When Parallel Lines Meet

July 13, 2025
Ladue Chapel Presbyterian Church
Luke 10:25-37
“When Parallel Lines Meet”
Douglas T. King

In Dostoevsky’s The Brothers Karamazov, many of you may remember that Ivan Karamazov is the character who is the most conflicted and confused about God.  My friend, Tom Long, writes about how Ivan, struggling in his faith, compares his struggle, oddly enough, to a geometry question.  He writes, “In Euclidian geometry, two parallel lines can never intersect, but Ivan goes on to say that there are some who ‘even dare to dream that two parallel lines which, according to Euclid can never meet on earth, may meet somewhere in infinity.  I, my dear chap, have come to the conclusion that if I can’t understand even that, then how can I be expected to understand God?” (Long, p. 275)

Dostoevsky’s Ivan is speaking of the difference between Euclidian geometry and projective geometry.   In Euclidean geometry parallel lines are defined as never intersecting.  However, in projective geometry, parallel lines are considered to meet at a juncture called the point at infinity, or the vanishing point. This concept helps to create a more complete and holistic geometric system.

Now at this point in the sermon you may wonder if this seemingly random tangent will ever intersect with this morning’s text from the Gospel of Luke.  I invite you to hang in there with me and I promise we will circle back.
 
This story, which we often refer to as the parable of the Good Samaritan, is encrusted with so many years of assumptions that I prefer not to preach on it.  The headline in our collective consciousness is simple and direct, we should help other people.  Can’t we find a more interesting text to think about this morning?
 
But the text is not quite that simple.  The Jewish scholar, Amy-Jill Levine writes this, “Mention  a priest and a Levite and any person who knows anything about Judaism will know that the third person is an Israelite…It’s automatic just as “Father, Son and…’ call for ‘Holy Spirit,’ and ‘Larry, Moe, and…calls for Curly,’ say ‘priest, Levite, and …’ and everyone knows what’s next, ‘an Israelite.’

“But there’s a surprise, and what happens is so startling, so outlandish that the plot becomes distorted and barely comprehensible.  The third character isn’t an Israelite at all, but a despised enemy: a Samaritan.  Instead of Larry, Moe and Curly…it goes “Larry, Moe, and Osama Bin Laden.” (Long, p. 272)

If the sole goal of Jesus sharing this parable was to tell us to be kind to one another he could have accomplished that by having an Israelite come along and help the wounded man.  But Jesus has bigger and deeper things on his mind.
 
As we remember, this text began with a lawyer seeking to challenge Jesus with a discussion of how to inherit eternal life.  Now, in most cases inheritances are possessions passed on by parents to children.  While not a guarantee it is common practice for parents to leave an inheritance to their children.  An inheritance is an often an expectation.  The lawyer challenging Jesus seems to be of the understanding that there is some formula he can follow that will set up a standard of expectation for eternal life.
 
If we flatten out this parable to suggest that if we are nice to people in need we have somehow earned eternal life as an expectation we will have missed the mark by far.  We would be thinking of our relationship with the divine in terms of Euclidian geometry and believing there is some formula to follow.  Instead of the Pythagorean formula that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides, we would be saying if we are nice to other people we earn a place in heaven.  

But this parable is speaking in terms of projective geometry.  Jesus is speaking in terms well beyond the boundaries of expectations by which we see the world and live in it.  In Jesus’ projective geometry even parallel lines meet at the point of infinity.  The assumptions and boundaries by which we view reality are not the deepest reality known and created by our God.  The lawyer in this text is looking for a simple contract.  If I, the party of the first part, do this, then God, the party of the second part will do that.  The risk for us is if we follow his line of thinking into the parable then we hear, if we are nice to other people we will get to go to heaven.

In Jesus’ projective geometry it is not about simple if/then formulas.  It is about a transformation so beyond the rules and order of this world that we will need to stretch our imaginations to begin to comprehend it.  Tom Long writes, “It is, rather, the arrival of another world, an unexpected and life-changing reality in which mercy overcomes alienation.” (Long, p. 275)

In Jesus’ projective geometry eternal life is not about some continuation of the status quo.  Just as it is difficult for us to imagine how parallel lines can ever intersect, so it is difficult for us to imagine how all of the disparate and conflicted elements of creation will come together in harmony.  Sworn enemies will be beloved to one another.  Everyone and everything will be healed and made whole.  Whatever is broken or lacking within us will be restored and filled.
 
In Jesus’ projective geometry Samaritans and Jews are one family.  In Jesus’ projective geometry there are no rivalries and no nations.  In Jesus’ projective geometry there are no different classes of people.  In Jesus’ projective geometry there are no political divides.  In Jesus’ projective geometry we do not hurt or wound each other in any way.  In Jesus’ projective geometry all of the hurts and wounds we carry within us are healed. In Jesus’ projective geometry all of our limitations and imperfections will be invited into a vanishing point; to a place of infinite perfection.
 
Tom Long shares a story he was told by a young pastor.  In this story of a tiny congregation, there are these two men, Jim and Robert. Jim and Robert had an argument many years ago that had become a lifelong grudge of enmity.  This division was so deep that every Sunday they sat on opposite sides of the sanctuary and made sure to exit in different directions.  Well Jim was scheduled to serve communion and his fellow server, Mary, had forgotten to show up.  Jim was standing up in the chancel ready to serve.

“The pastor quickly scanned the congregation and, without thinking, said, ‘It looks like Mary is not here this morning.  I wonder if I could get you to help serve today, Robert.’
 
“The pastor realized he had made an error when he saw a stricken look on Robert’s face and heard Robert, before he could censor himself, blurt out loudly, ‘Oh sh**!”

“But there was no respectable way for Robert to decline, so grudgingly he came forward and stood beside Jim at the communion table.  The two men exchanged a darting glance. But then the power of the Lord’s Supper and the power of the Spirit began to be felt as Jim and Robert gave bread to each other and passed the cup, looking each other in the eye as they said, ‘The bread of life’ and ‘The cup of salvation,’ and as the pastor prayed, ‘By your Spirit make us one with Christ, one with each other, and one in ministry to all the world, until Christ comes in final victory and we feast at his heavenly banquet.’

“It would be too much to say that, when the service ended, the two men rushed to embrace each other in forgiveness.  That would come much later. But they did stay to do what they had been unable and unwilling to do for years: to begin talking to each other and to allow the oil and wine of healing to begin to be poured over the old wounds.  Two stubbornly parallel lines, lines that would never have intersected by the rules of this world, had been curved by the force of eternal life and, at the table of mercy, touched each other.” (Long, p.276)

Projective geometry, indeed.
Thanks be to God.  Amen. 
 
Long, Thomas G., Proclaiming the Parables, WJK, Louisville,
 2024.