o/V. Then the equation of the curve of pursuit, when n is not equal to unity, is For example, let the dog run twice as fast as the hare, or n = 1, then the equation of the curve is The length of an element of the curve being dx V1 + p2, where p = dy/dx, the length of the curve from A to P is found to be algebraically expressed, thus: When x = 0 then y = 3 2/3 a, and the length of the curve is 4/3 a. Here the dog has run double the distance that the hare has run, and it catches the hare at the point x = 0, y = 2/3 a. When n is equal to or greater than unity, the dog can never catch the hare. When n is less than unity the dog will catch the hare. The student in calculus may find it profitable to solve the following problem: Let the dog run 10 feet per second and the hare 8 feet per second, and let a = 720 feet; prove that the dog will catch the hare in 6 minutes and 40 seconds from the instant when the hare starts at O and the dog starts at A. CHAPTER VII ASTRONOMY AND THE CALENDAR 120 STRONOMY is probably the most ancient of the physical sciences, the first facts being observed by shepherds who watched their flocks at night. The historian Josephus, in his Antiquities of the Jews, begins with the creation of the world and follows closely the biblical narrative. Speaking of Phaleg, fourth in descent from Noah and of his son Tera, who was the father of Abraham, he says: "God afforded them a longer life on account of their virtue and the good use they made of it in astronomical and geometrical discoveries." Speaking of the sojourn of Abraham among the Egyptians, he says, "He communicated to them arithmetic and delivered to them the science of astronomy; were unacquainted with those parts of learning, for that science came from the Chaldeans into Egypt and from thence to the Greeks." they 121 The order of the twelve constellations of the zodiac may be remembered by the following ancient lines: The Ram, the Bull, the Heavenly Twins, Next the Crab, the Lion shines, The Virgin, and the Scales, The Scorpion, Archer, and the Goat, In memorizing this it is well to note, that the word shines should rhyme with Twins, and Pot with Goat. The order here is from west toward east; when the Ram is setting in the west the Scales are rising in the east, when the Scales are setting in the west the Fish are rising in the east. This is a rough statement only, for at certain seasons of the year less than one-half of these constellations are above the horizon, while at other seasons more than onehalf of them are visible at one time. Unfortunately the artist who put several of the constellations of the zodiac on the ceiling of the grand concourse in the Grand Central Station in New York, reversed this order, for there we see Aquarius in the east while the Crab is in the west. The copy from which he worked evidently had been incorrectly made; perhaps he took it from a celestial globe and then turned it around so as to interchange east and west. 122 The greatest of all optical instruments was the reflecting telescope of William Herschel, which was finished in 1789. The tube was forty feet in length, five feet in diameter, and weighed 60,000 pounds. With this telescope, magnifying 6450 times, he discovered two new moons circling around the planet Saturn, and recorded hundreds of new double stars and nebulæ. His sister, Caroline Herschel, was his constant companion in all his astronomical labors. William Herschel died in 1822. In 1839 his celebrated son, John Herschel, took down the great telescope, which had then become a victim to the ravages of time and could no longer be used. The long tube was carefully laid upon three stone pillars where it could be preserved as a relic of the past. In the Christmas holidays of that year, John Herschel, his wife, and their six children held a family feast in the great tube, and there they sang a song written by him in honor of the occasion: In the old telescope's tube we sit, And the shades of the past around us flit, His requiem we sing with shout and din As the old year goes out and the new year comes in. And make the old telescope rattle and ring. Full fifty years did he laugh at the storm, And the blast could not shake his majestic form. Now prone he lies where he once stood high And searched the heavens with his broad bright eye. Here watched our father the wintry night And his gaze was fed by pre-adamite light; And united they strained their vision above. 123 Galileo was the first man who looked at the heavenly bodies through a telescope. It was in 1610 that he saw four satellites moving around the planet Jupiter, and this demolished the theory that the earth was the center around which the planets revolved. These four satellites of Jupiter were the only ones known until 1892, but since then four smaller ones have been discovered. The earth has one moon, Mars has two, Jupiter has eight, and Saturn has nine or ten. The inner moon of Mars is near the planet and has such a high velocity that it rises in the west and sets in the east, while both new and full moon can be observed in a single night. All other known satellites, like our own moon, rise in the east and set in the west. 124 BOTANY AND ASTRONOMY If we examine the leafy stem of a plant we shall find the leaves upon it arranged in a symmetrical order and in a way uniform for each species. If a line be drawn around the stem from the base of one leaf stalk to that of the next, and so on, this line will wind around the stem as it rises, and on any particular plant there will be the same number of leaves for each turn around the stem. In the basswood, the Indian corn, and all the grasses, we have the two-ranked arrangement; the second leaf starting on exactly the opposite side of the stem from the first, the third opposite the second and hence directly over the first, so that all the leaves are in two vertical ranks, one on one side of the stem and one on the other. Next is the three-ranked arrangement such as is seen in sedges; here the second leaf is onethird of the way around the stem, the third one two-thirds, and the fourth one directly over the first. Then in the apple, cherry, and most of our common shrubs, the leaves are arranged in five vertical ranks, and the spiral winds twice around the stem before it reaches a leaf directly over the first one; here the distance between any two ranks is two-fifths of the circumference of the stem. Then in the common plantain there are eight ranks, and three turns around the stem, so that the distance between any two ranks is three-eighths of the circumference. Now if we express these arrangements by figures, we have the fractions 1/2, 1/3, 2/5, 3/8, in which the denominator expresses the number of ranks and the numerator the number of turns of the spiral line around the stem before it reaches a leaf directly above the one from which it started. |