The whys and wherefores of navigation

A thorough understanding of time, one of the most important elements in navigation, clears the way to a better idea of the theory of finding one’s position at sea; there is, in the minds of many, considerable fog hanging about certain portions of this subject, and it is hoped that this explanation will clear some of this away.

Worcester’s Dictionary defines time as measured duration. It is the interval between events. It flows ceaselessly and with uniformity, yet the mortal mind is unable to conceive its beginning or its end. Man, in order to measure his activities, has blocked it off into different denominations convenient for his uses. Of these, the navigator uses the following in his determinations and reckonings: years, months, days, hours, minutes and seconds.

Certain astronomical phenomena were naturally enlisted by the ancient astronomers to furnish standards for time measurements; the value of a year was determined by the time necessary for a complete revolution of the earth around the sun, while the length of a day was fixed by the time of a rotation of the earth on its axis. The precision with which these evolutions are accomplished gives the required accuracy. The revolution of the earth governs the change of seasons, while the rotation is responsible for the alternating periods of day and night. With the exception of the month, the other measurements of time mentioned above are denominations of these standards. The month, one-twelfth of a year, is measured by the revolution of the moon around the earth.

Solar time, as its name implies, is measured by the apparent diurnal movement of the sun. It is the variety of time in universal use by which is regulated the daily activities of life; and this is indeed quite natural, for of necessity the work and play of the world depend upon the light and darkness that this body serves out to us.

While we are unconscious of the earth’s rotation, its effect is seen in the apparent daily course of the sun across the heavens, caused by our turning past it, yet in common practice the sun is assumed to revolve around the earth, and is usually thus spoken of for the purpose of simpler explanation.

The time at each meridian is necessarily different from that of every other, as only one of them holds the same position relative to the sun at the same time or putting it in another way, only one meridian can cross the sun at the same time, determining local noon for those places located upon it. It is forenoon for that part of the world westward of the sun and afternoon for that portion eastward of it. As the earth turns from west to east, the places or meridians to the eastward are first favored with the sun’s light, and those meridians cross this body before those to the westward. The sun apparently moves from the eastward to the westward, crossing each meridian in succession until in a few hours it is afternoon for places to the eastward and noon with us. The sun is now in our meridian, and it is forenoon for people to the westward of us. For example, at 7 A.M., 75th meridian time, it is noon in England and dead of night in our Pacific Coast; at our noon (75th), it is late afternoon in England and breakfast time in California.

It requires 24 hours, solar time, for the sun to make its apparent revolution around the earth, this course being a circle; it contains 360° of arc. It follows that in one hour it passes over 15° of arc, while 4 minutes are required for 1° to be traveled. Thus it is evident that any arc of the circumference of the earth, or difference of longitude, which is the same thing, has an equivalent time value and vice versa. That is, the arc comprised between the meridian of Greenwich and the 60th meridian west, for instance, besides being measured as 60° W., is equal to 4 hours of time. Again 4 hours of Greenwich time indicates that the sun has crossed the Greenwich meridian 4 hours ago and is at that particular instant crossing the meridian 60° west of Greenwich. If the arc were between Greenwich and a place 60° E., the equivalent time interval would also be 4 hours, because 60° of arc is everywhere equal to 4 hours of time; but the time at Greenwich, with sun on the 60th meridian east, is 20 hours of the previous day, or 8 A.M. of the present day. Thus: May 14, 20 hours, or May 15, 8 A.M.

The meridians extend from pole to pole, and it matters not what parallel you may be on, whether north or south latitude, 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 itself. If the time at Greenwich is carried, and the local time of any other meridian is desired, turn the difference of longitude into time and apply it with regard to signs: - if west of Greenwich and + if east. The local time at any place can thus be calculated; or to go farther, if the time of any meridian is at hand, the time of any other place can be readily found.

Every meridian carries a time of its own, and the instant of the click of a telegraph key may be recorded all over the world in the local time of each locality, yet the interval between this and a subsequent click has an absolute value which is the same at every place, regardless of whether it is expressed in solar, sidereal or lunar time, and its actual value is invariable.

For convenience, on land, our country is blocked off into belts of standard time, 15° wide, each carrying the time of its central meridian. For instance, 75th meridian time is used by the eastern states, while just westward the clocks’ faces show an hour earlier time, that of the 90° belt, and so on.

It is a good rule to remember in reckoning all kinds of time that the clock’s face shows earlier time to the westward, and from this it is easy to deduce the proper application of a correction.

There are two kinds of solar time used in navigation; the first to be considered is apparent time, the kind shown by the sun dial, or measured by the sun as we see it. It is noon of the apparent day when the sun is seen with the sextant to dip while taking a meridian altitude. It is at the moment of dipping that the navigator announces 12 o’clock, and with the striking of eight bells begins a new apparent day on shipboard.

The Day Lost and the Day Gained.—The fact that the sun seems to travel from east to west, determining the local time for successive meridians or places along the way, causes an interesting condition in reckoning time aboard ship. A vessel steaming westward on a parallel sails with the sun; in the forenoon she is sailing away from it, at noon the sun overhauls the vessel and they race together, but it becomes a hopeless chase for the steamer during the afternoon. In consequence of their similar course, however, the vessel will hold the sun longer, and the length of daylight will be increased over that time allotted a stationary position in proportion to the speed of the vessel. On the other hand, a vessel steaming eastward each hour advances to meet the sun; at noon the effect is as if they pass each other, and during the remainder of the day they are moving in opposite directions, hence this vessel has a shorter term of light and is deprived of its full share of sunshine.

In practice these facts require the continuous setting back of the ship’s clock, keeping apparent time on a westbound vessel. Take a concrete case for illustration: to-day assume we are at sea on the 45th meridian west and set the clock at the dipping of the sun, apparent noon; the vessel is westbound, steaming along the equator, and rolls along at a good 15-knot clip. In 24 hours by the clock we will cover 360 miles, or 360´ of arc on the equator, which is equal to 6° difference of longitude. (Should the easting or westing be made in higher latitudes, the difference of longitude will be increased proportionately.) So when the ship’s clock shows noon we will be 6° farther west than at the preceding noon, or in 51° west. The navigator, should he observe the sun, would find it had not reached its highest altitude (the meridian), and he would be obliged to wait (approximately) 24 minutes, the equivalent in time of 6°, before the sun would dip. The clock is carrying 45th meridian time, and we are now determining noon for the 51st meridian. He sings out 8 bells, but the clock shows 12.24 P.M. The ship has gained 24 minutes by sailing with the sun, and the clock is set back and a fresh start is made.

A vessel sailing east has the opposite experience. The navigator, if guided by the ship’s clock, would find that the sun had dipped some 24 minutes before noon if a run similar to the above mentioned was made eastward. In this case the apparent time of the 51st meridian is shown by the clock, while the ship has moved on to the 45th, and the time of noon is correspondingly approximately 24 minutes earlier than the clock admits.

In the above example, the clock in the first instance is 24 minutes fast and is set back that amount to correct it for the time of the 51° meridian W.; but this time cannot be thus arbitrarily thrown away without some subsequent reckoning. There is just so much time all over the world, and there are no gaps or extra intervals; it is absolute in its uniform flow. Therefore, there must be a way of squaring ourselves with Old Father Time.

But let us follow the voyage farther and see what transpires: Continuing the course westward and ignoring for convenience all intervening land, each day it becomes necessary to set the clock back 24 minutes until we have circumnavigated the earth. Suppose we took our departure from the Greenwich meridian and kept our log throughout the voyage with great care, expecting, according to our reckoning, to arrive on a Saturday, we would indeed be mystified on arrival to hear the ringing of church bells and find that it was Sunday. We have lost a whole day according to our log, by throwing away 24 minutes at a time. The time of the world goes on just the same, regardless of how we juggle the hands of the clock. Now, if we try a similar voyage eastward around the earth, we will be setting the clock ahead 24 minutes each day, and when the anchor is dropped on our return, we will discover that it is Friday instead of Saturday. The ship’s clock has skipped this 24 minutes each day, and our log is a day ahead of what it should be.

In order to prevent this difference of date, it was decided years ago to establish an international date line, which should correspond approximately, with the 180th meridian. The logs of vessels going west around the earth will be a day behind the calendar when they reach Greenwich, so a day is dropped from the reckoning when crossing the 180th meridian; that is, if it is Monday, the next day in the log will be Wednesday. On the contrary vessels bound eastward will be a day ahead when they reach their destination of Greenwich, so the date of crossing the date line is entered twice in the log, as for instance, there will appear two Mondays. By this method the accumulated errors of chasing local time, are in a measure straightened out, and ship’s logs are kept in agreement with the calendar of those at home. 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 180th meridian.

In slow cargo steamers and sailing vessels, particularly when the course creates but little departure, 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 noon. But with the development of the modern steamer, speed has increased to such an extent that the easting or westing of certain day’s runs correspond to a considerable amount of time, and to correct the clock to local time, all at once, would be a source of inconvenience and a bother. 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 longitude, thereby increasing the difference of longitude over a similar day’s run in lower latitudes. Hence, in order to more equally distribute its error, the longitude at noon is anticipated by the navigator and the clocks set at 8 A.M. for the local time of the approaching noon meridian.

When the clocks are set at noon, they are correct only for the moment and then start an accumulating error, depending in amount upon the rapidity of the easting or westing made. But by anticipating the longitude at noon, the forenoon watch will experience a decreasing error instead of one accumulated for twenty hours, and still increasing. It serves to keep the time of day more nearly correct.

In the transatlantic service, where high speed is maintained and the courses result in a large amount of easting and westing, another method is used for convenience. The navigator estimates the noon position of the next day and accordingly divides the error into thirds. The amount of the first third is applied at 11 P.M., the second at 3 A.M. and the last third at 5 A.M. By this method the error is distributed between the “first,” “mid” and “morning” watches. It is a matter of considerable moment, and no joke, to the hard-working stokers to have the clock set back on them the full amount of the day’s run all at one time; and likewise going east, it would be giving an unfair advantage to those on duty to set the clock ahead nearly an hour during the morning watch.

The apparent or real sun is not a very accurate timekeeper and its days are unequal in length. The aborigines and even our ancestors were content with the time of day indicated by the sun dial, but as the generations have passed, each bringing increased development, time has become valuable; 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 second.

It is impossible to construct a clock that will follow the irregularities of the apparent sun, so an imaginary sun has been devised which is assumed to make its revolution at a uniform speed along the celestial equator with exactly 24 hours between its transits of the same meridian. This interval is the average of all apparent days in one year.

The varying rate of the sun’s apparent motion is due to several causes which will be subsequently discussed under the Equation of Time.

The time measured by the transit and progress of the mean sun is called mean time; if at Greenwich, it is Greenwich mean noon and Greenwich mean time (G. M. T.); if it represents the time of the observer’s place or meridian, it is local mean time (L. M. T.). It is mean time that is shown by all clocks and chronometers used in every day life.

The distance between the apparent and mean suns, expressed in time, is known as the Equation of Time, and the application of this correction depends upon which sun is ahead. It is tabulated in the nautical almanac for every two hours of Greenwich mean time, with hourly differences, so it can be reduced for longitude in time to any meridian, or corrected to any intermediate Greenwich time. It is applied according to the sign accompanying it, and can be used to change apparent into mean, or mean into apparent time.

The progress of the mean sun across the sky with reference to the meridian is measured by the angle at the pole (expressed in time), between the meridian and the hour circle passing through the mean sun. This is the hour angle of the mean sun as well as the local mean time.

Civil Time is a variety of mean time, and is reckoned through 12 hours from midnight to noon, and again 12 hours from noon to midnight, dividing the day into the well-known periods of A.M. and P.M. With this kind of time, the day begins at midnight and the hour angle until noon is measured eastward through 180° of the revolution and westward through the remaining half from noon to midnight. In other words, 4:00 P.M. signifies that the sun has a westerly hour angle of 4 hours, while 8:00 A.M. indicates that the sun is 4 hours eastward of the meridian.

Astronomical time is reckoned westward through the whole 24 hours of the day, 0 hours being noon. From noon to noon is an astronomic day. Thus 5 P.M. civil is the same as 5 hours astronomical time, while 5 A.M., May 14th is the same as May 13th, 17 hours.

In every solar observation for time the real or apparent sun is observed and hence the time derived from the sight must be local apparent; to which the equation of time must be applied to convert it into local mean time. It has already been made clear that the longitude is equal to the difference between the local mean time and Greenwich mean time, or between local apparent time and Greenwich apparent time.

Sidereal Time

Sidereal is derived from the Latin word sidus, meaning of or belonging to the stars. Sidereal time is measured by the apparent diurnal revolution of the stars, resulting from the rotation of the earth. By their use the conditions which render the sun inaccurate as a timekeeper are eliminated; for the period of rotation of the earth is so regular that the passages of the stars across the meridian occur with great precision. This exactness enables the astronomer to keep the observatory clock checked to a remarkable degree of accuracy. These observatory clocks carry sidereal time, and for convenience it is customary to divide their faces into 24 instead of 12 hours.

Sidereal time is the bedrock of all time; 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 chronometers. Sidereal time is not practicable for every day use as its noon occurs, without regard to light or darkness, at every hour of day or night during a year. In March, at the time of the vernal equinox, it agrees with the solar clocks, but in September at the autumnal equinox, its noon occurs at the solar midnight.

While the sun is employed as the object of reference in solar time, it may appear strange that no particular star is thus used in sidereal; but in lieu of a definite stellar object by which to measure the sidereal movement of the heavens, we refer to the celestial vernal equinox.

This point was located in the constellation of Aries centuries ago, and hence its popular name—The First Point of Aries; but this has become a misnomer, for the point has long ago moved westward into another constellation, as discussed under the Precession of the Equinoxes. Navigators still cling to the name, however, and the equinox continues to serve its purpose, regardless of its slow drift westward.

This imaginary point of reference crosses the observer’s meridian much as the stars do, with the difference that it is always on the celestial equator and acquires no declination. 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 orbit. In other words, the distance of the equinox being infinite, lines drawn from perihelion and aphelion, respectively, to it, fail to produce an appreciable angle.

In explanation of this statement, it must be understood that for all uses on the earth the terrestrial system of direction (that is, using the bearing of the north pole as a standard, with east to the left and west to the right when the back is toward the pole) is entirely adequate, but when dealing with the direction of celestial bodies, a broader standard must be considered. North and South both have a definite place in the heavens, being the points of the extended axis of the earth, but east and west are only relative expressions. To demonstrate this: it is possible for a man, traveling westward on the Trans Siberian Railroad, to see from the rear platform, in the evening, a certain star bearing eastward. At the same moment it is possible for an officer of a transpacific liner in the early morning, to be taking a sight of this same star bearing westward. In terms of absolute direction that star bore the same from both sides of the world.

On the 21st of March the earth, sun, and celestial vernal equinox are in range, with the sun between the earth and the equinox. For a place in north latitude on the meridian of the terrestrial equinox, the sun as usual bears south at noon this day, and hence the range mentioned above bears south at that time.

This coincidence of bearing is only momentary, for the earth with its onward motion immediately moves out of range and forms an angle between the sun and the celestial equinox. At noon on the day succeeding the equinox the sun bears to the left of the so-called First Point of Aries (celestial vernal equinox). The sun according to terrestrial direction always bears south by true compass at noon, yet the First Point of Aries being at an infinite distance always bears the same by absolute direction. If this point could be seen and a bearing of it taken by compass simultaneously with the sun, it would be, perhaps, S. 1° W. and so on widening the angle, roughly speaking, a degree each day.

The interval between two successive transits of the sun across the meridian constitutes a solar day, and likewise the period required for a certain star to return to the same meridian is a sidereal day, but these two days are not of the same length. The solar day we know is 24 hours long, but its sidereal contemporary has a length of only 23 hours, 56 minutes (approximately) solar time. The sidereal clocks, however, are geared to show 24 hours, sidereal time, in 23 hours, 56 minutes, solar time. By this it will be seen that in any given period the face of the sidereal clock will show more hours than the solar clock.

Both the solar and sidereal clocks start even at the vernal equinox, about March 21st, but from then on, the sidereal clock gains on the solar time clock about 4 minutes a day until in a year it is a full 24 hours ahead, showing that there is one more day in a sidereal year than in a solar year. 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 15 days after March 21st, or two hours for each month.

In order to aid in a simpler explanation, let us again follow the earth around its orbit and note the conditions that distinguish sidereal from solar time. Let us once more assume it to be the time of the vernal equinox, the clocks, both sidereal and solar, now show 0 hours, and the sun, the earth and the First Point of Aries are in range. The earth immediately moves out of line by virtue of its onward motion, and the sun correspondingly appears to move eastward; this is imperceptible at first, however, and not noticeable without a careful measurement, as it seems to be swallowed up in the contrary (westward) diurnal movement.

After 24 hours of rotation from the instant of the equinox the earth turns the meridian until it causes the First Point of Aries to transit, marking sidereal noon of the first day. The sidereal clock at this moment reads 24 hours, but a glance at the solar clock shows 11 hours 56 minutes A.M., about 4 minutes short of (solar) noon. 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 noon, and the earth must turn this extra degree before the sun will be brought to the meridian, thus occupying the 4 minutes mentioned above. In other words, the earth turns 360° in a sidereal day but must turn about 361° in a solar day.

Three months after the vernal equinox, the angle between the First Point of Aries and the sun becomes, in round numbers, 90°, and it requires 6 hours for the earth to bring the sun to the meridian after the passage of the First Point of Aries. In plainer language, when the First Point of Aries crosses the meridian (sidereal noon) the sun is about 90° to the left—about rising in the eastern sky; the earth must make a quarter turn, or 6 hours, before it will be solar noon. Thus it will be seen that at this point sidereal time is 6 hours ahead of solar time.

In six months, when the First Point of Aries is on the meridian the noonday sun is shining on the antipodes, and it lacks 12 hours of solar noon. The difference between the sidereal and solar clocks has now reached 12 hours and through a continuation of the same process the interval between their readings, widens throughout the remainder of the year.

When the 21st of March comes around again, and the meridian presents itself to the sun and the First Point of Aries in range, a careful count of the number of times this latter point has crossed the meridian during the year, discloses 366¼ transits. That is, the earth has actually turned about its axis 366¼ times. The sun is found to have passed the meridian only 365¼ times. Counting the rotations of the earth by the number of the sun’s transits while we are revolving around him, causes the apparent loss of a day due to the earth unwinding itself once, so to speak, during the year. The accumulated difference amounts to one sidereal day. Hence it will be seen that a year contains 366¼ sidereal days of 23 hours, 56 minutes each, and 365¼ solar days of 24 hours each.

Now for a recapitulation of the subject of time.

The rotation of the earth is the real standard of measuring time intervals; the period required for this rotation does not vary. It has been suggested that the tide waves have a minute effect on the regularity of this movement, but the construction of our clocks is such, that if any variation exists we are unable to detect it. Whether we use the passages of the stars, or the transits of the sun to reckon our time, it falls back in either case upon the diurnal rotation of the earth.

Apparent time is measured by the seeming progress of the actual sun. The time of its transit of the meridian is irregular, but is always shown by its “dip,” culmination, with the sextant.

Mean time is reckoned by the revolutions of a fictitious sun, called the mean sun, and the length of one of these revolutions is the average of a year of apparent days. This, owing to its uniformity, is the time used for the everyday purposes of life. The difference between apparent and mean time is called the equation of time, and, by applying it according to signs given in the Nautical Almanac, one can be converted to the other as desired.

Sidereal time is indicated by the position of the First Point of Aries relative to the meridian; it is star time. The stars make a complete revolution of the heavens in 4 minutes less time than is required by the mean sun. Therefore the sidereal day is that amount shorter than the solar day.

The point of the celestial vernal equinox or First Point of Aries is a sort of celestial “bench mark;” besides indicating sidereal time, it serves as a point from which right ascension is measured eastward. This subject has been discussed previously, but as it is intimately associated with sidereal time, perhaps it may be made clearer since the latter has been so fully explained.

Right ascension is measured on the celestial equator, precisely the same as longitude on the earth, excepting that it is always measured eastward through the full 24 sidereal hours, contrary to the diurnal movement of heavenly bodies. Moreover, the meridian passing through the vernal equinox is called the celestial prime meridian, and sometimes the Greenwich of the heavens. There is another point of distinction, however, between this prime meridian of the sky and our meridian of Greenwich, which, while it does not effect practical navigation, has to receive consideration in the long run; our longitude values on the earth remain at all times constant, but owing to the precession of the equinoxes the celestial prime meridian is slowly moving westward, thus causing the right ascensions measured from it to become very slowly in error (50´´ yearly).

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 westward.

With all these important facts well in mind we will go ahead under a slow bell, through a few more statements which may be found a little perplexing. However, a careful study of the Time Diagram will, no doubt, drive away the haziness so often surrounding the subject of time.

The hour angle of the mean sun is the local mean time, and the hour angle of the First Point of Aries is the local sidereal time. 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 time. The right ascension of the meridian is the same thing, exactly, as the hour angle of the First Point of Aries, and both of these are identical with the local sidereal time. The sidereal time of Greenwich mean noon is the same as the right ascension of the mean sun at that time. The hour angle of a star plus the right ascension of the same star is equal to the local sidereal time.

Difference of longitude can be represented by an interval of sidereal time or by a difference of right ascension, precisely the same as by a difference of solar time. Thus with the local sidereal time calculated from an observation of a star, and the corresponding Greenwich sidereal time taken from the Nautical Almanac, the longitude is at hand, by turning their difference into arc. 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 time, has no influence on the resulting difference of longitude. The number of degrees in any arc can be the same, yet vary in linear measurement, but the same number of hours of solar and sidereal time represent the same proportionate part of a circle. It was just stated that the hour angle of a star plus its right ascension is the same as the local sidereal time; now this is also true of the sun. The hour angle of the mean sun plus the right ascension of the mean sun is equal to the local sidereal time; by means of this equality we are able to find the Greenwich sidereal time on any occasion. It is necessary to have this element in order to compare it with the local sidereal time, which we find by observation of a star, to obtain difference of longitude (in time). In page 2 of the Nautical Almanac will be found the Right Ascension of the Mean Sun at Greenwich Mean Noon; this must always be taken out for the preceding noon. We now have a measure of sidereal time to be added to a measure of mean time, but it will be remembered in early arithmetic that an apple and a peach can not be added together any more than ½ can be added to ⅓. The only course to steer is to reduce the quantities to a common denominator, or like quantities. So in handling these two varieties of time, solar time must be accelerated by adding a correction to it, or sidereal time retarded by subtracting an amount necessary to make it equal to a corresponding value of solar time. The tables for the conversion of one of these varieties of time to the other are found in the American Ephemeris Tables II and III, Bowditch Tables 8 and 9, and at the foot of pages 2 and 3, Nautical Almanac.

A practical illustration of this may make a clearer impression. In the early evening of April 8th, secured a sight of Regulus; the chronometer showed 8 days, 9 hours, 56 minutes, and 0 seconds (corrected), with other necessary elements given, the sight is worked as usual to find the star’s hour angle, which proves to be:

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 Almanac. The result is the Greenwich Sidereal Time.

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 Time.

As Greenwich mean time is the most used and is the best understood it is a very convenient practice to carry G. M. T. on the navigator’s watch. It is readily converted into any other time with ease but serves more purposes as it is without conversion. A stop watch is an excellent instrument for taking time sights where great accuracy is essential. By setting it at 0 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 G. T. of observation.

The most expeditious way to convert time into arc is to multiply the hours by 15 and add the number of minutes divided by 4 to get the degrees; multiply the remaining minutes by 15 and add the seconds divided by 4 to get the minutes; multiply the remaining seconds by 15 to get the seconds in arc:

Thus:

To change arc into time divide the degrees by 15 to get the hours; multiply the remainder by 4 and divide the minutes by 15 and add to get the number of minutes (m.); multiply the remainder of minutes (´) by 4 and divide the seconds (´´) by 15 carrying the division of tenths if desired, adding the result to get the seconds (s.):

Thus:

This may appear complicated at first but is much the quickest way of conversion. However, Table No. 7, Bowditch is always available if desired.

In getting an understanding of any time problem, that is such as changing mean time into sidereal time; obtaining the hour angle of a star or planet; in seeking the local time from the chronometer, or any time values that are found perplexing, always draw a diagram. Make the circle on the plane of the equator, with the pole as the center, and meridians radiating from it towards the circumference of the circle. Now imagine for the moment that you are at the north pole, and the date is the 21st of September. The sun is traveling in the horizon, and if the direction of the Greenwich meridian is known, this body serves as a time piece, for the angle between this meridian (direction) and the sun is the Greenwich time. This angle corresponds with twice the angle between XII hours on the watch and the hour hand; or would (disregarding equation of time) coincide with it if the watch’s face was divided into 24 hours. Likewise when the vernal equinox or First Point of Aries lies in the direction of Greenwich it is Greenwich sidereal noon, and the subsequent angle that appears through the rotation of the earth, shows the Greenwich sidereal time.

Equation of Time

It is necessary in considering this subject to reiterate some of the statements made in the preceding talk on time, but, as they are very important, no time is wasted by further impressing them on the mind. Let it be understood that the apparent orbit of the sun is actually due to the earth’s revolution around him, yet for simpler explanation it is considered to be the sun’s own revolution.

The apparent movement of the real sun is not of uniform speed and, in consequence, it has become necessary to devise a fictitious sun whose assumed revolutions around the earth are at all times regular in their rate.

The equation of time is the difference between these two suns and, as they are at times in conjunction and at other times attain a distance from each other of 16 minutes 20 seconds, and, moreover, as the real sun is sometimes ahead and again in the wake of the mean sun, it becomes evident that the equation of time is an ever varying quantity.

The irregularity of the sun’s apparent movement as compared with the uniformity of the mean sun, is subject to two causes: First, the earth travels in an ellipse, and, as the length of a degree varies in the different parts of the circumference, the motion would appear to be irregular, that is, if the sun actually traveled at a uniform rate, it would, from the above fact, appear to us to be variable in its motion; furthermore, the laws of forces only allow a body traveling in a circle the privilege of a uniform speed so the earth, owing to its varying distance from the sun, experiences a corresponding change in the amount of attraction exerted upon it by the sun and its velocity, actually becomes variable. Thus, during the winter, December and January, when they are nearest each other, the attraction is strongest and the earth increases its speed in revolution; while in June and July the earth is at its greatest distance from the sun and the attraction is less, resulting in a slowing down in the rate of the onward movement. As the sun appears to us to take on movements corresponding to those of the earth, these variable movements of the latter are seen in the apparent motion of the sun. Second, the plane of the earth’s orbit is inclined at an angle to that of the equator, which makes the sun appear to be traveling at a variable speed along the ecliptic.

With these two errors combined influencing the apparent sun, he becomes unreliable for regulating timepieces. The mean sun, which was originated to obviate these irregularities, is assumed to travel in a circle with the earth located in the centre, which disposes of the first reason for an apparent variable motion; and again, the mean sun revolves in the plane of the equator, thus eliminating the second obstacle in the way of uniform time.

Now we will continue a little farther into the explanation of the reasons for the irregular movement of the real sun. A law discovered by Kepler, and named for him, provides that a radius from the sun to the earth covers sectors of equal areas in equal times; a sector equal in area to any sector covered in the same time. That is, when the earth is in that part of the orbit near the vernal equinox, the radius of the orbit will in a given time, say a week, sweep over a certain area; the earth proceeds toward aphelion and when in the vicinity of that point, the radius becomes greatly increased in length. Now in a week with this longer radius, a far greater area would be covered if the earth maintained the same rate of speed as at the equinox, but Kepler’s law says, “equal areas in equal times,” so in order to conform with the law, the earth’s speed of revolution must be reduced. The earth does not slow down just for the sake of obeying Kepler, 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 little.

At the time of the autumnal equinox, an area similar to that of the vernal equinox is covered. As the earth approaches perihelion, 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 correspondingly. At the increased speed, the shorter radius sweeps the same area in a week as at other parts of the orbit and Kepler’s law still holds good.

As the apparent motion of the real sun corresponds exactly with the real motion of the earth, it is evident from the above that the real sun apparently moves at different rates of speed along the ecliptic, faster in winter and slower in summer than the mean sun.

The value of a mean solar day is the average of a year of apparent days, or in other words, there is the same number of mean solar days in a year as there are apparent days.

In considering the effect of the variable motion of the earth in its orbit, we will recall the conditions used when defining sidereal and solar days. The former comprises the interval between two successive passages of a certain star across the meridian, or perhaps better, between two successive passages of the meridian over a star. 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 days.

Now it requires about 3 minutes 56 seconds longer for the meridian to sweep around from sun to sun than from star to star, owing to the fact that the mean sun moves uniformly eastward that amount daily, thereby requiring the meridian, after reaching its position of yesterday noon, to overhaul the mean sun this 3 minutes 56 seconds of eastward movement. The mean sun maintains this uniform difference between its days and the length of the sidereal day. Without this daily easting of the sun, the sidereal and solar day would be the same.

But, in considering the apparent sun, we find the length of its days continually varies from both that of the sidereal and mean day. This is explained by the fact that the eastward movement of the apparent sun is due to the movement of the earth in its orbit, and as this movement becomes faster or slower the eastward movement of the sun becomes correspondingly faster or slower. Thus, we readily see that with the apparent sun moving eastward faster or slower at times, the length of the apparent day must vary accordingly and we cannot establish a uniform difference between it and the sidereal day, as in the case of the latter and the mean day. The apparent days exceed the mean days in length, between September and March, while the earth is traveling fastest in its orbit. Beginning at the autumnal equinox with the apparent sun eight minutes behind the mean sun, the former gains slowly at first but with increasing rapidity. About the end of December, at perihelion, it overhauls the mean sun and they are coincident as regards this correction only. Leaving perihelion, the apparent sun rapidly takes the lead but with a gradually decreasing amount until at the equinox in March, reaches its maximum lead of 8 minutes. Entering that portion of the year March to September, we find the earth traveling slower and the mean sun gaining on the apparent sun; between the vernal equinox and aphelion, the mean sun gains until both are together at the summer solstice and then forging ahead the mean sun attains a lead of 8 minutes in September.

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 time. The other portion is due to the obliquity of the orbit, or its inclination to the equator.

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 equator. In considering this phase of the question, we will ignore entirely, for the time being, the error of eccentricity described above.

The error of equation of time due to the obliquity of the orbit is a simple one to see, but like many simple things it is easier to show it by a diagram than to explain in words, so the reader is referred to the accompanying figure, that a study of it may be made before proceeding.