QB 4-4 IC-NRLF SUGGESTIONS TO TEACHERS DESIGNED TO ACCOMPANY A TEXT-BOOK OF ASTRONOMY BY GEORGE C. COMSTOCK DIRECTOR OF THE WASHBURN OBSERVATORY AND PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF WISCONSIN OF THE UNIVERSITY NEW YORK D. APPLETON AND COMPANY 1901 TWENTIETH CENTURY TEXT-BOOKS SUGGESTIONS TO TEACHERS DESIGNED TO ACCOMPANY A TEXT-BOOK OF ASTRONOMY BY GEORGE C COMSTOCK DIRECTOR OF THE WASHBURN OBSERVATORY AND PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF WISCONSIN NEW YORK D. APPLETON AND COMPANY 1901 COPYRIGHT, 1901 BY D. APPLETON AND COMPANY ASTRONOMY SUGGESTIONS TO TEACHERS THE suggestions contained in the following pages are intended primarily for the inexperienced teacher who is endeavoring for the first time to use the author's Text-Book of Astronomy. The experienced teacher may be left to follow his own methods, but the beginner is advised at the outset to read carefully the whole Text-Book before attempt- ing to teach any part of it. Cultivate your own interest in and knowledge of the subject by reading also other works e. g., at least a por- tion of those named in the Bibliography at the end of the Text- Book. Your function as a teacher is to awaken in your pupils an interest in the subject quite as much as to give them instruction in it, and neither of these purposes will be satisfactorily accomplished without enthusiasm on your own part and a wider information than is contained in any one elementary text. In your own reading of the text, note how the point of view changes as you pass from beginning to end. The initial chapters are devoted to things in the sky which can be seen with the unaided vision, and which the pupil should be taught to see for himself. Then follows reasoning about these things, using in some degree elementary mathemat- ical methods, much as the science was developed among primitive people some thousands of years ago. Then fol- lows in Chapters IV to VII an elementary statement of ~84374 2 ASTRONOMY some of the results attained by a more profound study of the phenomena which the pupil has learned to observe. But modern science deals with much more complicated data than have been presented thus far, and Chapter VIII, upon instruments, shows how the senses may be helped to obtain from the stars impressions and information that are quite beyond the reach of unaided vision. Subsequent chapters of the book are devoted mainly to a description of the heavenly bodies as they are made known to us by the new sources of information. An attempt has been made to bring before the pupil, by means of illustrations, the kind of information and evidence which the astronomer obtains at first hand from the tele- scope, camera, and spectroscope, and to draw from these illustrations the conclusions presented in the text. The teacher's attention is especially invited to this feature of the book which, if properly used, may be made to add greatly to..tjhe educational value of the work. Seek to train the pupil's perceptions and reasoning faculty as well as his memory. Beware of putting too much stress upon learn- ing facts as the substance of work in science. Although a certain basis of fact is essential, its acquisition from a book has little educational value, and emphasis ought not to be placed here, but upon the relation of facts to each other, and upon the ability to combine and make use of them. With this end in view numerous exercises and problems have been introduced into the text, and many facts inter- esting and important to a complete view of the science of astronomy have been omitted in order to make room for such work. Some of the problems involve numerical calculations that would be tedious if carried out by the ordinary arith- metical processes, but which are made short and simple by the use of logarithms. To facilitate this work a table of three-figure logarithms is given at the end of this manual. For convenient use in the class room it may be removed SUGGESTIONS TO TEACHERS 3 from the book and mounted upon a piece of cardboard. Show it to the pupils, and without going far into the the- ory of logarithms, explain its use in problems of multi- plication, division, and the extraction of roots. Be espe- cially careful in your explanations of interpolation and the process of finding the number corresponding to a given logarithm. The out-of-door work to be done by pupils in connec- tion with their study of astronomy depends so much upon conditions of climate and weather that few precepts of general application can be laid down regarding it, but the pupil should be taught to look at the stars whenever he is out of doors on a clear evening. Much may be accom- plished in this way without appreciable expenditure of time. Take advantage of the fact that the subject-matter of astronomy lies at your own door and require - no expe- dition to the country in order to find it. In addition to the suggested observation of the sky, the text contains a considerable amount of pure 1 ^ laboratory work with protractor and scale, in the measurement of pho- tographs, and in the graphical treatment of data. This is available at all times, and should be made prominent in the pupil's study. Assign specific problems of this kind to be worked out in good form, entered in a note book, and pre- served as a part of the pupil's work. It will of course be expedient to give different pupils slightly different data in order to avoid the temptation to collusion in their work. Make the note book count as a part of your final exami- nation. Before beginning this work, examine each pupil's pro- tractor and, if necessary, trim it with a pair of shears ex- actly to the line printed along the straight edge. Neglect of this precaution is sure to result in bad work e. g., in Exercise 2, ASTRONOMY CHAPTER I 1-3. Do not be satisfied with one lesson on this chapter. Come back to it once or twice a week, and repeat the exer- cises until the pupils acquire considerable facility and some pride in doing their work with precision. Exercises 4 and 5 may be practiced in a window facing south. If possible have the window open, so that the glass shall not interfere with the sun's rays. With reasonable care the sun's altitude may be measured in this way to within a small fraction of a degree. 5. Point out to the pupil that in Fig. 5 the time of the sun's greatest altitude, llh. 50m., comes near noon but not exactly at noon, and that the explanation of this discrep- ancy is to be found in the equation of time, 53. 6. Solve the problem at the end of this section by meas- uring the distance of the curve above the base line at the point corresponding to 12h. and 2.5h. e. g., 30 millime- ters and 23.3 millimeters and using the proportion 30.0 : 23.3 : : 57 : the required altitude. 7. This section, although not marked as an .exercise, should be treated as one, and the prescribed operations exe- cuted by the pupils. The method here outlined is, of course, available for determining the diameter of any heav- enly body which presents a sensible disk, but can not be applied to the earth. CHAPTER II The plumb-line apparatus which is introduced in this chapter should be kept standing in the recitation room, and pupils should be encouraged to use it at odd moments as they can find time. Employ it in connection with Ex- ercises 4 and 5 after the methods of Chapter I have become familiar. SUGGESTIONS TO TEACHERS 5 Before using the apparatus for observing stars, the pupil should have acquired some familiarity with the prin- cipal constellations. This may be done without any severe tax upon his time if he is taught to watch the sky when- ever he is out of doors on a clear evening. The habit, if once acquired, will be a lifelong source of pleasure, and the teacher who does not inculcate it ignores one of the best elements in the study of astronomy. Study the star maps. 9. The star maps are to be found at pages 124 and 190, instead of at beginning and end of book. 10. Answers. A straight line. Because they point toward Polaris. The letter W. 11. The rate at which the stars turn about the pole is 360 in 23h. 56m. 3.5s. 12. An ordinary clock may be turned into a sidereal clock by moving the regulator so as to make it gain 3m. 56.5s. per day. 13. Each trail subtends at the center of the picture an angle of 15, and the exposure therefore lasted one hour. 360 : 15 : : 24h. : Ih. 14. 15. The purpose of these sections is to connect the circular motion of stars near the pole with the diurnal mo- tion of sun and moon, and to show that the latter motion is like that of the stars, save that it takes place in a larger circle, a part of which dips below the horizon. The Pleiades are too far from the north pole to show in Plate I. 16. Pronounce, Ant-ar-es. 18. Note that to an observer at the north pole, Polaris would stand very near the zenith. As the observer travels toward the south the star will apparently move down toward the north horizon, will reach the horizon when the observer reaches the equator, and will become invisible as he goes farther south. Pronounce, Gas-tor, SpT-ca, Al-tar. 19. The meridian line may be equally well laid out at any convenient hour when Polaris is visible, regardless of 6 ASTRONOMY its place in the diurnal path by means of a method given in the periodical, Popular Astronomy, Vol. IX, No. 5. 20. " The time " always means the hour angle of some selected point of the sky, the vernal equinox for sidereal time and the sun for solar time. 21. It is well to put considerable emphasis upon the matter of exact definitions. If possible get the pupils to criticise the definitions of the text, as well as the defini- tions offered by their fellows. 22. Answers. Right ascension and declination are not changed by the diurnal motion. The hour angle of any star is when its altitude is greatest, and the same is very approximately true for the sun. It would be exactly true if there were no progressive change in its declination. After sunset the sun's altitude is negative. The north pole is always north from every other part of the sky. The intersection points are north and south in the case of the meridian, east and west in case of the equator. CHAPTER III The work of learning the principal constellations, which was commenced in Chapter II, should be continued through- out the entire period given to the study of astronomy, and in this connection the construction of star maps will be found a very useful exercise, but there is some difficulty in getting it well done. Select for a beginning some constel- lation of the zodiac, such as Taurus, Leo, or Virgo, and as a preparation for 28-31 have the pupil learn from the sky and the star maps the sequence in which the twelve zodiacal constellations make up the circuit of the sky. See Figs. 16 and 17. Continue this work, adding one new con- stellation each evening. Note carefully the relation of each new constellation to the old ones. 23. Pronounce, Al-deb-ar-an. SUGGESTIONS TO TEACHERS f 24. The line joining the moon's horns is nearly perpen- dicular to the ecliptic. The greater the angular distance of the moon from the sun, the greater is its visible area of illuminated surface. See Chapter IX for answers to other questions. 27. If the orbit of Mars coincided with that of the earth, the planet would always travel along the ecliptic instead of moving north and south of it, as shown in Fig. 14. EXEKCISE 16. Answers are to be obtained by direct measurement from Fig. 16. For the date of opposition of Jupiter, note from the figure that on January 1, 1906, the earth is not on line between the sun and Jupiter, nor is it on line at the dates represented by II, III, etc. At XII it is nearly on line, but Jupiter is then near the part of its orbit marked 1907 ; and since the earth will require several days to move up to this position, the planet's opposition will not occur until near the end of December. The exact date is December 27th. The date of conjunction, June llth, may be determined in the same way. Mark upon Fig. 16, for any given date, the positions of Jupiter and the earth, and imagine yourself standing upon the position of the earth and looking toward the sun. If Jupiter appears to the left of the sun, the planet will be visible in the evening hours ; if to the right of the sun, it may be seen in the morning sky. The same method may be applied to any other planet. From Fig. 16 the constellation opposite to the sun on January 1st of each year is Gemini, and this constellation will then reach the meridian at midnight, and so will every other constellation in the same right ascension e. g., Canis Major. See the star map. Note that the margins of Figs. 16 and 17 show the zodiacal constellations, and not the so-called signs of the zodiac. 30, 31. The exercises upon Figs. 16 and 17 should be 2 8 ASTRONOMY carefully worked out with reference to numerical accuracy. The parallel lines drawn from the sun are required, because the figures show correctly the directions of the constel- lations from the sun, but do not give their directions cor- rectly for any other point in the figures. With reasonable care the positions of the planets in the sky may be found so closely that there can be no doubt as to their identification. Point out in Fig. 17 how far the sun is away from the cen- ters of the orbits of Mercury and Mars. On July 4th, the sun is in the constellation Gemini. Sagittarius comes to the meridian at midnight, and from the star maps it may be seen that Aquila, Lyra, Draco, etc., come to the meridian at the same time with Sagittarius, since they have the same right ascension. As an exercise let the table of epochs (page 43) be ex- tended two or three years beyond 1910 by adding to the last date in each column multiples of the planet's periodic time e. g., for Venus 1910 June 28th, -f 224.7d. = 1911 February 8th, and this date increased by 2 X 224.7d. = 1912 May 3d, etc., which are the epochs for the respective years. Treat the other planets in the same way. 31. Answers. Mars comes into opposition, but Mercury and Venus do not. The maximum angular distance of Venus from the sun (elongation) may be found from Fig. 17 by measuring the angle between two lines drawn from the position of the earth, one to the sun, the other tangent to the orbit of Venus. This angle is approximately 45. From the same figure it is apparent that the shortest dis- tance between the orbits of Mars and the earth is on line toward the constellation Aquarius, and Fig. 16 shows, by the number IX, that the earth is in this part of its orbit at the beginning of September, which is therefore the month of nearest approach and maximum brightness. But also note from the figure that Mars does not come to this point of its orbit in every September. From Fig. 17, Venus may approach nearer to the earth SUGGESTIONS TO TEACHERS 9 than any other planet. The earth comes to the same longi- tude on approximately the same day of each year, because the interval between successive returns has been adopted as the unit of time a year. The intelligent teacher will be able to devise for him- self numerous additional exercises to be worked out in con- nection with Figs. 16 and 17 e. g., When will Mercury and Venus be simultaneously visible as evening stars ? When will they be at their greatest angular distance (elongation) from the sun ? When will Jupiter and Saturn appear close together in the sky (conjunction), etc.? CHAPTEK IV Much that is contained in this chapter will be familiar to pupils who have studied physics, and may be treated as a review. The exercises, however, should be worked out in order to furnish a clear understanding of the principles. In the presence of the class obtain from Fig. 16 the peri- odic time of Jupiter, 11. 9y., and from 134 its mean dis- tance from the sun, 5.2. Cube one of these numbers, square the other, and show that the quotient a 3 -f- T* is in fact very nearly unity. Ask the class to work out the cor- responding quotient for other planets. 37. Illustrate the difference between mass and weight. A bullet fired upward from a gun weighs less at the summit of its path than at the muzzle of the gun, because farther away from the earth, but its mass is unchanged. 38. Fig. 20. A body at P moving in the opposite direc- tion to that shown in the figure would describe toward the right a continuation of a conic section similar to those there shown. 40. The mathematical symbols refer to the first equa- tion on page 55. 42. Fig. 23 is intended to make clear the formation of a tide upon that side of the earth opposite the moon. This 10 ASTRONOMY is usually a difficult matter for the pupil to understand, and his attention should be especially directed to the be- havior of the blocks numbered 1 and 3 in the figure. Also note in Fig. 24 that while there are tidal waves on opposite sides of the earth, they do not stand directly under the moon, but are swept along for a certain distance by the earth's rotation. CHAPTER V In connection with this chapter, have the pupil read Coleridge's Ancient Mariner with reference to its numer- ous astronomical allusions. Trace out by means of these as nearly as may be the path of the ship. 45. Fig. 26 is intended to illustrate only the principles involved in determining the mass of the earth. The actual application of these principles must be made by means of much more elaborate apparatus. In the case suggested in the text, the displacement of the plumb bob would be approximately 0.00001 inches, as may be seen by solving for d the last equation on page 73. If the pupils have a knowledge of logarithms, let them solve by their aid the last two problems of 45. Otherwise omit these, as the computations are tedious when made by the ordinary pro- cesses of multiplication. 46. The subject-matter of this section is frequently called " precession of the equinoxes," but, as here devel- oped, the stress is to be laid upon the motion of the earth's axis ; the resultant effect upon the position of the equinox is a subordinate matter. To illustrate the precession, hold a coin so that its plane makes an angle of 23 with the top of a table, and keeping this angle unchanged, turn the coin around so that it shall face in succession north, east, south, and west. The table represents the plane of the ecliptic, the coin the plane of the earth's equator, a pencil held per- pendicular to the coin will then represent the axis of the SUGGESTIONS TO TEACHERS H earth and its motion relative to the ecliptic, while the line in which the plane of the coin intersects the plane of the table indicates by its successive position the motion of the equinox. In the case of a top the tipping force tends to bring the axis down into a horizontal position ; in the case of the earth the tipping force tends to straighten the axis up, per- pendicular to the ecliptic ; hence the wobble of the top and the precession of the earth's axis take place in opposite directions, one with the spin, the other opposed to it. Vega will be the pole star 12,000 years hence, and a Dra- conis was the pole star about 2700 B. c. After a lapse of 26,000 years the earth's axis will have made one complete " wobble," and will again point in the same direction, and the stars will again have very nearly the same right ascen- sions and declinations as now. The pole is now moving away from the Big Dipper, will never come exactly to Polaris, but after a lapse of 12,000 years will be distant about 48 from it i. e., a little more than the diameter of the circle along which it moves. 48. Point out that an increase in the obliquity of the ecliptic would cause the sun to rise higher in the sky dur- ing summer and to run lower during the winter months, and would therefore exaggerate the extremes of heat and cold. The precession has little effect upon climate, but if the earth's axis were directed toward Arcturus, for example, the obliquity of the ecliptic would be changed, with result- ing marked effects upon the seasons. Fig. 27. Insist upon the fact that the peculiar shape of the sun's reflected image shows that the surface of the water is not flat, but is curved in some way. Neglecting the effect of refraction, evening twilight lasts at the north pole from the time the sun passes the autumnal equinox until it reaches a declination of 18 south i. e., from September 23d to November 13th, 12 ASTRONOMY CHAPTEE VI Practice determining the equation of time for different dates from Fig. 30, and compare the results with the " sun fast " or " sun slow " furnished by an almanac. 52. The statement in the last line is wrong and must be corrected. For longer, read shorter. 58. Compare the watch with the telegraph signals be- fore and after observing the sun, and allow for its varia- tion between the comparisons. 61, 62. The pupil should not be asked to learn these formulae, but rather to use them with the book before him. The numerical factors and divisors in the formula of 62 of course come from the number of days in the week and from the numbers given in the leap-year rule. CHAPTEE VII Fig. 33. The new moon, at Q, would produce a partial solar eclipse visible in the regions about the north pole. 64. The theorem suggested at the end of the section is true. 65. Query. The clouds would cut off much of the light and the eclipse would be a dark one. The moon might disappear altogether. 68. The essence of a Solar Eclipse Limit is that when the earth is too far away from the node the new moon will stand so far north or south of the plane of the earth's orbit that the lunar shadows will fail to strike the earth. ?0. Eclipse prediction for Chicago, Fig. 35. Chicago lies inside, west of, the curve marked Begins at lh., and distant from it about one third the space between the curves for lh. and 2h. The eclipse therefore began about one third of an hour before one o'clock, Greenwich Time i. e., at 6h. 40m. A. M. Central Standard Time. Similarly it SUGGESTIONS TO TEACHERS 13 ended a very little after three o'clock e. g., 9h. 2m. Cen- tral Time. Fig. 36. Note that the next total solar eclipse visible in the United States occurs on June 6, 1918, and will be visible from Oregon to Georgia. 72. The following table shows all the eclipses that oc- curred during the years 1884-'89, and in connection with the saros it will suffice for predicting the eclipses of the years 1902-'07. Have the pupils predict the eclipses of next year, giving their dates, character, whether solar or lunar, partial, total, or annular, and the parts of the earth in which they are visible. Note that any given eclipse may extend over a much larger part of the earth's surface than is shown in the table, which gives approximately the center of the eclipse e. g., the first eclipse of the table was visible from England to Siberia and the north pole. Table of Eclipses 1884 March 27th Partial solar Norway. April 9th-10th Total lunar Pacific Ocean. April 25th Partial solar South Atlantic Ocean. October 4th Total lunar Africa. October 18th Partial solar Alaska. 1885 March 16th-17th Annular solar North America. March 30th. Partial lunar Indian Ocean. September 8th-9th Total solar South Pacific Ocean. September 24th Partial lunar East Pacific Ocean. 1886 March 5th-6th Annular solar Pacific Ocean. August 29th Total solar Atlantic Ocean. 1887 February 8th Partial lunar Pacific Ocean. February 22d-23d Annular solar South Pacific Ocean. August 2d Partial lunar Africa. August 19th Total solar Asia, 14: ASTRONOMY 1888 January 28th-29th Total lunar Northern Africa. February llth-12th. . . Partial solar South Pacific Ocean. July 9th Partial solar Indian Ocean. July 23d Total lunar South Pacific Ocean. August 7th Partial solar Arctic Ocean. 1889 January 1st Total solar California. January 17th Partial lunar West Indies. June 28th Annular solar Madagascar. July 12th Partial lunar Madagascar. December 22d Total solar Atlantic Ocean. To apply the saros to the first of these eclipses, March 27, 1884, we note that in the 18 years following this date there are only 3 leap years, while usually there are either 4 or 5. The length of the saros must therefore be taken as 18 years 12^ days, and adding this to the given date we find for the first eclipse of 1902 the date April 8th. This will be a partial solar eclipse visible in about the latitude of Norway, but one third of the way around the earth toward the west i. e., in British America and Alaska. Note how the eclipses of the preceding table illustrate the statement made at the end of 69. CHAPTER VIII This chapter is written from the standpoint of physical optics, and in teaching it the wave front instead of the ray of light is to be given prominence. Insist upon the anal- ogy between spherical waves in the ether and circular waves propagated along the surface of water. Especial care must be given at the outset to obtaining clear ideas of wave lengths. The analogies offered by sound are here very advantageous. Consult any good text-book of physics. 76. The principle involved in the case of the elliptical mirror is that the sum of the distances from the two foci of SUGGESTIONS TO TEACHERS 15 an ellipse to any point on its circumference is the same as the sum of the corresponding distances for any other point on the ellipse. 79. Try making a telescope with a reading glass for ob- jective and a pin hole for eyepiece, and note how it shows objects upside down, as do all telescopes used for astro- nomical purposes. Be sure not to get the reading glass too near the pin hole. Try also a second reading glass or other lens for an eyepiece, and note that with this combination a bit of paper or dirt stuck to the objective can not be seen through the eyepiece simultaneously with a distant object, thus illustrating the statement made near the end of 80. 80. Figs. 41-44. In picking out corresponding parts of these equatorial mountings, begin with the polar axis (b) and take next the declination axis (c). Fig. 49. The spectroscope is here shown at the extreme left of the figure. The prism is placed at the elbow, where the axis of the telescope tube meets the part projecting obliquely upward and to the left. The principles set forth in 85 and 88 should be thoroughly learned, as they are of frequent application in subsequent parts of the text. Compare the pictures of the stellar spectra shown in Chapter XIII with the diagram of the sun's spectrum (Fig. 50). Note that all of these are absorption spectra, while those shown in Figs. 47 and 48 are emission (bright-line) spectra. CHAPTEE IX 91. Question the pupil rather closely upon this section to determine whether his own observation corresponds to the statements here made. A foundation for this should have been laid during the preceding weeks by directing his attention to the moon and suggesting things to look for. Fig. 53 will repay careful study. Measure with the protractor the angle at the earth between moon and sun on 16 ASTRONOMY the different dates, and determine from this angle the time at which the moon will come to the meridian e. g., on June 30th the angle is 0, and sun and moon come to the meridian together, at noon. On July 3d the angle is 37.5 = 2h. 30m., and the moon crosses the meridian at 2.30 P. M. Observe also the relation of the moon's phase to the time at which it crosses the meridian, new moon corresponding to noon, full moon to midnight, etc. 94. Other methods of determining the mass of the moon are also employed by astronomers, but the one here given is sufficient for illustration. 98. Librations. Make sure that the pupils measure the positions of craters or other markings in Fig. 55, to show that there is a real libration between the two parts of the picture. Also make sure that the coin experiment (Fig. 54) is actually performed. In genera], whenever references are made to features of the moon's surface shown in Fig. 55, the same features may also be found in the plate facing page 150. Make use of this plate in working out the exercises contained in the remainder of the chapter. 102. The area of Lake Superior is 32,000 square miles. 104. The equation at the top of page 171 may be derived from that on page 170 by expanding the binomial and dropping h 2 as being insignificant in comparison with the much larger term 2Rh. Beware that the pupil does not infer from Fig. 61 that the moon has an atmosphere. The figure is intended to show effects that would be produced if there were an at- mosphere, and whose absence proves that there is none. CHAPTEE X 110. The Ode to Darkness should be read entire by the pupil. 111. The number 333,000 here given as the reciprocal SUGGESTIONS TO TEfcMfJfRS < "V x 17 of the earth's mass differs from the 329,000 of 40 because the latter corresponds to the sum of the masses of earth and moon. Use logarithms for the numerical solution of the equations in this section. Endeavor by the use of con- crete illustrations to give the pupil a good idea of parallax as a means of measuring distances e. g., Fig. 3, where the moon's distance is determined by its parallax, as seen from N^orth America and South America. 116. Don't ask the pupil to commit to memory this list of elements. It is given for illustration only. 119. Figs. 66-69. Have the pupil j)lot carefully upon some one of these figures the position of the group of spots as shown in each of the others. A smooth curve drawn through the positions thus plotted will show the path of the spot across the sun's disk. The position of the sun's rotation axis, shown by the straight line in the figure, is determined by drawing a diameter of the sun's disk per- pendicular to the path of the spot. 121. The dimensions of the sun spots are to be deter- mined from Figs. 67 and 68 in the same manner as the dimensions of lunar craters were found. 126. Enlarge upon the hand-pump experiment, and show by feeling of different parts of the apparatus that the heat comes from compression of the air, and not from ordi- nary mechanical friction. 129. The most recent determination of the sun-spot period, by Newcomb, makes its average length 11.13 years, and fixes as the probable epoch of the next maximum, December, 1904. 130. Draw a circle of the size of Fig. 82, and copy upon it from that figure the lines showing the position and fre- quency of sun spots for the year 1879. Construct a similar figure for each of the other years shown in Fig. 82, and arrange them in sequence, so as to show the progressive variation in the distribution of sun spots, 18 ASTRONOMY CHAPTEK XI In Fig. 85 E represents the earth. Bode's law. Point out that Uranus and the asteroids fall in with Bode's law, although unknown at the time it was first published, while Xeptune's distance from the sun does not at all agree with the law. Ask the pupil for an independent judgment of Bode's law. Is it probably a real law of Mature or a chance coincidence ? The opinions of astronomers upon this subject are divided. The order in which the planets are treated is quite un- conventional, but is based upon the sequence in stages of development presented by them, Jupiter standing nearest to the sun, while the small planets, Mercury and Mars, are farthest removed in respect of physical condition. It also seemed desirable to deal with Mars immediately before taking up the question of life upon the planets. 136. It is shown in 148 that the phase of a planet de- pends upon the angle at the planet between earth and sun. From Fig. 16 it is easy to find that in the case of Jupiter this angle can never much exceed 12, and $ is therefore the greatest amount of dark area that Jupiter's face can ever turn toward the earth. 142. Have the pupils determine the phase of Saturn's ring for the date of the next opposition e. g., for 1902 Saturn will be in Capricornus (see Fig. 16), and at that time (Fig. 92) the northern side of the ring will be visible, but the ring will not be opened out to its full extent. 145. The radius of the outer edge of the 'ring is 86,000 miles, and by Kepler's Third Law we have for the par- ticles composing it and for the satellite, Titan, whose peri- odic time is 383 hours, the proportion (86,000) 3 _ _ ~T* (383) SUGGESTIONS TO TEACHERS 19 which when solved furnishes for the rotation time of the outer edge of the ring jT=14.2h. Determine the inner radius of the ring by measurement from Fig. 93, and apply the above method to determine its periodic time. 148. Phases of V 7 enus. The distance of Venus from the earth is of course inversely proportional to its apparent diameter, and the distances scaled from Fig. 17 should con- firm this. In Fig. 17 the angle at Venus between the earth and sun is about 172, and this number divided by 180 gives as a quotient 0.955, from which it appears that on July 4, 1900, Venus presented to the earth a very narrow crescent of light, more than 95 per cent of its visible hemisphere being dark. 150. Next favorable opposition of Mars. In Fig. 17 try whether the planet approaches near the earth in the year 1907. 151. Determine from Fig. 17 the season in the northern hemisphere of Mars at the time of its next opposition. In 1902 Mars crosses the initial line, F, on March 16th ; the earth 196 days later, on September 23d. Following the numbers on the orbit of Mars and the earth, we find oppo- site the constellation Virgo a place where the numbers for Mars are 196 greater than for the earth i. e., the place at which the earth will have caught up with Mars, and where the planet will appear in opposition. But Virgo is just opposite to Pisces, and the sun as seen from Mars will be in Pisces at the opposition of April, 1903, and it will be summer in the northern hemisphere of Mars. 154. The name Schiaparelli is pronounced Ske-ap-ar- el-ly. 157. From Fig. 84 the force of gravity upon Mars is 0.44 times as great as on the earth. A man who weighs 150 pounds here would weigh 66 pounds there ; he could jump more than twice as far, etc. Have the pupils work this out for -both Mars and Venus. 20 ASTRONOMY CHAPTER XII Give especial heed to Figs. 109 and 110 as illustrating the motion of the comet. The angle between the plane of the comet's orbit and the plane of the earth's orbit is 38, and the fact that the two orbits do not lie in the same plane should be impressed upon the pupil. 165. Have the pupils watch for meteors on some suit- able night, and count the number that can be seen per hour. If several persons can work together, each keeping watch upon a limited part of the sky, the results will be more satisfactory. Impress upon the pupil that a meteor is not very different from an ordinary stone, and that a dozen average meteors might be carried in one's pocket without inconvenience. But some meteors are very much too large for this, and weigh hundreds if not thousands of pounds. 170. Construct a figure showing the sun, the earth, and the direction of the earth's orbital motion and rotation about its axis. It will be apparent from this figure that the part of the earth at which it is morning is on the front side. Fig. 113. Note that the straight lines which represent the meteor paths are in part heavy, corresponding to the time in which the meteor was incandescent, and in part light, corresponding to the time before and after its visi- bility. The chief purpose of the cut is to show how these parallel paths, as projected back against the sky, appear to radiate from the point a. This radiant shares in the diur- nal motion of the sky. No meteor which belongs to the shower in question can cross the radiant. The radiant is not strictly a point, but a small area e. g., in some cases twice as big as the full moon showing that the paths of the meteors are not strictly parallel, although nearly so. SUGGESTIONS TO TEACHERS 21 Fig. 115. The earth encounters these meteor showers about the middle of August and November respectively ; the latter is the Leonid shower, and the former have their radiant in the constellation Perseus, and are called Per- seids. Their radiant points are north of the ecliptic (above in the figure), and the meteors are best seen in the morning hours, although some of them are visible in the evening. 177. Any comet moving in a parabola would require 195 years, (150)^ -=- (89)*, to recede from the sun to a distance 150 times as great as that of the earth. 178. See Clerke, History of Astronomy in the Nine- teenth Century, for an account of Encke's Comet and its impeded motion. CHAPTER XIII 186. The exercises at the end of this section are to be solved by means of the equation there given. Thus to compare Spica with a sixth-magnitude star, m = l, n = 6, 6-1 B 100 5 "= 100 i. e., the one star is a hundred-fold brighter than the other. The number of seventeenth magnitude stars required to make one of the sixth magnitude is 100^ =100 2 X 100* = 100 s X 2.5119 = 25,119. The required number of full moons is 100 26 '" 12 = 100 2 - 9 = 630,100 i. e., more than three times as many as could be packed into the whole visible sky from zenith to horizon. 187. Assuming the stars whose magnitudes lie between 5.0 and 6.0 to have on the average the magnitude 5.5, we 22 ASTRONOMY 5.6-1 find from the equation B = 100 5 that a first-magnitude star gives as much light as 63 such stars. It would there- fore require 45 first-magnitude stars to give as much light as the 2,843 stars in question i. e., these faint stars col- lectively give more light than do all the 39 stars brighter than the second magnitude. By an entirely similar process we find for the number of tenth-magnitude stars required to give the same amount of light, a little more than 179,000. 188. The number 206,265 is the number of seconds con- tained in an arc equal in length to the radius of the circle to which the arc belongs. 189. Find on the star maps a few of the stars shown in Fig. 122, using for this purpose the right ascensions and declinations given in- the table e. g., show in this way that two such stars as Nos. 33 and 34, which appear near each other in Fig. 122, are really widely 'separated in the sky. Their apparent proximity in the figure comes fr*m the unavoidable neglect of their declinations in its construc- tion. 191. Insist that the pupil shall trace out upon the sky some of the allineations given in this section. 192. In Fig. 113 the apparent proper motions of the meteors are shown by the herring-bone markings upon the semicircle which represents the sky. 193. Show by a graphical construction like that of 7 how the angular proper motion of a star, given in seconds per year, can be turned into linear measure, miles per year, when the distance is known. The proper motion corre- sponds to the angle subtended by the window. 194. The question at the end of this section, of course, does not admit of a direct answer. It is inserted as an ap- peal to the pupil's imagination. 196. Putting m = 6 in the equation for solar stars, we find D = 23 X 2 3 = 184 years, which is the time required by light to come from an average sixth-magnitude star of SUGGESTIONS TO TEACHERS 23 the solar type. For an average Sirian star of the sixth magnitude D 416 years. 197. The average distance of Jupiter from the earth is 5.2 ; the distance of a Centauri ( 189) is 0.27 X 1,000,000 = 270,000, and the ratio of these numbers is easily found by means of logarithms to be 10 4 ' 72 , or 51,880 i. e., a Cen- tauri is 51,880 times as remote as Jupiter. But since each tenfold increase of distance changes the brightness by five magnitudes, Jupiter's brilliancy at the distance of a Cen- tauri would be 4.72 X 5 = 23.6 magnitudes, less than at present, and its stellar magnitude would therefore be 1.7 + 23.6 = 21.9. 200. The periodic times in the table are expressed in years. 201. The relative brightness of Sirius and its companion is found by applying the equation of 186. For Sirius B = 10,000, and for Procyon B = 25,000. 203. The problem in this section is readily solved if we note that the periodic time in seconds multiplied by the velocity is equal to the circumference of the orbit. Xote that one of the components of a spectroscopic binary may be a dark star, and that in the case of the stars Spica, Algol, and others, this is actually the case. The binary nature of the star is then shown by the variable velocity of its bright component in -the line of sight, in- stead of by a doubling of the lines. See Fig. 130. 204. Have the pupil look up Algol and Mira in the sky, and observe their change of brightness. This may require some considerable diligence and patience, but ultimate suc- cess may be expected. 205. The 43 days specified in this section is very nearly a multiple of Algol's periodic time, and its magnitude at the end of 43 days will therefore be very nearly the same as at the beginning. 206. /? Lyrge will have a maximum of brilliancy when the two components are so placed, relative to the earth, 24 ASTRONOMY that neither interferes with the light of the other i. e., when they have turned in their orbital motion 90 around from the position shown in the figure. 208. The temporary star of 1901, Nova Persei, although not so bright as that of Tycho Brahe, exceeded in brilliancy any new star during the preceding three centuries. Dis- covered on February 21, 1901, as a second-magnitude star, it rapidly increased in brilliancy, and on February 24th was brighter than Capella i. e., it had a negative stellar magnitude, 0.5. After February 24th it faded for a time with unusual rapidity, losing more than 95 per cent of its light within a fortnight. CHAPTER XIV 211. The essential fact to be insisted upon here is that the spectroscope shows differences in the kind of light which different stars emit, and by means of these differ- ences the stars are classified into three groups, which are not sharply marked off but blend into each other, just as men may be classified with respect to stature into tall, me- dium, and short. There is a real difference with no sharp dividing line between the classes. 213. The cuts in his and the following sections should receive especial attention. Trace out in them the features to which attention is directed in the text. 217. If we had two Orion nebulae of exactly the same appearance but one twice as far off as the other, the second one must be twice as long and twice as wide in order to look equally large i. e., its volume must be four times as great, but, owing to its double distance, each unit of mass would attract only one fourth as strongly as a similar unit of the first nebula. Hence the total attraction of the two nebulae, for the sun would be equal if their thickness and density were equal. SUGGESTIONS TO TEACHERS 25 218. Figs. 148-150. Point out that each tiny fleck of light in these pictures is a separate and independent star. 219. If possible obtain a field glass, and have each mem- ber of the class examine the Milky Way with it. Compare its brighter and fainter parts. 220. The questions at the beginning of this section are intended as a mental stimulus, and of course do not admit of direct answers. 222. Note that if the sun did not lie inside the galac- tic stratum the Milky Way would be a small circle of the sky. 225. That starlight is probably absorbed by dust and meteoric particles which fill the universe, is an important concept which should be emphasized, and the pupil made to realize the futility of all attempts to explore the whole universe. CHAPTEK XV A proper mental attitude toward this chapter is very important, and it must be the teacher's task to secure that attitude in the pupil. The subject-matter is inevitably thrust upon every philosophic mind that studies the prob- lems of astronomy, but the best results which can be at- tained rest upon reasoning that falls far short of demon- stration. On the other hand, the views here presented, although speculative, are not to be classed with the myths about the origin of things that fill so large a place in folk- lore, or with the idle tales about the " end of the world " issued by modern sensational writers. The chapter is an attempt to represent the best tendencies of modern thought upon great problems that have found only imperfect solu- tion, but which may be made valuable in education, not so much from the information they impart as from their stimulus to the imagination and the reasoning faculty. 26 ASTRONOMY Whenever possible, call out criticism upon the reasoning employed, and always seek to give the pupil a mental pic- ture of the things discussed. Don't allow the recitation to degenerate into a mere rehearsal of the text. For collateral reading upon this topic consult the works of Ball, Thom- son, and Newcomb, named in the Bibliography, page 384. A Table of Logarithms No. Log. No. Log. No. Log. 10 1 000 40 1.602 70 1.845 11 .041 41 .613 71 .851 12 .079 42 .623 72 .857 13 .114 43 .633 73 .863 14 .146 44 .643 74 .869 15 1.176 45 1.653 75 1.875 16 .204 46 .663 76 .881 17 .230 47 .672 77 .886 18 .255 48 .681 78 .892 19 .279 49 .690 79 .898 20 1.301 50 1.699 80 1.903 21 .322 51 .708 81 .908 22 .342 52 .716 82 .914 23. .362 53 .724 83 .919 24 .380 54 .732 84 .924 25 1.398 55 1.740 85 1.929 26 .415 56 .748 86 .934 27 .431 57 .756 87 .940 28 .447 58 .763 88 .944 29 .462 59 .771 89 .949 30 1.477 60 1.778 90 1.954 31 .491 61 .785 91 .959 32 .505 62 .792 92 .964 33 .519 63 .799 93 .968 34 .531 64 .806 94 .973 35 1.544 65 1.813 95 1.978 36 .556 66 .820 96 .982 37 .568 67 .826 97 .987 38 .580 68 .833 98 .991 39 .591 69 .839 99 .996 40 1.602 70 1.845 100 2.000 OF THE UNIVERSITY C 27 TWENTIETH CENTURY TEXT-BOOKS. The closing years of the nineteenth century witnessed a remarkable awak- ening of interest in American educational problems. There has been elaborate discussion in every part of our land on the co-ordination of studies, the bal- ancing of contending elements in school programmes, the professional training of teachers, the proper age of pupils at different stages of study, the elimina- tion of pedantic and lifeless methods of teaching, the improvement of text- books, uniformity of college-entrance requirements, and other questions of like character. In order to meet the new demands of the country along these higher planes of educational work, the Twentieth Century Text-Books have been prepared. At every step in the planning of the series care has been taken to secure the best educational advice, in order that the books may really meet the in- - creasing demand from academies, high schools, and colleges for text-books that shall be pedagogically suitable for teachers and pupils, sound in modern scholarship, and adequate for college preparation. 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