,,the ,,whys ,,and ,,wherefores ,,of ,,navigation ,,the ,,whys ,,and ,,wherefores ,,of ,,navigation ,,by ,,gershom ,,bradford #b;d ,navigating ,officer and ,senior ,instructor ,new ,york ,state ,nautical ,schoolship 8,newport10 ,late ,nautical ,expert1 ,hydrographic ,office1 ,u4 ,s4 ,n4 ,,illustrated ,illustration3 colophon ,,new ,,york ,d4 ,,van ,,nostrand ,,company #be ,,park ,,place #aiah ,copyright1 #aiah1 by ,d4 ,,van ,,nostrand ,,company ,,press ,,of ,,braunworth ,,co4 ,,book ,,manufacturers ,,brooklyn1 ,n4 ,y4 ,,to ,,the ,,american ,,sub-,,patrol ,,whose ,,war ,,zone ,,record ,,of ,,skill ,,and ,,daring ,,has ,,the ,,author',s ,,profound ,,respect ,,and ,,admiration ,,preface ,when taking into consideration the number of excellent works now published on navigation there would seem to be a small chance of finding a different angle from which to discuss the subject4 ,the purpose of the majority of such books is to give instruction to beginners4 ,this book1 however1 is written particularly for the men who do their work mostly by rote and wish to know more of the reasons2 or1 perhaps1 for schoolship graduates who may here find a chance to extend their horizons4 ,i have not considered it desirable to avoid repetition and in order to closely follow a line of thought have freely repeated many points already taken up4 ,this has in a number of cases avoided the distraction of seeking a page of reference elsewhere4 ,i have had in mind that it serve as supplementary reading to the ,american ,practical ,navigator1 ,bowditch1 that great bulwark of navigation which for over a hundred years has protected ,american ships through every deed of valor and every commercial adventure4 ,it is placed beyond criticism by its venerable name and its remarkable record and1 as a reference book for the navigator1 it stands without a peer1 but as a text book it founders the student4 ,if to such mariners as these a little insight is given to the 8,whys and ,wherefores0 of their work1 ,i shall be well repaid for the work of many watches below4 ,these discussions appeared several years ago1 in a less extended form1 in the ,master1 ,mate and ,pilot1 the magazine formerly published by the ,american ,association of ,masters1 ,mates and ,pilots4 ,i have freely consulted the following standard works3 ,american ,practical ,navigator1 ,bowditch2 ,wrinkles in ,practical ,navigation1 ,lecky2 ,the ,theory and ,practice of ,navigation1 ,dunraven2 ,general ,astronomy1 ,young2 ,navigation and ,compass ,adjustment1 ,muir2 ,guide to the ,marine ,board's ,examinations1 ,reed4 ,i have received and am grateful for very valuable help and suggestions from ,mr4 ,george ,w4 ,littlehales1 ,hydrographic ,engineer1 ,u4,s4,n41 ,mr4 ,felix ,riesenberg1 ,c4,e41 ,commander ,new ,york ,state ,schoolship ,newport1 and ,george ,a4 ,collie 7deceased71 ,nautical ,expert1 ,hydrographic ,office1 ,u4,s4,n4 ,g4 ,b4 ,,new ,,york1 ,april #ae1 #aiah4 ,,contents ,,page ,,chapter ,i ,,introductory ,,remarks #a ,,chapter ,,ii ,,nautical ,,astronomy #e ,,chapter ,,iii ,,declination ,,and ,,right ,,ascension1 ,,including ,,precession #ae ,,chapter ,,iv ,,time #cc ,,chapter ,v ,,corrections ,,for ,,observed ,,altitudes #fh ,,chapter ,,vi ,,latitude #gh ,,chapter ,,vii ,,azimuths ,,and ,,amplitudes #ic ,,chapter ,,viii ,,longitude #aja ,,chapter ,,ix ,,sumner ,,method1 ,,including ,,new ,,navigation #ajg ,,chapter ,x ,,the ,,moon #aci ,,chapter ,,xi ,,charts #adf ,,index #afa ,,the ,,whys ,,and ,,wherefores ,,of ,,navigation ,,chapter ,i ,introductory ,remarks ,in embarking upon the study of navigation familiarity with the compass is the first logical step3 the quick mental conversion of a course or direction given in points to the same direction in degrees expressed in quadrants as ,s4 #ce ,e41 and again into a system by which direction is indicated by degrees from #j to #cfj4 ,a mariner will encounter all three of these systems and will find constant conversion necessary back and forth for various purposes4 ,the #j to #cfj system is the most up-to-date and the simplest form of handling direction4 ,following the compass may be taken up the use and description of other nautical instruments with which every mariner is supposed to be familiar4 ,dead reckoning is the first calculation to appear and this involves the correction of the compass courses back and forth between true1 magnetic and compass directions4 ,this is dealt with under ,azimuths and ,amplitudes4 ,in practical navigation a vessel commences her voyage and attempts to sail in a certain direction1 but the well-known elements of compass error1 variation and deviation1 current1 wind1 seas and poor steering all divert the vessel from the projected course4 ,in dead reckoning a navigator strives to keep track of his position by keeping a record of actual courses steered and distances run4 ,he then is obliged to guess at the amount the vessel has diverted both in direction and distance until an astronomical observation sets him straight again4 ,it is here particularly shown that navigation becomes an art of estimating position and the better the navigator's bump of locality1 the greater his success4 ,this is a peculiar gift and usually is born in the man1 at least it cannot be learned from books4 ,the process of finding latitude and longitude by dead reckoning is supposed to be already well known to the reader and will not be detailed at length4 ,however1 every course angle is laid off from a meridian 7which is true ,n4 and ,s47 and terminates in a parallel of latitude4 ,this meridian and parallel intersect at right angles2 hence these with the distance run 7which is the hypothenuse7 form a plane right angle triangle1 plane because the curvature of the earth is not considered in short distances4 ,to solve this triangle1 we have the course angle and one side--the distance run4 ,with these the other two sides are easily found by computation1 but more easily by tables ,no4 #a and #b1 ,bowditch4 ,the side along the meridian is represented by the column headed ,lat4 7difference of latitude7 and the side lying in the parallel is in the column headed ,dep4 7departure74 ,thus the values of the sides of the triangle are given in miles and tenths1 showing the distance good made ,n4 or ,s4 and ,e4 or ,w4 ,it will be noted that at the top of the pages of these tables are four different courses and at the bottom are likewise four courses making the same page serve for eight different courses4 ,this is accomplished by the fact that triangles formed by these particular eight courses are the same in shape4 ,thus ,n4 #cj ,e41 for instance1 makes a similar triangle to ,n4 #cj ,w4 7#ccj72 ,s4 #cj ,e4 7#aej72 or ,s4 #cj ,w4 7#baj74 ,they have identically the same difference of latitude and departure4 ,if this fact is not clear draw a diagram and be convinced4 ,in the cases of ,n4 #fj ,e4 7#fj72 ,n4 #fj ,w4 7#cjj72 ,s4 #fj ,e4 7#abj72 ,s4 #fj ,w4 7#bdj71 the same shaped triangle as above is found1 but reversed in that what was the difference of latitude side now has become the departure side4 ,the values of these sides are read from the bottom of the page and are found in the reverse columns to fit the reverse triangle4 ,the latitude value read from the top of the page as #cj becomes a departure value when read from the bottom with #fj4 ,the subject of ,sailings is one of the early problems confronting the student of navigation and will be considered briefly4 ,the above remarks on dead reckoning cover the principle of plane sailing1 the simple method where the spherical surface of the earth is ignored and a flat ocean substituted4 ,this method will not serve for anything but short distances of a few hundred miles without sufficient error to render it impracticable4 ,traverse sailing is a series of plane sailing courses made1 for instance1 by a sailing vessel beating to windward4 ,in parallel sailing the vessel pursues a true ,e4 or ,w4 course and runs along a parallel of latitude4 ,thus all her progress is in the terms of departure with no difference of latitude4 ,as all meridians converge from the equator towards the poles the length in miles of a degree of longitude keeps on diminishing as the poles are approached and1 conversely1 miles of departure have an increasing value in degrees of longitude4 ,so in parallel sailing what we desire to know is what is the value in the particular latitude of our course of our departure 7miles7 in of longitude4 ,having this and applying it to the longitude left will give the longitude in4 ,middle latitude sailing is very similar to parallel sailing in that it is assumed1 for the purpose of getting the difference of longitude1 that the whole departure of the course or courses sailed has been made in the mean or middle latitude1 because the greater value 7in the northern hemisphere7 in difference of longitude of a mile northward of the middle latitude is counteracted by the corresponding lesser value southward of the middle latitude4 ,mercator sailing is perhaps the most extensively used1 as the ,mercator principle is employed almost universally in the construction of navigational charts4 ,it is described under ,charts in this book4 ,also under ,charts is a description of the ,gnomonic ,chart which is also called the ,great ,circle ,chart and used in ,great ,circle ,sailing1 also referred to in those pages4 ,the young navigator is counselled never to know where his vessel is1 lest through over confidence he be led into close and dangerous quarters4 ,,chapter ,,ii ,nautical ,astronomy ,the sun is the center of the solar system1 with all the planets1 including the earth1 revolving around it1 some with orbits greater and others less than that of the earth4 ,the planets in some instances have satellites revolving in turn around them1 as the moon in the case of the earth4 ,the movements of the earth will be1 perhaps1 more readily understood by assuming a position at the ,north ,pole2 here beneath the observer the earth is rotating left-handed--against the hands of a watch1 once each day2 while at the same time it is speeding onward through space in a left-handed curve1 which in the course of a year resolves itself into a complete revolution around the sun4 ,the sun and stars are considered to be stationary for all navigational purposes1 the apparent movements of these bodies being entirely due to the motions of the earth4 ,the result of our daily rotation from west to east is1 that the heavenly bodies march past our meridian in a majestic procession for #bd hours1 after which the performance is repeated4 ,the uninitiated are here forewarned against becoming confused by the assumption used for convenience by navigators that the heavens revolve around the earth4 ,the planets and moon join the sun and stars in the daily parade past our meridian1 but their apparent movements are not entirely an illusion1 for they have motions of their own that somewhat affect the precision of their daily revolution4 ,this is readily observed in the moon's hour of rising1 which is very perceptibly later each evening4 ,her actual revolution around the earth1 being from west to east1 is contrary to the apparent diurnal motion and thereby each evening finds her farther to the eastward and consequently rising later4 ,as a result of this change in the time of rising1 the moon must of necessity rise in every hour of both day and night in the course of a month--the time she requires to revolve around the earth4 ,the onward movement of the earth in its orbit as we face the sun in latitudes north of the tropics1 is toward our right1 and this causes the sun to apparently move slowly eastward or to the left among the stars1 corresponding exactly to our movement westward4 ,this movement is opposite to his daily course across the heavens4 ,as a year is required for the earth to accomplish our revolution around the sun1 it follows that this same length of time is consumed by the sun in making its apparent eastward revolution of the heavens4 ,the movements of the planets are more complex4 ,they all revolve around the sun in the same direction as the earth1 but as their orbits are of vastly different sizes1 they will be found in various positions relative to the sun2 they overhaul and pass each other1 but owing to their uniform direction of revolution they never meet4 ,the planetary system is like the horse race at a county fair2 the pole horse has the advantage1 but the varying speeds of the contestants soon place them at various parts of the track4 ,from the earth the movements of the planets1 aside from the diurnal movement1 are composed of their own actual movement around the sun1 combined with an apparent motion1 due to the earth's onward movement precisely as described above in the case of the sun4 ,the combined movement of a planet may be noted by reference to the fixed stars beyond it4 ,the positions of heavenly bodies are determined by two measurements--coordinates--the distance north or south of the celestial equator1 called the declination1 and the distance east from the prime celestial meridian taken as a reference1 called the right ascension1 each of which will be subsequently treated at length under its individual heading4 ,the movement of the planets eastward or westward as described1 constantly changes their right ascension2 and as their orbits are inclined at different angles to the celestial equator1 they are always changing their declination4 ,the planets whose orbits are smaller than that of the earth are called inferior1 while those whose orbits are of greater dimensions are known as superior planets4 ,mercury and ,venus are inferior planets and consequently are always nearer the sun2 their comparatively close proximity making them appear to us as morning and evening stars4 ,in fact1 ,mercury is so close that it is unavailable1 owing to the brilliancy of the sun1 for observation with a sextant2 while ,venus1 on the other hand1 a little more remote1 is an excellent body to observe1 and is always found in the east or west1 conveniently near the prime vertical1 the most favorable place for a time sight for longitude4 ,the twilight or dawn which usually prevails at the time of a ,venus sight gives the navigator a good horizon to observe upon4 ,mars1 ,jupiter and ,saturn are superior planets and their travels are so extended that they may be found almost anywhere in the heavens within the limits of their declinations4 ,the earth's orbit is slightly elliptical1 with the sun located a little out of center--a little nearer one end4 ,should a line or axis be drawn through the long diameter1 its intersection with that part of the orbit nearest the sun is called the ,perihelion while the opposite point is known as the ,aphelion4 ,the former is used as a point of reference from which the earth's position can be located in terms of angular measurement from time to time4 ,this angle1 known as the anomaly1 is formed by the line from the sun to the ,perihelion and that drawn from the sun to the earth4 ,the latter distance is called the radius vector of the earth4 ,we 7the earth71 are at the ,perihelion about ,january #a1 and consequently this angle at that date is #j1 but from this time on1 the angle increases approximately one degree a day throughout the year4 ,the plane of the earth's equator makes at all times an angle of about #bc #bh with the plane of its orbit4 ,this is a highly important angle to mankind1 for upon it depends the climate of the world4 ,the axis of the earth1 if we can conceive it as represented by a slender imaginary staff1 extends through the unlimited distance to a point in the heavens--the celestial pole2 this point is in the zenith for a person at our north pole4 ,since the distance between these points is mathematically infinite1 any number of lines parallel to this 8staff0 will appear to penetrate the sky at the single point of the celestial pole4 ,thus the parallel positions of the axis corresponding to the earth's various positions1 even those at opposite sides of the orbit1 converge into this common point4 ,to be clearer1 the parallel lines representing the different positions of the axis during the year according to our geometry form a group of separate points on the heavens1 but the distance being beyond all reckoning1 our limited conceptions fail to identify the group of points and it resolves into one point4 ,by the same line of reasoning the plane of the earth's equator remains parallel in all its positions throughout the yearly cruise around the sun1 and its projection marks but one celestial equator upon the sky4 ,while the direction of the axis and corresponding position of the equator are constant for all practical purposes1 there is1 nevertheless1 an extremely slow circular movement of the axis1 called the precession of the equinoxes1 a subject which is reserved for subsequent discussion4 ,,coordinates ,in nautical astronomy the earth is assumed to be the center of space with the heavens forming a globular shell around it1 known as the celestial sphere4 ,all fixed stars are assumed to lie on its concave surface from the earth regardless of their actual distances4 ,the tracks of all other bodies moving1 or appearing to move1 across the sky are considered to be on the surface of this sphere4 ,it is necessary1 in order to conveniently define the position of heavenly bodies to mark this celestial sphere with imaginary circles to serve as coordinates1 as we mark the earth with meridians of longitude and parallels of latitude4 ,before going into the explanation of these coordinates1 it may be well to consider a few facts concerning circles4 ,a great circle is of course understood to be one whose plane passes through the center of a sphere1 dividing it into two equal parts4 ,there can be an infinite number of these circles whose planes cut the sphere at every possible inclination as long as they pass through its center4 ,a circle may be a great circle of either the celestial sphere1 the earth1 or even of a baseball4 ,the poles of a great circle are the points on the surface of its sphere1 penetrated by the diameter perpendicular to the plane of the great circle4 ,as for example1 the poles of the earth are connected by the diameter that is perpendicular to the plane of the equator4 ,an angle at any pole is measured on the great circle which subtends it4 ,for instance1 angles at the poles of the earth are measured on the equator2 angles at the zenith on the horizon4 ,with these facts well in mind we will proceed1 showing the scheme of circles employed in laying off the surface of the heavens4 ,there are three systems of circles1 each designed to fulfil a different requirement4 ,the first system depends upon the position of the observer and changes its whole imaginary structure to correspond with his movements4 ,the plumb-line1 if extended to the heavens overhead1 will determine the zenith1 the point of origin of this system on the celestial sphere4 ,the corresponding point directly beneath us is known as the nadir4 ,the great circle of the celestial sphere everywhere equally distant from both the zenith and the nadir is the horizon4 ,it is plain that a new zenith and new horizon are created with every movement of the observer4 ,the facts that man is on the surface and not at the center of the earth1 and that his eye is elevated above its surface1 each creates another horizon4 ,the rational horizon is marked by a plane1 perpendicular to a plumb-line and passing through the earth's center2 while the sensible horizon is determined by a plane1 also perpendicular to the plumb-line1 but passing through the eye of the observer4 ,it will therefore be seen that these two parallel horizons are some #djjj miles apart1 the semi-diameter of the earth2 but this distance when projected on the celestial sphere becomes insignificant when compared with the infinite distance of this sphere from the earth1 and the rational and sensible horizons shrink into a single line so far as we can perceive4 ,while this statement is true when dealing with the stars1 it needs modifying when dealing with the sun and moon1 and in very accurate observations of planets1 as their distances are insufficient to eliminate the angle formed between the line from the body to the center of the earth1 and that from the body to the observer4 ,this is allowed for when observing these bodies by applying the correction of parallax to the observed altitude4 ,the visible horizon is the boundary seen between the sea and sky4 ,if the observer's eye were at the level of the sea1 his visible horizon would coincide with the sensible horizon1 defined above2 but the elevation above the surface from which sights are taken causes the line of vision1 tangent to the sea1 to be depressed below the plane of the sensible horizon making an angle with it called the dip of the horizon4 ,in practice all altitudes of heavenly bodies taken from a vessel are measured to the visible horizon and corrected for the dip to reduce them to the sensible horizon1 then again corrected for parallax to obtain the true altitude of the body above the rational horizon2 or what is the same thing1 the altitude as observed at the center of the earth4 ,from the zenith1 an infinite number of great circles1 known as vertical circles1 sweep around the celestial sphere1 cutting the horizon at right angles and passing through the nadir4 ,the one which cuts the north and south points is called the celestial meridian1 and is evidently a projection of the terrestrial meridian4 ,the vertical circle passing through the east and west points is called the prime vertical1 and has a distinction above other vertical circles by virtue of its being the most favorable position for a body in observations for longitude4 ,the heavens are further swept by an infinite number of parallels of altitude which are1 as their name implies1 parallel to the horizon4 ,the azimuth of a body is its angular distance from the north or south points of the horizon1 determined by the angle formed at the zenith1 or by the arc of the horizon between the meridian of the observer and the vertical circle passing through the body4 ,amplitude is the angle at the zenith formed by the prime vertical and the vertical circle passing through the body or it is the angular distance from the east or west points1 measured on the horizon1 and is measured similarly to the azimuth4 ,the system of laying off the heavens just described is well enough for the momentary locating of a body1 which is a very important feature in navigation1 but for some purposes a more stable point than the zenith1 which on shipboard is ever changing1 is needed1 from which to form a system that is constant the world over4 ,to meet this demand we take the point in the sky penetrated by the prolongation of the earth's northern axis--the celestial pole--and from this point meridians and parallels are developed upon the celestial sphere as has been done on the earth4 ,in fact1 it is as though these terrestrial coordinates were projected to the heavens where they hold the same relative positions as upon the earth4 ,the northern celestial pole is in the zenith at our north pole4 ,the same is true of the south pole4 ,the celestial equator1 or equinoctial1 is a great circle1 which is midway between the poles and everywhere #ij from them4 ,it marks the termination of the plane of our equator extended to the celestial sphere1 or in other words1 it is always directly over our equator4 ,the parallels of the celestial sphere evolved by this system1 corresponding to those of latitude upon the earth are called parallels of declination1 while the celestial meridians having as their point of origin the prolongation of the earth's axis1 are known as hour circles4 ,the particular hour circle passing through the zenith is one and the same with the celestial meridian4 ,it will be seen that this circle must pass through the zenith1 nadir and the poles4 ,the angle formed at the pole by the hour circle passing through a body and a local meridian is the hour angle of that body1 and is measured westward through #bd hours1 although ,a4,m4 hour angles of the sun are reckoned eastward through #ab hours4 ,at the north pole where the zenith is identical with the celestial pole1 the vertical circles1 parallels of altitude and rational horizon are coincident with the hour circles1 parallels of declination and the equator1 respectively2 but departing from this point they form angles with each other corresponding to the degrees of latitude from the pole2 at the equator the angle reaches #ij4 ,the system of circles described above is by far the most extensively used1 and positions determined by its coordinates are comparatively constant1 but there is still a third system of circles which was used and handed down to us by the ancients4 ,in the place of the celestial equator1 a similar great circle is used1 known as the ecliptic4 ,this circle is determined by the extension of the plane of the earth's orbit to the celestial sphere4 ,the poles of the ecliptic everywhere #ij from this circle are the points from which meridians depart as upon the earth4 ,the prime meridian of this system passes through the intersection of the celestial equator1 and the ecliptic--the vernal equinox or ,first ,point of ,aries4 ,celestial latitude and longitude are the coordinates used with this system1 but navigators universally prefer to use the well-known declination and right ascension4 ,hence the path of usefulness of the former seldom leads beyond the observatories4 ,,chapter ,,iii ,declination and ,right ,ascension ,owing to the important place that declination holds in nautical astronomy1 a detailed explanation will appropriately follow closely in the wake of the preceding remarks4 ,it must be made clear1 before getting under way1 that declination is the distance1 in degrees1 minutes and seconds1 of a body north 77 or south 7-7 of the celestial equator measured on the hour circle passing through the body4 ,this distance is identical with the latitude of the place in the zenith of which the body happens to be4 ,what declination is to a body in the heavens1 latitude is to the place on the earth directly beneath it4 ,the declination of fixed stars changes very slowly from month to month1 but the planets meander about on the celestial sphere in a way that is liable to puzzle anyone other than an astronomer4 ,this element1 however1 is worked out in the observatory and given in the nautical almanac in a way that relieves the navigator of worry concerning the complex movements of these latter bodies4 ,the same may be said of the moon1 but the subject will be treated1 somewhat superficially though sufficiently for the needs and desires of the practical mariner1 in a special talk on the moon4 ,this eliminates all the celestial bodies except the sun1 the most important2 and for this reason the facts relative to its declination will be considered at some length4 ,as has already been stated1 the sun is stationary1 but our movements around it to the right causes it to appear to move to the left2 precisely as you see1 when under way1 an anchored vessel's masts move to the left along the land behind her1 while you move on to the right4 ,we have no landmarks behind the sun by which to observe his apparent movements1 so in lieu of such ranges1 we resort to the fixed stars1 which serve as excellent marks to get a bearing on ,old ,sol and keep tab on him as he moves eastward among them4 ,this movement must in no way be confounded with his apparent daily motion westward4 ,as an illustration1 we may see ,orion--a familiar friend--swinging high in the western sky in the early evening2 some weeks later he is riding low1 and yet a little later still1 he is swallowed up in the brilliancy of the setting sun4 ,in other words1 the sun and ,orion have approached and passed each other4 ,we know ,orion does not move1 for he is composed of fixed stars1 and this seeming westward movement of his is in reality the apparent eastward marching of the sun1 which is due to the earth's movement of revolution4 ,the sun in this apparent movement eastward follows a course at a rate equal to that of the earth1 along a great circle of the celestial sphere called the ecliptic1 a circle that plays an important part in the explanation of declination1 particularly that of the sun4 ,the ecliptic is marked by the extension of the earth's orbit to the celestial sphere4 ,a few more words concerning great circles will be introduced here1 and the following statements1 while they apply to great circles in general1 especially fit the relationship of the equinoctial or celestial equator to the ecliptic4 ,these two great circles cut each other at an angle of #bc #bh4 ,great circles always bisect each other1 and hence any two great circles of the celestial sphere1 regardless of the angle they may take with the celestial equator1 must intersect each other at exactly opposite points1 #ahj apart4 ,what is true in this regard of the celestial sphere is equally true of the great circles of the earth4 ,a vertex of a great circle is the point which departs the greatest distance from the equator--the highest point of the circle reached in declination4 ,there are two vertices #ahj apart with the two points of intersection #ij in either direction4 ,the declination or latitude of either vertex is equal to the angle at which the circles intersect each other4 ,the intersections are called the equinoxes1 and it may be well to say here that the word equinox has several meanings in navigation1 often rendering it necessary to judge by the text which is intended4 ,the vernal equinox1 for instance1 refers to a certain time of year--,march #bast4 ,the sun is that day directly overhead at the intersection of the equator and the terrestrial ecliptic and this point is sometimes called the vernal equinox4 ,again1 the sun at the same time occupies a point on the heavens also known as the vernal equinox1 which is at the intersection of the celestial equator and the ecliptic4 ,the point in the orbit occupied by the earth at this time is also spoken of as the vernal equinox4 ,the reader is now asked to arouse his imagination and if possible to conceive himself a passenger in an aeroplane equipped with some remarkable power capable of carrying him to a position in space1 above1 yet a little outside1 the earth's orbit1 near the ,perihelion1 and there to heave to and view awhile an astronomical picture4 ,spread out before his unrestricted vision will be the earth1 its orbit1 and the sun4 ,it is to be hoped that the imagination of the reader is still sufficiently supple to suppose the plane of the orbit to be the surface of an infinite ocean stretching away beyond human conception of distance and 8breaking0 against the celestial sphere2 the 8surfline0 there marks the ecliptic2 the 8ocean's0 surface representing the great plane of the ecliptic4 ,the sun will be seen as if at anchor in his proper place within the orbit4 ,the earth is 8underway10 half submerged1 and listed #bc #bh toward our point of vantage4 ,this inclination1 or direction of the axis1 is in a general way toward the perihelion1 and within a few degrees of being parallel with the long diameter of the orbit4 ,the earth maintains this nearly parallel position of its axis with the long diameter throughout the period of its revolution2 a fact of importance to remember4 ,it will be readily seen that during the encircling of the sun there must be one position where the northern axis is inclined directly toward that body1 another opposite where it is headed away from him1 and two positions midway where the bearing of the axis 7projected on the plane of the orbit7 is at right angles to the bearing of the sun from the earth2 another feature to be 8salted down0 in the memory4 ,if the earth revolved on an even keel1 the equator and the 8waterline0 would be coincident1 but fortunately this is not the case1 and owing to the inclination of the axis another great circle is defined by the 8waterline10 called the terrestrial ecliptic1 being directly beneath its celestial namesake4 ,the inclination of the northern pole being in a general way toward the perihelion1 correspondingly depresses or 8submerges0 that half of the equator below the plane of the ecliptic1 represented by the 8water surface10 and at the same time the opposite side rolls the equator above it4 ,at two points 7the equinoxes7 on opposite sides of the earth1 and at right angles to the direction of its inclination1 the equator and terrestrial ecliptic cross each other at the 8water's edge40 ,the sun is always exactly overhead for that point of the earth which is nearest to it4 ,this is an essential fact to remember in navigation4 ,bearing in mind that the sun is stationary and ignoring for a time the rotation of the earth1 each advance in its orbit brings about a change of bearing of the sun and a new position becomes the nearest point1 and thereby directly beneath the sun4 ,the constant changing of the sun's bearing continues throughout the year1 or one revolution1 and a circle of these overhead positions is marked upon the earth1 which is coincident with the terrestrial ecliptic--the visionary 8waterline40 ,it is obvious that the vertical rays of the sun must apparently follow this line1 for it can only be overhead for places that are in the same plane1 and this again is the level of the 8ocean40 ,this circle of overhead positions projected on the celestial sphere marks the ecliptic--the 8margin0 of the infinite ocean1 and the path that the sun seems to follow eastward among the stars4 ,the above paragraphs show us that the sun in following this line around the earth crosses the equator twice1 and twice he attains a distance of #bc #bh from it1 and so must be on the equator twice and reach a declination of #bc #bh north and #bc #bh south in the course of one year4 ,returning to our imaginary illustration1 we will now follow the peregrinations of the earth for a year and note the effect of its inclination in the different parts of the orbit upon the declination of the sun4 ,it will be assumed that it is the #bast of ,march and from our airy position we see the earth away on our right nearly #ij from the ,perihelion4 ,as this is the vernal equinox1 there are a number of interesting points to be considered3 ,the direction of the earth's axis1 projected on the plane of the orbit1 is at right angles to the bearing of the sun from the earth2 the sun is directly over the intersection of the equator and terrestrial ecliptic1 and being overhead for this point on the equator1 the declination must be #j4 ,moreover1 a line drawn from this intersection1 or terrestrial vernal equinox1 through the center of the sun and extended to the celestial sphere would strike the corresponding intersection of the ecliptic and the equinoctial or celestial equator--the celestial vernal equinox4 ,the arrival of the earth at this position is the signal of spring for the northern hemisphere1 likewise it announces the advent of autumn to our southern neighbors below the 8,line40 ,the sun this day rises in the east 7approximately7 and passing through the zenith1 sets in the west for those living on the equator4 ,the explorer at the north pole is cheered by the first light as the sun appears in the horizon1 while the south pole becomes enshrouded in the long ,antarctic night4 ,without lingering for ceremonies over the change of seasons1 the earth continues steadily on its way toward the aphelion2 the sun's vertical rays leave the intersection of the equator and the terrestrial ecliptic1 and follow along the latter1 thus widening its distance from the equator as the earth proceeds4 ,as the ecliptic in this half of the orbit is above1 or north1 of the equator the former is in north latitude and the sun1 following along it1 is thereby also in north declination4 ,a line from any place having the vertical rays1 through the sun to the celestial sphere1 always terminates on the celestial ecliptic1 all being in the same plane1 and shows the corresponding celestial position of the sun on it4 ,its declination distance from the celestial equator1 in degrees1 minutes and seconds1 is identical with that of the place on the earth directly beneath it relative to our equator4 ,so by showing the course of the sun's overhead positions on the earth its celestial positions are1 at the same time1 indicated4 ,the overhead position of the sun on the terrestrial ecliptic gradually departs from the equator culminating about ,june #bast1 the summer solstice1 in a declination of #bc #bh at a point near the aphelion in the orbit1 #ij 7approximately7 from the equinox4 ,the positions in the orbit of the summer and winter solstices are reached by the earth several days before the points of the aphelion and perihelion4 ,these respective positions would be in conjunction were it not for a slow and remarkable motion of the earth's axis before spoken of1 and later to be described1 called the precession of the equinoxes4 ,the summer solstice is the great half-way point of the earth's annual circumnavigation of the sun2 it is a matter of moment all over the world1 and another great change of seasons is at hand4 ,the sun is overhead for places along the parallel of #bc #bh ,n4 and bears north #bc #bh from the zenith at noon from places on the equator4 ,at the north pole1 since its appearance on the horizon on ,march #bast1 the sun has mounted to an altitude of #bc #bh and to nearly #fg at places on the ,arctic circle4 ,the earth's northern axis is1 in this position1 inclined #bc #bh directly toward the sun1 which pours its rays continuously upon the northern regions1 uninterrupted even by the earth's daily rotation4 ,it is on this day that the whole ,arctic zone enjoys the full glory of the midnight sun4 ,the earth's continuous movement of revolution does not allow a delay of this favorable season in northern latitudes1 but continues to make the sun's vertical rays follow the terrestrial ecliptic as before on its way toward the intersection with the equator #ij away4 ,on this leg of the journey1 the sun is traveling on the upper one of two converging lines and thereby gradually lessening its distance from the other--the equator--or1 in other words1 reducing its declination4 ,this continues until ,september #bast when the autumnal equinox is reached and the sun's declination becomes #j4 ,the sun now being overhead at the intersection of the equator and the terrestrial ecliptic1 is on the opposite side of the earth from the intersection of ,march #bast4 ,in fact the conditions are similar1 but now the earth is on the opposite side of the sun1 and the change of seasons is the entrance of spring for the dwellers in southern latitudes4 ,the sun has dropped lower and lower in the sky at the north pole since ,june1 until on this day it is in the horizon and it is time for the ,esquimos to seek their igloos and prepare to hibernate during the long ,arctic night now ushered in4 ,the sunshine at the time of the equinoxes is equally distributed over the northern and southern zones1 and the zenith distance of the sun at noon at any place is1 theoretically1 equal to the latitude of the place 7except a small error due to change of declination accumulated subsequent1 or previous1 to the instant of the equinox74 ,the conditions during the next six months are reversed as the earth proceeds into that half of the orbit containing the perihelion4 ,now the sun following the terrestrial ecliptic enters southern latitudes or south declination1 for in this part of the orbit the equator is above 7or north7 the plane of the ecliptic4 ,the sun's diverging course from the equator leads it farther and farther southward until on or about ,december #bast it arrives at the winter solstice with a culmination of #bc #bh south declination4 ,at this point the earth is but a few degrees from the perihelion as it was from the aphelion at the summer solstice4 ,the earth's north pole is now inclined directly away from the sun and its rays have entirely forsaken the ,arctic for the ,antarctic zone2 notwithstanding the earth's daily rotation1 which brings alternating light and darkness to the greater part of the world1 the northern polar regions are in a continuous shadow1 and no sunlight reaches these remote parts4 ,at this time of the year the northern hemisphere above the tropic of ,cancer1 is in an unfavorable position relative to the sun1 and as a result places situated on parallels less remote than the ,arctic are having long nights and short days in proportion to their latitude north4 ,on the other hand1 in the southern hemisphere the days are longer and the nights shorter1 as the southern latitude increases until at the ,antarctic circle night disappears and the sunshine is uninterrupted4 ,it is seen that this is an exact reversal of the conditions at the summer solstice4 ,the earth enters the last quadrant of the great ellipse of its orbit1 the sun now approaches the equator as the earth nears the vernal equinox4 ,the south declination diminishes until on ,march #bast it becomes #j and the earth has completed its revolution4 ,we will now go on another tack and instead of considering only the effects of declination due to the earth's revolution1 will assume that the earth has been halted in its onward course of revolution and is making its daily rotation in the same position4 ,the earth turning from west to east causes the sun to appear to proceed from east to west in its diurnal motion4 ,each rotation1 requiring #bd hours1 marks upon the earth a circle of overhead positions parallel to the equator and hence without change of declination4 ,the result of such a remarkable condition would be1 no change of seasons and no change in the length of the days and nights4 ,in reality1 however1 we are saved from such monotony1 for both the motion of rotation and revolution of the earth are acting together and giving a compound effect on the apparent movements of the sun4 ,this alters the daily circles just mentioned to a fine spiral of overhead positions1 ever changing in declination4 ,the daily difference of the sun's declination shown in the ,nautical ,almanac is equivalent to the distance between two threads of this spiral4 ,the change of declination is most directly seen and felt in the polar regions1 where the activities of the denizens are mostly limited to the favorable phases of this change4 ,at the north pole1 after the sun has appeared above the horizon1 this spiral of declination can be continuously followed4 ,the sextant will disclose a constant increase in altitude as the sun circles round and round the sky1 winding itself up and finally culminating at #bc #bh4 ,the process is then immediately reversed4 ,the stars here make daily circles of equal altitudes as their change of declination is insignificant2 but the circles of the planets and the moon are converted into spirals1 the fineness of which is in proportion to the rate of their change of declination4 ,the fact that the sun reaches an altitude of #bc #bh at the pole at the summer solstice with its declination of a like amount and that on ,march #bast1 when the sun is in the horizon with the altitude #j1 it is directly over the equator with #j declination1 shows that at this place 7the pole7 the altitude is equal to the declination4 ,should an explorer travel southward #a1 his sextant would show an altitude #a greater than at the pole1 yet moving about does not affect the declination at a given time4 ,it follows by taking his altitude at noon the explorer in the polar regions may readily learn his distance from the pole by subtracting the declination in the ,nautical ,almanac from his sextant reading4 ,it may not generally be known that the southern summer is shorter than the summer of the northern hemisphere1 but such is the case by approximately eight days4 ,the reason of this inequality lies in the fact that the sun is nearer one end of the orbital ellipse1 and the short diameter passing through this body divides the orbit into unequal parts4 ,the smaller part being that traveled by the earth during the southern summer4 ,furthermore the nearer proximity of the sun causes an accelerated motion which further tends to lessen the time spent by the earth in this part of the orbit4 ,,right ,,ascension ,declination and right ascension being used together as coordinates1 we will not separate them4 ,it will be remembered that the equator and the terrestrial ecliptic cross each other on opposite sides of the earth2 that on or about ,march #ba1 the sun is overhead at the intersection that is the vernal equinox4 ,now if at this intersection on this day a plumb-line were carried upward1 it would at length reach the sun1 and continued to infinity and projected on the celestial sphere would locate a point called the celestial vernal equinox1 known by many as the ,first ,point of ,aries4 ,this point is one of the most important celestial 8landmarks0 used in astronomy and navigation1 but1 unfortunately1 no heavenly body marks its place4 ,however1 as its relative position among the neighboring stars is well known1 its exact location is easily ascertained4 ,the hour circle which passes through this point is known as the equinoctial colure1 and may be considered the prime meridian of the heavens1 for from it is measured the right ascension of all bodies4 ,right ascension of a body is the angle at the celestial pole between this meridian of reference and the hour circle passing through the body4 ,it is always measured eastward through #bd hours of sidereal time 7#cfj74 ,the angle is measured by the arc intercepted on the celestial equator4 ,for example1 a star #ae east of the equinoctial colure has a right ascension of #a hour or #ae1 but1 if the star is #ae west1 its right ascension is #bc hours or #cde4 ,the positions of heavenly bodies are defined by right ascension and declination exactly as positions upon the earth are expressed by longitude and latitude1 right ascension corresponding to longitude and declination to latitude4 ,in the discussion of ,time1 to follow1 more facts concerning right ascension will be found4 ,,precession ,,of ,,the ,,equinoxes ,a comparison of the present positions of the fixed stars with their places as recorded in ancient times shows a great discrepancy4 ,the celestial latitudes1 which were reckoned from the ecliptic1 show no appreciable change2 but in the declinations and right ascensions there is a great departure from the old positions4 ,the error of right ascension was found by the old ,greek astronomer1 ,hipparchus1 to appear as a uniform eastward movement of all the stars1 which led him to reason that1 instead of the stars themselves changing1 their point of reference was moving westward1 thus lengthening all right ascensions4 ,the famous astronomer after further reasoning decided that the position of the celestial pole was changing1 in fact that the line of the earth's axis was describing a circle on the heavens1 which was left-handed or against the hands of a watch as viewed from the north pole of the earth4 ,this movement was found to be extremely slow1 requiring #be1hjj years to complete the circle which has as its radius the amount of the inclination of the earth's axis--#bc #bh4 ,if a match is put through a piece of cardboard about the size of a half dollar to the distance of inch1 and spun1 the motion of the cardboard just as it staggers through loss of speed1 gives some idea1 although exaggerated1 of the precession movement of the plane of the equator1 which is of course infinitely slower4 ,the movement of the top of the match is a semblance of the corresponding motion of the vanishing point of the axis on the celestial sphere4 ,the earth1 as already explained1 points its axis at practically the same spot in the heavens throughout the year1 and if it were not for this annual precession of #ej it would for all intents and purposes hold a permanent direction4 ,about ,december #ba1 the winter solstice1 while the earth is still some degrees from the perihelion1 its northern axis1 is inclined directly away from the sun4 ,each year this distance from the perihelion is becoming greater1 widening this angle between the direction of the axis1 projected on the plane of the orbit1 and the major diameter of the orbit1 until in time the north pole will be headed directly away from the sun in that part of the orbit which the earth now occupies in ,september1 and so on4 ,illustration3 ,this diagram shows the successive positions of the earth at the ,vernal ,equinox 7,march #bast7 due to the revolution of the axis and the consequent westward movement of the ,first ,point of ,aries4 ,,fig4 #a4 ,in the year ,a4,d4 #abej the winter solstice occurred at perihelion and in the year #fdjj ,a4,d4 the vernal equinox will occur at this point of the orbit4 ,that is1 the axis of the earth was inclined directly away from the sun at perihelion in the former year but in the latter year the inclination will have changed about #ij backward against the earth's course about the sun1 and it will be the beginning of spring 7the vernal equinox7 when the earth is at perihelion instead of the first of winter as in #abej ,a4,d4 ,since #abej ,a4,d4 the inclination has changed an equivalent of about #aa days for now the earth is at perihelion about ,january #ast1 and the solstice occurring about ,december #bast1 shows the present relative situation4 ,in other words1 the vernal equinox is slipping back in the orbit towards the perihelion1 and as the solstices maintain their positions at #ij from the equinoxes they must likewise be 8slipping a cog0 each year4 ,the vernal equinox was situated many centuries ago in the first part of the constellation of ,aries1 and was known as the ,first ,point of ,aries1 but owing to the movement of precession it has dropped back or westward 7as we face our southern horizon7 #ej a year until it has left that constellation entirely and is now about leaving the constellation of ,pisces1 some #cj from the position used by ,hipparchus in his calculations4 ,the majority of navigators still call this point of the celestial vernal equinox the ,first ,point of ,aries4 ,holding these facts in mind1 it may be clear that as the earth approaches that part of the orbit where the vernal equinox occurs it has turned its pole1 and correspondingly points on the equator1 #ej to the right or west during the year2 thus causing the point of the terrestrial equinox to meet 7or come under7 the sun that much sooner4 ,in other words1 referring to the effect as seen on the heavens1 the celestial equinox was advanced to the westward that much to meet the sun in its eastward movement among the stars and will become the nearest point to the sun1 #ej before the position of the equinox of last year4 ,as the points in the orbit where the vernal equinox occurs year by year works back toward the perihelion1 the range line through the sun to the heavens beyond must each year correspondingly edge its way westward along the celestial ecliptic through different constellations4 ,this is what is known as the precession of the equinoxes4 ,the course of the celestial pole in the heavens is shown by a circle drawn about the pole of the ecliptic using #bc #bh as a radius4 ,this path will pass #a from our pole star and this position marks the present termination of our extended axis2 half way around the circle it passes the first magnitude star ,vega close aboard1 thus making this the future pole star some #ab1jjj years hence4 ,if there be such creatures as navigators in those far-away days1 latitude by ,vega will no doubt be a popular sight among them4 ,the cause of this remarkable movement of the earth is due to the fact that the earth is not a true sphere1 and the influence of the sun is not exerted equally upon its mass4 ,its flattening at the poles is attended by a corresponding bulging along the equatorial belt4 ,when the earth is in the vicinity of the perihelion1 leaning away from the sun1 the half of this ring of extra matter on the side towards the sun is above the plane of the ecliptic or orbit4 ,the tendency of the added attraction exerted upon it1 is to draw the earth to an upright position1 or in other words1 at this time the sun is pulling stronger on the northern or upper side than on the lower4 ,again1 when near the aphelion and summer solstice1 leaning towards the sun1 that part of the ring of extra matter on the side towards the sun is below the level of the orbit1 and the attraction is again tending as before to pull the earth upright4 ,at the equinoxes there is an equal amount of this extra matter above and below the plane of the orbit evenly distributing the attraction4 ,the effect of this influence would in time bring the earth's equator and the plane of the ecliptic into coincidence and the earth's pole would be directly beneath the pole of the ecliptic1 were it not for its rotation4 ,the two forces acting upon the earth result in the slow revolution of the axis4 ,the exact effect of these forces is rather complex but it is a demonstration of the principle of the gyroscope4 ,the movement of the axis is affected very slightly by other influences than that of the sun1 the most notable of which is the moon1 whose monthly revolutions around the earth produce a similar influence in the bulging mass within the tropics1 but as its revolutions are so rapid1 it has but a slight effect on the precession movement of the earth4 ,it is sufficient1 however1 to cause the extended axis to nod slightly and make a waved circle of precession on the heavens4 ,this is called 8,nutation10 from the ,latin word nuto1 meaning to nod4 ,,chapter ,,iv ,time ,a thorough understanding of time1 one of the most important elements in navigation1 clears the way to a better idea of the theory of finding one's position at sea2 there is1 in the minds of many1 considerable fog hanging about certain portions of this subject1 and it is hoped that this explanation will clear some of this away4 ,worcester's ,dictionary defines time as measured duration4 ,it is the interval between events4 ,it flows ceaselessly and with uniformity1 yet the mortal mind is unable to conceive its beginning or its end4 ,man1 in order to measure his activities1 has blocked it off into different denominations convenient for his uses4 ,of these1 the navigator uses the following in his determinations and reckonings3 years1 months1 days1 hours1 minutes and seconds4 ,certain astronomical phenomena were naturally enlisted by the ancient astronomers to furnish standards for time measurements2 the value of a year was determined by the time necessary for a complete revolution of the earth around the sun1 while the length of a day was fixed by the time of a rotation of the earth on its axis4 ,the precision with which these evolutions are accomplished gives the required accuracy4 ,the revolution of the earth governs the change of seasons1 while the rotation is responsible for the alternating periods of day and night4 ,with the exception of the month1 the other measurements of time mentioned above are denominations of these standards4 ,the month1 one-twelfth of a year1 is measured by the revolution of the moon around the earth4 ,solar time1 as its name implies1 is measured by the apparent diurnal movement of the sun4 ,it is the variety of time in universal use by which is regulated the daily activities of life2 and this is indeed quite natural1 for of necessity the work and play of the world depend upon the light and darkness that this body serves out to us4 ,while we are unconscious of the earth's rotation1 its effect is seen in the apparent daily course of the sun across the heavens1 caused by our turning past it1 yet in common practice the sun is assumed to revolve around the earth1 and is usually thus spoken of for the purpose of simpler explanation4 ,the time at each meridian is necessarily different from that of every other1 as only one of them holds the same position relative to the sun at the same time or putting it in another way1 only one meridian can cross the sun at the same time1 determining local noon for those places located upon it4 ,it is forenoon for that part of the world westward of the sun and afternoon for that portion eastward of it4 ,as the earth turns from west to east1 the places or meridians to the eastward are first favored with the sun's light1 and those meridians cross this body before those to the westward4 ,the sun apparently moves from the eastward to the westward1 crossing each meridian in succession until in a few hours it is afternoon for places to the eastward and noon with us4 ,the sun is now in our meridian1 and it is forenoon for people to the westward of us4 ,for example1 at #g ,a4,m41 #geth meridian time1 it is noon in ,england and dead of night in our ,pacific ,coast2 at our noon 7#geth71 it is late afternoon in ,england and breakfast time in ,california4 ,it requires #bd hours1 solar time1 for the sun to make its apparent revolution around the earth1 this course being a circle2 it contains #cfj of arc4 ,it follows that in one hour it passes over #ae of arc1 while #d minutes are required for #a to be traveled4 ,thus it is evident that any arc of the circumference of the earth1 or difference of longitude1 which is the same thing1 has an equivalent time value and vice versa4 ,that is1 the arc comprised between the meridian of ,greenwich and the #fjth meridian west1 for instance1 besides being measured as #fj ,w41 is equal to #d hours of time4 ,again #d hours of ,greenwich time indicates that the sun has crossed the ,greenwich meridian #d hours ago and is at that particular instant crossing the meridian #fj west of ,greenwich4 ,if the arc were between ,greenwich and a place #fj ,e41 the equivalent time interval would also be #d hours1 because #fj of arc is everywhere equal to #d hours of time2 but the time at ,greenwich1 with sun on the #fjth meridian east1 is #bj hours of the previous day1 or #h ,a4,m4 of the present day4 ,thus3 ,may #ad1 #bj hours1 or ,may #ae1 #h ,a4,m4 ,the meridians extend from pole to pole1 and it matters not what parallel you may be on1 whether north or south latitude1 your distance can always be measured to the ,greenwich meridian in arc or time precisely as well as though you were on the exact parallel of ,greenwich itself4 ,if the time at ,greenwich is carried1 and the local time of any other meridian is desired1 turn the difference of longitude into time and apply it with regard to signs3 - if west of ,greenwich and if east4 ,the local time at any place can thus be calculated2 or to go farther1 if the time of any meridian is at hand1 the time of any other place can be readily found4 ,every meridian carries a time of its own1 and the instant of the click of a telegraph key may be recorded all over the world in the local time of each locality1 yet the interval between this and a subsequent click has an absolute value which is the same at every place1 regardless of whether it is expressed in solar1 sidereal or lunar time1 and its actual value is invariable4 ,for convenience1 on land1 our country is blocked off into belts of standard time1 #ae wide1 each carrying the time of its central meridian4 ,for instance1 #geth meridian time is used by the eastern states1 while just westward the clocks' faces show an hour earlier time1 that of the #ij belt1 and so on4 ,it is a good rule to remember in reckoning all kinds of time that the clock's face shows earlier time to the westward1 and from this it is easy to deduce the proper application of a correction4 ,there are two kinds of solar time used in navigation2 the first to be considered is apparent time1 the kind shown by the sun dial1 or measured by the sun as we see it4 ,it is noon of the apparent day when the sun is seen with the sextant to dip while taking a meridian altitude4 ,it is at the moment of dipping that the navigator announces #ab o'clock1 and with the striking of eight bells begins a new apparent day on shipboard4 ,the ,day ,lost and the ,day ,gained4--,the fact that the sun seems to travel from east to west1 determining the local time for successive meridians or places along the way1 causes an interesting condition in reckoning time aboard ship4 ,a vessel steaming westward on a parallel sails with the sun2 in the forenoon she is sailing away from it1 at noon the sun overhauls the vessel and they race together1 but it becomes a hopeless chase for the steamer during the afternoon4 ,in consequence of their similar course1 however1 the vessel will hold the sun longer1 and the length of daylight will be increased over that time allotted a stationary position in proportion to the speed of the vessel4 ,on the other hand1 a vessel steaming eastward each hour advances to meet the sun2 at noon the effect is as if they pass each other1 and during the remainder of the day they are moving in opposite directions1 hence this vessel has a shorter term of light and is deprived of its full share of sunshine4 ,in practice these facts require the continuous setting back of the ship's clock1 keeping apparent time on a westbound vessel4 ,take a concrete case for illustration3 to-day assume we are at sea on the #deth meridian west and set the clock at the dipping of the sun1 apparent noon2 the vessel is westbound1 steaming along the equator1 and rolls along at a good #ae-knot clip4 ,in #bd hours by the clock we will cover #cfj miles1 or #cfj of arc on the equator1 which is equal to #f difference of longitude4 7,should the easting or westing be made in higher latitudes1 the difference of longitude will be increased proportionately47 ,so when the ship's clock shows noon we will be #f farther west than at the preceding noon1 or in #ea west4 ,the navigator1 should he observe the sun1 would find it had not reached its highest altitude 7the meridian71 and he would be obliged to wait 7approximately7 #bd minutes1 the equivalent in time of #f1 before the sun would dip4 ,the clock is carrying #deth meridian time1 and we are now determining noon for the #east meridian4 ,he sings out #h bells1 but the clock shows #ab4bd ,p4,m4 ,the ship has gained #bd minutes by sailing with the sun1 and the clock is set back and a fresh start is made4 ,a vessel sailing east has the opposite experience4 ,the navigator1 if guided by the ship's clock1 would find that the sun had dipped some #bd minutes before noon if a run similar to the above mentioned was made eastward4 ,in this case the apparent time of the #east meridian is shown by the clock1 while the ship has moved on to the #deth1 and the time of noon is correspondingly approximately #bd minutes earlier than the clock admits4 ,in the above example1 the clock in the first instance is #bd minutes fast and is set back that amount to correct it for the time of the #ea meridian ,w42 but this time cannot be thus arbitrarily thrown away without some subsequent reckoning4 ,there is just so much time all over the world1 and there are no gaps or extra intervals2 it is absolute in its uniform flow4 ,therefore1 there must be a way of squaring ourselves with ,old ,father ,time4 ,but let us follow the voyage farther and see what transpires3 ,continuing the course westward and ignoring for convenience all intervening land1 each day it becomes necessary to set the clock back #bd minutes until we have circumnavigated the earth4 ,suppose we took our departure from the ,greenwich meridian and kept our log throughout the voyage with great care1 expecting1 according to our reckoning1 to arrive on a ,saturday1 we would indeed be mystified on arrival to hear the ringing of church bells and find that it was ,sunday4 ,we have lost a whole day according to our log1 by throwing away #bd minutes at a time4 ,the time of the world goes on just the same1 regardless of how we juggle the hands of the clock4 ,now1 if we try a similar voyage eastward around the earth1 we will be setting the clock ahead #bd minutes each day1 and when the anchor is dropped on our return1 we will discover that it is ,friday instead of ,saturday4 ,the ship's clock has skipped this #bd minutes each day1 and our log is a day ahead of what it should be4 ,in order to prevent this difference of date1 it was decided years ago to establish an international date line1 which should correspond approximately1 with the #ahjth meridian4 ,the logs of vessels going west around the earth will be a day behind the calendar when they reach ,greenwich1 so a day is dropped from the reckoning when crossing the #ahjth meridian2 that is1 if it is ,monday1 the next day in the log will be ,wednesday4 ,on the contrary vessels bound eastward will be a day ahead when they reach their destination of ,greenwich1 so the date of crossing the date line is entered twice in the log1 as for instance1 there will appear two ,mondays4 ,by this method the accumulated errors of chasing local time1 are in a measure straightened out1 and ship's logs are kept in agreement with the calendar of those at home4 ,thus it will be seen that it is the accumulation of time thus gained or lost that obliges navigators to add or drop a day to or from their logs when crossing the #ahjth meridian4 ,in slow cargo steamers and sailing vessels1 particularly when the course creates but little departure1 the change of time due to difference of longitude is not sufficiently large to cause much inconvenience and can be taken care of by setting the clock back or ahead at noon4 ,but with the development of the modern steamer1 speed has increased to such an extent that the easting or westing of certain day's runs correspond to a considerable amount of time1 and to correct the clock to local time1 all at once1 would be a source of inconvenience and a bother4 ,this is especially true where a fast steamer covers much easting and westing in high latitudes where the convergence of the meridians has shortened the degrees of longitude1 thereby increasing the difference of longitude over a similar day's run in lower latitudes4 ,hence1 in order to more equally distribute its error1 the longitude at noon is anticipated by the navigator and the clocks set at #h ,a4,m4 for the local time of the approaching noon meridian4 ,when the clocks are set at noon1 they are correct only for the moment and then start an accumulating error1 depending in amount upon the rapidity of the easting or westing made4 ,but by anticipating the longitude at noon1 the forenoon watch will experience a decreasing error instead of one accumulated for twenty hours1 and still increasing4 ,it serves to keep the time of day more nearly correct4 ,in the transatlantic service1 where high speed is maintained and the courses result in a large amount of easting and westing1 another method is used for convenience4 ,the navigator estimates the noon position of the next day and accordingly divides the error into thirds4 ,the amount of the first third is applied at #aa ,p4,m41 the second at #c ,a4,m4 and the last third at #e ,a4,m4 ,by this method the error is distributed between the 8first10 8mid0 and 8morning0 watches4 ,it is a matter of considerable moment1 and no joke1 to the hard-working stokers to have the clock set back on them the full amount of the day's run all at one time2 and likewise going east1 it would be giving an unfair advantage to those on duty to set the clock ahead nearly an hour during the morning watch4 ,the apparent or real sun is not a very accurate timekeeper and its days are unequal in length4 ,the aborigines and even our ancestors were content with the time of day indicated by the sun dial1 but as the generations have passed1 each bringing increased development1 time has become valuable2 the crude timepieces have been forced aside by more reliable instruments until to-day we figure down at times to a one hundredth part of a second4 ,it is impossible to construct a clock that will follow the irregularities of the apparent sun1 so an imaginary sun has been devised which is assumed to make its revolution at a uniform speed along the celestial equator with exactly #bd hours between its transits of the same meridian4 ,this interval is the average of all apparent days in one year4 ,the varying rate of the sun's apparent motion is due to several causes which will be subsequently discussed under the ,equation of ,time4 ,the time measured by the transit and progress of the mean sun is called mean time2 if at ,greenwich1 it is ,greenwich mean noon and ,greenwich mean time 7,g4 ,m4 ,t472 if it represents the time of the observer's place or meridian1 it is local mean time 7,l4 ,m4 ,t474 ,it is mean time that is shown by all clocks and chronometers used in every day life4 ,the distance between the apparent and mean suns1 expressed in time1 is known as the ,equation of ,time1 and the application of this correction depends upon which sun is ahead4 ,it is tabulated in the nautical almanac for every two hours of ,greenwich mean time1 with hourly differences1 so it can be reduced for longitude in time to any meridian1 or corrected to any intermediate ,greenwich time4 ,it is applied according to the sign accompanying it1 and can be used to change apparent into mean1 or mean into apparent time4 ,the progress of the mean sun across the sky with reference to the meridian is measured by the angle at the pole 7expressed in time71 between the meridian and the hour circle passing through the mean sun4 ,this is the hour angle of the mean sun as well as the local mean time4 ,civil ,time is a variety of mean time1 and is reckoned through #ab hours from midnight to noon1 and again #ab hours from noon to midnight1 dividing the day into the well-known periods of ,a4,m4 and ,p4,m4 ,with this kind of time1 the day begins at midnight and the hour angle until noon is measured eastward through #ahj of the revolution and westward through the remaining half from noon to midnight4 ,in other words1 #d3jj ,p4,m4 signifies that the sun has a westerly hour angle of #d hours1 while #h3jj ,a4,m4 indicates that the sun is #d hours eastward of the meridian4 ,astronomical time is reckoned westward through the whole #bd hours of the day1 #j hours being noon4 ,from noon to noon is an astronomic day4 ,thus #e ,p4,m4 civil is the same as #e hours astronomical time1 while #e ,a4,m41 ,may #adth is the same as ,may #acth1 #ag hours4 ,in every solar observation for time the real or apparent sun is observed and hence the time derived from the sight must be local apparent2 to which the equation of time must be applied to convert it into local mean time4 ,it has already been made clear that the longitude is equal to the difference between the local mean time and ,greenwich mean time1 or between local apparent time and ,greenwich apparent time4 ,,sidereal ,,time ,sidereal is derived from the ,latin word sidus1 meaning of or belonging to the stars4 ,sidereal time is measured by the apparent diurnal revolution of the stars1 resulting from the rotation of the earth4 ,by their use the conditions which render the sun inaccurate as a timekeeper are eliminated2 for the period of rotation of the earth is so regular that the passages of the stars across the meridian occur with great precision4 ,this exactness enables the astronomer to keep the observatory clock checked to a remarkable degree of accuracy4 ,these observatory clocks carry sidereal time1 and for convenience it is customary to divide their faces into #bd instead of #ab hours4 ,sidereal time is the bedrock of all time2 for it is by converting it into solar time and sending it throughout the country by telegraph and radio that the people of the world get the standard by which to set their clocks and chronometers4 ,sidereal time is not practicable for every day use as its noon occurs1 without regard to light or darkness1 at every hour of day or night during a year4 ,in ,march1 at the time of the vernal equinox1 it agrees with the solar clocks1 but in ,september at the autumnal equinox1 its noon occurs at the solar midnight4 ,while the sun is employed as the object of reference in solar time1 it may appear strange that no particular star is thus used in sidereal2 but in lieu of a definite stellar object by which to measure the sidereal movement of the heavens1 we refer to the celestial vernal equinox4 ,this point was located in the constellation of ,aries centuries ago1 and hence its popular name--,the ,first ,point of ,aries2 but this has become a misnomer1 for the point has long ago moved westward into another constellation1 as discussed under the ,precession of the ,equinoxes4 ,navigators still cling to the name1 however1 and the equinox continues to serve its purpose1 regardless of its slow drift westward4 ,this imaginary point of reference crosses the observer's meridian much as the stars do1 with the difference that it is always on the celestial equator and acquires no declination4 ,the value of this point becomes further enhanced by the fact that it always lies in the same direction regardless of our position in the orbit4 ,in other words1 the distance of the equinox being infinite1 lines drawn from perihelion and aphelion1 respectively1 to it1 fail to produce an appreciable angle4 ,in explanation of this statement1 it must be understood that for all uses on the earth the terrestrial system of direction 7that is1 using the bearing of the north pole as a standard1 with east to the left and west to the right when the back is toward the pole7 is entirely adequate1 but when dealing with the direction of celestial bodies1 a broader standard must be considered4 ,north and ,south both have a definite place in the heavens1 being the points of the extended axis of the earth1 but east and west are only relative expressions4 ,to demonstrate this3 it is possible for a man1 traveling westward on the ,trans ,siberian ,railroad1 to see from the rear platform1 in the evening1 a certain star bearing eastward4 ,at the same moment it is possible for an officer of a transpacific liner in the early morning1 to be taking a sight of this same star bearing westward4 ,in terms of absolute direction that star bore the same from both sides of the world4 ,on the #bast of ,march the earth1 sun1 and celestial vernal equinox are in range1 with the sun between the earth and the equinox4 ,for a place in north latitude on the meridian of the terrestrial equinox1 the sun as usual bears south at noon this day1 and hence the range mentioned above bears south at that time4 ,this coincidence of bearing is only momentary1 for the earth with its onward motion immediately moves out of range and forms an angle between the sun and the celestial equinox4 ,at noon on the day succeeding the equinox the sun bears to the left of the so-called ,first ,point of ,aries 7celestial vernal equinox74 ,the sun according to terrestrial direction always bears south by true compass at noon1 yet the ,first ,point of ,aries being at an infinite distance always bears the same by absolute direction4 ,if this point could be seen and a bearing of it taken by compass simultaneously with the sun1 it would be1 perhaps1 ,s4 #a ,w4 and so on widening the angle1 roughly speaking1 a degree each day4 ,illustration3 ,the acceleration of ,sidereal over ,mean ,time ,,fig4 #b4 ,the interval between two successive transits of the sun across the meridian constitutes a solar day1 and likewise the period required for a certain star to return to the same meridian is a sidereal day1 but these two days are not of the same length4 ,the solar day we know is #bd hours long1 but its sidereal contemporary has a length of only #bc hours1 #ef minutes 7approximately7 solar time4 ,the sidereal clocks1 however1 are geared to show #bd hours1 sidereal time1 in #bc hours1 #ef minutes1 solar time4 ,by this it will be seen that in any given period the face of the sidereal clock will show more hours than the solar clock4 ,both the solar and sidereal clocks start even at the vernal equinox1 about ,march #bast1 but from then on1 the sidereal clock gains on the solar time clock about #d minutes a day until in a year it is a full #bd hours ahead1 showing that there is one more day in a sidereal year than in a solar year4 ,the approximate relation of the times shown by these clocks is readily calculated by allowing a gain in the sidereal clock of one hour for each #ae days after ,march #bast1 or two hours for each month4 ,in order to aid in a simpler explanation1 let us again follow the earth around its orbit and note the conditions that distinguish sidereal from solar time4 ,let us once more assume it to be the time of the vernal equinox1 the clocks1 both sidereal and solar1 now show #j hours1 and the sun1 the earth and the ,first ,point of ,aries are in range4 ,the earth immediately moves out of line by virtue of its onward motion1 and the sun correspondingly appears to move eastward2 this is imperceptible at first1 however1 and not noticeable without a careful measurement1 as it seems to be swallowed up in the contrary 7westward7 diurnal movement4 ,after #bd hours of rotation from the instant of the equinox the earth turns the meridian until it causes the ,first ,point of ,aries to transit1 marking sidereal noon of the first day4 ,the sidereal clock at this moment reads #bd hours1 but a glance at the solar clock shows #aa hours #ef minutes ,a4,m41 about #d minutes short of 7solar7 noon4 ,an observation will show that the sun has apparently moved about a degree eastward of the hour circle passing through the ,first ,point of ,aries since the preceding noon1 and the earth must turn this extra degree before the sun will be brought to the meridian1 thus occupying the #d minutes mentioned above4 ,in other words1 the earth turns #cfj in a sidereal day but must turn about #cfa in a solar day4 ,three months after the vernal equinox1 the angle between the ,first ,point of ,aries and the sun becomes1 in round numbers1 #ij1 and it requires #f hours for the earth to bring the sun to the meridian after the passage of the ,first ,point of ,aries4 ,in plainer language1 when the ,first ,point of ,aries crosses the meridian 7sidereal noon7 the sun is about #ij to the left--about rising in the eastern sky2 the earth must make a quarter turn1 or #f hours1 before it will be solar noon4 ,thus it will be seen that at this point sidereal time is #f hours ahead of solar time4 ,in six months1 when the ,first ,point of ,aries is on the meridian the noonday sun is shining on the antipodes1 and it lacks #ab hours of solar noon4 ,the difference between the sidereal and solar clocks has now reached #ab hours and through a continuation of the same process the interval between their readings1 widens throughout the remainder of the year4 ,when the #bast of ,march comes around again1 and the meridian presents itself to the sun and the ,first ,point of ,aries in range1 a careful count of the number of times this latter point has crossed the meridian during the year1 discloses #cff transits4 ,that is1 the earth has actually turned about its axis #cff times4 ,the sun is found to have passed the meridian only #cfe times4 ,counting the rotations of the earth by the number of the sun's transits while we are revolving around him1 causes the apparent loss of a day due to the earth unwinding itself once1 so to speak1 during the year4 ,the accumulated difference amounts to one sidereal day4 ,hence it will be seen that a year contains #cff sidereal days of #bc hours1 #ef minutes each1 and #cfe solar days of #bd hours each4 ,now for a recapitulation of the subject of time4 ,the rotation of the earth is the real standard of measuring time intervals2 the period required for this rotation does not vary4 ,it has been suggested that the tide waves have a minute effect on the regularity of this movement1 but the construction of our clocks is such1 that if any variation exists we are unable to detect it4 ,whether we use the passages of the stars1 or the transits of the sun to reckon our time1 it falls back in either case upon the diurnal rotation of the earth4 ,apparent time is measured by the seeming progress of the actual sun4 ,the time of its transit of the meridian is irregular1 but is always shown by its 8dip10 culmination1 with the sextant4 ,mean time is reckoned by the revolutions of a fictitious sun1 called the mean sun1 and the length of one of these revolutions is the average of a year of apparent days4 ,this1 owing to its uniformity1 is the time used for the everyday purposes of life4 ,the difference between apparent and mean time is called the equation of time1 and1 by applying it according to signs given in the ,nautical ,almanac1 one can be converted to the other as desired4 ,sidereal time is indicated by the position of the ,first ,point of ,aries relative to the meridian2 it is star time4 ,the stars make a complete revolution of the heavens in #d minutes less time than is required by the mean sun4 ,therefore the sidereal day is that amount shorter than the solar day4 ,the point of the celestial vernal equinox or ,first ,point of ,aries is a sort of celestial 8bench mark20 besides indicating sidereal time1 it serves as a point from which right ascension is measured eastward4 ,this subject has been discussed previously1 but as it is intimately associated with sidereal time1 perhaps it may be made clearer since the latter has been so fully explained4 ,right ascension is measured on the celestial equator1 precisely the same as longitude on the earth1 excepting that it is always measured eastward through the full #bd sidereal hours1 contrary to the diurnal movement of heavenly bodies4 ,moreover1 the meridian passing through the vernal equinox is called the celestial prime meridian1 and sometimes the ,greenwich of the heavens4 ,there is another point of distinction1 however1 between this prime meridian of the sky and our meridian of ,greenwich1 which1 while it does not effect practical navigation1 has to receive consideration in the long run2 our longitude values on the earth remain at all times constant1 but owing to the precession of the equinoxes the celestial prime meridian is slowly moving westward1 thus causing the right ascensions measured from it to become very slowly in error 7#ej yearly74 ,the hour angle of a body is the angle formed at the pole between the meridian and the hour circle passing through the body measured westward4 ,with all these important facts well in mind we will go ahead under a slow bell1 through a few more statements which may be found a little perplexing4 ,however1 a careful study of the ,time ,diagram will1 no doubt1 drive away the haziness so often surrounding the subject of time4 ,illustration3 ,this diagram represents the plane of the equator looking down upon the ,north ,pole4 ,the #ge ,w meridian is chosen as that of the observer and local time reckoned therefrom4 ,the arrows on the outer circumference indicate the directions of the earth's rotation2 the other smaller arrows indicate the direction in which each element is reckoned4 ,,fig4 #c4 ,the hour angle of the mean sun is the local mean time1 and the hour angle of the ,first ,point of ,aries is the local sidereal time4 ,the local mean time and longitude in time accelerated by ,table ,,iii ,nautical ,almanac plus right ascension of the mean sun is equal to the local sidereal time4 ,the right ascension of the meridian is the same thing1 exactly1 as the hour angle of the ,first ,point of ,aries1 and both of these are identical with the local sidereal time4 ,the sidereal time of ,greenwich mean noon is the same as the right ascension of the mean sun at that time4 ,the hour angle of a star plus the right ascension of the same star is equal to the local sidereal time4 ,difference of longitude can be represented by an interval of sidereal time or by a difference of right ascension1 precisely the same as by a difference of solar time4 ,thus with the local sidereal time calculated from an observation of a star1 and the corresponding ,greenwich sidereal time taken from the ,nautical ,almanac1 the longitude is at hand1 by turning their difference into arc4 ,the fact that the actual time interval is longer in the case of solar time than in an interval of the same number of hours of sidereal time1 has no influence on the resulting difference of longitude4 ,the number of degrees in any arc can be the same1 yet vary in linear measurement1 but the same number of hours of solar and sidereal time represent the same proportionate part of a circle4 ,it was just stated that the hour angle of a star plus its right ascension is the same as the local sidereal time2 now this is also true of the sun4 ,the hour angle of the mean sun plus the right ascension of the mean sun is equal to the local sidereal time2 by means of this equality we are able to find the ,greenwich sidereal time on any occasion4 ,it is necessary to have this element in order to compare it with the local sidereal time1 which we find by observation of a star1 to obtain difference of longitude 7in time74 ,in page #b of the ,nautical ,almanac will be found the ,right ,ascension of the ,mean ,sun at ,greenwich ,mean ,noon2 this must always be taken out for the preceding noon4 ,we now have a measure of sidereal time to be added to a measure of mean time1 but it will be remembered in early arithmetic that an apple and a peach can not be added together any more than can be added to 4 ,the only course to steer is to reduce the quantities to a common denominator1 or like quantities4 ,so in handling these two varieties of time1 solar time must be accelerated by adding a correction to it1 or sidereal time retarded by subtracting an amount necessary to make it equal to a corresponding value of solar time4 ,the tables for the conversion of one of these varieties of time to the other are found in the ,american ,ephemeris ,tables ,,ii and ,,iii1 ,bowditch ,tables #h and #i1 and at the foot of pages #b and #c1 ,nautical ,almanac4 ,a practical illustration of this may make a clearer impression4 ,in the early evening of ,april #hth1 secured a sight of ,regulus2 the chronometer showed #h days1 #i hours1 #ef minutes1 and #j seconds 7corrected71 with other necessary elements given1 the sight is worked as usual to find the star's hour angle1 which proves to be3 h4 m4 s4 ,h ,a ,regulus #a #ei #dg bearing west4 ,r ,a #aj #c #jb ,,rams #a h4 #f m4 #g s4 -- -- -- ,,gmt #i h4 #ef m4 #j s4 ,l4 ,s4 ,t4 #ab #jb #di ,accel4 #i h4 #a m4 #bi s4 ,g4 ,s4 ,t4 #aa #jc #de ,accel4 #ef m4 #i s4 -- -- -- ----- ----- ----- ,long4 in time #ei #jd ,g4 ,s4 ,t4 #aa h4 #c m4 #de s4 ,this being a star sight we obtain from it the sidereal time at place of observation and as the chronometer carries ,greenwich mean time we seek the corresponding sidereal time by adding this and its acceleration for a sidereal interval to the right ascension of the mean sun taken from ,nautical ,almanac4 ,the result is the ,greenwich ,sidereal ,time4 ,it is occasionally required to find the sidereal time at the ship in which case it is only necessary to apply the longitude in time to the ,greenwich ,sidereal ,time4 ,as ,greenwich mean time is the most used and is the best understood it is a very convenient practice to carry ,g4 ,m4 ,t4 on the navigator's watch4 ,it is readily converted into any other time with ease but serves more purposes as it is without conversion4 ,a stop watch is an excellent instrument for taking time sights where great accuracy is essential4 ,by setting it at #j minutes a man can observe alone starting the watch as he makes contact with the horizon and when subsequently comparing with the chronometer subtract the reading of the watch to get the ,g4 ,t4 of observation4 ,the most expeditious way to convert time into arc is to multiply the hours by #ae and add the number of minutes divided by #d to get the degrees2 multiply the remaining minutes by #ae and add the seconds divided by #d to get the minutes2 multiply the remaining seconds by #ae to get the seconds in arc3 ,thus3 ,long4 in time #b hrs4 #db m4 #bc s4 #cj #cj #de #aj #e -------------------- ,long4 in arc #dj #ce #de ,to change arc into time divide the degrees by #ae to get the hours2 multiply the remainder by #d and divide the minutes by #ae and add to get the number of minutes 7m472 multiply the remainder of minutes 77 by #d and divide the seconds 77 by #ae carrying the division of tenths if desired1 adding the result to get the seconds 7s473 ,thus3 ,long4 in arc #dj #ce #de #b hrs4 #dj m4 #bj s4 #b #c ------------------- ,long4 in time #b hrs4 #db m4 #bc s4 ,this may appear complicated at first but is much the quickest way of conversion4 ,however1 ,table ,no4 #g1 ,bowditch is always available if desired4 ,in getting an understanding of any time problem1 that is such as changing mean time into sidereal time2 obtaining the hour angle of a star or planet2 in seeking the local time from the chronometer1 or any time values that are found perplexing1 always draw a diagram4 ,make the circle on the plane of the equator1 with the pole as the center1 and meridians radiating from it towards the circumference of the circle4 ,now imagine for the moment that you are at the north pole1 and the date is the #bast of ,september4 ,the sun is traveling in the horizon1 and if the direction of the ,greenwich meridian is known1 this body serves as a time piece1 for the angle between this meridian 7direction7 and the sun is the ,greenwich time4 ,this angle corresponds with twice the angle between ,,xii hours on the watch and the hour hand2 or would 7disregarding equation of time7 coincide with it if the watch's face was divided into #bd hours4 ,likewise when the vernal equinox or ,first ,point of ,aries lies in the direction of ,greenwich it is ,greenwich sidereal noon1 and the subsequent angle that appears through the rotation of the earth1 shows the ,greenwich sidereal time4 ,,equation ,,of ,,time ,it is necessary in considering this subject to reiterate some of the statements made in the preceding talk on time1 but1 as they are very important1 no time is wasted by further impressing them on the mind4 ,let it be understood that the apparent orbit of the sun is actually due to the earth's revolution around him1 yet for simpler explanation it is considered to be the sun's own revolution4 ,the apparent movement of the real sun is not of uniform speed and1 in consequence1 it has become necessary to devise a fictitious sun whose assumed revolutions around the earth are at all times regular in their rate4 ,the equation of time is the difference between these two suns and1 as they are at times in conjunction and at other times attain a distance from each other of #af minutes #bj seconds1 and1 moreover1 as the real sun is sometimes ahead and again in the wake of the mean sun1 it becomes evident that the equation of time is an ever varying quantity4 ,the irregularity of the sun's apparent movement as compared with the uniformity of the mean sun1 is subject to two causes3 ,first1 the earth travels in an ellipse1 and1 as the length of a degree varies in the different parts of the circumference1 the motion would appear to be irregular1 that is1 if the sun actually traveled at a uniform rate1 it would1 from the above fact1 appear to us to be variable in its motion2 furthermore1 the laws of forces only allow a body traveling in a circle the privilege of a uniform speed so the earth1 owing to its varying distance from the sun1 experiences a corresponding change in the amount of attraction exerted upon it by the sun and its velocity1 actually becomes variable4 ,thus1 during the winter1 ,december and ,january1 when they are nearest each other1 the attraction is strongest and the earth increases its speed in revolution2 while in ,june and ,july the earth is at its greatest distance from the sun and the attraction is less1 resulting in a slowing down in the rate of the onward movement4 ,as the sun appears to us to take on movements corresponding to those of the earth1 these variable movements of the latter are seen in the apparent motion of the sun4 ,second1 the plane of the earth's orbit is inclined at an angle to that of the equator1 which makes the sun appear to be traveling at a variable speed along the ecliptic4 ,with these two errors combined influencing the apparent sun1 he becomes unreliable for regulating timepieces4 ,the mean sun1 which was originated to obviate these irregularities1 is assumed to travel in a circle with the earth located in the centre1 which disposes of the first reason for an apparent variable motion2 and again1 the mean sun revolves in the plane of the equator1 thus eliminating the second obstacle in the way of uniform time4 ,now we will continue a little farther into the explanation of the reasons for the irregular movement of the real sun4 ,a law discovered by ,kepler1 and named for him1 provides that a radius from the sun to the earth covers sectors of equal areas in equal times2 a sector equal in area to any sector covered in the same time4 ,that is1 when the earth is in that part of the orbit near the vernal equinox1 the radius of the orbit will in a given time1 say a week1 sweep over a certain area2 the earth proceeds toward aphelion and when in the vicinity of that point1 the radius becomes greatly increased in length4 ,now in a week with this longer radius1 a far greater area would be covered if the earth maintained the same rate of speed as at the equinox1 but ,kepler's law says1 8equal areas in equal times10 so in order to conform with the law1 the earth's speed of revolution must be reduced4 ,the earth does not slow down just for the sake of obeying ,kepler1 but at this part of the orbit it is at its greatest distance from the sun and hence the reduced attraction causes the earth to lag a little4 ,at the time of the autumnal equinox1 an area similar to that of the vernal equinox is covered4 ,as the earth approaches perihelion1 the radius is gradually shortened by the eccentricity of the sun in this part of the orbit and the increased attraction causes the earth to speed up correspondingly4 ,at the increased speed1 the shorter radius sweeps the same area in a week as at other parts of the orbit and ,kepler's law still holds good4 ,as the apparent motion of the real sun corresponds exactly with the real motion of the earth1 it is evident from the above that the real sun apparently moves at different rates of speed along the ecliptic1 faster in winter and slower in summer than the mean sun4 ,the value of a mean solar day is the average of a year of apparent days1 or in other words1 there is the same number of mean solar days in a year as there are apparent days4 ,in considering the effect of the variable motion of the earth in its orbit1 we will recall the conditions used when defining sidereal and solar days4 ,the former comprises the interval between two successive passages of a certain star across the meridian1 or perhaps better1 between two successive passages of the meridian over a star4 ,this is the true length of the earth's rotation and is the standard to which we may refer the length of the mean or apparent solar days4 ,now it requires about #c minutes #ef seconds longer for the meridian to sweep around from sun to sun than from star to star1 owing to the fact that the mean sun moves uniformly eastward that amount daily1 thereby requiring the meridian1 after reaching its position of yesterday noon1 to overhaul the mean sun this #c minutes #ef seconds of eastward movement4 ,the mean sun maintains this uniform difference between its days and the length of the sidereal day4 ,without this daily easting of the sun1 the sidereal and solar day would be the same4 ,but1 in considering the apparent sun1 we find the length of its days continually varies from both that of the sidereal and mean day4 ,this is explained by the fact that the eastward movement of the apparent sun is due to the movement of the earth in its orbit1 and as this movement becomes faster or slower the eastward movement of the sun becomes correspondingly faster or slower4 ,thus1 we readily see that with the apparent sun moving eastward faster or slower at times1 the length of the apparent day must vary accordingly and we cannot establish a uniform difference between it and the sidereal day1 as in the case of the latter and the mean day4 ,the apparent days exceed the mean days in length1 between ,september and ,march1 while the earth is traveling fastest in its orbit4 ,beginning at the autumnal equinox with the apparent sun eight minutes behind the mean sun1 the former gains slowly at first but with increasing rapidity4 ,about the end of ,december1 at perihelion1 it overhauls the mean sun and they are coincident as regards this correction only4 ,leaving perihelion1 the apparent sun rapidly takes the lead but with a gradually decreasing amount until at the equinox in ,march1 reaches its maximum lead of #h minutes4 ,entering that portion of the year ,march to ,september1 we find the earth traveling slower and the mean sun gaining on the apparent sun2 between the vernal equinox and aphelion1 the mean sun gains until both are together at the summer solstice and then forging ahead the mean sun attains a lead of #h minutes in ,september4 ,it must be borne in mind that this error is caused only by the eccentricity of the orbit and is but a component part of the whole correction of the equation of time4 ,the other portion is due to the obliquity of the orbit1 or its inclination to the equator4 ,this error is introduced through the fact that the apparent sun moves in the ecliptic and the mean sun is assumed to proceed along the celestial equator4 ,in considering this phase of the question1 we will ignore entirely1 for the time being1 the error of eccentricity described above4 ,the error of equation of time due to the obliquity of the orbit is a simple one to see1 but like many simple things it is easier to show it by a diagram than to explain in words1 so the reader is referred to the accompanying figure1 that a study of it may be made before proceeding4 ,illustration3 ,,fig4 #d4 ,at the equinoxes1 where the ecliptic and equator cross and the solstices--the vertices of the ecliptic--that is1 four times a year1 the true and mean sun are together1 but departing from these points they do not travel with the same right ascension1 remembering that right ascension is measured on the equator4 ,taking1 for example1 the earth in that quadrant of the orbit comprised between the vernal equinox and summer solstice1 the apparent sun in the heavens would be by cause of obliquity alone1 to the right or to the westward of the mean sun1 and thus it will be seen that with the earth rotating from right to left the apparent sun will cross the meridian first2 consequently between ,march #bast and ,june #bast that part of the equation of time due to the obliquity of the orbit bears a minus sign when mean time is desired from the apparent time4 ,this correction reaches its maximum half-way or #de from the equinox1 amounting at that point to nearly #aj minutes4 ,illustration3 ,,fig4 #e4 ,now the reason for this difference between the mean and apparent sun when each 7so far as this problem is concerned7 moves along its respective path--the equator and the ecliptic--at the same rate1 is this3 suppose the equator between the equinox and the solstice is divided into an equal number of parts and an hour circle drawn through each point of division4 ,beginning from the equinox 7the common apex of the triangle7 the arc of each hour circle between the equator and the ecliptic1 forms the altitude of a right-angled triangle1 while the equator and the ecliptic are base and hypothenuse respectively4 ,thus it will be seen that each portion of the equator 7base7 is shorter than the corresponding part of the ecliptic as defined the hour circle1 to the extent of the ratio of the base to the hypothenuse4 ,this amount increases with the increasing size of the triangle1 but a new element enters to counteract its effect4 ,with the increasing divergence of the ecliptic and equator1 the divergence of the hour circles becomes a factor and as the solstice is approached the divisions on the equator1 are represented on the ecliptic by gradually decreasing spaces between the hour circles4 ,the combination of these effects produces the error due to the obliquity of the orbit4 ,the error has the opposite effect in the next quadrant1 that is1 from ,june to ,september2 and in opposite quadrants it is the same4 ,so it will be seen that error due to the eccentricity causes the apparent sun to lead the mean sun from ,december #cast to ,june #cjth1 reaching a maximum of #h minutes about ,april #b;d4 ,this sun then falls astern until ,december1 again attaining a maximum of -#h minutes about ,october #ast4 ,the error due to the obliquity of the orbit accumulates between the equinoxes and solstices for at these points the two suns are together and there is no error1 but about the #fth of ,may1 ,august1 ,november1 and ,february1 it reaches a maximum of #aj minutes2 in ,august and ,february1 the mean sun takes the lead4 ,these two errors of equation of time combined algebraically will result in the plain line of the diagram4 ,,calendar ,the ancients1 in order to keep track of past events and to anticipate the future1 devised a calendar1 which1 while not identical with the one now in use1 was of itself a remarkable production4 ,they chose the revolutions of the moon as their basis of measuring the passage of time1 and1 as a lunar month contains only #bg days1 the #ab months used in this early calendar comprised a year of about #ced days4 ,it became apparent as the world progressed that use of the moon was very unsatisfactory for this purpose1 as the calendar became complicated and confusion resulted1 owing to the difference between the lunar and solar years4 ,this condition remained until the reign of ,caesar1 when that monarch determined to establish a more satisfactory method of reckoning time4 ,with the aid of an eminent astronomer1 he completely revised the calendar1 using the revolution of the earth around the sun as the standard for the length of a year4 ,the time required for this is #cfe days1 approximately1 and as it was inconvenient to include the day in the year1 it was allowed to accumulate for #d years1 when as a whole day1 it was added to the end of ,february4 ,caesar1 evidently proud of this accomplishment1 honored himself by naming his astronomer's invention the ,julian ,calendar and in order to further immortalize himself he changed the name of the seventh month to ,july4 ,augustus1 his successor1 apparently envious of the honor his predecessor derived from this source1 and determined not to be thus outdone in perpetuating his name1 changed the month ,sextilis to ,august4 ,the commonly accepted #cfe days as the length of year is only an approximation1 however1 and the small difference between this and the actual length of a year began to accumulate until this weak point in the ,julian ,calendar became a matter of moment4 ,the exact length of a year from vernal equinox to vernal equinox is #cfe days #e hours #dh minutes and #df seconds1 which lacks just #aa minutes #ad seconds of #cfe days4 ,this caused the dates of the equinoxes and solstices to keep slipping back #aa minutes each year and when considerable time had passed the difference became large enough to cause inconvenience2 the date of the vernal equinox having dropped back to ,march #aath in #aehb4 ,in this year ,pope ,gregory1 acting under the advice of an astronomer by the name of ,clavius1 modified the ,julian ,calendar4 ,he first added #aj days to restore the date1 and then to forestall a further retrogressing of the calendar1 provided that only those century years divisible by #djj should be considered leap years4 ,in this way the #aa minutes #ad seconds was prevented from causing further mischief4 ,this calendar known as the ,gregorian ,calendar is now in almost universal use1 though at first it was adopted only by ,catholic countries4 ,it is interesting to note that the time consumed by the sun in making his apparent yearly revolution from a certain star back to that star again is a sidereal year of #cfe days #f hours #i minutes #i seconds4 ,the tropical year--the one in common use--is shorter1 being the time required for the sun to leave and return to the vernal equinox1 or ,first ,point of ,aries4 ,this point1 it will be remembered1 is moving westward about #ej annually1 and it will be seen that while the sun starts its eastward revolution among the stars1 the equinox is very slowly moving westward to meet him1 thus making the tropical year about #bj minutes shorter than the sidereal year4 ,while discussing the calendar it is an opportune time to explain a matter concerning the dates of the equinoxes and solstices4 ,it has of course been noticed by everyone that the vernal equinox occurs one year on ,march #bjth and another on ,march #bast1 or the summer solstice on ,june #bb;d and yet another year on ,june #bast and so on4 ,aside from the slight change due to the dropping back of the seasons in the orbit by the precession of the equinoxes1 the actual time of the equinoxes and solstices may be considered as constant1 yet the dates vary a few hours4 ,the year in common use--the tropical year--contains approximately #cfe days1 yet we take account of only #cfe days1 the extra day being laid aside for future reckoning4 ,during the next year this #f hours will be augmented by #f more2 the next by another #f1 making #ah hours ahead of the calendar4 ,the fourth year this amount reaches #bd hours1 and a full day1 the #bith of ,february1 is added to the calendar for that year1 and we are square again4 ,but the different equinoxes and solstices occur at just #cfd days2 taking for example the vernal equinox1 it occurs on the following 7approximate7 dates1 which it will be noticed are #f hours later each year3 #aijh--,march #bj1 #ab hours 7leap year74 #aiji--,march #bj1 #ah hours4 #aiaj--,march #bj1 #bd hours or ,march #bast noon4 #aiaa--,march #ba1 #f hours4 #aiab--,march #bj1 #ab hours 7leap year74 ,it is evident by the above that the insertion of the extra day just previous to the equinox in the leap years1 sets the date of the equinox back a day by the calendar4 ,juggling the day as shown above causes the change in the calendar dates of these phenomena4 ,,chapter ,v ,corrections for ,observed ,altitudes ,the observed altitude of a heavenly body1 as measured with the sextant1 requires treatment for numerous errors to reduce it to the true altitude and make it ready for use in working any of the navigational observations4 ,the amount of error varies in different bodies1 the moon requiring the maximum and the fixed star the minimum correction4 ,all the errors are not common to all bodies1 that is1 with some1 certain errors become so insignificant that they are cast aside4 ,these errors comprise the index error of the sextant1 refraction1 dip1 semi-diameter1 and parallax4 ,in ,table #df1 ,bowditch1 will be found the combined corrections 7index error excepted7 to be applied to an observed altitude of a star or planet and to that of the sun's lower limb4 ,a supplementary table furnishes an additional correction to be applied to the semi-diameter of the sun when accuracy is desired4 ,these corrections will now be discussed in the order named3 ,,index ,,correction ,the index and horizon glasses of a sextant are supposed to be parallel when the zero of the vernier and the zero of the limb are in one1 and with this the case1 the true and reflected images seen in the horizon glass should exactly coincide4 ,any difference between them is the index error4 ,it is seldom that a sextant is so well adjusted that no index error exists1 but it is not desirable to keep tampering with the instrument with an attempt to eliminate this error1 for it will in time injure its accuracy4 ,by testing the sextant at each sight1 the error can be closely watched and allowance made for it in correcting altitude4 ,the easiest and most accurate method of ascertaining this error is by using a star in the following manner3 ,set the zero of the vernier a little to either side of the zero of the limb1 and observe a #b;d or #c;d magnitude star--move the reflected image past the real and note if they pass directly over one another4 ,if not1 the horizon glass is not perpendicular and needs adjustment4 ,bring the reflected star in exact conjunction with the real star1 and read off the index correction--if the zero of the vernier is to the left of the zero of the limb--on the arc--the difference is minus 7-7 and subtracted from the observed altitude2 and if to the right--off the arc--it is plus 77 and added4 ,a well-known rule of thumb expresses it thus3 if it's on it's off1 and if it's off it's on4 ,the sea horizon is also available for determining this correction and serves the purpose with fairly accurate results4 ,,semi-,,diameter ,in measuring the altitude of certain bodies for navigational purposes1 it is necessary to determine the distance of the center of the body above the horizon4 ,to accomplish this in an accurate manner the lower edge or limb is brought down to the horizon and the semi-diameter applied to this measured altitude4 ,when the lower limb is used1 as is the usual practice1 the correction for semi-diameter is obviously plus 774 ,the upper limb can be resorted to1 however1 should the lower side of the body become veiled by cloud1 and in this case the correction is minus 7-74 ,semi-diameter of the sun is obtained readily from the ,nautical ,almanac for each ten-day period1 for it must be remembered that the sun is continually changing its distance from the earth1 and consequently the diameter of the former is increased and lessened slightly at different times of the year4 ,for instance3 ,on ,january #b;d1 when the earth is near perihelion and we are at our nearest point to the sun1 the semi-diameter is #af #ag4ij1 while on ,july #b;d1 when we are in the remote parts of the orbit1 the semi-diameter is only #ae #de4fi1 making a difference of over #cb4 ,the moon being such a near neighbor of ours gives more trouble in determining her diameter4 ,besides being greatly affected by her rapidly changing distance from the earth1 a further correction is occasioned by the fact that our position on the surface is nearer the moon at times than is the center of the earth4 ,that is1 when the moon is in the zenith we are #djjj miles1 the earth's radius1 nearer that body than when she is in our horizon4 ,it is evident that the direction of the moon in our sensible horizon is at right angles to a perpendicular erected at our place of observation and passing through the earth's center1 and this again makes it evident that the moon is about equally distant from the earth's center and our position on the surface2 but as she ascends the heavens she comes nearer our position until in the zenith the distance has been reduced by #djjj miles and the diameter appears correspondingly larger4 ,draw a diagram and see for yourself4 ,this ,augmentation of the ,semi-diameter1 due to the altitude1 is found tabulated in ,bowditch1 ,table #ah4 ,the semi-diameter becomes too small to consider in ordinary navigation when observing any of the planets1 and of course fixed stars are beyond its scope4 ,,refraction ,everyone knows that the blade of an oar when dipped in the water appears to be bent in a remarkable manner at the surface4 ,this is a clear case of refraction4 ,should the oar1 however1 be held under everywhere at an equal depth1 a square look downward at it would fail to show any refraction4 ,so it becomes evident that refraction is caused by the rays of light passing obliquely from a rarer to a denser medium or vice versa4 ,a ray of light coming from a heavenly body to the earth passes through a medium of gradually increasing density1 from the thin outer air to the denser atmosphere at the surface of the earth4 ,the ray of light consequently becomes curved downward and reaches the earth at a point nearer the heavenly body than would be the case if the light ray traveled in a straight line4 ,the effect of this to the observer is that the body appears higher than it really is4 ,the difference between the actual direction of the ray of light unaffected by the air1 and our line of vision as we see the body1 is the refraction4 ,the amount of refraction ordinarily affecting an observed altitude depends upon the distance of the body above the horizon4 ,at the zenith1 the rays of light1 entering our atmosphere perpendicularly1 are not deflected and refraction is nil4 ,but1 on the other hand1 when the body is near the horizon1 the rays of light pass through the atmosphere at a sharp angle and are consequently subject to the greatest bending1 thus giving us our maximum refraction4 ,in fact1 this element becomes so unreliable in low altitudes that it is not advisable to observe a body when less than #aj or #ac above the horizon4 ,this in no way concerns bearings taken of bodies in the horizon for amplitude1 as refraction affects the altitude and not the azimuth of a body4 ,,dip ,it is a well-known fact to every seaman that by going aloft1 he can pick up a light sooner than on deck2 that is1 the higher his elevation the wider his horizon becomes4 ,the horizon of a man in a small boat is only about #c miles away1 but1 if he climbs to the bridge of a steamer some #fj feet above the water1 he finds that the horizon has receded until he has a range of about #i miles4 ,the fact that the horizon can be altered by changing the altitude should appeal to every navigator as a possible means of getting a horizon in foggy weather1 by going aloft or getting as low as possible1 provided the fog bank is lying above or close to the water4 ,the altitude of a body is measured to the visible horizon1 yet the measurement must be adjusted to the sensible horizon before the true altitude can be obtained4 ,this correction is accomplished by applying to the observed altitude the amount of the angle formed at the observers eye by the planes of the sensible and visible horizons4 ,the angle is known as the dip of the horizon4 ,it is readily seen that this angle always makes the observed altitude too large1 for the eye if located at the exact surface of the sea1 theoretically sees the sensible and visible horizons in one1 while at every elevation above the surface it depresses the visible horizon correspondingly4 ,it is1 therefore1 always necessary to apply the dip as found in ,table #ad1 ,bowditch1 with a minus 7-7 sign4 ,an inspection of the table of dip will show that the rate of increase of this error becomes more rapid as the height of the eye is diminished4 ,to illustrate3 ,the reader will note that between an elevation of #d feet and one of #i feet there is a difference of #a minute in the dip1 while higher up1 say between #bf feet and #ch feet1 a difference of #a minute is likewise found1 yet in the first instance there was a range of #e feet and in the second a range of #ab feet4 ,this fact in itself is an argument in favor of observing altitudes at a good height above the water4 ,in calculating a meridian altitude1 an error in the dip directly affects the result by a corresponding amount1 so extra care should be exercised in this respect4 ,in this instance1 we endeavor to locate the body relative to our zenith and anything that affects its altitude directly affects the latitude4 ,in a time sight1 a different principle is involved4 ,here the position of the body as defined by the latitude locates the apex of one angle of the astronomical triangle and hence a small error in the altitude will very likely cause a greater effect on the longitude4 ,an allusion was made under the caption of ,refraction to the displacement of the visible horizon by terrestrial refraction to detect which requires watchfulness on the navigator's part4 ,the familiar 8loom0 seen along the coast is an example of the workings of variable refraction4 ,now imagine this distortion less aggravated with no land to show its existence and you have a good illustration of this error4 ,refraction of this nature is usually found during light airs and calms when the different layers of air arrange themselves according to their temperatures4 ,the heated air over land below the horizon in hot weather will displace the intervening horizon2 moreover1 when the air is warmer than the sea1 the horizon is elevated above the normal and1 when the conditions are reversed1 the horizon is unduly depressed4 ,thus lights become visible a little sooner after a hot day ashore4 ,the ,red ,sea1 ,gulf ,stream1 mouth of the ,amazon1 and other large rivers are places where the horizon should be especially distrusted4 ,capt4 ,lecky1 in his famous ,wrinkles in ,practical ,navigation1 refers to an experience he once had with this error4 ,the latitude had been found 8by an excellent meridian altitude of the sun to be as much as #ad in error4 ,the time was mid-winter--the day a clear cloudless one--the sea smooth1 and the horizon clean-cut4 ,five observers at noon agreed within the usual minute or half minute of arc1 nevertheless1 on making ,long ,island 7,u4 ,s4 ,a47 in less than two hours afterwards1 the latitude was found wrong to the amount stated4 ,many such cases have come under the writer's notice1 but this one alone is cited on account of the magnitude of the phenomenon40 ,what ,captain ,lecky said in his work on navigation is reliable and this should serve to make an impression as to the dangers of such occurrences4 ,in clear weather this displacement of the horizon may be lessened somewhat by observing from aloft4 ,by extending the horizon1 such disturbing influences as the motion of the vessel and an irregular horizon caused by rough sea are minimized4 ,in hazy weather1 however1 it is recommended to observe low1 bringing the horizon as close as possible4 ,,parallax ,in calculating the true altitude of a body the distance of its center above the horizon is supposed to be measured from the center of the earth1 or what is the same thing1 the altitude above the rational horizon4 ,the application of semi-diameter adjusts the measured angle with the center of the body1 while parallax corrects the error due to our observing from the surface of the earth to the sensible horizon1 instead of from the center to the rational horizon4 ,parallax1 in other words1 is the angle formed at the body by the lines drawn from the observer's position1 and from the center of the earth1 respectively4 ,this angle is subtended by the radius of the earth1 and it is obvious that the farther away a body is1 the smaller the angle1 and consequently the less the parallax4 ,so when dealing with planets or fixed stars1 it becomes insignificant and no parallax is considered4 ,the moon1 on the contrary1 is so close aboard that the angle of parallax reaches a value of nearly #a2 as a minute of altitude means a minute of latitude and in turn a mile1 so with this body the error due to parallax must be carefully determined4 ,in the case of the sun1 however1 it is somewhat of a waste of time to bother with parallax1 for it never exceeds #h or #i and such fine calculation is uncalled for in ordinary navigation where so many greater errors must be kept in sight4 ,however1 we desire to eliminate every known element of error without undue figures1 so it is recommended that ,table #bj;,b1 ,bowditch1 be used1 where without extra trouble the parallax may be found conveniently combined with the refraction4 ,when a body is in the sensible horizon1 the parallax is greatest4 ,the angle of parallax subtended by the radius of the earth is then an acute angle of a right-angled triangle and is as large as it can possibly be with the body at the same distance4 ,as the body obtains altitude above the horizon1 the right angle of the triangle 7at the observer7 becomes obtuse and our acute angle of parallax becomes smaller and smaller until the body reaches an altitude of #ij--;in our zenith1 when the obtuse-angled triangle has resolved itself into the perpendicular line that passes through our position and the earth's center4 ,the angle of parallax here disappears4 ,when a body is in the horizon1 its parallax is known as ,horizontal ,parallax in contra-distinction to ,parallax of ,altitude4 ,the latter has become generally known among navigators merely as parallax4 ,our position on the surface causes a body to appear lower than if viewed from the center of the earth1 so the error of parallax is added to the observed altitude2 when1 however1 it is combined with refraction it is subtracted in an observation of the sun1 but added when the moon is used4 ,the parallax of the moon is excessive because the radius of the earth becomes a considerable amount when compared with the close proximity of the body1 and causes a considerable angle at the body between the lines drawn from the observer and that drawn from the center of the earth4 ,the change in parallax is so great that it becomes necessary in order to preserve accuracy to correct the observed altitude for index correction1 dip and semi-diameter1 to secure an approximate corrected altitude before attempting to correct for parallax4 ,the horizontal parallax1 which is the angle subtended by the earth's radius when the moon is in the horizon1 is taken from the ,nautical ,almanac1 and with this and the approximate altitude as arguments1 enter ,table #ah1 ,bowditch1 and pick out1 having regard for correction tables at the right1 the parallax and refraction combined4 ,the usual corrections to the observed altitude of the sun or stars can be picked out at once from ,table #df1 ,bowditch1 where they are all combined for a quick correction4 ,,chapter ,,vi ,latitude ,,meridian ,,altitude ,it is surprising to us1 in these advanced days of nautical science1 to read of our adventurous ancestors of a century ago navigating their ships to all parts of the known and unknown world with nothing to guide them but their dead reckoning and the latitude crudely obtained by the method of meridian altitude4 ,many of our finest ships1 as late as the first decade of the nineteenth century1 sailed to ,china and back with no knowledge of their longitude save what the master guessed it to be4 ,even in later days much navigating has been done in the less lucrative trades by mariners who had no knowledge of the method of finding longitude4 ,it required more time and distance to navigate by latitude and dead reckoning only1 as it was not always safe to lay a course from an indefinite position directly for the coast4 ,it was the custom in the old days to keep off soundings until on the latitude of the port of destination1 then steer due west1 and whatever the longitude might turn out to have been the master would sooner or later make the land in the vicinity of his port4 ,the first step in obtaining the latitude by meridian altitude is the measurement with the sextant of the sun's altitude4 ,this is done when it reaches its highest point in its course across the sky2 this occurs when it bears due ,n4 or ,s4 true and this moment is local apparent noon4 ,a few minutes before this time the image of the sun should be brought to the horizon1 and by swinging the lower part of the instrument the image will be made to swing likewise in an arc2 the lowest point of its lower edge 7limb7 should then be brought in contact with the horizon as closely as the circumstances will permit4 ,the image will keep rising from the horizon1 but by using the tangent screw it can be continually brought back to contact4 ,at noon it will hang1 and dip below2 the reading of the sextant at this moment is the meridian altitude4 ,in working the problem three quantities are used and the navigator must be familiar with them3 ,the first is the zenith distance 7z71 which as its name implies is the sun's 7or stars7 distance from the zenith4 ,zenith is #ij from the horizon1 so the true altitude of the body subtracted from #ij is z the quantity desired4 ,the second element is the declination 7d71 which is the distance in degrees1 minutes1 seconds1 of the body either north or south of the equator4 ,this is taken from the ,nautical ,almanac4 ,the third and resulting quantity is the latitude1 which is the distance in degrees1 minutes1 seconds1 of the ship either north or south of the ,equator4 ,the altitude observed taken with the sextant at noon is corrected for semi-diameter1 parallax1 dip1 refraction and instrument error 7if any exists74 ,these corrections are explained in detail in ,corrections for ,observed ,altitudes4 ,the declination of the sun is constantly changing between #bc ,n4 and #bc ,s4 ,this is given in the ,nautical ,almanac for each two hours of ,greenwich mean time with the difference for each hour given for each day4 ,so it becomes necessary to ascertain the declination at the moment of observation1 namely1 at local noon4 ,this anywhere in the ,atlantic will occur subsequent to ,greenwich noon1 as the sun 7apparently7 passes around the world from the eastward to the westward once a day--#bd hours--which corresponds to #cfj of longitude4 ,the rate of travel is therefore equivalent to #ae in an hour4 ,hence if the sun crosses ,greenwich meridian and five hours later crosses the meridian of the ship1 say in #ge ,w41 the interval is #ge divided by #ae1 or #e hours4 ,during this interval the sun has changed in declination northward or southward and should be picked out of the ,almanac for #e hours ,greenwich mean time4 ,when the zenith distance and declination are at hand the latitude is obtained by a mere algebraic addition1 which is1 z d latitude2 where1 if the body bears south the z is marked 1 if north it is marked -2 if the declination is south it is marked - and if north it is 4 ,the result of the addition if - indicates south latitude1 if north latitude4 ,the meridian altitude of a star1 planet or moon is found in a similar manner4 ,the formula of z d latitude1 having regard to signs named as above1 is applicable to each4 ,the declination and the correction of the observed altitude are picked out of the ,almanac and ,bowditch tables in a somewhat different manner peculiar to each body4 ,it is found by many navigators to be more convenient to observe a body for meridian altitude by time than in waiting for the 8dip40 ,the altitude is taken at exactly local apparent noon in case of the sun and the time of meridian passage in the cases of other bodies4 ,this expedient is especially desirable in observing stars1 as the horizon is not as distinct and the 8dip0 not so easily detected as with the sun4 ,in order to secure the local mean time 7,l4 ,m4 ,t47 of a star's transit1 the ,g4 ,m4 ,t4 of the star's transit over the ,greenwich meridian is found in the ,nautical ,almanac 7p4 #if7 for the first day of the month and correct for the day by table on next page 7,n4 ,a474 ,the ship's mean time of transit will be the same1 as both sun and star hold their relative positions as the star moves from ,greenwich to the ship's meridian except for the small retardation of the sun's movement over the star's movement4 ,this is best found at the foot of page #b1 ,nautical ,almanac1 where the longitude in time gives the correction to be subtracted from 7,g4 ,m4 ,t47 of transit which will give the local mean time of transit at ship--the time to observe the star4 ,an observation of a planet is similarly handled4 ,the moon is somewhat unreliable owing to its rapid changes in position and the large corrections necessary to correct the altitude1 and is consequently rather an unpopular body to observe4 ,however1 there are times when she might prove valuable in giving position when much needed4 ,in the case of the sun the time of transit is local apparent noon1 by applying the longitude in time gives ,greenwich apparent time of local noon1 and corrected for equation of time gives ,greenwich mean time of transit4 ,it is often necessary to report the latitude at noon very quickly to the master4 ,this can be accomplished by calculating the problem to a point where the addition or subtraction of the observed altitude is all that is necessary to give the latitude4 ,the corrections are applied in advance by the estimated altitude1 and declination corrected by the estimated longitude4 ,art4 #cbe1 ,bowditch1 gives the constants to be used in four different situations4 ,,circum- ,,or ,,ex-,,meridian ,,altitude ,it frequently happens1 especially in the higher latitudes1 that an aggravating mass of cloud drives over the sun or other objects that you are chasing1 with the tangent screw1 and it is lost from view together with all hope of a meridian altitude4 ,but such an unfortunate occurrence as the loss of the mid-day latitude may be averted by employing the ,circum-meridian sight or ,ex-meridian1 as our ,english cousins call it4 ,the mariner accustomed to its use 8shoots0 the sun and notes the time by chronometer or watch4 ,or on cloudy days1 he would be standing by1 near apparent noon watching for a chance to catch a glimpse of the object through a rift of cloud1 and thereby forestall the loss of his latitude4 ,the theory of this observation is extremely simple1 being merely to add to the observed altitude1 taken before or after apparent noon when the sun is being considered1 the amount of rise or fall between the time of sight and the time of culmination1 and proceed with this amended altitude as in an ordinary meridian altitude sight4 ,the use of this method of obtaining the latitude is restricted to certain limits4 ,those who use ,bowditch ,tables will find themselves restricted to #bf minutes from the time of transit and a declination of #fc1 while ,brent's ,ex-meridian ,tables allow a greater scope and their limit of #gj of declination includes many stars that would be otherwise unavailable4 ,a good guide is to never allow the number of degrees in the zenith distance to be exceeded by the number of minutes from noon4 ,in very high altitudes circum-meridians are not to be recommended1 and the higher the altitude1 the more accurate must be the time used4 ,this is plain when it is realized that the lower the sun's altitude at noon1 the more nearly its diurnal path approaches the line of the horizon2 with the lessening curve of its course1 comes a lessening rise near noon1 hence less accuracy is needed in the exact time of sight from that of transit4 ,in the tropics1 however1 where high altitudes of the sun prevail1 the clouds do not offer such an element of bother as they do farther north or south1 and there this problem as applied to the sun loses its popularity4 ,in practice the use of the tables of ,bowditch makes this problem an exceedingly simple one1 requiring but few figures4 ,table #bg contains the value of rise of the body for one minute1 but as this rise varies as the square of the interval from noon1 it becomes necessary to resort to another table 7#bf7 of constants for a multiplier1 in lieu of the number of minutes from noon4 ,that is1 if we should multiply the amount of rise or fall for #a minute by the number of minutes from noon1 we would not be taking into account the decreasing rapidity of rise or the increasing rapidity of fall as the body approaches or leaves the meridian4 ,but ,table #bf provides a multiplier which reconciles this inequality and gives the proper correction to apply to the observed altitude4 ,this quantity is added in every case where the upper transit is observed but subtracted when a sight is taken below the pole where the conditions are reversed4 ,there are several pamphlets and books on the market from which the correction to the observed altitude may be obtained4 ,all are simple in form and with their explanations are readily understood4 ,notable among these ,ex-meridian ,tables are those by ,capt4 ,armisted ,rust1 ,u4 ,s4 ,n4 ,the circum-meridian is a reliable method of finding the latitude1 but the time used should be accurate to produce satisfactory results4 ,if1 however1 the conditions be favorable1 it is not necessary to discard this observation even if the correctness of the time is somewhat in doubt1 for in ,towson's ,ex-meridian ,tables is found this note3 8,if equal altitudes be taken before and after the meridian passage1 half the elapsed time may be employed as the hour angle for determining the reduction4 ,or1 when the altitudes before and after noon differ by only a few minutes1 the mean of the two may be reduced by employing half the elapsed time as the hour angle for reducing the mean altitude40 ,in practicing this suggestion it is necessary1 in order to preserve accuracy1 to put the vessel on the nearest east or west course during the run between these equal altitude observations4 ,this is imperative in a swift vessel4 ,the stars and planets offer themselves for use in this problem as in all others1 and here they possess special advantages of which the mariner may well avail himself4 ,indeed1 it may be said in truth that when a horizon can be obtained the latitude is always available through this problem4 ,and right here should be impressed upon the navigator the great advantage of becoming familiar with the stars1 not merely those of greatest brilliancy1 but the 8lesser lights0 that can be observed4 ,among the latter1 especial acquaintance should be sought with those whose right ascensions place them in the gaps between the larger stars1 thus almost the entire heavens are included in the scope of operations1 making the latitude and longitude practically always available1 provided again there is a horizon4 ,star charts1 planispheres1 and globes are for sale everywhere and no study is more interesting than that of the ways of these celestial travelers4 ,they appear and pass each day1 year after year1 until you consider them as old friends1 and1 as you come on deck for the mid-watch1 you look for ,orion1 for instance1 the same as you look for the members of your watch at their proper stations4 ,but we are off our course4 ,the increasing popularity of the circum-meridian and its undoubted accuracy when used with time obtained from a carefully rated chronometer1 is breaking the hold of the time-honored meridian altitude4 ,there is no waiting with cold fingers1 perhaps1 for the body to dip for this sight1 just shoot the star1 note the time and duck for the chart room to work it up4 ,the most favorable position of a body for a circum-meridian altitude is one in which the rise and fall near the meridian are slow4 ,in the case of the sun1 it was explained that a low altitude proved the best1 but1 in the case of the stars1 we find another condition2 those near the pole1 or in other words1 of large declinations1 describe such small diurnal circles that here also the change in altitude is correspondingly small1 thus fulfilling a desired condition for the successful working of this problem4 ,to illustrate this point the reader is referred to ,polaris4 ,now this star has an extremely small diurnal circle and it will be remembered that the altitude is for all practical purposes the same for a half hour either side of the meridian1 showing the extreme slowness of its movement of revolution4 ,the stars are used in the same way as the sun except1 of course1 that the distance from the meridian becomes the star's hour angle instead of local apparent time4 ,this is readily obtained as follows3 ,adding to the ,greenwich mean time the sidereal time of the preceding ,greenwich mean noon 7,nautical ,almanac71 together with the acceleration of ,greenwich mean time 7,table #i ,bowditch71 gives the right ascension of the meridian4 ,taking the difference between this latter quantity and the right ascension of the star1 we have the star's hour angle1 west1 if the right ascension of the meridian is greater than that of the star1 and east1 if contrary conditions exist4 ,the circum-meridian1 as well as the straight meridian altitude1 is available for use of stars near the meridian below the pole1 and1 as one proceeds into higher latitudes1 the pole becomes more and more elevated1 offering thereby more opportunities for practicing this phase of the problem4 ,the only feature to be remembered in this case is that the body is higher at the time of a circum-meridian than when it transits1 so the correction to be applied to the observed altitude must be subtracted 7-7 in order to obtain the meridian altitude4 ,the planets1 too1 are used by the ex-meridian altitude method1 but being wanderers in the heavens their right ascensions and declinations must be determined for the ,greenwich date from the ,nautical ,almanac4 ,the amended altitude of any body is assumed to be the meridian altitude and is used in the familiar formula z d latitude 7see ,latitude by meridian altitude72 but it must be borne in mind that the result is not the latitude at noon but at the time of sight4 ,if the observation was made say #i minutes before noon and the latitude considered to be the position at local apparent noon as in an ordinary meridian altitude1 there would be an error of #c miles from the correct position for a #bj-knot steamer4 ,another point to be guarded against is that when taking several altitudes and their corresponding times their mean cannot be obtained in the ordinary way1 but each altitude must be separately reduced and the mean taken of the results4 ,it is again necessary to diverge from the subject to impress on the mariner an urgent warning against anything but the most untiring vigilance in the care of his chronometer1 and the keeping of accurate time4 ,if this element cannot be depended upon there will be many hours of anxiety coming to him and probably sooner or later downright disaster4 ,the almost universal establishment of time signals in all good-sized sea ports of the world together with radio time signals sent broadcast allows but little excuse for not obtaining a good rate by the time a vessel is ready for sea4 ,every well-known work on navigation deals with the subject of rating chronometers and so no space will here be given to it4 ,after reading this talk on one of the most important and up-to-date observations where so much depends upon the accuracy of the time1 the reader cannot fail to appreciate this earnest admonishment4 ,,polaris ,the process of finding the latitude by means of ,polaris is valuable1 comparatively short and the result1 if the conditions are favorable1 is accurate4 ,we will consider it first in a general way4 ,the imaginary line representing the earth's axis1 if extended indefinitely1 is presumed to pierce the celestial sphere at the celestial pole1 therefore for an observer standing at our north pole this imaginary point would be exactly in the zenith and hence #ij from the horizon just as the pole is #ij from the equator1 these amounts evidently bear a relation to one another4 ,should the person at the pole leave his frigid surroundings and proceed toward the equator1 he would note that the pole had dropped lower and lower in the heavens1 precisely in proportion to his progress southward1 until at length1 when the equator 7latitude #j7 was reached1 the pole would be observed to be exactly in the horizon 7altitude #j74 ,from this it is easy to deduce the statement that the altitude of the celestial pole is equal to the latitude of the place of observation4 ,the object of this problem then is to obtain the altitude of the celestial pole4 ,this point1 unfortunately1 is marked by no star of which a direct altitude may be observed to aid the navigator in reaching this desired result4 ,there is1 however1 a star of the #b;d magnitude1 called ,polaris 7because of its proximity to the pole7 with a polar distance of only #a4 ,as all fixed stars are apparently revolving in circles around the celestial pole1 this star joins the grand procession with its little radius of #a4 ,it is plain that at no time can this star be more than the amount of this radius 7#a7 from the pole1 and when on the meridian either above or below the pole the full amount of the radius is subtracted from or added to the corrected altitude of the star to obtain the true altitude of the pole4 ,when the star is on a line passing through the pole and parallel to the horizon at its elongations as it is called1 the altitude is then equal to the latitude1 for its elevation is the same as that of the pole4 ,it requires #bd hours for this star to complete the small circle of revolution1 the same time required by every star2 its movement is necessarily very slow4 ,by computing its hour angle1 we can locate its position on this circle1 and hence by applying a correction to its altitude1 subtracting or adding according to the position of the star above or below the pole1 we will obtain the altitude of the pole4 ,a rough estimate of the position of the pole may be secured by noting the position of the ,big ,dipper1 the second star in the handle1 called ,mizar1 is approximately in line with ,polaris and the pole4 ,we will now proceed to show the method by which the hour angle is obtained3 ,in the talk on ,time1 it was stated that the local 7astronomical7 mean time plus the right ascension of the mean sun is equal to the local sidereal time2 and again1 that the right ascension of a star plus its hour angle equals local sidereal time4 ,with these facts as a basis1 the formula for latitude by ,polaris given in the ,nautical ,almanac will be followed in explanation4 ,illustration3 ,,fig4 #f4 ,the time of observation must be noted by chronometer and converted into local 7astronomical7 mean time2 this must be corrected by ,table ,,iii 7,nautical ,almanac7 in order to change this solar interval into a sidereal time interval2 to this converted time must be added the ,greenwich sidereal time of mean noon 7page #b72 that is1 the hour angle of the ,first ,point of ,aries1 or what is the same thing1 the right ascension of the mean sun2 to this sum must be applied a correction for longitude1 in time1 taken from the foot of page #b1 ,n4 ,a4 ,the sum is the local sidereal time4 ,the reason for the correction of longitude is this3 ,the difference between the right ascension of the mean sun at noon on two successive days is #c m4 #ef s41 the same as the accumulated difference between solar and sidereal time in #a day4 ,now we take from the ,nautical ,almanac this element for ,greenwich mean noon1 yet the sun has since covered the distance equal to the longitude1 and during the interval required to do this1 the sidereal time has accelerated over the solar an amount which bears the same ratio to the #c m4 #ef s41 that the longitude in time bears to #bd hours4 ,the ,nautical ,almanac handles the terms of this proportion in tabular form at the foot of page #b4 ,it is stated that the sun has traveled from the meridian of ,greenwich to the local meridian1 and it might be suggested that at the time of observation the sun has covered this amount plus the local hour angle or the local astronomical mean time4 ,this is true but the amount of local hour angle has been previously accelerated to sidereal time by the correction to local astronomical mean time4 ,with the local sidereal time enter ,table ,i 7,nautical ,almanac7 and pick out the correction to be applied according to sign to the altitude4 ,it is probably needless to say that the observed altitude must be corrected for index error1 dip and refraction before applying this latter correction1 which converts it into latitude4 ,this is called the ,nautical ,almanac method and is sufficiently accurate for navigational purposes1 but should a greater refinement be desired there are tables of further corrections given in the ,american ,ephemeris and ,nautical ,almanac4 ,it is always advantageous to get an observation of a star near twilight or dawn1 in order that a well-defined horizon may be available2 but1 in taking a sight of ,polaris1 another important feature is to be considered4 ,when the star's hour angle is at or near #f or #ah hours1 that is1 near that part of its orbit cut by a line passing through the pole and parallel to the horizon1 it is rising or falling most rapidly1 with the result that a small error in time will produce a considerable error in the hour angle1 an error of #c minutes introducing a difference of #a in the latitude4 ,it is quite worth while1 therefore1 to select a time for the observation of ,polaris when this star is near either of its culminations1 its highest or lowest positions1 where the time need not be especially accurate2 but by carefully noting the time it is possible to get good results at other times when the horizon is defined4 ,by using the position of the star ,mizar1 as suggested above1 however1 the navigator will be greatly aided in selecting the most propitious time for observing ,polaris4 ,,chapter ,,vii ,azimuths and ,amplitudes ,of all the navigational instruments now in practical use1 there are few1 if any1 that exceed the mariner's compass in usefulness to mankind4 ,the part it has played in the development of the world has been most important1 and its utility is no less to-day than in the past1 for the intercourse of nations is still guided by the compass needle4 ,with such a responsibility depending on this instrument1 it would naturally be supposed that its indications must be very accurate1 but1 on the contrary1 the needle is swayed by the slightest magnetic influence and points ,north only on rare occasions2 and in steel vessels only by mere chance4 ,the needle is drawn from true north first by the direction of the earth's magnetic force which is not coincident with the meridian owing to the position of the magnetic poles4 ,the north magnetic pole being in the extreme northern part of ,canada1 all the lines of force in the northern hemisphere converge toward this locality4 ,the needle when otherwise undisturbed lies in the direction of these lines of force and takes an angle with the meridian depending on the locality4 ,the amount of divergence from the true north1 or variation1 as it is called1 differs in different localities but is readily obtained by a glance at the chart where each compass rose shows the amount of variation at that place4 ,from a magnetic course1 or bearing1 the true course1 or bearing1 is readily found by the proper application of this variation1 which may be either easterly or westerly4 ,the true course is to the right of the magnetic course1 when considered from the center of the compass1 in easterly variation2 ,t4 ,r4 ,e4--,true-,right-,easterly4 ,remember these three words and the whole lesson is learned1 for if true is to the right in easterly variation it must be to the left in westerly2 and if true is to the right in easterly1 the magnetic course must be to the left of true course in easterly and to the right in westerly4 ,in this way the true and magnetic courses are converted one to the other at will4 ,if we were to sail always in an entirely wooden ship1 our compass troubles would be very few1 for the above would include every phase of the situation4 ,as wood is non-magnetic the compass would be uninfluenced by outside disturbances4 ,wood1 as a ship-building material1 having been so much displaced by iron and steel1 the use of these metals has brought many problems to solve in connection with the deflection of the compass needle4 ,the effect on the compass needle of the magnetism in a vessel and her cargo is known as deviation and is very complicated1 owing to many influences which are at work at all times giving an ever-varying value to this element of error4 ,the causes of deviation and its treatment in the way of compensating the compass are subjects much too extensive for this little book2 furthermore1 they are carefully dealt with in a half dozen of the well-known works on navigation1 so we will touch only on the every-day side of compass work4 ,the deviation changes with every alteration of the vessel's head1 owing to the change in position of prominent parts of the vessel's hull relative to each other1 to the compass1 and to the terrestrial lines of force 7magnetism74 ,as a result of these influences on the compass needle1 the mariner has three courses to deal with4 ,the first is the true course1 which is based on a compass whose needle points true north4 ,the second1 the magnetic course1 is taken from a compass affected by variation alone and therefore pointing to the magnetic pole4 ,the third is the compass course1 or that course actually shown by an ordinary standard compass in a steel ship1 affected by the error of variation combined with the error of deviation4 ,the combination of the deviation and the variation is the compass error and is obtained by adding the deviation and variation if both are of the same name1 the compass error taking that name2 for instance suppose we have a variation of #b ,w4 and deviation of #aj ,w41 the combined error is #ab ,w4 ,if1 however1 the variation and deviation are of different names1 it becomes necessary to find the difference between the two and name the result after the greater quantity2 thus1 with a deviation of #d ,e4 and a variation of #aj ,w41 the error is #f ,w4 ,the compass error is applied to compass course to obtain true course and vice versa by the same rule as for variation4 ,the navigator in planning his course between two positions lays the parallel rulers on these positions on the chart and carries this direction to the nearest compass rose4 ,this may be a true rose1 in which event he remembers his ,t4 ,r4 ,e4 rule1 reversing it in this case1 and with the variation given on the chart secures the magnetic course4 ,in an iron or steel vessel1 the deviation for that course must be ascertained from the deviation card by trial or from a ,napier ,diagram direct and applied to the magnetic course in order to obtain the compass course4 ,this is accomplished precisely as in finding the magnetic from the true course 7to the left if deviation is easterly and to the right if westerly74 ,the course by standard compass is now at hand by which we can steam from one selected point to the other4 ,the deviation as has been said is an ever-varying error1 and consequently it is quite impracticable to depend wholly on a fixed deviation card4 ,we may take aboard some magnetic cargo or change our latitude to a great extent1 the vessel may be pounded excessively by heavy seas1 a stroke of lightning or by stranding2 all these are causes liable to affect the deviation more or less4 ,in order to forestall the serious consequences that are liable to attend such a derangement of the normal and expected deviation1 the careful navigator takes azimuths or amplitudes on every course when practicable4 ,azimuths and amplitudes are nothing more nor less than astronomical bearings of heavenly bodies2 they indicate the true bearing of the body1 and the difference between this bearing and the bearing taken simultaneously by standard compass is the compass error4 ,the azimuth of a body is the angle at the zenith between the meridian and the vertical circle passing through the body4 ,it is customary1 however1 to consider the azimuth as measured by the arc of the horizon rather than by the angle at the zenith4 ,it is measured from the north or south point according to the latitude1 toward either the east or west point1 through #ahj4 ,an amplitude1 unlike an azimuth1 is restricted as to time of observation1 for the body must be on the horizon either rising or setting2 and should be observed when the sun is about its own diameter above the horizon and with a not excessive height of eye4 ,the amplitude is measured from the east or west point through #ij to the north or south point4 ,if the body observed has a south declination and is rising1 the amplitude will be ,east so much ,south2 if declination is north1 ,east so much ,north1 for a body rises in the ,east point when its declination is #j--on the equator4 ,the principle of the amplitude lies in the solution of a right-angled spherical triangle1 whose sides are the body's polar distance1 the co-latitude1 and the zenith distance which is #ij4 ,we desire the complement of the angle at the zenith4 ,it is unnecessary to compute an amplitude1 for in ,table #ci1 ,bowditch1 will be found the desired bearings for different latitudes and declinations4 ,the sun will be found the most satisfactory of the heavenly bodies to utilize for amplitudes4 ,there are two methods of calculating an azimuth1 one known as the time azimuth and the other as the altitude azimuth4 ,the former is the most popular owing to the tables that have been compiled1 an inspection of which facilitates the navigator in quickly obtaining the true azimuth of a body4 ,before entering the tables1 it is necessary to have as arguments the latitude and declination1 and1 if using the sun1 the local apparent time1 or for stellar work the hour angle4 ,should the star's hour angle exceed #ab hours1 #ab hours should be subtracted from it1 and the remainder used as ,p4,m4 time4 ,a planet may be employed precisely in the same manner as a star4 ,one of the simplest and most expeditious methods of securing the azimuth is by means of a diagram4 ,upon this convenient invention the bearing of a body can very quickly be taken off with a pair of rulers4 ,weir's ,azimuth ,diagram is sold by the ,hydrographic ,office for a very small sum4 ,the only argument that can be used against its use is that it requires a small table to lay it upon4 ,simple and complete directions are printed on the diagram4 ,the altitude azimuth is often computed at the same time as the ordinary ,a4,m4 and ,p4,m4 time sights1 utilizing the altitude of the body for both operations4 ,the principle involved in computing both an altitude azimuth and a time azimuth is the solution of the same astronomical triangle for the same angle1 but in the case of the altitude azimuth three sides are given 7the co-latitude1 the zenith distance1 and polar distance7 to find the angle at the zenith4 ,in the time azimuth1 two sides and the included angle are given 7the polar distance1 co-latitude and local apparent time or hour angle7 to find also1 the angle at the zenith4 ,the azimuth found by computation should be named ,north if in north latitude1 or ,south if in south latitude4 ,it has been customary to add up the logs1 divide by #b and the cosine will be half the azimuth named from the elevated pole1 but a more expeditious way is after adding the logs seek in the log haversines and find the azimuth directly but named from the opposite pole to the latitude4 ,with the correct bearing of the sun1 and its simultaneous bearing by standard compass at hand1 the compass error is found by merely taking their difference4 ,now this error1 as said before1 is composed of the sum or difference of the deviation and the variation1 so1 if either is subtracted from their sum1 or added to their difference the remainder is the other quantity4 ,the variation being always known is subtracted from or added to the compass error to obtain the deviation1 thus checking the deviation card for that particular course the vessel was steering at the time of observation4 ,with the compass error at hand1 many students become perplexed as to the proper manner of dealing with this error1 and finding from it the deviation4 ,the compass error is first named1 by considering the two bearings 7compass and true7 from the center of the compass2 if the true is to the right of the compass bearing1 the error is easterly1 if to the left1 westerly4 ,now should the variation happen to be identical with the compass error1 both in amount and in name1 there is no deviation2 if the variation is #j1 then the whole error is deviation4 ,if by chance the compass error is #j1 it indicates that the variation and deviation are equal in amount and opposed to each other in their influence on the needle4 ,the deviation1 in such a case1 naturally takes the opposite name from the variation4 ,in separating the variation from the compass error1 it is necessary to exercise a little thought and to consider what deviation applied to the given variation will produce that compass error4 ,this will be readily seen after a little practice4 ,there are1 however1 some rules which are here given1 by which the deviation can be obtained mechanically4 ,the deviation is the difference between the variation and the compass error if they are of the same name or adding them if of different names4 ,it is given the same name as the compass error unless the compass error is subtracted from the variation1 when the deviation takes the opposite name4 ,or a diagram in which the error is shown by its particular number of degrees east or west of the true north line may be drawn and the variation likewise properly shown east or west of true north4 ,if the error is to the left of the variation the deviation is west and if to the right the deviation is east4 ,,chapter ,,viii ,longitude ,the longitude of any position on the earth is its distance east or west from the meridian of ,greenwich1 which has been chosen as the meridian of origin4 ,longitude is measured on the equator eastward and westward through #ahj1 completing in this way the whole circumference of the earth4 ,the circumference of every circle comprises #cfj1 whether it is a great circle of the earth or any of the parallels which range in size from a point at the poles to a great circle at the equator4 ,there are always #cfj but the length of each degree is determined by the size of the circle4 ,thus a degree of longitude on the equator is #fj miles1 while on the #ejth parallel of latitude it is only about #ci miles1 owing to the decreasing size of the parallels of latitude4 ,a minute of longitude on the equator1 like a minute of latitude1 is equal to one mile1 but the difference between the meridians in actual distance decreases toward the poles gradually lessening the linear value of a degree of longitude4 ,thus it will be seen that when it is desired to represent a difference of longitude in distance1 it must be done in terms of departure 7miles7 corresponding to the particular parallel of latitude of the position4 ,the sun apparently moves around the earth in its diurnal motion1 covering #cfj in #bd hours1 whether the declination is north or south1 and a little simple division shows that in one hour he passes over #ae of longitude1 whatever the latitude4 ,this reduced shows that #a is passed over every #d minutes4 ,as the standard time1 the world over1 is reckoned by the movements of the sun1 it is plainly seen that when considering longitude1 a definite relation exists between time and arc 7--74 ,owing to this relation1 time and arc become interchangeable by a simple process of conversion4 ,so it follows that if we have the time at ,greenwich by a chronometer1 and through a trigonometrical calculation we determine the local mean time at the ship1 the difference in time between ,greenwich and the ship's meridian represents the longitude in time1 which is readily converted into arc4 ,the calculation involved in determining the local mean time is the solution of the astronomical triangle1 or in other words it is a problem in spherical trigonometry4 ,this triangle has its apex at the pole with one side as the polar distance 7#ij - declination of the observed body71 another side the co-latitude 7#ij - dead reckoning latitude7 and the third side the zenith distance 7#ij - the corrected altitude of the body74 ,it is one of the principles of trigonometry that with any three elements given in a triangle any of the remaining elements may be computed2 that is1 any angle or side is obtainable4 ,the solution of the astronomical triangle for various elements includes the finding of the zenith distance and from this the altitude1 which forms the main feature of the problem involved in the ,new ,navigation4 ,it also provides us with the angle between co-latitude and the zenith distance1 which is the azimuth of the body1 by which the mariner is able to ascertain the error of his compass4 ,the most important feature of the astronomical triangle is the angle at the pole1 known as the hour angle1 which when found secures for the navigator his local time4 ,the problem presents itself in the form of three sides being given to find one angle4 ,it is found by the time sight formula1 which is too well-known to need any discussion here4 ,the shape of the triangle is determined by the declination of the body1 its altitude and the latitude of the vessel1 and the polar or hour angle2 and it stands to reason that a formula will not produce the same accuracy in the hour angle with every shape of the triangle4 ,for instance1 in high latitudes or when the body has a declination approaching #ij1 the accuracy of the time sight formula becomes effected4 ,another very important point to bear in mind when observing a body with the view of computing its hour angle1 is its azimuth4 ,when the bearing is nearly east or west1 on the prime vertical1 the body is rising or falling faster than at any other time1 and an error in altitude or latitude will produce the least error in the resulting longitude4 ,the necessity for close attention to this point is increased with the latitude4 ,observations for time taken when the body has an azimuth of less than #de or over #ace are wholly unreliable4 ,the sun does not always cross the prime vertical in his daily track across the heavens1 for under certain conditions1 say during the northern winter1 he will rise southward of east and set southward of west4 ,under these adverse conditions1 the calculation of longitude is not dependable1 and the best a navigator can do when using the sun is to observe as soon as he has an altitude sufficient to clear the excess refraction existing near the horizon4 ,it is under such circumstances that star sights are of incalculable value1 for a star can always be found in a suitable position with but little waiting1 or we may employ the ,new ,navigation method1 where the azimuth of the sun is as good one place as another4 ,in order that a body will cross the prime vertical1 the latitude must be of the same name and greater than the declination4 ,in conditions cited above the declination of the sun is south and the latitude is north1 hence the body will never be on the prime vertical4 ,if the latitude is less than the declination1 the sun's diurnal track is tilted toward the zenith1 instead of away from it as when the latitude is greater1 and the result is that the sun1 while never on the prime vertical1 approaches it for a time after rising1 then recedes again4 ,it should be observed when at its nearest point to the bearing of east of west4 ,the bearing of various bodies can be readily found by an inspection of ,hydrographic ,office ,azimuth ,tables ,nos4 #ga and #abj1 the declination and latitude being used as arguments4 ,there is a method of finding the longitude known as the equal altitude method1 but it is not valuable4 ,the conditions are exacting where accurate results are required and when these conditions exist the ordinary time sight is available and at its best advantage1 so longitude by equal altitudes is not popular4 ,to secure good results1 the body must have an altitude above #gj and near the prime vertical2 and1 furthermore1 the ship must be kept on an east or west course or remain stationary4 ,the theory of the problem is simplicity itself1 and for this reason is very alluring1 but the best use that equal altitudes can be put to is the determination of chronometer error ashore1 and in these days of radio time signals even this use is almost obsolete4 ,the rule is as follows3 ,observe the sun's altitude1 simultaneously noticing the time by chronometer and clamp the sextant to prevent any chance of the altitude becoming disturbed4 ,when the sun has fallen to the same altitude as of the forenoon sight1 note the time again by the chronometer4 ,the mean of the two times1 corrected for chronometer error1 equation of time1 and the equation of equal altitudes due to change in declination1 in case of the sun1 is the ,greenwich apparent time corresponding to our local noon or our longitude in time1 which should then be converted into arc4 ,the stars and planets are available as well as the sun for the finding of longitude and when there is a distinct horizon1 stellar sights have many advantages4 ,the problem depends upon the solution of the astronomical triangle by the same formula as with the sun4 ,there are a few points of difference between a time sight of the sun and one of a star or planet needing explanation4 ,in the case of the former body1 we naturally compare the solar time of ,greenwich with the solar time of the local meridian1 but in stellar work we employ for this comparison stellar time1 or1 as it is more popularly called1 sidereal time4 ,so it becomes necessary to turn the ,greenwich mean time of the chronometer into ,greenwich sidereal time and compare it with local sidereal time4 ,the difference1 as in mean time1 is the longitude in time1 which is converted into arc in precisely the same way4 ,the ,greenwich mean time is turned into sidereal time by adding to it the right ascension of the mean sun1 taken from the ,nautical ,almanac and the acceleration for the ,greenwich mean time 7,table #i1 ,bowditch74 ,the local sidereal time is the result of an addition of the star's right ascension and the star's hour angle1 the right ascension is taken from the ,nautical ,almanac without correction if a fixed star is being considered and the computation of a time sight gives the star's hour angle4 ,the local sidereal time being the right ascension of the meridian1 it follows that the angle from the vernal equinox to the star plus the angle from the star to the meridian is what we desire2 hence the above rule for obtaining the local sidereal time4 ,should the star bear east of the meridian1 the local sidereal time may be found by subtracting the 7easterly7 hour angle from the star's right ascension or adding them as above and subtracting #bd hours4 ,reference to the ,time ,diagram1 ,fig4 #c1 will make these points clear also4 ,it is customary to add up the familiar logs of time sight--sec4 lat41 cosec4 p4 d41 cos sum1 sin1 remainder--divide by #b and seek the ,h4 ,a4 7hour angle7 in the ,a4,m4 or ,p4,m4 column of ,table #dd1 ,bowditch1 using the log as a sin2 but a more expeditious way is to use the sum of the logs as the log haversine in ,table #de and pick out the hour angle directly4 ,,chapter ,,ix ,sumner ,method ,every mariner who has reached a position in the profession where he is intrusted with the responsibilities of navigating a vessel is under obligation to the late ,capt4 ,thomas ,h4 ,sumner1 of ,boston4 ,this shipmaster discovered and developed the principle of the so-called ,sumner or ,position ,lines1 a principle which has proved of inestimable value and which1 with its subsequent improvements1 has well-nigh revolutionized the methods of navigation4 ,the discovery was purely accidental and for that reason is interesting4 ,here1 in ,capt4 ,sumner's own words1 is how it occurred3 8,having sailed from ,charleston1 ,s4 ,c41 #beth ,november1 #ahcg1 bound for ,greenock1 a series of heavy gales from the westward promised a quick passage4 ,after passing the ,azores1 the wind prevailed from the southward1 with thick weather1 after passing longitude #ba ,w41 no observation was had until near the land1 but soundings were had not far1 as was supposed1 from the edge of the bank4 ,the weather was now more boisterous1 and very thick1 and the wind still southerly4 8,arriving about midnight1 #agth ,december1 within #dj by dead reckoning1 of ,tuskar light1 the wind hauled ,s4,e4 7true71 making the ,irish coast a lee shore4 ,the ship was then kept close to the wind and several tacks made to preserve her position as nearly as possible until daylight1 when1 nothing being in sight1 she was kept on ,e4,n4,e4 under short sail1 with heavy gales4 ,at about #aj ,a4,m4 an altitude of the sun was observed1 and chronometer time noted2 but having run so far without any observation1 it was evident that the latitude by dead reckoning was liable to error and could not be entirely relied upon4 8,however1 the longitude by chronometer was determined1 using the uncertain ,d4 ,r4 latitude1 and the ship's position fixed in accordance4 ,a second latitude was then assumed #aj to the north of the last and working with this latitude a second position of the ship was obtained and again a third position by means of a third latitude still #aj further north4 8,on picking off these three positions on the chart it was discovered that the three points were all disposed in a straight line lying ,e4,n4,e4 and ,w4,s4,w41 and that when this line was produced on the first-named direction it also passed through the ,smalls ,light4 ,the conclusion arrived at was that the observed altitude must have happened at all three points1 at the ,smalls ,light1 and at the ship at the same instant of time4 ,the deduction followed that1 though the absolute position of the ship was doubtful1 yet the true bearing of the ,smalls ,light was certain1 provided the chronometer was correct4 ,the ship was therefore kept on her course1 ,e4,n4,e4 and in less than an hour the ,smalls ,light was made bearing ,e4 by ,n4 ,n4 and close aboard4 ,the latitude by ,d4 ,r4 turned out to be #h in error40 ,if the captain had worked more time sights using different latitudes1 he would have added new positions on the line to which he refers1 each placed upon it according to the latitude used4 ,had he cared to pursue his experiments farther1 and used latitudes very wide of his dead reckoning position1 he would have discovered that the resulting positions instead of lying in a straight line1 were in a curve and an arc of a circle4 ,the principle involved is very crudely illustrated in the following experiment3 ,let the reader consider himself aboard ship lying at anchor--say a full-rigged ship1 so as to insure a foremast of good height4 ,lower the dinghy and take along a sextant4 ,we start with a series of measurements to determine the angle1 as read from the sextant in the dinghy1 between the truck and the waterline about the vessel4 ,as a result of these measurements1 we discover that this angle becomes smaller as the distance from the vessel increases4 ,carrying our tests farther1 suppose when the sextant shows the altitude of the fore truck above the waterline to be #gj1 that the distance to the vessel be determined4 ,with this distance as a radius and the foremast as the center1 we row in a circle around the vessel1 the sextant will continue to read #gj all around the circle4 ,it is thus demonstrated that a circle surrounds that foremast upon which the altitude of its truck is everywhere #gj--;a circle of equal altitudes4 ,not being quite sure of this interesting fact1 perhaps1 another angle is selected by moving a little farther from the ship4 ,the sextant shows the fore truck to have an altitude of #ej2 the distance to the vessel is established1 whereupon the dinghy is rowed around the vessel with this distance as a radius4 ,again the sextant reveals no change from #ej and it is clearly shown that we have moved about on a circle of #ej elevation of the truck4 ,we can continue experimenting in this way until the distance from the ship becomes so great that some physical condition prevents our reading the angle of the truck's altitude4 ,these investigations show that there is a system of concentric circles of equal altitude about every elevated object like the little undulations we have seen so many times produced by the splash of a stone thrown into a pool4 ,these circles of equal altitudes surround not only elevated terrestrial objects but also celestial bodies1 as will now be shown4 ,as the sun is the most convenient body for this illustration1 let us substitute it for the fore truck of the foregoing experiments1 while for the waterline of the vessel we will use the point on the earth touched by a plumb-bob suspended from the center of the sun4 ,this point will fall on the equator on the #bast of ,march or thereabouts1 as the sun coming up from his southern declination crosses the equator into north declination at this time4 ,the instant of the transit is the vernal equinox4 ,now this point will be found an excellent one from which to study this problem1 but1 as this takes some time and the sun is ever on the move1 we will imagine ourselves endowed with the power of ,joshua to command the sun and moon1 which will enable us to study this phenomenon while free from the restlessness of the ,universe4 ,first of all1 it must be understood that the sun shines on one hemisphere of the earth at all times2 it matters not how the earth is tipped in relation to him1 one half of the world is always enjoying sunshine4 ,the center of the lighted area is the spot directly beneath the sun where the plumb-bob touches and about this point lies the system of concentric circles of equal altitudes of the sun4 ,under the conditions shown above1 the sun is in the zenith of the terrestrial vernal equinox1 shining on the earth for a distance of #ij in every direction2 but its altitude diminishes in direct proportion with the distance of the observer from the point of the equinox4 ,on the great circle everywhere #ij from the equinox the sun is in the horizon with an altitude of #j 7provided we disregard dip and refraction74 ,suppose the members of some intrepid expedition have reached the northern or southern pole2 they would1 at the time being considered1 see the sun in the horizon and in the direction of the meridian passing through the vernal equinox4 ,eastward along the equator #ij of longitude from the vernal equinox1 the inhabitants are just resting from the toils of the day1 for with them the sun is setting in their western horizon1 while away to the westward #ij the people are showing signs of activity1 for it is just sun-up in their eastern horizon4 ,so all around the world just #ij from this selected position and at this appointed time is a circle of equal altitudes1 namely #j1 for is not the sun seen in the horizon at all points on this circle8 ,the altitude of the sun is #ij at the point of observation and #j on its outer circle of altitude2 these are the two extremes and between them lies an infinite number of concentric circles of equal altitude for navigators to utilize4 ,the zenith distance1 derived by subtracting the altitude from #ij1 indicates the distance of each circle from the center or sun's position4 ,thus if an observation was taken by some bewildered mariner in which the altitude was found to be #hj1 the corresponding zenith distance of #aj multiplied by #fj would indicate that the altitude was taken #fjj miles from the sun's position1 or to put it in another way1 the circle of equal altitudes upon which the observer was located in this case had a radius of #fjj miles4 ,what is true of the sun on the equator regarding the principle of the circles of equal altitudes holds good throughout its range of declination1 the whole system moving north and south with the continuous change of declination and from east to west with its apparent diurnal motion4 ,in the quoted article1 ,capt4 ,sumner shows a method by which the position of a vessel may be established on some particular circle of equal altitude2 it matters not where the observed body happens to be at the time1 for with the ,nautical ,almanac and chronometer it can be located should we care to know4 ,the navigator1 however1 cares to deal ordinarily only with a very small arc of the circle embraced within his immediate whereabouts4 ,should he be somewhat uncertain of these he would simply require the use of a longer line to extend beyond the limits of his possible position4 ,except when in a latitude that differs but little from the declination of the observed body1 the circle of equal altitude will be sufficiently large to allow the mariner to represent its arc in his vicinity by a straight line4 ,thus the lines of position used to plot a vessel's position on the chart are in reality chords or tangents of the circle of equal altitude4 ,in geometry it will be remembered that we used to study about circumscribed and inscribed polygons and here we have a practical application of their use4 ,if we consider the line of position to be a tangent1 it is one side of a great polygon with a vast number of sides circumscribed about the circle of equal altitude2 and if we consider it to be a chord1 it is likewise a side of a great polygon inscribed within the circle of equal altitude4 ,it matters not1 however1 if the line or curve of position is considered a straight line1 except in the ill-chosen condition of the body near the zenith when the radius of the circle will be proportionately small4 ,if exactly in the zenith there will be no circle of equal altitude at all and the sextant will measure an altitude of #ij4 ,it is comparatively rare1 however1 that such a condition will embarrass the use of this method4 ,another point to be remembered in connection with the inscribed and circumscribed polygon propositions and one which has a practical application in the use of position lines1 is that the tangent or chord of a circle is at right angles to the radius passing through the point of tangency or center of the chord4 ,it follows that the sub-celestial or terrestrial position of the observed body1 being at the center of the circle1 is always at right angles to a line of position4 ,this important fact gives the navigator an opportunity to check his compass error each time he establishes a position line1 by comparing a compass bearing of the body taken simultaneously with the measurement of the altitude1 with the true bearing4 ,to establish a position line as ,capt4 ,sumner did and as it was done for years afterwards1 by assuming two latitudes usually #aj each side of the dead reckoning latitude1 and drawing the line through the two resulting longitudes1 is known as the chord method4 ,the two longitudes being positions on the circle a line drawn between them is a chord of the circle4 ,the work of computing a time sight is more or less laborious to everyone and with some seafarers forms their most arduous mental exercise4 ,at any rate no one wants to work any more than is necessary to insure accurate results4 ,so when establishing a position line it will often be found convenient to use the short cut known as the tangent method4 ,with the latitude by account work the observed altitude as in the ordinary time sight1 instead of assuming two latitudes4 ,seek the true azimuth in the tables or on diagram1 using the latitude and declination employed in the time sight and the local apparent time gained from it1 as the arguments4 ,the true azimuth1 it will be remembered1 always bears at right angles to the position line4 ,hence if the azimuth is laid down through the position furnished by the time sight1 the position line may also be readily plotted at right angles to the line of azimuth at the time sight position4 ,the navigator now-a-days is expected to think in position lines when he is clear of the land1 as a pilot thinks in shore bearings and marks4 ,that is1 he must see these imaginary lines of the different visible bodies1 and keep track of their availibility for his particular use4 ,it is easy to get into the habit of this1 for they are simply astronomical bearings instead of bearings on distant terrestrial objects1 with the distinction that the celestial bearing allows of a #ij correction to produce a position line4 ,the morning sun on the prime vertical with a sufficient altitude to avoid any dangerous refraction1 will produce a north and south line of position4 ,during the forenoon as the sun passes toward the meridian1 the northern end of the position line will move in direct proportion with the body's change in azimuth to the eastward and the southern end to the westward1 until at noon with the sun on the meridian we have an east and west position line4 ,it will be seen that at one moment of the day it is a very easy matter to establish a line of position2 the mere working of a meridian altitude does this4 ,this simple expedient of finding a position line was utilized a great deal as a means of making a landfall in the days before chronometers were perfected4 ,in those good old days1 before the clipper ship era1 time was not held at such a premium as in the present hustling period1 and a few days more or less at sea mattered but little4 ,the shrewd shipmasters then would keep well offshore until in the latitude of ,boston or the ,virginia capes1 as the case might be1 when they would haul due west and let her go1 making1 no doubt1 first rate landfalls1 if the old pig yoke was in good working order4 ,the value of a position line was demonstrated to the writer some years ago when bound in from the eastward and running into a heavy and very extensive fog bank somewhere southeastward of ,halifax4 ,during a break in the prevailing conditions the navigator succeeded in securing an ex-meridian sight and fortunately got a fairly good idea of the latitude4 ,the vessel was under sail and making but slow progress1 and as a result of the protracted period of overcast sky the longitude became considerably a matter of guess work4 ,the vessel1 however1 was kept on a west course with a careful allowance made for the set4 8,sir ,william ,thompson0 was kept going at regular intervals and it was surprising to see the soundings check up with the chart as the vessel approached1 crossed1 and left astern the ,roseway ,bank1 southward of ,cape ,sable4 ,one felt as sure of the position as did the old ,nantucket sailor in crossing 8,marm ,hackett's garden40 ,in cases where the soundings do not check so precisely as in this instance1 it will sometimes be found a great help to lay off to scale the depths obtained on the edge of a piece of draftman's transparent linen4 ,place it on the chart in the line of the course1 and1 should the soundings fail to agree1 move the scale forward and back or to either side1 always preserving the direction of the course1 until a position is found where the soundings on the scale agree with the depths given by the chart4 ,progress has been made in the science of navigation as in all other sciences1 and the modern shipmaster is not obliged to hold aloof from ,nantucket ,shoals and ,georges ,bank under ordinary conditions as our ancestors were compelled to do1 for with a correct chronometer and a knowledge of the position line such outlying dangers have been robbed of many of their anxiety-producing elements4 ,before showing the method of working around such places another point of value of the position line is called to the reader's attention4 ,a line of position extended until it reaches the land or some danger will indicate to the mariner the bearing of that particular point of the coast or danger4 ,if it so happens that this point is not the place of destination1 the navigator1 not being able to lay a course direct for his objective port through inability to determine the vessel's distance offshore1 overcomes the difficulty by sailing a sufficient distance at right angles1 then hauling on to a new position line parallel to the original one4 ,this is similar to what our ancestors did in the simple way cited above4 ,if the line lies in the direction of an off-lying or isolated shoal that is dangerously near the course1 an offset like that shown above will allow a course parallel to the position line to be sailed in safety4 ,here is an example to show its useful application3 ,a steamer sailing from ,st4 ,john1 ,n4 ,b41 for ,new ,york proceeded but about #aj hours on her voyage1 when she ran into a terrific gale4 ,the master was soon forced to heave his vessel to and ride it out as best he could4 ,the driving snow and mountainous seas occupied the attention of the officers in their efforts to save the steamer and in this way the dead reckoning position became a matter of mere guesswork4 ,the wind after some #bj hours in the northeast quadrant hauled to the northward1 at length blowing out in the northwest with clearing weather4 ,it was the master's intention to pass through the ,south ,channel1 between ,georges ,bank and ,nantucket ,shoals1 but as he had lost his reckoning to such an extent he hesitated about laying a course through such a danger-strewn locality4 ,in the late twilight immediately following the clearing sky1 the master succeeded in catching the altitude of a star bearing #cjj and established a line1 the direction of which led close westward of ,cultivator ,shoal 7a #f-;foot spot on ,georges ,bank74 ,so to be on the side of prudence and give this shoal a good berth1 the master steamed #h miles at right angles to this position line4 ,the course or direction of the new position parallel to the first was found to lead directly into the range of visibility of ,nantucket ,lightship4 ,so the master's mind was put at rest as he laid his course along the second position line1 knowing he would at length make the lightship4 ,it often happens that a distant mountain peak is visible and the sun is in a suitable position to establish a set of cross bearings1 using the mountain for one object and the sun for the other4 ,now with what has previously been stated1 it is hardly necessary to remind the reader that a 8line of position0 obtained from observations of the sun will be at right angles to the sun's true bearing2 therefore1 in order to judge whether these objects are properly placed to give a good intersection1 due consideration must be given to the relative bearings of the objects4 ,it is evident that the sun must bear by compass nearly in the direction of the mountain or in the opposite direction to have the position line and the line of bearing of the mountain cut at nearly right angles4 ,of course1 as with any set of cross bearings the angle of intersection may still be effectual if the lines cut at #ej to #fj1 but the nearer a #ij cut the more accurate the resulting position4 ,a position line is liable to displacement through a variety of causes among which is an inaccurate altitude and through incorrect ,greenwich mean time4 ,in the former instance1 an error of #a will displace the position one mile2 if the altitude is #a too large1 the correct position of the line will be #a mile directly away from the bearing of the body and vice versa4 ,the effect of an error in time upon a position line is to displace it bodily eastward or westward the amount of arc corresponding to the error in the chronometer2 the direction of the line is1 however1 unaltered4 ,the sun carries his system of circles of equal altitude with him from east to west as he travels along a certain parallel of latitude corresponding to his declination 7neglecting the slight change in declination74 ,it is quite evident that any arc of a selected circle1 will1 if its position is plotted on a small scale chart--say every #bj minutes--be found continuously parallel with itself4 ,and the intervals between each two plotted positions of the arc will be #e 7of arc7 the corresponding value of #bj minutes4 ,thus the displacement of the position line due to an error of time is explained4 ,if the time was slow1 the line was too far to the eastward1 if fast1 it was too far to the westward4 ,the value of a position line has been demonstrated1 yet with all it does not positively establish the position of a vessel4 ,the mariner in locating his vessel in a harbor does not usually stop after he has taken one bearing1 but proceeds to find another object whose bearing will make a favorable 8cut0 with the first1 and thus at their intersection determines his position4 ,as a further check against possible error a third object may be chosen and1 if the three bearings plot without forming a triangle at their intersection1 a very reliable fix will be obtained4 ,what applies to terrestrial objects thus employed may be used as an illustration to be followed in taking celestial bearings4 ,if the mariner establishes a position line and knows his vessel is located at a point somewhere along it1 let him look about for another body so placed that the position line derived from it will make a good intersection with the first line2 if all data are correct this point will indicate the position of the vessel4 ,when the sun is used this is seldom possible but in lieu of another body the sun can again be employed to establish the second position line after it has moved sufficiently in azimuth to make a good cut4 ,the thought no doubt immediately arises as to the effect of the vessel's change in position during the interval4 ,this is easily taken care of by means of the course and distance run during the interval between the sights4 ,the first position line must be considered carried bodily by the vessel without change of bearing from its first position to the position of the second observation4 ,that is1 if at #i ,a4,m4 a position line was established bearing in a #ae-aie direction1 and the vessel then steamed and made good a #dj-;course for #f hours and #aj knots an hour1 when another position line was established1 the #ae-aie line of #i ,a4,m4 would be moved bodily in a #dj direction #fj miles2 where its intersection with the second line would indicate the position of the vessel at #c ,p4,m4 ,the determination of position at sea by employing two position lines of a body with the run between sights is called ,sumner's double altitude problem4 ,it has already been shown that one body1 notably the sun1 can be used to get an intersection of two of its lines of position by waiting a sufficient time between observations for the body to change its bearing at least #cj1 the nearer #ij the better4 ,the relationship between the interval of time and the amount of change of bearing varies greatly1 depending upon the latitude of the observer and declination of the body4 ,for example1 let us consider the two extreme cases3 ,suppose a mariner to be observing the sun on the equator on ,march #bast1 he will note practically no change in azimuth during the whole forenoon4 ,yet another mariner in the ,polar sea1 whose latitude differs about #ij from that of the former1 will have the sun encircling his horizon making the whole amount of the sun's movement a corresponding change in azimuth4 ,therefore it will be seen that with a low-riding sun 7or other body7 the change of azimuth is greater in a given time1 and for this reason the position lines derived from the sun are more advantageously practiced in higher latitudes1 especially in winter4 ,this is a point of great value in view of the fact that the sun's diurnal course is such that it is never on the prime vertical in northern latitudes during the winter months1 making longitudes derived from chronometer sights very unreliable4 ,but to go back to the mariner on the equator whose latitude and sun's declination so nearly agree4 ,he is in a predicament should he persist in the plan to determine his whereabouts by position lines of the sun4 ,in such an unusual case1 it would be well to resort to some other method or wait until evening and determine the ship's position by establishing the position line of some star or stars4 ,it will be but a few days before the ship's progress will cause the sun to leave his right course across the sky and take the hour circles at an angle4 ,take a case when the sun at noon has a zenith distance of #aj1 the change of azimuth during the forenoon is still small1 but suppose the bearing was noted #a hour1 or even less1 before noon and again in similar amount after noon1 a change will be found of perhaps #ij1 the difference of moving from the southeast quadrant 7if declination is south of latitude7 to the southwest quadrant4 ,in this way1 a remarkably good cut may be had within a comparatively short time4 ,the foregoing will convince the reader that he must be governed by the change of bearing and not by time elapsed1 in predicting the value of the cut of his position lines4 ,in the use of position lines1 it is necessary to bear in mind1 that1 when the body's altitude begins to approach the zenith1 or1 what is the same thing1 when the ship is getting close to the body's sub-celestial position1 the circle is getting proportionately smaller4 ,under such conditions the arcs of the circle of equal altitudes can no longer be shown as a straight line4 ,the double altitude as it is ordinarily practiced is here impracticable4 ,and even outside this impracticable area1 discretion must be shown4 ,the dead reckoning position must be proportionately accurate1 and the assumed latitudes must be brought correspondingly close together1 in order to have a shorter line of position1 because the curvature of the circle is getting sharper as the sub-celestial point is approached4 ,to put it in another way1 a smaller arc must be used in order to avoid the error due to excessive curvature4 ,very good results can be obtained by noting the time of observation by chronometer 7,g4 ,m4 ,t47 and correcting it for equation of time in order to get ,greenwich apparent time4 ,this1 if converted into arc1 is the longitude of the sub-solar position4 ,by using the ,greenwich mean time to correct the declination taken from the ,nautical ,almanac for that day1 the latitude of the sub-solar position may be obtained4 ,plot this position on the chart and use it as the center of a circle2 then with the zenith distance 7#ij - altitude7 as a radius1 draw an arc in the probable position of the vessel4 ,somewhere along this arc is the ship's position4 ,the bearing of the sun 7rather hard to get so nearly overhead7 corrected for compass error1 reversed1 and laid off from the sub-solar position will give a fair idea of the position of the vessel4 ,now by waiting a sufficient time for the sun to change its azimuth enough to make a good cut and using its new sub-solar position as a center with the zenith distance of a second observation as a radius1 an arc may be drawn which will intersect the first arc at the position of the vessel4 ,the run between the sights will1 of course1 require the first arc to be carried forward as the first position line in the ordinary double altitude problem4 ,,johnson',s ,,method ,it is not always found convenient to plot the position lines of a set of observations on a chart2 perhaps for lack of a chart of proper scale or possibly for want of the chart itself4 ,again many navigators do not take kindly to the graphic method1 but prefer to solve their latitudes or longitudes by computation4 ,in any event ,johnson's ,method comes as a relief to such persons1 saving them from the arduous duty of establishing a set of position lines by the chord method of assuming two latitudes to get two longitudes4 ,johnson's ,method can be practiced in both the double altitude problem of the sun1 where the first sight1 or position line1 is brought forward to the second sight by correcting it for the intervening run1 or where stars are used simultaneously4 ,chief among its merits is the saving of figures4 ,it is only necessary to compute two 7instead of four7 chronometer sights in order to find the ship's position1 thus obtaining a mathematically accurate result by a short cut4 ,but also a great advantage in the ,johnson ,method is that the resulting longitude is obtained by calculation and it is not necessary to plot the lines upon the chart to secure the position4 ,in using ,johnson's method it is not absolutely necessary to observe two stars simultaneously as the quick work of a good man is sufficiently close for the practical purposes of navigation4 ,it becomes evident to anyone reading the foregoing pages that every ordinary time sight places the vessel on a circle of equal altitude1 the longitude resulting from the computation1 depending on the latitude1 by dead reckoning1 used4 ,now rather than work two sights employing two assumed latitudes on either side of the supposed position1 make the calculation only once1 using the latitude by account4 ,suppose by way of explanation that the altitude of a star bearing ,s4 #ee ,e4 is observed simultaneously with that of another star bearing ,s4 #be ,e4 ,the longitudes derived by working the time sight of each should be identical1 provided the altitudes are the true altitudes1 the ,greenwich time is without error1 and the latitude used is correct4 ,a combination of accuracy1 indeed1 and one not likely to be experienced often in actual practice4 ,however1 a skillful navigator should find no great difficulty these days in always having the correct ,greenwich time at hand4 ,there is always1 of course1 an opportunity for the display of skill in measuring altitudes1 refraction particularly being an illusive element and not always easy to detect4 ,but if care has been taken to eliminate the errors as much as possible from the time and the altitude1 it is safe to consider any discrepancy between the resulting longitudes as accountable to an error in the dead reckoning latitude4 ,the method of obtaining the ship's position from the difference in the longitudes1 derived from double or simultaneous observations1 was originated by ,a4 ,c4 ,johnson1 ,r4 ,n41 and its many advantages have for years made it the most popular form among progressive shipmasters4 ,the working of this problem involves the application of a correction to each calculated longitude in such a way as to bring them into agreement4 ,the tables 7,bowditch ,tables #dg and #dh7 furnish this correction1 which is known as the longitude factor and is symbolized by the letter ,f4 ,it constitutes the change in longitude due to a change of #a in latitude4 ,this quantity changes directly with the change of azimuth of the body2 for example1 the change in longitude is nil if the change in latitude is made on a due north or south line1 and change in longitude increases as the change in latitude is made on lines bearing more and more eastward or westward4 ,so it is necessary in order to obtain these corrections to have the true azimuths of the bodies at the moment of observation to use as an argument in the table of longitude factors4 ,these are readily taken from the ,azimuth ,tables or diagram using the data furnished by the time sight4 ,the two longitudes obtained from time sights in which the same dead reckoning latitude is used1 lie on the parallel of this latitude1 but 7unless the two longitudes happen to be coincident7 the ship's position is either north or south of this parallel according to the error existing in the dead reckoning latitude4 ,if the observed azimuth of the body 7or bodies7 fall within the same quadrant or in opposite quadrants1 the correct longitude will be found to the eastward or the westward of both calculated longitudes4 ,this is clearly shown in ,fig4 #g2 both azimuths are between south and east4 ,if the observed azimuth of the body 7or bodies7 fall in adjacent quadrants say1 one between south and east and the other between south and west1 the ship's position will be found between the two calculated or erroneous longitudes4 ,the position of this true longitude is determined by means of the before-mentioned factors4 ,the factor of a longitude is the distance of the true longitude east or west of the meridian passing through the calculated or erroneous longitude1 assuming the latitude to be in error #a4 ,the moment of this factor1 it will be seen1 depends on the azimuth of the body1 which in turn determines the direction of the position line4 ,illustration3 ,,fig4 #g4 ,the combination of the two factors1 by adding if the bodies are in the same or opposite quadrants or vice versa1 is the combined error in difference of longitude due to #a of error in latitude4 ,it now becomes a matter of proportion by which to obtain the error in the dead-reckoning latitude4 ,as the combined error in difference of longitude for #a of latitude1 is to #a of latitude1 so is the difference between the two calculated longitudes1 to the error in latitude4 ,the longitude factors are based upon an error of #a1 so if the error is more than #a it becomes necessary to multiply the factor by the error in order to obtain the correction to the calculated or erroneous longitude4 ,illustration3 ,,fig4 #h4 ,an altitude may be taken of any body and after a suitable change in bearing has taken place 7not less than #cj7 a second altitude may be taken and the first longitude advanced for the run during the interval to the parallel of the latitude by dead reckoning at the time of second sight4 ,in the usual event of a disagreement in the calculated longitudes the rule of procedure is as follows3 ,with the body's true azimuth at each observation1 the difference between the longitudes and the latitude by dead reckoning1 used at second sight1 enter ,table #dg1 ,bowditch1 and take out the corresponding numbers4 ,if the azimuths are in adjacent quadrants1 these quantities should be added1 but if in the same or opposite quadrants1 they must be subtracted4 ,the result in each case gives the combined error in difference of longitude for an error of #a in latitude4 ,it is now only necessary to divide the difference between the two longitudes by this combined error and we have the error between the correct latitude and the latitude by dead reckoning4 ,now multiply the error in latitude by the number taken from ,table #dg corresponding to the first longitude1 to obtain the correction to that longitude1 and by multiplying the same error in latitude by the number corresponding to the second longitude we have the correction for that longitude4 ,the application of these corrections should bring the two calculated longitudes into agreement at the position of the true longitude4 ,some difficulty may be experienced in learning how to apply these corrections to the calculated longitudes1 but it is always easy to make a rough diagram if at all in doubt4 ,a horizontal line representing the parallel of latitude by dead reckoning at second sight may be drawn with the two longitudes plotted upon it2 establish the position lines through these longitudes by drawing them at right angles to the sun's 7or star's7 azimuth4 ,the intersection of the two position lines indicates the true longitude and a glance shows how to apply the corrections to each calculated longitude to get the true4 ,in ,fig4 #h the westerly longitude requires the correction to be applied to the east and the easterly longitude correction to the west in order to arrive at the true longitude4 ,without the use of a diagram a rule easy to remember in deciding whether to apply the correction in longitude to the eastward or westward is here given3 ,if the error in latitude is of the same name as the first letter of the bearing1 the change in longitude is contrary in name to that of the second letter and vice versa4 ,for example take the case just cited4 ,when a body's azimuth is less than #de1 it is wiser1 and insures more accurate results to work by ,new ,navigation1 or1 if sufficiently close to the meridian1 as an ex-meridian4 ,in the case of the latter the corrections in such a case are taken from the table of latitude factors 7,bowditch1 ,table #dh71 the problem being the same in principle and in solution as that described above4 ,good results are even obtained by using two ex-meridians1 one on each side of the meridian4 ,the corrections in the latitude may be applied according to the following rule1 if it is preferred to the rough diagram method3 if the error of longitude is of the same name as the second letter of the bearing1 the change in latitude is of the contrary name to the first letter1 and vice versa4 ,,the ,,new ,,navigation ,in every branch of science and industry since time immemorial a continuous process of simplification and increased accuracy has been taking place1 and amid this general evolution of working systems the science of navigation will not be found an exception4 ,even now there is a tendency to displace the time-honored chronometer sight1 together with a long list of more or less bewildering ways of obtaining latitude and longitude4 ,the advance method is popularly known as the ,new ,navigation1 yet its principles were originally brought forward by ,marcq ,st4 ,hilaire1 a ,french admiral1 nearly #dj years ago4 ,it is not a new method of finding position1 but rather an improved way of establishing a ,sumner line4 ,like many innovations it has taken all these years for navigators to become reconciled to the change and break away from the more familiar forms4 ,in order to facilitate a simple explanation of ,new ,navigation it will be brought to mind that every heavenly body has a corresponding point on the earth directly beneath it1 which bears the same relation in latitude and longitude to the earth1 that the body does in declination and right ascension to the celestial sphere4 ,to an observer at such a sub-celestial point the body is in the zenith with an altitude of #ij2 and about him lies a system of concentric circles of equal altitude1 which extends over a hemisphere of the earth #ij in every direction from the point of origin4 ,this point1 through the apparent diurnal revolution of the body1 carries this whole system of circles around the earth each day and northward and southward with the body's change in declination4 ,on the outer limit of this system of circles1 the altitude of the body is #j4 ,thus it is seen that the altitude of the body decreases and its zenith distance 7#ij - altitude7 correspondingly increases in direct proportion as the observer departs from the sub-celestial point1 and vice versa4 ,if1 for instance1 an observer is #ajj miles 7nautical7 from this point1 the zenith distance is #ajj or #a #dj and the altitude of the body is #hh #bj2 at #bgjj miles #bgjj_/fj #de of zenith distance1 and #ij - #de of altitude4 ,a feature is now introduced that has a close bearing upon the principle under discussion1 to serve as an opening view of the subject3 ,a navigator fortunate enough to have a body reasonably near his zenith1 say #e1 has at hand an extremely simple way of graphically finding his ship's position4 ,this situation has previously been described1 but is repeated to make clear the principle of ,new ,navigation4 ,the sub-celestial position of the body at the moment of observation is readily ascertained by noting the time by chronometer and recourse to the ,nautical ,almanac for its declination4 ,with the point thus established as a center1 and the zenith distance derived from the observed altitude as a radius1 swing a circle upon the chart4 ,the ship's position is somewhere on the circumference of this circle of equal altitudes4 ,this circle is now carried forward the amount and direction of the run of the vessel between this observation and a subsequent one similarly taken4 ,during this interval the bearing of the body should have changed sufficiently to make a good intersection of the circles4 ,the ship being on both circles must be at one of the two intersections1 between which the mariner can readily decide4 ,the conditions cited are comparatively unusual but show the practical use of a circle of equal altitude in its simplest form4 ,the zenith distance is ordinarily too large to become a radius for such use on the chart4 ,the circles of equal altitude are in practice so large that #aj to #dj mile arcs in the vicinity of the vessel are treated by her navigator as straight lines1 known as ,sumner or position lines4 ,these lines are1 theoretically1 chords or tangents according to the method employed in establishing the line1 but in practice the divergence from the circle is negligible1 excepting always when the body is too close to the zenith4 ,the establishment of the position line has been done in several ways for many years until the advent of this new and more expeditious method4 ,the altitude of a body at any selected time for an assumed position can be readily calculated4 ,if this altitude does not agree 7and it seldom does7 with the altitude measured simultaneously with the sextant1 corrected for the usual errors1 the assumed position is not coincident with the actual position of the vessel4 ,the navigator now proceeds to lay off from the assumed position1 the line of azimuth of the body taken from the azimuth tables1 ,weir's ,azimuth ,diagram1 or determined by observation4 ,on this line the distance between the observed and computed altitude1 expressed in minutes of arc1 is measured1 towards the body if the observed altitude is greater1 and away from it if less1 than the computed altitude4 ,the point thus indicated is a position on a circle of equal altitudes1 the arc in the immediate vicinity of the computed point being1 approximately1 the position line4 ,this line is at right angles to the azimuth for the reason that a tangent is at right angles to the radius of a circle at a given point4 ,it is now known that the ship is somewhere on this line of position1 and it is necessary to cut it with another such line to determine definitely her position4 ,if the sun is the body being observed1 it becomes necessary1 in order to provide a good angle of intersection1 to wait until the azimuth changes at least #cj1 when the observation is repeated1 a second line established1 and the first line brought forward in exact accordance with the ship's run4 ,the interval required naturally depends upon the latitude of the ship and the declination of the sun4 ,the intersection of the lines will be the position of the ship at the time of the second sight4 ,the use of stars has a decided advantage in that there are always some of these bodies available for observation lying in various azimuths2 it is practicable1 with a well-defined horizon1 to observe simultaneously two or more of these bodies whose bearings show that they would produce desirable position lines4 ,from the resulting intersections the position of the ship is secured4 ,this obviates waiting for the second line1 a feature that is always inconvenient and sometimes1 perhaps1 dangerous4 ,the calculation of the altitude is accomplished by the solution of the spherical triangle in which we have given the co-latitude 7#ij - assumed latitude71 the polar distance and the hour angle of the meridian of the assumed position4 ,thus with two sides and the included angle1 the third side or the zenith distance 7#ij - altitude7 is easily determined by either of several formulas4 ,with the use of this method all the formulas that formerly1 and still1 often puzzle the navigator to remember can be reduced to this one sight4 ,one of the most important features it possesses is that it can be utilized regardless of the altitude of the body 7except when very high71 its azimuth1 or its hour angle1 all of which are elements that have to be used under certain favorable circumstances in order to get accurate results from the older forms4 ,the navigator is now given a greater freedom in choosing bodies to observe than is found in any other method4 ,the mariner to-day has been almost entirely relieved from the labor of computing position at sea1 should he care to avail himself of a set of altitude tables1 several excellent ones have made their appearance on the market1 among them ,hydrographic ,office ,publication ,no4 #bjj4 ,from them the altitude can be selected corresponding to the conditions of any particular observation4 ,with a set of these tables a navigator is no longer required to be a mathematician or to remember the forms of a half dozen sights4 ,thus in this wonderful age the mariner's utopian dream of obtaining position at sea by inspection1 is1 in a way1 realized4 99 99 99 99 99 ,in order to illustrate the practical working of a problem by this method1 the following example is taken up point by point3 ,early on the morning of ,may #ba1 #ahii1 while in the assumed position of latitude #ee #jj ,n41 longitude #aab #jh ,e4 observed the true altitude of the star ,arcturus to be #cg #ad #ej1 bearing west of the meridian4 ,the chronometer carrying ,greenwich mean time read #bj d4 #f h4 #bj m4 #jc s4 ,the observer desired his position4 ,the problem by the ,st4 ,hilaire method resolves itself into the solution of the spherical triangle shown in ,fig4 #aj1 where two sides and an included angle are given3 ,polar distance #ij - declination 7,nautical ,almanac74 ,co-latitude #ij - latitude 7by dead reckoning74 ,hour angle of star4 ,see figure and solution below4 ,the hour angle of the sun is more readily found than that of a star4 ,it is accomplished by applying the longitude 7in time7 of the assumed position to the ,greenwich time shown by the chronometer at the time of sight4 ,this hour angle of the mean sun must be corrected by the equation of time to obtain the hour angle of the actual sun4 ,illustration3 ,,fig4 #i4 ,illustration3 ,,fig4 #aj4 ,the cosine-haversine formula serves the purposes of this problem very satisfactorily3 ,hav z hav 7,l d7 cos ,l cos d hav h which is derived from the well-known expression3 ,cos z sin ,l sin d cos ,l cos d cos h where z zenith distance2 ,l the latitude2 and h the hour angle4 ,,solution ,dec4 ,arcturus #ai #db #bi4 ,lat4 #ee #jj ,n4 7,assumed74 ,g4 ,m4 ,t4 #bj d4 #f h4 #bj m4 #jc s4 ,r4 ,a4 ,m4 #c #ea #db ,acceleration #a #jb -------------------------------------- ,lat4 #ee #jj #jj4 ,g4 ,s4 ,t4 #aj #ab #dg ,dec4 #ai #db #bi ,long4 #g #bh #cb ------------------------ -------------------------------------- ,l d #ce #ag #ca ,l4 ,s4 ,t4 #ag #da #ai ,r4 ,a4 #ad #aa #jc -------------------------------------- ,h4 ,a4 #c #cj #af ,w4 7,observer7 #eb #cd #jj ,lat4 #ee #jj #jj cos4 #i4gehei ,dec4 #ai #db #bi cos4 #i4igcgh ,h4 ,a4 #eb #cd #jj hav4 #i4bibdd -------------------- #i4jbdha nat4 hav4 4#ajehh nat4 hav4 #ce #ag #ca 4#jiahi -------- z #eb #dh #ce nat4 hav4 4#aiggg #ij #jj #jj ------------ ,computed altitude #cg #aa #be ,observed altitude #cg #ad #ej ------------ ,altitude difference #c #be4 ,illustration3 ,,fig4 #aa4 ,a ship's position is usually obtained by plotting the lines of azimuth and the position lines much in the manner shown in the chartlet4 ,the azimuth of the body at the moment of observation is readily taken by inspection from the azimuth tables or better still from ,weir's ,azimuth ,diagram1 both published by the ,u4 ,s4 ,hydrographic ,office4 ,in order to get an intersection of two lines of position and thereby ascertain the latitude and longitude at once it is assumed that the observer took an observation of another star bearing ,s4 #de ,e41 simultaneously with ,arcturus4 ,when ordinary ,a4,m4 time sights are taken the resulting longitude establishes a north and south sumner line but the latitude is by ,d4 ,r42 at noon the latitude by meridian altitude establishes an east and west line but the longitude is by ,d4 ,r4 so it is with a sumner line a position is established upon it but the position along it is by ,d4 ,r4 the latitude and longitude1 however1 can be obtained by a slight calculation without drawing the lines on the chart2 that is1 the most probable position4 the altitude difference having been determined enter table #b1 bowditch1 using the azimuth1 or its reciprocal as the case may be1 as the course1 and with the altitude difference as the distance1 pick out the difference of latitude and the departure and apply them to the dead reckoning latitude and longitude as is the usual practice4 the result is the most probable position 7according to the ,d4 ,r47 on the sumner line4 ,,chapter ,x ,,the ,,moon ,the moon is the most interesting of the heavenly bodies not only from a romantic viewpoint1 but from the astronomical as well4 ,looking at the practical side1 it is due mostly to the moon's influence of attraction on the waters of the earth that we have the highly important phenomena of the tides4 ,the moon is our nearest neighbor in the heavens2 in fact1 she is a satellite1 that is1 revolves around the earth4 ,this movement is from west to east at an average rate of #ea minutes each day4 ,the moon's orbit is elliptical with the earth lying a little out of center1 not unlike the situation of the sun in the earth's orbital ellipse but more pronounced4 ,when the moon is at the nearest point to the earth she is said to be at 8perigee0 and the point where she is most remote is called the 8apogee40 ,the moon is a non-luminous body and gives off nothing but reflected sunlight4 ,the lunar hemisphere facing the sun is therefore the only illuminated portion of the body1 and as she turns on her axis precisely as the earth does1 the same side is always towards us4 ,the astronomers have seen but one side of our satellite4 ,this solar illumination accounts for the various interesting phases of the moon which we see each month4 ,when this body in her monthly revolution around the earth passes between us and the sun1 the illuminated side is towards the sun and the dark side towards us4 ,we see no moon at this time and call it new moon4 ,two weeks later1 she has completed one-half of her revolution and is now on the other side of the earth and we are between the moon and the sun4 ,the illuminated face of the moon is now directly towards us and we call it full moon4 ,at the time of new moon1 the eastward movement quickly brings her out of range with the sun and in a couple of days we are able to see a fine crescent in the western sky4 ,this is the very edge of the illuminated face--we can see around the corner just that much4 ,day after day the moon's lighted surface becomes larger and larger until in about a week she is near our meridian at sunset and therefore at1 roughly 7depending on the time of the year71 #ij from the bearing of the sun4 ,the moon now presents to us a face one-half dark and one-half light4 ,this is called the quadrature4 ,this term also applies to the similar condition occurring a week after full moon when she is again bearing at right angles to the sun4 ,these occasions are also called the first and last quarter1 respectively4 ,the movements of the moon are very rapid4 ,she makes her revolution around the earth in #bg days1 making a change in right ascension of #cfj or #bd hours in this interval1 a change of over two minutes each hour4 ,the declination passes through its whole cycle of change from north to south and return also in #bg days2 the sun requires a year to pass through its extremes of declination and return4 ,the change of the moon's declination averages about #i per hour4 ,these facts demand careful attention when employing the moon in navigation4 ,it is a very curious and happy circumstance that in the higher latitudes when the short days of the winter sun occur1 the moon at full rides its highest declinations1 and consequently gives extra long nights of moonlight2 and that in the summer1 when the sun is in higher declination and the days are long1 the moon at full is in low declination and there is less moonlight when it is least needed4 ,the reason of these conditions is that the full moon occurs when on the opposite side of the earth from the sun and at the winter solstice when the earth's north pole is inclined away from the sun she must be inclined towards the moon passing that body in high declination4 ,the reverse conditions exist at the summer solstice4 ,another fortunate provision for lovers of moonlight nights is the fact that the plane of the moon's orbit is not in the same plane as that of the earth's orbit1 for if such were the case each time the three bodies1 the earth1 sun and moon1 came in range there would be an eclipse4 ,the new moon coming in between the earth and the sun would cause an eclipse of the sun1 and at full moon when the earth is between the sun and moon1 there would be an eclipse of the moon4 ,therefore1 there would be an eclipse twice a month4 ,this fortunately is avoided by the angle of #e that the plane of the moon's orbit takes with that of the earth4 ,as a result they only come in exact range occasionally when the moon at new and full happens to be on the ecliptic--the earth's orbit4 ,if1 to repeat1 this occurs at full there is an eclipse of the moon1 if it occurs at new1 there is an eclipse of the sun4 the moon moves eastward through the heavens on her monthly course of revolution2 it then becomes apparent that she must return to the meridian later and later each day the amount of 8retardation10 as it is called4 ,this retardation is a variable quantity dependent upon the moon's irregular change in right ascension4 ,it is caused by the moon's motion in her elliptical orbit and at the inclination which her orbit takes with the celestial equator4 ,these causes are precisely the same in character as those producing the equation of time in the conditions relative to the sun and the earth's orbit1 but those of the moon are much greater4 ,the errors causing a variation in the right ascension of the sun requiring a year where the similar conditions in the moon are brought about in a month1 which accounts for the marked changes in the moon's rate of eastward motion4 ,the average daily retardation1 or average later time in arriving at the meridian1 is very close to #ea minutes4 ,yet the extremes of retardation range from #ch to #ff minutes4 ,the average of #ea minutes daily retardation is also noticed in the later rising and setting of the moon4 ,the extreme times between successive risings or settings during the year1 while they average #ea minutes like the crossing of the meridian1 they do not maintain the same extremes1 changing on account of the latitude of the observer as well as upon her own motions4 ,at #da north the retardation on successive risings and settings ranges between #bc minutes and #a hour and #ag minutes4 ,as the vessel proceeds farther north the range is greater until near #ff north when the moon is in her average greatest declination north she does not set at all becoming circumpolar for a certain time each month4 ,in the duration of a month the moon changes her right ascension #bd hours1 where the sun takes a year to accomplish this amount as it 7apparently for navigational purposes7 moves eastward around the earth4 ,this shows the much more rapidly increasing change in right ascension in the case of our satellite4 ,thus again the moon's rapid motions are accounted for4 ,the moon's orbit around the earth is not coincident--does not lie in the same plane as the earth's orbit 7the ecliptic7 but takes an angle of about #e #h with it4 ,the point of intersection between the moon's orbit and the ecliptic are called nodes 7corresponding with the equinoxes74 ,the point crossed by the moon as it passes from southern to the northern side of the ecliptic is called the ascending and the other the descending node4 ,the moon's axis is very slowly describing a circle in the heavens similar to that of the earth2 and in consequence the nodes are slowly moving westward along the ecliptic year by year4 ,just as is the equinox by the movement of precession1 but at a much greater rate 7see remarks on precession elsewhere74 ,the moon's axis completes its revolution in about #ai years1 while the earth requires #bf1jjj years4 ,this is called the lunar cycle4 ,at the time in the lunar cycle when the ascending node of the moon's orbit is in range with the vernal equinox the moon has her greatest range of declination--about #eg from extreme north to extreme south4 ,she is then #bc north1 the amount that the ecliptic is from the equator and #e more1 the amount that the moon's orbit is above the ecliptic4 ,about #i years later when the moon's axis has listed in the opposite direction and the descending node coincides with the vernal equinox1 the moon's maximum declination equals #bc minus #e or #ah north or south1 a range of only about #bf4 ,illustration3 ,this figure is viewed from the sun looking towards the earth on march #bast--the vernal equinox4 ,,fig4 #ab4 ,in the autumn1 there occurs an interesting phenomenon regarding the moon called the ,harvest ,moon4 ,this is the time of unusually fine moonlight nights in which the moon rises for three or four evenings at about the same time instead of the usual rapid retardation4 ,the time the sun or moon is above our horizon depends upon its declination and our latitude4 ,as the sun moves northward in declination from ,march to ,june1 our days lengthen by the sun rising earlier and farther in the northeast1 and setting later and farther to the northwest4 ,similarly the moon in ,september is moving northward in declination very rapidly and would be rising earlier each evening were it not for its own eastward movement of revolution which causes her to slip eastward an average of #ea minutes daily and causes her later rising at night4 ,the result is that these two influences at work almost counteract each other and cause the moon to rise at about the same time for several days giving us three or four glorious moonlight nights called the ,harvest ,moon4 ,it will be seen by the foregoing that great care must be exercised in having the time of observation accurately determined owing to those rapid movements of the moon4 ,it is also a matter of great difficulty to correct the observed altitude of this body on account of numerous errors that become considerable in amounts due to her proximity to the earth4 ,and for these reasons this body is not popular for observations with the general run of navigators4 ,in the case of the semi-diameter1 considerable error is apparent and is fully described1 with parallax1 which is excessive1 under 8,corrections for ,observed ,altitudes40 ,,chapter ,,xi ,charts ,a difficulty was encountered when the early cartographers attempted to represent the earth's spherical surface on a flat sheet4 ,it can not be done1 of course1 without distortion being introduced in some manner4 ,there are various methods of taking care of this error and one is adopted for one certain purpose while another scheme is used in some other work4 ,these methods of caring for the error or distortion are known as projections1 the principal being the ,mercator1 the gnomonic and polyconic4 ,the ,mercator projection is almost universally used for navigational purposes2 the gnomonic projection facilitates the use of great circle sailing1 and the polyconic is used for surveying sheets4 ,the ,mercator chart represents the earth as though it were a cylinder instead of a sphere4 ,if we take the skin of one-half an orange1 and assume it to represent the northern hemisphere of the earth1 an attempt to forcibly bring it flat upon a table will result in the tearing or stretching of the skin4 ,it can1 however1 be brought flat to the table in a regular uniform way by cutting it in a saw tooth fashion from the stem 7pole7 to the edge 7equator71 as shown in the diagram4 ,illustration3 ,,fig4 #ac4 ,the shaded portions represent actual earth's surface and the blank parts show the error introduced by using this method4 ,in this form it is useless as a chart1 so the real parts are stretched or extended each way to the dotted lines making a complete chart4 ,it is now1 however1 without a vestige of accuracy in representing the bodies of land and water as they really exist4 ,the result would be that if a round island should be in the latitude of the top of the chart it would be stretched into an elongated island lying east and west giving a very erroneous inaccurate idea of it1 as shown by the east and west shading4 ,if1 however1 the island had been on the equator where no east and west stretching would have occurred the island would appear in its natural shape1 but the farther north or south it lies just in proportion to the latitude will it be stretched in an east and west direction4 ,such a condition will not serve the purposes of navigation and it becomes necessary to extend the degrees of latitude1 making them appear longer and longer as the equator is departed from4 ,this stretches the elongated east and west island in a north and south direction and brings it back approximately to its actual shape of a round island4 ,if there was a round island in #aj ,n4 and a similar-sized similar-shaped island in #ej ,n4 the ,mercator chart would show the northern one to be almost twice as large owing to this artificial distortion4 ,but its relative shape would remain practically correct4 ,it will be seen that the latitude scale on the sides of the chart carries an increasing value towards the north--on a chart where a degree is about long at the equator it would be about long in #fj ,n4 or ,s4 ,a minute of latitude is equal to a mile on this scale1 but it becomes necessary to use it in the latitude in which the measurement is taken4 ,if a course runs ,n4 #fj ,e4 from latitude #cj ,n4 to #dj ,n4 and the distance is desired1 take at the middle latitude at the side of the chart a convenient multiple of distance1 say #cj1 on the dividers and step off the distance4 ,or the whole course can be taken off at once and with the points of the dividers at equal distances north and south of the middle latitude read off the number of minutes of latitude lying between them4 ,in very high latitudes the ,mercator chart is not reliable4 ,the distortion becomes excessive and bearings taken will not plot correctly4 ,all the meridians on a ,mercator chart are parallel and cut the equator at right angles4 ,they all lie in a true north and south direction4 ,the parallels of latitude all lie east and west and are parallel to each other and at right angles to the meridians4 ,the degrees of longitude on the globe grow smaller and smaller as the pole is approached due to the actual convergence of the meridians1 but as all meridians are parallel on the ,mercator chart the length of a degree must be shown the same length at the top as well as at the bottom of the chart4 ,in just the proportion that the degrees of longitude have been lengthened artificially beyond their true length must the degrees of latitude be lengthened in each latitude4 ,this amount is shown in ,table #c1 ,bowditch1 reckoned as the distance in miles each parallel is from the equator by the ,mercator projection4 ,thus in latitude #dj ,n4 the distance is #bdjj or miles1 the table shows that in the construction of a ,mercator chart this parallel should be increased artificially to #bfjg4f4 ,these are called the meridional parts4 ,on a ,mercator chart the ship's course is represented by a straight line and cuts each meridian at the same angle and is called a rhumb line4 ,for all practical purposes on short runs this rhumb line is the best to use1 but it is not the shortest distance between two points4 ,should you be able in a course a thousand miles long1 to see your port of destination your rhumb line course at the outset would not head your ship for it1 but 7in northern latitude7 to the southward of it4 ,however1 as you proceeded the ship's head would gradually draw towards the port and you would eventually arrive4 ,what appears to be a straight line on this chart is really a curve on the sphere of the earth4 ,your line of actual vision is a great circle1 and in order to follow such a bee line you must constantly change your compass course 7on a long run7 and describe a curve on the ,mercator chart unless the ship is headed north or south or east or west along the equator1 in which cases she is sailing on a great circle4 ,the well-known ,hydrographic ,office ,pilot ,charts are on the ,mercator projection and show all steamship tracks as curves1 for they are great circles4 ,the gnomonic chart is based on a projection of the earth's surface upon a plane tangent to any chosen point which is to be the center of the chart4 ,the eye is assumed to be at the center of the earth looking outward to the point of tangency4 ,it will be seen that the surface of the earth adjacent to the point of tangency will be very accurately shown on the chart1 but becomes distorted gradually from the center1 the sides of which show the land in such an unnatural shape that it is hardly recognizable4 ,with a chart on this projection great circle sailing is much simplified4 ,the straight line between two points indicates the great circle to follow1 and the course and distance is obtained by following the directions and illustrated example given on each chart4 ,they are constructed for the different oceans and are for sale by the ,hydrographic ,office4 ,the course can be transferred to a ,mercator chart by taking successive positions from the gnomonic chart and plotting them according to latitude and longitude1 and joining by straight or curved lines4 ,in setting out on a voyage the port of destination could it be seen ahead would indicate the great circle course1 and in order to continue to head directly for it1 the course must be continually changed4 ,while in the ,north ,atlantic bound for ,europe the course must be changed constantly to the east 7right7 in order to remain on the great circle--the straight and shortest distance4 ,the course and distance can be computed by the form given in ,bowditch1 in which two sides and an included angle are given to find the other side and the 7course7 angle at the point of departure4 ,the co-latitudes of the points of departure and destination and the angle between them at the pole1 are respectively the sides and the included angle4 ,however1 the gnomonic chart gives the course and distance graphically4 99 99 99 99 99 ,when a chart is purchased or received from the ,government offices the date of issue stamped upon it should be carefully noted4 ,it can safely be taken for granted that the chart has been corrected up to that date and it is incumbent upon the navigator or master to seek in all ,notices to ,mariners subsequent to this date for any that affect the chart4 ,if a ,notice contains information requiring a correction the number of the chart appears in boldface type4 ,the alterations should be made neatly with ,india waterproof ink1 and if by the nature of the information it is impracticable to make the changes a note should be made in a conspicuous place4 ,charts are printed from copper1 zinc or aluminum plates and small changes easily made by hand are not changed on the plate until an accumulation of errata make it necessary1 or sweeping changes of a more extensive nature takes place such as a new survey1 dredged channels1 etc4 ,a chart under extensive correction is brought up to date in every particular1 including the latest geographic spelling1 new docks and public works4 ,the date is noted on the right of the center margin and the dates of smaller hand corrections are indicated at the lower left corner2 the figures denote the number of the weekly ,notice to ,mariners1 in which the information is found1 and the year4 ,the different scales of charts range from those of the world to a harbor plan4 ,there are charts of oceans2 general coasts1 such as from the ,st4 ,lawrence to below ,new ,york2 intermediate coasts1 as from ,eastport1 ,maine to ,cape ,ann2 and approaches to ports1 say from ,cape ,ann to ,cape ,cod for the port of ,boston2 and lastly there are harbor plans4 ,those covering large areas are known as small-scale charts while harbor charts are called large-scale charts4 ,a chart depends on the surveys that furnished its data1 and its accuracy and reliability rests upon that survey4 ,even with the most careful surveys1 where the lead is used to ascertain depths1 there are many instances where pinnacle rocks have escaped detection by coming between the casts of the lead taken by the surveying party4 ,these isolated rocks become points of great danger to vessels of deep draft1 and it becomes a measure of safety to avoid rocky coasts and offshore patches by giving them a wide berth4 ,spaces devoid of soundings may well be viewed with suspicion if in reasonably shallow water1 for it would appear to indicate a lack of thoroughness in the survey1 at least a lack of soundings4 ,the wire drag1 a device used to sweep important areas to a certain depth1 is the only sure way of discovering all the dangers of the bottom4 ,the aids to navigation shown on the charts are described by symbols and abbreviations as fully as possible with the limited space4 ,all symbols are placed in the location of the aid1 but in some cases the actual position may be in doubt by the nature of the symbol1 for instance a buoy's location is denoted by the ring that accompanies the symbol and not the triangle2 a light vessel by the position of the dot of the light1 or between them if there are two dots 7lights74 ,buoys and light vessels often drag their moorings or go adrift entirely1 especially in the winter season4 ,it is therefore the part of wisdom to check a ship's position by shore marks when possible and be prepared to find buoys out of position4 ,the mechanism of a light buoy is often disarranged through various causes4 ,the characteristics of all lights are briefly given with the visibility and height above the sea4 ,the charted visibility is the distance they should be seen from a vessel's deck on which the height of the eye is fifteen feet above the sea1 so1 from the deck of an ordinary power boat a light will not be seen until well within the range of visibility as published1 while from the deck of a large steamer the light will be seen outside its charted visibility4 ,this refers to high-powered lights where the curvature of the earth has to be given consideration4 ,a flashing light is one in which the flash is of less duration than the eclipse1 while an occulting light has an eclipse equal to or less than the period of light4 ,flashes and eclipses are often grouped and receive the name of group flashing or group occulting4 ,an alternating light is one in which two colors are shown each for an equal interval with no intervening eclipse1 but if an eclipse separates the color flash from a white flash1 for instance1 it becomes a flashing white light varied by a red flash4 ,it is a very common practice to insert sectors of different colors into the arc of visibility of a light in order to cover a dangerous shoal or to indicate a channel4 ,bearings defining these sectors are taken from seaward and not from the light4 ,the term luminous range will be met with1 and indicates the distance the power of the light can carry the visibility irrespective of an intervening horizon4 ,a light may have a luminous range much in excess of its visibility which is limited by the horizon but in a haze or fog its penetrating power will greatly exceed that of a light of similar visibility but less luminous range4 ,the power of a light is more commonly shown by units of a thousand candle power1 thus1 #e4f indicates a power of fifty-six hundred candles4 ,the catoptric 7,c47 light employs the reflecting1 and the dioptric 7,d47 the refracting principle4 ,there is a large amount of useful information given on every chart that the average mariner allows to escape his notice4 ,this failure on his part is mostly due to familiarity1 or reliance on pilots with local knowledge4 ,the first important feature of a chart to be considered is the shoreline1 which is shown as a continuous line representing the high-water mark4 ,this1 it must be borne in mind1 is much changed at low water1 and where the range of tide is large the shoreline is proportionately in error4 ,again1 where the water is shallow the change is more marked than where the shores are steep-to4 ,if account is not taken of the stage of the tide it is easy to be very much deceived4 ,in approaching a strange harbor the chart should be scrutinized for prominent marks1 and these identified as soon as possible1 then the lesser objects can be picked up by their relative positions with the already identified landmarks4 ,among the lesser marks may be found cliffs1 boulders1 sandy beaches1 vegetation1 buildings 7particularly church spires and houses with cupolas74 ,prominent elevations of land always serve to identify a locality4 ,the chart shows these elevations clearly by contour lines4 ,a twenty-foot contour line1 for instance1 shows the line of the cut1 should the hill be sawed off twenty feet above the sea4 ,when contour lines are wide apart the land has a gradual slope1 and as the grade becomes steeper the contour lines come closer proportionately4 ,the chart1 wherever possible1 represents the earth as seen from overhead1 but in the case of vertical objects1 they are of necessity shown horizontally4 ,one of the notable cases of this is in the representation of cliffs1 which in order to show their height are drawn with the side view as seen from the water4 ,numbers seen on the land show the height above the high water4 ,from the topographical features of a chart we turn to those of hydrography4 ,all the depths indicated on a chart are those existing at mean low water1 on the ,atlantic ,coast1 and mean lower low water on the ,pacific ,coast4 ,all the ,british ,admiralty and most of the ,hydrographic ,office charts are reduced to the level of low water at ordinary spring tides4 ,while there is usually more water to be expected than shown1 it must be remembered that when the plane of mean low water is used1 the low waters that1 roughly speaking1 come between the moon's first and last quarters1 will fall below the soundings on the charts4 ,illustration ,illustration ,the effect of an abnormal barometer and high winds must at times be borne in mind1 for a continuous northwest wind will make a vital difference in the depths along the ,atlantic seaboard1 especially in the ,delaware and ,chesapeake ,bays4 ,very often there is information that can not be symbolized on the chart and is placed in italics in the form of a note4 ,these are always important and should be read carefully4 ,sailing directions are written to supplement the charts and preserve for the mariner a mass of information which otherwise would not reach him4 ,it is always well to pay attention to the current arrows1 as they are a means by which the strength and direction of the tidal stream may be ascertained4 ,the symbol of a tide rip should not be ignored by one in a small boat1 as the conditions might be right to make them dangerous4 ,on the water areas of the chart we find contours of depths as on the land are contours of height4 ,it is a good scheme1 if one wanted to take the trouble to run a contour line indicating a depth a few feet greater than the draft of his boat1 and tint the shallow water with a brush4 ,large vessels of deep draft upon approaching the coast are guided by the ten-fathom curve unless the shore is very steep-to and the water very deep4 ,the masters of such vessels would remain outside that curve until their position was well established4 ,lighter draft vessels are guided by the five-fathom curve in a similar manner4 ,when approaching the land and the landmarks are not available1 the character of the bottom further assists the mariner as he sounds slowly towards the land4 ,the kind of bottom is indicated by abbreviations which are obvious on almost every portion of the charts4 ,the three-fathom curve is the most important to the greatest number of navigators and for this reason is made the most pronounced4 ,in the majority of charts it is shown by a 8sanded0 area within it1 but in many new charts it is heavily tinted4 ,in changing from one chart to another while working in an unfamiliar locality1 take especial note whether the soundings are in feet or fathoms4 ,it is an excellent practice when a vessel is brought to anchor and cross bearings taken1 to estimate the radius of her swinging circle by adding the amount of chain out to the length of the vessel and with this describe a circle on the chart and note if there is any danger of tailing into shoal water at any quarter4 99 99 99 99 99 ,this has been a long voyage and ,i am glad to tie up and let the printers take charge4 ,if any of my readers see places where ,i have stood into the shallow water of inaccuracy1 ,i will be grateful for a passing hail that ,i may shift helm and get out with as little damage as possible4 ,,index ,a ,acceleration for longitude1 #ia ,amplitude1 #ab1 #ig ,aphelion1 #h ,apogee1 #aci ,astronomical triangle1 #ajb ,azimuth1 #ab1 #if altitude1 #ih ,weir's diagram of1 #ih ,azimuths and amplitudes1 #ic ,c ,calendar1 #fd ,celestial latitude1 #ad longitude1 #ad ,chart1 contours1 #aee corrections1 #aea scales1 #aeb ,charts1 #adf datum1 #aee diagrams of symbols1 #aef fathom curves1 #aeh information on1 #aed symbols1 #aeb ,chronometer time1 necessity of accuracy in1 #hg ,circles of equal altitudes1 #aaj ,circum-meridian altitude1 #hb ,compass1 #ic deviation of1 #id error of1 #id naming of1 #ii ,coordinates1 #i ,corrections for observed altitudes1 #fh ,d ,day lost and day gained1 #cg ,dead reckoning1 #a ,declination1 #ae daily change of1 #bd ,deviation1 naming of1 #ajj ,dip1 #gb ,double altitude problem1 #abj ,e ,earth's orbit1 #h revolution around the sun1 #bj ,eclipses1 #ada ,ecliptic1 #ad ,equal altitude circle1 #aji ,equal altitude method1 #ajd ,equation of ,time1 #ef diagram of1 #fb error due to obliquity1 #fj ,equinoctial colure1 #bf ,ex-meridian altitude1 #hb bodies below the pole1 #hf planets1 #hf stars1 #hf ,f ,first point of ,aries1 #ad1 #bf ,g ,gnomonic chart1 #aej ,great circles1 #aj1 #ag ,great-circle sailing1 #aej ,greenwich mean time on navigator's watch1 #ed ,h ,horizon1 rational1 #aa1 #ge sensible1 #ab1 #gc visible1 #aa1 #gc ,hour circles1 #ac ,i ,index correction1 #fh ,j ,johnson's method1 #abc diagrams1 #abf ,k ,kepler's ,law1 #ef ,l ,latitude1 #gh by ,polaris1 #hh factor1 #abi formula for finding1 #hj ,laying a course1 #if ,lights1 characteristics1 etc41 #aec ,longitude1 #aja and time1 #ajb by equal altitudes1 #ajd factor1 #abe prime vertical1 #ajc stars and planets1 #aje ,lunar cycle1 #adc ,m ,mean time into sidereal1 #ajf ,mercator ,projection1 #adf diagram1 #adg ,meridian altitude1 #gh time to observe a star1 #ha to report at noon1 #ha ,meridional parts1 #adi ,middle latitude sailing1 #d ,mizar1 #hi ,moon1 #aci full1 #adj harvest1 #adc new1 #adj ,moon's declination1 #adc diagram1 #add nodes1 #adc retardation1 #adb revolution1 #adj right ascension1 #adb ,n ,nadir1 #aa ,napier's diagram1 #if ,nautical astronomy1 #e ,new navigation1 #ajb1 #abi advantages of1 #acc diagrams of1 #ace1 #acg finding position without plotting on chart1 #ach illustrated examples1 #acd ,notices to mariners1 #aea ,nutation1 #cb ,p ,parallax1 #ge horizontal1 #gf moon1 #gg ,parallel sailing1 #c ,perigee1 #aci ,perihelion1 #h ,plane sailing1 #b ,planets1 #f-g ,polaris1 #hh diagram1 #ij time to take sight1 #ib ,precession of the equinoxes1 #bg causes of1 #ca ,prime vertical1 #ab observations on1 #ajc ,q ,quadrature1 #adj ,r ,refraction1 #ga ,rhumb line1 #adi ,right ascension1 #bf1 #ej ,s ,sailings1 #c ,seasons1 the1 #bj ,semi-diameter1 #fi ,ship's time1 #dj ,sidereal time1 #dc 7local7 how to obtain1 #eb longitude from1 #ec year1 #ff ,solar system1 #e ,summer1 length of northern and southern1 #be lines1 as chords and tangents1 #aab when under the sun1 #aba ,sumner1 ,capt41 his experience1 #ajg line1 displacement of1 #aah tangent method1 #aad value of1 #aae method1 #ajg ,sun's change in azimuth1 #abj eastward movement1 #af ,t ,time diagram1 #ea ,tropical year1 #ff ,v ,vega as a pole star1 #ca ,vernal equinox1 #bj dates of1 #fg ,vertical circles1 #ab ,w ,weir's azimuth diagram1 #ih ,z ,zenith1 #aj ,,the ,,literature of ,,naval ,,and ,,marine ,,science ,on our shelves is the most complete stock of technical1 industrial1 engineering and scientific books in the ,united ,states4 ,the technical literature of every branch of naval architecture1 construction1 ordnance1 marine engineering1 seamanship and navigation is well represented1 as well as are the various other related subjects4 ,a large number of these we publish and for an ever increasing number we are the sole agents4 ,illustration3 ,printer's logo ,all ,our ,inquiries are ,cheerfully and ,carefully ,answered and ,complete ,catalogs as well as ,special ,lists are ,sent ,free on ,request 33 33 ,d4 ,,van ,,nostrand ,,company ,publishers and ,booksellers #be ,,park ,,place ,,new ,,york ,,van ,,nostrand',s ,,nautical ,,manuals #cej ,pages #e #g ,postpaid1 #c4jj ,the ,men on ,deck ,,master1 ,,mates ,,and ,,crew ,their ,duties and ,responsibilities ,a ,manual for the ,american ,merchant ,service ,by ,,felix ,,riesenberg ,commanding ,schoolship ,newport ,new ,york ,state ,nautical ,school ,,master ,chief ,,mate ,second ,,mate ,concise statement ,third ,,mate of ,,watch ,,officer their duties and ,junior ,officer responsibilities ,cadet and the laws under ,quartermaster which they work ,carpenter and live ,boatswain ,foremast ,hands ,,discipline--on board ship ,the ,books of the ,sea ,points out the things the various members of the deck department of an ocean or ,great ,lakes steam vessel may reasonably be expected to know1 and the things they may be required to do4 ,the book does not pretend to tell ,,how1 but shows ,,what a modern ,american seaman ought to know1 and to do4 ,,van ,,nostrand',s ,,naval ,,books #gcj ,pages #f #i ,postpaid #f4ej #aei ,full ,page ,plates #aj ,color ,plates ,modern ,seamanship ,by ,admiral ,austin ,m4 ,knight1 ,u4,s4,n4 ,,seventh ,,edition ,,revised ,,and ,,enlarged ,,contents--,the ,hull and ,fittings of a ,ship4 ,rope4 ,knotting and ,splicing4 ,mechanical ,appliances on ,shipboard4 ,blocks and ,tackles4 ,handling ,heavy ,weights4 ,compass4 ,log and ,lead4 ,submarine ,signals4 ,boats4 ,handling ,boats in a ,surf4 ,ground ,tackle4 ,carrying ,out ,anchors4 ,the ,steering of ,steamers4 ,the ,rules of the ,road4 ,manoeuvering to ,avoid ,collision4 ,piloting4 ,handling a ,steamer ,alongside a ,dock4 ,placing a ,ship in a ,dry ,dock4 ,weather and the ,laws of ,storms4 ,handling ,steamers in ,heavy ,weather4 ,the ,handling of ,torpedo ,vessels4 ,keeping ,stations and ,manoeuvering in ,squadron4 ,towing4 ,rescuing the ,crew of a ,wreck4 ,man ,overboard4 ,stranding4 ,hints for ,junior ,officers ,doing ,line ,duty4 ,appendix4 ,the official text book at the ,u4 ,s4 ,naval ,academy and in use throughout the service4 ,describes the handling of all vessels1 men-of-war1 merchantmen1 naval auxiliaries and small boats4 ,transcriber's ,note3 ,words may have multiple spelling variations or inconsistent hyphenation in the text4 ,these have been left unchanged unless indicated below4 ,obsolete and alternative spellings were left unchanged4 ,words and phrases in italics are surrounded by underscores1 like this4 ,those in bold are surrounded by equal signs1 like this4 ,obvious printing errors1 such as duplicate words1 backwards1 upside down1 or partially printed letters and punctuation1 were corrected4 ,final stops missing at the end of sentences and abbreviations were added4 ,the following items were changed3 8access0 to 8excess0 8,manoeuvring0 to 8manoeuvering0 8effect0 to 8affect0