Mathematical Instruments
Book VI. Ch. II.

Of the Construction and Uses of the Micrometer.

The Micrometer is an Instrument of great Use in Astronomy, and principally in measuring the apparent Diameters of the Planets, and taking small Distances not exceeding a Degree, or Degree and a half. This Instrument is composed of two rectangular Brass Frames, one of which, viz. ABCD, is commonly 2$$\frac{1}{2}$$ Inches long, and 1$$\frac{1}{2}$$ broad, having the Sides AB and CD, divided into equal Parts, about four Lines distant from each other (for this is according to the Turns of the Screw, as shall be hereafter explained), but in such manner, that the Lines drawn thro’ each Division be perpendicular to the Sides AB and CD, and having human Hairs strained from Division to Division, fastened with Wax to the Places 2, 2, &c.

Now because Hairs are subject to divers Accidents by Heat, and otherwise, therefore M. de la Hire proposes a very thin and smooth piece of Glass to be used instead of them, adjusted in Grooves made in the Sides of the Frame, and having very fine parallel Lines drawn thereon, which produce the same Effect: as the parallel Hairs. All the Difficulty consists in chusing a very fine and well polished Piece of Glass, and drawing the Lines extremely nice; for the Defaults will grossly appear, when the said Lines are perceived in a Telescope.

Note, These Lines must be very lightly drawn upon the Glass with a small Diamond, whose Point is very fine.

This Instrument is joined to a Telescope, by means of the prominent Pieces L, L, which slide in a Kind of parallelogramick Tin-Box, at the two Sides of which are two Circular Openings, wherein are soldered two short Tubes; that on one Side being to receive the Tube carrying the Eye-Glass; and that on the other Side, the Tube carrying the Object-Glass, so that the Micrometer may be in the Focus of the said Object-Glass.

Use of the Micrometer

In order to use this Instrument, a lively Representation of Objects appearing thro’ the Telescope must be made in the Point whereat the parallel Hairs are placed; therefore if the Object-Glass be placed at it’s Focal Distance from the Micrometer, more or less, according to the Nature and Constitution of the Eyes of the Observator, the Objects and the parallel Hairs will appear distinctly in the said Focus.

If then the Focal Length of the Object-Glass be measured in Lines or 12th Parts of Inches, or, which is all one, the Distance from the Center of the Object-Glass to the parallel Hairs of the Micrometer, be measured, this Distance is to the Length of four Lines, which is the Interval between two fixed parallel Hairs nighest each other, as Radius is to the Tangent of the Angle, subtended by the two nearest parallel Hairs. This is evident from Dioptricks: for the Distance between the Object and the Observator’s Eye, is supposed to be so great, that the focal Length of the Object-Glass, compared therewith, is of no consequence; so that the Rays proceeding from the Points of the Object directly pass thro’ the Center of the Object-Glass in the same Manner, as tho’ the Observator’s Eye was placed in the said Object-Glass. This may be shewn by Experience thus:

Draw two black Lines parallel upon a very smooth and white Board, whose Interval let be such, that at the Distance of 200 or 300 Toises, they may be met with or embraced by two parallel Hairs of the Micrometer. This being done, remove the Table in a convenient Place (there being no Wind stirring), so far from the Telescope, until the Lines drawn thereon, which must be perpendicular to a Right Line drawn from the Table to the Micrometer, be catched by two fixed parallel Hairs of the Micrometer; and then the Distance between the Table and the Object-Glass will have the same Proportion to the Distance between the Lines on the Board, as Radius is to the Tangent of the Angle subtended by two Hairs of the Micrometer.

Now move the Frame EFGH, by means of the Screw, ’till it’s Hair exactly agrees with one of the parallel Hairs of the other Frame; and when this is done, observe the Situation of the Index of the Screw; then turn the Screw until the said Hair of the Frame EFGH agrees with the next nearest fixed Hair of the other Frame; or, which is the same thing, move the Frame EFGH the Length of four Lines, or one third of an Inch, which may be easily known by means of the Object-Glass, which magnifies Objects, and count the Revolutions and Parts of the Screw, compleated in moving the said Frame that Length. Finally, make a Table, shewing how many Revolutions, and Parts of a Revolution of the said Screw, are answerable to every Minute and Second, by having the Angle subtended by the two black Lines on the Board given, and taking the Revolutions proportional to the Angles; that is, if a certain Number of Revolutions give a certain Angle, half this Number will give half the Angle, &c. And this Proportion is exact enough in these small Angles.

Now the Manner of taking the apparent Diameters of the Planets, is thus: Having directed the Telescope, and it's Micrometer, towards a Planet, dispose the Hairs, by the Motion of the Telescope, in such a Manner, that one of the fixed parallel Hairs do just touch one Edge of the Planet, and turn the Screw ’till the moveable Hair just touches the opposite Edge of the said Planet. Then, by means of the Table, you will know how many Minutes or Seconds correspond to the Number of Revolutions or Parts, reckoning from the Point of the Plate over which the Index stood when the fixed Hair touched one Edge of the Planet, to the Point it stands over when the moveable Hair touches the opposite Edge; and consequently, the apparent Diameter of the said Planet will be had. And in this manner may small Angles on Earth be taken, which may be easier done than those of the Celestial Bodies, because of their Immobility.

This Method is convenient enough for measuring the apparent Diameters of the Planets, if the Body of any one of them moves between the parallel Hairs. Yet it ought to be observed, that the Sun and Moon’s Diameters appear very unequal upon the account of Refraction; for in small Elevations above the Horizon, by the Space of 30 Minutes, the vertical Diameters appear something lesser than they really are in the Horizon, and the horizontal Diameters cannot be found, unless with much Trouble, and several repeated Observations; as likewise the Distance between two Stars, or the Horns of the Moon, because of their Diurnal Motions, which appear thro’ the Telescope very swift.

If two Stars of different Altitudes pass by the Meridian at different Times, the Difference of their Altitudes will be the Difference of their Distances from the Equator towards either of the Poles, which is called their Difference of Declination; and by their Difference of Time in coming to the Meridian, the Difference of their Distance from a determinate Point of the Equator, that is, the first Degree of Aries will be had; and this is their Difference of Right Ascension.

If the two Stars are distant from each other, we have Time enough, in the Interval of their Passage by the Meridian and Micrometer, to finish the Operations regarding the first, before proceeding to those of the second; but if they be very near each other, it is extremely difficult to make both the Observations at the same Time, that so the two Stars may be precisely catched in the Meridian. But M. de la Hire shews how to remedy this Inconveniency, by only using the common Micrometer: for the Observation of the Passage of Stars between, or upon the Hairs of the Micrometer, will give, by easy Consequences, their Difference of Right Ascension and Declination, without even supposing a Meridian known or drawn.

But if the Difference of Declination and Right Ascension of two Stars that cannot be taken in between the Hairs of the Micrometer be required, this may be found in the following Manner.

We adjust a Cross-Hair to the Micrometer, cutting the parallel ones at Right Angles, which we fasten with Wax to the Middle of the Sides AC and BD. Then the Telescope, and it’s Micrometer, being fixed in a convenient Position, so that the Stars may successively pass by the parallel Hairs, as the Stars A and S, in Figure 10, we observe, by a second Pendulum Clock, the Time wherein the first Star A touches the Point in which the aforementioned Cross-Hair AS erodes some one of the parallel Hairs, as Ad. The Micrometer being disposed for this Observation, which is not difficult to do, reckon the Seconds of Time elapsed between the Observations made in the Point A, and the arrival of the said Star to the Point B, being the Concourse of another parallel Hair BD. We likewise observe the Time wherein the other Star S meets the Cross-Hair at the Point S, and then at the Point D of the parallel Hair BD. Note, It is the same thing if the Star S first meets the parallel Hair in D, and afterwards the Cross-Hair in S.

Now as the Number of Seconds the Star A is moving through the Space AB, is to the Number of Seconds the Star S is moving through the Space SD; so is the Distance AC, known in Minutes and Seconds of a Degree in the Micrometer, to the Distance CS, in Minutes and Seconds of a Degree. But the Horary Seconds of the Motion through the Space AB, must be converted into Minutes and Seconds of a great Circle, by the Rule of Proportion.

Having first converted the Seconds of the Time of the said Motion from A to B, which may be here esteemed as a Right Line, or an Arc of a great Circle, into Minutes and Seconds of a Circle, in allowing 15 Minutes of a Circle to every Minute of an Hour, and the same for Seconds: We say, by the Rule of Proportion, As Radius is to the Sine Complement of the Stars known Declination, So is the Number of Seconds in the Arc AB also known, to the Number of Seconds of the same Kind contained in the Arc CA, as an Arc of a great Circle.

Moreover, in the Right-angled Triangle CAB, the Sides CA, and AB being given, as likewise the Right Angle at C, we find the Angle CAB; and supposing CPR perpendicular to the Line AB, AB will be to CA as CA is to AP.

But in the Right-angled Triangle CAP, we have (besides the Right Angle) the Angle A, as likewise the Side CA given; therefore As Radius is to CA, So is the Sine of the Angle CAP, to CP. And as the Number of horary Seconds of the Motion from A to B, is to the horary Seconds in the Motion from S to D, so is CP to CR. Then taking CR from CP, or else adding them together, according as AB or SD is next to the Point C, and we shall have the Quantity of PR in parts of a great Circle, which will be the Difference of the two Stars Declinations. We have no regard here to the Difference of Motion through the Spaces AB and SD, caused by the Difference of Declination, because it is of no Consequence in the Difference of Declinations, as they are observed by the Micrometer.

Finally, As AB is to AP, So is the Number of horary Seconds of the observed Motion of the Star A through the Space AB, to the Number of Seconds of the Motion of the said Star through the Space AP. Wherefore the Time when the Star A comes to P, will be known. But as the Number of Horary Seconds of the Motion through the Space AB is to the Number of Horary Seconds of the Motion through the Space SD; So is the Number of Horary Seconds of the Motion through the Space AP, to the Number of Horary Seconds of the Motion through SR.

Moreover; The Time when the Star S is in S is known, to which if the Time of the Motion through SR be added, when A and S are on the same Side the Point C, or substracted if otherwise, and the Time when the Star is in R will be had. Now the Difference of Time between the Arrivals of the Stars in P and R, that is, the Difference of the Times wherein they come to the Meridian, will be the Difference of their Right Ascensions, which by the Rule of Proportion may be reduced into Degrees and Minutes. Note, We have no regard here to the proper Motion of the Stars.

From hence it is easy to know how, instead of the parallel Hair AB, to use another parallel one, passing thro’ A, or any other, as also a moveable Parallel, provided that they form Similar Triangles, as will be easily conceived by what hath been already said.

The aforesaid Operation may yet be done by another Method. For the parallel Hairs of the Micrometer being so disposed, that the first of the Stars may move upon one of them; and if the Time wherein the said Star crosses the Cross-Hair of the Micrometer be observed, and if moreover the Time wherein the other Star crosses the said Cross-Hair be observed, and at the same Time the moveable parallel Hair be adjusted to the second Star, no ways altering the Micrometer; we shall have, by means of the Distance between that parallel Hair, the first Star moved upon, and the moveable parallel Hair, the Distance between two parallel Circles, to the Equator, passing thro’ the Places of the said Stars, which is their Difference of Declination. And if moreover, the Difference of the Times between the Passages of each of the Stars by the Cross-Hair of the Micrometer be converted into Minutes and Seconds of a Degree, the said Stars ascensional Difference will be had. This needs no Example.

But if this be required between some Star, and the Sun or Moon; as for Example, Mercury moving under the Sun’s Disk; place the Micrometer so, that the Limb of the Sun may move along one of the parallel Hairs, and observe the Times when the Sun’s antecedent and consequent Limbs, and the Center of Mercury, touch the Cross-Hair; then the Difference of Mercury’s Declination, and the Sun’s Limb, by means of the moveable Hair, will be had, the Micrometer remaining fixed. And if to the Time of the Observation of the Sun’s antecedent Limb, half the Time elapsed between the Passages of the antecedent and consequent Limb be added, we shall have the Time of the Passage of the Sun’s Center by the Cross-Hair of the Micrometer; and by this means the Difference of the Times between the Passage of the Sun’s Center and Mercury over the Cross-Hair, that is, by the Meridian, will be obtained. And this Difference of Time being converted into Degrees and Minutes, will give their ascensional Difference.

Moreover, since the Sun’s Center is in the Ecliptick, if in the same Time as the said Center passes over the Cross-Hair (the Sun’s true Place being otherwise known), you seek in Tables, the Angle of the Ecliptick with the Meridian, you will likewise have the Angle that the Ecliptick makes with the Sun’s Parallel, as in Fig. 11. the Angle OCB, of the Ecliptick OCB, and of the Parallel to the Equator RC. Let PC be the Meridian, Mercury in M, the Center of the Sun in C, MR parallel to PC, and CR the Difference of Right Ascension between the Center of the Sun C, and Mercury in M. Now the Minutes of the Difference of the Right Ascension CR in the Parallel, being reduced to Minutes of a great Circle, say, As Radius is to the Sine Complement of the Sun’s or Mercury’s Declination; So is the Number of Seconds of the Difference of Right Ascension, to the Number of Seconds CR, as the Arc of a great: Circle. Then in the Triangle CRT, Right-angled at R, we have the Side CR (now found); as also the Angle RCT, viz. the Difference between the Right Angle, and the Angle made by the Ecliptick and Meridian; whence the Hypothermic CT, and the Side RT may be found. And if RT be taken from MR, which is the Difference of Declination of Mercury in M, and the Center of the Sun in C, there will remain TM. Again, As CT is to TR, So is TM to TO; MO will be the Latitude of Mercury at the Time of Observation: And adding TO to the Side CT, we shall have CO, the Difference of Longitude between Mercury and the Sun’s Center. Therefore the Sun’s Longitude being known, that of Mercury’s may also be found.

If moreover, two or three Hours after the first Observation of Mercury in M, the Difference of Declination and Right Ascension thereof be again observed, when he is come to N, we shall find, as before, NQ the Latitude of Mercury, and CQ the Difference of Longitude of him and the Sun’s Center C; whence the Place of the apparent Node of Mercury will be had. But note, The Point of Concourse A, in the Right Line MN, with the Ecliptick CB, is not the Place of the said Node, with regard to the Point C, because between the Observations made in the Points M and N, the Sun by it’s proper Motion is moved a few Minutes forwards, according to the Succession of Signs, which notwithstanding we have not regarded in the Observations. Therefore say, As the Difference of the Latitudes MO and NQ, to OQ, minus the proper Motion of the Sun, between the Observations made in M and N; So is MO to the Distance OA, whence the true Distance from the Sun’s Center C to Mercury’s Node A will be had. Note, The proper Motion of the Sun between the Observations must be taken from OQ, because during that Time Mercury is retrograde; but if it’s Motion had been direct, the Sun’s Motion must have been added to OQ.

In the Observations of Mercury’s Passage under the Sun’s Disk, we have had no regard to the proper Motion of the Sun, as being of small consequence; but if it is required to be brought into Consideration, CO and CQ must be diminished by so much of the Sun’s proper Motion, as is performed in the Interval of Time between the Passage of the Sun’s Center and Mercury, by the Meridian.

By the same Method, the Distances of Planets from each other, or from fixed Stars near the Ecliptick, may be observed; nevertheless, excepting some Minutes, not only upon the account of the proper Motions of the Stars, but also because of their Distance from the Ecliptick or too great Latitude. Note, This second Method for finding the Difference of Declination and Right Ascension is not exacter than the former, altho’ it is performed with less Calculation: for it is so difficult to dispose the Hairs of the Micrometer according to the Parallel of the Diurnal Motion, that it cannot be done, but by several uncertain trials.

M. de la Hire hath also invented another Micrometer, whose Construction is easy; for it is only a Pair of proportional Compasses, whose Legs on one Side, are, for Example, ten Times longer than those on the other Side. The shortest Legs of these Compasses must be put thro’ a Slit made in the Tube of the Telescope, and placed so in the Focus of the Object-Glass, that the two Points, which ought to be very fine, may be applied to all Objects represented in the said Focus. Then if the Angle subtended by the distance of two Objects in the Focus of the Object-Glass be required to be found by means of these Compasses, you must shut or open the two shortest Legs ’till their Points just touch the Representations of the Objects; and keeping the Compasses to this Opening, if the longest Legs be applied to the Divisions of a Scale, the Minutes and Seconds contained in the Angle subtended by the Distance of the aforesaid Objects will be had. The Manner of dividing the said Scale, is the same as that for finding the Distances of the parallel Hairs of the other Micrometer, in saying by the Rule of Proportion, As the Number of Lines contained in the Focal Length of the Object-Glass, is to one Line; So is Radius to the Tangent of the Angle subtended by one Line in the Focus: therefore if the longest Legs be ten Times longer than the others, ten Lines on the Scale will measure the said Angle subtended by one Line, which being known, it will be easy to divide the Scale for Minutes and Seconds.

This Micrometer may be used for taking the apparent Diameters of the Planets; as also to take the Distances of fixed Stars which are near each other, and measure small Distances on Earth.