of longitude of dial-plane 25° 20' = 1° 41'.

The dial may now be constructed as horizontal, and for latitude 17° 20'; and since the meridian line lies to the east of the XII. o'clock line, the hour lines of 28° 41', 3° 41', &c., reckoned from the substyle of the dial, must Dialling-be found, and made the hour lines of XL, X., &c. in the forenoon. Also, the hour line of 41° must be made the hour line of I. The hour line of II. will be on the other side of the substyle, and will correspond to 18°, that of III. to 1°, 18°, and so on.

Of the Time when the Sun begins or ceases to shine on a Dial on a given Day.

75. The solution of this problem is wanted, that it may be known what hours should be inscribed on a dial. It has various cases; but as all are to be resolved on the same principles, it will be sufficient if we consider one.

Retaining the construction of the fig. of art. 73, let PI be the hour circle passing through the sun when he is in the plane of the dial on a given day. In the spherical triangle HAP, we have found AP, the measure of the angle contained by the axis and substyle; and HPA, the difference of longitude. Again, in the spherical triangle IAP, right-angled at A, besides AP, there is known IP, the sun's distance from the pole on the given day. Hence the angle IPA may be found by this proportion:

\[ \tan PI : \tan PA = \text{rad.} : \cos API. \]

The angle API, expressed in time, is half the period the sun shines on the plane; and the hour angle HPI, in time, is the interval between noon and the sun's leaving the plane.

76. Taking the dial constructed in art. 73 as an example, it will be found that when the sun is in the northern tropic, that is, when PI = 66° 30', the angle HPA = 82° 12'. Now (art. 74), HPA = 25° 30'; therefore, when the sun ceases to shine on the plane, the horary angle IPH from noon is 107° 32' = 7 hours 10 minutes, the sum of the two arcs; and when he begins to shine on it, the horary angle is 56° 52' = 3 hours 47 minutes, their difference. Hence it will be needless to trace on the dial any hour line earlier than VIII. in the morning, or later than VIII. in the evening.

Of the Line described by the extremity of a Shadow on a Plane.

77. Sometimes the line which is the boundary of the space passed over by the shadow of the axis of a dial on given days of the year is traced on its plane. The path of the extremity of the shadow when the sun enters the different signs of the ecliptic is an elegant appendage to a dial, and the geometrical problem which determines it is interesting.

78. Setting aside the considerations of the change in the sun's declination, the apparent diurnal path of the sun in the heavens is, in the theory of dialling, considered to be a circle parallel to the equator; therefore any fixed point that projects a shadow throughout a day may be considered as the vertex of a cone, the base of which is the diurnal circle described by the sun on that day, and the space bounded by the shadow will be the same cone continued beyond its vertex. Thus it appears that the shadow of a fixed point in space generates the surface of a cone whose axis passes through the pole of the world; and hence the path of the shadow on any plane will be a conic section, because it is the line in which the surface of the cone meets the plane.

Let O be the centre of a horizontal dial, OF the axis, O XII. the meridian line, and OB the shadow of the axis at any time; join FB, draw FA perpendicular to O XII., and AD perpendicular to OB, meeting OB in D; and join FD. Because FA is perpendicular to the plane BO XII., the plane of the triangle FAD is perpendicular to that plane. Now, by construction, BD is perpendicular to their common section DA, therefore it is perpendicular to the plane... of the triangle FAD, and consequently FD is perpendicular to OB.

Let \(a = OF\) be the length of the axis; \(r = OB\) be the length of its shadow; \(v = \) the angle BOA contained by the shadow and the meridian; \(z = \) the angle FOB contained by the shadow and the axis; \(L = \) the angle FOA, the latitude; \(D = \) the angle OFB, the sun's polar distance; in the triangles ODF, ODA, right-angled at D, \(OD : OF = \cos FOD : rad.\) \(OA : OD = \) rad. : \(\cos AOD,\) therefore \(OA : OF = \cos FOD : \cos AOD;\) but in the triangle OAF, \(OA : OF = \cos FOA : rad.\) therefore \(\cos FOA : rad. = \cos FOD : \cos AOD.\)

From the triangle FOB we get this other proportion, \[\sin (F + O) \text{ or } \sin B : \sin F = OF : OB.\]

The last two proportions in symbols are \[\cos L : \text{ rad.} = \cos v : \cos y,\] \[\sin (D + v) : \sin D = a : r.\]

By the first proportion, we may determine \(v\) from \(y\), the angle the shadow makes with the meridian; and by the second we get \(r\), the length of the shadow. We may also express its length by the hour angle from noon; for if we put \(x\) to denote that angle, then (art. 23), \[\text{rad.} : \sin L = \tan x : \tan y.\]

From the first and last of these three proportions, it is easy to infer, by spherical trigonometry, that the arcs \(L\) and \(y\) are the sides of a right-angled spherical triangle, of which \(v\) is the hypotenuse, and that \(x\) is the angle opposite the side \(y\); hence it follows that \[\cos x : \tan L = \text{ rad.} : \tan v.\]

To determine the length of the shadow, we have now these formulæ, \[\tan v = \frac{\text{rad. tan } L}{\cos x},\] \[1\] \[\cos v = \frac{\cos L \cos y}{\text{ rad.}},\] \[2\] \[r = \frac{a \sin D}{\sin (D + v)}.\] \[3\]

By these, and the formula \[\tan y = \frac{\sin L \tan x}{\text{ rad.}},\] the position of the shadow, and its length, may be found at any time on any given day; the sun's declination being taken from the table which concludes this article.

79. It has been shown (art. 78) generally, that the line described by the shadow is a conic section: the formulæ just found serve to determine its nature in any given case.

By the calculus of sines (Algebra, art. 239), \(\sin (D + v) = \sin D \cos v + \cos D \sin v\); now \(\sin v = \sqrt{(1 - \cos^2 v)}\), therefore, from formulæ (2) and (3), putting \(\text{ rad.} = 1\), we have

\[r = \frac{a \cos L \cos y + \cot D \sqrt{1 - \cos^2 L \cos^2 y}}{\cos L \cos y + \cot D \sqrt{1 - \cos^2 L \cos^2 y}}.\]

This is the polar equation of the line which limits the shadow; it may also have this form,

\[a - r \cos L \cos y = \cot D \sqrt{r^2 - r^2 \cos^2 L \cos^2 y}.\]

Supposing now \(t\) and \(u\) to be rectangular co-ordinates, which have their origin at \(O\); we have \(t = r \cos y\), and \(r^2 = t^2 + u^2\); therefore, by substitution, and taking the squares of both sides, we find

\[(\cot^2 D \sin^2 L - \cos^2 L) t^2 + 2at \cos L + u^2 \cot^2 D - a^2 = 0,\]

an expression which by the calculus of sines may be transformed to

\[\ell^2 \sin (D + L) \sin (D - L) - 2at \cos L \sin^2 D = 0,\]

\[-u^2 \cos^2 D + a^2 \sin^2 D = 0.\]

This is the equation of the line described by the shadow referred to axes perpendicular to each other; it contains the first and second powers of the co-ordinates and constants, therefore the line is, in general, a conic section.

80. If \(L = D\), then \(\sin (L - D) = 0\). In this case the term which contains \(\ell\) vanishes, and the remaining terms indicate that the curve is a parabola.

If \(L\) is greater than \(D\), so that the sine of \(D - L\) is negative, the equation belongs to an ellipse; but if \(L\) be less than \(D\), the equation belongs to a hyperbola. In each case the meridian line is the transverse axis of the curve.

The path of the shadow is an ellipse at any place within the polar circle on the days when the sun does not set; it is a parabola at that place on the day that the sun just touches the horizon at midnight; it is a straight line at all places of the earth on the equinoctial days; and in every other case it is a hyperbola.

81. The points in which the curve crosses its axis may be found from its polar equation by making \(u = 0\), and \(v = 180^\circ\). If \(r'\) and \(r''\) denote the distances from the centre, when \(u = 0\), then \(r' = \frac{a \sin D}{\sin (D + L)}\); \(r'' = \frac{a \sin D}{\sin (D - L)}\).

The first of these is the length of the shadow at noon. The vertices of the curve lie on the same side of the centre of the dial when it is a hyperbola, but on opposite sides when it is an ellipse. The other elements of the curve may be found in the same way, and the curve may be described as a conic section. In practice it will however be sufficient to find the points in which the curve crosses the hour lines of the dial, and then trace the curve through them by hand. The intersection of the curve and any hour line may be found by this construction.

Let \(O\) be the centre of the dial, and \(O XII\) the meridian line. Take any two lines, \(OM, ON\), in the proportion of the cosine of the latitude to radius; and about \(O\) as a centre, with these distances, describe circles. Take \(OF\) in the meridian line equal to the axis of the dial, and make the angle \(OFH\) equal to the sun's distance from the pole. Let K be the intersection of any hour line and the lesser circle. Draw KL perpendicular to the meridian, meeting the greater circle in G. Draw OG, meeting FH in H. In the hour line OY, take OB equal to OH, and B is the point in which the hour line meets the path of the shadow.

For, by trigonometry, \[ \frac{OG}{OK} = \sin OKL : \sin OGL = \cos KOL : \cos GOL; \] that is, because \( KOL = YOL = y, \) \[ \text{rad.} : \cos L = \cos y : \cos HOL. \]

Hence the angle HOL = v (formula 2 of art. 78); now HFO = D, and OF = a, and sin H: sin F = OF : OH, that is, \( \sin (D + v) : \sin D = a : OH; \) therefore OH = OB is the length of the shadow (formula 3).

Whatever has been shown regarding the shadow of a horizontal dial, will apply to any dial whatever, if L be put for the latitude of the place where the dial would be horizontal, and the substyle be taken for the meridian line.

**Dials with variable Centres.**

82. Dials of this kind are not common; yet they are deserving of attention, because of the elegance of their geometrical theory. Their construction depends on this principle.

*It is possible to determine a system of hour points on a plane, such, that if a style be placed in the plane of the meridian at certain points, to be found, corresponding to the days of the year, and making with the horizon any given angle, its shadow shall pass through the hour points at the times they indicate, and in this way show the time of the day.*

83. To establish the truth of this proposition, the following problem is to be resolved.

Having given the sun's declination, the time from noon, and the latitude of a place; to find the angle which the shadow of a style in the plane of the meridian, and inclined at a given angle, makes with the meridian line on a horizontal plane.

Let LMN be the horizon, LPN the meridian, P the pole, CQ the style, which, being produced, meets the celestial meridian in Q. Let S be the sun in the hour circle PS, and MSQA a great circle passing through S and Q, and cutting the horizon in the line MCA.

Put L = PN, the given latitude; \[ E = QN, \] the given elevation of the style; \[ D = PS, \] the sun's distance from the pole; \[ x = QPS, \] the hour angle; \[ z = \angle LQM; \] \[ y = \angle ACN, \] or arc LM, which is to be found.

In the spherical triangle PSQ, by spherics, \[ \cot PS \cdot \sin PQ = \cot Q \cdot \sin P + \cos PQ \cdot \cos P; \] and in the spherical triangle QLM, right-angled at L, \[ \cot LQM = \cot LM \cdot \sin QL. \]

Now, in the first of these formulae, \[ PS = 90^\circ - D, \quad PQ = E - L, \quad Q = 180^\circ - z, \quad P = z; \] and in the second, \[ LQM = z, \quad LM = y, \quad \sin QL = \sin QN = \sin E; \] therefore, by substituting the symbols in the formulae, we have \[ \tan D \cdot \sin (E - L) = -\cot z \cdot \sin x + \cos (E - L) \cdot \cos x; \quad (1) \] \[ \cot z = \cot y \cdot \sin E. \quad (2) \]

Let the value of \( \cot z \) be substituted instead of it in the first equation, then, deducing from the result the value of \( \cot y, \) we find \[ \cot y = \frac{\cos (E - L) \cdot \cos x - \sin (E - L) \cdot \tan D}{\sin E \cdot \sin x}. \]

This formula gives the value of \( y, \) the angle made by the shadow and the meridian of the dial, which was required.

84. Fig. 13 of Plate CLXXXVII. represents a dial with a moveable centre, O XII., being the meridian line, and the hour points as shown in the figure. Let C be the variable centre, which is the position of the bottom of the style on any given day, A any one of the hour points, and AC the position of the shadow at that hour on the given day; then, \( x \) denoting the hour angle from noon, and \( y \) the hour line angle AC XII., the preceding formula gives the value of \( y. \)

Draw AB perpendicular to the meridian, put OB = t, AB = u, t and u being the co-ordinates of an hour point; and put r for OC, the variable distance of the bottom of the style from the fixed point C in the plane of the dial; then BC = t - r. And because \( \cot C = \frac{Be}{BA}, \) that is, \[ \cot y = \frac{t - r}{u}, \] therefore \[ \frac{t - r}{u} = \frac{\cos (E - L) \cdot \cos x - \sin (E - L) \cdot \tan O}{\sin E \cdot \sin x}; \] and hence, deducing the value of \( r, \) \[ r = \frac{\sin E \cdot \sin x - u \cdot \cos (E - L) \cdot \cos x}{\sin E \cdot \sin x} + u \cdot \sin (E - L) \cdot \tan D. \]

By the nature of the dial, the position of the style must depend entirely on the sun's declination, and be altogether independent of the hour of the day. These conditions will be satisfied if we make \[ t \cdot \sin E \cdot \sin x - u \cdot \cos (E - L) \cdot \cos x = 0; \] \[ \frac{u \cdot \sin (E - L)}{\sin E \cdot \sin x} = a, \] a constant quantity;

for by this assumption \( t \) and \( u \) are independent of \( D, \) the sun's declination; and \( r = a \cdot \tan D \) independent of \( x. \)

By resolving the equations, we obtain \[ t = a \cdot \cot (E - L) \cdot \cos x; \quad (1) \] \[ u = a \cdot \frac{\sin E}{\sin (E - L)} \cdot \sin x; \quad (2) \] \[ r = a \cdot \tan D. \quad (3) \]

These equations express the nature of every dial of this kind.

85. To construct the dial, we must assume a line of any convenient length, as a scale on which the co-ordinates of the hour points may be measured. The values of \( t \) and \( u \) are now to be computed by making \( x = 15^\circ \) for the hours of XI. and I.; again, \( x = 30^\circ \) for the hours of X. and II., and so on. A graduated scale must be formed along the meridian line, proceeding both ways from O, that point being the position of the style at the equinoxes; and since it appears from the figure in art. 83, that for any given hour angle \( x, \) the angle \( y \) at the variable centre ought to increase as the sun approaches the north pole, the scale of declination for the north side of the equator... Dialling must lie on the north side of O; and that for the south, on the opposite side. Lastly, the months and days of the year ought to be placed on the scale opposite to the degrees of declination to which they correspond.

86. We may investigate, from the equations, the nature of the curve that is the locus of the dial points, or which has \( t \) and \( u \) for rectangular co-ordinates. This will be obtained by eliminating the angle \( x \).

Put \( m = a \cotan (E - L), n = a \frac{\sin E}{\sin (E - L)} \);

we have now

\[ \frac{t^2}{m^2} + \frac{u^2}{n^2} = \cos^2 x + \sin^2 x = 1; \]

hence it appears that the locus of the hour points is an ellipse, of which the axes are \( m \) and \( n \).

Azimuth or Analemmatic Dial.

87. This dial, represented by fig. 16, Plate CLXXXVII., is of the kind having a variable centre. Its style is vertical, therefore, by what has been shown in art. 84, the equations of the hour points on the dial will be

\[ t = a \tan L \cos x; \quad u = a \sec L \sin x; \quad r = a \tan D. \]

The dimensions of the ellipse which is the locus of the hour points are these:

\[ m = \text{semiconjugate or merid. axis} = a \tan L, \] \[ n = \text{semitransverse axis} = a \sec L, \] \[ \text{Eccentricity} = a. \]

These values of \( t, u, r \), serve to find the hour points by calculation. They may also be found by this construction, fig. 15.

Draw two straight lines \( Aa, Bb \), intersecting at right angles in \( O \); and in \( OA \), one of these lines, take \( OD \) of any suitable length for the eccentricity of the dial; draw \( DB \) so as to make with \( DO \) an angle equal to the latitude of the place; then \( CB \) will be half the lesser axis, and \( B \) the twelve o'clock hour point.

In \( OD \), take \( OA \) and \( Oa \) each equal to \( DB \); and \( Aa \) will be the hour points for six o'clock in the morning and evening.

On \( O \) as a centre, with \( OA \) and \( OB \) as radii, describe circles, and divide the quadrants in the same angle, each into six equal parts. From \( K \), any one of the divisions of the outer circle, draw \( KL \) perpendicular to \( Oa \); and from \( k \), the corresponding point in the inner circle, draw \( kN \) parallel to \( Oa \), meeting \( KL \) in \( N \); this will be one of the hour points, and in the same way may all the others be found.

At the point \( D \) make angles \( ODE, ODe \), each \( 23\frac{1}{2} \) degrees, the sun's greatest declination; and \( E, e \) shall be the positions of the bottom of the style at the solstices, that of the summer being on the north of \( O \), the middle of the dial. Describe a circle with \( DO \) as a radius, and find the tangents of the series of arcs \( 1^\circ, 2^\circ, 3^\circ, \ldots \) of that circle, and lay them down as a scale from \( O \) to \( E \) and \( e \), on each side. Find, in the tables which conclude this article, the sun's declination on the first day of every month, and mark the beginning of the month on the scale \( E \) opposite to its corresponding degree of declination. As many of the intermediate days may in like manner be laid down as there is room for.

The style must now, by some contrivance, be placed over the scale, so as to admit of sliding along it, and being set to any day, and the dial is finished.

If a dial of this kind be united to a horizontal dial, as they can only show the same hour when their meridian lines are in their true position, the compound dial may be set to show time without the help of a compass or meridian line.

Lambert's Dial.

88. This is a particular case of the class of dials with a variable centre. It was given by M. Lambert in the Berlin Ephemerides for 1777.

It has been found (art. 86), that \( L \) being put to denote the latitude, and \( E \) the angle which the style is to make with the meridian line, the semiaxes of the ellipse, which is the locus of the hour points, are

\[ a \cotan (E - L), \quad a \frac{\sin E}{\sin (E - L)}. \]

Now these will be equal, if the cosine of \( E - L \) be equal to the sine of \( E \), that is, if \( E - L + E = 90^\circ \), or \( E = \frac{90^\circ + L}{2} \). When \( E \), the elevation of the style, has this value, the hour points are in the circumference of a circle. At London \( E \) would be \( 70^\circ 45' \).

Lambert's dial may be constructed geometrically thus: (see fig. 14 of Plate CLXXXVIII.)

Take a straight line \( OD \) of any length, and draw a perpendicular \( OB \). At the point \( D \), make the angle \( ODB \) equal to half the sum of \( 90^\circ \) and the latitude of the place, and \( OB \) will be the radius of the dial.

Describe a circle about \( O \) as a centre, with the radius \( OB \), and divide the quadrants each into six equal parts, and the points of division will be the hour points on the dial.

Draw the lines \( DE, De \), and make a scale of tangents of the sun's declination from \( O \) to \( E \) and \( e \), and against the divisions of the scale write the days of the month exactly as directed in the Analemmatic dial. Also place the style over the meridian, so that it may be adjusted to the time of the year, and make with the meridian line an angle equal to \( BDO \), and the dial will be constructed.

Notes.—The style must be on the north or south of the point \( O \), according as the sun is on the north or south side of the equator.

In these dials, instead of making the style moveable, it might be fixed, and different sets of hour points found for different days of the year. This, however, would make the construction laborious.

Thus far we have given the theory of the dials, along with their construction. In what follows, we shall, for the sake of brevity, simply give their construction. In general, the theory is just an application of what has been already delivered.

Portable Dial on a Card.

89. This dial, represented by fig. 11, Plate CLXXXV., has been called the Capuchin, because of a supposed resemblance it has to the head of a capuchin friar with his cowl inverted. It may be constructed as follows:

Draw a straight line \( ACB \) parallel to the top of the card (fig. 10), and another \( DCE \) bisecting the former at right angles; on \( C \) as a centre, with any convenient radius \( CA \), describe a semicircle, and divide it into twelve equal parts, at the points \( r, s, t, u, v, \ldots \). From the points \( r, s, t, \ldots \) draw lines perpendicular to \( ACB \), the diameter of the circle; and these will be the hour lines, viz. the line through \( r \) will be the hour line of XI. I. (see fig. 11), that through \( s \) the hour line of X. II. &c., the hour of XII. noon being at \( A \). The half hours and quarters may also be laid down by subdivision. At \( A \), the extremity of the diameter, draw a line \( AD \), making with \( AC \) an angle equal to the latitude of the place, and meeting the six o'clock line in \( D \), through which draw a line \( FDG \) at right angles to \( AD \). At the point \( A \), draw lines \( AF, AG \), to make with \( AD \), on opposite sides, angles of \( 23\frac{1}{2} \) degrees, the sun's greatest Dialling.

91. This dial is represented by fig. 4, Plate CLXXXVI. It shows the hour of the day, the sun's place in the ecliptic, and his altitude at the time of observation. The dial is constructed by tracing the lines on a rectangular piece of paper (fig. 5), and pasting it on the surface of the cylinder. The lines may be traced by the following rules. Having formed on paper a rectangle ABDC, of such a size that, when wrapped round the cylinder, its opposite sides AC, BD, may just coincide; produce the side BA to a, any convenient distance, and on a as a centre, with the distance AA, describe the quadrant arc AE, and divide it into ninety equal parts for degrees; draw lines from the centre through as many divisions of the quadrant as there are degrees of the sun's altitude in the longest day of the year at noon of the place for which the dial is to serve, which at London is 62°, and continue these lines until they meet the tangent line AC. From the points of meeting draw lines across the rectangle parallel to AB, as shown in the figure, and these will be the parallels of the sun's altitude, in whole degrees, from sunrise to sunset, on all the days of the year.

Divide AB, CD, the top and bottom sides of the rectangle, into twelve equal parts for the signs of the ecliptic, and draw straight lines, joining the points of division, on which, at the bottom, place the characters of the signs, as in the figure. The twelve divisions should be subdivided by parallel lines into halves, and, if there be room, into quarters.

At the top of the rectangle make a scale of the months and days of the year, so as that the division of the scale for each day may stand over the division for the sun's place on that day in the scale of signs. The sun's place in the ecliptic each day of the year 1830, the second after leap year, is given at the end of this article, and may serve for this scale. Compute the sun's altitude for every hour in the latitude of the place, when he is in the beginning and middle of each sign, and in the upright parallel lines make marks for these computed altitudes among the parallels of altitude, reckoning them downward according to the order of the numeral figures set to them along the side BD, answering to the divisions of the quadrant on the opposite side; and through these marks draw the curve hour lines, and set the hours to them, as in the figure, reckoning the forenoon hours downwards, and the afternoon hours upwards. The sun's altitude should also be computed for the half hours; as to the quarter hour points, they may be put nearly in their proper place by estimation with the eye.

The scales and hour lines being constructed, the part of the paper on which the quadrant was drawn is to be cut off along the line AC, and that on the opposite side along BD. The superfluous parts at the top and bottom also are to be removed, and it will be fit for being pasted round the cylinder.

The cylinder, reduced in size, is represented by ABCD, fig. 4. It should be hollow, to hold the style DEe when it is not used. The crooked end of the style is put into a hole in the top AD of the cylinder, which fits on it tightly, and may be turned round, like the lid of a paper snuff-box. The style must stand straight out, perpendicular to the side of the cylinder, just over the line AB, round the top from which the parallels of the sun's altitude are reckoned; and the length of the style, projecting beyond the cylinder, must be exactly equal to AA, the radius of the quadrant in fig. 5.

To use the dial, place the base of the cylinder on a level table where the sun shines, and turn the top round till the style stand directly over the day of the month; then turn Dialling.

The cylinder about on the table till the shadow falls on it parallel to those upright lines which divide the signs, that is, till the shadow be parallel to the (supposed) axis of the cylinder; and then the point or lowest shadow will fall upon the time of the day among the curve hour lines, and will show the sun's altitude at that time, among the cross parallels of his altitude which go round the cylinder; at the same time it will indicate the sign of the ecliptic in which the sun is.

The dial may also, when used, be suspended by the ring F in its top. Dials of this kind, of rather rude construction, are sometimes used by the rural population in France.

92. The construction of this dial requires that the sun's altitude be found for any given hour of the day, knowing the latitude of the place and the sun's declination.

Let \( D \) be the sun's declination, \( L \) the latitude, \( x \) the hour angle at the pole, \( z \) the sun's altitude.

Then \( 90^\circ - D, 90^\circ - L, 90^\circ - z \), are the three sides of a spherical triangle, and \( x \) is the angle opposite to the last of these sides; hence, by spherics,

\[ \sin z = \sin L \sin D + \cos L \cos D \cos x \ldots \ldots \ldots \ldots (1) \]

This is equivalent to

\[ \sin z = \sin L \sin D (1 + \cot L \cos D \cos x) \ldots \ldots \ldots \ldots (2) \]

From this formula, by giving different values to \( x \), the hour from noon, there will be found corresponding values of \( z \), the sun's altitude.

Ex. To find the sun's altitude at ten in the forenoon or two in the afternoon on the 21st of May at London, in latitude \( 51^\circ 31' \), the sun then nearly entering the sign Gemini. In this case, from the table at the end, \( D = 20^\circ 5' \); we have also \( L = 51^\circ 31' \) and \( x = 2h = 30^\circ \).

\[ \begin{align*} \cot L & = 990035 \\ \cot D & = 1048550 \\ \cos x & = 993753 \end{align*} \]

\[ \begin{align*} 1-878 & = 0-27368 \\ 2-878 & = 1 + \cot L \cot D \cos x \\ z & = 50^\circ 51' \end{align*} \]

The formula supposes that the latitude and declination are of the same name, that is, both north or both south. When one is north and the other south, then the first term will be negative, and we have for this case

\[ \sin z = \sin L \sin D (\cot L \cos D \cos x - 1) \ldots \ldots \ldots \ldots (3) \]

and when the hour angle exceeds \( 90^\circ \), it must be remembered that \( \cos x \) is a negative quantity, so that the expression in which it occurs is subtractive.

Ring Dial.

93. This kind of dial is shown in figure 7. It is formed of a brass ring or rim, usually about two inches in diameter, and one third of an inch in breadth. In a point of this rim there is a hole, through which the sun's light passes, and forms a lucid speck on the opposite concave surface, on which the hours of the day are marked.

These divisions are made by describing a circle to represent the ring, and drawing a horizontal chord \( EF \); with this as a radius describe a quadrant \( FD \), and divide it into degrees, and through those which mark the sun's altitude at every hour of the day, at the time of either equinox, draw lines from \( E \) to the opposite side of the circle, and there put the corresponding figures. Thus for a place whose latitude is \( 51^\circ 32' \), the XII. o'clock line will pass through \( 38^\circ 33' \) the colatitude, and the XI. and I. line through \( 36^\circ 56' \), the X. and II. through \( 32^\circ 36' \), the IX. and III. through \( 26^\circ 5' \), the VIII. and IV. through \( 18^\circ 8' \), and the VII. and V. through \( 9^\circ 17' \). The dial, thus constructed, would only show time at the equinox; but to make it perform the whole year, the hole is made moveable, and the signs of the ecliptic or the days of the month are marked on the convex surface of the ring by taking \( ET \) and \( ET \), on each side of \( E \), equal to twice the sun's declination when he enters any particular sign, as Taurus and Pisces, and there marking the character of the sign or corresponding month, and so for all the others; and by these the dial is rectified for the time of the year.

From the figure it appears that \( E XII. \delta \) is equal to \( FE XII. \) or the altitude of the sun in the equinox; but \( T XII. E \) is equal to the sun's declination in Taurus, because it is an angle at the circumference standing on an arc which is double the declination; therefore \( T XII. \delta \) is equal to the meridian altitude when the sun enters Taurus, and a ray passing through the hole at \( T \) will mark the XII. o'clock hour of that day. But this dial will not show the other hours exactly, because \( T III. \delta \) exceeds \( E III. \delta \), the equinoctial hour, by the angle of declination, and the same holds of the rest. To remedy this defect, the concave surface of the ring has sometimes been made broader, and seven circles described upon it, the middle one to represent the equinoctial, and the extremes the tropics; and on these circles the hours have been marked as shown in a table of altitudes.

To use the dial, set the moveable hole to the day of the month, or the degree of the sun's place in the ecliptic; then, suspending it by the ring, turn it towards the sun till his rays point out the hour among the divisions in the inside.

Universal or Astronomical Ring Dial.

94. This is represented in figure 6 of Plate CLXXXVI. It serves to show the time of the day in any part of the earth, whereas the former is only adapted to a particular latitude. It consists of two rings or flat circles from two to six inches in diameter. The outer ring A represents the meridian of any place. Two of its opposite quadrants are divided into \( 90^\circ \), serving, the one for north and the other for south latitude. The inner ring represents the equator, and turns within the outer on two pivots at the extremities of a diameter, where the hour XII. is marked. A thin reglet or bridge goes across the circles, and in its middle there is an opening, along which there slides a cursor C, having a small hole in it for the sun to shine through. The middle of this bridge represents the axis of the celestial sphere, and its extremities the poles; on one side of it the signs of the zodiac are drawn, and on the other the days of the month. A piece, to which a ring H is fixed, slides along the meridian; and the dial when used is suspended by the ring.

In this dial, the divisions laid down on the axis on either side of the centre are the tangents of the angles of the sun's declination, to a radius equal to that of the brass circle, which represents the equator. These may be laid down from a scale of equal parts, of which 1000 answer to the length of the semiaxes from the centre to the equinoctial ring; and then the extreme divisions on the axis will be at the distance of \( 43^\circ \) of these equal parts (the tangent of \( 23^\circ 29' \)) from the division at the centre.

To use this dial, place the line marked on the sliding piece (opposite to the supporting ring) over the degree of the latitude of the place, and put the line which crosses the hole of the cursor to the degree of the sign or to the day of the month. Turn the inner ring on its pivots, and put it at right angles to the outer; and suspend the instrument by the ring H, so that the middle of the bridge (which is the axis of the dial) may be parallel to the axis of the world; then turn its flat side towards the sun, so that his rays passing through the little hole in the cursor may fall exactly on a line drawn round the middle of the concave surface of the inner ring; and in this posi- The bright spot shows the hour of the day on that surface.

The hour of XII. is not shown by this dial, because the outer circle, being then in the plane of the meridian, hinders the sun's rays from falling on the inner. Nor will it serve to show the hour on the equinoctial days, because his rays then pass parallel to the plane of the dial.

Universal Dial on a Cross.

95. This dial, represented by figure I of Plate CLXXXVI., is moveable on a joint C for elevating it to any latitude on the quadrant CO 90, as it stands on the horizontal board A. The arms of the cross stand perpendicularly to the middle part, and the top of it from a to n is of equal length with either of the arms m, n.

Having set the board A level, and the line ut (on the middle of a side of the cross) to the latitude of the place on the quadrant, and the point N on the compass north by the needle, allowing for the variation; the plane of the cross will then be parallel to the plane of the equator, and the dial rectified so as to show time.

The morning hours from III. till VI. will be shown by the upper edge kl of the arm io casting a shadow on a face of the arm cm; from VI. to IX. the lower edge i of the arm io will cast a shadow on the hours of the side og; from IX. to XII. noon, the edge ab of the upper part an will cast a shadow on the arm nf; from XII. to III. in the afternoon, the edge cd of the top part will cast a shadow on the hours of the arm hlm; and from III to VI. the edge gh will cast a shadow on the hours on the part pn; and from VI. to IX. the shadow of the edge ef will show the time on the top part on.

The breadth of each part ab, cf, &c. must be so great as never to let the shadow fall entirely without the part or arm on which the hours are marked, when the sun is at his greatest declination from the equator. To determine the breadth of the sides of the arms which contain the hours, make an angle ABC (fig. 2) of 23° 16', the sun's greatest declination; and in Be take Bd, equal to the intended length of each arm, from the side of the long middle part, which is also the length of the top part above the arms, and draw the perpendicular adf. Then, as the edges of the shadow from each arm will be parallel to BA when the sun's declination is 23° 16', it is plain that if the length of the shadow be Bd, the least breadth it can have to keep the edge Be of the shadow Befd from going off the side of the arm before it comes to its end, must be de or Bd'; but in order to keep the shadow within the quarter divisions of the hours when it comes near the end of the arm, the breadth thereof should be still greater, so as to be almost doubled, on account of the distance between the tips of the arms.

To place the hours right on the arms, lay down the shape and size of the cross abed on paper (fig. 3), and on a as a centre, with ae as a radius, describe the quadrantal arc ef. Divide this arc into six equal parts, and through the divisions draw lines ag, ah, &c. continuing three of them to the arm ae, which are all that fall on it; and they will meet the arms in the points through which the hour-lines are to be drawn right across it.

Divide each arm in the same manner, and set the hours to their proper places, as shown in the figure; each of the hour spaces should be divided into four equal parts for the half hours and quarters in the quadrant ef; and straight lines should be drawn through the division marks of the quadrant to the arms of the cross, in order to de-

An universal Dial, showing the Hours by a Globe, and by several Gnomons.

96. The dial is represented in fig. 9, Plate CLXXXVI. It may be made of a thick square piece of wood or hollow metal. The sides are cut into semicircular hollows, on which the hours are placed, the style of each coming out from its bottom as far as the ends of the hollow projects. The corners are cut into angles, in the insides of which the hours are also marked; and the edge at the end of each side of the angle serves as a style for casting a shadow on the hours marked on the opposite side. In the middle of the uppermost plane there is an equinoctial dial, in the centre of which an upright wire is fixed for casting a shadow on the hours of that dial, and supporting a small terrestrial globe on its top. The whole dial stands on a pillar rising out of a circular base, in which there is a compass and magnetic needle for placing the meridian style towards the south. The pillar has a joint with a graduated quadrant on it (supposed to be hid from sight under the dial in its representation) for setting the dial to the latitude of any given place. The equator of the globe is divided into twenty-four equal parts, and the hours are laid down on it: the time of the day may be shown by these hours when the sun shines on the globe.

To construct the dial: On a square piece of wood or metal of proper thickness draw the lines ae, bd (fig. 8) at a distance equal to the intended thickness of the style abed; and in the same manner trace out the form of the three other styles efgh, ihlm, knpq, all directed towards the centre of the square. On a and b as centres, with such a radius as will leave sufficient strength of stuff, when the distance KI is equal to Ao, describe the quadrantal arcs Ag, Bd. In like manner, with the same radius describe arcs in all the quadrantal openings, leaving room, however, for the equinoctial dial in the middle. Divide each quadrant into six equal parts for as many hours, as in the figure, and subdivide these for half hours and quarters, numbering the whole eight quadrants as in the figure. To lay down the hours in the angular spaces in the corners: On K and I as centres, with a radius equal to KI, which is equal to Ao, describe the arcs Kl, Il, meeting in t. Divide each arc into four equal parts, and from the centres through the points of division draw the right lines 13, 14, 15, 16, 17, and K2, Kl, K12, K11; and they will meet the sides of the angle where the hours should be marked. These hour spaces should also be subdivided into quarters. Do the like for the other three angles, determining in this way the hours shown as in the figure, in which it will be observed that the dotted lines which are parallel are all directed to the same hour of the day. The angular and quadrantal spaces should now be cut out, quite through the solid material of which the dial is formed, and the gnomons inserted in their proper position, observing that these should be as broad as the dial is thick, and this breadth and thickness such as to keep the shadows of the gnomons from ever falling quite out of the hollows, even when the sun's declination is greatest. Lastly, construct the equinoctial dial in the middle, agreeably to its theory in art. 19; and the dial will be finished.

To rectify and use the dial: Place it with its base level, and the gnomons of the quadrants directed to the cardinal points by means of the compass; then bend the pillar in the joint, till the axis be inclined to the horizon at an

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1 Place dashes (omitted in the engraving) over the letters d and e in the perpendicular to Be, thus making them d', e'; and in fig. 3 observe that the letter a ought to be over the figure 3 on the adjoining line. Also in fig. 1 write b on the top opposite to c. Dialling angle equal to the latitude of the place. The plane of the equinoctial dial will then be parallel to the equator, the axis of the globe directed to the pole of the heavens; and when the sun shines on the dial, the hours will be indicated by all the parallel edges which cast a shadow, as well as by the axis of the globe and the globe itself, which will show to what places of the earth the sun is rising, setting, and in the meridian.

Universal Mechanical Dial, Plate CLXXXVII. Fig. 11.

96. This is an equinoctial dial, which may be applied to the construction of a dial on any kind of plane, by a process requiring no calculation. Suppose, for example, the plane ABCD is horizontal, and that GEF is the meridian line, assumed if the plane be moveable, or found if it be fixed. Place the equinoctial dial H with its axis GI directed to the pole, or, in the present case, so as to make with the meridian line an angle equal to the latitude of the place, with the twelve o'clock hour line over the meridian line. If now EF, the lower edge of the upright triangular support KEF, lie along the meridian, and the edge EK make with EF an angle equal to the latitude, the dial H will be in its proper position when its axis GI is parallel to EK. Supposing now the whole apparatus to be in a dark place, let a lighted candle be carried round GI, the axis of the equinoctial dial, and let the position of the shadow on the horizontal plane be noted when it falls on the successive hours of the equinoctial dial, then lines from G through these positions will be the hour lines on the horizontal dial. In this way a dial may be described on any plane whatever.

Dials on three different Planes, Plate CLXXXVII. Fig. 12.

97. This combination of dials shows the hour at the same time on the equinoctial dial IK, the horizontal dial ABC, and the vertical south dial; the same axis or gnomon FHG serving all the three. The figure sufficiently explains its construction and application. The method of tracing the hour lines proposed in the preceding article may be applied to produce the horizontal and south dials from the equinoctial dial.

Babylonian, Italian, and Jewish Hours.

The dials which have been described in this article are all intended to show astronomical or apparent time, of which the hours are equal, if we set aside the equation of time. Such are the hours by which we reckon time; but besides these, there are Babylonian, Italian, and Jewish hours.

The Babylonian hours are reckoned from sun rising to sun rising, and are twenty-four equal hours, nearly of the same length as the common hours, only they are differently numbered.

The Italian hours begin at sunset, and are numbered to twenty-four at sunset next day. They also are equal, and nearly the same as the common hours.

The Jewish hours, otherwise called the old unequal planetary hours, are reckoned from sunrise; and the day, from sunrise to sunset, is divided into twelve equal parts or hours. The hours of one day, however, will not be equal to those of another, at least in our climate, because of the inequality of the length of the days.

The older writers on dialling teach how to construct dials which shall show time according to all these ways of dividing the day. They are, however, mere matters of curiosity, and therefore we think it sufficient, in this work, to have thus briefly noticed them. They are fully explained by Emerson in his Dialling; see also Ferguson's Lectures.

Table of the Sun's Longitude and Declination, and the Equation of Time, for every day in the year.

The construction of some of the dials which have been described requires that the sun's place in the ecliptic, that is, his longitude reckoned from the beginning of the sign Aries, and his declination for the different days of the year, should be known. And in order to convert apparent time into true time, that is, the time shown by the sun into the time shown by a good clock or watch going uniformly, the equation of time is wanted for every day of the year. None of these, however, can be exactly the same on the same day of every year at any given place. Thus, at Greenwich, for four succeeding years beginning with 1820, they were at noon on the first of March as follows:

| Years | Sun's Longitude | Declination | Equation of Time | |-------|-----------------|-------------|------------------| | 1820 | 11 | 52 | 39 | 7 29 44 | 12 36-2 | | 1821 | 11 | 10 | 38 | 10 | 7 35 13 | 12 39-1 | | 1822 | 11 | 10 | 23 | 38 | 7 40 43 | 12 41-7 | | 1823 | 11 | 10 | 8 | 49 | 7 46 18 | 12 43-4 | | 1824 | 11 | 10 | 54 | 55 | 7 28 50 | 12 36-1 |

There are various reasons why the above three elements are not the same on the same day of every year: the principal one is, because the sun requires nearly six hours more than 365 days to complete a revolution in the ecliptic; and this annual deficiency being compensated by the intercalary day in every fourth year, after this interval the elements return nearly, but not exactly, to the same value on the same day of the year. Our table corresponds to 1830, the second after leap year. This will be sufficiently accurate for the construction of dials, and finding true time by them for a considerable number of years to come.

The columns of the table containing the sun's longitude and declination require no explanation. The numbers in that for the equation of time have the signs of addition (+) and of subtraction (−) annexed to them. Thus the sign + joined to 3m. 50s., the equation for the first of January, is understood to belong to all the days from the beginning of the year to the 15th of April inclusive, and to indicate that the equation is to be added to the time shown by the dial to get true time, or that shown by the clock. Again, the sign − belongs to all days from 16th April to 15th June inclusive, and shows that the equation must be subtracted from the time shown by the dial, and so on throughout the year. ### TABLE showing the Sun's Longitude and Declination, and the Equation of Time, for every day of the second year after leap year.

#### JANUARY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 9 10 39 | 23 2 S. | 3 59+ | | 2 | 9 11 40 | 22 57 | 4 19 | | 3 | 9 12 42 | 22 51 | 4 47 | | 4 | 9 13 43 | 22 45 | 5 14 | | 5 | 9 14 44 | 22 39 | 5 41 |

#### MARCH

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 9 15 45 | 22 32 | 6 8 | | 2 | 9 16 46 | 22 24 | 6 34 | | 3 | 9 17 47 | 22 17 | 7 0 | | 4 | 9 18 49 | 22 8 | 7 25 | | 5 | 9 19 50 | 22 0 | 7 49 |

#### MAY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 9 20 51 | 21 50 | 8 13 | | 2 | 9 21 52 | 21 41 | 8 57 | | 3 | 9 22 53 | 21 31 | 8 59 | | 4 | 9 23 54 | 21 21 | 9 22 | | 5 | 9 24 55 | 21 10 | 9 43 |

#### FEBRUARY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 10 12 13 | 17 9 S. | 13 57+ | | 2 | 10 13 13 | 16 52 | 14 4 | | 3 | 10 14 14 | 16 34 | 14 11 | | 4 | 10 15 15 | 16 17 | 14 17 | | 5 | 10 16 16 | 15 58 | 14 22 |

#### APRIL

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 10 17 17 | 15 40 | 14 26 | | 2 | 10 18 17 | 15 22 | 14 29 | | 3 | 10 19 18 | 15 3 | 14 32 | | 4 | 10 20 19 | 14 44 | 14 34 | | 5 | 10 21 19 | 14 24 | 14 35 |

#### JUNE

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 10 22 20 | 14 5 | 14 35 | | 2 | 10 23 21 | 13 45 | 14 35 | | 3 | 10 24 21 | 13 25 | 14 33 | | 4 | 10 25 22 | 13 5 | 14 31 | | 5 | 10 26 22 | 12 44 | 14 29 |

#### JULY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 10 27 23 | 12 23 | 14 25 | | 2 | 10 28 23 | 12 3 | 14 21 | | 3 | 10 29 24 | 11 42 | 14 16 | | 4 | 11 0 24 | 11 20 | 14 11 | | 5 | 11 1 25 | 10 59 | 14 5 |

#### AUGUST

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 11 2 25 | 10 37 | 13 53 | | 2 | 11 3 26 | 10 16 | 13 51 | | 3 | 11 4 26 | 9 54 | 13 43 | | 4 | 11 5 26 | 9 32 | 13 34 | | 5 | 11 6 27 | 9 9 | 13 25 |

#### SEPTEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 11 7 27 | 8 47 | 13 15 | | 2 | 11 8 27 | 8 24 | 13 4 | | 3 | 11 9 27 | 8 2 | 12 53 |

#### OCTOBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 11 10 27 | 7 39 S. | 12 42 + | | 2 | 11 11 28 | 7 16 | 12 30 | | 3 | 11 12 28 | 6 53 | 12 17 | | 4 | 11 13 28 | 6 30 | 12 4 | | 5 | 11 14 28 | 6 7 | 11 50 |

#### NOVEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 11 15 28 | 5 44 | 11 36 | | 2 | 11 16 28 | 5 21 | 11 22 | | 3 | 11 17 27 | 4 58 | 11 7 | | 4 | 11 18 27 | 4 34 | 10 51 | | 5 | 11 19 27 | 4 11 | 10 36 |

#### DECEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 11 20 27 | 3 47 | 10 20 | | 2 | 11 21 27 | 3 24 | 10 4 | | 3 | 11 22 27 | 3 0 | 9 47 | | 4 | 11 23 26 | 2 36 | 9 30 | | 5 | 11 24 26 | 2 13 | 9 13 |

#### JANUARY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 12 25 | 1 15 | 16 28 | | 2 | 12 26 | 1 16 | 16 45 | | 3 | 12 27 | 1 17 | 17 1 | | 4 | 12 28 | 1 18 | 17 8 | | 5 | 12 29 | 1 19 | 17 34 |

#### MARCH

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 12 30 | 1 20 | 17 49 | | 2 | 12 31 | 1 21 | 18 4 | | 3 | 12 32 | 1 22 | 18 20 | | 4 | 12 33 | 1 23 | 18 34 | | 5 | 12 34 | 1 24 | 18 49 |

#### MAY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 12 35 | 1 25 | 19 3 | | 2 | 12 36 | 1 26 | 19 17 | | 3 | 12 37 | 1 27 | 19 43 | | 4 | 12 38 | 1 28 | 19 56 | | 5 | 12 39 | 1 29 | 20 8 |

#### FEBRUARY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 20 | 1 30 | 20 26 | | 2 | 13 21 | 1 31 | 20 43 | | 3 | 13 22 | 1 32 | 20 59 | | 4 | 13 23 | 1 33 | 20 76 | | 5 | 13 24 | 1 34 | 20 93 |

#### APRIL

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 25 | 1 35 | 21 10 | | 2 | 13 26 | 1 36 | 21 27 | | 3 | 13 27 | 1 37 | 21 44 | | 4 | 13 28 | 1 38 | 21 51 | | 5 | 13 29 | 1 39 | 21 58 |

#### JUNE

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 30 | 1 40 | 21 65 | | 2 | 13 31 | 1 41 | 21 72 | | 3 | 13 32 | 1 42 | 21 79 | | 4 | 13 33 | 1 43 | 21 86 | | 5 | 13 34 | 1 44 | 21 93 |

#### JULY

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 35 | 1 45 | 22 10 | | 2 | 13 36 | 1 46 | 22 17 | | 3 | 13 37 | 1 47 | 22 24 | | 4 | 13 38 | 1 48 | 22 31 | | 5 | 13 39 | 1 49 | 22 38 |

#### AUGUST

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 40 | 1 50 | 22 45 | | 2 | 13 41 | 1 51 | 22 52 | | 3 | 13 42 | 1 52 | 22 59 | | 4 | 13 43 | 1 53 | 22 66 | | 5 | 13 44 | 1 54 | 22 73 |

#### SEPTEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 45 | 1 55 | 22 79 | | 2 | 13 46 | 1 56 | 22 86 | | 3 | 13 47 | 1 57 | 22 93 | | 4 | 13 48 | 1 58 | 23 00 | | 5 | 13 49 | 1 59 | 23 07 |

#### OCTOBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 50 | 2 0 | 23 14 | | 2 | 13 51 | 2 1 | 23 21 | | 3 | 13 52 | 2 2 | 23 28 | | 4 | 13 53 | 2 3 | 23 35 | | 5 | 13 54 | 2 4 | 23 42 |

#### NOVEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 13 55 | 2 5 | 23 49 | | 2 | 13 56 | 2 6 | 23 56 | | 3 | 13 57 | 2 7 | 23 63 | | 4 | 13 58 | 2 8 | 23 70 | | 5 | 13 59 | 2 9 | 23 77 |

#### DECEMBER

| Days | Sun's Longitude | Sun's Declination | Equation of Time | |------|-----------------|-------------------|------------------| | | | | | | 1 | 14 00 | 3 0 | 23 84 | | 2 | 14 01 | 3 1 | 23 91 | | 3 | 14 02 | 3 2 | 23 98 | | 4 | 14 03 | 3 3 | 24 05 | | 5 | 14 04 | 3 4 | 24 12 | | JULY | SEPTEMBER | NOVEMBER | |------|-----------|----------| | Days | Sun's Longitude | Equation of Time | Days | Sun's Longitude | Equation of Time | Days | Sun's Longitude | Equation of Time | | 1 | 3 9 4 | 23 9 N. | 1 | 5 8 30 | 6 24 N. | 1 | 7 8 34 | 14 22 S. | | 2 | 3 10 1 | 23 5 | 2 | 5 9 23 | 6 2 | 2 | 7 9 34 | 14 41 | | 3 | 3 10 58 | 23 0 | 3 | 5 10 26 | 7 40 | 3 | 7 10 35 | 15 0 | | 4 | 3 11 56 | 22 55 | 4 | 5 11 24 | 7 18 | 4 | 7 11 35 | 15 19 | | 5 | 3 12 53 | 22 50 | 5 | 5 12 22 | 6 56 | 5 | 7 12 35 | 15 38 | | 6 | 3 13 59 | 22 44 | 6 | 5 13 20 | 6 33 | 6 | 7 13 35 | 15 56 | | 7 | 3 14 47 | 22 38 | 7 | 5 14 19 | 6 11 | 7 | 7 14 35 | 16 14 | | 8 | 3 15 44 | 22 32 | 8 | 5 15 17 | 5 48 | 8 | 7 15 36 | 16 31 | | 9 | 3 16 41 | 22 25 | 9 | 5 16 15 | 5 26 | 9 | 7 16 36 | 16 49 | | 10 | 3 17 39 | 22 18 | 10 | 5 17 14 | 5 3 | 10 | 7 17 36 | 17 6 | | 11 | 3 18 36 | 22 10 | 11 | 5 18 12 | 4 40 | 11 | 7 18 37 | 17 23 | | 12 | 3 19 33 | 22 2 | 12 | 5 19 10 | 4 17 | 12 | 7 19 37 | 17 39 | | 13 | 3 20 30 | 21 54 | 13 | 5 20 9 | 3 54 | 13 | 7 20 38 | 17 55 | | 14 | 3 21 23 | 21 45 | 14 | 5 21 7 | 3 31 | 14 | 7 21 38 | 18 11 | | 15 | 3 22 25 | 21 36 | 15 | 5 22 6 | 3 8 | 15 | 7 22 39 | 18 27 | | 16 | 3 23 22 | 21 26 | 16 | 5 23 5 | 2 45 | 16 | 7 23 39 | 18 42 | | 17 | 3 24 19 | 21 16 | 17 | 5 24 3 | 2 22 | 17 | 7 24 40 | 18 57 | | 18 | 3 25 17 | 21 6 | 18 | 5 25 2 | 1 59 | 18 | 7 25 41 | 19 12 | | 19 | 3 26 14 | 20 53 | 19 | 5 26 0 | 1 35 | 19 | 7 26 41 | 19 28 | | 20 | 3 27 11 | 20 44 | 20 | 5 26 50 | 1 12 | 20 | 7 27 42 | 19 40 | | 21 | 3 28 9 | 20 33 | 21 | 5 27 58 | 0 49 | 21 | 7 28 43 | 19 53 | | 22 | 3 29 6 | 20 21 | 22 | 5 28 57 | 0 25 | 22 | 7 29 43 | 20 6 | | 23 | 4 0 3 | 20 9 | 23 | 5 29 55 | 0 2 | 23 | 7 30 44 | 20 19 | | 24 | 4 1 1 | 19 57 | 24 | 5 30 4 | 0 02 S. | 24 | 8 1 45 | 20 32 | | 25 | 4 1 58 | 19 44 | 25 | 5 31 5 | 0 45 | 25 | 8 2 45 | 20 44 | | 26 | 4 2 55 | 19 31 | 26 | 5 32 5 | 1 3 | 26 | 8 3 46 | 20 55 | | 27 | 4 3 53 | 19 18 | 27 | 5 33 1 | 1 32 | 27 | 8 4 47 | 21 7 | | 28 | 4 4 50 | 19 4 | 28 | 5 34 0 | 1 55 | 28 | 8 5 48 | 21 17 | | 29 | 4 5 47 | 18 50 | 29 | 5 34 9 | 2 19 | 29 | 8 6 49 | 21 28 | | 30 | 4 6 45 | 18 36 | 30 | 5 35 8 | 2 42 | 30 | 8 7 49 | 21 38 | | 31 | 4 7 42 | 18 22 | 31 | 5 36 7 | 3 0 | 31 | 8 8 50 | 21 48 |

| AUGUST | OCTOBER | DECEMBER | |--------|---------|----------| | 1 | 4 8 40 | 16 7 N. | 1 | 6 7 47 | 3 5 S. | 1 | 8 5 59 | 21 48 S. | | 2 | 4 9 37 | 17 51 | 2 | 6 8 46 | 3 29 | 2 | 8 9 51 | 21 57 | | 3 | 4 10 34 | 17 36 | 3 | 6 9 45 | 3 52 | 3 | 9 10 52 | 22 6 | | 4 | 4 11 32 | 17 20 | 4 | 6 10 44 | 4 15 | 4 | 9 11 53 | 22 14 | | 5 | 4 12 29 | 17 4 | 5 | 6 11 43 | 4 38 | 5 | 9 12 54 | 22 22 | | 6 | 4 13 27 | 16 43 | 6 | 6 12 42 | 5 1 | 6 | 9 13 55 | 22 29 | | 7 | 4 14 24 | 16 31 | 7 | 6 13 42 | 5 24 | 7 | 9 14 56 | 22 36 | | 8 | 4 15 22 | 16 15 | 8 | 6 14 41 | 5 47 | 8 | 9 15 57 | 22 43 | | 9 | 4 16 19 | 15 57 | 9 | 6 15 40 | 6 10 | 9 | 9 16 58 | 22 50 | | 10 | 4 17 17 | 15 49 | 10 | 6 16 40 | 6 33 | 10 | 9 17 59 | 22 57 | | 11 | 4 18 15 | 15 22 | 11 | 6 17 39 | 6 56 | 11 | 9 18 0 | 23 0 | | 12 | 4 19 12 | 15 5 | 12 | 6 18 39 | 7 19 | 12 | 9 19 1 | 23 5 | | 13 | 4 20 10 | 14 46 | 13 | 6 19 33 | 7 41 | 13 | 9 20 2 | 23 9 | | 14 | 4 21 8 | 14 23 | 14 | 6 20 33 | 8 4 | 14 | 9 21 3 | 23 13 | | 15 | 4 22 5 | 14 10 | 15 | 6 21 37 | 8 26 | 15 | 9 22 4 | 23 17 | | 16 | 4 23 3 | 13 51 | 16 | 6 22 37 | 8 48 | 16 | 9 23 5 | 23 20 | | 17 | 4 24 1 | 13 32 | 17 | 6 23 36 | 9 10 | 17 | 9 24 6 | 23 24 | | 18 | 4 24 58 | 13 12 | 18 | 6 24 36 | 9 32 | 18 | 9 25 7 | 23 28 | | 19 | 4 25 56 | 12 53 | 19 | 6 25 36 | 9 54 | 19 | 9 26 8 | 23 32 | | 20 | 4 26 54 | 12 33 | 20 | 6 26 35 | 10 16 | 20 | 9 27 9 | 23 36 | | 21 | 4 27 52 | 12 13 | 21 | 6 27 35 | 10 37 | 21 | 9 28 10 | 23 40 | | 22 | 4 28 50 | 11 53 | 22 | 6 28 35 | 10 59 | 22 | 9 29 11 | 23 44 | | 23 | 4 29 48 | 11 33 | 23 | 6 29 35 | 11 20 | 23 | 9 30 12 | 23 48 | | 24 | 5 0 46 | 11 13 | 24 | 6 30 35 | 11 41 | 24 | 9 31 13 | 23 52 | | 25 | 5 1 43 | 10 52 | 25 | 6 31 34 | 12 2 | 25 | 9 32 14 | 23 56 | | 26 | 5 2 41 | 10 31 | 26 | 6 32 34 | 12 15 | 26 | 9 33 15 | 23 60 | | 27 | 5 3 39 | 10 11 | 27 | 6 33 34 | 12 43 | 27 | 9 34 16 | 23 64 | | 28 | 5 4 37 | 9 49 | 28 | 6 34 34 | 13 13 | 28 | 9 35 17 | 23 68 | | 29 | 5 5 35 | 9 28 | 29 | 6 35 34 | 13 23 | 29 | 9 36 18 | 23 72 | | 30 | 5 6 33 | 9 7 | 30 | 6 36 34 | 13 43 | 30 | 9 37 19 | 23 76 | | 31 | 5 7 31 | 8 45 | 31 | 6 37 34 | 14 3 | 31 | 9 38 20 | 23 80 |