Of the Equation of Time

Time is the Measure of Motion, especially that of our Earth, from whence our Idea of Time proceeds; and if this were equable and uniform, there would be no such Thing as Equation; that is, if the Earth’s annual Orbit was a perfect Circle concentric to the Sun, and if the Earth moved an equal Part or Portion of that Circle every Day, and if at the same Time the Axis of her daily Motion were to be always perpendicular to the Plain of her Orbit, Time would never be equal.

But as the Motion of the Earth, is both unequal in its Orbit, and oblique upon its Axis, Time as entirely depending upon it, must also be unequal and subject to an Equation.

The causes from whence the Equation may proceed are many, but the chief, or most sensible, such as have already come under our Observation are only two. The first and greatest is this.

The Sun’s apparent mean diurnal Motion in the Ecliptic, is, taking one Day with another about 59^min. 8^2ds. but this Quantity in the Earth’s Orbit, from the Obliquity of the Ecliptic, produces an unequal Quantity in the Equinotial, and renders the Sun’s Appulse to the Meridian unfixed to any certain Time. And from this daily Interval of Motion arising from the Earth’s Advance in her annual Orbit, being various, not only from the difference from the Sun’s daily right Ascension, but also from the unequal Distance of the Sun in Longitude, and right Ascension from the nearest equinoctial Point, springs the first Equation thus. See Fig. V.

The second Cause of Equation, properly proceeds from an Equation of the first Equation, which hitherto we have consider’d, as the Effect of an equal Motion in an oblique Direction. But now we must look upon it to be unequal, and the Reason why it is so, will appear from what follows.

From the Theory of the Earth’s primary Motion, we prove that the Sun’s progress thro’ the Southern Signs, is made much sooner than thro’ the Northern ones, as thus.

Anno Dom. 1732, being Bissextile,

And in the common Year, 1733,

Hence ’tis plain, the Earth’s Motion must be alternately faster and slower in her elliptical Orbit, according to her Distance from the Sun, and will occasion a second Inequality of Time, which will be this.

Let CDE represent half the Earth’s elliptical Orbit; let A be the Center, to which the mean or middle Motion of the Earth is referr’d, and P the other Focus, representing the Sun. Here ED will be the Line of the Absides, and respectively E and D the Aphelion and Perihelion Points.

Now if the Earth be at G it is easy to conceive from the Figure and parallel Position of the Line aA, the true Motion of the Earth will go before and move faster than the Mean; In this Case Clocks will be too slow for the Sun: At C this Equation will be the greatest, to which, from the Absis D, the Quantity increase; and from which, to the Abiss E, the Quantity decreases; but on either Side, as at Y and K, it is very inconsiderable. In the first six Signs of the Anomalia, this Equation must be added to the apparent Times to gain the equal, but in the latter fix it must be substracted; its greatest Quantity is about 7^min. 44^2ds. and how it affects Time is shewn at bc and e.

Composed for common Use; shewing how many Minutes the mean or equal Time, is faster or slower than the apparent, or Sun’s Time, every Day in the Year.

The reason why the Seconds are not added in this Table, must be obvious to every Astronomer, since no Table of this Kind, depending upon the Sun’s Place, can be near true to Seconds, for more than eon Year in four, and that but to one time of the Day.

But in order to be able to keep a just Regulation of Time, and know how to distinguish at any time the mean or equal, from the apparent Time; a Movement should be made, whose Pendulum should vibrate true Seconds of equal Time, and that all the Oscillations may be as near equal to one another as possible, it is necessary that the Librations should be made betwixt and against two Cycloidical Curve Surfaces, as BZD and BE, whose Semi-chord, or Subtense, as is BD. And the Distance from the Point of Suspension B to the Center of Motion W, should be to different Parts of the World, as in the following Table.

Their Differences arise from the Figure of the Earth. See Sir Isaac Newton’s Principia.