As a freshman in college, I was taught that parallel lines never cross. This is the mathematical layman's way of summarizing the fifth postulate of Euclid's Elements of Geometry. Euclid's plane geometry, in which space has no curvature, is the standard geometry that we use to explain the world as we experience it. In a Euclidean world, all triangles are 180 degrees, for instance. Euclidean geometry is beautiful and regular; its elegance and symmetry helps to explain our everyday experience of the world, and for many centuries it was assumed that Euclidean geometry was an absolute explanation of reality.

Mathematicians, however, including Euclid himself, were not happy with the fifth postulate because it was neither self-evident nor provable. This led, in the 19th century, to the development of non-Euclidean geometries, which reject the fifth postulate. As a senior in college, I was introduced to Lobachevsky's non-Euclidean geometry, which allows for parallel lines to cross at some point in the infinite distance. According to the Britannica Concise Encyclopedia, "Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity."

I was reminded of the differences between Euclidean and non-Euclidean or Lobachevskian geometry as I read "Communion, Covenant, and Our Anglican Future," the recent statement by archbishop of Canterbury Rowan Williams. In it he writes:

22....[T]here is at least the possibility of a twofold ecclesial reality in view in the middle distance: that is, a ‘covenanted’ Anglican global body, fully sharing certain aspects of a vision of how the Church should be and behave, able to take part as a body in ecumenical and interfaith dialogue; and, related to this body, but in less formal ways with fewer formal expectations, there may be associated local churches in various kinds of mutual partnership and solidarity with one another and with ‘covenanted’ provinces.

23. This has been called a ‘two-tier’ model, or, more disparagingly, a first- and second-class structure. But perhaps we are faced with the possibility rather of a ‘two-track’ model, two ways of witnessing to the Anglican heritage, one of which had decided that local autonomy had to be the prevailing value and so had in good faith declined a covenantal structure. If those who elect this model do not take official roles in the ecumenical interchanges and processes in which the ‘covenanted’ body participates, this is simply because within these processes there has to be clarity about who has the authority to speak for whom.

24. It helps to be clear about these possible futures, however much we think them less than ideal, and to speak about them not in apocalyptic terms of schism and excommunication but plainly as what they are – two styles of being Anglican, whose mutual relation will certainly need working out but which would not exclude co-operation in mission and service of the kind now shared in the Communion. It should not need to be said that a competitive hostility between the two would be one of the worst possible outcomes, and needs to be clearly repudiated. The ideal is that both ‘tracks’ should be able to pursue what they believe God is calling them to be as Church, with greater integrity and consistency. It is right to hope for and work for the best kinds of shared networks and institutions of common interest that could be maintained as between different visions of the Anglican heritage. And if the prospect of greater structural distance is unwelcome, we must look seriously at what might yet make it less likely.

What got me thinking of geometry was the archbishop's helpful suggestion that we think of these two emerging ways of being Anglican as being two tracks. The question that arose in my mind was: are these two tracks parallel to each other in a Euclidean sense or a non-Euclidean sense?

In a Euclidean world, these two tracks would be parallel but never converge, not in the time and space of our everyday ecclesial existence, not in the space of the imagination, and not in the infinitude of God. In a Euclidean world, ACNA and TEC might be parallel but always in competition and hostility, like two siblings in perpetual reactive rivalry. But in a non-Euclidean world, these two tracks might converge, where and when one cannot say, but at least there is the possibility. At the very least, we might concede that it is possible that both tracks have the potential of converging in the Kingdom of God. We might even say that they converge in the realm of the One Holy Catholic and Apostolic Church in the here-and-now, though those who hold to the Euclidean notion that ACNA and TEC, for instance, are preaching "separate gospels" could not accept such a non-Euclidean postulate. In a Euclidean world, parallel tracks never meet. This is the fifth postulate of an ecclesiology where schism is the necessary result of unacceptable differentiation.

By contast, the archbishop seems to be setting out a rather non-Euclidean vision for the Church, because he does not see this sort of "structural differentiation," as he terms it in the final paragraph, as necessarily destructive of the Church, but leading, rather, to greater clarity and integrity on the part of each "track." Indeed, as Williams writes, "All of this is to do with becoming the Church God wants us to be, for the better proclamation of the liberating gospel of Jesus Christ."

I'm glad the archbishop reminds us of the main thing. This should be the main thing whether our ecclesiology is Euclidean or non-Euclidean. All too often, however, it gets lost in the equation.