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.Pierrette trudged back through Citharista and beyond it.She was tired and herfeet were sore, but she pressed on toward the cape.It was always impossibleto predict, from the course of the sun or stars"outside" what it would be "inside." She did not think the old man himselfknew he just declared it"night" whenever he felt need of a nap and that made it so.She hoped thatwhen she got there she could sleep, and not speak with Anselm until she hadrested.Crossing the natural stone bridge, all that remained of the old roadway, shenoticed that another chunk had fallen from the span, leaving a raw, red scar.What would happen when the rest fell, leaving the gate a mere hole in a barePage 50ABC Amber Palm Converter, http://www.processtext.com/abcpalm.htmlcliff, impossible to enter by? Would Anselm starve, without bread and oil,when the last of his magic died?"I must find out why the magics have changed.I don't want to lose anotherperson I love."Her first opportunity had not long to wait.Anselm (who was not sleepy, anddid not give Pierrette time to rest) wished to teach her a whole new way oflooking at the world."Geometry!" he announced."You'll be able to measure the height of a tree, acliff.the moon and stars themselves.without lifting your prettylittle bottom from your bench."Pierrette was not sure how she felt about such allusions to her body.She waspleased that Anselm wasaware she was a girl, almost a woman, and was flattered to be called "pretty,""delectable," even"succulent" which sounded grown-up and slightly dangerous.But Anselm's wordsstirred uneasy feelings she did not wish to acknowledge; had she read Ovid, orcertain lascivious Saracen writers, she would have called them erotic.She hadspent more years inside the keep than outside, but was less comfortable withwomanly things than any other girl her physical age would be.Anselm did not press his unconscious suit beyond occasional words; it was easyenough to let them pass unremarked.Besides, his enthusiasm for his main topicwas contagious."We start with a point," he was saying, "a spot that has nosize, only position.""Like zero!" Pierrette blurted.Anselm looked strangely at her."I never thought of that," he remarked.Hemade a charcoal-dot on her smooth board."That is a point." He made another."Another point," he said."Now what have we?""Two points.""That's not what I meant," he said, annoyed."What is between the points?"Pierrette stared at the two dots, but saw nothing."Well then, how many points can I fit between those dots?"Pierrette almost said, "About fifty," after estimating the size of the dots,but instead asked a question."The dots have no size? I mean, your dots only symbolize much smaller dots?Infinitely small ones?" He nodded.She had only heard the words "infinite" and "eternal" a week before, and theconcepts foreverness, something without limits, a whole universe of tiny,perfectly real numbers all crowded between zero and one made her head spin."Then," she said, "there are an infinite number of.of points." Onlyfifty dots or so, but an infinity of tiny, sizeless points.She took thecharred grapevine and drew all the points at once, with one sweep of her hand."There they all are," she said."All infinity of them."Anselm drew her from her chair, then capered around her room, holding herhands and pulling her along."You're brilliant," he said."You understand."Of course she did.It was self-evident, once you knew about zero and infinity."Now what is that?" Anselm asked, leaning on the table, panting, pointingbetween the two original dots."It is an infinity of.no, it is.a line.A line of points.""Exactly!" he blurted."And this?" He took the charcoal which broke in hishand and made two more dots.She said it was both two dots, an infinity ofinvisible dots, and also a line."A postulate!" he crowed."Two points define a line.Now three points, likethis.""Three lines a triangle.and three infinities of dots.can there bethree infinities?"Page 51ABC Amber Palm Converter, http://www.processtext.com/abcpalm.html"You may have as many as you want, child.An infinity of infinities."Pierrette's first geometry lesson was off to a grand start.The core of whatshe learned that first day was less obvious to Anselm than to her: that allknowledge and understanding is related.Not just zero, learned in one context,and dimensionless points in another; every fact, every possible observation,was linked in some way to every other.Over several working and sleeping periods she learned how postulates initialassumptions like the existence of points could be combined, and manipulatedinto proofs or theorems, grander statements of derived reality."Kewayday!"she exclaimed when she had proven, to Anselm's satisfaction, the number ofpoints required to define a circle."What?" he blurted."Kewayday.Quod erat demonstrandum," she said."That is like `amen' at the end of a prayer, or`abarakat' after a magical spell.""Bah! What makes you think so? Nonsense! This is a logical proof, not a prayeror.or a spell." He eyed her suspiciously."Have you been dabbling withprayers? Spells?""Only those my mother left me," she replied, "and I haven't `dabbled.' Istudied them and wrote them down.""Well, cease doing so.They won't work right, and you'll get hurt.""Prayers and spells and mathematical theorems aren't so different," she mused."I'm not sure they're not exactly the same.""Nonsense! Have you been eating mushrooms? If you're going to waste time withcrazy daydreams, do so in your father's grove.I'm trying to teach youworthwhile things."Pierrette did not bring the subject up again, but she did not stop thinkingabout it.The geometry ofEuclid seemed little different than well-written incantations.When she progressed into trigonometry, she upset Anselm no end."The sum ofthe angles of a triangle,"she said one day, "must be at least 180 degrees.""At least? That's wrong.The sum of a triangle's included angles is always 180degrees.No more, no less.""It's immutable? Just like magic, Master? Is that what you mean?""It is not! I mean that if a geometric postulate changed, everything thatproceeded from it would also, and the proofs would have to be redone, with newconclusions.That's nonsense.The universe would fall apart.""That's what I mean," Pierrette insisted, carefully forming her argument asdid philosophers in books [ Pobierz całość w formacie PDF ]