Mathematical Instruments
Book VI. Ch. IV.

# Of the Construction and Use of an Instrument shewing the Eclipses of the Sun and Moon, the Months and Lunar Years, as also the Epacts.

This Instrument was invented by M. de la Hire, and is composed of three round Plates of Brass, or Pieces of Paste-board, and an Index which turns about a common Center upon the Face of the upper Plate, which is the least. There are two circular Bands, the one blue, and the other white, in which are made little round Holes; the outward of which shews the New Moons, and the Image of the Sun; and the inward ones, the Full Moons, and the Image of the Moon. The Limb of this Plate is divided into 12 lunar Months, each containing 29 Days, 12 Hours, 44 Minutes; but in such manner, that the End of the 12th Month, which makes the Beginning of the second lunar Year, may surpass the first New Moon by the Quantity of 4 of 179 Divisions s denoted upon the middle Plate.

Upon the Limb of this Plate is fastened an Index, one of whose Sides, which is in the fiducial Line, makes part of a right Line, tending to the Center of the Instrument; which Line also passes thro’ the Middle of one of the outward Holes, shewing the first New Moon of the lunar Year. Note, The Diameter of the Holes is equal to the Extent of about 4 Degrees.

The Limb of the second Plate is divided into 179 equal Parts, serving for so many lunar Years, each of which is 354. Days, and about 9 Hours. The first Year begins at the Number 179, at which the last ends.

The Years accomplished are each denoted by their Numbers 1, 2, 3, 4, &c. at every fourth Division, and which make four times a Revolution to compleat the Number 179, as may be seen in the Figure of this Plate. Each of the lunar Years comprehend four of the aforesaid Divisions: So that in this Figure they anticipate one upon the other four of the said 179 Divisions of the Limb.

Upon the Limb of the same Plate, under the Holes of the first, there is a Space coloured black, answering to the outward Holes, and which shews the Eclipses of the Sun, and another red Space, answering to the innermost Holes, shewing the Eclipses of the Moon. The Quantity of each Colour appearing through the Holes, shews the Bigness of the Eclipse. The Middle of the two Colours, which is the Middle of the Moon’s Node, answers on one Side to the Division marked 4$$\frac{2}{3}$$ of a Degree; and on the other Side it answers to the opposite Number.

The Figure of the coloured Space is shewn upon this second Plate, and it’s Amplitude or Extent shews the Limits of Eclipses.

The third and greatest Plate, which is underneath the others, contains the Days and Months of common Years. The Divisions begin at the first Day of March, to the End that a Day may be added to the Month of February, when the Year is Bissextile. The Days of the Year are described in form of a Spiral, and the Month of February goes out beyond the Month of March, because the lunar Year is shorter than the solar one; so that the 15th Hour of the 10th Day of February answers to the Beginning of March: But after having reckoned the last Day of February, you must go back again to have the first of March. There are thirty Days marked before the Month of March, which serve to find the Epacts.

Note, That the Days, as they are here taken, are not accomplished pursuant to the Use of Astronomers, but as they are vulgarly reckoned, beginning on a Minute, and ending at the Minute of the following Day. Therefore every time that the first Day, or any other of a Month is spoken of, we understand the Space of that Day marked in the Divisions; for we here reckon the current Days according to vulgar Use.

In the Middle of the upper Plate are wrote the Epochs, shewing the Beginning of the lunar Years, with respect to the solar Years, according to the Gregorian Calendar, and for the Meridian of Paris. The Beginning of the first Year, which must be denoted by 0, and answers to the Division 179, happened in the Year 1680 at Paris, the 29th of February at 14$$\frac{1}{2}$$ Hours. The End of the first lunar Year, being the Beginning of the second, answers to the Division marked 1, which happened at Paris in the Year 1681, the 27th of February, at 23$$\frac{1}{2}$$ Hours, in counting successively 24 Hours from one Minute to the other. And left there should be an Error in comparing the Divisions of the Limb of the second Plate with the Divisions of the Epochs of the lunar Years answering them, we have put the same Numbers to them both.

We have set down successively the Epochs of all the lunar Years, from the Year 1700 to the Year 1750, to the End that the Use of this Instrument may more easily serve to make each of the aforesaid solar and lunar Years agree together. As to the other Years of our Cycle of 179 Years, it will be easy to render it compleat, in adding 354 Days, 8 Hours, 48$$\frac{2}{3}$$ Minutes for each lunar Year.

The Index extending itself from the Center of the Instrument to the Limb of the greatest Plate, serves to compare the Divisions of one Plate with those of the two others. And if this Instrument be applied to a Clock, a perfect and accomplished Instrument in all it’s Parts will be had.

The Table of Epochs, which is fitted for the Meridian of Paris, may easily be reduced to other Meridians; if for the Places eastward of Paris, the Time of the Difference of Meridians be added; and for Places westward, the Time of the Difference of Meridians be substracted.

It is proper to place the Table of Epochs in the Middle of the upper Plate, to the End that it may be seen with the Instrument.

## How to make the Divisions upon the Plates.

The Circle of the greatest Plate is so divided, that 368 Deg. 2 Min. 42 Sec. may comprehend 354 Days, and something less than 9 Hours; from whence it is manifest, that the Circle must contain 346 Days, 15 Hours, which may without sensible Error be taken for $$\frac{2}{3}$$ of a Day. Now to divide a Circle into 346$$\frac{2}{3}$$ equal Parts, reduce the whole into third Parts, which in this Example make 1040; then seek the greatest Multiple of 3 less than 1040, which may be halved. Such a Number will be found in a double Geometrical Progression, whose first Term is 3; as for Example, 3, 6, 12, 24, 48, 96, 192, 384, 768.

Now the 9th Number of this Progression is the Number sought. Then substract 768 from 1040, there will remain 272, and find how many Degrees, Minutes, and Seconds, this remaining Number makes; by saying, As 1040 is to 360 Deg. So is 272 to 94 Deg. 9 Min. 23 Sec.

Therefore take an Angle of 94 Deg. 9 Min. 23 Sec. from the said Circle, and divide the remaining Part of the Circle always into half, after having made 8 Subdivisions, you will come to the Number 3, which will be the Arc of one Day; by which likewise dividing the Arc of 94 Deg. 9 Min. 23 Sec. the whole Circle will be found divided into 346$$\frac{2}{3}$$ Days; for there will be 256 Days in the greatest Arc, and 90$$\frac{2}{3}$$ Days in the other. Each of these Spaces answer to 1 Deg. 2 Min. 18 Sec. as may be seen in dividing 360 by 346$$\frac{2}{3}$$ and ten Days make 10 Deg. 23 Min. And thus a Table may be made, serving to divide the Plate.

Those Days are afterwards distributed to each of the Months of the Year, according to the Number corresponding to them, in beginning at the Month of March, and continuing on to the 15th Hour of the 10th of February, which answers to the Beginning of March, and the other Days of the Month of February go on farther above March.

The Circle of the Second Plate must be divided into 179 equal Parts; to do which, seek the greatest Number which may be continually bisected to Unity, and be contained exactly in 179: you will find 128 to be this Number, which take from 179, and there remains 51. Now find what part of the Circumference of the Circle the said Remainder makes; in saying, As 179 Parts is to 360 Deg. So is 51 Parts to 102 Deg. 34 Min. 11 Sec.

Therefore having taken from the Circle an Arc of 102 Deg. 34 Min. 11 Sec. divide the remaining Part of the Circle always into half; and after having made seven Subdivisions, you will come to Unity: whence this part of the Circle will be divided into 128 equal Parts; and then the remaining 51 Parts may be divided, by help of the last Opening of the Compasses. Wherefore the whose Circumference will be found divided into 179 equal Parts, every of which answers to 2 Deg. 40 Sec. as may be seen in dividing 360 by 179.

Lastly, To divide the Circle of the upper Plate, take one fourth of it’s Circumference, and add to it one of the 179 Parts or Divisions of the Limb of the middle Plate; the Companies opened to the Extent of the Quadrant thus augmented, being turned four Times over, will divide the Circle in the Manner as it ought to be: for in subdividing every of the Quarters into three equal Parts, one will have twelve Spaces for the twelve Lunar Months, in such manner, that the End of the 12th Month, which makes the Beginning of the Lunar Year, exceeds the first New Moon by 4 of the 179 Divisions, marked upon the middle Plate.

## Use of this Instrument.

A Lunar Year being proposed, to find the Days of the Solar Year corresponding to it, in which the New and Full Moons, together with the Eclipses, ought to happen.

For Example; Let the 24th Lunar Year of the Table of Epochs be proposed, which answers to the Division 24 of the middle Plate. Fix the Fiducial Line of the Index on the Upper Plate, over the Division marked 24, in the middle Plate, wherein the Beginning of the 25th Lunar Year is; and seeing by the Table of Epochs, that that Beginning tails upon the 14th Day of June, of the Year 1703, at 9 Hours, 52 Minutes, turn the two upper Plates together, in the Position they are in, till the Fiducial Line of the Index, fastened to the upper Plate, answers to the 10th Hour, or thereabouts, of the 4th of June, denoted upon the undermost Plate; at which time, the first New Moon of the proposed Lunar Year happens: for then the Fiducial Line passes thro’ the middle of the Hole of the first New Moon of the said Lunar Year.

Afterwards, without changing the Situation of the three Plates, extend a Thread from the Center of the Instrument, or the moveable Index, making it pass thro’ the middle of the Hole of the first Full Moon; and the Fiducial Line will answer to the beginning of the 29th Day of June, at 4 Hours and a Quarter; which is the time that that Full Moon was totally eclipsed, as appears by the red Colour quite filling the Hole, shewing the Full Moon.

By the same means we may know, that at the time of the Full Moon, which happened about the third Hour in the Morning, of the 14th of July, there was a partial Eclipse of the Sun.

If we proceed farther, the Eclipses may be known which happened in the Month of December, in the Year 1703, and towards the beginning of the following Year. But because the 10th New Moon goes out beyond the 28th Day of February, having brought the Index to the 28th Day of February, move the two upper Plates backwards, conjointly with the Index (in the Posture they are found in) until the Fiducial Line happens over the beginning of March; whence moving the Index over all the Holes of the New and Full Moons, and the last Plate will shew the times in which the Eclipses ought to happen.

But because the 13th New Moon is the first of the succeeding Lunar Year, which answers to the Number 25 of the Divisions of the middle Plate, leave the two undermost Plates in the posture they are found, and move forwards the upper Plate ’till the Fiducial Line meets with the Number 25 of the middle Plate, at which Point it will shew upon the greatest Plate, the first New Moon of the 26th Lunar Year, according to the order of our Epoch, which happened the 2d Day of June, 18 Hours 40 Minutes of the Year 1704; and afterwards moving the Index over the middle of the Holes of the New and Full Moons, it will shew upon the last Plate the Days they happened on, as well as the Eclipses to the End of February: after which, the same Operation must be made for the preceding Year, that is, that after having come to the last Day of February, you must proceed backwards to the first Day of March.

We might likewise find the Beginnings of all the Lunar Years without using the Table of Epochs; but since it is not possible to adjust the Plates and the Index so exactly one upon another, as that some Error may not happen, which will augment itself from Year to Year, the said Table of Epochs will serve to rectify the Use of this Instrument.

In placing the Fiducial Line of the Index upon the Moon’s Age, between the Days of the Lunar Months, denoted upon the Limb of the upper Plate, the correspondent Days of the common Months will be shewn, and the Hours nearly, upon the Limb of the lower Plate.

Note, That the Calculations of the Table of Epochs are made for the mean Time of the Full Moons, which supposes the Motions of the Sun and Moon always equable; from whence there will be found some Difference between the apparent Times of the New Moons, Full Moons, and Eclipses, as they appear from the Earth, and the times found by that Table.

The proper Motions of the Sun and Moon, as well as those of the other Planets, appear to us sometimes swift, and sometimes slow; which apparent Inequality in part proceeds from their Orbits being not concentric with the Earth, and in part from hence, that the equal Arcs of the Ecliptick, which are oblique to the Equator, do not always pass thro’ the Meridian with the equal Parts of the Equator. Astronomers, for the ease of Calculation, have fitted a Motion which they call mean or equable, in supposing the Planets to describe equal Arcs of their Orbits, in equal Times. That Time which they call true or apparent, is the measure of true or apparent Motion, and mean Time is the measure of mean Motion. They have likewise invented Rules for reducing mean Time to true or apparent Time, and contrariwise, for reducing true or apparent Time to mean Time.

## To find by Calculation whether there will happen an Eclipse at the time of the New or Full Moon.

For an Eclipse of the Sun, multiply by 7361, the Number of Lunar Months accomplished from that which begun the 8th of January, 1701, according to the Gregorian Calendar, to that which you examine, and add to the Product the Number 33890; then divide the Sum by 43200; and after the Division, without having regard to the Quotient, examine the Remainder, or the difference between the Divisor and the Remainder: for if either of them be less then 4060, there will happen an Eclipse of the Sun.

But to find an Eclipse of the Moon, likewise multiply by 7361, the Number of Lunar Months, accomplished from that which begun the 8th of January, 1701, to the New Moon preceding the Full Moon examined; add to the Product 37326, and divide the Sum by 43200. The Division being made, if the Remainder, or the difference between the Remainder and the Divisor, be less than 2800, there will be an Eclipse of the Moon,

Note. An Eclipse of the Sun or Moon will be so much the greater, as the Remainder or Difference is lesser; and contrariwise.

## Example of an Eclipse of the Sun.

It is required to find, whether at the New Moon of the 22d of May, in the Year 1705, there happened an Eclipse of the Sun.

From the 8th of January, 1701. to the 22d of May, 1705, there were accomplished 54 Lunations. Multiply, according to the Rule, the Number 54 by 7361, and add to the Product 33890: the Sum being divided by 43200, there will remain 42584, which is greater than 4060; and the Difference between the Remainder 42584, and the Divisor 43200, is 616, which is less than 4060: therefore there was then an Eclipse of the Sun.

## Example of an Eclipse of the Moon.

It is required to find whether the Full Moon of the 27th of April, in the Year 1706, was eclipsed.

From the 8th of January, in the Year 1701, to the New Moon preceding the Full Moon in question, there were 65 Lunar Months accomplished; therefore having multiplied, according to the Rule, the Number 65 by 7361, and added to the Product 37326, the Sum will be 515791; which being divided by 43200, without having any regard to the Quotient, the Remainder will be 40591, greater than 2800. The Difference between the Divisor and the Remainder is 609, which is less than 2800; therefore there was an Eclipse of the Moon the 27th day of April, 1706.