Mathematical Instruments
Book VI. Ch. III.

Of making Celestial Observations.

Observations of the Sun, Stars, &c. made in the Day-Time with long Telescopes, are easy, because the Cross-Hairs in the Focus of the Object-Glass may then be distinctly perceived; but in the Night the said Cross-Hairs must be enlightened with a Link, or Candle, that so one may see them with the Stars, thro’ the Telescope: and this is done two ways.

First, We enlighten the Object-Glass of the Telescope, in obliquely bringing a Candle near to it, that so it’s Smoke or Body do not hinder the Progress of the Rays coining from the Star. But if the Object-Glass be something deep in the Tube, it cannot sufficiently be enlightened, without the Candle’s being very near it, and this hinders the Sight of the Star; and if the Telescope is above six Feet long, it will be difficult sufficiently to enlighten the Object-Glass, that so the cross Hairs be distinctly perceived.

Secondly, We make a sufficient Opening in the Tube of the Telescope near the Focus of the Object-Glass, thro’ which we enlighten with a Candle the cross Hairs placed in the Focus.

But this Method is subject to several Inconveniences, for the Light being so near the Observator’s Eyes, he is often incommoded thereby. And moreover, since the cross Hairs are by that Opening uncovered and exposed to the Air, they lose their Situation, become slack, or may be broken.

Besides this, the said second Method is liable to an Inconveniency for which it ought to be entirely neglected; and that is, that it is subject to an Error, which is, that according to the Position of the Light illuminating the cross Hairs, the said Hairs will appear in different Situations: because, for Example, when the Horizontal Hair is enlightened above, we perceive a luminous Line, which may be taken for the said Hair, and which appears at it’s upper Superficies. And contrariwise, when the said Hair is enlightened underneath, the luminous Line will appear at it’s lower Superficies, the Hair not being moved; and this Error will be the Diameter of the Hair, which often amounts to more than six Seconds. But M. de la Hire hath found a Remedy for this Inconveniency. For he often found, in Observations made in Moonshine Nights, in Weather a little foggy, that the cross Hairs were distinctly perceived; whereas, when the Heavens were serene, they could scarcely be seen: whence he bethought himself to cover that End of the Tube next to the Object-Glass with a Piece of Gawze, or very fine white silken Crape; which succeeded so well, that a Link placed at a good Distance from the Telescope so enlightened the Crape, that the cross Hairs distinctly appeared, and the Sight of the Stars was no way obscured.

Solar Observations cannot be made without placing a smoked Glass between the Telescope and the Eye, which may thus be prepared. Take two equal and well polished round Pieces of flat Glass, upon the Surface of one of which, all round it’s Limb, glew a Pasteboard Ring; then put the other Piece of Glass into the Smoke of a Link, taking it several times out, and putting it in again, for fear left the Heat of the Link should break it, until the Smoke be so thick thereon, that the Link can scarcely be seen thro’ it: but the Smoke must not be all over it of the same Thickness, that so that Place thereof may be chosen answering to the Sun’s Splendor. This being done, this Glass thus blackened, must be glewed to the before-mentioned Pasteboard Ring, with it’s blackened Side next to the other Glass, that so the Smoke may not be rubbed off.

Note, When the Sun’s Altitude is observed thro’ a Telescope, confiding of but two Glasses, it’s upper Limb will appear as tho’ it were the lower one.

There are two principal Kinds of Observations of Stars, the one being when they are in the Meridian, and the other when they are in vertical Circles.

If the Position of the Meridian be known, and then the Plane of the Quadrant be placed in the Meridian Circle, by means of the plumb Line suspended at the Center, the Meridian Altitudes of Stars may be easily taken, which are the principal Operations, serving as a Foundation to the whole Art of Astronomy. The Meridian Altitude of a Star may likewise be had by means of a Pendulum Clock, if the exact Time of the Star’s Passage by the Meridian be known. Now it must be observed, that Stars have the same Altitude during a Minute before and after their Passage by the Meridian, if they be not in or near the Zenith; but if they be, their Altitudes must be taken every Minute, when they are near the Meridian, which we suppose already known, and then their greatest or least Altitudes will be the Meridian Altitudes sought.

As to the Observations made without the Meridian in Vertical Circles, the Position of a given Vertical Circle must be known, or found by the following Method.

First, The Quadrant and it’s Telescope remaining in the same Situation wherein it was when the Altitude of a Star, together with the Time of it’s Passage by the Intersection of the cross Hairs in the Focus of the Object-Glass, was taken, we observe the Time when the Sun, or some fixed Star, whole Latitude and Longitude is known, arrives to the Vertical Hair in the Telescope; and from thence the Position of the said Vertical Circle will be had, and also the observed Star’s true Place.

But if the Sun, or some other Star, does not pass by the Mouth of the Tube of the Telescope, and if a Meridian Line be otherwise well drawn upon a Floor, or very level Ground, in the Place of Observation, you must suspend a Plumb-Line to some fixed Place, about three or four Toises distant from the Quadrant, under which upon the Floor must a Mark be made in a right Line with the Plumb-Line. This being done, you must put a thin Piece of Brass, or Pasteboard, very near the Object-Glass, in the Middle of which there is a small Slit vertically placed, and passing thro’ the Center of the Circular Figure of the Object-Glass. Now by means of this Slit, the beforementioned Plumb-Line may be perceived thro’ the Telescope, which before could not be seen, because of its Nearness thereto. Then the Plumb-Line must be removed and suspended, so that it be perceived in a right Line with a vertical Hair in the Focus of the Object-Glass, and a Point marked on the Floor directly under it. And if a right Line be drawn thro this Point, and that marked under the Plumb-Line before it was removed, the said Line will meet the Meridian drawn upon the Floor; and so we shall have the Position of the vertical Circle the observed Scar is in with respect to the Meridian, the Angle whereof may be measured in assuming known Lengths upon the two Lines from the Point of Concourse; for if thro’ the Extremities of these known Lengths, a Line or Base be drawn, we shall have a Triangle, whole three Sides being known, the Angle at the Vertex may be found, which will be the Angle made by the Vertical Circle and Meridian.

The Manner of taking the Meridian Altitudes of Stars.

It is very difficult to place the Plane of the Quadrant in the Meridian exactly enough to take Meridian Altitude of a Star; for unless there be a convenient Place and a Wall, where the Quadrant may be firmly fastened in the Plane of the Meridian, which is very difficult to do, we shall not have the true Position of the Meridian, proper to observe all the Stars, as we have mentioned already. Therefore it will be much easier, and principally in Journeys, to use a portable Quadrant, by means of which the Altitude of a Star must be observed a little before its Passage over the Meridian, every Minute, if possible, until its greatest or least Altitude be had. Now, tho’ by this means we have not the true Position of the Meridian, yet we have the apparent Meridian Altitude of the Star.

Altho’ this Method is very good, and free from any sensible Error, yet if a Star passes by the Meridian near the Zenith, we cannot have its Meridian Altitude, by repeated Observations every Minute, unless by chance; because in every Minute of an Hour the Altitude augments about fifteen Minutes of a Degree: and in these kind of Observations, the inconvenient Situation of the Observator, the Variation of the Star’s Azimuth several Degrees in a little time, the Alteration that the Instrument must have, and the Difficulty in well replacing it vertically again, hinders our making of Observations oftner than in every fourth Minute of an Hour; during which Time the Difference in the Star’s Altitude will be One Degree. Therefore in these Cases it will be better to have the true Position of the Meridian, or the exact Time a Star passes by the Meridian, in order to place the Instrument in the said Meridian, or move it so that one may observe the Altitude of the Star the Moment it passes by the Meridian.

Of Refractions.

The Meridian Altitudes of two fixed Stars, which are equal, or a small matter different, the one being North, and the other South, being observed, and also their Declination otherwise given; to find the Refraction answering to the Degrees of Altitude of the said Stars, and the true Height of the Pole, or Equator, above the Place of Observation.

Having found the apparent Meridian Altitude of some Star near the Pole (by the aforegoing Directions) if the Complement of the said Star’s Declination be added thereto, or taken therefrom, we shall have the apparent Height of the Pole. After the same manner may also the apparent Height of the Equator be found, by means of the Meridian Altitude of some Star near the Equator, in adding or substracting its Declination.

Then these Heights of the Pole and Equator being added together, their Sum will always be greater than a Quadrant; but 90 Degrees being taken from this Sum, the Remainder will be double the Refraction of either of the Stars observed at the same height: and therefore taking the said Refraction from the said apparent Height of the Pole, or Equator, we shall have their true Altitude.


Let the Meridian Altitude of a Star observed below the North Pole, be 30 Deg. 15 Min and the Complement of its Declination 5 deg. whence the apparent Height of the Pole will be 35 Deg. 15 Min. Also let the apparent Meridian Altitude of some other Star, observed near the Equator, be 30 Deg. 40 Min. and its Declination 40 Deg. 9 Min. whence the apparent Height of the Equator will be 54 Deg. 49 Min. Therefore the Sum of the Heights of the Pole and Equator thus found, will be 90 Deg. 4 Min. from which substracting. 90 Deg. and there remains 4 Min. which is double the Refraction at 30 Deg. 28 Min. of Altitude, which is about the middle between the Heights found: therefore at the Altitude of 30 Deg. 15 Min. the Refraction will be something above 2 Min. viz. 2 Min. 1 Sec. and at the Altitude of 30 Deg. 40 Min. the Refraction will be 1 Min. 59 Sec.

Lastly, If 2 Min. 1 Sec. be taken from the apparent Height of the Pole 25 Min. 15 Sec the Remainder 35 Deg. 12 Min. 59 Sec. will be the true Height of the Pole; and so the true Height of the Equator will be 54 Deg. 47 Min. 1 Sec. as being the Complement of the Height of the Pole to 90 deg.

Note, The Refraction and Height of the Pole found according to this way, will be so much the more exact, as the Altitude of the Stars is greater; for if the Difference of the Altitudes of each Star should be even 2 Deg. when their Altitudes are above 30 Deg. we may by this Method have the Refraction, and the true Height of the Pole, because in this Case the Difference of Refraction in Altitudes differing two Degrees, is not sensible.

Another Way of observing Refractions.

The Quantity of Refraction may also be found by the Observations of one Star only, whose Meridian Altitude is 90 Deg. or a little less; for the Height of the Pole or Equator above the Place of Observation being otherwise known, we shall have the Star’s true Declination, by it’s Meridian Altitude; because Refractions near the Zenith are insensible.

Now if we observe by a Pendulum the exact Times when the said star comes to every Degree of Altitude, as also the Time of it’s Passage by the Meridian, which may be known by the equal Altitudes of the Star being East and West, we have three things given in a spherical Triangle, viz. the Distance between the Pole and Zenith, the Complement of the Star’s Declination, and the Angle comprehended by the aforesaid Arcs; namely, the Difference of mean Time between the Passage of the Star by the Meridian and it’s Place, converted into Degrees and Minutes; to which must, be added, the convenable proportional Part of the mean Motion of the Sun in the Proportion of 59 Min. 8 Sec. per Day: therefore the true Arc of the Vertical Circle between the Zenith and the true Place of the Star may be found.

But the apparent Arc of the Altitude of the Star is had by Observation, and the Difference of these Arcs will be the Quantity of Refraction at the Height of the Star. By a like Calculation the Refraction of every Degree of Altitude may be found.

The same may be done by means of the Sun, or any other Star, provided it’s Declination be known, to the End that at the Time of Observation the true Distance of the Sun or Star from the Zenith may be found.

The Refractions of Stars being known, it will then be easy to find the Height of the Pole; for having observed the Meridian Altitude of the Polar Star, as well above as below the Pole, the same Day, and having diminished each Altitude by it’s proper Refraction, half of the Difference of the corrected Altitudes, added to the lesser Altitude corrected, or substracted from the greater Altitude thus corrected, will give the true Height of the Pole.

M. de la Hire has observed with great Care for several Years the Meridian Altitudes of fixed Stars, and principally of Sirius, and Lucida Lyra, with Astronomical Quadrants very well divided, and very good Telescopes at different Hours of the Day and Night, and at different Seasons of the Year; and he assures us, that he never found any Difference in their Altitudes, but what proceeded from their proper Motion.

And because Sirius comes to about the 26th Degree of the Meridian, we might doubt whether in the lesser Altitudes the Refractions in the Winter would be greater than those in the Summer; hence he also observed, with the late M. Picard, the lesser Meridian Altitudes of the Star Capella, which is about 4\(\frac{1}{2}\) Degrees at several different Times of the Year.

Having compared these different Observations together, and made the necessary Reductions, because of the proper Motion of that Star, there was scarcely found one Minute of Difference, that could proceed from any other Cause but Refraction. Therefore he made but one Table of the Refraction of the Sun, Moon, and the Stars, for all Times of the Year, conformable to the Observations that he made from them.

Notwithstanding this, one would think that Refractions nigh the Horizon are subject to divers Inconstancies, according to the Constitution of the Air, and the Nature of high or low Grounds, as M. de la Hire has often found; for observing the Meridian Altitudes of Stars at the Foot of a Mountain, which seemed to be even with the Top of it, they appeared to him a little higher, than if he had observed them at the Top: But if the Observations of others may be depended upon, Refractions are greater, even in Summer, in the frozen Zones, than in the temperate Zones.

How to find the Time of the Equinox and Solstice by Observation.

Having found the Height of the Equator, the Refraction and the Sun’s Parallax at the same Altitude, it will not afterwards be difficult to find the Time in which the Center of the Sun is in the Equator; for if from the apparent Meridian Altitude of the Center of the Sun, the same Day as it comes to the Equinox, be taken the convenient Refraction, and then the Parallax be added thereto, the true Meridian Altitude of the Sun’s Center will be had. Now the Difference of this Altitude, and the Height of the Equinoctial, will shew the Time of the true Equinox before or after Noon: and if the Sum of the Seconds of that Difference be divided by 59, the Quotient will shew the Hours and Fractions which must be added or substracted from the true Hour of Noon, to have the Time of the true Equinox.

The Hours of the Quotient must be added to the Time of Noon, if the Meridian Altitude of the Sun be lesser than the Height of the Equator about the Time of the vernal Equinox; but they must be substracted, if it be found greater. You must proceed contrariwise, when the Sun is near the autumnal Equinox.

Example. The true Height 41 Deg. 10 Min. of the Equator being given, and having observed the true Meridian Altitude 41 Deg. 5 Min. 15 Sec. of the Sun, found by the apparent Altitude of it’s upper or lower Limb, corrected by it’s Semidiameter, Refraction, and Parallax, and the Difference will be 4 Min. 45 Sec. or 285 Seconds, which being divided by 59, the Quotient will be 4\(\frac{49}{59}\), that is, 4 Hours 48 Minutes, which must be added to Noon, if the Sun be in the vernal Equinox, and consequently the Time of the Equinox will happen 4 Hours 48 Minutes after Noon. But if the Sun was in the autumnal Equinox, the Time of the said Equinox would happen 4 Hours 48 Minutes before Noon, that is, at 12 Minutes past Seven in the Morning.

As to the Solstices, there is much more Difficulty in determining them than the Equinoxes, for one Observation only is not sufficient; because about this Time the Difference between the Meridian Altitudes in one Day, and the next succeeding Day, is almost insensible.

Now the exact Meridian Altitude of the Sun must be taken, 12 or 15 Days before the Solstice, and as many after, that so one may find the same Meridian Altitude by little and little; to the End that by the proportional Parts of the alteration of the Sun’s Meridian Altitude, we may more exactly find the Time wherein the Sun is found at the same Altitude, before and after the Solstice, being in the same Parallel to the Equator.

Now having found the Time elapsed between both the Situations of the Sun, you must; take half of it, and seek in the Tables the true Place of the Sun at these three Times. This being done, the Difference of the extreme Places of the Sun must be added to the mean Place, in order to have the mean Place with Comparison to the Extremes; but if the mean Place found by Calculation, does not agree with the mean Place found by Comparison, you must take the Difference, and add to the mean Time, the Time answering to that Difference, if the mean Time found by Calculation be lesser; but contrariwise, it must be substracted if it be greater, in order to have the Time of the Solstice.

Example. The last Day of May, the apparent Meridian Altitude of the Sun was found at the Royal Observatory, 64 Deg. 47 Min. 25 Sec. and the 22d Day of June following, the apparent Meridian Altitude was found 64 Deg. 28 Min. 15 Sec. from whence we know, by having the Difference of Declination at those Times, that the Sun came to the Parallel of the first Observation, the 22d of June, at 4 Hours 12 Minutes in the Morning; and consequently the mean Time between the Observations, was on the 22d of June, at 2 Hours 6 Minutes in the Morning.

Now by Tables, the true Place of the Sun at the Time of the first Observation, was 2 Signs 18 Deg. 5S Min. 23 Sec. and at the Time of the last it was 3 Signs, 11 Deg. 4 Min. 52 Sec. and in the middle Time 3 Signs, 1 Min. 56 Sec. But the Difference of the two extreme Places is 22 Deg. 6 Min. 29 Sec. half of which is 11 Deg. 3 Min 15 Sec. which added to the mean Place, makes 3 Signs, 1 Min. 38 Sec. which is the mean Place with comparison to the Extremes. Again, The Difference between the mean Place, by Calculation 3 Signs, 1 Min. 56 Sec. and the mean Place by Comparison, is 18 Seconds, which answers to 7 Min. 18 Sec. of Time, which must be taken from the mean Time, because the mean Place by Calculation is greater than the mean Place by Comparison. Therefore the Time of the Solstice was the nth of June, at 1 Hour, 58 Min. 18 Sec. in the Morning.

Note, The Error of a few Seconds, in the observed Altitude of the Sun, will cause an alteration of an Hour in the true Time of the Solstice; as in the proposed Example, 10 Seconds, or thereabouts, in Altitude, will cause an Error of an Hour; whence the true Time of the Solstice cannot be had but with Instruments well divided, and several very exact Observations.

Observations made in the Royal Observatory at Paris, about the Time of the Solstice for finding the Height of the Pole, and the Sun’s greatest Declination or Obliquity of the Ecliptick.

Deg. Min. Sec.
The apparent Meridian Altitude of the upper Limb of the Sun at the Time of the Summer Solstice, gathered from several Observations, is found 645524
Refraction to be substracted 000033
Parallax to be added 000001
True Altitude of the upper Limb of the Sun 645452
Semidiameter of the Sun 001549
True Meridian Altitude of the Sun’s Center 643903
Deg. Min. Sec.
At the Time of the Winter Solstice, the apparent Meridian Altitude of the upper Limb of the Sun 180024
Refraction to be substracted 000312
Parallax to be added 000005
True Altitude of the upper Limb of the Sun 175717
Semidiameter of the Sun 001621
True Meridian Altitude of the Sun’s Center 174056
Deg. Min. Sec.
Then the true Distance of the Tropicks is 46587
The half, of which the greatest Declination of the Sun, is 23293\(\frac{1}{2}\)
The Height of the Equator above the Observatory 410959\(\frac{1}{2}\)
Its Complement, which is the Height of the Pole 485000\(\frac{1}{2}\)

Observations of the Polar Star.

By divers Observations of the greatest and least apparent Meridian Altitudes of the Polar Star, which is in the end of the Tail of the Little Bear, it is concluded that the apparent Altitude of the Pole, as M. Picard his denoted it in his Book of the Dimensions of the Earth, between St James’s and St Martin’s Gates (about S. Jaques de la Boucherie, at Paris) is 48 Deg. 52 Min. 20 Sec.

Deg. Min. Sec.
The Reduction being made according to the Distance of the Places, the apparent Height of the Pole at the Royal Observatory will be 465102
The Convenable Refraction to that Height 000104
Then the true Height of the Pole at the Observatory 484958
For which let us take 485000
And consequently the Height of the Equator will be 411000

The true apparent Time in which a Planet or fixed Star passes by the Meridian, being given, to find the Difference of Right Ascension between the fixed Star, or Planet, and the Sun.

The given Time from Noon to or from the time of the Passage of the Star or Planet by the Meridian, must be converted into Degrees, and what is required will be answered.

Example. Jupiter passed by the Meridian at 10 Hours, 23 Min. 15 Sec. in the Morning, whose Distance in time from Noon, which is 1 Hour, 36 Min. 45 Sec. being converted into Degrees of the Equator, will give 24 Deg. 11 Min. 15 Sec. for the Difference of Right Ascension between the Sun and Jupiter, in that Moment the Center of Jupiter passed by the Meridian.

In this, and the following Problem, we have proposed the true or apparent Time, and not the mean Time; because the true Time is easier to know by Observations of the Sun, than the mean Time. We shall explain what is meant by mean Time, as likewise true or apparent Time, in the next Chapter.

The true Time between the Passages of two fixed Stars by the Meridian being given, or else of a fixed Star and a Planet, to find their Ascensional Difference.

The given Time between their Passages by the Meridian must be converted into Degrees of the Equator, and the Right Ascension of the true Motion of the Sun answering to that time, must be added thereto; then the Sum will be the Ascensional Difference sought.

Example. Suppose between the Passages of the Great Dog, called Sirius, by the Meridian, and the Heart of the Lion named Regulus, there is elapsed 3 Hours, 20 Min. of time, and the Right Ascension of the true Motion of the Sun, let be 7 Min. 35 Sec.

Whence converting 3 Hours, 20 Min. into Degrees of the Equator, and there will be had 50 Deg. to which adding 7 Min. 35 Sec. and the Sum 50 Deg. 7 Min. 35 Sec. will be the Ascensional Difference between Sirius and Regulus.

You must proceed thus for the Ascensional Difference of a fixed Star and a Planet, or of two Planets; yet note, if the proper Motion of the Planet or Planets be considerable between both their Passages by the Meridian, regard must be had thereto.

How to observe Eclipses.

Amongst the Observations of Eclipses, we have the Beginning, the End, and the Total Emersion, which may exactly enough be estimated by the naked Eye, without Telescopes, except the Beginning and the End of Eclipses of the Moon, where an Error of one or two Minutes may be made, because it is difficult certainly to determine the Extremity of the Shadow. But the Quantity of the Eclipse, that is, the eclipsed Portion of the Sun and Moon’s Disk, which is measured by Digits, or the 12th parts of the Sun and Moon’s Diameter, and Minutes, or the 60th parts of Digits, cannot be well known without a Telescope joined to some Instrument. For an Estimation made with the naked Eye is very subject to Error, as it is easy to see in the History of ancient Eclipses, altho they were observed by very able Astronomers.

The Astronomers who first used Telescopes furnished with but two Glasses, namely, a Convex Object-Glass, and a Concave Eye-Glass, in the Observations of Eclipses, observed those of the Sun in the following manner. They caused a hole to be made in the Window-shutter of a Room, which Room in the Day-time, when the Shutters were shut, was darkened thereby; thro’ which Hole they put the Tube of a Telescope, in such manner, that the Rays of the Sun, passing thorough the Tube, might be received upon a white piece of Paper, or a Table-Cloth, upon which was first described a Circle of a convenable bigness, with five other Concentric Circles, equally distant from one another, which, with the Center, divided a Diameter of the outward Circle into 12 equal Parts. Then having adjusted the Table-Cloth perpendicular to the Situation of the Tube of the Telescope, the luminous Image of the Sun was cast upon the Table-Cloth, which would full be greater according as the Table-Cloth was more distant from the Eye-Glass of the Telescope; whence by moving the Tube forwards and backwards, they found a Place where the Image of the Sun appeared exactly equal to the outward Circle, and at that Distance they fixed the Table-Cloth, with the Tube of the Telescope, which composed the Instrument for the said Observation. Afterwards they moved the Tube according to the Sun’s Motion, to the End that the luminous Limb of it’s Disk might every where touch the outward Circle described upon the Table-Cloth, by which means the Quantity of the eclipsed Portion was seen, and it’s created: Obscurity measured by the Concentric Circles; they denoted the Hour of every Phase, by a Second Pendulum Clock, rectified and prepared for that purpose. The same Method is still observed by many Astronomers, who use also a Circular Reticulum, made with six Concentric Circles upon very fine Paper, which must be oiled, to render the Sun’s Image more sensible. The greatest of the Circles ought exactly to contain the Image of the Sun in the Focus of the Object-Glass of a Telescope of 40 or 60 Feet; the six Circles are equally distant, and divide the Diameter of the Sun in twelve equal digits. When the Paper is placed in the Focus of a great Telescope, the enlightened part of the Sun will very distinctly be seen; then the Eye-Glass is not used.

There are others who use a Telescope furnished with two Convex-Glasses, from whence the same Effect; follows. But altho’ the Use of a Telescope in this manner be very proper to observe Eclipses of the Sun, yet it is not fit to observe Eclipses of the Moon, because it’s Light is not strong enough. Lastly, Others place a Micrometer in the common Focus of the Convex Lenses. Besides the Quantity of the Phases of the Eclipses of the Sun and Moon (easily known by the said Micrometer), we may have the Diameters of the Luminaries, and the Proportion of the Earth’s Diameter to the Moon’s, as well by the obscure Portion of it’s Disk, as by the luminous Portion and the Distance between it’s Horns.

The Method of observing Eclipses by means of the Micrometer will be much better, if the Divisions to which the parallel Hairs are applied be made so, that six Intervals of the Hairs, may contain the Diameter of the Sun or Moon. For the moveable Hair posited in the Middle of the Distance between the immoveable ones (which is not difficult to do), will shew the Digits of the Eclipse.

The same Telescope and Micrometer may serve for all the other Observations, and to measure Eclipses; as, to observe the Passage of the Earth’s Shadow over the Spots of the Moon, in Lunar Eclipses.

There yet remains one considerable Difficulty, and that is, to make a new Division of the Micrometer serving as a common Reticulum for all Observations; for it scarcely happens in an Age in two Eclipses, that the apparent Diameters of the Sun and Moon are the same.

Therefore M. de la Hire has invented a new Reticulum, which having all the Uses of the Micrometer, may serve to observe all Eclipses, it being adapted to all apparent Diameters of the Sun and Moon, and it's Divisions are firm and solid enough to resist all the Vicissitudes of the Air, altho’ they are as fine as Hairs.

The Construction and Use of this Reticulum is thus: First, Take two Object Lenses of Telescopes of the same Focus, or nighly the same, which join together. As for Example, The Focus of two Lenses together of eight Feet, which is the fit Length of a Telescope for observing Eclipses, unless the Beginning and the End of Solar ones, which require a longer Telescope exactly to determine them.

Secondly, We find from Tables, that the greatest Diameter of the Moon at the Altitude of 90 Deg. is 34 Min. 6 Sec. To which adding 10 Sec. and there will arise 34 Min. 16 Sec. Therefore say, As Radius is to the Tangent of 17 Min. 8 Sec. (the half of 34 Min. 16 Sec.) So is 8 Feet, or the focal Length of the two Lenses to the Parts of a Foot, which doubled will subtend an Angle of 34 Min. 16 Sec. in the Focus of the Telescope, and this will be the Diameter of the said Circular Reticulum.

Thirdly, Upon a very flat, clear, and well polished Piece of Glass, describe lightly with the Point of a Diamond, fastened to one of the Legs of a Pair of Compasses, six Concentric Circles, equally distant from each other. The Semidiameter of the greatest and last let be equal to the fourth Term before found. Likewise draw two Diameters to the greatest Circle at Right Angles. The flat Piece of Glass being thus prepared and put into the Tube, of which we have before spoken, and in the Focus of the Telescope, will be a very proper Reticulum for observing Solar and Lunar Eclipses, and it will divide all the apparent Diameters into twelve equal Parts or Digits, as we are now going to explain.

It is manifest from Dioptricks, that all Rays coming from Points of a distant Object; after their Refraction by two convex Lenses, either joined or something distant from each other, will be painted in the common Focus of the said Lenses, which mil appear so much the greater, according as the Lenses be distant from one another; 10 that they will appear the smallest when the Lenses are joined together. Therefore if the Object-Glasses used in this Construction, be each put into a Tube, and one of these Tubes Aides within the other; then the said Lenses being thus joined, the Image of a distant Object, whose Rays fall upon the Lenses under an Angle of 34 Min. 16 Sec. will exceed the Moon’s greatest apparent Diameter by 10 Sec. Therefore in moving the Lenses by little and little, such a Position may be found, wherein the Diameter of the greatest Circle on the Reticulum posited in the Focus, will answer to an Angle of 34 Min. 16 Sec. For the Image of an Object perceived under a less Angle, may be equal to the Image of the same Object perceived under a greater Angle, according to the different Lengths of the Foci. But the Reticulum is in a separate Tube, and so it may be removed at a Distance at pleasure from the Object-Glasses. We now proceed to lay down two different Ways of finding the Positions of the Lenses and Reticulum, proper to receive the different Diameters of the Sun and Moon.

First, In a very level and proper Place for making Observations with Glasses, place a Board, with a Sheet of Paper thereon, directly exposed to the Tube’s Length, having two black Lines drawn upon it parallel to each other, and at such a Distance from each other, that it subtends an Angle of 34 Min. 6 Sec. so that the Distance of the said two Lines, represented in the Focus of the Object-Glasses, may likewise subtend an Angle of 34 Min. 6 Sec. And this may be found in reasoning thus, (as we have already done for the Micrometer). As Radius is to the Tangent of 17 Min. 3 Sec. So is the Distance from the Tube of the Object-Glasses to the Broad, To half the Distance that the parallel Lines on the Paper must be at. And thus we shall find by Experience the Place of each Object-Glass, and the Reticulum in the common Focus, in such manner that the Representation of the two black Lines on the Paper, embraces entirely the Diameter of the greatest Circle of the said Reticulum. Now we set down 34 Min. 6 Sec. upon the Tubes, in each Position of the Lenses and their Foci, or the Reticulum, that so the Lenses and Reticulum may be adjusted to their exact Distance, every time an Angle of 34 Deg. 6 Min. is made use of.

Again, Let the said Board and white Paper be placed further from the Tube, in such manner, that the Distance between the parallel Lines on the Paper subtend, or is the Base of an Angle of 33 Min. for example, whose Vertex is at the Lenses of the Telescope: which may be done, in saying, As the Tangent of 16 Min. 30 Sec. is to Radius; So is half the Interval of the parallel Lines on the Paper, To the Distance of the Board from the Lenses. Now in this Position of the Telescope and Board, the Position of the Lenses and Reticulum between themselves must be found; so that the Representation of the parallel Lines, which appear very distinctly in the Focus of the Lenses, occupies the whole Diameter of the greatest Circle on the Reticulum. This being done, the Number 33 Min. must be made upon the Tubes, in the Places wherein each of the Lenses and Reticulum ought to be. Proceed in this manner for the Angles of 32 Min. 31 Min. 30 Min. and 29 Min.

If the Distances, denoted upon the Tubes between the different Positions of the Lenses and the Reticulum, answering to a Minute, be divided into 60 equal Parts, we shall have their Positions for every Second; and by this means the same Circle of the Reticulum may be accommodated to all the different apparent Diameters of the Sun and Moon, and the Diameter of the greatest Circle being divided into 12 equal Parts, it will serve to measure the Quantities of all solar and lunar Eclipses.

The second Method taken from Opticks, being not founded upon so great a Number of Experiments as the former, may perhaps appear easier to some Persons; for the Foci of both the Lenses being known, say, As the Sum of the focal Lengths of the Lenses (whether they be equal or not) less the Distance between the Lenses, is to the focal Length of the outward Lense, less the Distance between the Lenses; So is this same Term, to a fourth, which being taken from the focal Length of the outward Lens, there remains the Distance from the outward Lens, to the common Focus of the Lenses, which is the Place of the Reticulum.

The Position of the common Focus of the Lenses may also be known by this Method; when they be joined, in using the aforesaid Analogy, without having any regard to the Distance between the Lenses. which is computed from the Places of the Lenses Centers; therefore in supposing several different Distances between the Object-Lenses, the Length of their Foci will he had, that is, the Place of the Reticulum, correspondent to each Distance.

Again, say, As the known focal Length is to the Semidiameter of the Reticulum, be it what it will; So is Radius, to the Tangent of the Angle answering to the Semidiameter of the Reticulum. By this Method we may likewise have the Magnitude of the said Reticulum, in saying, As Radius is to the Tangent of an Angle of 17 Min. 3 Sec. So is the focal Length of the Lenses, to the Semidiameter of the outward concentrick Circle. Having thus found the Minutes and Seconds subtended by the Diameter of the greatest Circle of the Reticulum, according to the different Intervals of the Lenses, they must be wrote upon each Tube of the Lenses and Reticulum, and the Distances between the Terms found, divided into Seconds, as is mentioned in the former Method. And thus may the Positions of the Lenses and Reticulum be soon found, which shall contain the apparent Diameters of the Sun or Moon, according as they appear. If it be found very difficult to draw exactly the concentrick Circles upon the Piece of Glass, you need but draw thirteen right Lines thereon with the Point of a Diamond, equally distant and parallel to each other, with another right Line perpendicular to them; but the Length of this Perpendicular between the two extreme Parallels, must be equal to the Diameter of the Reticulum, found in the manner aforesaid. This Reticulum may be used instead of one composed of Hairs.

A plain thin Piece of Glass, having Lines drawn thereon with a very fine Point of a Diamond, may likewise be used in an Astronomical Telescope, &c. for if it be adjusted in its proper Frame, in the manner as is directed in the Micrometer, the Lines drawn thereon may be used instead of the parallel Hairs. I am of opinion, that the aforesaid Reticula are very useful in practical Astronomy, they not being subject to the Inconstancies of the Air, of being gnawed by Insects, Or to the Motions of the Instrument, which the Hairs are.

There are those who prefer Hairs, to Lines drawn upon a Piece of Glass, whose Surface may cause some Obscurity to the Objects, or if it be not very flat, there may some Error arise: but if they have a mind to avoid these Difficulties, which are of no consequence, as we know by Experience, they may use straight Glass-Threads, instead of Hairs: for some of these may be procured as fine as Hairs, and of Strength enough to resist the Inconstancies of the Air.

Although the Phases or Appearances of the Eclipses of the Moon, apply’d by Astronomers to Astronomical and Geographical Uses, may be observed much easier and exacter by our Reticulum, than by the ancient Methods; yet it must be acknowledged, that the Immersions into, and Emersions of the Moon’s Spots out of the Earth’s Shadow, may more conveniently be observed, because of their great Number, than the Phases, and that there is less Preparation is using a Telescope, which need be only six Feet in length: and in order for this, a Map of the Moon’s Disk, when it is at the full, must be procured, wherein are denoted the proper Names of the Spots, and principal Places appearing on its Disk. This may be found in the reformed Astronomy of R. P. Riccioli, &c.

There are great Advantages arising from Observations of Eclipses, for if the exact Time of the Beginning of an Eclipse of the Moon, of it’s total Immersion in the Shadow, of it’s Emersion and it’s End, as likewise of the Passage of the Earth’s Shadow by the Spots on it’s Surface, be observed, we shall have the Difference of Longitude of the two Places wherein the Observations are made; this is known to all Astronomers. But since Lunar Eclipses seldom happen, so as that the Difference of Longitude may thereby be concluded, the Eclipses of Jupiter’s Satellites may be observed instead of them; but principally of the first, whose Motion about Jupiter being very swift, one may make several Observations thereof during the Space of one Year; and from thence the Difference of Longitude of the two Places, wherein the said Observations are made, may be had.

Nevertheless you must take notice, that Lunar Eclipses may much easier be observed, than the Eclipses of Jupiter’s Satellites, which cannot be easily and exactly done without a Telescope of twelve Feet in length; whereas the Phases of the Beginning or End, or of the Immersion and Emersion of Lunar Eclipses, may be observed without a Telescope, and the Immersions and Emersions of its Spots with one of an indifferent length.

M. Cassini, a very excellent Astronomer of the Academy of Sciences, published in the Year 1693, exact Tables of the Motions of Jupiter’s Satellites; therefore in comparing the Times of the Immersion or Emersion of Jupiter’s first Satellite, found by the Tables fitted for the Observatory (at Paris) with the Observations thereof made in any other Place, we shall have, by the Difference of Time, the Difference of Longitude of the Observatory, and the Place wherein the Observations were made: which may be confirmed in observing the same Phænomena in both Places.

It is proper here to inform Observators of one Case, which often hinders an exact Observation of Jupiter’s Satellites; which is, that in a serene Night, we often find the Light of Jupiter and its Satellites, observed thro’ the Telescope, to diminish by little and little, so that it is impossible to determine exactly the true Times of the Immersion and Emersion of the Satellites. Now the Cause of this Accident proceeds from the Object-Glass of the Telescope, which is covered over with Dew, and thereby a great Number of Rays of Light, coming from Jupiter and its Satellites, is hinder’d from coming thro’ the Object-Glass to the Eye. A very sure Remedy for this, is, to make a Tube of blotting Paper; that is, a Tube about two Feet long, and big enough to go about the End of the Tube of the Telescope next to the Object-Glass, must be made, in rolling two or three Sheets of sinking Paper upon each other. This Tube being adjusted about the Tube of the Telescope, will suck in or drink up the Dew, and hinder its coming to the Object-Glass; and by this Means we may make our Observations conveniently.

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