Tif/.l J?iQ_ 2 .tins'* A. S.J'iintiersm X. Telescopic -7zew- o£ -fke itijl Mx> on.. • S.TeLescoprc vi.-w of Saturn '-'..is xiug*. cLo of a^gmt of the Moon mesa? cp^iaili-atiLL'p '. 4. do of Jiwttpr &■!««. "NTooois. AN INTRODUCTION TO ASTRONOMY DESIGNED AS A TEXT-BOOK FOR THE USE OF STUDENTS IS COLLEGE. BY DEOTSON OLMSTED, LL.D., PROFESSOR OP ASTRONOMY IN TALE COLLEGE, AND E. S. SNELL, LL.D., PROFESSOR OP MATHEMATICS IN AMHERST COLLEGE. THIRD STEREOTYPE EDITION Carefully revised, with additions. ^ NEW YORK: COLLINS & BROTHER, 414 BROADWAY. V Q3* 3 JLatared according to Act of Congress, in th^, yet- la-Mt By DENISON OLMSTED, in tne Clerk's Office of the District Court of Cot lectiofflt Revised Edition. Entered according to Act of Congress, in the year 18&I, By JULIA M. OLMSTED, For the Children of Demson Olmsted, deceased, Ls 3he Clerk's Office of the District Court of the District of Ckmasettaftg Thibd Stereotype Edition. Enteied according to Act of Congress, in the year 1866, By JULIA M. OLMSTED, For the Children of Demson Olmsted, deceased, Li iko Clark s Office of the District Court jf the Dfcfcr? * at Coa&eetlMk Third Stereotype Edition. Carefully revised, with, additions; Copyright, 1883, By JULIA M. OLMSTED. PREFACE TO THE EDITION OF 1883. The late discoveries made in Astronomy, principally by the aid of the spectroscope, require that something be added to the descriptive parte of this work. In the present edition,, therefore, information of this nature, accompanied with illus- trations, is given in an Appendix, with references to and from the corresponding articles in the text. The mean equatorial Horizontal Parallax of the Sun, adopted from Professor Newcomb's "Investigation of the Distance of the Sun and the Elements which depend on it," is 8". 848. This number is founded upon a discussion and combination (with their relative weights) of the results given by all the different methods of obtaining the parallax, and therefore is as near an approximation to the truth as can be made at present. The distances and magnitudes throughout the work are reduced to conform to this value. This edition contains the latest emendations of Professor Snell ; and also various numerical corrections, in accordance with the best authorities, for which the Publishers are indebted to Professor Selden J. Coffin, Lafayette College. Professor Coffin has also added to Art.. 264, Appendix M, and has enlarged and thoroughly revised Tables II, IV, and V. August, 1883. CONTENTS CHAPTER I. PA«» Astronomy.— Its subject. — Globular form of the earth proved.— Modes of measuring the earth. — The terrestrial equator. — The horizon and seconda- ries. — The celestial equator. — The ecliptic. — The diurnal motion. — Its phe- nomena. — Problems on the globes 1-14 CHAPTER II. Parallax. — Diurnal parallax.— Its variation.— To find the parallax of the moon. — Atmospheric refraction. — Illumination of the sky. — Twilight 15-28 CHAPTER III. The observatory. — The transit-instrument. — The astronomical clock. — Measur- ing right ascension. — The mural circle. — Measuring declination. — Altitude and azimuth instruments.— The sextant. — Spherical problems 24-3$ CHAPTER IV. Observations of the sun's place. — The ecliptic and zodiac. — The annual mo- tion. — The change of seasons. — Arrangement of heat and cold. — Form of the earth's orbit. — Mode of determining it 88-47 CHAPTER V. The sidereal and solar day. — Mean and apparent solar time. — Reasons why solar days are unequal. — The equation of time. — The calendar 47-54 CHAPTER VI. • Projectile, centripetal, and centrifugal forces. — Laws of centrifugal force. — Its effects on the earth. — Loss of weight. — Spheroidal form. — Proofs of diurnal motion 54-63 CHAPTER VII. The sun. — Its form. — Its distance. — Its dimansions. — Its rotation. — Solar spots —Theory of spots. — Condition of the sun's surface. — The zodiacal light 62 -68 CONTENTS. CHAPTER VIII. PAGB Kepler's laws, — Law of areas proved. — Law of gravity proved. — Its prevalence throughout the system. — The paths of projectiles. — Effect of an impulse on one body of a system 69-8Q CHAPTER IX. Precession of equinoxes. — Consequent motion of the poles. — Cause. — Compo- sition of rotations. — The tropical and sidereal years. — Nutation. — Aber- ration of light. — Velocity of light discovered. — Advance of apsides. — Its cause. — How to find the sun's true place 80-88 CHAPTER X. The moon. — Its distance and size. — Its motion round the earth. — Its orbit. — Librations. — Its path about the sun. — Its phases. — The harvest moon. — The moon's surface. — Measurement of its mountains. — Appearance of the earth from the moon 88-102 CHAPTER XL The moon's motion disturbed by the sun. — Gravity to the earth diminished. — Equations for finding the moon's place. — Equation of the center. — Evec- tion. — Variation. — Annual equation. — Advance of apsides. — Eetrogradation of nodes. — Periodical and secular equations 102-109 CHAPTER XII. Eclipses. — Their cause. — Eclipse months. — The earth's shadow. — Its dimen- sions computed. — To find beginning, middle, and end of a lunar eclipse. — Eclipse of the sun. — Dimensions of the moon's shadow. — Its velocity over the earth. — The Saros. — Phenomena of a solar eclipse 109-124 CHAPTER XIII. (ftethods of determining longitude. — By the chronometer. — By eclipses. — By the lunar method. — By the telegraph. — Change of days in going round the earth 125-1 29 CHAPTER XIV. Tides. — Form of equilibrium under the action of the moon. — Joint action of sun and moon. — Diurnal inequalities. — Effect of coasts. — Tides in seas and lakes 180-185 CHAPTER XV. Planets grouped. — Distances from the sun. — Revolutions. — Dimensions. — Masses and densities. — Mercury. — Its motions. — Its phases. — Its transits. — Venus. — Its transits. — Parallax of the sun found. — Mars. — Its motions 135-151 CONTENTS. CHAPTER XVI. PA6S The planetoids. — Jupiter. — Its belts. — Its satellites. — Their eclipses and oo- cultations. — The velocity of light found by them. — Saturn. — Its rings. — Their disappearances. — The satellites of Saturn. — Uranus. — Its satellites. Neptune. — Its discovery 152— Kit CHAPTER XVII. Moments of a planetary orbit. — Method of finding the first. — The second. — The third. — The fourth. — The fifth and sixth. — The masses of the planets found. — Perturbations. — In the positions of orbits. — In their forms.— Sta- bility of the system. — Relations of the planets 165-180 CHAPTER XVIII. . Comets. — Their number. — Effects of eccentricity of orbit, — Dimensions of comets. — Their masses. — How to find their orbits. — Halley's comet. — Comets of short period. — A resisting medium. — Remarkable comets. — Shooting stars. — Meteoric showers. — Aerolites 180-194 CHAPTER XIX. The stellar universe. — Classifications of stars. — Constellations. — Annual par- allax. — Stars whose distance is known. — Nature of fixed stars. — Proper motions. — Double stars. — Binary stars. — Their orbits. — Their masses.— Periodic stars. — Clusters. — Nebulae. — The galaxy. — The Nebular hypothe sis 194-218 Appendix A to M 214-224 Tables I.— The Calendar 226-227 II.— Elements of the Planets 228 III.— Elements of the Satellites 229 IV.— Mean Places of Principal Stars 230 Y.— Planetoids . . . i 231-233 Plates.— Spectroscope 225 Chronograph 234 Comet of 1843. Comet of 1858.— Nebulae. Part of Galaxy.— Double Stars. Clusters.— "Nebulae. ASTRONOMY. CHAPTEE I. GENERAL FORM AND DIMENSIONS OF THE EARTH. — THE DIURNAL MOTION. — ARTIFICIAL GLOBES. 1. General definitions. — Astronomy is the science which treats of the heavenly bodies — that is, of the sun, the planets and their satellites, the comets, and the fixed stars. The sun, planets, satellites, and comets constitute the solar system, which is so called because the sun is the principal body belonging to it, and controls the movements of all the others. The fixed stars are the bodies situated at vast distances out side of the solar system, and which, on account of that distance, exhibit little or no change of position with respect to each other. 2. The Copernican system. — This name is given, in honor of Copernicus, to the science of astronomy as now established by demonstration, in distinction from the erroneous systems of the ancients. It explains the diurnal and annual motions of the heavens, by supposing the earth to rotate each day on its axis, and to revolve once a year around the sun. 3. The globular form of the earth. — That the earth is nearly if not exactly a sphere, is indicated in several ways. 1. It is one of the planets. And, as we see the other planets to be nearly spherical, we reason from analogy that the earth is 6pherical also. 2 DIP OF THE HORIZON. 2. Iii a lunar eclipse, whichever side is turned toward the moon, the outline of its shadow, projected on that lody, is always circular. 3. Its convexity, by which it wholly or partially conceals distant objects, as a lighthouse or a ship at sea, appears to be equally great on all parts of the ocean. 4. An arc of a given number of miles, measured on any part of the earth, is found always to subtend an angle of nearly equal size at the center ; showing that the curvature is every- where nearly the same. 5. The depression, or dip of the horizon, is equally great at every place, and on every side of the observer, provided his elevation above the ocean level is the same. This will be un- derstood by the next article. Fig.l. 4. Dip of the horizon. — If the eye were at A (Fig. 1) on the surface of the earth, the vault of the heavens would be lim- ited by a plane touching the earth at A, and would therefore be just a hemi- sphere. But if the eye is elevated, as to O, and tangent lines are drawn from that point to the earth on every side, then more than a hemisphere of the sky is visible. Let ZC be the direc- tion of a plumb-line, and let HOR represent a plane perpendicular to it ; then there would be a celestial hemi- sphere in view above this plane, and the remotest visible points on the earth would be depressed below the plane by the angle HOD or ROE. This angle is called the dip of the horizon. If AO is a given height, it is found that the angle HOD is sensibly equal on whatever side of the station, or on whatever part of the earth, the measurement is made. It follows from this that the earth is very nearly a sphere. At the height of 100 feet, the depression is about 10', and varies nearly as the square root of the height. The word down expresses the direction in which a plumb- DIMENSIONS OF THE EARTH. 3 line hangs, or a body falls — that is, toward the center of the earth. Hence, on different parts of the earth, " down" denotes all possible directions. So " up," or from the center, is in every direction ; and the direction which is down at one place, is up at a place on the opposite side of the earth. Fig. 2. 5. Dimensions of the earth. — The semi-diameter of the earth may be approximately fonnd by measuring the height oi the station AO (Fig. 1), and the length of the tangent line OD. If O were the summit of a mountain, then D would be the most distant point from which it could be discerned. In Fig. 2, suppose that the height of the mountain BD, and the distance to the point where it is just seen in the hori- zon AD, have been measured. Let BD = A, and AD = d, and the radius, AC or BO = x. Then a? 2 + d 2 ={x + hy ■x* + 2 hx + h\ Hence, 2 hx = d 2 — h% and x = 2A Thus, the semi-diameter of the earth is found in terms of k and d. The magnitude of the earth may be more accurately found, by measuring the arc of a meridian. Let a line be carefully measured due north on the earth's surface, and the correspond- ing difference of latitude be observed, as indicated by the change in the elevation of the stars. Then, the surveyed line is the same part of the earth's circumference, which the differ- ence of latitude is of 360°. Thus, if the arc is 1° 30 r , its length is found to be about 103.5 miles. Hence, 1° 30' : 360° :: 103.5 : 24,840; which is nearly the number of miles in the circumference oi the earth. By a comparison of the most accurate measure ments, it is ascertained that The circumference of the earth = 24,857 miles. The diameter (24,857 -f- 3.14159+) = 7,912.4 miles. One degree of the circumference = 365,000 feet. One second = about 100 feet. 4 SECONDARIES OF THE EQUATOR. 6. Inequalities of surface. — Although the surface of the earth is uneven, and there are high mountains and deep valleys in many parts of it, yet these are very minute compared with the magnitude of the entire earth ; so that the spherical form is not disturbed by their existence. Mountains, four or five miles high on the earth, are relatively no more than are the particles of dust which adhere to a globe one foot in diameter. Thin writing-paper, pasted upon such a globe in the form ot the continents, would be sufficiently thick to represent their general elevation above the oceans. 7. The diurnal rotation. — The earth revolves continually from west to east, on an imaginary line drawn through its cen- ter, called the earth? s axis. The time occupied in completing a revolution is called a day, which is divided into twenty-four hours. A great circle of the earth, perpendicular to the axis, is called the equator. In the diurnal rotation, every particle of the earth describes a circle, whose plane is either parallel to the equator or coincident with it. The extremities of the axis are called respectively the north and south poles. 8. Secondaries of the equator. — All great circles passing through the poles, and therefore perpendicular to the equator, are called meridians. Such a circle may be supposed to pass through any place whatever on the earth, and is called the me- ridian of that place. As all great circles of a sphere which are perpendicular to a given great circle, are called its secondaries^ the meridians are secondaries of the equator. The latitude of a place is its distance north or south from the equator, measured on the meridian of that place, in degrees, minutes, and seconds. Parallels of latitude are small circles of the earth, parallel to the equator. The longitude of a place is the distance of its meridian in degrees, minutes, and seconds, east or west from ^ome standard meridian, as that of the observatory of Greenwich. The people of different nations usually reckon longitude from s^Oie import- ant observatory of their own country. Thus, the X lench reckon from Paris, and the Americans from Washington. Any place on the earth is determined by giving its latitude and longitude THE HORIZON AND ITS SECONDARIES. 5 9. The celestial sphere. — The earth is called the terrestrial sphere. The celestial sphere is that apparent vault, called the sky, which surrounds the earth on every side, and to which all the heavenly bodies seem to be attached. The center of the earth is regarded as the center of the celestial sphere also. Bat the distance of nearly all the heavenly bodies is so immense, that it is immaterial from what point of the earth they are viewed. Hence, for most purposes of astronomy, the eye of the observer may be considered as the center of the celestial sphere. 10. The horizon and its secondaries. — If the plumb-line (usually called the vertical), at any place on the earth, is sup- posed to be extended till it intersects the celestial sphere, it marks the zenith above the place, and the nadir below it. And a plane passed through the center of the earth, perpendic- ular to the vertical, is called the rational horizon of that place. This is a great circle of the celestial sphere, and divides it into -upper and lower hemispheres. The sensible horizon is parallel to the rational horizon, and passes through the place on the earth's surface. The planes of these two horizons are therefore near 4,000 miles apart ; but so great is the distance of the heavenly bodies, that the two planes seem to unite in the same great circle of the heavens. If the observer is at all elevated above the earth's surface, the boundary line between sky and water is a little lower than the horizon, so that somewhat more than half of the celestial sphere is in view (Art. 4). The secondaries of the horizon intersect each other in the vertical line, and are called vertical circles. One of them is the meridian of the place. The inter- sections of the meridian and horizon are the north and south points of compass. The vertical circle at right angles to the meridian is called the prime vertical. This intersects the hori- zon in the points called east and west. The altitude of a heavenly body is its elevation above the horizon, measured on the vertical circle passing through the body. The zenith distance of a body is the distance between it and the zenith, and is therefore the complement of its Altitude. 6 CELESTIAL EQUATOR. The azimuth of a heavenly body is an arc of the horizon > measured from the meridian to the vertical circle, which passes through the body. The amplitude is measured from the verti cal circle passing through the body to the prime vertical, and is therefore the complement of the azimuth. The altitude, or zenith distance of a heavenly body, along with its azimuth or amplitude, determines its place in the visible heavens. 1 1 . The celestial equator and its secondaries. — If the axis on which the earth revolves is produced to the heavens, it becomes the axis of the celestial sphere, and marks the north and south poles of that sphere. The north pole is at present in the con- stellation of Ursa Minor. If the plane of the equator be ex- tended in like manner, it becomes the celestial equator. The secondaries to this circle are called meridians, as on the earth* They are also called hour-circles, because the arcs of the equator intercepted between them are used as measures of time. Fig. 3. ZD Let n (Fig. 3) represent the north pole of the earth, s its bo nth pole, eqthe equator (projected in a straight line), o a given THE ECLIPTIC. 7 place whose north latitude is eo. Then N, S, are the poles of the celestial sphere, EQ is the celestial equator, Z is the zenith of the place o, R is its nadir, and HO its rational horizon. oesqn is the terrestrial meridian of the same place, and ZESQK is its celestial meridian, or hour-circle. 1 2. The ecliptic. — Besides the equator, there is an import- ant circle of the celestial sphere, called the ecliptic. It is that in which the sun appears to make its annual circuit around the heavens. It is inclined to the equator at an angle of nearly 23J°, crossing it in two opposite points, called the equinoctial points, or equinoxes. The word " equinoxes" is used also to express the times at which the sun crosses the equator, because at those times the nights are equal to the days. The vernal equinox is the time when the sun passes the equator from south to north, as it occurs in the spring, about March 20th. The autumnal equinox occurs on or near September 22d, when the sun returns to the south of the equator. The solstitial points, or solstices, are those points of the ecliptic, which are furthest north or south from the equator, situated therefore midway between the equinoxes. They are so named, because there the sun stops in his advance north- ward or southward, and begins to return. The summer solstice is the point where, and also the time when the sun is furthest north, about the 21st of June. He passes the winter solstice on or near the 21st of December. The equinoctial colure is that secondary to the equator which passes through the equinoxes. The solstitial colure is that which passes through the solstices. They are therefore at right angles to each other, and the latter is a secondary to the ecliptic, as well as to the equator. 13. Signs of the ecliptic— The ecliptic is divided into 12 equal parts of 30° each, called signs, which, beginning at the vernal equinox, succeed each other eastward, in the follow ing order : DIUKNAL MOTION OF THE HEAVENS. Northern. Southern. 1. Aries . . . f 7. Libra . . , =Cb 2. Taurus . . . 8 8. Scorpio m 3. Gemini . . n 9. Sagittarius * 4. Cancer . . . © 10. Capricornus V3 5. Leo . . . . $1 11. Aquarius . AW 6. Yirgo . . . T02. 12. Pisces . . X The vernal equinox being at the first point of Aries, the sum mer solstice is at the first of Cancer, the autumnal equinox at the first of Libra, and the winter solstice at the first of Capricorn. 14. R'ght ascension and declination. — The right ascen- sion of a heavenly body is the angular distance of its meridian from the vernal equinox, measured eastward on the equator. The declination of a body is its angular distance north or south from the equator, measured on the meridian of the body. The equator is the plane of reference for right ascension and declination on the celestial sphere, as it is for latitude and longitude on the terrestrial. But terrestrial longitude is reck- oned both east and west, while right ascension is reckoned only to the east. 15. Celestial longitude and latitude. — On the celestial sphere, longitude and latitude are referred to the ecliptic, not to the equator. Suppose a secondary to the ecliptic to pass through a heavenly body ; the distance of the body from the ecliptic, measured on the secondary, is its latitude ; and the dis- tance of this secondary from the vernal equinox, measured eastward on the ecliptic, is its longitude. Eight ascension and longitude are reckoned only eastward, from 0° to 360°, the first on the equator, the other on the ecliptic. - 1 6. Apparent diurnal motion of the heavens. — As the earth revolves from west to east on the axis ns, an observer, not being conscious of this motion, sees the heavenly bodies appa- rently revolving in the opposite direction — that is, from east to west, about the axis NS. The sun, moon, and every planet, comet, and star, is observed to pass over from the eastern part of the sky toward the western, with a regular motion, reap- pearing again in the east, after the lapse of about one day, in the same, or nearly the same place. The fixed stars describe circles, which are exactly parallel to the equator, and in pre- cisely the same length of time. But the other bodies vary somewhat in their paths, and the periods of describing them, thus indicating that they are affected by other motions besides the diurnal rotation. 17. Rising, setting, and culmination. — In Fig. 3, AB, DO, FG, etc., drawn parallel to EQ, represent the diurnal circles of stars, projected in straight lines. Some of these circles intersect the horizon HO. These intersections are the points of rising or setting. Thus, a star describing the circle GF, rises in the northeast quartei, and sets in the northwest, at points which are both represented by r. The star, whose diurnal circle is IK, rises in the southeast, and sets in the south- west, at t. A star on the equator rises exactly in the east, and sets in the west, at the point G. The points, in which these circles cut the meridian, are called the points of culmination. Thus, the star on FG makes its upper culmination at F, arid its lower one at G. On AB, both the upper and lower culminations are above the horizon ; on MP, they are both below. If both culminations of a star are above the horizon, it is always in view; if both below, it never comes in sight. The number of stars which do not rise and set, depends on the position of the celestial poles in relation to the horizon — that is, on the latitude of the place. By the culmination of a body, in the ordinary use of the word, is meant its upper culmination. 18. Relations of the horizon to the diurnal circles. — Every change of position on the earth changes the horizon. If an observer moves eastward, all the heavenly bodies which rise and set, rise earlier, and also culminate and set earlier. If he moves westward, they rise, culminate, and set later. If he moves toward the nearer pole of the earth, the corresponding pole of the celestial sphere becomes more elevated, and the 10 THE PARALLEL SPHERE. other more depressed ; and the contrary, if he moves from the nearer pole — that is, toward the equator. In all north latitudes, the north pole is elevated, and the south pole depressed ; and the reverse in south latitudes. And the elevation of one pole, and the depression of the other, equals the latitude. For (Fig. 3) NO, the elevation of one pole (=HS, the depression 01 the other), equals EZ, since each is the complement of ZN But EZ=&9, the latitude, because they subtend the same angl atC. The elevation of the celestial equator equals the complement of latitude. For EH is the complement of EZ, which equals eo, the latitude. Hence, the angle by which all the circles of diurnal motion are inclined to the plane of the horizon, equals the complement of latitude, since they are parallel to the equator. On account of this change of inclination between the horizon and the diurnal circles, the aspect of the diurnal rotation is very different in different parts of the earth. 19. Tht right sphere. — This name is given to those posi- tions, in which the diurnal circles cut the horizon at right angles. All points of the equator are so situated. As the latitude is zero, the poles, having no elevation or depression (Art. 18), are both in the horizon ; the celestial equator passes through the zenith, thus coinciding with the prime vertical; and all the paths of daily motion, being parallel to the equator, are perpendicular to the horizon. Every heavenly body, unless situated exactly at one of the poles, rises and sets during each revolution, and continues above the horizon just as long as it remains below it. If a star rises in the east, it sets in the west, and culminates in the zenith and nadir. 20. The parallel sphere. — This term expresses the appear- ance of the heavens at those points of the earth where the circles of daily rotation are parallel to the horizon. This aspect can be presented only at the poles. For, at those points, the latitude being 90°. one pole must be elevated 90° — that is, to the zenith — and the other depressed 90°, or to the nadir. Hence. U?e diurnal circles, being perpendicular to the axis, must Iks ARTIFICIAL GLOBES. 11 horizontal, and the equator must coincide with the horizon. Every star in view passes around the sky, maintaining the same elevation at every point of its path. JSTo one of the iixed stars ever rises or sets, and every point of a diurnal circle may be regarded as a point of culmination, since it is on a meridian passing through the observer's place. At the north pole, that half the year in which the sun is north of the equator, is uninterrupted day ; during the other half, the sun being south of the equator, it is constant night. In the right sphere, the whole sky is seen, and every part of it just half the time ; in the parallel sphere, only one-half the sky is ever seen, but it is seen the whole time. 2 I . The oblique sphere. — At all latitudes, except 0° and 90°, the circles of daily motion are oblique to the horizon, since they incline at an angle equal to the complement of the lati- tude. Thus, at latitude 42° N., the celestial equator is elevated 48° above the southern horizon, and all the diurnal circles have the same inclination, as shown in Fig. 3. The circle OD, whose distance from the elevated pole equals its elevation, just touches the horizon at the lower culmination, and is the limit of that part of the sky which is always in view. This is called the circle of perpetual apparition. The circle HL, at the same distance from the depressed pole, also touches the horizon, and is called the circle oft. perpetual occultatio?i, since it limits that part of the sky which is always concealed. The horizon HO, bisects the equator EQ. Hence, a body on the equator is as long above the horizon as below it, in every part of the earth. But all bodies between the equator and the elevated pole are longer above the horizon than below, while on the opposite side they are longer below than above. 22. Artificial globes. — They are of two kinds, terrestrial and celestial. The terrestrial globe is a miniature representa- tion of the earth, having also the equator and several meridians and parallels of latitude traced upon it. The celestial globe exhibits the principal fixed stars in their relations to each other, and to the equator and ecliptic. The artificial globe is suspended in a strong brass ring by an Cl PROBLEMS ON THE GLOBES. axis passing through the north and south poles, Dn which it is free to revolve. This ring represents the meridian of any place, and is supported vertically within a horizontal wooden ring which stands upon a tripod. The wooden ring represents the horizon. The brass ring may be slid around in its own plane, so as to elevate or depress either pole to any angle with the horizon. It is graduated from the equator each way to the poles, for measuring latitude and declination ; while the horizon ring has near its inner edge two graduated circles, one for azimuth, and the other for amplitude. On this ring also, for convenient reference, are delineated the signs of the ecliptic, and the sun's place in it for every day of the year. Around the north pole is a small circle, marked with the hours of the day ; and at the same pole, a brass index is attached to the meridian, which can be set at any hour of the circle. The quadrant of altitude is a flexible strip of brass, graduated into 90 parts, each equal to a degree of the globe. This can be used for measuring angular distances in any direction on the sphere ; and when applied to a vertical circle of the celestial globe, it determines the altitude, or zenith distance of a heav- enly body. To adjust either globe for any place on the earth, elevate the corresponding pole to a height equal to the latitude. The axis will then form the proper angle with the horizon. And if the globe is turned (the celestial westward, or the terrestrial east- ward), the diurnal motion will be truly represented. 23. Problems on the terrestrial v 1. To And the latitude and longitude of a place. Turn the globe so as to bring the place to the brass meridian ; then the degree and minute on the meridian over the place shows its latitude, and the point of the equator, under the meridian, shows its longitude. Example. "What are the latitude and longitude of New York? 2. To find a place by its given latitude and longitude. Find the given longitude on the equator, and bring it to the meridian; then under the meridian, at the given latitude, will be found the required place. PROBLEMS ON THE GLOBES. 13 Ex. What place is in latitude 39° 2SL, and longitude 77° W. ? 3. To find the beaiing and distance of one place from another. Adjust the globe for one of the places, and bring it to the meridian ; screw the quadrant of altitude directly over the place, aud bring its edge to the other place. Then the azimuth will be the bearing of the second place from the first, and the number of degrees between them, multiplied by 69J, will give their distance apart in miles. Ex. Find the bearing of New Orleans from New York, and the distance between them. 4. To find the difference of time at different places. Bring to the meridian the place which lies west of the other, and set the hour-index at XII. Turn the globe westward, until the other place comes to the meridian, and the index will show the hour at the second place when it is noon at the first. The hour thus found ie the difference required. Ex. When it is noon at New York, what time is it at London ? 5. The hour being given at any place, to find what hour it is at any other place. Find the difference of time between the two places, as in (4) ; then, if the place, whose time is required, is east of the other, add this difference to the given time ; but if west, subtract it. Ex. What time is it in Boston, when it is 2 p. m. in Paris ? 6. To find the antiscii, the perioeci, and the antipodes of a given place. Bring the given place to the meridian ; then, under the meridian, in the opposite hemisphere, in the same degree of latitude, are found the antiscii. Set the index to XII., and turn the globe until the other XII. is under the index ; then, the perioeci will be at the same point of the meridian as the given place was, and the antipodes will bo where the antiscr. wpra 14 PROBLEMS ON THE GLOBES. Ex. Find the antiscii, the perioeci, and the antipodes of Lake Superior. To the antiscii, the hour of the day is the same as at the given place, but the season is reversed. To the perioeci, the season is the same, but the hour opposite. To the antipodes, both hour and season are opposite 24. Problems on the celestial globe. 1. To find the right ascension and declination of a heav enly body. Bring the place of the body to the meridian ; then the point directly over it shows its declination ; and the point of the equator under the meridian, its right ascension. Ex. Find the right ascension and declination of a Lyras. Also, of the sun on the 3d of May. 2. To represent the appearance of the heavens at any time. Adjust the globe for the place. (Art. 22.) On the wooden horizon find the day of the month, and against it is given the sun's place in the ecliptic. On the ecliptic find the same sign and degree, and bring the point to the meridian. The globe then presents the positions of the stars at noon. Set the hour-index at XII., and turn the globe till the index points to the required hour. The aspect of the heavens at that hour is then represented. Ex. Required the aspect of the stars at Lat. 51°, Dec. 5th, at 10 p. m. 3. To find the time of the rising and setting of any heav- enly body, at a given place. Having adjusted for the latitude, bring the sun's place in the ecliptic to the meridian, and set the index at XII. Turn the globe eastward, and then westward, till the given body meets the horizon, and the index will show the times of rising and setting. The times of the surfs rising and setting may be found in the same manner, on the terrestrial globe, since the ecliptic is usually represented on it. PARALLAX DEFINED. 15 Mb. At what time does the sun rise and set on the 4th of July? Find the time of the rising and setting of Arcturua on the 10th of November. i. To find the altitude and azimuth of a star for a given latitude and time. Adjust the globe for the aspect of the heavens (2) screw the quadrant of altitude to the zenith, and direct it through the place of the star. Then, the arc between the star and the horizon is the altitude ; and the arc of the horizon between the quadrant of altitude and the meridian, is the azimuth. Ex. Find the altitude and azimuth of Sirius, Dec. 25th, at 9 p. m. Lat. 43°. ft To find the angular distance between two stars. Lay the quadrant of altitude across the two stars, so that the zero shall fall on one of them ; then, the degree at the other will show their distance from each other. Ex. Find the distance between Arcturus and a Lyrse. 6. To find the sun's meridian altitude for a given latitude and day. Find the sun's place, and bring it to the meridian. The degree over it will show its declination. If the declination and latitude are both north or south, add the declination to the co-latitude ; if not, subtract it. Ex. Find the sun's meridian altitude at noon, Aug. 1st. Lat. 38° 30' K CHAPTER II. PARALLAX. — ATMOSPHERIC REFRACTION. — TWILIGHT. 25. Parallax defined. — When a person changes his place, objects about him in general appear in different directions from him. This change of direction is called parallax. If, for ex- ample, he moves north, an object, which was directly west or 16 DIURNAL PARALLAX. him, is moved by parallax towards the southwest • and an object which was east, now appears in the southeast quarter. The direction of every thing is more or less altered, except those objects which are in the line of his motion. 26. Diurnal parallax. — While a person therefore travels over the earth, or is carried about it by the diurnal rotation, the heavenly bodies mnst in the same way suffer some paral- lactic change. By the true place of a heavenly body, is meant that which it would seem to occupy if viewed from the center of the earth. At the surface, therefore, it appears generally displaced from its true position ; and this displacement is called the diurnal parallax. Thus, the true place of the body M (Fig. 4.), is in the direction CK ; but at A it appears in the line AH ; and the parallax is the angle AMC. So, the true place of M' is Q, its apparent place is P, and the parallax is AM'C. But the body W" appears at Z, whether viewed from A or C, and the parallax in this case is zero. Since the earth's radius, in each instance, subtends the angle of parallax, we have the following definition : The diurnal parallax of a body is the angle at that body subtended by the semi-diameter of the earth. Fig. 4. 27. On what diurnal parallax depends. — In the triangle ACM', let AC=r, CM'=^, and the parallax, AM'C=p. Let the zenith distance of the body, ZAM' = z ; then, the angle CAM' is the supplement of z. Hence, sin^> : sins :: r : d: r sin z ... smi , = ___ Since p is always very small, sinj? varies nearly as^? itsell PARALLAX OF THE MOON. 17 Therefore, regarding r as constant, p is usually constant ; and if its parallax, at a certain elevation, has been obtained, its horizontal parallax is found by the variation, p oo sin z. At the horizon, s = 90°, and sin z = rad. If, when the zenith distance is 53°, the moon's parallax is found by observation to be 45', then sin 53° : rad : : 45' : 56' 21", which is its horizontal parallax. 29. To correct for parallax. — The effect of parallax is to cause a body to appear lower than its true place. Hence, the true altitude of a body is obtained by adding the parallax to its apparent altitude. As parallax is a depression on a vertical circle, then, if a body is on the meridian, the parallax affects its declination just as much as its altitude, since the meridian is also a vertical ; bu in other cases, the vertical circle being oblique to the equator the parallax can be resolved into two components, one of which, parallel to the equator, is parallax in right ascension ; the other perpendicular to the equator, is parallax in declination. 30. To find the parallax of the moon. — Let A and B (Fig. 5) be two stations on the same meridian, taken as far apart as possible. The latitude of each place being known, the arc AB — that is, the angle ACB — is known. When the moon crosses the meridian, let its zenith distance be observed at each station. The observer A sees the moon projected in the sky at Y, and the zenith distance is the angle ZAY, while that at B is Z BY'. The supplements of these angles, MAC, MBC, are therefore known. In the isosceles triangle ABC, obtain the angles A and B, and the side AB ; subtract the angles from 18 ATMOSPHEEIC REFRACTION. Fig. 5. MAC and MBC respectively, then MBA, MAB are known, which, with the side AB, will give AM and BM. Finally, in the triangle AMC, the angle A and sides including it will fur- nish the angle AMC, which is the parallax sought for the station A, at the zenith distance ZAY. From this the horizontal paral- lax can be obtained, as in Art. 28. The horizontal paral- lax of the moon is much greater than that of any other heavenly body. Its mean value is about 57', and is correctly repre- sented by the angle EMC, in Fig. 6. The above method has also been employed for two or three of the planets, when they come near to the earth. But, with these exceptions, all the heavenly bodies are so far from us, that their horizontal parallax is too small to be obtained in this way with sufficient accuracy. The parallax of the sun is less than 9" ; that of nearly all the planets is much smaller than this ; and as to bodies outside of the solar system, they afford not the slightest indication of any diurnal parallax. E Fig. 6. M 31. Atmospheric refraction. — Before the true place of a body can be found by observation, a correction must also be applied for the refraction of its light by the atmosphere. While parallax depresses bodies below their true places, more or less according to their distance, refraction elevates them, the near and the distant alike. The earth's atmosphere may be conceived to consist of an ATMOSPHERIC REFRACTION. 19 indefinite n amber of strata, bounded by spherical surfaces, as AA, BB, etc. (Fig. 7), these strata being more dense according as the j are nearer the earth. Light from a star S, entering the air at a, is bent toward the perpendicular to its surface (which Fig. 7. is the earth's radius produced to that point), and describes ab f instead of ax. For the same reason, it is again bent into ho, and then into cO ; and therefore the star appears in the direc- tion of cO produced, at S', higher than its true place. The path of the ray from a to O is in reality not a broken line, as in the figure, but a curve, because the changes of density occur at every point. A body at the zenith is not moved out of place, because its light strikes the surfaces perpendicularly. The refraction at the horizon is about 35'. This is the greatest of all, since the angle of incidence there is the greatest possible. From the zenith to the horizon the refraction constantly in creases, — slowly at great elevations, but very rapidly near the horizon, as shown in the following table. Elevation. Eefraction. Elevation. Eefraction. 90° 0' 0" 20° 2' 37" 80 10 10 5 16 60 33 5 9 47 45 58 2 18 09 40 1 09 1 24 25 30 1 40 U 54 The true size of the largest angle of refraction is seen in 20 METHODS OF MEASURING REFRACTION. Fig. 8. AB is a portion of the surface of the earth, ah the surface of the atmosphere, AC, BC portions of the radii of the earth ; S is the true place of a star, S' the place as elevated by horizontal refraction. Fig. 8. 32. Measurement of refraction, — At latitudes greater than 45°, stars which culminate in the zenith make their lower culminations above the horizon. Such a star is observed at both culminations, and its distance from the pole is measured at each. These polar distances are really equal, but appa- rently unequal, because below the pole the star is elevated by refraction, while at the zenith it is not displaced. The differ- ence of the apparent polar distances, therefore, gives the amount of refraction at the place of lower culmination. The refraction within several degrees of the zenith is so slight, and its change so uniform, that observations may be made in the same way on stars which culminate several degrees north or south of the zenith ; and thus, by applying a small correction, the refraction may be measured at many different altitudes. 33. General method of measuring refraction. — A star, vrhose declination is known, may be used for determining re- fraction at any altitude, in the following manner. Let m n (Fig. 9) be the path of diurnal rotation of a star, whose declination xr is known. When the star is at x, let its apparent altitude be measured, and let the exact time also be observed. When it culminates at m, observe the time again. The difference of these times, allowing 15° for an hour, will give the angle at the pole ZPx. The co-latitude of the place, ZP, and the co-declination of the star, P#, being known in the TABLES OF REFRACTION. 21 spherical triangle ZP#, the side Za? can be computed. Its complement xy is the true altitude. This, subtracted from the apparent altitude before observed, gives the refraction at that elevation. Fig. 9. 34. Tables of refraction. — It is demonstrated, that except near the horizon, the mean refraction varies as the tangent ol the zenith distance. Tables of atmospheric refraction are cal- culated in accordance with this law, for all zenith distances less than 80°. They are, however, extended beyond that limit down to the horizon, being calculated for the last 10° by a different and more complex law, and the results of calculation being more uncertain. On this account, all astronomical measurements are made, so far as is possible, within 75° of the zenith. In order to obtain the place of a body with the utmost accuracy, tables of refraction are accompanied with means of correcting for the state of the barometer and the thermometer at the time of observation. 35. Time of rising and setting affected by refraction. — Since any heavenly body at the horizon is considerably elevated by refraction, it therefore appears to rise earlier and set later 22 TWILIGHT. than it would do if there were no atmosphere. The angular breadth of the sun is about 32 ', while horizontal refraction is a little more than this— 35'. Therefore, the sun appears just above the horizon, when, in truth, it is wholly below. This adds at least four minutes to the day, two in the morning and two at evening. 36. Distortion of the sun's and moon's disk by refrac- tion. — The change in the amount of refraction is so rapid near the horizon, that when the sun has just risen, or is just about to set, the lower limb is elevated more than the upper, by a very perceptible quantity. Its form, therefore, does not appear cir- cular, but nearly elliptical, the vertical diameter being shortened about 5 7 or 6'. The lower half, however, appears more flat- tened than the upper half, because the difference of refraction between the lower limb and the center is greater than that, between the center and the upper limb. 37. Illumination of the shy. — During the day, the atmos- phere is illuminated by the light of the sun, which penetrates every part of it, and is reflected in all directions. If there were no air, the sky, instead of appearing luminous by day, would exhibit the same blackness as by night, and the stars would be visible alike at all times. We should, in that case, lose a great part of that generally diffused light which illuminates the interior of buildings, and other places screened from the direct rays of the sun. The earth's surface, and all terrestrial objects, on which the sunlight falls directly, would indeed, by radiant reflection, cause a degree of illumination, but it would be far less than we now enjoy. It has been observed, in ascending to great heights, either on mountains or in balloons, where, of course, the air which is most dense and reflects most abun- dantly is left below, that the sky assumes a very dark hue, and the general illumination is greatly diminished. 38. Twilight. — The illumination of the sky begins before the sun rises, and continues after it sets : it is then called twi- light. More or less of it is visible, as long as the sun is not more than 18° vertically below the horizon. Those parts of the DtrBATION OF TWILIGHT. 23 atmosphere are most luminous, which lie nearest to the direc« tion of the sun. Thus, in Fig. 10, let A be a place on the earth, where the sun is just setting. The whole sky, IEFH, is illuminated. But, to a place further east, as B, the twilight extends from E to H, — the part of the sky, IIK, remote from the sun, being in the shadow of the earth. At C, only FH is illuminated, and HL is dark. At D, the twilight is entirely gone. Fig. 10. Though the twilight terminates at H, there is no abrupt transition from light to shade at that point, since the reflection from those high and rare parts of the air is exceedingly feeble ; and also, because the thickness of the illuminated segment, through which we look, diminishes gradually to that limit, as is obvious from an inspection of the figure. 39. Duration of twilight. — To an observer at the equator, at those times of the year when the sun is on the celestial equator, the twilight continues lh. 12m. For, in the diurnal motion, 15° are described in an hour, and therefore 18° in l T 3 jh. = lh. 12m. This is the shortest duration possible. For, if the sun were on a parallel of declination, the degrees of diurnal motion would be shorter than those on a great circle. And, if the observer were on some parallel of latitude, the circles of daily motion would be oblique to his horizon, and the sun must therefore pass over more than 18°, in order to move 18° verti eally. An extreme case occurs at the poles, where twilight lasts several months. 24 THE TRANSIT INSTRUMENT. CHAPTEE III. THE OBSERVATORY AND ITS INSTRUMENTS.- PROBLEMS. -SPHERICAL 40. The observatory. — Accurate knowledge of the motions of the heavenly bodies is mostly obtained by observing their relations to the diurnal rotation. The observatory is furnished with several instruments by which such observations are made. 41. The transit instrument. — This is a telescope so mount- ed as to observe a heavenly body, at the instant when it cul- minates^ — that is, makes a transit of the meridian. AD (Fig. 11) represents the telescope supported by a horizontal axis, which consists of two hollow cones, placed base to base, so as to combine lightness and strength. The ends of the axis rest in sockets, Attached to two stone piers, E and W. That ADJUSTMENT OF TRANSIT INSTRUMENT. 25 the instrument may receive no tremors from the building, the piers stand on a firm foundation in the ground, passing through the floor without contact. The axis being placed east and west horizontally, the telescope, which is perpendicular to it, will, when turned, revolve in the plane of the meridian. A graduated circle, n, is attached to one end of the axis, for marking altitudes or zenith distances. The whole instrument can be raised from the sockets, and the axis inverted, so that the east end shall rest on the pier W, and the west end on the pier E. 42. Adjustments of the transit instrument. — The visual axis of the telescope, AD, is called the line of collimation^ and is marked by the intersection of two exceedingly fine wires in the focus of the eye-glass. One of these wires is horizontal, fh (Fig. 12), the other vertical, d e ; the latter visibly marks the direction of the meridian, when the instrument has been prop- erly adjusted. The sockets, in which the ends of the axis rest, are so connected with the stone piers, that one of them can be raised or lowered by a screw, and the other can, in a similar manner, be moved north or south. By the spirit-level, L, which hangs on the axis, it can be seen whether the axis is horizon- tal. If not, raise or lower the end which admits of vertical motion. To find whether the line of collima- tion is perpendicular to the axis of revolution, observe whether a distant terrestrial object, which is on the vertical wire, remains on it after the ends of the axis have been inverted in their sockets. If not, move the plate which carries the wires laterally, till the vertical wire bisects the distance between the two positions of the object. And finally, to determine whether the axis is east and west, observe if a circumpolar star occupies the same length oi 26 TO OBSERVE RIGHT ASCENSION. time in passing from the upper to the lower culmination, as from the lower to the upper ; and if not, move the end of the axis horizontally, till the intervals are equal. For fuller instructions on adjustment, see Loomis's Practical Astronomy. 43. The astronomical clock. — The transit instrument marks the event of crossing the meridian ; the clock must be used in connection with it, to fix the time of the transit. The clock of the observatory is made to keep sidereal time, — that is, it marks off 24 hours in the interval between two successive transits of a star, instead of the sun. This interval is called a sidereal day, and is about 4 minutes less than a solar day. The sidereal day begins when the vernal equinox transits the meridian. At that instant, the clock is at Oh. Om. Os. ; and any hour of the clock shows how long a time has elapsed since the equinox culmi- nated. 44. Error and rate of clock, — The uniform movement of the clock is its most important excellence. This may be tested by the transit instrument, and a list of right ascensions of stars. If it does not indicate Oh. Om. Os. when the vernal equinox cul- minates, the difference is called its error. If it marks any more or less than 24 hours between two successive transits of a star, this gain or loss is called its rate. If both error and rate are known, then the true time is known ; and generally it is not best to alter the clock, but only to keep a record of error and rate. 45. To observe the right ascension of a heavenly body. — Having elevated the telescope to the altitude of the body at the time of culmination, notice the exact instant when it appears on the vertical wire de (Fig. 12). This is its right ascension, which may be given either in time or in arc. Thus, if the clock is at 13h. 46m. 32s. when a star passes the wire, its right ascension is 13h. 46m. 32s. ; or, at the rate of 15° for each hour, 206° 38' 0". To secure greater accuracy, several equidistant wires are placed parallel to de, an equal number on each side, as in Fig THE CHKO.N0GKAPH. 27 12. The time of passing each wire is noted, and the average of all obtained for the time of crossing the central one. To observe the right ascension of the snn or a planet, the transit of each limb must be noticed, and the mean of all the times will be the right ascension of the center of the disk. In order to render the wires visible by night, the field of view is faintly illuminated by a lamp, placed at one end of the hollow axis, the light of which, after entering the telescope, is reflected toward the eye-piece. 46. Transits recorded by the chro?iograph. — To observe the time of a star-transit, the eye must discern the instant of its bisection by the wire, and the ear must hear the beat of the clock, — the seconds being counted from the last completed minute before the observation began. If the bisection occurs between two beats, as it commonly does, the observer needs much practice to be able to divide the second accurately into tenths, and decide at which of them the transit takes place. Transits are now generally observed and recorded with much greater ease and accuracy by the use of the galvanic circuit. Fig. 13. The pendulum of the observatory clock is arranged to close the circuit of a battery and break it again, at the beginning of every beat. The closing of the circuit gives a small lateral motion to the registering pen, under which the paper is ad- vancing on a revolving cylinder, about an inch per second. Thus the seconds are all permanently recorded by notches one inch asunder in a straight line, as a, b, c, d (Fig. 13). The mark at the beginning of each minute has some peculiarity by which it may be distinguished from the rest. The observer has under his hand a key, which, by a quick touch, will also close and break the circuit. Whenever a star is on one of the wires of the transit instrument, he touches the key, the pen is moved aside, and indents the line as at A, and the observation is thus recorded ; and the place where this motion commenced between the second-marks can afterward be carefully examined. Thus, 28 THE MUEAL CIECLE. without the distraction of attending to the clock, he can record the transits of all the wires ; and if he only notices within what minute the work begins, he can read the entire record with accuracy to the T ^ or even the too of a second. Since the general adoption of this method, the number of wires has been increased, sometimes to 30 or 40, so as to obtain the mean of more numerous observations on the same star. The instru ment, as above described, is known as the chronograph. 47. The mural circle. — The circle of the transit instrument is used principally for finding a body whose altitude is known, and is too small for accurate measurement of arcs on the meri* Fig. 14. uian. For measuring meridian arcs, the mural circle is em- ployed ; so called, because it revolves by the side of a vertical wall. It consists of a circle usually six or eight feet in diame- ter, and a telescope attached to its face. It is made so large, in THE VERNIER. 29 order that very small angles may be measured by the divisions on its limb. Fig. 14 represents the instrument attached to the meridian wall. Its radii are hollow and of conical form. The axis, which is on one side only, is firmly set in the wall ; and the circle and telescope revolve upon it. The graduations are made on the rim, and not on the face of the circle, and are read by means of microscopes attached to the wall. 48. Subdivisions of the graduated limb. — The reading of a. graduated arc can always be carried much lower than the divisions actually marked on it. This is sometimes accom- plished by the vernier, and sometimes by the reading micro- scope. 49. TJie vernier. — This contrivance, so named from the in- ventor, is a short graduated arc, which slides along the limb of the circle that is to be subdivided. For example, AB (Fig. 15)^ is a vernier for dividing the 12' spaces of the arc on its right into portions of 1' each. For this purpose, the vernier consists of 12 parts, which together are equal to 11 of the divisions of the limb. Since 12 parts of the vernier are less than 12 divisions of the arc by a whole division, one part of the vernier is less than one division of the arc by yV of a division ; two are less than two by ^ °f a division, and so on. Now, in the figure, the zero of the vernier has passed 10° 24/ ; and in order to find how many twelfths of the next space it has passed, it is only necessary to look along the vernier, and observe the number of the division line, which coincides with a line of the arc. In this case we find it to be the 8th. Hence, the 8 parts of the vernier from to 8 are less than the cor- responding S divisions of the arc by T 8 2 ; that is, zero is T 8 2 of 12' beyond 10° 24\ Therefore the reading is 10° 32'. The vernier is sometimes made, so that a given number of parts equals one more, instead of one less, than the same number on the limb. Rut the principle of making subdivisions is the same. Fig. 15. 13° -12° ■ir ■io c 30 TO FIND DECLINATION. 50. The reading microscope. — This is a compound micro- scope, having in the focus of its eye-piece a pair of spider-linea intersecting each other, and in the same field of view are the magnified divisions of the arc. The intersection of the spider- iines is moved laterally from one division line of the arc to another by a screw. If the divisions, for example, are equal to 5' each, then the screw is so made as to move the intersection from one line to another by five revolutions, and therefore each revolution indicates a motion of V. A circle is attached to the axis of the screw, having its circumference divided into 60 equal parts. As each revolution ■ of the screw can thus be divided into 60 equal parts, so each minute of the arc can be divided into seconds. One of these reading microscopes is represented at A (Fig. 14) ; and the places of others are marked at B, C, D, E, F, 60° from each other. Six are used, instead of one, for the purpose of obtaining a more accurate result, by taking a mean of the seconds in the several readings. 51. To find the declination of a heavenly body. — This may be done by measuring its meridian altitude. Let the mural circle be adjusted in altitude, so that, at the instant when the body crosses the vertical wire of the telescope, it is on the horizontal wire also. The graduation of the limb shows its altitude. The latitude of the observatory being known, the elevation of the equator is known ; and the difference between the altitude of the body and the elevation of the equator, is the decimation sought. In northern latitudes, if the altitude of the heavenly body exceeds the elevation of the equator, the differ ence is a northern declination ; if it is less, the decimation is south. Before altitudes can be measured, the horizontal position ol the telescope must be determined. This may be done by bisecting the angle between the direction of a fixed star, as seen at culmination, and its apparent direction, when seen at another culmination in a mirror of liquid mercury, called the artificial horizon. By a law of optics, the apparent depression below the horizon equals the elevation above it, sc chat the whole angle equals twice the altitude. ALTITUDE AND AZIMUTH INSTEUMENT. 31 5 2. The transit circle. — Sometimes the circle of the transit ir. '.trument is made of much larger size than is represented in Fig. 11, in order that declinations as well as right ascensions may be observed by it. This combination of the transit instru- ment and mural circle is called the transit circle, and is con- sidered by some practical astronomers to possess an advantage over the mural circle in the steadiness of its axis. 53. The altitude and azimuth instrument. — The essential parts of this instrument are, a telescope and two graduated circles, one vertical, the other horizontal. Fig. 16 presents one of its more simple forms. The telescope AB is movable on a Fig. 16. horizontal axis at the center of the vertical circle dbc, and also on a vertical axis, passing through the center of the horizontal circle EFGL The levels g and A, placed at right angles to each other, show when the circle EFG- is brought to a horizontal position by the tripod screws. The tangent screws, d and — Fig. 17. *8— and a vernier at the extremity, D. The horizon glass, H, is silvered only on one-half of its surface. When the zero oi the vernier coincides with that of the arc at F, the mirrors are precisely parallel. If now we direct the telescope to a star, it may be seen in the transparent part of the horizon glass, and its image in close contact with it, in the silvered part. This is owing to the fact, that a heavenly body is so far dis- tant, that the rays from it to the two mirrors are sensibly par- allel to each other. SPHERICAL PROBLEMS. 33 55. To measure an angle by the sextant. — Let it he required to measure the angular distance between the star S and the moon M. The telescope being directed to S, and the sextant being held so that the plane of reflection shall pass through the two objects, turn the index from F toward E, until the image of the moon is brought to the star, its nearer limb just touch- ing S. Now, according to an optical principle, the angular distance between the moon and its image is just twice that be- tween the mirrors. Therefore, by reading the vernier at D, we obtain the angular distance between the star and the moon's nearer limb. Again, bring the further limb to the star, and find its distance. Half their sum is the angular distance be- tween the moon's center and the star. In like manner, the altitude of a body may be found, by bringing its image to coincide w r ith the image of the same body seen in the artificial horizon. One-half the angle read from the vernier is the altitude of the body. The graduation on the limb of the sextant, for convenience, corresponds, not to the actual length of the arc passed over by the vernier, but to the angular motion of the body, which is twice as rapid. Hence, on the arc of 60°, the graduation reaches 120° ; and all angles not greater than this can be measured by the instrument. The two instruments just described are sometimes conven- ient at the observatory, but their chief use is elsewhere. The altitude and azimuth instrument is of great value in trigono- metrical surveying. The sextant is important for the naviga- tor, since a stationary instrument cannot be employed at sea. Fi S- 18 - 56. Spherical problems. — I. To compute the sun's right ascension, declination, or longi- tude, or the obliquity of the eclip- tic to the equator, when any two of the others are given. Let PEP' (Fig. 18) represent the solstitial colure, PP' the axis, EQ the equator, E'C the ecliptic, 3 34 SPHERICAL PROBLEMS. and PSP' a secondary of the equator passing through the sun S. Then SAR is the obliquity of the ecliptic, and RS the dec- lination of the sun. And if its longitude is less than 90°, AS is its longitude, and AR its right ascension. If its longitude is more than 90°, AS and AR are the supplements of longitude and right ascension. In both cases the declination is north. When the sun's place is represented by S', and its longitude is between 180° and 270°, then the longitude = 180° + AS', and the right ascension — 180° -f- AR'. But if its longitude is more than 270°, longitude — 360° — AS', and right ascen- sion = 360° — AR'. In each case the declination is south. The triangle ARS is right-angled at R ; and by Napier's rule, any one of the parts may be found, whea two others are given. Ex. 1. When the sun's right ascension is 53° 38', and its dec- lination, 19° 15' 57", required its longitude, and the obliquity of the ecliptic. 1. Rad . cos AS =cos AR . cos RS. 2. Rad . sin AR — tan RS . cot A. Ans. Long. = 55° 57' 43". Obi. = 23° 27' 501". Ex. 2. On March 31st, the sun's declination was observed to be 4° 13' 31|", and the obliquity was 23° 27' 51" ; required the sun's right ascension. Ans. 9° 47' 59". Ex. 3. What is the sun's longitude in November, when its declination is 21° 16' 4", and its right ascension is 16h. 14m. 58.4s. % Ans. 245° 39' 10". The above data show that the sun's longitude is more than 180° and less than 270°, and the declination south. The tri- angle for computation is AR'S'. Ex. 4. The sun's longitude being 8 s 7° 40' 56", and the obliquity 23° 27' 42£" ; required right ascension in time. Ans. lt)h. 23m. 34s. II. Given the latitude of a place, and the declination of the sun, to find the time of its rising and setting. Let PEP' (Fig. 19) be the meridian of the place, Z its zenith, and HO its horizon. Let LL' be the diurnal circle of the sun ; RS is its declination, S the place of its rising and setting, and LS the arc described between either and midnight. But LS, in degrees, equals QR, the complement of AR. The angle SPHEEICAL PROBLEMS. 35 Fig 19. S AE = E AH, which is measured by EH, the co-latitude, and E is a right angle. Therefore, rad . sin AE = cot A . tan ES. Ex. 1. Eequired the time of sun- rise at latitude 52° 13' 1ST., when the sun's declination is 23° 28' K We find AE=34° 3' 21*"; .\ QE = 55° 56' 38f /, = (in time) 3h. 43m. 46±s. This is the time of sun- rise. The same subtracted from 12h., gives 8h. 16m. 13Js. for the time of sunset. Ex. 2. Eequired the time of sunrise at latitude 57° K, when the sun's declination is 23° 28' K Arts. 3h. 11m, Ex. 3. How long is the sun above the horizon in latitude 58° 12' K, when its declination is 18° 40' S. ? Ana. 7h. 35m. 52s. In a similar manner, if the declination of any heavenly body be given, the interval of time between its culmination, and its rising or setting, can be computed. III. Given the latitude of a place, and the declination of a heavenly body, to compute its altitude and azimuth, when on the six o'clock hour-circle. Let PEP 7 (Fig. 20) be the meridian of the place, and P the elevated pole. Then PP r rep- 2' 54 49s. resents the six o'clock hour- circle, which is at right angles to the meridian, and therefore projected in a straight line. Let the body cross it at S, and let ZSB be the vertical circle passing through it. In the tri- angle ASB, AS is the declina- tion, SB the altitude, AB the amplitude or complement to the azimuth OB, and B is a right angle. Ex. 1. What were the altitude Fig. 20. and azimuth of Arcturus, 36 SPHERICAL PROBLEMS. when on the six o'clock hour-circle, latitude 51° 28' 40" N.", its declination being 20° 6' 50" N. I ^n*. Altitude 15° 36' 27" ; Azimuth 77° 9' 4". Ex. 2. In latitude 62° 12' 1ST. the altitude of the sun at sis o'clock, a. m., was observed to be 18° 20' 23 // . Eequired its declination and azimuth. Am. Declination 20° 50' 12" 1ST. ; Azimuth 79° 56' 4". LV. Given the latitude of a place and the sun's declination. to find the time a. m. when it will cease shining on the north side of a building, or the time p. m. when it will begin to shine upon it. Let PEP 7 (Fig. 21) be the meridian of the place, ZAK the prime vertical, and S the place where the sun crosses it, and thus ceases to shine on the north side of a vertical wall. Let PSB be the hour-circle through the sun at S. BS is the sun's declination, BAS (=EZ) is the lati- tude, and AB, changed into time, will show how long after six o'clock a. m., or before six p. m., the sun transits the prime vertical. Ex. 1. In latitude 42° 22' 17" K, when the sun's declination is 23° 27' 36" K., at what times does the sunshine begin and end on the north and south sides of a building? Ans. 7h. 53m. 3Ss. a. m., and 4h. 6m. 22s. p. m. Ex. 2. How long does the sun shine on the south side of a vertical wall, in latitude 20° 30' N., when the sun's declination is 20° N? Ans. lh. 45m. 48s. Y. The latitude and the sun's declination being given, to find the time of day by the sun's altitude. Let Z (Fig. 22) be the zenith of the place, P the pole, and 8 the place of the sun. Measure ZS, the zenith distance of the sun, and correct it for refraction and parallax. PZ is the co-latitude of the place, and PS the co-declination of the sun. Therefore, the sides of the spherical triangle PZS are all known, and the angle ZPS can SPHERICAL PROBLEMS. 37 Fiar. 23. be computed ; which, changed to time, shows how long before or after noon the observation was made. VI. Given the latitude and the sun's declination, to find the time when twilight begins and ends. The twilight begins or ends when the sun is about 18° below the horizon (Art. 38). Let Z (Fig. 23) be the zenith, P the pole, and S the place of the sun at the beginning or ei.d of twilight. ZS = 108°, ZP = co-lat, PS = co-decl. The three sides of ZPS are given, to find the hour-angle ZPS. This may be done by dropping the perpendicular arc P/>, and using the proportion (Sph. Trig.) tan J ZS : tan \ (PS +ZP) : : tan £ (PS - ZP) : tan \ (Bp — Zp). Having obtained Zp and Sp, compute the angles at P, and add them together. Ex. In lat. 42° 22', when does twi- light begin and end, at midsummer, the sun's declination being 23° 2S / ? Ans. 2h. 6m. 20s. a. m. VII. Given the right ascension and declination of a body, to find its longitude and latitude. Let EQ (Fig. 24) be the equator, and P its north pole, E'O the ecliptic, and B. its pole, and S the place of the body. Join PS and ES, and draw the arc SB perpendicular to PC. PS, the complement of declination, is known ; likewise RP, which equals EE', the obliquity. As A is the vernal equinox, SPQ is the complement of right as- cension, and therefore known. SRC is the complement of lon- gitude, and PS is the comple- ment of latitude. In the right-angled triangle PSB, PS and P being known, find PB. Then KB(=RP+PB) is known. Then (Sph. 9h. 53m. 40s. p. m. Fig. 24. 38 THE sun's eight ascension. Trig.) sin KB : sin PB : : tan P : tan E. Thus R, the com plement of longitude, is found. Then, in the right-angled triangle BSB, BB and the angle K enable us to find BS, the complement of latitude. Ex. 1. The right ascension of a planet was observed to be 82° 7', and its declination 24° 26' K Calling the obliquity 23° 27' 20", what were the longitude and latitude of the planet ? Ans. Long. 82° 49' 30" ; Lat. 1° 10' 27" K Ex. 2. What are the longitude and latitude of the star, whose right ascension is 4h. 40m. 49s., and its declination 66* 6' 37" K % Ans. Long. 79° V 8" ; Lat. 43° 24' 5" K CHAPTEB IV. THE EARTH S ANNUAL MOTION ABOUT THE SUN. — THE SEASONS. — FIGURE OF THE EARTH'S ORBIT. 57. Ohservatums of the suns place. — If we employ the in- struments of the observatory in measuring from day to day the right ascension and declination of the sun, at the moment of its crossing the meridian, it will be discovered that these quantities are constantly changing ; or, in other words, that the sun is constantly shifting its place in relation to the stars. 58. Its right ascension. — By the transit instrument and clock, it is found that the sun's right ascension is always in- creasing by a quantity which is not quite uniform, but which amounts to nearly one degree every day. So that, in about 36 5 days, it describes the whole 360° of right ascension, and appears again in the same place among the stars. This is the apparent annual motion of the sun, by which it seems to pass round the heavens from west to east once in a year. THE TROPICS AND POLAR CIRCLES. 39 59. Its declination. — But while thus passing round, it also moves alternately north and south. For, by measuring the declination each day by the mural circle, it is found that after passing the vernal equinox, March 20th, its declination is north, and increases to the summer solstice, June 21st, when it reaches nearly 23^° ; from that point it diminishes to zero at the autumnal equinox, September 22d. The declination then becomes south, increasing to the winter solstice, December 21st, when it is 23J°, and thence diminishing to nothing at the vernal equinox, on March 20th of the following year. 60. The ecliptic. — The apparent annual path of the sun is found by the foregoing observations to lie in a plane, cutting the celestial sphere in a circle called the ecliptic (Art. 12), and inclined to the plane of the equator at an angle of about 23° 27'. This plane maintains almost a constant position among the stars, and is used far more than any other circle of the sphere as a plane of reference. The obliquity of the equator to the ecliptic in 1850 was 23° 27' 31", and diminishes at the rate of 46" in a century. 6 1 . The zodiac. — This name is given to a zone of the heav- ens, 16° wide, extending along the circle of the ecliptic, 8° on each side of it. The paths of the principal planets lie within this zone. Its length is divided into 12 signs of 30° each, having the same names and arranged in the same order as those of the ecliptic (Art. 13), though not coincident with them. The signs of the zodiac are distinguished from each other by the stars which occupy them. 62. The tropics and polar circles. — Through the two points of the ecliptic most distant from the equator, called the sol- stices (Art. 12), we imagine circles to be drawn parallel to the equator, called the tropics. The northern circle, passing through the first of Cancer on the ecliptic, is called the tropic of Cancer ; the southern one, for a like reason, is called the tropic of Capricorn. Two other parallels to the equator, passing through the poles of the ecliptic, and therefore 23 d 27' from the .poles of the equator, are called the polar circles. 40 ANNUAL MOTION. 63. Terrestrial zones. — On the terrestrial sphere, a similar system of circles divides the earth's surface into the well-known zones of geography, called the torrid, temperate, and frigid zones. The tropics are the limits of vertical sunshine in mid- summer. The polar circles are the limits within which the sun makes a diurnal revolution in midsummer and mid-winter, without rising or setting 64. The annual motion observed without instruments. — If the stars were visible in the daytime, we should perceive the sun making progress among them toward the east, by a dis- tance equal to nearly twice its own breadth every day, since the apparent diameter of the sun is a little more than half a degree. But, as they are invisible by day, we detect the same fact, when we notice that at a given hour of the night, all the stars are further west than on a previous night. For example, at 9 o'clock p. m. — that is, 9 hours after noon — it is easily observed that there is, from one evening to another, a regular progress of all the stars westward, as long as we choose to watch them. In other words, the sun is at the same rate advancing eastward relatively to the stars. Fig. 25. 65. The annual motion is a motion of the earth, not of the mn. —There is abundant evidence that the motion of the sun CHANGE OF SEASONS. 41 around the earth, above des3ribed, is only apparent, and results from a real motion of the earth about the sun. Thus, suppose the earth to pass around the sun S (Fig. 25) in the orbit ABPC, in the order of the signs ; if we were unconscious of this motion, the sun would appear to us to move about the earth in the same order of the signs, though, at any given moment, in a contrary direction. When the earth is at B (in the sign T, as seen from the sun), we should see the sun in the sign =a= ; when we reach tf , the sun is seen in ^l ; and so on. 66. Cause of the change of seasons. — The changes of the seasons are due to the fact, that the two revolutions of the earth, one on its axis, and the other around the sun, are in dif- ferent planes ; in other words, that the equator and the ecliptic make an angle with each other. In Fig. 26, let ABCD repre- sent the ecliptic (seen obliquely), and suppose the earth to pass around in the order of the letters, occupying the position A on the 20th of March, B on June 21st, C on Sept. 22d, and D on Dec. 21st. In every position of the earth, the equator eq, is inclined the same way, and always at the angle of 23^° with the ecliptic. The axis ns, being perpendicular to the equator, is everywhere parallel to itself. When the earth is at A , the vernal equinox, the line of inter- section of the ecliptic and equator, passes through the sun S, and the light just reaches the poles n and s ; so that, as the earth rotates on ns, every place is one-half of the time in the light, and the other half in darkness. The days and nights are therefore equal. As the earth passes on toward B, the light readies beyond 42 HEAT IN SUMMER AND COLD IN WINTER. the north pole more and more, till at B, the summer solstice, it extends 23-J- beyond n, and falls as much short of s, the sun being now north of the equator eq. As the earth now rotates on ns, all places north of eq are in the light longer than in the shade, and the reverse is true of all places south of eq. It is summer in the northern hemisphere, and winter in the southern. On the 22d of September, the earth arrives at C, the autum- nal equinox ; the intersection of the two planes again passes through the sun, the light once more reaches the poles, and the days and nights are equal. At D, the winter solstice, the north pole n is turned as far as possible into the shade, and s into the light. Every place north of eq is in the light a shorter time than in the darkness, and the reverse south of eq. It is now winter in the northern hemisphere, and summer in the southern. If the equator were in the same plane with the ecliptic, the case would be represented by Fig. 26a. The axis ns would then be perpendicular to the ecliptic as well as to the equa- tor, the circle of illumination would always reach just to the poles n and s, and in the daily rotation, every place would be half the time in the sunlight, and half in the darkness. There would, therefore, be no inequality of day and night, and no change of seasons. 67. Causes of heat in summer and cold in winter. — These are two. 1st. The length of the aay compared with the night. The heat of the earth is passing off by radiation during the whole time, whether the sun shines or not But the earth receives GREATEST HEAT AND GREATEST COLD. 43 beat from tlie sun, only while the sun is above tho horizon. Hence, the longer the period of sunshine, compared with the time of a diurnal revolution, the greater the heat. For this reason, therefore, the summer is warmer than the winter. 2d. The different inclination of the rays to the general sur face of the earth. The number of rays falling on a given sur- face, varies as the sine of inclination. Let AB (Fig. 27) be the breadth of the surface. If the rays fall on it at the angle ABC, the perpendicular breadth of the beam is AC ; if at the angle ABD, the breadth of the beam is AD ; while, if they fall perpendicularly, the breadth of the beam is AB itself. Now, the number of rays in the beam obviously varies as its perpendicular breadth. But these breadths, AC, AD, and AB, are as the sines of the several inclinations. In summer, the sun rises to a greater elevation each day than at other seasons, and therefore sheds a greater quantity of heat on that part of the earth. Ex. 1. What is the relative quantity of direct heat from the Bun at noon, on two equal horizontal areas, one in latitude 75° N., the other 30° K, when the sun's declination is 19° 1ST. ? Ans. As 100 : 175J. Ex. 2. Find the ratio, as in Ex. 1, in latitude 50° !N". and latitude 45° S., when the sun's declination is 15° 45' S. Ans. As 100 : 212.4. 68. Why the greatest heat is later than the summer solstice, and the greatest cold later than the winter solstice. — If the sun sheds on a given surface more heat each day than the surface loses by radiation, then the heat accumulates from day to day. This is the case during the long days of summer ; and more heat is gained than lost, till a month or more after the summer solstice. For a like reason, during the middle hours of the day, heat is received from the sun more rapidly than it is lost by radiation, so that the hottest hour is 2 or 3 o'clock p. m. 44 GREATEST CHANGES OF SEASON. In the winter, on the contrary, the loss by radiation exceeds the quantity received from the sun, during all the shortest days, so that the temperature descends till many weeks after the winter solstice. If loss by radiation were at a uniform rate at all tempera- tures, and the temperature of successive years should remain constant, as it now is, then the greatest heat would be near the autumnal equinox, and the greatest cold near the vernal equi- nox, the times when the surface receives heat at the mean rate. On the contrary, if the existing amount of loss by radiation were distributed so as to be exactly proportional to the acces- sions received from the sun, there would be no change of tem- perature at the different seasons of the year or the different hours of the day. But the radiation of heat follows neither of these laws ; the quantity radiated is greater, when the quantity received is greater, but it does not vary at so rapid a rate. 69. No change of seasons, if there were no obliquity. — The angle between the planes of the two motions of the earth being the cause of the change of seasons, it follows that there would be no such change if those motions were in the same plane. If, while the earth advances in its orbit about the sun, it should rotate in the same direction on its axis, then the sun would always be in the plane of the equator, and would, every day, describe the equator as its diurnal circle, rising exactly in the east, culminating at a zenith distance equal to the latitude of the place, and setting exactly in the west. At the equator, the sun w r ould always follow the prime vertical, and at either pole it would always be passing round in the horizon. See Fig. 26a. 7 0. The greatest changes of season, if the obliquity were 90°. — If, while the earth revolves on its axis from west to east, it should pass around the sun in a plane lying north and south, then the ecliptic would pass through the north and south poles, and the solstices would be at the poles. Hence, at a station on the equator, the sunw T ould, during the year, describe the prime vertical and various small circles parallel to it, down to the oorth and south points of the horizon, where it would be FORM OF THE EARTH S ORBIT. 45 stationary alternately at the times of the solstices. At the equator, therefore, there would be an alternation from summer to winter, or the reverse, every three months. At either pole there would be but one summer and one winter in a year; but the extremes would be far more intense. For the sim, in describing diurnal circles parallel to the horizon, would occupy six months in ascending to the zenith and returning to the horizon, and the remaining six months in performing corresponding revolutions below the horizon. At intermediate places, the extremes of the seasons would also be intermediate. 7 1 . Mode of determining the form of the earth? s orbit. — The earth's orbit is an ellipse described about the sun, which is situated in one of its foci. This is ascertained by observing the changes in the sun's apparent diameter throughout the year^ When the sun appears smallest, it is most distant ; and when largest, it is nearest. And its distance, in all cases, varies in- versely as its apparent diameter. Therefore, if the sun's angu- lar diameter be accurately measured as frequently as possible, the reciprocals of those angles express the relative distances ; and these distances determine the form of the orbit. Thus, suppose the earth to be at E (Fig. 28), and that the sun's apparent diameter is measured when in the di- rection E#. After it has advanced eastward some days, so as to be seen in the direction E&, let another measurement be made ; and so on, at every opportunity through the year. Then let E«, E5, Ec, etc., be made proportional to the recipro- cals of the apparent diame- ters, and be laid down at angles equal to the angular changes of the sun's place. Fig. 28. A line, a b m v, passing through their extremities, shows the form of the sun's apparent orbit 46 LINE OF APSIDES, about the earth, and therefore the form of the earth's real orbit about the sun. In this manner, even while ignorant of the size of the orbit, we learn that its form is an ellipse, and that the sun occupies one of its foci. 72. Definitions relating to a planetary orbit. — Let E be the focus occupied by the sun, and am the major axis of an ellipti- cal orbit described about it ; the nearest point, a, is called the perihelion, and the most distant point, m, the aphelion. The two p©ints a and m are also called the apsides. The point a is sometimes called the lower apsis, and m the higher apsis. The varying distance, E&, E5, E^, etc., is called the radius vector. If the major axis, am, is bisected in C, the ratio of EC to the semi-major axis, aC, is called the eccentricity of the orbit. The less EC is, compared with aC, the less is the eccentricity, and the nearer does the ellipse approach to a circle. If E coincides with C, the eccentricity is nothing, and the orbit is a circle. 73. The earthus orbit very nearly circular. — The eccen- tricity of the earth's orbit in 1850 was 0.01677, and is very slowly diminishing. This fraction is about -g^, — that is, EC (Fig. 28) is eV of aC. As aC, in this figure, is about one inch long, EC should be only ^ of an inch, in order to represent correctly the proportions of the earth's orbit. If it were thus drawn, it could not be distinguished from a circle in its appear- ance ; for the minor axis, as may be easily computed, would be shorter than the major axis by only to ff o o of an inch. 74. Position of the line of apsides. — The direction of the major axis of the earth's orbit, or the line of apsides, is slowly changing ; but at present it passes through the 1 0th degree of Cancer and Capricorn, as represented in Fig. 25. The earth is at perihelion on the 1 st of January, and at aphelion on the 1st of July. We are therefore nearest to the sun in the winter of the northern hemisphere, and furthest from it in the summer. 7 5 , Distance from the sun, as affecting the seasons. — The SIDEREAL TIME. 47 intensity of the sun's heat at the earth, as well as that of its light, varies inversely as the square of our distance from it. On this account, the intensity of heat at perihelion is to that at aphelion as 61 2 : 59 2 , which is nearly as 31 : 29. Therefore, so far as distance is concerned, the earth receives ^ more heat on the 1st of January than on the 1st of July. This produces a slight effect to mitigate the severity of cold in winter and o* heat in summer, in the northern hemisphere, and to aggravate the same in the southern hemisphere. But, on account oi changes going on in the places of the equinoxes and apsides, this modifying effect will be reversed after the lapse of about 10,000 years. CHAPTEK Y. SIDEREAL TIME. — MEAN AND APPARENT SOLAR TIME. — THE CALENDAR. 76. The sidereal day. — This is the interval of time which elapses between two successive culminations of a star (Art. 43). The length of this interval appears to be invariable, whatever star is observed, or in whatever season or year the observation is made. On this account, the sidereal day is regarded as the true period of the earth's rotation on its axis. In order to reckon by sidereal time, the moment chosen for the beginning of each sidereal day is the moment when the vernal equinox culminates. The sidereal clock, if correct, then points to Oh. 0m. 0s. Each sidereal day is divided into 24 sidereal hours, each hour into 60 sidereal minutes, and each minute into 60 sidereal seconds. 7 7 . The w.ean solar day.- -This is the mean interval between two successive culminations of the sun. It will be shown pres- ently, that these intervals vary throughout the year. As the sun, by the annual motion, is advancing eastward continually among the stars, the solar day must always be longer than the 48 INEQUALITY OF SOLAR DATS. sidereal day. For, if the sun and a star were 01 the rnendiafi of a place together, then, while that place passes around east- ward till its meridian meets the star again, the sun has ad- vanced eastward nearly a degree, and the place must revolve nearly a degree more than one revolution before its meridian will reach the sun. This will require nearly 4 minutes of time; for, in the diurnal motion, 15° correspond to one hour, and therefore 1° to y 1 - of an hour — that is, four minutes. 78. The relation of sidereal time to mean solar time. — As the sun, in its apparent annual motion, describes 360° in 365.24 days, it will, in one day, on an average, pass over 360° -7- 365.24 = 59' 8.35", or nearly 1°, as before stated. But, by the diurnal motion, a given place on the earth in one solar day describes 360° plus the above arc. Therefore, 360° 59' 8.35" : 59' 8.35" : : 24h. : 3m. 55.9s. of solar time. This is the excess of the mean solar day above a sidereal day. And one sidereal hour, minute, or second is to one solar hour, minute, or second as 360° : 360° 59' 8.35",— that is, as 1 : 1.0027379. Therefore, to reduce a given period of time from the mean solar to the sidereal reckoning, multiply by 1.0027379 ; and to reduce side- real time to mean solar time, divide by the same number. 79. The apparent solar day. — This is the actual interval between two successive culminations of the sun. And this in- terval changes its length from day to day through the entire year, being sometimes greater, and sometimes less than the mean solar day. In keeping solar time by clocks and watches, it is customary, for convenience, to aim to keep the mean rather than the apparent time, and to regard the sun as going alternately too fast and too slow. 80. First cause of inequality in apparent solar days. — One* cause of inequality of days, as measured by the sun, is found in the elliptical form of the earth's orbit, and the consequent unequal increments of longitude made by the sun from day to day. At P (Fig. 25) the sun is nearest to us, and at A it is most distant. The motion in the parts of the orbit near P INEQUALITY OF SOLAR DATS. 49 would therefore appear greater than in the parts near A, even if it were uniform in all parts. But, besides this, as will be shown in Chapter Till , the motion is really greatest at P and least at A. For both these reasons, then, the sun, while in the nearer half of its orbit, passes over the longest arcs each day in the ecliptic — that is to say, in longitude — and the shortest arcs, in the half most distant from us. The sun, in fact, occu- pies nearly 8 days more time in describing the remote hah than the nearer one. Eecollecting, now, that a solar day consists of a sidereal day, plus the time of describing diurnally the arc which the sun, in the mean time, advances annually, it is clear that if this daily arc is longer, the solar day is longer ; and if shorter, the solar day is shorter. So far as this cause is concerned, therefore, the longest solar day would be the 1st of January, and the shortest, the 1st of July ; and about half-way from P to A, and from A to P, the apparent days would have their mean length. 81. Second cause of inequality in apparent solar days. — But the solar days are unequal for another reason — the ob- liquity of the ecliptic to the equator. Time is measured by arcs of the equator. But the sun's daily advance toward the east is made in the ecliptic. Even if the daily increments of the sun's longitude were equal, those of its right ascension would be unequal, and therefore the solar days unequal. Let Fig. 29 repre&ent a portion of the celestial sphere, AF a part of the equator projected in a straight line, OH a cor- responding part of the ecliptic, Q the vernal equinox, S the summer solstice, and P the north pole. Draw through P a few meridians, dividing that part of the ecliptic near Q into short arcs, to represent the daily increments of the sun's longi- tude on CH, and of its right ascension on AF. These meri- dians are oblique to CD, but perpendicular to AB. Hence, as AQC is a right-angled triangle, QC is longer than AQ ; so also, DQ is longer than BQ ; and thus each part of CD is longer than the corresponding part of AB ; — that is, the increment! of the sun's right ascension, near the equinoxes, are less than those of its longitude. The obliquity, therefore, by short- 4 50 INEQUALITY OF SOLAR DAYS. ening these increments of right ascension, shortens the soja? days. Bat if meridians are drawn to that part of the ecliptic near S, the arcs GH and EF are abont parallel to each other, and the increments on the equator are not shortened, as they are at Q But, on the other hand, the divergency of the meridians causes EF to be longer than GH, and each part of EF longer than the corresponding part of GH. At the solstices, therefore, the increments of right ascension are lengthened by the divergency of the meridians, and hence the solar days are lengthened also. About midway between the equinox and solstice, the two effects just described neutralize each other, and the daily arcs of right ascension, so far as this cause is concerned, are at their mean value. Fig. 29. 82. Location of extreme and mean solar days from each cause. — Suppose the first cause alone in operation, and that the sun and a uniform clock agree with each other at P (Fig. 25), on the 1st of January. Then, as the solar days are longer than their mean, the sun becomes slower, compared with the clock, from day to day, for about three months, when the days will have reached their mean length, at a point near half-way from P to A. Afterward, the days being diminished below the mean, the sun slowly gains on the clock, and catches up with it at A, July 1st. But the days now being shortest of all, the sun is immediately in advance of the clock, and most of all at a point half-way frcm A to P. The gain and loss compensate EQUATION OF TIME. 52 each other from A to P, as they did from P to A. Thus mean and apparent time would agree twice in a year, at intervals of six months, if eccentricity of orbit were the only cause of irregularity. Again, if the second cause alone existed, and we suppose the sun and clock to agree at the equinox Q (Fig. 29), then the sun gains on the clock every day, on account of the short arcs of right ascension near Q. In about 1| months, however, the days reach their mean length, the sun begins to lose what it has gained, and at S, June 21st, the sun and clock are again to- gether. But the sun is now losing, falls behind the clock, and is furthest behind midway between the solstice and the next equinox. The autumnal equinox and the winter solstice are, in like manner, points of time at which the clock and sun agree with each other. Thus, if the second were the only cause of ^regularity, the mean and apparent time would agree four times in a year, at intervals of about three months each. 83. The equation of time. — The difference between mean time and apparent time, on any given day, is the equation of time for that day. If the sun is slow, the equation must be added to the apparent time ; if fast, it must be subtracted from it, in order to give mean time. We have seen by the two preceding articles that, on account of eccentricity of orbit, the equation would be reduced to zero twice in a year ; and, on account of obliquity of ecliptic and equator, it would be zero four times in a year. The joint effect of these two causes is, to reduce the equation to zero four times in a year, at unequal intervals of time. 84. The equation of time represented graphically. — The ordinates of the curves in Fig. 30 exhibit to the eye the equation of time as depending on each cause by itself, and on the two conjointly. The relative lengths of the ordinates above and below AB show the positive and negative equations, as caused by eccentricity, and those on CD the equations as caused by obliquity ; while the algebraic sum of these on each vertical line, gives the resultant effect on the line EF. The figure shows that the equation reaches its first maximum, + 14 minutes, on the 11th of February; its first minimum, — 4 52 THE JULIAN CALENDAR. minutes, May 14th ; its second maximum, + 6 minutes, July 2 6 tli ; and its second minimum, — 16 minutes, November 2d. The four times of agreement, when the equation is zero, are shown by the intersections ; they occur April loth, June 15th, September 1st, and December 24th. The sign + shows that the sun is on the meridian after mean noon, the sign — before mean noon. Jan. Feb. Mar. April. May. Fig June. . 30. July. Aug. Sept Oct Ncv. Dec. | A -"" C + 14 + 6 ^-4^ 16 85. Civil and astro?wmical time. — The mean solar day, when employed for civil purposes, is supposed to begin and end at mid- night, and is divided into hours, numbering from 1 to 12 a. m., and then from 1 to 12 p. m. But the astronomical day (which is also the mean solar day) begins and ends at noon, 12 hours later than the corresponding civil day, and its hours are counted from 1 to 24. Thus, the astronomical date, April 12d. 20h., is the same as the civil date, April 13th, 8 o'clock a. m. S6. The Julian calendar. — The period in which the sun passes from the vernal equinox to the same point again, is called the tropical year. In that period the round of the seasons is exactly completed. The length of the tropical year is 365d. 5h. 48m. 46.15s. This is so near 365i days, that in the adjustment of the calendar made by Julius Caesar (hence called the Julian calendar), three successive years were made to contain 365 days each, and the fourth 366 days. The addi- tional day is called the intercalary day. In this calendar it was introduced by reckoning twice the 6th day before the THE GKEGOKIAN CALENDAR. 53 Kalends of March ; and hence the year containing this addi- tional day was called the bissextile. The intercalary day is now the 29th of February , and the year containing such a day is called leap-year, 87. The Gregorian calendar. — By calling the tropical year 865^ days, the Julian calendar makes it more than 11 minutes too long, and the intercalation of one day in four years is therefore too great. This excess amounts to more than 18 hours in a century. Hence, by dropping the intercalary day three times in four centuries, the adjustment is nearly complete. The Julian calendar, thus amended, is called the Gregorian calendar, because adopted under Pope Gregory XIII. At that time, 1582, the vernal equinox, by the error of the Julian cal- endar, had fallen back to March 11th. To bring the equinox to its proper date, 10 days were first dropped (the 5th being •called the 15th), and then the following system was adopted. Every year, not exactly divisible by 4, has 365 days. Every year, divisible by 4, and not by 100, has 366 days. Every year, divisible by 100, and not by 400, has 365 days. Every year, divisible by 400, has 366 days. The Gregorian calendar will not be correct perpetually, but the error will not amount to a day in 4,000 years. The nation of Eussia has not yet adopted the Gregorian cal- endar, so that there is now a discrepancy of 12 days between their dates and those of other nations. The reckoning still used by them is known as old style, and is distinguished by appending the letters O. S. to every date. 88. How to compare days of the month and of the week in passing from one year to another. — A common year of 365 days contains 52 weeks and one day ; a leap-year contains 52 weeks and two days. Hence, a year usually begins a day later in the week than the year previous. And, generally, any day of any month is one day later in the week than the same day of the preceding year. Thus, July 4th, 1884, falls on Friday; 1885, on Saturday; 1886, on Sunday. But, in leap year, this rule applies only till the end of February. From that time to the same date in the year following, every day of a 54 CENTRIFUGAL FORCE. month falls two days later in the week than in the previous year. Thus, July 4th, 1883, is Wednesday ; 1884, Friday. And February 2d, 1884, is Saturday; 1885, it is Monday. Table I., at the end of the volume, contains a complete cal- endar for 77 centuries. CHAPTEE YI. CURVILINEAR MOTION. — SPHEROIDAL FORM OF THE EARTH. — ITS DENSITY. — PROOFS OF ITS ROTATION ON AN AXIS. 89. Projectile and centripetal forces. — Motion in a curve line is always the effect of two forces ; one, an impuhe which, acting alone, would have caused a uniform motion in a straight line, and whose influence is always retained in the curve motion ; the other, a continued force, which constantly urges the moving body toward some point out of the original line ol motion. The "first is called the projectile force, the other the centripetal force. If the action of the latter were to cease at any moment, the body by its inertia would from that moment continue uniformly in the direction in which it was then mov- ing. Such motion in the tangent may be regarded as the effect of an impulse first given in the direction of that tangent. This supposed impulse is the projectile force for the moment in question ; but it is in truth the resultant of the original im- \ ulse, and the infinite series of actions already produced by the centripetal force. The centripetal force may be resolved into two components one in the direction of the tangent, the other perpendicular to it. The tangential component will accelerate or retard the motion in the curve according as it acts with the projectile force or in opposition to it. When the body moves in the cir- cumference of a circle, the tangential component of the cen- tripetal force is 0, and hence the motion is un form. 90. Centrifugal force, — When a body moves in a curve, CENTRIFUGAL FORCE. 55 since by its inertia it tends to proceed in the tangent at that point, there is a continual outward pressure directed from the center of force: this is called the centrifugal force. It is always opposed to the centripetal force, and in circular motion is always equal to it. It must not be viewed as a third force introduced to explain curvilinear motion, but as that compo- nent of the projectile force which acts in opposition to the cen- tripetal force. 01. First law of centrifugal force in circular motion. — When a body moves in a circular path, its centrifugal (or cen- tripetal) force varies as the square of the velocity divided by the radius. Let Ab (Fig. 31) = v, the space passed over in one second. The projectile force is then represented by AB, and the body would move in that line uniformly, were it not for the centripetal force acting toward E, and thus deflecting it into Ab. Aa being the distance through which the body falls in one second, 2A# or c represents the centripetal force. Let AE = r. Then Aa : v Ab : : Ab : AD : or ^-c : v : : v : 2r, and c = —. As the centripetal and centrifugal forces are equal in circular motion, c may represent either in value, though they are opposite in direction. Hence, in a given circle, where r is constant, the force either toward or from the center varies as v 2 } the square of the velocity. In whirling a ball, for in- stance, with a string of given length, if the velocity is doubled, the strain upon the string (the centrifugal force) is four times as great, and the strength of the string (the centripetal force) needs also to be four times as great. So, if a train of cars goes round a curve with a velocity 1J times that which is intended, its tendency to be thrown from the track is increased 2J times. 93. Second law of centrifugal force in circular motion. — When the path of a body is circular, its centripetal or centrif- ugal force varies as the radius of the circle divided by the square of the time of revolution. Let t = the time of describing the whole circumference 2jtr ; 56 LOSS OF WEIGHT. and let the velocity per second %nr 4tt 2 r 2 t r a * 2rr 2 r varies as Therefore 2nr = i>£, and v = * 2 £ 2 Aa or ^c But (Art. 91) c = f = 4tT 2 7' which Hence, if the time of revolution is the same, the attraction to the center must be increased as the radius is increased ; for then coor. Thus, if a string is twice as long, it must have twice the strength, in order to whirl a ball at the same rate oi revolution. 93. Centrifugal force on the eartKs surface. — As the earth makes its diurnal rotation, all free particles upon it are in- fluenced by the centrifugal force. Let NS (Fig 32) be the axis, and A a particle describing a circle with the radius A(X If AB, in the plane of that circle, represent the centrifugal force, resolve it into AD on CA produced, and AF, tan- gent to the meridian 1STQS. The effect of AD is to dimin- ish the weight of the particle, e while the effect of AF is to urge it horizontally toward the equator. If the surface, then, consists of yielding matter, as water, the spherical form can not be retained, but the parts about the poles, N and S, will be depressed, and those about the equator, EQ, will be elevated. At each point between the pole and the equator, a particle is. held in equilibrium, by that component, AF, of the centrifugal force which urges it toward the equator, and that component of gravity which urges it down the inclined surface toward the pole. 94. Loss of weight at the equator caused by rotation. — Let the weight of a body, w, be taken to express the force ot gravity, and let \g (= 16^ feet) be the distance fallen through by this body in one second. Kow, c is the force by which Aa LOSS OF WEIGHT AT EQUATOR. 57 2n 2 r (Fig 31) is described in one second ; and Aa = — ^— (Art. 92} Hence, w : c :: \g: 2ttV .\ c = w X j2 « Using the values of the letters in the fraction, we obtain c, the centrifugal force, in terms of w, the weight of the body. The equatorial radius of the earth, r, is 3962.8 miles = 20,923,584 feet. The earth makes one rotation in 24 sidereal hours = 86,400 sidereal seconds. Reducing this to solar seconds (Art. 78), we find c = w X t = 86,164s. Hence, 4 x 3.14159 2 x 20,923,58 4 = _w_ 321 x 86,164 2 ~ 289* And, since the centrifugal force at the equator acts directly from the center, a body at the equator loses ^ig of its weight by the rotation of the earth. 95. Loss of weight by rotation at other latitudes. — Since c varies as r (Art. 92), the centrifugal force is greatest at the equator, and zero at the poles, and the force at the equator is to that at any latitude A (Fig. 32) as QC : AO — that is, as rad : cos lat. But, except at the equator, the centrifugal force does not directly oppose gravity. If AB is the whole centrif- ugal force at A, AD is the component of it which acts agains* gravity. But AB : AD : : AC : AO : : rad : cos lat. So thai the loss of weight is diminished again in the same ratio as be- fore. Tlierefore, the loss of weight at the equator is to that at any given latitude, as rad 2 : cos 2 of latitude. 96. Whole loss of weight at the equator. — It is found by observations made with the pendulum, that the weight of a body at the equator is y^- less than that at the poles. But the 58 SPHEROIDAL FORM OF THE EARTH, loss from centrifugal force is only ^ig. Subtracting this from j-jL^, the remainder is very nearly ^ 6 , a loss of weight at the equator which must be ascribed to some other cause. This cause is the oblateness itself, by which the equator is more distant from the center than the poles are. 97. Spheroidal form of the earth found by measurement — Not only is the oblate form of the earth inferred from its rota- tion on its axis, but the measurement of the length of a degree of latitude, at various distances from the equator, proves that the meridians of the earth are ellipses, whose major axes are in the plane of the equator, and their common minor axis a line joining the poles. If the meridians were circles, all the degrees of latitude would be of the same absolute length, but it has been ascertained, by numerous and most accurate trigometri- cal surveys, that the length of a degree of latitude is least at the equator, and increases toward the poles. But if the degree lengthens as we go toward the pole, then the radius must lengthen in the same proportion, and therefore the curve, belong- ing to a larger circle, must become more flattened. And this change of curvature belongs to an ellipse, not to a circle. Thus, at Q (Fig. 33) the degree is shortest, longer at K, still longer at L, and so on to the pole. The center of the arc Q is at A, nearer than the center of the earth ; the center of K is B, of L is D, and of the polar arc it is F, beyond the cen- ter C. Thus, the cen- ters of curvature of the elliptical quadrant Q~N lie on the curve ABDF, which is the evolute of that quadrant. Each meridian quadrant is in like manner the involute of a curve, and their four evolutea form the figure AFG-H about the center. Eo part of a meri- dian has its center of curvature at the center of the earth. MASS OF THE EAKTH. 59 The following numbers express both the size and the form of the earth : Equatorial diameter Polar diameter Mean diameter Difference of diameters 7925.604 miles. 7899.100 " 7912.357 " 26.504 " The difference of diameters is ^^-g of the equatorial diani eter ; this is called the compression of the poles, or the ellip- iicity of the earth. So slight is the oblateness above described, that an exact model of the earth could not be distinguished by sight or touch from a perfect sphere. The volume of the earth = (7912.357) 3 x g = 259,400,000,000 cubic miles. 98. The equatorial belt. — If we imagine a sphere con- structed on the polar diameter of the earth, the difference be- tween the sphere and spheroid will be a sort of shell or ring, thirteen miles thick at the equator, and growing thinner on every side to the poles. This is sometimes called the equa- torial ring or belt of the earth, and it produces sensible effects on the earth's relations to the moon and sun. 99. Weight and density of the earth. — The earth's mass, and therefore its density, can be obtained by comparing the effects produced upon a plumb- line, by the earth and a moun- tain of known weight. Let M (Fig. 34) be an abrupt moun- tain situated alone on a plain, and let a station, B, be selected on the north side of it, and an- other, D, in the same meridian, on its south side, for measuring the zenith distances of stars. If Fig. 34. * B- 60 DIUKtfAL KOTAT10N. the mountain were not present, the plumb-line of the zenith sector would hang in the lines B and D, and would mark E and G as the zeniths of the stations. But the attraction ui the mountain draws the plumb-line toward it, so as to joint to the false zeniths E' and G'. When the star S, therefore, culminates, its apparent zenith distance, SE', is measured at one station, and at another culmination, SG' is measured. The difference, SE' — SG', is the distance between the apparent zeniths. The distance, EG, between the true zeniths, is the same as the difference of latitude between the stations B and D. Let a trigonometrical survey, therefore, be made around the mountain, and thus the arc BD, or its equal EG, be found. E'G' — EG = the sum of the two angles by which the plumb- line is drawn from a vertical position at the two stations. The volume and density of the mountain being measured, and the angle being found, as above, by which it draws a plumb- line from a true vertical, we have the means of determining the mass of the earth. And, as its volume is known, its density is inferred. Observations of this kind were made near Mount Schehallien, Scotland, by Dr. Maskelyne, who found the deviation of the plumb-line to be a little more than 6". The mean density of the earth, as deduced from a great number of results, obtained by this and other methods, is 5.46, — that is, the earth, as a whole, is 5.46 times the weight of the same volume of water. Calling the weight of a cubic foot of water 62^ lbs., the weight of the earth is somewhat more than 6,000,000,000,000,000,000,000 tons. 1 OO. Proofs of the earth? s diurnal rotation. 1. To suppose the earth to rotate eastward on its axis, is the only reasonable way of explaining the fact, that all the mill- ions of fixed stars, at various and immense distances from us, in large and in small circles of the sphere, perform their ap- parent revolutions about us in precisely the same length of time — viz., one sidereal day. 2. "Without supposing the earth to rotate on its axis, we can not account for the oblate form, of the waters of the ocean. Whatever form the solid parts might have, the movable portion would be spherical, if the earth were at rest. Moreover, the ROTATION OF THE EARTH. 61 degree of oblateness is exactly that which is required on a sphere having the diameter and mass of the earth, if it be sup- posed to rotate once in 24 hours. 3. The weight of a body at the equator, compared with that at the poles, is too small to be wholly accounted for by in- creased distance. Centrifugal force, arising from rotation, can alone explain the remaining difference. 4. A body dropped from a great height strikes further east than the vertical line in which it began to fall. If the earth rotates, the top of a tower moves faster than the base ; and therefore a body let fall from the top, retaining the east- ward motion of that point, will strike further east than the- base. At the equator, this distance would be near 2 inches, for a fall of 500 feet. Numerous experiments on the fall of bodies through great distances have been very carefully made by different individuals, and in different latitudes. And they all concur in proving that a body in falling deviates from a vertical line toward the east. 5. It is jxroved by the vibrations of a pendulum that the earth rotates eastward. Let us suppose a weight to be sus- pended by a long fine wire, and then made to vibrate in a plane. The plane in which the wire and weight move is ver- tical, and passes through the point of suspension. The weight itself may be considered as describing a straight horizontal line. On account of inertia, the weight tends to keep always in the same line, or (if the point of suspension be moved) in a line parallel to itself. And it will always remain strictly parallel tc itself, provided it can at the same time remain horizontal, and in a vertical plane passing through the point of suspension. Thus, if at the equator the weight be made to vibrate north and south — that is, in the plane of a meridian— it will continue to do so without deviation, as the earth rotates eastward, be- cause it will thus remain moving horizontally in a plane which passes through the point of suspension, though that plane is continually changing. In this case, the lines in which the weight vibrates are all parallel among themselves. If the experiment be tried at the pole, and the weight be made to vibrate in the plane of a certain meridian, the point of suspension does not move from its place, but only revolves in -62 FORM OF THE SUN. it ; and while the earth revolves 15° per hour, the weight pre- serving its own plane of vibration, will seem to shift that plane 15° per honr in the contrary direction, keeping pace with the stars in their diurnal motion. At localities between the equator and the pole, the line oi vibration remaining horizontal, and in a vertical plane which passes through the point of suspension, can not at the same time preserve its parallelism. But it will come as near fulfilling this condition as possible. Its north extremity will deviate eastward from the meridian more or less, according as it is nearer the pole or the equator. It is proved that the deviation per hour is to 15° as the sine of latitude to radius. When experiments are performed with sufficient care, it is found that the pendulum actually deviates eastward from the meridian, and at a rate corresponding well with the calculated result. The pendulum thus furnishes evidence that the earth rotates on its axis. The above is known as Foucault's experiment. 6. It will be seen hereafter that the motion of the equinoc tial points toward the west, called the precession of the equi noxes, affords an independent proof of the earth's diurnal motion. CHAPTEK VII. THE SUN. — SOLAR SPOTS. — CONDITION OF THE SUN'S SURFACE. — THE ZODIACAL LIGHT. 101. The form of the sun. — The disk of the sun is always circular. And, as it presents all sides toward us in its rotation, we infer that its form must be spherical. But since it rotates on an axis, and its surface is hi a fluid state, it might be ex- pected to reveal a spheroidal form. The reasons why it does not are, that the force of gravity on the sun is very great, and, in consequence of the slowness of its rotation, the centrifugal force is small. It appears by calculation that the angle sub tended by the equatorial and the polar diameters can not differ DIMENSIONS OF THE SUN. 63 from each other, except by a small fraction of a second. Its oblateness is, therefore, too slight to be perceived. 102* Distance of the sun, and size of the earth's orbit.—* The sun's horizontal parallax is 8."848. Therefore, the distance of the sun from the earth is found (Fig. 4) by the proportion. sin 8."848 : rad : : 3962.802 : 92,381,000-; which is the distance in miles from the earth to the sun. The circumference of the earth's orbit, or the distance trav eled by the earth each year, is 92,381,000 x 2?r = 580,447,000 miles. 1 03. Velocity of the earth on its axis and in its orbit coin- pared. — In the diurnal motion, a place on the equator describes nearly 25,000 miles in 24 hours — that is, more than 1,000 miles per hour, or about 17 miles in a minute. In the annual motion, the earth describes 580,447,000 miles in 365J days, thus passing over a distance of 1,589,000 miles each day; which is about 1,103 miles in a minute, or 18.393 miles in a second. The earth's velocity in its orbit is about 65 times as great as that of the equator in the diurnal motion. 1 04. To find the dimensions of the sun. — The angle sub- tended by the sun's diameter may be measured by instruments. Let AES (Fig. 35) equal one-half the measured angle. Then we have rad : sin AES : : ES : AS, the semi-diameter of the sun. As the sun's mean apparent semi-diameter is 16' 2", and ES is 92,381,000 miles, we find the sun's radius near 430,855, and therefore its diameter 861,710 miles. Fig. 35. The sun's diameter is about 109 times that of the earth. And, since spheres vary as the cubes of their diameters, tho volume of the sun to that of the earth is as 109 3 : l 3 : : 1,295,000 : 1, nearly. 64 DIURNAL ROTATION OF THE SUN. 1 05. The surfs mass and density. — It is found, by methodi to be described hereafter, that the sun does not exceed the earth in mass nearly so much as it does in volume. While the volumes are as 1,295,000 : 1, the masses are about as 326,800 : 1. The density of the sun, therefore, is to that of the earth a? 326,800 : 1,295,000 : : 1 : 4, nearly. 106. Force of gravity at the surface of the sun. —When the relative masses and diameters of bodies are known, it is easy to find the relative force of gravity on their surfaces. For G oo -^- (TS"at. Phil., Art. 16), where G represents gravity, Q the mass of the body, and D its semi-diameter. Let W repre- sent weight at the earth, and W at the snn, and we have W : W : : -1 : S *^° : : 1 : 27.5. Hence, the weight of a body at the sun is 27.5 times as great as at the earth, and a body would fall 442 feet in the first second of its descent. 107. Diurnal rotation of the sun. — By observations on the solar spots, it is found that the snn rotates on its axis nearly in the same direction in which the earth revolves about the sun. In general, a spot which appears on the edge of the disk passes across, then disappears, and afterward reappears in the same place as at first in 27^ days. If the earth were at rest, this wonld be the period of the sun's ^ 36 rotation on its axis. But, as the earth revolves in nearly the same direction in its orbit, the appa- rent rotation of the snn is longer than its real rotation. In Fig. 36, suppose the earth to be sta- tionary at E, and that a spot on the sun appears on the diek at A. Then, after passing through B, D, H, it will appear again at A, at the end of one revolution. "^e — "^ F But, if the earth in the mean time moves on to F, the*> the APPEARANCE OF THE SOLAR SPOTS. 65 spot must pass over AB, in addition to one revolution, before it will be seen on the edge of the disk. As EC is perpendicu- lar to AD, and FC to BH, the corresponding arcs on the two circles are obviously similar. Therefore, EGE + EF : EGE : i ADA + AB : ADA. Instead of the arcs, we may use the times of describing them ; and then we have 1 year -f 27^ days : 1 year : : 2T| days : 25 days, 8-J hours, which is the period of the sun's rotation. Appendix A. 108. Position of the sun's equator. — If the solar spots al ways described their paths across the disk in apparent straight lines, it would be inferred that the sun's equator coincides with the plane of the ecliptic. But these lines appear straight only twice in the year, near the middle of June and of December. At other times, they appear as semi-ellipses, having the greatest breadth in March and September. The earth, therefore, passes the plane of the sun's equator in June and December. The inclination of the sun's equator to the plane of the ecliptic is found to be about 7i°. 109. — Appearance of the solar spots. — On examining the sun's disk with a telescope, there is usually seen a greater or a less number of dark spots, differing from each other in form and size, and each spot generally consisting of two distinct parts, called the macula, or nucleus, and the umbra. The macula is black, of irregular form, and commonly surrounded by the umbra, which has a lighter shade. The two parts of the spot do not often shade into each other, but are each marked by a sharp, though irregular outline. If watched from day to day, they are seen not only to move slowly across the disk, as already stated, but they change their form and general appearance. A large spot sometimes divides into two or more smaller ones ; and again a group unites into a single large spot. Sometimes a spot diminishes and disappears, first the macula, then the umbra. The reverse also happens — a spot is seen in the midst of the disk, where there was none the day before. Though only a few are commonly in sight at once, yet they have been, in some instances, counted by tens and even hundreds. Very rarely a spot is so large as to be seen by the naked eye. 5 bb RELATION OF SPOTS TO SURFACE LEVEL. Figure 37 (lower part) shows two views of the same grou|; as seen July 9th and 11th, 1 844. Fig. 37. 110. The spots are at the surface, and limited to a northern and a southern zone. — Each spot appears on the disk during one- half the time of its entire revolution. It must, therefore, be at the surface, and not at any distance from it. For, if it revolved at any distance from the surface, as in the orbit abc (Fig. 35),, then it would be seen on the disk only from a to h, which is less than half its orbit. But the spots do not pass across all portions of the disk; their paths are limited to a zone which extends not more than 35° on each side of the equator ; and with very few exceptions,, they lie in the outer, rather than the central parts of this zone. Spots are very rarely seen within the zone lying between 10° of north and south latitude; and still more rarely in the polar zones above latitudes 35° north and 35° south. The macular zones, as they are sometimes called, are represented in Figure 37, limited by the dotted lines, EQ being the equator. 111. Relation of the spots to the surface level. — If the spots were flat surfaces on the same level with the general surface of the sun, then all their parts would be foreshortened alike, THE RECEIVED THEORY. 67 when near the edges of the disk. If they were elevated ob- jects, as mountains, rising above the solar atmosphere, then the umbra nearest the edge of the disk would be hidden by the darker part, and on the edge the spot would appear as a pro- tuberance. But it is proved, by multiplied observations, that the spots must be degressions below the general surface, and the macula a deeper depression than the umbra. For, as a spot approaches the edge, while it is foreshortened by perspective, the umbra furthest from the edge disappears first, and then the macula itself, while that part of the umbra nearest the edge is still in sight. As a spot comes from the edge toward the central part of the disk, the order of appearances is* reversed. These changes are indicated in Fig. 37, upper zone. Appendix B. IIS. The general surface, — The luminous part of the sun's surface is not uniform, nor at rest. Every portion of it is mi- nutely mottled by spots and streaks of unequal illumination. These are called facidce. And continued observation shows that these faint inequalities are also undergoing incessant changes. The faculse are most strongly marked, and indicate the greatest agitation of surface, where a spot is about to ap- pear, or where one has recently disappeared. Appendix C. 113. The received theory. — No theory so well explains the telescopic appearances of the sun, as that which in substance was proposed by Sir William Herschel, in 1801. Whatever may be the condition of the central mass, the external surface, called the photosphere, consists of gas in an incandescent state, while below it, within the solar atmosphere, is a cloudy stratum, less luminous than the outer surface. Whenever, from any cause, a rent is made in the photosphere, the less luminous stratum below is seen through it, as the umbra of a spot; and a smaller rent in the lower stratum reveals the denser and darker part of the sun, as the macula of the same spot. The strata in which the rents occur are in a gaseous condition ; for the constant motions going on in the outlines of the spots, forbid the supposition that they consist of solid matter ; and the extreme rapidity of these motions, often more 68 THE ZODIACAL LIGHT. than 1,000 miles per day, is inconsistent witA the idea thai they are liquid. 114. The tody of the sun not necessarily dark. — The very dark appearance of the macula may be due to its strong con trast with the intense illumination of the general surface. For it is found by experiment that the brightest artificial light which has been produced, if placed between the eye and the sun, appears as a dark spot compared with the solar surface. 115. Cause of the spots. — Sir John Herschel has suggested that there are reasons for considering the equatorial regions of the sun to be more heated than the other portions, so that there are currents in the solar atmosphere analogous to the trade- winds on the earth. Resulting from these currents, he sup- poses that occasional local winds are produced, rotating on a vertical axis, and rending the atmosphere and clouds by their centrifugal force. The ruptures thus occasioned are the spots on the sun. This supposition derives considerable plausibility from the considerations, that the spots are limited to narrow zones a little distance from the equator; that they sometimes differ from each other in their motions across the disk ; and that, in a few instances, they have shown signs of rotation about their own centers. Appendix D. 116. Periodicity of the spots. — The number and size of spots vary exceedingly in different years. Sometimes for days and weeks none are to be seen ; and again, for many months, the disk is never free from them. It is noticed, of late years, that their frequency alternately increases and decreases during a period of 10 or 11 years. The years in which the greatest number has been seen of late, were 1870, 1882. And those in which there were fewest, were 1867, 1878. Appendix E. 117. The zodiacal light. — This name is given to a faint, ill- defined light, extending along the zodiac, either in the west, after sunset, or in the east, before sunrise. It so much resem- bles the twilight, that it is not ordinarily noticed, because it KEPLER'S LAWS. 69 appears as a mere upward extension of it. It is projected on the sky as a triangle, inclined to the horizon at the same angle as the ecliptic (Fig. 38). In the evening it is best seen at the season when the ecliptic is most nearly perpendicular to the ho- rizon, after twilight has ceased. It is therefore most conspicu- ous at evening in the month of February. When the air is clear, and there is no moon, it is visible till after 9 o'clock. For a like reason, the best time for seeing it before morn- ing twilight is the month of October. The apparent extent breadth and increased by of it, both in height, is much indirect vision. 118. Its nature. — There has been much speculation rela- tive to the nature of the zodiacal light. But astronomers gen- erally regard it as a nebulosity attending the sun, and extend- ing beyond the orbits of Mercury and Venus, and even beyond the orbit of the earth. CHAPTER VIII. KEPLER S LAWS. — THE LAW OF GRAVITATION. 119. Statement of Kepler's laws. — From a long and labo- rious examination of the recorded observations of Tycho Brahe, Kepler deduced three laws relating to the movements of the planets about the sun. They are hence called Kepler's laws, and may be stated as follows. 1. The areas described about the sun by the radius vector of an orbit, vary as the times of describing them. 70 LAW OF AREAS. 2. The orbit of every planet is cm ellipse, having the sup. in' one focus. 3. The squares of the periodic times of the several planets vary as the cubes of their mean distances. To render the language of the third law strictly correct, the cube of the distance should be divided by the sum of the masses of the sun and planet. But the mass of even the largest planet is so small, compared with the sun, that the omission intro- duces an error which is scarcely appreciable. Kepler established these three laws as facts in the solar system ; but Newton afterward demonstrated, by mathematical reasoning, that they are necessarily involved in the laws of in- ertia and gravitation. 120. Areas described by the radius vector. — Whatever path a body describes under the influence of a projectile and a cen- tripetal force, the areas described about the center of force vary, as the times of describing them. Let S (Fig. 39) be the center of attraction, and suppose the projectile force in the line YE, to be such as to cause the body to pass over the equal spaces PQ, QR, etc., each in a certain LAW OF VELOCITY IK AN ORBIT. 71 emit of time. When the body reaches Q, let the action toward S be sufficient to move it over QY in the same time in which by the original impulse it would describe QR. Then it will in the same time describe the diagonal QC of the par- allelogram. Jo£n ES and CS. The triangles QSC and QSR are equal ; but QSR = QSP ; .\ QSC = QSP ,— that is, the areas described in the first and second units of time are equal. In like manner, by supposing a second action toward S to occur at C, a third at D, etc., it is proved that QCS, CDS, DES, etc., which are described in equal times, are equal. This is true, however small the unit of time between the successive actions toward S, and is therefore true when the central force acts incessantly and causes curvilinear motion. As all the areas are equal, which are described in the several units of time, therefore the areas vary as the times. As the diagonal of each parallelogram is in the same plane with its two sides, it is obvious that the whole orbit lies in one and the same plane. Conversely, if areas described about a point vary as the times, the deflecting force acts toward that point. For PSQ = QSE, as before (Fig. 39) ; and by supposition, PSQ = QSC ; .% QSC - QSR ; hence CR is parallel to QS, and QC is the diagonal of a parallelogram, whose side QY, in which the deflecting force acts, is directed toward S. Since it is an established fact, agreeably to Kepler's first law, that the radius vector of each planetary orbit describes areas about the sun, which vary as the times; therefore, the cen- tripetal force, acting on the planets, is directed toward the sun. 121. The law of velocity in an orbit. — The velocity at any point varies inversely as the perpendicular from the center of force to the tangent at that point. Let ST (Fig. 39) be perpendicular to PQ ; then the area rSPQ =iPQ x SY, which varies as PQ x SY; ,\ PQ oo ^$. But PQ oo Y, the velocity at P ; and the area SPQ is constant; •"*- "V* °° oyj or the velocity varies inversely as the perpendicu- 72 LAW OF GKAVITATICXN. lar from S, upon the line in which the body is moving; in other words, upon the tangent of its path, if it describes a curve. 122. Law of gravitation in an orbit, as related to dis- tance. — If a body describes an elliptical orbit, by a centripetal, force which acts toward the focus, that force varies inversely a& the square of the distance. Fig. 40. Let the body be at M (Tig. 40), and MF the radius vector at that point. Let MO be the radius of curvature at M, and. therefore perpendicular to the tangent ; and suppose M^N" to be an infinitely small arc described in a given small portion o£ time. Draw FP perpendicular to the tangent MP, KK to FM, and NH to MO ; then PFM, MHI, KNI are similar tri angles. ME", considered as a straight line, is described by the joint action of the centripetal force in the line MI, and the projectile force which is parallel to IN. The motion in MI may be regarded as uniformly accelerated, because in the in- finitely small time of describing it, the centripetal force may be considered constant. Hence, 2MI may be taken as the measure of the centripetal force f (ISTat. Phil., Art. 28). Therefore, f c© MI. It is to be proved that MI qo r^. LAW OF GKAVITATION. 73 123. By similar triangles, MI : MH : : XI : NK; NI flow, the chord MN is a mean proportional between tha TVTNT 2 versed sine MH and the diameter 2MO ; or MH = oiuTj » Nil 2 but, as the arc is infinitely small, NH = MIST ; .\ MH = o]v/r(V Again, the versed sine MH, and therefore HI, is infinitely small compared with NH, and NI may be substituted for NH ; ••• MH = Sro- 124. Now it is shown in conic sections, that r mKJ 2 V FP/ ' FM NI therefore, since by similar triangles -^p = -Sxt?? M0 =*ffiY 2VJSTK/ Substituting this for MO in the equation for MH above, we have MH = m . ^> . NI Hence, in the equation for MI we have EK 3 NI 1_ T _■ MI = — ^ T x ^^ = - NK. j? . ]NI M jp Now, the sector FMN is measured by JFM . NK ; . • . NK = * Jackson's Conic Sections. The same may be derived from Coffin's Conic (FM MV) 3 Sections, Pr. V., Curvature, R 2 or MO = v ■— — - , a and b being tlie semi- 1 3 3 axes; .'. MO = — x (FM. MV) 2 . Multiply by (& 2 ) 2 , and divide by its equal (FP VL)s ; JkenMO = L (^^) 2 = °- (|^j 2 , since FMP and VML are similar. But - = |; ,. MO = f (— )* = f ( p --) . 74 LAW OF GRAVITATION. 2FMN , AT1Z2 4FMN 2 __. 4FMN 2 _ t ■_ -^j- ; and NK 2 = -j^-; .-. MI = — w , But as the areas described by the radius vector vary as the times, FMN ia constant. Therefore, MI(=/)co^; that is, the centripetal force in the orbit varies inversely as the square of the distance. 125. Applicable to every conic section. — It is thus proved that, in any elliptical orbit described about the focus as the center of attraction, the intensity of that attraction varies in- versely as the square of the radius vector. As there is nothing in the foregoing demonstration to limit the conclusion to the orbits which are nearly circular, like those of the planets, we are at liberty to apply it to orbits of extreme eccentricity, as those of the comets. And it is proved by Newton, in his Prin- cipia, that the same law of force is necessary, in order that a body may describe any one of the conic sections about its focus as the center of attraction. 126. Law of gravitation as to distance, in different or- bits. — And not only does this law prevail in all parts of any one orbit, but it is true also that all the different bodies of a system, describing orbits about the same center of force, are urged toward that center by attractions which vary, from one orbit to another, inversely as the square of the distance. Let a be the semi-major, and b the semi-minor axis of any elliptic orbit. Then a is the mean distance of all points of the orbit from the focus. By a rule of mensuration, the area of the ellipse = nab. If s = the area described by the radius vector in a unit of time, as one second, and t = the number of seconds in the whole period of revolution, then the ellipse also = ts. Therefore, nab = is ; and t = — ; and f = — — . By 27 2 7 2 Kepler's third law (Art. 119), tfao a 3 ; .*. —5- 00 a z ; ,\ — op s\ s a But, because the semi-parameter ^- is a third proportional to LAW OF GRAVITATION. 75 the semi-axes a and £, — = ^ ; foes. Hence, substituting % for s 2 ,— that is, FMN 2 ,— in the equation for MI (Art. 124), we find MI 4FMN 2 = 2/y 2 ^.FM^FM 2 ' .-./.oo Or, the jp . FM 2 jp. FM 2 FM 2 ' " J " FM 2 * force varies inversely as the square of the distance, in different orbits, as well as in different parts of the same orbit. The satellites which revolve about the planets are found to conform to Kepler's laws, and therefore the force which urges them toward their respective primaries varies in each case in- versely as the square of the distance. 127. Law of gravitation within small distances. — But the inquiry still remains, does the law of gravity, as demonstrated in the foregoing articles, hold good at the smallest distances also ? For example, do the tendencies of bodies resting on the earth, and of those elevated in the air, and of the moon toward the earth's center, come under the same general law ? This is the very question which presented itself to the mind of New- ton, after he had discovered that the force which deflects the planets from their lines of motion toward the sun, varies in- versely as the square of their distance from it. As he noticed the fall of an apple, the inquiry arose, may not this fall be of the same nature as the lending of the moon's path toward the earth, and may not the force in the two cases be as the squares of the distances inversely % The distance through which the moon actually descends in one second may be represented by Ka (Fig. 41), A o being the arc described in the same time. For, as the moon was going toward B, it would not have deviated from the line AB, if some force had not turned it aside. This influence must be directed toward the earth, E, because it is about E that the radius is known to describe areas propor- tional to the times (Art. 120). There*- 76 LAW OF GKAVITATTON. fore, Bh, or the versed sine Aa (which may be considered equal to it), is the distance fallen through in one second. Now, the circumference of the moon's orbit, divided by the number of seconds occupied in describing it, gives the arc Ah. This arc and its chord may be considered the same, and by geometry we have 2 AE : Ah : : Ah : Aa = 0.0535 of an inch. At the surface of the earth, a body falls 1 6^ feet in the first second. On the supposition that gravity varies inversely as the square of the distance, we find the fall in one second at the moon, by the proportion, the square of the moon's dis- tance : square of the earth's radius :: 16^ feet : 0.0536 of an inch, agreeing very accurately with the distance which the moon actually falls from a tangent in one second. Therefore, a body falling at the surface of a planet, and a satellite revolv- ing about it, are both subject to the same law of centripetal force. 128. The law prevails throughout the solar system. — As will appear hereafter, there are numerous disturbances pro- duced upon the motion of each body in the system by the attraction of every other. Every one of these disturbing influ- ences is measured, by applying the law of distance already men- tioned. If a planet or comet moves toward a plauet for a certain length of time, it is accelerated ; and its acceleration is greater, as the square of the distance is less ; and it is retarded* according to the same law, when departing from it. 129. The law of gravitation, as related to the quantity of matter. — The force of gravity varies directly as the quantity ot matter. In Mechanics, we infer the existence of this law from the fact that all bodies, light and heavy, and of every kind ot material, fall with equal velocity toward the earth. So, in the solar system, a planet and all its satellites, when at equal dis- tances from the sun, are urged toward it by forces proportional to their masses, or they could not maintain their mutual relations as they do. And it is found that every disturbing influence in the system is accounted for only by applying both parts of the law of gravity — that it varies directly as the quantity of matter % and, inversely as the square of the distance. PATHS OF PROJECTILES. 77 130. Paths of projectiles considered as orbits. — When a 6tone is thrown, or a ball is fired, its path (undisturbed by the atmosphere) is part of an elliptic orbit, one of whose foci is at the center of the earth. In Mechanics, the path of a projectile is proved to be a parabola (Nat. Phil., Art. 44) ; but, in that demonstration, the vertical lines were assumed to be parallel to each other, and the force of gravity a constant force, neither of which is strictly true. Knowing the distance and period of the moon, the time in which a projectile would complete its revolution if. found by Kepler's third law. Any force, which man could apply, would carry the lower extremity of the orbit so little beyond the center of the earth, that the mean distance might be. called one-half the radius of the earth. Therefore, calling the moon's distance 60 radii, and its period 27J days, we have (60) 3 : (£) 3 :: (27J) 2 : x% from which x is found to be about 31 minutes. Every projectile, then, if it were free to complete its orbit unobstructed, and according to the law of gravity which prevails outside of the earth, would make an en- tire revolution, and return to its place, in about half an hour. 131. Effect of increased velocity of projection. — Suppose that P (Fig. 42) is a point near the earth, ADE, and that the velocity of projection, in the direction PB, is so greatly in- creased that the projectile strikes the earth at D. By a still greater increase of velocity it might meet the earth at E. In these cases the earth's center would be in the most remote focus of the orbit. But if we suppose the velo- city so much increased that the centrifugal force just equals the force of gravity, then the body would de- scribe the circular orbit PFG (Art. 90). As the mean dis- tance now equals the radius of the earth, the time of revolution is found, by Kepler's third law, to be lh. :4m. 39s. Any increase of the velocity of pro- 78 MOTIONS OF SUN AND PLANET. jection beyond this will again produce an ellipse, as PK. whose nearer focus is at the earth's center. And we can imagine the velocity increased till the ellipse becomes one ol extreme eccentricity, and then changes into the branch of a parabola, and then of a hyperbola, in which last cases the body will never commence a return toward the earth. 132. Orbit motion and diurnal rotation by one impulse.— If we suppose the projectile motion of the earth, or any other planet, to have been produced by a single impulse, that im- pulse may also have caused the diurnal rotation of the body. If the impulse had been directed in a line passing through the center of gravity of the planet, then it would have caused a progressive motion without rotation on an axis. But, if the line of impulse did not pass through the center of gravity, there would be rotation as well as progression. It has been calculated that the two existing rotations of the earth might have been produced by one impulse, applied in a line which passes 24 miles from the earth's center, on the side most remote from the sun. Had it been directed through a point lying on the side nearest the sun, the diurnal motion would obviously have been retrograde. 133. Motions of sun and planet, resulting from an impulse given to the planet. — Suppose that the sun at S (Fig. 43), and the earth at E, mutually attract each other, and that an im- pulse is given to E in a line perpendicular to ES. S can not remain stationary and E revolve about it ; for it is proved (Nat. Phil., Art. 89) that their center of gravity will move precisely as the sum of the bodies would move if united at the center, and the same impulse were applied to them. Suppose, for the sake of simplicity, that the weights of the bodies and the strength of the impulse are so related that the center, C, will pass over each unit of space, Ga, ab, bo, etc., while E advances 45° in a circle about the moving center. Then, when the center is at a, E is at 1, 45° from a perpendicular at a. But S must be on the opposite side of a, and as far from it as from before. Therefore, by the impulse given to E, and the mutual A PLANET AT APHELION OR PERIHELION. 79 Attraction between E and S, the latter has been drawn along from S to 1/. Again, when the center is at b, E is at 2, and S at 2'. While E was on the upper side of CA, S was drawn toward that line, and now crosses it, and by its inertia con- tinues upward, although E is now below the line. In this manner the bodies revolve about the moving center, describing circles relatively to that, but curves of a totally different char- acter in space. These curves are always some variety or other of the class of curves called epicycloids. In the case repre- sented in the figure, the planet describes an epicycloid which forms a series of loops, intersecting its own path at every revo- lution, while the path of the heavier body is of a waving form. The body E retrogrades on the lower part of the loop from 3 to 5, while S advances continually, but with unequal velocities, each body being alternately drawn forward and held back by the other. Fig. 43. -E The only way in which two separate bodies could be made to rotate about a fixed center of gravity, would be to give an equal impulse to each body, and in opposite directions. Two such forces would constitute a couple (Nat. Phil., Art. 54), whose effect is to produce rotation merely. 134. Why a planet at aphelion begins to return, or at peri- helion begins to depart. — It might be thought that a planet at 80 PRECESSION OF EQUINOXES. its aphelion, C (Fig. 44), being less attracted toward the sun than at any other point, wonld continue to withdraw, instead of commencing to return ; and that when at its perihelion, G-, being more attracted than else- where, it would continue to ap- proach till it falls to the sun. The reason why a planet begins to re- turn after reaching the aphelion is to be found in its diminished ve- locity. As the plauet recedes through. H, K, and A, the centrip- etal force toward S draws it back, and causes continual retardation, till at C the velocity is so much diminished that the attraction of S, though less than elsewhere, is still sufficient to curve the path so that it falls within a circle about the centre S, and the planet begins to approach the sun. Again, as the planet passes through D, E, and F, the at- traction toward S partly conspires with its inertia, and it is continually accelerated, till, at G, its velocity has become so great that its path strikes outside of a circle about the center, S, and it begins again to depart as before. CHAPTEK IX. PRECESSION OF EQUINOXES. — NUTATION. — ABERRATION OF LIGHT. — APSIDES OF THE EARTH'S ORBIT. 135. Precession of equinoxes described. — The points in which the equator intersects the ecliptic on the celestial sphere are not stationary, but have a slow retrograde movement — that is, they revolve from east to west. The sun, therefore, in its annual progress eastward, crosses the equator each year a little further west than it did the year previous This motion is CAUSE OF PEECESSION. 8 J ealled the precession of the equinoxes, either because the time of the equinoxes precedes the time in which the sun would have passed them if they had remained at rest, or because, in the daily transit of the meridian, the equinoxes precede those stars which crossed at the same time with them the preceding year. The equinoctial points retrograde about 50i A ' each year. At this rate, it will require 25,800 years to make a complete circuit of the heavens. 136. Signs of the ecliptic displaced from the signs of the zodiac. — The want of coincidence between the signs of the ecliptic and the signs of the zodiac was noticed (Ait. 61). They coincided at the time the division was made, about 2,000 years ago ; and the precession daring this period has moved the equi- noxes backward 2,000 x 50i" = 28°, nearly. Hence, Aries of the zodiac almost coincides with Taurus of the ecliptic, Taurus of the zodiac with Gemini of the ecliptic, etc. 137. Motion of the north and, south poles. — Considering the plane of the ecliptic as fixed, its poles of course occupy fixed positions among the stars. But this is not true of the poles of the equator. Their distance from the polefi of the ecliptic is equal to the obliquity of the two circles — that is, 23° 27 / . As this angle remains nearly constant, and the points of intersection move around westward, the poles of the equator must likewise move round those of the ecliptic in the ^ame direction, and occupy the same period, 25,800 years in com- pleting their revolution. The north pole of the equator is row near the star in Ursa Minor, known as the pole-star. Accord- ing to the earliest catalogues, the pole was 12° distant from the pole-star. It is now somewhat more than 1° distant, and will, at the nearest, pass within \° of it. In about 13,000 years the pole will be on the opposite side of the pole of the ecliptic, near the bright star a Lyrse, which will then be the pole-star. 138. Cause of precession. — The precession of the equinoxes is a disturbance produced by the sun's and moon's attraction upon the equatorial ring of the earth, as it rotates on its axis. 6 82 CAUSE OF PKECESSION. The sun being in the ecliptic, while the equatorial ring is inclined 23° 27' to it, the sun's attraction is oblique to the plane of the ring; and one component of this force is perpendicular to the ecliptic. In most positions of the ring in relation to the sun, this component acts on one part to press it towards the ecliptic, and on another part to move it from the ecliptic. But the first is in excess ; so that, on the whole, the ring tends to turn on the line of equinoxes towards the plane of the ecliptic. And this tendency, compounded with the inertia of the ring in its diurnal rotation, moves the equinoxes backward. Fig. 45. Let EC (Fig. 45) represent the plane of the ecliptic, ana QR the equatorial ring of matter. A particle, A, of the ring, by its inertia of rotation, tends to move toward T in the plane QR. Let AB represent this force, and AF the pressure toward EC, produced by the sun ; then the resultant will be the diag- onal AD, shifting the equinox back to T'. All the particles are subjected to this influence, except at the moment (each day) of crossing T and — , so long as the sun itself is not in the line T=^ produced, which occurs in March and September. The effect is then interrupted for a time. As the moon is always near the ecliptic — sometimes on one side of it, and sometimes on the other — its action on the whole conspires with that of the sun. And as it is compar- atively near, though it is so small a body, its effect is more than twice as great as that of the sun. The planets produce a THE TROPICAL AND SIDEREAL YEAR. 83 very minute effect on the ring, tending to diminish the amount of precession. The joint effect of all the bodies mentioned is, as stated above, 50J". 139. Law of composition of rotations. — The case of pre- cession of equinoxes is classed under the general law for the composition of two rotations, which is analogous to that for the composition of two rectilinear motions (Nat. Phil., Art. 38). It may be stated thus : if two forces are applied to a body, which, separately, would cause rotation on two different axes, their joint action will produce rotation on a third axis lying in the plane of the other two, and making angles with them, whose sines are inversely as the forces. In precession, the earth rotates on the diurnal axis by one force, and the sun and moon tend to rotate it on the line of the equinoxes. As the latter force is minute compared with the other, the new axis is shifted by a very small angle each year from the diurnal axis toward the line of equinoxes. And this line slides along the ecliptic, so that the two axes remain perpetually at right angles with each other. The rotascope, a modification of Foucault's gyroscope, may be used to exhibit a very perfect illustration of the precession of equinoxes. 140. Cause of the slowness of precession. — If the equatorial ring were a separate body rotating about the earth in its own plane, its points of intersection with the ecliptic would retro- grade very rapidly by the action of the sun and moon. The reason why the precession is exceedingly slow is, that while the disturbing action is exerted only on the ring, the force around the diurnal axis consists of the inertia of the entire earth. The ring can not move by itself, but must carry the whole mass of the earth with it. 141. The tropical and sidereal year. — The fact of preces- sion shows that the year has two different values, according as we reckon from a star or from an equinox. Hence, the side- real year is defined to be the period occupied by the sun in £4 NUTATION. passing eastward around the heavens from a star to the same star again ; and the tropical year, the time of passing around from an equinox to the same equinox again (Art. 86). As the equinox moves westward, the sun reaches it sooner than if it were stationary, and thus makes the tropical year shorter than the sidereal, by the time required to pass over 50y, which is 20m. 22.9s. As the tropical year is 365d. 5h. 48m. 46.15s. (Art. 86), the sidereal year, therefore, is 365d. 6h. 9m. 9s. Though the sidereal year is the true period of the earth's revolution about the sun, yet the tropical year possesses by far the greatest interest, because it is the period in which the seasons are completed. 142. Nutation. — By precession alone, the pole of the equator would move in the circumference of a circle about the pole of the ecliptic. But this motion is modified by a minute vibration from side to side, as it advances, so that the line described Fi g- 46. by the pole is a delicate wave lying along on the circumference, as rep- resented in Fig. 46, where P repre- sents the pole of the ecliptic, and MN the path of the pole of the equator around it. This vibratory motion is called nutation. It is principally due to the unequal ac- tion of the moon upon the equa- torial ring. The moon's action, at any given time, tends to revolve the ring into the plane of its orbit. But, on account of the retrograde motion of its nodes, the angle between the ring and the moon's orbit varies * from 1 8J° to 28^-°, going through all the changes every nine- teen years. Owing to these changes of position, the equinoxes vrill recede sometimes faster, and sometimes slower ; while the inclination of the equator to the ecliptic will also increase and decrease, causing the poles of the equator to oscillate, a? stated ABERRATION OF LIGHT. 85 above. The amount, by which the pole of the equatoi moves to and from the pole of the ecliptic is IS". The waves in the figure are exceedingly exaggerated The arc MN being about T V of the circumference, the waves, if truly represented, would be small enough to cross the arc 270 times. 143. Aberration of light. — The heavenly bodies suffer a minute apparent displacement, on account of the progressive motion of light, combined with the earth's motion in its orbit. Suppose the earth to move from C to E (Fig. 47), while the light, coming from S, describes the line I)E. If they arrive together at the point E, the impulse on the retina of the eye will not be in the same direction as if the observer had been at rest ; but the light will appear to come in the direction S'E, the body being apparently thrown forward from S to T , S'. For, make EA = DE, and complete the parallelogram CA ; and suppose, according to the principle of equal action and reaction, that the light has the motion EC given to it, in place of the earth's motion, CE ; then the two motions, EA and EC, will produce the resultant, EB, as though the light had come from S' instead of S. 144. Aberration illustrated. — The apparent direction of any kind of impulse is modified in the same way, by the motion of the person who receives it. For instance, if the wind drives drops of rain in a person's face, at a certain inclina- tion, while he is standing still, when he comes to move toward the wind, they will strike him at a less inclination with the horizon, as though the source of the drops was further forward. For, when the person moves, the effect is the same as if he remained at rest, and the wind were to receive an increment of velocity equal to his motion. 145. Greatest and least aberration. — The greatest aberra- tion occurs when the body, from which the lis;ht comes, is in a 86 ADVANCE OF APSIDES. direction at right angles to the line of the earth's motion The displacement is then 20". 5. When the earth is moving directly toward or directly from the body, the aberration is zero. Therefore, a star in the plane of the ecliptic is seen in its true place once every six months ; bnt three months before and three months after either of those times, it is displaced 20".5 in opposite directions, making the total arc of displace- ment 41". But a star at the pole of the ecliptic, being always thrown forward of its true place by 20". 5, will seem to de- scribe each year a circle, whose diameter is 41". Between the ecliptic and its poles, the apparent orbit of aberration is an ellipse, whose major axis is 41", and whose minor axis increases with the latitude of the body. 146. Velocity of light computed by aberration. — In the triangle AEB (Fig. 47), AB represents the velocity of the earth, AEB the observed aberration, and EAB the angle between the line of the earth's motion and the direction of light. When EAB=90°, the aberration is found to be 20".4451. Therefore, tan 20".4451 : rad : : 18.393 miles : 185,600 miles per second, which is about the velocity of light. 147. Advance of the apsides of the earth's orbit. — It was intimated in Art. 74 that the line of apsides is not stationary. If the exact place of the perihelion among the stars be noted, it will be found the next year 11 ".5 further east — that is, the apsides advance 11". 5 per year. But in longitude, the advance- is much faster, since the vernal equinox, from which longitude is reckoned, retrogrades 50^" per year. The perihelion, there- fore, increases its longitude nearly 62" each year. As the longitude of the perihelion in 1800 was 279° 30' 8" (that is, 9° 30' 8" past the winter solstice) it must have been just at the solstice in the year 1247. For, 9° 30' 8" -f- 61f " = 553 years; and 1800 — 553 = 1247. In a similar manner, it is found that the perihelion will be at the summer solstice in the year 11741. In the course of many centuries, the length and temperature of the seasons are modified by these slo^v movements of the equinoxes and the apsides (Art. 75). LONGITUDE OF THE SUN. 87 1 48. Cause of the advance of apsides. — The apsides of the earth's orbit are made to advance by the attraction of the heavy planets, whose orbits are outside of it. The entire re- sultant of the attractions of these planets upon the earth, is to diminish a little the earth's tendency to the sun. Hence, as the earth approaches one of its apsides, its path is not suffi- ciently drawn in by the sun to meet the former line of apsides at right angles. But it makes right angles with a radius vec- tor a little further on, which becomes, therefore, the new line of apsides. 1 49. Sun's anomaly. — The sun's longitude is his distance eastward on the ecliptic from the vernal equinox (Art. 15). Its anomaly is its distance eastward, on the ecliptic, from perihelion. The reason for reckoning motion from the peri- helion is, that the angular velocity depends on it ; so that, to find the true longitude of the sun at any time, we need to know how far it is from the perihelion. 150. How to find the true longitude of the sun at a given time. — It is first supposed that the sun moves uniformly in a circle. And by knowing what its mean motion is, and how long it is since it passed the vernal equinox, we have its mean longitude at once. But this needs correction on account of the variable motion in the ellipse. Let E (Fig. 48) be the earth ; PCA, the elliptic orbit of the sun ; and BCF, the sup- 8S THE MOuN'S DISTANCE. posed circular orbit whose area equals that of PCA. Suppose the sun's mean place to be at S', and the vernal equinox at °P ; then its mean longitude is TDS', already obtained. The angle BES' is its mean anomaly. But as the .sun has been passing through the nearest part of its orbit, its true place is further advanced, as at S. The angle PES is the true anomaly, and the difference between them — that is, S'ES — is called the equa- tion of the center. This equation, or correction, being found in tables of the sun's motions, and applied to the mean longi- tude, gives the true longitude. If the mean and true places are considered as agreeing at P, then the equation of the center immediately becomes positive, and increases to its maximum at C ; after which it diminishes, and the mean and true places agree again at A. After that, the sun falls behind its mean place, and the equation is neg- ative, till the sun reaches P, the greatest value being at D. The eccentricity of the earth's orbit is so small, that the sun's mean and true places never differ so much as 2°, the greatest equation of the center being 1° 55' 27". 151. The anomalistic year. — The perihelion is another point from which to measure the revolution about the sun. The time of passing round from perihelion to perihelion again is called the anomalistic year. It is 4m. 40s. longer than the Bidereal year, or 365 d. 6h. 13m. 49s. CHAPTEE X. T^¥ MOON. — ITS REVOLUTIONS. — ITS PHASES- -THE CONDITION OF ITS SURFACE. 1 52. Distance and dimensions of the moon. — The moon is a satellite of the earth, revolving about it within a compara- tively small distance, and accompanying it in its orbit around the sun. The mean horizontal parallax of the moon at the MONTHS. 89- earth's equator being 57' 2". 7, its mean distance is found by the proportion (Fig. 4), sin 57' 2". 7 : rad : : 3962.8 : 238,820m. The moon's angular diameter is 31' 6"; therefore, rad : sin 15' 33" : : 238,820 : 1080.3 ; which is the moon's semi-diameter in miles. Hence, the moon's diameter is 2,160.6 miles. The surfaces of the earth and moon being as the squares of their radii, are as 13 : 1. The volumes of the earth and moon being as the cubes of their radii, are as 4:9 : 1, nearly. But the moon's density is so> small (3.4), that the masses are nearly as 81 : 1. The force of gravity on the earth to that on the moon is as- W5Voow ::6:1 ' nearly - 153. Revolution about the earth. — The slightest attention, to the position of the moon, from night to night, shows that it moves eastward, among the stars, several degrees every day. If the instruments of the observatory be employed to measure its right ascension and declination, as in the case of the sun (Arts. 58, 59), it is ascertained that the moon describes nearly a great circle, inclined about 5° to the ecliptic, and occupies- 27.32 days in returning to the same place among the stars. The inclination of the moon's orbit to the ecliptic va~ ries from 5° 20' 6" to 4° 57' 22" ; but its mean value is 5° 8' M". 154. Months. — The period just mentioned, in which the- moon makes a revolution from a star to the same star again, is called the sidereal month. The time occupied in making a revolution relatively to the sun, instead of a star, is called a synodical month. This is more than two days longer than the sidereal month ; for the moon's daily progress is about 13° ; and during the 27 days of its revolution, the sun, at the rate of 1° per day, will advance 27°, requiring more than two addi* tional days for the moon to overtake it. The mean length of the synodical month is 29.53 days. 90 moon's orbit. 155. Node*. — The points where the moon's }ath cats the circle of the ecliptic are called the moon's nodes. The ascend- ing node is the one through which the moon passes from the south to the north side of the ecliptic ; the other, 180° from it, is called the descending node. 156. The moon? 8 positions in relation to the sun. — The moon is said to be in conjunction with the sun, when both bodies have the same longitude ; in opposition, when their longitudes diifer by 180°. The conjunction and opposition are called by the common name of syzygies. When the longitude of the moon is 90°, or 270° greater than that of the sun, it is said to be in quadrature. The points midway between syzygies and quadratures are called octants. The period in which the moon passes from any one of these points to the same point again — that is, a synodical month — is also called a lunation. 157. To find the synodical month. — The synodical month is best obtained by comparing ancient and modern eclipses. An eclipse of the sun takes place at the time of conjunction. If then, the whole interval between the recorded date of a solar eclipse, which occurred before the Christian era, and the time of another, which occurred recently, be divided by the number of intervening lunations, the quotient is a very accu- rate expression of the mean synodical month. The mean synodical month, as thus obtained, is 29d. 12h. 44m. 3s. = 29.5306 days. 158. To find the sidereal month. — Dividing 360° by 365.25635, the number of days in a sidereal year, we have 0°.9856, the mean daily progress of the sun. Multiplying this by 29.53, the number of days in a synodical month, we find 29°.105, the arc passed over by the sun in that time. Now, the moon passes over 360° + 29°.105 in a synodical month, but only 360° in a sidereal month. Hence, we have the pro- portion, 360° + 29°.105 : 360° :: 29.53d. : 2T.32d. The sidereal month, more exactly, is 27d. 7h. 43m. lis. LIBRATION TN LONGITUDE. 91 159. Form of the moon's orbit. — It is ascertained by the