Paper and Pringles and Balls... and Saints!

Comments from our friend "Wild Goose" have brought me to mentioning such esoteric terms as "Euclidean" and "non-Euclidean" geometry in the comment-boxes of the ACS blogg. Hee hee. So I have decided to stuff the envelope even more, and explain why such things relate to the great feast of next week.

First, remember that Euclid was a great Greek mathematician who lived in the 4th century before Christ. His postulate 5 in Book I might be phrased thus: "through a point not on a line, there is exactly one line parallel to the given line". This leads to important results, such as the fact that the sum of the angles in a triangle is exactly 180 degrees (alluded to even by GKC!) Despite many attempts by great mathematicians over two millennia to prove this important postulate - that is, to turn it into a theorem based on other, more basic postulates - it remains a postulate.

Which means one might postulate an alternative (just as GKC hinted, as one writes a story!)

Like that there can be more than one such line - that is, more than one line can go through the point and not intersect the given line. This can happen if the "lines" are actually curves within the strange surface called a "hyperbolic paraboloid" - which any mathematical snack-food enthusiasts will recognize as the shape of a Pringle. This means that when one draws a "triangle" on a Pringle, the sum of its angles will be less than 180 degrees. This is hard to do, because the Pringle will probably break, so try it if you like, or you can just trust me on this one.

Or, on the other hand, there can be less than one such line - that is, every line through the point intersects the given line. This happens when the "lines" are curves (actually called "great circles") on an even more unusual surface called a sphere, for which some people use the term "ball". Now, in this case, triangles can have more than 180 degrees. This is easily seen from the terrestrial globe, and it's lots of fun, so get out your flying saucer or broom or aircraft, get a GPS or other locating device and try it:

1. Start on the Equator on the Greenwich meridian. This point is in the Atlantic Ocean off the coast of Africa, south of Ghana, west of Gabon.
2. Go west along the equator until you reach the 90 degree west meridian, and turn right, or north. You will be in the eastern end of the Pacific Ocean, just about where the Galapagos Islands are.
3. Go north, passing over Guatemala, New Orleans and Memphis, Ontario and Hudson Bay, finally reaching the Arctic Circle and Santa! (Ahem) I mean the North Pole, which is at 90 degrees North.
4. Again turn right, and go south. You will pass over England, France, Spain, and Algeria, finally coming to the Equator and the point at which you started.

Thus you will have made a triangle of three right angles, totalling 270 degrees, you non-Euclidean globe-trotter! What fun.

Ahem. Now I have lectured for quite some space about a rather esoteric idea in mathematics, and not connected it at all with the last term in my equation, er I mean title. What does the idea of Euclid and postulates and ways of drawing triangles have to do with saints?

Clearly, there is something fundamental about sainthood. It is as indescribable, and as inarguable, as a postulate, and as everlasting - and yet there are many ways of getting to be a saint. And yet there is something common to all saints: their love of God, and love of neighbor. And, like mathematics, there is an order to this love - according to St. Augustine, the term for order in love is virtue. I can think of no better exemplification of this order than the "Paradiso" of Dante's Divine Comedy, where (by the "math" of St. Thomas Aquinas) he arranges the Choirs of Saints into orders by virtue. (For the life of me, I can never understand why people seem to be so much more interested in the orders of punishment in "Inferno" or of correction in "Purgatorio".) Rather than attempt some mathematical study of this order, I will turn to GKC and see how his writing suggests the same idea, even if (like the Mathemagician) I start with a number.

There are hundreds of appearances of "saint" and "St." in the GKC collection I have. Here is just an attempt to collect GKC's mentions of the various saints:

Larger works:
1. Mary, Queen of Saints (Queen of Seven Swords, a book of poems; she is mentioned in many books)
2. St. Francis of Assisi (biography; an early poem)
3. St. Thomas Aquinas (biography)
4. St. Anthony the Hermit (play in CW11:207 et seq)
5. St. George (in the play "The Turkey and the Turk" CW11:216 et seq; patron of England)
6. St. Barbara ("The Ballad of St. Barbara" in Collected Poems; the patroness of artillery)
7. St. Pius X (obituary in ILN Aug 29 1914)
8. St. Joseph (fragments of a play "Dialogue Between Our Lord and St. Joseph" in CW11:33 et seq)

Chance mentions:
Sts. Agatha, Agnes, Albert, Andrew, Anselm, Anthony of Padua, Athanasius, Augustine; Sts. Bartholomew, Benedict, Bernard, Bonaventure, Boniface, Brandon; Sts. Catherine (martyr), Catherine of Siena, Charles Borromeo, Clare, Cecilia, Crispin, Cuthbert; Sts. Denys, Dominic, Dorothy, Dunstan, Edmund, Edward the Confessor, Etheldreda; Sts. Faith, Francis de Sales, Francis Xavier, George of Cappadocia, Giles, Helena, Hilarion, Hugh; Sts. James, Januarius, Jerome, Joan of Arc, John, John the Baptist, John Chrysostom, John of the Cross; Sts. Julian, Lawrence, Louis, Matthew, Mark, Mary Magdalen, Michael; St. Mungo (yes; this is not in Harry Potter but in Father Brown!); Sts. Nicholas, Patrick, Paul, Perpetua, Peter, Philip, Sebastian, Sernin (Saturninus) of Toulouse; Sts. Simeon Stylites, Stephen, Telemachus, Teresa (of Avila), Theresa (of Lisieux), Thomas; Sts. Thomas Becket, Thomas More, Urban, Valentine, Veronica, Vincent de Paul.

(He also mentions St. Pancras, a real saint, but also a railroad station in London; other saints also appear as location-names.)

I will conclude with two quotes. The first is one of GKC's earliest ponderings on the idea of sainthood. It is profound because, though it seems to apply only to martyrs, it actually applies to every kind of saint, because it is the artistic expression of this idea of order, virtue, and love. And though his comments are actually just an "aside" from another discussion, they are by no means a passing thought, as you shall see:

...those figures of virgin martyrs that may be seen in the old illuminations - virgin martyrs each of whom carries a gigantic axe or a portable rack or a gridiron on which she has been grilled at a previous stage of her career. But in that case the saint carries the weapon of her enemies. It was certainly one of the boldest and most picturesque of the revolutions made by Christianity, this idea that the things used against a man became a part of him; that he could not only kiss the rod, but use it as a walking-stick. It was felt, I suppose, that when a red-hot spear had been driven clean through a gentleman's body - it became in some sense his property. Torture itself was turned into a decoration; as if we were to make an artistic wall-paper pattern out of gibbets and cats-o'-nine-tails. [GKC, ILN Oct 21, 1905 CW27:40]

The second is was written 24 years later: again almost an aside in an argument about the "monkey trial lawyer" Clarence Darrow. It will show his consistency in what might be called "GKC's formula of sainthood for the Common Man":

One of [Darrow's] arguments against immortality is that people do not really believe in it. And one of his arguments for that is that if they did believe in certain happiness beyond the grave, they would all kill themselves. He says that nobody would endure the martyrdom of cancer, for instance, if he really believed (as he apparently assumes all Christians to believe) that in any case the mere fact of death would instantly introduce the soul to perfect felicity and the society of all its best friends. A Catholic will certainly know what answer he has to give. But Mr. Clarence Darrow does not really in the least know what question he has asked. ... A Catholic does not kill himself because he does not take it for granted that he will deserve heaven in any case, or that it will not matter at all whether he deserves it at all. He does not profess to know exactly what danger he would run; but he does know what loyalty he would violate and what command or condition he would disregard. He actually thinks that a man might be fitter for heaven because he endured like a man; and that a hero could be a martyr to cancer as St. Lawrence or St. Cecilia were martyrs to cauldrons or gridirons. The faith in a future life, the hope of a future happiness, the belief that God is Love and that loyalty is eternal life, these things do not produce lunacy and anarchy, if they are taken along with the other Catholic doctrines about duty and vigilance and watchfulness against the powers of hell. They might produce lunacy and anarchy, if they were taken alone. [GKC The Thing CW3:306-8]

You do not have to be a martyr - you don't even have to know about triangles! - but you still can be a saint. Make sure YOUR story ends correctly!