A Tangled Tale

TO MY PUPIL.  Beloved pupil! Tamed by thee,  Addish-, Subtrac-, Multiplica-tion,  Division, Fractions, Rule of Three,  Attest thy deft manipulation!    Then onward! Let the voice of Fame  From Age to Age repeat thy story,  Till thou hast won thyself a name  Exceeding even Euclid's glory!    This Tale originally appeared as a serial in The Monthly Packet, beginning in April, 1880. The writer's intention was to embody in each Knot (like the medicine so dexterously, but ineffectually, con-cealed in the jam of our early childhood) one or more mathematical questions "in Arithmetic, Algebra, or Geometry, as the case might be" for the amusement, and possible edification, of the fair readers of that Magazine.  LEWIS CARROLL      "Goblin, lead them up and down."      The ruddy glow of sunset was already fading into the sombre shadows of night, when two travellers might have been observed swiftly-at a pace of six miles in the hour-descending the rugged side of a mountain; the younger bounding from crag to crag with the agility of a fawn, while his companion, whose aged limbs seemed ill at ease in the heavy chain armour habitually worn by tourists in that district, toiled on painfully at his side.    As is always the case under such circumstances, the younger knight was the first to break the silence.  "A goodly pace, I trow!" he exclaimed. "We sped not thus in the ascent!"    "Goodly, indeed!" the other echoed with a groan. "We clomb it but at three miles in the hour."  "And on the dead level our pace is--?" the younger suggested; for he was weak in statistics, and left all such details to his aged companion.  "Four miles in the hour," the other wearily replied. "Not an ounce more," he added, with that love of metaphor so common in old age, "and not a farthing less!"    "'Twas three hours past high noon when we left our hostelry," the young man said, musingly. "We shall scarce be back by supper-time. Perchance mine host will roundly deny us all food!"  "He will chide our tardy return," was the grave reply, "and such a rebuke will be meet."  "A brave conceit!" cried the other, with a merry laugh. "And should we bid him bring us yet another course, I trow his answer will be tart!"