fitted if their education provides them with widening and inspiring subjects for contemplation when they reach maturity, nor, indeed, is such fitness the sole end of life.

PARALLEL STRAIGHT LINES AND THE
METHOD OF DIRECTION.*

The absence of any direct contact in England between the critical movement of higher Pure Mathematics and the more recent movement for the reform of geometrical teaching in schools has not been altogether favourable to the latter. In no instance has this unfavourable result appeared more widely, and more dangerously to both real reform and true education, than in the various attempts made to replace Euclid's treatment. of parallels. So long as such attempts were put forward privately, private criticism seems to have been sufficient on the whole to preserve the reform movement from serious errors; but in England the conditions are no longer private, for in March 1909, the Board of Education issued, under the title "Teaching of Geometry and Graphic Algebra in Secondary Schools," a Circular (No. 711), which contains an attempt to base the treatment of parallels on the notion of direction, and thus openly encourages teachers to adopt a method which has been often and rightly condemned.

The exact words of Circular 711 (pp. 3 and 4) are as follows:

"The next concept, direction, is seldom as well treated, with the result that trouble arises later on with angles and parallels.

It is impossible to discuss, much more to define, direction in the abstract-just as it would be to discuss colour.

But just as children gain the general idea of colour from recognising and naming colours-red, blue, green, so they gain the idea of direction (and gain it accurately) by recognising and naming certain standard directions-vertical, north, south, etc.

"At the outset it is essential to keep away from pencil and paper, and the introductory question should be 'Show me a vertical line?' then How would you test whether it is really vertical or not?' so introducing the familiar knowledge of the plumb-line. Then "Show me a horizontal line.' 'How would you test it?' The test must be independent of the vertical, i.e., the spirit level must be suggested.

"Then, to clear things up further, Can you draw a vertical line on the wall, on the desk, on the floor?' and similarly with horizontal lines.

"Then 'How many vertical lines can be drawn through a point? ' 'How many horizontal?' so passing on to the different horizontal directions (north, south, east, and west). Boys should know how the windows of their room face and how they themselves face as they sit.

"Then the master may ask, 'Are all vertical lines in the same direction?' 'Are all horizontal lines?' thus leading to a clear conception of parallels as lines in the same direction.”

* NOTE.-Owing to unforeseen causes the author was unable to complete this paper in the time intended; and the date at which it was in fact received has rendered it impossible for the Advisory Committee to obtain a contribution discussing the question of Parallels, from another standpoint.

† Board of Education, Circular 711, March 1909. (London: Wyman and Sons.)

A denial may at once be given to the statement of the last paragraph; for it is impossible to give anyone a clear conception of parallels as lines in the same direction. Young boys may readily acquire a clear conception of direction, as, for instance, illustrated by the different directions given by the magnetic compass" the Points of the Compass "-which correspond to the different directions given by straight lines which intersect at a common point. From this clear notion, it is obvious that lines which coincide have the same direction, and lines which intersect have different directions. Neither Circular 711 nor any author before or since its appearance, suggests any possible intuitive way of leading from this clear notion of the different directions of intersecting lines, and of the same direction of coincident lines, to any notion of non-secant lines as lines in the same direction. That there is no logical basis for this attempt has been shown often, and evidence of this is quoted later.

The appeal made to intuition in the Circular is the basis of its suggestion; and the questions referring to horizontal, vertical, plumb-line, spirit-level, imply a reference to physical geography. In these days of correlation in the teaching of the various subjects. of school education, is it possible to avoid teaching that plumblines point towards the centre of the earth? Is a clear notion of parallels possible to a boy, who is told in the geometry lesson that vertical lines are in the same direction and so are parallel, but who, half an hour later, in a geography or practical mechanics. lesson, learns that vertical lines all point towards the same point--the centre of the earth? Will such a boy always avoid thinking that straight lines pointing towards, and so passing through, the same point are in the same direction? The comparison of the spherical surface of an actual ball or globe with the surface of a large blackboard, and in imagination of the earth's surface with an ideal plane surface, is of real educational value to all pupils; but it makes an appeal to geometrical perception quite fatal to the course of procedure advocated in Circular 711. On the surface of a sphere there can, of course, be no straight lines; if great circles (the shortest sailing routes for ocean steamships) are taken as elements replacing straight lines, then there are no parallels. It is a wise step in geometrical teaching to educate geometrical perception to this point at least, that pupils may realise that the existence of a parallel line, whether established by assumption or deductive proof, involves the admission of a new kind of surface where the notions of vertical and horizontal no longer necessarily apply. The failure to make this comparison and contrast between spherical and plane surfaces may long retard the growth of sound geometrical instinct.

Again it is quite necessary in dealing with parallels to make quite sure that some assumptions must be made as a basis for any discussion or proof. As a practical illustration of the

plausibility of the possible untruth (i.e., lack of generality) of Playfair's axiom, we suggest the following:

Let SS' be a finite plane surface, of any size. A B a straight line drawn on it, and P a given point. It is always possible to draw as many straight lines as desired passing through P, but not cutting A B within the limits of the given surface. Produce A B to the boundary of the surface. Choose points C, D on the boundary nearer P. Join C P, DP, and produce to C', D' respectively. Then as many straight lines as desired, e.g., K P L, MP N, may be drawn through P, not cutting A B within the given surface. A similar construction is obviously possible, however large SS' may be, so long as it remains finite.

The restrictive aspect of Euclid's fifth postulate (or Playfair's axiom) may be illustrated by reference to suitable rules in well-known games, e.g., to the offside rule in Association football, which prohibits play after a forward pass under certain conditions, and so has the ultimate effect of limiting the shots between the posts, finally counted as goals, to those scored in a particular way, and of not considering the others. The game was formerly played without this offside rule, when such successful shots did count as goals. Similarly, Euclid's postulate restricts further discussion on the subject of two parallels, to one only of the logically infinite number of straight lines through P, non-secant to A B; or, in other words, the parallel to A B through P must be conceived as drawn in a particular way, and the other non-secant lines are not considered. In the Geometry of Lobachewski and of Bolyai such non-secant lines are considered in dealing with parallels.

The notion of straight lines which do not intersect may rightly and ought to be derived from sensory experience; but a clear concept of parallels is impossible without comparison and analysis of such geometric perceptions, and without some appeal to infinity.

Even as a tentative method of approach for the youngest pupils, this suggestion of Circular 711 seems without definite value; for no simple practical test is given to determine when any two straight lines are in the same direction. Indeed, no one has succeeded in stating a criterion for sameness of direction without introducing (usually by suppressed assumption) some property of parallels already known, because proved by Euclid.

Early History.-The controversy with regard to parallels is at least as old as Aristotle; possibly even this very fallacy of "direction" is referred to in the passage: "for they unconsciously assume such things as it is not possible to demonstrate if parallels do not exist." In later times, according to Heath, Killing has traced the origin of the direction theory to Leibniz. However, there is no doubt that Gauss brought his great influence to bear against the fallacy. In the extracts from the correspondence between Gauss and Schumacher, edited by Houel, no specific mention of the "direction method" occurs, although Gauss enters minutely into the other unexpressed assumptions of Schumacher. But this passage from Heath's

Euclidt seems to be decisive :

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The idea of parallels being in the same direction perhaps arose from the conception of an angle as a difference of direction (the hollowness of which has already been exposed); sameness of direction for parallels follows from the same 'difference of direction' which both exhibit relatively to a third line."

But this is not enough. As Gauss said (Werke, iv., p. 365):

"If it [identity of direction] is recognised by the equality of the angles formed with one third straight line, we do not yet know without an antecedent proof whether this same equality will also be found in the angles formed with a fourth straight line,'

(and any number of other transversals); and in order to make this theory of parallels valid, so far from getting rid of axioms such as Euclid's, you would have to assume as an axiom what is much less axiomatic, namely, that

"straight lines which make equal corresponding angles with a certain transversal do so with any transversal."‡

De Morgan. The opinion of this great teacher was adverse to the direction method, and was strongly expressed in a review (“Athenæum," July 18, 1868) of Wilson's Geometry, 1868. Large extracts from this review were reprinted in Appendix II. of Dodgson's "Euclid and his Modern Rivals"; and from this source we quote the following:

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There is in it one great point, which brings down all the rest if it fall, That point is the treatment of the angle, which amounts to this, that certain notions about direction, taken as self-evident, are permitted to make all about angles, parallels and all, immediate consequences.

"What direction' is we are not told, except that 'straight lines which meet have different directions.' Is a direction a magnitude? Is one direction greater than another? We should suppose so; for an angle, a magnitude, a thing which is to be halved and quartered, is the difference of the direction' of 'two straight lines that meet one another.'

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Parallels, of course, are lines which have the same direction. It is stated, as an immediate consequence, that two lines which meet

* Taken from Heath's Euclid, Vol. I., p. 191.

+ Op. cit., p. 194.

1885.)

Dodgson: "Euclid and his Modern Rivals," p. 128. (Macmillan & Co.,