LAT. XI. A.M. X. A.M. IX. A.M. VIII. A.M. VII. A.M. VI. A. M.
1111. P.M.
I. P.M. II. P.M. III. P.M. V. P.M. VI. P.M
50° 0' 11° 36' ~3° 51' 37° 27 53° 0' 70° 43, 90° 0'
50 30 II 41 24 I 37 39 53 12 70 51 90 0
51 0 II 46 24 10 37 51 53 23 70 59 90 0
51 30 II 51 24 19 38 3 53 35 71 6 90 0
52 0 II 55 24 28 38 14 53 46 71 13 90 0
52 30 12 0 24 37 38 25 53 57 71 20 90 0
53 0 12 5 24 45 38 37 54 8 71 27 90 0
53 30 12 9 24 54 38 48 54 19 71 34 90 0
54 0 12 14 25 2 38 58 54 29 7i 40 90 0
54 30 12 18 25 10 39 9 54 39 71 47 90 0
55 0 12 23 25 19 39 19 54 49 71 53 90 0
55 30 12 27 25 27 39 30 54 59 71 59 90 0
56 0 12 31 25 35 39 40 55 9 72 5 90 0
56 30 12 36 25 43 39 50 55 18 72 II 90 0
57 0 12 40 25 50 39 59 55 27 72 17 90 0
57 30 12 44 25 58 40 9 55 36 72 22 90 0
58 0 12 48 26 5 40 18 55 45 72 28 90 0
58 30 12 52 26 13 40 27 55 54 72 33 90 0
59 0 12 56 26 20 40 36 56 2 72 39 90 0
59 30 13 0 26 27 45 45 56 II 72 44 90 0
Vertical South Dial.—Let us take again our imaginary transparent sphere QZPA (fig. 4), whose axis PEp is parallel to the earth's axis. Let Z be the zenith, and, consequently, the great circle QZP the
meridian. Through E, the centre of the sphere, draw a vertical plane
facing south. This will cut the sphere in the great circle ZMA,
which, being vertical, will pass through the zenith, and, facing south,
will be at right angles to the meridian. Let QMa be the equatorial
circle, obtained by drawing a plane through E at right angles to the
axis PEp. The lower portion Ep of the axis will be the style, the
vertical line EA in the meridian plane will be the XII o'clock line,
and the line EM, which is obviously horizontal, since M is the inter
section of two great circles ZM, QM, each at right angles to the vertical
plane QZP, will be the VI o'clock line. Now, as in the previous
problem, divide the equatorial circle into 24 equal arcs of 15° each,
beginning at a, viz. ab, bc, &c.,—each quadrant aM, MQ, &c., con
taining 6,—then through each point of division and through the
z
A FIG. 4.
axis Pp draw a plane cutting the sphere in 24 equidistant great circles. As the sun revolves round the axis the shadow of the axis will successively fall on these circles at intervals of one hour, and if these circles cross the vertical circle ZMA in the points A, B, C, &c., the shadow of the lower portion Ep of the axis will fall on the lines EA, EB, EC, &c., which will therefore be the required hourlines on the vertical dial, Ep being the style.
There is no necessity for going beyond the VI o'clock hourline on each side of noon; for, in the winter months.the sun sets earlier than 6 o'clock, and in the summer months it passes behind the plane of the dial before that time, and is no longer available.
It remains to show how the angles AEB, AEC, &c., may be calculated.
The spherical triangles pAB, pAC, &c., will give us a simple rule. These triangles are all rightangled at A, the side pA, equal to ZP, is the colatitude of the place, that is, the difference between the latitude and 90°; and the successive angles ApB, ApC, &c., are 15°, 30°, &c., respectively. Then
tan AB =tan 15° sin colatitude;
or more simply,
tan AB =tan 15° cos latitude,
tan AC =tan 30° cos latitude,
&c. &c.
and the arcs AB, AC so found are the measure of the angles AEB, AEC, &c., required.
In this case the angles diminish as the latitudes increase, the opposite result to that of the horizontal dial.
Inclining, Reclining, &c., Dials.—We shall not enter into the calculation of these cases. Our imaginary sphere being, as before supposed, constructed with its centre at the centre of the dial, and all the hourcircles traced upon it, the intersection of these hourcircles with the plane of the dial will determine the hourlines just as in the previous cases; but the triangles will no longer be rightangled, and the simplicity of the calculation will be lost, the chances of error being greatly increased by the difficulty of drawing the dial plane in its true position on the sphere, since that true position will have to be found from observations which can be only roughly performed.
In all these cases, and in cases where the dial surface is not a plane, and the hourlines, consequently, are not straight lines, the only safe practical way is to mark rapidly on the dial a few points (one is sufficient when the dial face is plane) of the shadow at the moment when a good watch shows that the hour has arrived, and afterwards connect these points with the centre by a continuous line. Of course the style must have been accurately fixed in its true position before we begin.
Equatorial Dial.—The name equatorial dial is given to one whose plane is at right angles to the style, and therefore parallel to the equator. It is the simplest of all dials. A circle (fig. 5) divided into 24 equal arcs is placed at right angles to the style, and hour divisions are marked upon it. Then if care be taken that the style point accurately to the pole, and that the noon division coincide with the meridian plane, the shadow of the style will fall on the other divisions, each at its proper time. The divisions must be markedon both sides of the dial, because the sun will shine on opposite sides in the summer and in the winter months, changing at each equinox.
To find the Meridian Plane.—We have, so far, assumed the meridian plane to be accurately known; we shall proceed to describe some of the methods by which it may be found.
The mariner's compass may be employed as a first rough approximation. It is well known that the needle of the compass, when free to move horizontally, oscillates upon its pivot and settles in a direction termed the magnetic meridian. This does not coincide with the true north and south line, but the difference between them is generally known with tolerable accuracy, and is called the variation of the compass. The variation differs widely at different parts of the surface of the earth, and is not stationary at any particular place, though the change is slow; and there is even a small daily oscillation which takes place about the mean position, but too small to need notice here (see MAGNETISM, TERRES FIG. 5.
TRIAL).
With all these elements of uncertainty, it is obvious that the compass can only give a rough approximation to the position of the meridian, but it will serve to fix the style so that only a small further alteration will be necessary when a more perfect determination has been made.
A very simple practical method is the following:
Place a table (fig. 6), or other plane surface, in such a position that it may receive the sun's rays both in the morning and in the afternoon. Then carefully level the surface by means of a spiritlevel. This must be done very accurately, and the table in that position made perfectly secure, so that there be no danger of its shifting during the day.
Next, suspend a plummet SH from a point S, which must be rigidly fixed. The extremity H, where the plummet just meets the surface, should be somewhere near the middle of one end of the table. With H for centre, describe any number of concentric arcs of circles, AB, CD, EF, &c.
A bead P, kept in its place by friction, is threaded on the plummet line at some convenient height above H.
Everything being thus prepared, let us follow the shadow of the bead P as it moves along the surface of the table during the day. It will be found to describe
a curve ACE ... FDB, approaching the point H as the sun advances towards noon, and reced  ing from it afterwards. (The curve is a conic section—an hyperbola in these regions.) At the moment when it crosses the arc AB, mark the point A; AP is then the direction of the sun, and, as AH is horizontal, the angle PAH is the altitude of the sun. In the afternoon mark the point B where it crosses the same arc; then the angle PBH is the altitude. But the rightangled triangles PHA, PHB are obviously
equal; and the sun has FIG. 6.
therefore the same alti
tudes at those two instants, the one before, the other after noon. It follows that, if the sun has not changed its declination during the interval, the two positions will be symmetrically placed one on each side of the meridian. Therefore, drawing the chord AB, and bisecting it in M, HM will be the meridian line.
Each of the other concentric arcs, CD, EF, &c., will furnish its meridian line. Of course these should all coincide, but if not, the mean of the positions thus found must be taken.
The proviso mentioned above, that the sun has not changed its declination, is scarcely ever realized; but the change is slight, and may be neglected, except perhaps about the time of the equinoxes, at the end of March and at the end of September. Throughout the remainder of the year the change of declination is so slow that we may safely neglect it. The most favourable times are at the end of June and at the end of December, when the sun's declination is almost stationary. If the line HM be produced both ways to the edges of the table, then the two points on the ground vertically below those on the edges may be found by a plummet, and, if permanent marks be made there, the meridian plane, which is the vertical plane passing through these two points, will have its position perfectly secured.
To place the Style of a Dial in its True Position.—Bef ore giving any other method of finding the meridian plane, we shall complete the construction of the dial, by showing how the style may now be accurately placed in its true position. The angle which thelstyle makes with a hanging plumbline, being the colatitude of the place, is known, and the north and south direction is also roughly given by the mariner's compass. The style may therefore be already adjusted approximately—correctly, indeed, as to its inclination—but probably requiring a little horizontal motion east or west. Suspend a fine plumbline from some point of the style, then the style will be properly adjusted if, at the very instant of noon, its shadow falls exactly on the plumbline,—or, which is the same thing, if both shadows coincide on the dial.
This instant of noon will be given very simply, by the meridian plane, whose position we have secured by the two permanent marks on the ground. Stretch a cord from the one mark to the other. This will not generally be horizontal, but the cord will be wholly in the meridian plane, and that is the only necessary condition. Next, suspend a plummet over the mark which is nearer to the sun, and, when the shadow of the plumbline falls on the stretched cord, it is noon. • A signal from the observer there to the observer at the dial enables the latter to adjust the style as directed above.
Other Methods of finding the Meridian Plane.—We have dwelt at some length on these practical operations because they are simple and tolerably accurate, and because they want neither watch, nor sextant, nor telescope—nothing more, in fact, than the careful observation of shadow lines.
The Pole star, or Ursae Minoris, may also be employed for finding the meridian plane without other apparatus than plumblines. This star is now only about 1 ° 14' from the pole; if therefore a plumbline be suspended at a few feet from the observer, and if he shift his position till the star is exactly hidden by the line, then the plane through his eye and the plumbline will never be far from the meridian plane. Twice in the course of the twentyfour hours the planes would be strictly coincident. This would be when the star crosses the meridian above the pole, and again when it crosses it below. If we wished to employ the method of determining the meridian, the times of the stars crossing would have to be calculated from the data in the Nautical Almanac, and a watch would be necessary to know when the instant arrived. The watch need not, however, be very accurate, because the motion of the star is so slow that an error of ten minutes in the time would not give an error of oneeighth of a degree in the azimuth.
The following accidental circumstance enables us to dispense with both calculation and watch. The right ascension of the star n Ursae Majoris, that star in the tail of the Great Bear which is farthest from the " pointers," happens to differ by a little more than 12 hours from the right ascension of the Pole star. The great circle which joins the two stars passes therefore close to the pole. When the Pole star, at a distance of about t ° 14' from the pole, is crossing the meridian above the pole, the star n Ursae Majoris, whose polar distance is about 40°, has not yet reached the meridian below the pole.
When n Ursae Majoris reaches the meridian, which will be within half an hour later, the Pole star will have left the meridian; but its slow motion will have carried it only a very little distance away. Now at some instant between these two times—much nearer the latter than the former—the great circle joining the two stars will be exactly vertical; and at this instant, which the observer determines by seeing that the plumbline hides the two stars simultaneously, neither of the stars is strictly in the meridian ; but the deviation from it is so small that it may be neglected, and the plane through the eye and the plumbline taken for meridian plane.
In all these cases it will be convenient, instead of fixing the plane by means of the eye and one fixed plummet, to have a second plummet at a short distance in front of the eye; this second plummet, being suspended so as to allow of lateral shifting, must be moved so as always to be between the eye and the fixed plummet. The meridian plane will be secured by placing two permanent marks on the ground, one under each plummet.
This method, by means of the two stars, is only available for the upper transit of Polaris; for, at the lower transit, the other star n Ursae Majoris would pass close to or beyond the zenith, and the observation could not be made. Also the stars will not be visible when the upper transit takes place in the daytime, so that onehalf of the year is lost to this method.
Neither could it be employed in lower latitudes than 400 N., for there the star would be below the horizon at its lower transit ;—we may even say not lower than 45° N., for the star must be at least 5° above the horizon before it becomes distinctly visible.
There are other pairs of stars which could be similarly employed, but none so convenient as these two, on account of Polaris with its very slow motion being one of the pair.
To place the Style in its True Position without previous Determination of the Meridian Plane.—The various methods given above for finding the meridian plane have for ultimate object the determination of the plane, not on its own account, but as an element for fixing the instant of noon, whereby the style may be properly placed.
We shall dispense, therefore, with all this preliminary work if we determine noon by astronomical observation. For this we shall want a good watch, or pocket chronometer, and a sextant or other instrument for taking altitudes. The local time at any moment may be determined in a variety of ways by observation of the celestial bodies. The simplest and most practically useful methods will be found described and investigated in any work on astronomy.
For our present purpose a single altitude of the sun taken in the forenoon will be most suitable. At some time in the morning, when the sun is high enough to be free from the mists and uncertain refractions of the horizon—but to ensure accuracy, while the rate of increase of the altitude is still tolerably rapid, and, therefore, not later than to o'clock—take an altitude of the sun, an assistant, at the same moment, marking the time shown by the watch. The altitude so observed being properly corrected for refraction, parallax, &c., will, together with the latitude of the place, and the sun's declination, taken from the Nautical Almanac, enable us to calculate the time. This will be the solar or apparent time, that is, the very time we require. Comparing the time so found with the time shown by the watch, we see at once by how much the watch is fast or slow of solar time; we know, therefore, exactly what time the watch must mark when solar noon arrives, and waiting for that instant we can fix the style in its proper position as explained before.
We can dispense with the sextant and with all calculation and observation if, by means of the pocket chronometer, we bring the time from some observatory where the work is done; and, allowing for the change of longitude, and also for the equation of time, if the time we have brought is clock time, we shall have the exact instant of solar noon as in the previous case.
In former times the fancy of dialists seems to have run riot in devising elaborate surfaces on which the dial was to be traced. Sometimes the shadow was received on a cone, sometimes on a cylinder, or on a sphere, or on a combination of these. A universal dial was constructed of a figure in the shape of a cross; another universal dial showed the hours by a globe and by several gnomons. These universal dials required adjusting before use, and for this a mariner's compass and a spiritlevel were necessary. But it would be tedious and useless to enumerate the various forms designed, and, as a rule, the more complex the less accurate.
Another class of useless dials consisted of those with variable centres. They were drawn on fixed horizontal planes, and each day the style had to be shifted to a new position. Instead of hourlines they had hourpoints; and the style, instead of being parallel to the axis of the earth, might make any chosen angle with the horizon. There was no practical advantage in their use, but rather the reverse; and they can only be considered as furnishing material for new mathematical problems.
Portable Dials.—The dials so far described have been fixed dials, for even the fanciful ones to which reference was just now made were to be fixed before using. There were, however, other dials, made generally of a small size, so as to be carried in the pocket; and these, so long as the sun shone, roughly answered the purpose of a watch.
The description of the portable dial has generally been mixed up with that of the fixed dial, as if it had been merely a special case, and the same principle had been the basis of both; whereas there are essential points of difference between them, besides those which are at once apparent.
In the fixed dial the result depends on the uniform angular motion of the sun round the fixed style; and a small error in the assumed position of the sun, whether due to the imperfection of the instrument, or to some small neglected correction, has only a trifling effect on the time. This is owing to the angular displacement of the sun being so rapid—a quarter of a degree every minute—that for the ordinary affairs of life greater accuracy is not required, as a displacement of a quarter of a degree, or at any rate of one degree, can be readily seen by nearly every person. But with a portable dial this is no longer the case. The uniform angular motion is not now available, because we have no determined fixed plane to which we may refer it. In the new position, to which the observer has gone, the zenith is the only point of the heavens he can at once practically find; and the basis for the determination of the time is the constantly but very irregularly varying zenith distance of the sun.
At sea the observation of the altitude of a celestial body is the only method available for finding local time ; but the perfection which has been attained in the construction of the sextant enables the sailor to reckon on an accuracy of seconds. Certain precautions have, however, to be taken. The observations must not be made within a couple of hours of noon, on account of the slow rate of change at that time, nor too near the horizon, on account of the uncertain refractions there ; and the same restrictions must be observed in using a portal !e dial.
To compare roughly the accuracy of the fixed and the portable dials, let us take a mean position in Great Britain, say 54° lat., and a mean declination when the sun is in the equator. It will rise at 6 o'clock, and at noon have an altitude of 36°,—that is, the portable dial will indicate an average change of onetenth of a degree in each minute, or two and a half times slower than the fixed dial. The vertical motion of the sun increases, however, nearer the horizon, but even there it will be only oneeighth of a degree each minute, or half the rate of the fixed dial, which goes on at nearly the same speed throughout the day.
Portable dials are also much more restricted in the range of latitude
for which they are available, and they should not be used more than 4 or 5 M. north or south of the place for which they were constructed.
We shall briefly describe two portable dials which were in actual use.
Dial on a Cylinder.—A hollow cylinder of metal (fig. 7), 4 or 5 in. high, and about an inch in diameter, has a lid which admits of tolerably easy rotation. A hole in the lid receives the style shaped somewhat like a bayonet; and the straight part of the style, which, on account of the two bends, is lower than the lid, projects horizontally out from the cylinder to a distance of i or 12 in. When not in use the style would be taken out and placed inside the cylinder.
A horizontal circle is traced on the cylinder opposite the projecting style, and this circle is divided into 36 approximately equidistant intervals.' These intervals represent spaces of time, and to each division is assigned a date, so that each month has three dates marked as follows:—January Io, 20, 31; February 10, 20, 28; March lo, 20, 31; April so, 20, 30, and so on,—always the loth, the loth, and the last day of each month.
Through each point of division a vertical line parallel to the axis of the cylinder is drawn from top to bottom_ Now it will be readily understood that if, upon one of these days, the lid be turned, so as to bring the style exactly opposite the date, and if the dial be then placed on a horizontal table so as to receive sunlight, and turned round bodily until the shadow of the style falls exactly on the vertical line below it, the shadow will terminate at some definite point of this line, the position of which point will depend on the length of the style—that is, the distance of its end from the surface of the cylinder —and on the altitude of the sun at that instant. Suppose that the observations are continued all day, the cylinder being very gradually turned so thzt the style may always face the sun, and suppose that marks are made on the vertical line to show the extremity of the shadow at each exact hour from sunrise to sunset—these times being taken from a good fixed sundial,—then it is obvious that the next year, on the same date, the sun's declination being about the same, and the observer in about the same latitude, the marks made the previous year will serve to tell the time all that day.
What we have said above was merely to make the principle of the instrument clear, for it is evident that this mode of marking, which would require a whole year's sunshine and hourly observation, cannot be the method employed.
The positions of the marks are, in fact, obtained by calculation. Corresponding to a given date, the declination of the sun is taken from the almanac, and this, together with the latitude of the place and the length of the style, will constitute the necessary data for computing the length of the shadow, that is, the distance of the mark below the style for each successive hour.
We have assumed above that the declination of the sun is the same at the same date in different years. This is not quite correct, but, if the dates be taken for the second year after leap year, the results will be sufficiently approximate.
When all the hourmarks have been placed opposite to their respective dates, then a continuous curve, joining the corresponding hourpoints, will serve to find the titne for a day intermediate to those set down, the lid being turned till the style occupy a proper position between the two divisions. The horizontality of the surface on which the instrument rests is a very necessary condition, especially in summer, when, the shadow of the style being long, the extreme end will shift rapidly for a small deviation from the vertical, and render the reading uncertain. The dial can also be used by holding it up by a small ring in the top of the lid, and probably the verticality is better ensured in that way.
Portable Dial on a Card.—This neat and very ingenious dial is attributed by Ozanam to a Jesuit Father, De Saint Rigaud, and probably dates from the early part of the 17th century. Ozanam says that it was sometimes called the capuchin, from some fancied resemblance to a cowl thrown back.
Construction.—Draw a straight line ACB parallel to the top of the
' Strict equality is not necessary, as the observations made are on the vertical line through each divisionpoint, without reference to the others. It is not even requisite that the divisions should go completely and exactly round the cylinder, although they were always so drawn, and both these conditions were insisted upon in the directions for the construction.
card (fig. 8) and another DCE at right angles to it ; with C as centre, and any convenient radius CA, describe the semicircle AEB below the horizontal. Divide the whole arc AEB into 12 equal parts at the points r, s, I. &c., and through these points draw perpendiculars to the diameter ACB; these lines will be the hourlines, viz. the line through r will be the XI . I line, the line through s the X . II line, and so on; the hourline of noon will be the point A itself ; by subdivision of the small arcs Ar, rs, st, &c., we may draw the hourlines corresponding to halves and quarters, but this only where it can be done without confusion.
Draw ASD making with AC an angle equal to the latitude of the place, and let it meet EC in D, through which point draw FDG at right angles to AD.
With centre A, and any convenient radius AS, describe an arc of circle RST, and graduate this arc by marking degree divisions on
it, extending from o° at S to 23i° on each side at R and T. Next determine the points on the straight line FDG where radii drawn from A to the degree divisions on the arc would cross it, and carefully mark these crossings.
The divisions of RST are to correspond to the sun's declination, south declinations on RS and north declinations on ST. In the other hemisphere of the earth this would be reversed ; the north declinations would be on the upper half.
Now, taking a second year after leap year (because the declinations of that year are about the mean of each set offoul years), find the days of the month when the sun has these different declinations, and place these dates, or so many of them as can be shown without confusion, opposite the corresponding marks on FDG. Draw the sunline at the top of the card parallel to the line ACB ; and, near the extremity, to the right, draw any small figure intended to form, as it were, a door of which a b shall be the hinge. Care must be taken that this hinge is exactly at right angles to the sunline. Make a fine open slit c d right through the car(' and extending from the hinge to a short distance on the door,—the centre line of this slit coinciding accurately with the sunline. Now, cut the door completely through the card; except, of course, along the hinge, which, when the card is thick, should be partly cut through at the back, to facilitate the opening. Cut the card right through along the line FDG, and pass a thread carrying a little plummet W and a very small bead P; the bead having sufficient friction with the thread to retain any position when acted on only by its own weight, but sliding easily along the thread when moved by the hand. At the back of the card the thread terminates in a knot to hinder it from being drawn through; or better, because giving more friction and a better hold, it passes through the centre of a small disk of card—a fraction of an inch in diameter—and, by a knot, is made fast at the back of the disk.
To complete the construction,—with the centres F and G, and
radii FA and GA, draw the two arcs AY and AZ which will limit the hourlines; for in an observation the bead will always be found between them. The forenoon and afternoon hours may then be marked as indicated in the figure. The dial does not of itself discriminate between forenoon and afternoon; but extraneous circumstances, as, for instance, whether the sun is rising or falling; will settle that point, except when close to noon, where it will always be uncertain.
To rectify the dial (using the old expression, which means to prepare the dial for an observation),—open the small door, by turning it about its hinge, till it stands well out in front. Next, set the thread in the line FG opposite the day of the month, and stretching it over the point A, slide the bead P along till it exactly coincide with A.
To find the hour of the day,—hold the dial in a vertical position in such a way that its plane may pass through the sun. The verticality is ensured by seeing that the bead rests against the card without pressing. Now gradually tilt the dial (without altering its vertical
W FIG. 9.
plane), until the central line of sunshine, passing through the open slit of the door, just falls along the sunline. The hourline against which the bead P then rests indicates the time.
The sunline drawn above has always, so far as we know, been used as a shadowline. The upper edge of the rectangular door was the prolongation of the line, and, the door being opened, the dial was gradually tilted until the shadow cast by the upper edge exactly coincided with it. But this shadow tilts the card onequarter of a degree more than the sunline, because it is given by that portion of the sun which just appears above the edge, that is, by the upper limb of the sun, which is onequartet of a degree higher than the centre. Now, even at some distance from noon, the sun will sometimes take a considerable time to rise onequarter of a degree, and by so much time will the indication of the dial be in error.
The central line of light which comes through the open slit will be free from this error, because it is given by light from the centre of the sun.
The carddial deserves to be looked upon as something more than a mere toy. Its ingenuity and scientific accuracy give it an educational value which is not to be measured by the roughness of the results obtained.
The theory of this instrument is as follows: Let H (fig. 9) be the point of suspension of the plummet at the time of observation, so that the angle DAHis the north declination of the sun,—P, the bead, resting against the hourline VX. Join CX, then the angle ACX is the hourangle from noon given by the bead, and we have to prove that this hourangle is the correct one corresponding to a north latitude DAC, a north declination DAH and an altitude equal to the angle which the sunline, or its parallel AC, makes with the horizontal. The angle PHQ will be equal to the altitude, if HQ be drawn parallel to DC, for the pair of lines HQ, HP will be respectively at right angles to the sunline and the horizontal.
Draw PQ and HM parallel to AC, and let them meet DCE in M and N respectively.
Let HP and its equal HA be represented by a. Then the following values will be readily deduced from the figure:
AD = a cos decd. DH =a sin decl. PQ =a sin alt.
CX = AC = AD cos lat. = a cos deck cos lat.
PN = CV = CX cos ACX = a cos decd. cos lat. cos ACX. N =MH=DH sin MDH=a sin decl. sin lat.
'. the angle MDH = DAC = latitude.)
And since PQ=NQ+PN,
we have, by simple substitution,
a sin alt. = a sin decl. sin lat. +a cos decd. cos lat. cos ACX ; or, dividing by a throughout,
sin altr=sin decd. sin lat.+cos decl. cos lat. cos ACX . . (I) which equation determines the hourangle ACX shown by the bead. To determine the hourangle of the sun at the same moment, let fig. io represent the celestial sphere, HR the horizon, P the pole, Z the zenith and S the sun.
From the spherical triangle PZS, we have
cos ZS=cos PS cos ZP+sin PS sin ZP cos ZPS
but ZS = zenith distance 90°—altitude
ZP =9o°— PR = 90°—latitude
PS= polar distance =9o°— declination,
therefore, by substitution
sin alt. sin decd. sin lat.+cos decl. cos lat. cos ZPS . (a) and ZPS is the hourangle of the sun.
A comparison of the two formulae (I) and (2) shows that the hourangle given by the bead will be the same as that given by the sun, and proves the theoretical accuracy of the carddial. Just at sunrise or at sunset the amount of refraction slightly exceeds half a degree. If, then, a little cross m (see fig. 8) be made just below the sunline, at a distance from it which would subtend half a degree at c, the time of sunset would be found corrected for refraction, if the central line of light were made to fall on cm.
End of Article: LAT 

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