the Place of the Sun when his right Afcenfion is 15°. In the right-angled Triangle V a, right-angled at a, are given the Angle at r 23°. 29', and the Bafea 15°, to find the Hypothenufe O, the Longitude of the Sun from the firft Point of Y, or Place in the Ecliptic; which is an Extreme conjunct, and the Hypothenufe is one of the Extremes;

whence, by Lord Nepier's fecond Rule, and its Note,

By which it appears, that, when the Sun's Place is 16°. 17', his right Afcenfion is but 15°; fo that, in that Time, there is a Difference of 1°. 17' of the Equinoctial, which answers to five Minutes and eight Seconds of Time. Thus, as the Sun goes on towards the Tropic, he gains, 'till he comes to 15° of 8, which is half Way; after which he loses, 'till he arrives at the Tropic, when his Place and right Afcenfion become equal. And, if the Equinoctial were divided in every fifteenth Degree, and Meridians drawn thro' each, they would interfect the Ecliptic in the Sun's Place, for each 15° of right Afcenfion, as below:

The Earth revolves round its Axis in twenty-four Hours, at the fame Time moving forward in its Orbit about fifty-nine Minutes and eight Seconds. By this Means, the whole Equinoctial, and fifty-nine Minutes, eight Seconds, paffes the Meridian of any Place in twenty-four Hours: But the Ecliptic,

interfecting the Equinoctial at an Angle of 23° 29′, does not pass the Meridian equally with the Equinoctial. Hence, the Time from the Sun's Appearance on the Meridian of any Place, to the Time of his Return to it again, is fometimes more, fometimes lefs, than twenty-four Hours; but Clocks and Watches keep equal Time, when they go right; and this is one Reason why good Clocks and Watches go fometimes fafter, fometimes flower, than the Sun. The greatest Variation, arifing from this Caufe, is when the Sun is in the Equinoctial and Tropics; for, when the Sun is in the Equinox, and going towards the Tropics, the Time from Noon to Noon will be but twenty-three Hours, fifty-nine Minutes, and forty Seconds; but, in returning from the Tropics towards the Equinox, the Day will be twenty-four Hours and twenty-two Seconds.

There is another Caufe of the Inequality of natural Days, arifing from the annual Motion of the Earth, which moves round the Sun in an Ellipfis, in one of whofe Foci the Sun is placed; and, from the Laws of Gravity, the Velocity of the Earth is continually changing in its Orb, by which the Earth moves thro' different Portions of the Orb in twenty-four Hours, which produces a Variation in the Length of natural Days. The Time marked by the Motion of the Sun is called apparent Time; the Time marked by Clocks, and other Timekeepers, is called true Time; and the Difference is called the Equation of Time.

Prob. 95. The Sun's right Afcenfion 75°.48', and his greatest Declination 23°. 29', (always known) to find his Place in the Ecliptic, and prefent Declination.

cutting the Ecliptic in O, by which will be formed the right

angled

angled Triangle Oa, right-angled at a; in which is given the Angle at Y 23°. 29', and the Bafe Y a 75°.48', to find the Hypothenufe V O, or the Sun's Place in the Ecliptic; which is an Extreme conjunct, and the Hypothenufe is one of the Extremes; wherefore, by Lord Nepier's fecond Rule, and its Note, Page 266,

As the Tangent of r a, 75°. 48',

Is to the Radius,

So is the Co-fine of Ora, 23°.29',
To the Co-tangent of V O, 76°.56',

Therefore the Sun's Place is 16°. 56′ of п.
Declination, the fame Things being given, it is
treme conjunct, and the Perpendicular a O, being
Extremes, may be found by the fame Proportion:

As the Co-tangent of TO, 23°. 29′,

Is to the Radius,

So is the Sine of ra, 75°. 48',

10.596813

10.

9.962453

9.365640

To find the ftill an Exone of the

10.362044

10.

9.986523

19.986523

10.362044

To the Tangent of a C, 22°. 50′, the Sun's 9.624479

prefent Declination,

The Perpendicular a O may be measured by Prob. 46, and the Hypothenufer may be measured by Prob. 43.

Prob. 96. The Sun's prefent Declination 22°. 56' North, increafing, and greatest Declination 23°. 29, given, to find his Place in the Ecliptic, and right Afcenfion.

of Half-Tangents, lay, by Prob. 42, 22°.56′ from Ō to x; then,

Rr

by

by Prob. 17, draw bxa, the Parallel of the Sun's present Declination, interfecting the Ecliptic in thro' O, P, and S, by Prob. 17, draw the Meridian POS, interfecting the Equinoctial in D, and forming the right-angled Triangle OOD, right-angled at D; in which is given the Angle at O 23°. 29, and the Perpendicular DO 22°. 56', to find the Hypothenufe O, the Sun's Place in the Ecliptic; which is an Extreme disjunct, and the Hypothenufe is one of the Extremes; wherefore, by Lord Nepier's fourth Rule, and its Note, Page 266,

As the Sine of OD, 23°. 29′,

Is to the Radius,

So is the Sine of D O, 22°. 56′,

Co. Ar. 0.399591

To the Sine of O O, 77°. 55', the Sun's Place,

10.

9.590686

9.990277

That is 17.55 of II. But, had the Declination been North, decreasing, fubtract 77° 55′ from 180°, and the Remainder 102°. 5' is the Sun's Place in the Ecliptic, that is, 12°.5' of .

The Hypothenufe OO may be measured by Prob. 43.

Now, for the right Afcenfion, the fame Things being given, to find DO, is an Extreme conjunct, and DO is the middle Part; wherefore, by Lord Nepier's firft Rule, and its Note, Page 265,

As the Radius

Is to the Co-tangent of OOD, 23°.29',
So is the Tangent of DO, 22°.56',

To the Sine of DO, 76°.51′,

Which is the Sun's right Afcenfion. But, if the Declination had been North, decreafing, fubtract 76°. 51′ from 180°, and the Remainder, 103°. 9, is the right Afcenfion; and, if the Declination had been South, increafing, 180° muft be added to the Afcenfion found by the Problem; laftly, if the Declination given is South, decreafing, take the Degrees found by the Problem from 360°, and the Remainder is the Sun's right Afcenfion.

The Perpendicular DO may be measured by Prob. 46.

Prob.

Prob. 97. The Latitude of the Place (as London) 51°. 32', and the Sun's prefent Declination 23°. 4′ North, given, to find the Altitude of the Sun, and Hour of the Day, when the Sun is due Eaft or Weft.

H

Conftruction. With the Chord of 60°, on O, as a Center, draw H ZRN the Meridian ; draw HOR the Horizon, and ZON the Prime Vertical, or Azimuth of Eaft and Weft, at right Angles to it; take 51°.32, the Latitude, and fet from H to P, and from R to S; Æ draw POS the Axis of the World, and EOQ at right Angles to it for the quinoctial; then lay 23°.4′ from Æ to a,

and from Q to b; alfo lay the Half-Tangent of 23°.4′ from O to c; and, by Prob. 17, draw the Parallel of Declination, a cb, interfecting the Prime Vertical in O, the Sun's Place when due Eaft or Weft; laftly, by Prob. 17, draw the Meridian POS, interfecting the Equinoctial in d, and forming the right-angled Triangle O Od, right-angled at d; in which we have given the Angle O Od 51°. 32', the Distance of the Equinoctial ftom the Zenith, equal to the Latitude of the Place, and the Side Od 23°.4', the Sun's present Declination, to find the Hypothenufe OO, the Sun's Altitude above the Horizon when due Eaft or Weft; this is an Extreme disjunct, and the Hypothenufe is one of the Extremes; wherefore, by Lord Nepier's fourth Rule, and its Note, Page 266,

As the Sine of d, 51°. 32',
Is to the Radius

So is the Sine of O d, 23°.4'

Co. Ar. 0.106255

IO.

9.593067

To the Sine of OO, 30°.2', the Sun's Altitude, 9.699322

Then, for the Hour of the Day, the fame Things being given, to find the Bafe Od is an Extreme conjunct, and the Bafe Od is the middle Part; wherefore, by Lord Nepier's first Rule, and its Note, Page 265,