Mathematical Instruments
Suppl. Ch. II.

# Of some more modern Instruments used at Sea, in taking the Degrees of the Altitude of the Sun or Stars, or the Degrees of their Distances.

Besides the Instruments for this Purpose already spoken of in our Treatise of Mathematical Instruments, there have been of late Years several others actually made, described, and used for this Purpose, indeed not essentially differing from one another, all of them being designed to be used where the Motion of the Objects, or Observer, or any thing occasioning an unsteadiness in the other Instruments when using, make the Observations difficult or subject to Error.

The first of these Instruments for taking the Moon’s Distance from the fixed Stars was invented long ago by Sir Isaac Newton, as appears in a Paper of Sir Isaac Newton’s own Hand-Writing, found amongst those of the late Dr Halley, and the very Instrument itself that Sir Isaac Newton either made himself, or caused to be made so long ago as when Dr Halley went about making the Catalogue of the fixed Stars in the South-Seas, which was in the Year 1672, was not long ago to be seen at Mr Heath’s the Mathematical Instrument-Maker in the Strand. See the Philosophical Transactions, Number 465, for the Year 1742.

Many Years after this, Instruments for doing the same thing, were published in the Philosophical Transactions, Number 420, and 425, under the Name of Mr Hadley’s Octants or Quadrants, not much differing from Sir Isaac’s. Two or three more of the like, not essentially varying from the Original one of Sir Isaac Newton, have also been made and published under the Name of Mr Caleb Smith, with Engravings of two of them described in a printed Sheet of Paper, entituled The Use of the new Instrument, or Sea Quadrant, for taking Altitudes of the Sun, Moon, and Stars, from the visible Horizon, by an Observation either forwards or backwards, as well as any other angular Distances, without Impediment or Interruption from the Ship’s Motion, whereby the Latitude at Sea may be obtained with greater Certainty and more frequently, than by any other Instrument commonly used for that purpose. Sold by Mr Heath, Mathematical Instrument-Maker in the Strand, and several other Mathematical Instrument-Makers, and by some Booksellers too.

But as Sir Isaac Newton has now been found to be the first Inventor of this Instrument, and as one of those not much different from his, called by the Name of Mr Hadley’s Sea-Octant or Quadrant, is now much in vogue and used at Sea, I shall therefore in this Chapter give only the Description and Use of Sir Isaac Newton’s Quadrant of this Kind, and that other called Mr Hadley’s, leaving Mr Smith’s to be seen in the Paper above described.

## I. The Description and Use of Sir Isaac Newton’s Sea Instrument or Quadrant.

The Figure PQ, RS denotes a Brass Plate accurately divided in the Limb DQ into half Degrees half Minutes, and $$\frac{1}{12}$$ Minutes by a Diagonal Scale; and the $$\frac{1}{2}$$ Degrees, and $$\frac{1}{2}$$ Minutes, and $$\frac{1}{12}$$ Minutes, counted for Degrees, Minutes, and $$\frac{1}{6}$$ Minutes. AB is a Telescope three or four Feet long, fixed on the Edge of that Brass Plate. G is a Speculum fixed on the said Brass Plate perpendicularly, so as to be inclined 45 Degrees to the Axis of the Telescope, and intercept half the Light which would otherwise come from the Telescope to the Eye. CD is a moveable Index turning about the Centre C, and with it's fiducial Edge shewing the Degrees, Minutes, and $$\frac{1}{6}$$ Minutes on the Limb of the Brass Plate PQ; the Centre C must be over against the middle of the Speculum G. H is another Speculum parallel to the former, when the fiducial Edge of the Index shewing the Degrees falls upon 00 Deg. 00 Min. 00 Seconds; so that the same Star may then appear through the Telescope in one and the same place, both by the direct Rays, and by the reflected ones; but if the Index be turned, the Star shall also appear in two places, whose Distance is shewed on the Brass Limb by the Index.

### The Use of this Instrument.

By this Instrument the Distance of the Moon from any fixed Star is thus observed: view the Star through the Telescope by the direct Light, and the Moon by the reflected Light, or the contrary, and turn the Index ’till the Star appears to touch the Edge of the Moon, and the Index will shew upon the Brass graduated Arch of the Instrument, the Distance of the Star from the Limb of the Moon; and though the Instrument shakes by the Motion of your Ship at Sea, yet the Moon and Star will move together as if they did really touch one another in the Heavens, so that an Observation may be made as exactly at Sea as at Land; and by the same Instrument may be observed exactly the Altitudes of the Moon and Stars by bringing them to the Horizon, and thereby the Latitude and Times of Observations may be, determined more exactly than by the Ways now in Use. Note, In the Time of the Observation, if the Instrument moves angularly about the Axis of the Telescope, the Star will move in a Tangent of the Moon’s Limb, or of the Horizon; but the Observation may notwithstanding be made exactly, by noting when the Line described by the Star is a Tangent to the Moon’s Limb, or to the Horizon; and to make the Instrument useful, the Telescope ought to take in a large Angle; and to make the Observation true, the Star must touch the Moon’s Limb, not on the outside of the Limb, but on the inside: thus far Sir Isaac Newton.

## The Description and Use of Mr Hadley’s Instrument for taking the Latitude, or other Altitudes at Sea.

The Instrument ABC (Fig. 6.) is an Octant having it’s Arch or Limb CB but 45 Degrees diagonally divided into 90 equal Parts, and these again subdivided into as many equal Parts as possible with distinction; each of the first mentioned equal Parts being Degrees, and the subdivisions Parts of a Degree, as serve to take any Altitude from the Horizon to the Zenith. The Octant is generally made of Mahogany Wood, and sometimes of Ebony, or Brass, the Radius of the Instrument being generally 20 Inches; the Thickness is half an Inch; the graduated Arch is usually of Box-Wood, and sometimes Brass about the Thickness of a Half-penny. About the Centre of the Instrument there is an Index D, freely but stiffly moving on a Pin, shewing the Degrees upon the Arch. Upon this Index, near the Centre, is fixed a plane Speculum E, or piece of Looking-Glass Quick-silvered on one side, having it’s middle E directly over the Centre of the Octant, which Speculum is perpendicular to the Plane of the Index, and whose Plane coincides with the fiducial Line ED drawn along the middle of the Index: though it may make some Angle with that Line in other Instruments of this kind. This Speculum is about 2 Inches by 2$$\frac{1}{8}$$. This Glass is to receive the first Image of the Sun, or other Object to be observed by, and to reflect it to another lesser plane Speculum, or Looking-Glass F, $$\frac{3}{4}$$ of an Inch square, fixed perpendicularly on a piece of Wood fastened to the Side AB, or Limb of the Octant, and having it’s Surface so situated, that when the Index is brought to mark the Beginning of the Divisions on the Limb (that is 0°), it may be exactly parallel to the Surface of the other Speculum. That part of this Speculum is Quick-silvered which is next to the Plane of the Instrument, and so set in it’s Brass Work, that it may at any time be see perpendicular to the Plane of the Instrument, if by any accident it should be removed from that posture. When it is in the Perpendicular, it can also be turned round, keeping still perpendicular, so as at pleasure to bring it to it’s true position with regard to the Glass fixed on the Index. This Glass serves for forward Observations with the Instrument, and from it the Eye receives the Image of the Object by a second Reflection. There is another such Speculum G placed on the side of the Octant, but farther from the Centre, and with the same Sort of Brass Work to set it perpendicular, and to it’s true Position.

There are two dark or smoaked Glasses H, set in Frames of Brass fixed to a Pin, the one lighter, and the other darker, so as to be turned at pleasure either of them, or both together, as the Sun’s brightness may require between the Speculum on the Index, and either of the two on the fide of the Instrument, according as the fore or back Observation is used. For which reason there are two Holes on the side of the Instrument, that they accordingly may be shifted from the one to the other. I being the Hole for them in the back Observation. Each of the Observations has a sight Vane K, of Box-Wood, about 2$$\frac{1}{2}$$ Inches from the Centre. In that for the fore Observation are made two Holes to direct the placing the Eye, the one being exactly as high above the Plane of the Instrument as is the middle of the unquicksilvered Part of the lesser Glass; the other the Height of the Edge of the Quick-silver itself, or a little lower. That for the back Observation requires but one Hole, which is placed exactly at the Height of the middle of the clear Part of that Glass. The Altitude of the Sun or Star above the Horizon, taken by this Instrument, is determined by the Inclination of the Planes of the two Glasses (F and E for the fore Observation, and F and G for the back Observation) to each other, when the Sun or Star appears in the Horizon. In the fore Observation, the double of this Angle of Observation is the Altitude sought, and is marked by the Index, on the Arch of the Instrument divided into half Degrees. But in the back Observation, twice the Difference of this Inclination from a right Angle gives the same Altitude, and is marked by the Index in the same manner, the same Scale of Degrees serving for both; so that when the Index stands at the beginning of the Scale, the Surface of the lesser Glass F used for the fore Observation, is parallel to the Surface of that one the Index, viz. E, but the Surface of the other Glass G perpendicular to it.

## The Use of this reflecting Octant as thus described, in taking the Altitude of the Sun, or a Star at Sea.

### I. To adjust the Glass for a fore Observation.

Set the Index D exactly to the Beginning of the Scale C, that is, close up to the Button on the Side of the Limb AC. Then holding the Instrument with your Left-hand, by the Index and Limb together, near the Arch BC, or over the Button, as upright as you can with the Arch downwards, and keeping the Index all the while touching the Button, look thro’ the lower Hole in the Sight-Vane, that is, that next to the Instrument, and see the Edge of the Sea (taken for the Horizon), through the Part of the Glass which does not reflect, and mark whether the Line of the Sea’s Edge, that is, the Horizon thus seen by direct vision coincides, or is the same, with the same Edge seen by the Reflection of the Quick-silvered part of that Glass. If not, turn that Glass by the Handle on the back side of the Instrument, ’till they do thus join or coincide. This being done,

### II. To take an Altitude by a fore Observation.

Set the Index to the Altitude as near as you can judge, if it be within 10 or 15 Degrees the Matter is not much. Hold the Instrument upright, as near as possible, with it’s Plane in a vertical Circle passing through the Zenith and Sun, or other Object, with the Arch downwards. And supposing the Object to be the Sun, placing your Eye at the upper Hole of the Sight-Vane K. Look at the Edge of the Sea (taken for the Horizon), just under the Sun by direct view, through the outer Part of the Speculum on the Limb not Quick-silvered over. Then by moving the Index, the Image of the Sun must be brought to appear as if it were really joined to the Edge of the Sea, that is, as if it were brought down to the Horizon, touching it where the vertical Circle passing through the Zenith and Centre of the Sun cuts the Horizon, and the Degrees and Minutes of the Arch marked by the Index will denote the Sun’s Altitude.

Note, If the Reflection be too bright for the Eye, turn up one or both the coloured Glasses H, according as you see occasion. But if the Image be too faint to appear on this Part of the Glass, or it be any other Object than the Sun, or a bright Moon, then look through the lower Hole of the Vane, and see it on the Quick-silvered part of the Speculum, and then you must judge when the Line of the Horizon, seen directly through the unquick-silvered part if produced, would pass through it.

Note also the Line of the Direction of the Sight, or the Line in which you see the Image when directly looking at it, must be kept as near as possible parallel to the Plane of the Instrument, in order to be very exact. Wherefore when you look through the upper Hole, take the Observation about the middle of the unquick-silvered Part of the Glass, and not too near the Edge of the Quick-silver, nor the outer Edge of the Glass; but, if you use the under Hole, bring the Image near the Edge of the Quick-silver, lean the Instrument a little awry to right and left by turns, by which the Image will seem to swing to and fro, and move the Index ’till you have brought the Image just to brush the Horizon in the lowest Part of it’s swing.

Note, For common use it will be well enough to observe by the Sun’s Centre. But this way is not so exact, as to do it by the Sun’s upper or under Edge; if the under Edge be used, in the fore Observation, you must add 16 Minutes to those pointed by the Index; if the upper, substract 16 Minutes.

### To adjust the Glasses for the back Observation.

Set the Index off so much before the beginning of the Scale of Degrees as is the double of the Dip of the Horizon, or Edge of the Sea (viz. 8 Minutes, for 44 Feet above the Surface of the Sea; 7′ for 34 Feet; 6′ for 25; 5′ for 17 Feet; 4′ for 11 Feet; 3′ for 6 Feet; and 2′ for 3 Feet). Look through the Hole of that Sight-Vane holding the Instrument upright, and see that the Edge of the Sea behind you, appearing by reflection either from the clear Part of that Glass, if it be discernable there, or else from the Quick-silver on both sides; join all along with that before you, seen through the same clear Part of the Glass. If those two Edges cross one another, the Instrument is not held upright, and the Adjustment will not be so certain.

Note, The Edge of the Sea seen by reflection will be inverted; the Water appearing above, and the Sky below.

### To take an Altitude by a back Observation.

Having set the Index to the Altitude as near as you can guess, look through the Hole in the Sight-Vane, and the middle of the clear Part of that Glass between the Quick-silvered Parts on both Edges, just over your own Shadow, and move the Index ’till you bring the Image of the Sun, &c. exactly to the Horizon; and the Degrees and Minutes marked on the graduated Arch will be the Altitude sought.

Note, To know whether you hold the Instrument upright, carry it to Right and Left by turns, keeping your Arms steady, and you will see the Image slide along the Edge of the Sea, if it be held upright; otherwise it will run in a Line cutting it. In this Observation, if you use the apparent under Edge of the Sun, substract 16 Minutes, otherwise add 16 Minutes, contrary to what is to be done in the fore Observation.

Note, The Dip of the Horizon set down above, in the back Observation, is to be added to the Degrees of Altitude marked upon the Arch, and not substracted as in the fore Observation.

### To observe the Altitude of a Star, by the fore Observation.

Look directly up at it first with the Index standing close to the Button, then move the Index forwards, and cause the Star to slide down to the Horizon, that you may not mistake it for another; and the Degrees marked by the Index will be that Star’s Altitude; and thus is the Altitude of a Star taken by the fore Observation. But when the Horizon is bright, and the Star faint, it may be bell to take the Altitude by a back Observation; in which case you are to look directly up at the Star, and bring the Horizon, or Edge of the Sea, behind you to touch the Star by reflection. As the Stars are used in the Night, Moon-light Nights are best for the Observation, either fore or back.

Note, The back Observation is not so easy as the fore one, and is useful in taking small Altitudes of the Sun when it’s Light may be troublesome to the Eye in the fore Observation. It is also useful in verifying the Truth of an Altitude of the same Object, taken at the same time, or nearly so, by a fore Observation with the same Instrument. That the back and fore Observation may agree accurately, in a lofty Ship especially, the Index must not stand exactly at the beginning of the Scale in the Adjustment for the fore Observation, but must be set off so much before the beginning of the Scale, as has been already said, as is the double of the Angle of the Dip or the visible Horizon, or Edge of the Sea, below the true Horizon, for which reason there is a few Minutes graduated before the beginning of the Arch.

The chief Advantage of this Instrument, says Mr Hadley, consists in this; that whereas by taking an Altitude by other Instruments now in use, a certain exact Posture of them is required. But the Motion of the Ship continually disturbs the Observer, by putting them out of that true Posture, even in moderate Weather, and in hollow Seas to that degree, that Observations taken by them cannot oftentimes be accurate enough to be depended upon. But with this Instrument, though the Ship rolls ever so much, provided the Instrument be kept in, or near, an upright Posture, though it be leaned forewards or backwards therein, yet the Image of any Object, when once brought by sliding the Index to appear on the Edge of the Sea, will there remain absolutely immovable, as long as the Index continues in the same Place, without being stirred, and the Observer has the same Advantage of making the Observation as if he took it in smooth Water, and the Instrument was held still without Motion. Thus far Mr Hadley.

Hence it is evident from what has been said, that whatever Improvement as to the Use and Exactness of these Instruments others may have found out, whether by the Matter, Figure, Weight, alteration of some Parts, He. Sir Isaac Newton most certainly must have the Honour of the Invention. All the Praise due to others, can only consist in the Degree of Facility of the Use, and the Exactness, whereby the Instrument, as contrived by them, exceeds that described by Sir Isaac Newton; and, indeed, whatever this be, Sir Isaac Newton at least must have been the original Cause for had he not shewn this his Invention to Dr Halley, it is probable that Nobody else might have thought of such a Contrivance, and consequently there would have been no such Instrument, and accordingly no Improvement upon it. Or supposing somebody else to have thought of such a Contrivance by himself, as well as Sir Isaac Newton; yet as Sir Isaac appears to have been the first Inventor, this other Inventor is allowed to have no share of the Honour of the Invention, and much less should those who only alter or improve some particular Part or Parts of the original Invention; perhaps (more to disguise and distinguish it from that of others, thereby the better to appropriate the whole to themselves) than render the original Invention more perfect and better suiting the Design of the original Thing itself. He who alters the Situation of a Door, Window, Chimney, or other lesser Part in a commodious well designed Building, has but a small Title to any of the Reputation acquired by the Architect of that Building. Columbus, who first discovered the West-Indies, is justly allowed by all to have the Honour of the Discovery, although Americus Vesputius, who sailed there after him, had the Reputation to have that Country called by his Name. So Mr Hadley’s and Mr Smith’s Quadrants, though they do not differ from that of Sir Isaac Newton as to the Main of the Invention, yet these Instruments are called by their Names.

The Inconveniences in the Use of this Instrument seem to be, 1ft, For him who wants to find the Altitude of the Sun by that Instrument, first to guess at that Altitude without such Instrument, at least within 10 or 15 Degrees, which is not easy to do by one who is not much used to the taking of Altitudes. Secondly, The too great Obscurity of the Objects when seen by a double Reflection, especially in hazy Weather. Thirdly, The uncertainty of seeing the true Horizon in hazy Weather or at Nights. Fourthly, The uncertainty of bringing a Star to the true Horizon in the Night, even when the Moon shines. The Horizon being then not easily to be guessed at, there being Vapours oftentimes on the Surface of the Sea causing the Refraction to be sometimes more and sometimes less according to the Weather, Time of the Year, &c. so unconstantly various, that no Rule, or Tables, by any one hitherto given for determining it’s Quantity, can be entirely depended upon. By which means the true Horizon, as to Situation, will sometimes appear higher or lower through an Instrument looking at it, that is, at the Line seperating the apparent Surface of the Sea from the Sky. And that according as the Medium that he looks at it through, is more or less dense, or the Mercury in the Barometer is higher or lower. For all which Reasons I am apt to think that this Newtonian Quadrant or Octant, is subject to uncertainty with regard to the Practice, as well as all the others that have been hitherto invented. But how far this Instrument exceeds any of them, it is not in my Power to say. Experience, and that a good deal, sufficient Judgment, and proper Caution, are the best Proofs. This may have the Advantage of some of them in some Particulars, and they of this in others. This I am certain, when the Sun is bright and the visible Horizon clear, the Altitude, all things else alike, will be best taken by this Instrument. But in the Night when the Moon does not mine, the meridian Altitude of a noted Star, as the Pole Star, which is chiefly wanted in order to find the Latitude, cannot be taken so exactly by this Instrument, because the Horizon cannot be distinctly seen, as it can by a Brass Astrolabe, or Ring, properly poised by a dexterous Person, who has been used to all the various Motions of a Ship, and where the sensible Horizon has nothing to do with the Operation. Besides, the Divisions of the Degrees upon this Astrolabe, are twice as big as those upon one of these Octants of the same Radius, which must, certainly be some advantage. Moreover, as there are but few who use this Octant who know the Reason why it gives the Altitude that they are finding by it, but, on the contrary, every body who understands the first Rudiments of Astronomy and Geography, immediately sees the Reason of his Operation by the Astrolabe; the latter has some advantage over the former upon this account, viz. with respect to simplicity. And this simplicity may be a Means to bring this Astrolabe more in use than it is at present, and I think should stir up some impartial, ingenious Persons to compare the Uses of the two Instruments, which are best, and if the Astrolabe be deficient herein, to render it more perfect by some additional Contrivances, which I think it is capable of. Telescopick Sights to the Index might be of use; and if the Wind should be so high as to disturbe the true poising of the Instrument, this may also be remedied In short, of all the Instruments for taking the Altitudes at Sea, this most ingenious Octant, as to contrivance, and that most Ample one the Astrolabe, I take to be the best of any others.

The Foundation of the Contrivance of the Newtonian Octant, is chiefly built upon the following Caloptrical Principles.

1. The Angle of Reflection is equal to the Angle of Incidence.
2. The Place of the Image of an Object, seen by the Reflection of a plane Speculum, will be behind the Speculum in a Perpendicular let fall from that Object, and as far behind the Speculum as that Object is before the Speculum (the Objects are supposed to be Points for the ease of Imagination).
3. The Image of an Object seen by the successive Reflections of one or more plane Speculums, will be seen in the second reflected Ray, by an Eye any where placed in that Ray, provided the Speculums themselves do not hinder the Sight. Hence,

If two plane Speculums bB, cC (Fig. 7.) inclined to one another under the Angle HRI, be at right Angles to some third Plane MAN, as suppose that of the Horizon MAN. And if a Ray of Light AF emitted from any Point A of an Object, be successively reflected by those two plane Speculums (that is, by the Speculum bB from F to G, and by the Speculum cC from G through L, the Angle AFB being equal to bFG, and the Angle CGF equal to the Angle cGL); the first Image of A being at M, as far behind the Continuation of the common Section RH of the Speculum bB and the Horizon, as A is before it; and the second Image N made by the Reflection of the second Speculum cC, as far beyond the Continuation RI of the common Section of the Plane of the second Speculum cC, as the Image M is on this Side of it. I say, the three Points R, M, N, will be at the same Distance from the common Section R of the Planes of the two Speculums with the horizontal Plane. And the Angle ARN contained under the two right Lines AR, NR, drawn from the Object A, and it’s second Image N, to the said common Section R, will be the double of the Angle HRI of the Inclination of the Planes of the two Speculums bB, cC, viz. their common Sections with the Plane of the Horizon.

For draw RA, join AM, MN, cutting RH, RI, in the Points P and S. Also let NG, AF, continued out, meet in L; then since all Reflection is made in the same Plane, with the incident Rays, the Points R, L, M, A, N, F, G, will be all in the same Plane, viz. that of the Horizon, now because PM = PA, and PF = PF, and the Angle AFH = RFG = MFH; the Triangles FMP, FAP will be equal to one another; wherefore FM = FA, and the Angle MFA will be twice the Angle AFH = MFH, and so the Arch AM = 2 Arch AH, or = 2 Arch MH. In like manner, since the Angle MGC = Angle LGR = NGI, and MS = SN, and GS = GS, therefore will the Triangles GMS, GNS be equal; wherefore GM = GN, and the Angle MGN = 2 Angle NGI = IGM, therefore the Arch MN = 2IM = 2lN. Hence the Triangles RMF, RAF will be equal, and so RM = RA; also the Triangles RMG, RNG will be equal to one another, wherefore RM = RA = RN = RI.

Again, make the Angle IRQ = IRH, of the Arch IG, IQ = IH, then will QH = 2IQ. But QH = AN, for since NQ = MH = AH. If AQ be added to NQ, and AH be added to NQ, there will be had AN = QH; wherefore AN = 2HI, or the Angle ARN = 2HRI.

Also the Angle NLA will be equal to the Angle NRA, and so the four Points A, N, R, L, will be all in one and the same Circle. For since the Angles ALN + GFL = FGN, and so ALN = FGNGFL. But FGN = 2FGI, and GFL = 2GFR, and consequently their Difference is = 2FRG = 2HRI, wherefore the Points A, N, R, L will be in the Circumference of the same Circle, the Angle NLA being equal to the Angle NRA. Hence, if the Eye be placed at L, in the second reflected Ray NGL, the Image N of the Point A of a visible Object will appear to be at N, supposing the Part cG of the Speculum cC to be away, viz. not hindering the Sight, or contrariwise, if N be a visible Point of an Object, the Eye at L will see it’s Image at A; if the Part bF of the first Speculum be away, so as not to hinder the direct Progress of the Rays AFL coming from A to L. Since the Radius AR of the Horizon MAN (or, indeed, of any great Circle in the Heavens), is infinite, the Points R, L, G, will coincide, or fall into one another; and in this Case, the Arch AN of the Horizon contained between the visible Point A, and it’s second Image N, will be exactly equal to one half the Arch HI of the Angle of the Inclination of the two Speculums bB, cC, to one another. Hence, if MAN be the Meridian, and A be a Point where that Meridian cuts the Horizon, and if N be supposed to be the Sun, in the Meridian, elevated above the Horizon by the angular Distance, or Arch AKN. If these two Speculums bB, cC be so inclined to one another, that the Image of the Sun N, appears to the Eye at R or L, by the Reflection of the Speculums to be cast upon the Point A of the Horizon. Twice the Angle of the Inclination of the Speculums will be equal to the Sun’s Meridian Altitude AN; and this is the main Foundation of the whole Contrivance of all these new Sorts of reflecting Sea Octants for the fore Observation. See a more particular Account of this Theory by Mr Hadley himself, in the Philosophical Transactions, Numb. 420, 425; as also in Dr Smith’s Opticks.

And the Reason of the Operation for a back Observation is derived from this Theorem, viz. twice the Difference of this Angle of Inclination of the Planes of the two Speculums, from a right Angle, is the Angle ARN of Altitude.