are perpendicular to the other, and that the perpendiculars are necessarily equal, as we have shown above. As usual, Saccheri missed the point, and went on a wild goose Ghase, but the decoys were planted for him before by Clavius and Giordano. In the case of our definition, the proof is quite different. If the sides of a quadrilateral as formed by two equal and corresponding triangles on opposite sides of the same base are equidistant as measured by the other two sides, they are equidistant at all other points, and will remain equidistant no matter how far the sides may be produced in either direction. In other words, we can prove the parallel condition for any two pairs of equidistant points. These equidistant points can be made any two points on one line, and therefore all points. We can also choose any two points on the other line having the same distance apart as the points on the first line, and, therefore, all points. We can thus either construct our parallel lines, or if the parallel lines are constructed, prove them equidistant. For instance, to construct the line parallel to AB through the point C take
any two points such as a and b on the line AB; join a and C and b and C; on bC as a base construct a triangle corresponding to the triangle abC, or the triangle bCd. Then Cd is parallel to ab, or AB. For, since the triangles are equal, the corresponding sides are equal, and hence aC equals bd; or AB and CD are equidistant at equidistant points. Suppose now that having thus constructed our parallel lines by means of two equal and corresponding triangles on opposite sides of the same base, we were asked to demonstrate than any other chosen points on these lines were also equidistant. The proof will be simple.
Let AB and CD be the parallel lines as constructed by the triangles acd and adb. Let us take the points a, b’ on line AB and take corresponding points d’ , d on the line CD. To prove that ad’ = b'd. Therefore cd’ = b'b (axiom 3). Or we can prove that, if two lines are parallel, any corresponding points we choose are equidistant. For instance,
let the lines AB and CD be two parallel lines. To prove that any points we choose are equidistant from two corresponding points on the second line. Let the points be a and b on the line AB; choose any two points c and d on the line CD such that their distance is equal to the distance between a and b. Join ac and bd, also the vertices b and c. The same can be done for any points. Therefore any corresponding points on two lines are equidistant from any two corresponding points on another of two parallel lines. There are certain legitimate corollaries from this legitimate definition and proved construction that can be used for further proof. Among them are: (l) Parallel lines never meet no matter how far they are produced. The reason is that they remain always equidistant. The definition of Euclid thus becomes a mere corollary. (2) Parallel lines cannot enclose a space; since they never meet. (3) Parallel lines and one transversal cannot enclose a space; for they cannot meet on either side of the transversal, that is to say, they cannot form a triangle. Thus it necessarily follows that a triangle must be formed of non-parallel lines. (4) The lines joining parallel lines and marking their equality of distance apart are themselves parallels. For they respond to the definition of parallel lines. (5) Through a given point only one parallel can be drawn to a straight line. For if two or more parallels could be drawn through the same point, they could not be everywhere equidistant, otherwise they would coincide. But this is contradictory to what we have proved. This last is known as “Playfair’s axiom.” This axiom has been used instead of Euclid’s to prove I, 29, and Euclid’s postulate itself. It can easily be deduced from the equidistance definition and construction. Without this proof, or some other, it is just as much of an assumption as that of Euclid. With this definition and the demonstration of its validity and the accompanying corollaries, there is sufficient doctrine on parallel lines to satisfy all the requirements of Euclidean geometry. We have now completed the demonstration of the definition of parallel lines and the postulate of Euclid, and we have shown the connection of the one with right angles, and of the other with an interior arrangement at the base of less than two right angles. We did this without using anything except what could be deduced from the other postulates and theorems universally accepted. Euclidean geometry is therefore completely established.
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