Mathematical Instruments
Suppl. Ch. X.

A short Account of some Instruments either of less general Use, whose full Descriptions are too long to insert in this Appendix, or such as with some Alterations or Additions of Parts are already described in this Book.

I. A Meridian Telescope.

This is a Telescope fixed at right Angles to an Axis, and turned about it In the Plane of the Meridian. Dr Halley had one of these; another of them was contrived by Mr Cotes, and the present Duke of Argyle has a very good one. See their Description in Dr Smith’s Opticks, Vol. II. Book iii. Chap. 6. The Use of these Instruments is for finding Time, by observing when the Sun or any Star has equal Altitudes on each Side of the Meridian; as also for finding the Time of their Appulses to the Meridian. And thence by means of a Clock, and the computed right Ascension of the Sun at that Time by the Clock, or that of a Star, we have the Difference between a Day reckoned by the Clock, and that of the Sun or Stars. Dr Halley at first made all his Observations of the right Ascensions of the Moon by such an Instrument; his Telescope was 5$$\frac{1}{2}$$ Feet long, and the transverse Axis above three Feet.

II. An Equatorial Telescope.

The ingenious Mr Short (in the Philosophical Transactions, at Numb. 493, for the Year 1749.) gives the Description and Uses of an Instrument consisting of a Combination of several Circles, with two Spirit-Levels, and a Reflecting Telescope, which will serve for a Dial; an equal Altitude Instrument, a Transit Instrument, a Theodolite, a Quadrant, an Azimuth Instrument, and a Level. He says, he has made three of these Instruments, one of which he sold to Count Bentinck, for the Use of the late Prince of Orange. That he had the other two by him. That he does not pretend to any thing new in the Combinations of the Circles of which the Instrument consists. The same Combinations having several Times before been made by way of Dial; but believes the putting so large a Telescope upon the Machinery, and applying it to the Uses he has done, is somewhat new.

III. The Micrometer.

Any Micrometer, of which there are several Sorts, well made, may be sufficient to measure small Angles subtended by remote Objects at the naked Eye; to do which is the chief Business of this Instrument.

The old Micrometer of Mr Townley’s, improved by Dr Hook (and long since described in the Philosophical Transactions, Numb. 29.) and it’s Uses in Numb. 25. appears to have been good, or else Sir Isaac Newton would not, at the Beginning of the third Book of his Mathematical Principles of Natural Philosophy, have ventured to set down the Measures of the Distances of the Satellites of Jupiter, from Jupiter’s Center, as taken by Mr Townley with that Micrometer. This to me is a sufficient Proof of the Exactness of that old Instrument. As to Facility of Use, and some other lesser Advantages, other more modern Micrometers have been made and described. And Dr Smith, in his Opticks, Vol. II. Chap. 8. Book 3. describes a Micrometer, which he calls one of the best Sort. The Micrometer, and it’s Use is also treated of by Mr Auzont, a Frenchman; by Mr Hevelius, in the Acta Eruditorum, Anno 1708; by Mr Balthasaris, and others. Scarcely, in my Opinion, deserving a particular Account here. What our Author and myself having already said upon this lnstrument appearing to me to be enough.

IV. Of Mr. Graham’s Astronomical Sector.

This Instrument is for observing the Place of a Planet or Comer, by taking the Differences of it’s right Ascension, and Declination, from those of a known fixed Star; when it cannot be done at all by a Micrometer, or not conveniently by a large Quadrant or Sextant, the Contriver in brief is thus.

Let AB (Fig. 21.) represent an Arch of a Circle containing 10 or 12 Degrees, having a long Plate CD for its Radius fixed to the middle of the Arch at D. Let this Radius be applied to the side of an Axis HFI, and be moveable about a Joint fixed to it at F so as the Plane of the Sector may be always parallel to the Axis HI, which being parallel to the Axis of the Earth, the Plane of the Sector will always be parallel to the Plane of some Hour Circle. Let a Telescope CE be moveable about the Centre C of the Arch AB, from one end of it to the other, by turning a Screw at G, and let the Line of Sight be parallel to the Plane of the Sector; then by turning the whole Instrument about the Axis HI, ’till the Plane of it successively directed, first to one of the Stars, and then to another, it is easy to move the Sector about the Joint F into such a Position, that the Arch AB when fixed, shall take in both Stars in their Passage by the Plane of it, if the Difference of their Declination does not exceed the Arch AB. Then having fixed the Plane of the Sector a little to the Westward of both the Stars, move the Telescope CE by the Screw G, and observe by a Clock the Time of each Transit over the cross Hairs; and also the Degrees and Minutes upon the Arch AB, cut by the Index at each Transit. Then is the Difference of the Arches, the Difference of the Declinations, and by the Difference of the Times, we have the Difference of the right Ascensions of the Stars. See a more particular Description of this Instrument in the ninth Chapter of the third Book of Dr Smith’s Opticks, Vol. II.

V. Of Mr Sisson’s Theodolite.

This is certainly the best, most complete, handsome, and well designed Instrument possible, not only serving as a Theodolite to take horizontal Angles, but likewise to take vertical Angles as a Quadrant; and besides, may be used to take the Levels of Places, all with Speed, Ease, and sufficient Exactness, for most of the common Purposes for which it is designed. See more particularly, concerning the same, in Mr Lawrence’s Surveying.

There is likewise another very good Theodolite, made by Mr Heath the Mathematical Instrument Maker, whose Uses are to be seen in Hammond’s Surveying, wrote in reality by the late very ingenious Mr Samuel Cunn (who being a Butcher, that kept a Butcher’s Shop in Newport Market) was also a very great Mathematician: One of the best Measurers of Artificers Works, Surveyors of Land, and Expounders of Euclid, and Apollonius, in the World.

VI. Mr Barston’s Universal Astronomical Quadrant, (as he calls it) is an Instrument contrived to take the Altitudes of the Sun, Moon, and Stars, without a visible Horizon, with the greatest Accuracy, as well as Speed, as the Author himself will have it. The Instrument consists of a Semicircle, with half its Arch, divided into 90 Degrees, having a Telescope upon the Diameter, and Wheel-work inclosed within the Instrument, which turns two small Hands about, pointing at the Divisions of two circular Arches, on that Radius of the Quadrant, or rather Semicircle at right Angles to the Diameter of the Semicircle, which Divisions give the Degrees and Minutes of Altitude. The Contrivance of such a Quadrant to take Altitudes at once is not new. I heard of one, it is now almost fifty Years ago, which was then called by the Name of a Ketch-Quadrant. They were well known, and commonly talked of amongst a Society of ingenious Mathematicians, that met together once a Week in London, from the Year 1710 to 1724. But I won’t say these Kerch-Quadrants were exactly the same as to Structure, with those of Mr Barston. His may possibly be better. He got a Patent for the sole making and selling them; and they were made and sold by Mr Joseph Turner, a Watch-Maker in Hatton Garden, London, with the Names of John Barston and Joseph Turner engraved upon the Limb.

VII. Dr Hook, Samuel Molyneux, Esq; and Dr Bradley, our present Royal Professor of Astronomy, had each an Instrument for trying to discover the annual Parallax of the fixed Stars. Mr Molyneux’s Instrument was a Telescope of 25 Feet long, with a square Tube, contrived to swing in a vertical Position by two polished Cylinders fixed near the top of the Tube, so that their common Axis, if produced through the Tube, would pass at right Angles to it’s Axis through a Point near the Centre of the Object-Glass. When the Telescope turned upon those Cylinders, it’s Axis of Vision moved like a Pendulum in the Plane or the Meridian, while a fine long Wire hung down by the side of the Tube, whose Loop was put over one of the Cylinders, was gently stretched by a Plummet immersed in a Vessel of Water, designed to retard it’s Vibrations. The lower End of this Wire played gently against the smooth side of a slender Brass Plate fixed cross-ways to the side of the Tube, so as to point Northwards and Southwards; and in the middle of this Plate was punched a very fine round Hole, a little broader than the Thickness of the Wire. The Telescope was gradually moved upon it’s Axis of Suspension, by the Pressure of a long Screw, divided into fine Threads, like that of a Micrometer, which worked in the Hole of a Plate fixed to the Wall of the House, and the Tube was made to bear against the end of this Screw by a Weight fixed to the end of a String passing over a Pulley having it’s other end tied to a Hook fixed in the side of the Tube. The opposite end of that long Screw was fixed like an Axis in the Centre of a Brass Wheel, whose Circumference was divided into a great Number of equal Parts, while an Index fixed to an upright Board pointed to the Divisions of the Wheel.

Now while the Wheel was gently turned, the part of the Wire which played against the Cross-Plate was viewed through a double Microscope, ’till the little Hole in this Plate appeared to be bisected by the Wire. The Telescope being by this means placed verticle, and rectified immediately before the beginning of every Observation of the Transit of the bright Star in the Head of the Dragon, which passes very near the Zenith of Kew, (for viewing of which this Instrument was contrived and set up in the Year 1725, at Kew near Richmond) and the Division of the Wheel over against the Index being then noted, the Wheel was turned again ’till the Intersection of the Wires in the Focus was brought to touch the Star at the instant of it’s Transit. Then by the Number of the Revolutions of the Wheel and it’s parts that had parted by the fixed Index, the angular Motion of the Axis of the Telescope was easily found by a Table of Minutes and Seconds, answering to those Revolutions; then the Differences of those Angles found at different Observations, are the Differences of the Star’s Declinations.

Dr Hook’s Instrument, which he fixed up at Gresham College, described by himself, was much such another, with a Telescope of 36 Feet long.

Dr Bradley’s Instrument, as he himself says in the Philosophical Transactions, Numb. 406. was erected at Wanstead in Essex, in the Year 1727. It’s Telescope was but 12$$\frac{1}{2}$$ Feet long. That it was upon the same Principles, and for the same Purpose, as those of Dr Hook and Mr Molyneux; that it’s Structure was contrived, and chiefly directed by Mr Graham. That Mr Molyneux’s Instrument being originally designed to try, whether the bright Star in the Head of the Dragon had any sensible Parallax, could not be altered, as to it’s Direction, more than seven or eight Minutes, and so could not well be used to make trial with other Stars. That his own Instrument had a divided Arch of 12$$\frac{1}{2}$$ Degrees, reaching to 6$$\frac{1}{4}$$° on each side the Zenith. And that he could be secure of it’s Situation to half a Second, viz. he could take by it a Zenith Distance of a Star to half a Second. Which I think to be scarcely possible in the actual Practice, however it may be made out by Theory. For an Arch of half a Second to a Radius of 12$$\frac{1}{2}$$ Feet will be but about the 2600th part of an Inch, which I take to be too small an Interval to be distinguished by the Sight, especially when it is liable to be disturbed by Refraction, (even in the Zenith, as it has been alerted) as well as other Impediments. Be this as it will, Dr Bradley could not find any annual Parallax of the fixed Stars. Nor could Mr Molyneux with his Instrument, although Dr Hook says he did with his. Dr Bradley suspects the Accuracy of Dr Hook’s Instrument, as well as the Truth of the Conclusions deduced from the Observations made with it; that is, that the fixed Stars have really no annual Parallax, though Dr Hook, by his Observations makes it out, that they have one of about 27 or 30″. See Dr Hook’s Treatise, called, An Attempt to prove the Motion of the Earth. So also Mr Flamstead will have it, that the Distances of the Pole Star from the Zenith varies, as he found by Observations made for seven Years with a Mural Quadrant of almost seven Feet Radius. His Letter to Dr Wallis, Anno 1698. Dec. 20. is to be seen at page 701. of the Third Volume of Dr Wallis’s Mathematical Works.

VIII. Artificial Magnets.

1. Whether artificial Magnets in general, have such strong and lasting lifting, and northpointing Powers, as good natural Magnets, and accordingly whether they are so proper to give the Needles of Sea-Compasses so strong and lasting directive Powers as natural Magnets?

Why I make a Question of this may be gathered from the following Passages of Mr Savery’s and Dr Knight’s Accounts, in thole Tranfactions of the Royal Society, mentioned above. Mr Savery says, he touched 37 large Steel Wires, 2.74 Inches long one by one, made a hexagonal Bundle of them, and bound them together with Armour, which then would lift a Key weighing 1125 Grains Troy by the South Pole; but, says he, when after each of those Wires had been placed separately in a magnetical Line for two Days, each of them had lost some Virtue; and again, that seven round Bars of Steel, all of a Size, $$\frac{3}{8}$$ of an Inch in Diameter, and Length above 12 Inches, being prepared, touched, and armed, the same as those Wires, lifted up about half a Year afterwards, more than one Pound; by which it should seem that Mr Savery meant they lifted more than one Pound at first.

Dr Knight produced, at a Meeting of the Royal Society, a compound artificial Magnet, consisting of 12 Steel Bars, which had, in an Experiment before the President, lifted up twenty-three Pound 2$$\frac{1}{2}$$ Ounces Troy, did here, under all the Inconveniences and Disadvantages of a crowded Room, still lift up 21 Pounds and 11 Ounces Troy. That a single armed Block of Steel, which had before lifted 14 Pounds and two Ounces, did here, under the same Disadvantages, as the former, lift thirteen Pounds and seven Ounces Troy Weight. Now I confess, that I cannot comprehend how a crowded Room should cause the Magnets to lift up less Weight than they did at first; but rather think they might have lost some of their Virtue.

2. Whether those artificial Magnets that have the greatest lifting Powers, do always give to touched Needles the greatest north-pointing Powers?
3. Whether the Conclusions from Experiments drawn from one of the Ways of measuring the Strength of the north-pointing Power of a touched Needle, by bringing a Magnet, &c. towards it, ’till the Magnet just begins to move the Needle from it’s fixed Place of pointing, be not truer and more to be depended upon than the other way, by the greater or less Number of horizontal Vibrations of the Needle, and that of their Times of Vibration?

From what has been said, it is evident to me, that artificial Magnets lose some of their Power in some Time, and if they are found in a longer Time to lose more, they may at last be so weak, as not to communicate to a Needle a North-pointing Power of sufficient Strength; that is, would be good for nothing. It might likewise be proper to observe, whether the Virtue of natural Magnets may not decrease in time, as well as that of artificial Magnets •, and which do this the least, for these last are certainly the best.

The natural Magnets of the greatest attractive Powers, that I ever heard of, are three One is an armed Terrella, belonging to the Royal Society, presented to them by the Earl of Abercorn, which, when his Lordship had it, is said to have lifted 40 Pounds weight. Another of the late Duke of Devonshire’s, some Years ago, at the House of the Royal Society, did at a Meeting of the Society left up near 100 weight, if I remember right. Lastly, I have been told, that the King of Portugal has got a Magnet that will lift up 6000 Ounces (but what Ounces I do not know), A Son of mine saw a Print and Description of it in the Hands of a Tradesman at Cork.

Mr Muschenbroeck, (see the Philosophical Transactions, Numb. 390.) had a Magnet that would act upon a touch’d Needle at the Distance of 14 Rhinland Feet.

The best Philosophers, who have endeavoured to explain the Manner in which the magnetical Virtue of the Loadstone acts, say, that the magnetick Matter of a Loadstone moves in a Stream from one Pole to the other internally, and is then carried back in curve Lines externally, ’till it arrives again at the Pole where it first entered, to be again admitted. The immediate Cause why two or more magnetical Bodies attract each other, is the Flux of one and the same Stream of magnetical Matter through them; and the immediate Cause of magnetical Repulsion, is the Conflux and Accumulation of the magnetical Matter. This is tolerably well made appear by Des Cartes, in his Treatise upon the Magnet. See page 270, 283. Part 4. of his Principles of Philosophy; and better by others after him.

I shall now proceed to mention a few Things in general, about the Variation of the magnetical Needle.

1. It has been found by Observations, both at Sea and Land, for near 200 Years last, that the touched Needle of a Sea-Compass does not point exactly to the North or South, but in a few Places, which all lie nearly in the same Meridian. At all other Places, in which Experiments have been made, the pointing of the Needle is not North, but to some other Degree of the Horizon, either towards the North-Well, North-East, South-West, or South-East, more or less. It has also been found, that the Variation of the Needle from the North at the same Place alters, more and more, by a very slow Motion. For Example, Mr Burrows, in the Year 1580, at Limehouse near London, observed the Variation to be 11° 11′ easterly. Mr Gunter, in 1612, 6° easterly. Mr Gellibrand, in 1633. 4° easterly. In the Year 1657. there was no Variation. In 1672. the Variation was 2° 30′ westerly. In 1692. it was 6° westerly. In 1723. 14°. 17′. And Mr George Graham (in the Philosophical Transactions, at Numb. 487. for the Year 1748.) concludes, that during the Course of 167 Years, viz. from 1580, to the End of the Year 1747. the magnetical Needle has moved westwards at London 28° 55′.

Hence, if the magnetical Needle at London moved equally towards the West, it would go westerly about 10′. 2″ every Year, and perform one Revolution in about 2079 Years. But it does not move equally to the West, according to these Observations; for because in the Year 1580. the Variation was E. 11°. 15′. and in the Year 1657. there was no Variation; in 77 Years, it moved 11°. 15′. And in the Year 1723. it was observed to be 14°. 17′. westerly; and so had moved in 66 Years (from 1657 to 1723.) 3°. 2′. more than it did before in 77 Years; consequently it’s Motion was slower from 1580 to 1657, than it was from 1657 to 1723.

Again, in the Philosophical Transactions, Numb. 64. Mr Hevelius at Dantzick, observed the Variation of the Needle to be westward. In the Year 1620, 1°. In the Year 1642, 3°. 5′. In the Year 1670, 7°. 20′. Hence, according to him, the Variation in 50 Years, viz. from 1620 to 1670. was 6°. 20′. wherefore, if the Motion were equable, a compleat Revolution by Mr Hevelius’s Observations, would be performed in about 2857 Years. He makes the Variation at Dantzick to increase every Year 9′. 6″. But how true this is I know not; it can only be found by Experience, whether it be so or not; and if it be true, the Variation at any given Time, past or to come, might easily be known without Observation.

2. The Variation of the Needle in the high Latitudes has generally been found to be much greater than in other Places, as in Hudson’s Streights, from 68 Degrees of Longitude to 81 Degrees, and Latitude 64°. The Variation was observed by Captain Middleton (see Numb. 393. of the Philosophical Transactions) to be 46°; and in Bassin’s Bay, in the Latitude of 80°. the Variation (see a Table in the Philosophical Transactions, Numb. 148.) was observed to be 57°. westerly in the Year 1668. Capt. Middleton says, the Compass in Hudson’s Streights, would scarcely traverse at all. He says also (at Numb. 449. of the said Transactions for the Year 1735.) when he was in the Ice in Hudson’s Bay, the Needle of his Compass would oftentimes lose so much of it’s magnetick Virtue, as that it would not traverse at all, any longer than the Quarter-Matter kept touching it, but by bringing the Compass to the Fire he recovered it’s virtue. The accurate Mr Joseph Harris (see the Philosophical Transactions, Numb. 428. for the Year 1733.) says, that being at Sea about Jamaica, and the Havannah, in the Year 1733. he thought the Virtue of the Needle of the Compass was not always of equal Strength. Sometimes several Observations would agree very well, at other Times the Card would stand indifferently any where, within a Degree or more of it’s Meridian; and he observed this several Times. He also says, the Card would differ from itself about 2°. sometimes between the Morning and the Evening of the same Day; and this Difference would continue regular, as it were, for several Days, and then vanish for a Week and more, and then would return as before. This ingenious Person also says (see Numb. 401. of the Philosophical Transactions) that in his Voyage from England to Vera Cruz they observed the Variation with a good Azimuth Compass; but he always found the best Observations they could make, when compared together, differed so much, that they could not be depended upon, to much less than three or four Degrees, and sometimes to half a Point of the Compass. Moreover, the ingenious Mr George Graham (see the Philosophical Transactions, Numb. 383.) from Observations he made at London, of the Variation of the magnetick Needle, found that three touched Needles would exactly settle and point to the same Place, after being removed, by a great Number of Trials made immediately one after another, but yet he found them at different Times of removal to differ considerably from their former Directions. After many Trials he found all the Needles he used would not only vary in their Directions upon different Days, but frequently upon different Times of the same Day; and this Difference would sometimes amount to more than half a Degree in the same Day, sometimes in a few Hours. He also found the Variation has generally been greatest for the same Day, between the Hours of twelve and four in the Afternoon, and the least about six or seven in the Evening, &c. This ingenious Person also, in the Philosophical Transactions, Numb. 389. for the Year 1723. gives an Account of a Dipping Needle of his, with Observations to try if the Dip and Vibrations were constant and regular.

All these Inequalities and Irregularities of the Variation of the magnetick Needle (and many others to be found in Authors) may in part be true, and proceed from true Causes assigned, and partly be not so, but let down as true from fallable, uncertain, erroneous Observations of those that made them, which might proceed,

1. From the Nearness of Iron.
2. The Weakness of the magnetick Virtue of the Needle disposing it to stand indifferently at any Point of the Compass it is set to.
3. The Friction of the Cap.
4. The Motion of the Ship.
5. The Resistance of the Air, which in high Latitudes may be so great in the cold Weather, as to considerably affect the true pointing of the Needle.
6. The great Refraction, in high Latitudes, causing an Error in finding the Meridian Line, which Line, in high Latitudes, cannot be determined with any great Certainty, by any Method whatsoever, in my Opinion; because in these Latitudes all the Points of the Compass begin almost to be nearly Meridians, (for at the Pole itself, every Point of the Horizon is South at the North Pole, and North at the South Pole,) and because the Amplitudes and Azimuths of the Sun or Stars, cannot be taken in those high Latitudes, without Error, if at all, by reason of the too great obliquity of the Arches of the diurnal Motion in those Latitudes; and since the meridian Lines cannot be found, but by an Amplitude or Azimuth observed, it is evident the Variation cannot be had to any great Exactness, if at all, in such high Latitudes.

Mr Wright, in his Treatise of the Errors of Navigation, first published before the Year 1600, gives a Table of the Variation of the Compass, as taken by himself, both at Sea and at Shore, in a Voyage of the Right Honourable the Earl of Cumberland, in the Year 1598. He also gives two other Tables of the Variation at the End of that Book, the one pretty long and extensive, with regard to the principal Places of the Earth or Sea; and the other a lesser Table, of Peter Plancius’s. Dr Halley (in the Philosophical Transactions, Numb. 148. for the Year 1683.) gives another Table of the Variation in different Places, at different Times. But, in my Opinion, this Table is much too short and imperfect for a Foundation to build a true Theory of the Variation upon, though the Doctor himself did take it to be sufficient in that Transaction, where he supposes the whole Globe of the Earth to be one great Magnet with four fixed magnetical Poles, near each Pole of the Equator two, and that in those Parts of the World which lie near adjacent to any of those magnetick Poles, the nearest Pole being always predominant over the more remote, &c. Upon this Supposition the Doctor ingeniouily enough accounts in general for the Variations and Alterations, when they are always the same at the same Place; but since they have been observed to be different at the same Place. The Doctor had laid aside, for some Years, this Theory of his of four fixed magnetical Poles, and (in the Philofophical Transaclions, Numb. 195.) re fumed it again with Alterations and Amendments, making two of the magnetical Poles moveable, &c. and conjectures, that these two moveable Poles perform one Revolution in about 700 Years, at the end or which Time all the Variations will be the same again in the same Places, &c. Mr Bond, in his Treatise of the Longitude found, pag. 7. says this Revolution will be once in 600 Years, and that the Earth has but two magnetick Poles, at 80$$\frac{1}{2}$$° Distance from the Poles of the Equator, &c. The Frenchmen who, in the Year 1737, went to measure a Degree of the Earth, at Tornea, in the Latitude 65°. 50′. found the Variation to be there 5°. 5′. westwardly. And in the Year 1695. it was found to be there 7° westwardly; hence, in 42 Years the Variation has moved 1°. 55′. and so one Revolution, supposing the Variation to alter equally, will not be performed ’till the end of about 8062 Years. Now neither Dr Halley’s, or as Body else Suppositions to account for the Causes of the Variation, can reconcile these several Differences of the Times of the Revolutions of the Variation that I have given above, and therefore their Hypotheses cannot be true.

In short, this Affair of the Variation is Very uncertain, it’s true Cause not rightly known; nor have we any true and fixed Rules how to compute what it is or will be at a given Place, at a given Time. Tables of the true Quantity thereof for several Years, at many different Places, under the same Meridians and Parallels, being wanted to build a true System upon. From such Tables only it can be gathered how the Variation at given Places alters, and at what Rate, and according to what Regularity, and whether any of these Things can be discerned or not.

Dr Halley’s, Chart of the Variation is very ingenious, but how true it is I cannot say. It true for the Time he made it, it cannot be so in future, because of the Alteration of the Variation at a given Place.

There are several Authors, our own Countrymen, who have wrote upon the attractive Virtue of the Magnet. As Gilbert de Magnete, Robert Norman, and Mr Burroughs’s new Attractive, wherein is an Account of the Variation. The Lord Paisley’s Treatise about the attractive Virtue of the Loadstone, with Calculations and Tables, &c. And Mr Muschenbroeck’s Dissertatio de Magnete. This last by an ingenious Dutchman, who has wrote well upon the Subject.

IX. A new contrived Mariner’s Compass.

This Instrument is described in the Philosophical Transactions, Numb. 495. it differs from the others that are commonly made, in having it’s Needle in shape of a Parallalepipedon. It’s Card, consisting of unstiffened Paper, and a light thin Circle of Brass, divided into Degrees, &c. An ivory Cap, turned so as to receive a small bit of Agate at the Top. The Point supporting the Card is a common sewing Needle. The very ingenious Dr Knight, who exhibited one of these Compasses before the Royal Society in the Year 1750. made by Mr Smeaton, will have these Alterations to be for the better, both for the true and well pointing of the Needle, and it’s long Continuance to do so. But although the great Number of Experiments the Doctor has tryed and made, many of which are to be seen in the Philosophical Transactions, at Numb. 474, in the Year 1744. is a plain Proof of his Skill in Magnetism; yet I am not quite satisfied with what he says of the Faults in the Compass Cards, and his Remedies. His Experiments about this to me seem to be too few, to deduce from them a general Rejection of the Make and Fashion of the old Cards, as being less perfect than those he recommends. Skilful Mariners (of which we have enough) who have used Compasses at Sea for many Years, may at least be allowed to be as competent Judges of the Goodness of their Compasses, as the Doctor; and I have always been apt to think, that real Improvements of Instruments that have been generally used for more than two Hundred Years, would have been thought of before the Year 1750. The Plants and Trees of the Gardens, of the Arts and Sciences, cultivated by the Dung of Ambition, and nourished with the Waters of Interest, are very subject to be blasted by the Winds of Error, and sometimes Hunted by the Weeds of Imposition.

X. The Use of an Azimuth Compass, of a new Contrivance, by Capt. Middleton, is to be found in the Philosophical Transactions, Numb. 450. for the Year 1738.

The Captain says, that the Variation of the magnetical Needle at Sea may be found by it, with greater Use and Exactness, than by any other Azimuth-Compass contrived before that Time, viz. the Year 1738. But as he conceals the Description of his Instrument, we don’t well know how to judge of it’s Excellency, he telling us, that those who have the Instrument before them, have no Occasion for a Description of it. That he himself found by Experience it was effectual. He only shews the Manner of using it; says that one Person may manage it, whereas the old Compass requires several Persons, which also makes it subject to many great Errors, as he will have it.

The Main of this new contrived Compass of the Captain’s, gathered from the short Account he gives of it’s Use, is the taking the Sun’s Altitude by Reflection, viz. bringing it down to the Horizon, in some such Manner as is done by Mr Hadley’s Sea-Quadrant. It carries a Telescope with a vertical Hair within it. It is very possible this is a very good Compass, but it’s dearness may be some Objection to it’s general Use.

XI. Celestial Globes somewhat improved.

The late ingenious Mr Senex, at Numb. 447. of the Philosophical Transactions for the Year 1738. shews how to make the Poles of the Diurnal Motion in a Celestial Globe pass round the Poles of the Ecliptic. His Contrivance is this; the Poles of the diurnal Motion do not enter into the Globe, but are affixed at one End to two Shoulders or Arms of Brass, at the Distance of 23$$\frac{1}{2}$$ Degrees from the Poles of the Ecliptick. These Shoulders at the other end are strongly fastened on to an iron Axis, passing through the Poles of the Ecliptick, and is made to move round, but with a very stiff Motion; so that when it is adjusted to any Point of the Ecliptick which you desire the Equator may intersect, the diurnal Motion of the Globe about it’s Axis will not be able to disturb it.

The Reason of such a Contrivance is to render the Use of the celestial Globes more perfect and lasting, than they can be without it; because the Places of the fixed Stars, put down upon celestial Globes, made at any given Time, will not be true but for some Years, by reason of the Procession of the equinoctial Points, viz. the Intersection of the Planes of the Ecliptick and Equator, causing the Distances of the fixed Stars from those Points continually to alter; this was observed by the antient Astronomers, who made the Sphere of the fixed Stars to move about the Poles of the Ecliptick with a slow Motion, so that all the fixed Stars in the Ecliptick or its Parallels, will go once round in the Space of 25920 Years, after which Time the Stars will again return to their former Places.

Hence the Longitude of the fixed Stars alter one Degree at the End of every 72 Years or 50″ every Year. Therefore celestial Globes made many Years ago, or stich as are new made, many Years hence, cannot be used with Certainty, the Places of the Stars set down upon them altering too much, at the Ends of those Times, and to remedy this in some measure is what Mr Senex proposes to do, by his Contrivance abovementioned.

Note, There is another Way to do the same Thing by Dr Latham, at Numb. 460. of the Philosophical Transactions, for the Year 1741.

The very ingenious Mr James Ferguson, (at Numb. 483. in the Year 1747. of the Philosophical Transactions,) has added another Improvement to the celestial Globe, chiefly for finding the Times of the Rising and Setting of the Moon, with her Time of coming to the South, tolerably exact, without an Ephemeris; but nearer the Truth, by having given the Moon’s Latitude for the Day by an Ephemeris,

His Contrivance is as follows: On the Axis of the Globe above the Hour circle is fixed the Arch A at one End by the Screw D (see Fig. 24.) so as to leave sufficient Room for turning the Hour Index. The other End at B, being always over the Pole of the Ecliptick, has a Pin fixed into it, whereon the Collets C and B, are moveable by their Wires F and G when the Screw E is slackened, and may be made fast at Pleasure by this Screw; so that the turning the Globe round will carry the Wires round with it, shewing thereby the apparent Motions of the Sun and Moon by the little Balls on their Ends at H and I. On the Collet C, in which the Sun’s Wire is fixed, there is also fixed the circular Plate L, whereon the 29$$\frac{1}{2}$$. Days of the Moon’s Age are engraven, which have their beginning just below the Sun’s Wire, consequently the Plate L cannot be turned without carrying the Sun’s Wire along with it, by which means the Moon’s Age is always counted from the Sun; and the Moon’s Wire being turned so as to be under the Day of her Age on this Plate, will set her at her due Distance from the Sun at that Time. These Wires being Quadrants from C to H, and from B to I, must still keep the Sun and Moon directly over the Ecliptick, because the Centre of their Motions at C and B is directly over the Pole of the Ecliptick, in the artick Circle. But because the Moon does not keep her Course in the Ecliptick, as the Sun does, having a Declination of 5$$\frac{1}{2}$$ Degrees on each side of it in every Lunation, she is made to screw on her Wire as far on both sides, as this her Declination or Latitude amounts to; for this purpose K is a small Piece of Paste-board to be applied over the Ecliptick at right Angles, the middle Line 00 standing perpendicularly thereon; from this Line there is maked 5$$\frac{1}{2}$$ Degrees on each side upon the outward Limb, which reaching to the Moon makes her to be easily adjusted to her Latitude at any Time. Note, The Horizon of the Globe should be supported by two semicircular Arches, instead of the usual Way of doing it by Pillars, because the Arches will not stop the Progress of the Balls, when they go below the Horizon in an oblique Sphere.

The Use.

To rectify the Globe. Elevate the Pole to the Latitude of the Place, then bring the Sun’s Place in the Ecliptick to the brazen Meridian, and set the Hour Index to XII at Noon. Keeping the Globe in this Position, slacken the Screw E, and set the Sun directly over his Place in the Meridian; which done, set the Moon’s Wire under the Day of her Age, for that Time, on the Plate D, and she will stand over her Place in the Ecliptick, for that Time, and you will see in what Constellation she then is. Lastly, fasten the Wires by the Screw E, and the Globe will be rectified; this done, turn it round in the usual Way, and you will see the Sun and Moon rise and set for that Day, on the same Point of the Horizon, as they do in the Heavens; the Times of their rising and letting are shewn by the Hour Index, which also shews the Time of the Moon’s passing over the Meridian. If you want to find the Times of the rising and setting of the Moon to greater Exactness, find her Latitude for that Day by the Ephemeris; and as it is South or North, screw her so many Degrees from the Ecliptick, measuring them by the Paste-board K, applying it to the Ecliptick, as abovementioned; and then turning the Globe round, you will see the Time of the Moon’s rising and setting by the Hour Index, and her Amplitude on the Horizon for that Time, as it is affected by her Latitude, which will sometimes be very considerable. Note, All the Phenomena of the Harvest-Moon become very plain by this additional Contrivance, which is curious enough. But some People may think the common way of finding the Time of the Moon’s coming to the South, (given in what is called the Julian Calendar, to be found in some Books of Navigation) which is easy and well known, and may be exact enough for many Purposes, may be sufficient, without such an additional Contrivance as this, to do the same Thing.

XII. Clocks, and other Time-Keepers.

The Instruments for measuring Time, are Sun-Dials, Water Dials, Sand-Dials, a single Pendulum, or Clocks and Watches. Sun-Dials have been enough treated of by Mr Bion. Water-Dials and Sand-Dials are now in no very great Esteem, although some of them are ingenious enough; they are treated of in several Writings, viz. in Vitruvius’s Architecture, Gesner’s Pandecticks, Martinelli of Elementary Clocks, Archangelo Maria Radi. of Sand-Dials, Schottus in his Technica Curiosa, &c.

The old indented Wheel-Clocks that were regulated by horizontal Balances., and generally used ’till the Year 1660, are now in no great Use, being far more imperfect than those which have Pendulums adapted to them to regulate their Motion; nor do Watches measure Time so equable and true as Pendulum-Clocks. Galileo’s Son was the first who applied a Pendulum to a Clock in the Year 1649. And the great Mr Huygens, in the Year 1673, published his famous Book, called Horologium Oscillatorium, explaining the Manner of doing it with a Figure of the Clock, being indeed the very same as that Clock described by Mr Bion.

There are several more Things about Clocks, treated of by Mr Huygens, in his Book abovementioned, the chief of which are what follow.

Mr Huygens, at page 13. of his Horologium, gives two Ways to regulate the Motion of a Clock. The first is by the Observations of the Transits or Passages of some of the known fixed Stars, and the other by means of the Times given by the. Shadow of the Sun upon a Dial, together with an Equation Table.

The Substance of what he says is this, choose some proper Place for the Eye, from which you may see several of the fixed Stars instantly disappear in passing behind high Buildings, and in that Place fix a Plate with a Hole in it to look through, of the Magnitude of the Pupil of the Eye, in order to bring it always to the same Point; this done, observe the Time shewn by the Clock the Moment you see a known fixed Star vanishing from, your Sight behind the Building, the Eye looking through that fixed Hole, and doing the same Thing the next Night, or rather some Nights after, if the Interval of Time between two such Observations of the same Star be only one Day, and the Time given by the Clock in the later Observation be less than that given in the former, by three Minutes and 56 Seconds, then it is evident the Clock goes right, because every Revolution of the fixed Stars is less than a mean solar Day by that Quantity. I say a mean solar Day, because solar Days, or the Intervals of the Sun’s Applies to the Meridian are unequal. But if the same Observations be repeated some Nights after, the Computation of that Difference must be made at each. For Example, suppose at the first Observation a Star vanishes at 9h 30′ 18″ by the Clock, and seven Days after the same Star appears to vanish by the Clock at 8h. 50′. 24″. the Difference of these Time is 39′. 54″. and a seventh Part of it is 5′. 42″ which shews, that the diurnal Period or the Clock is 5′. 54″. longer than a sidereal Day, whereas a solar Day of a mean Length is but 3′. 6″. longer, and by Consequence the diurnal Period of the Clock is 1′. 46″. too long. Note, The Intervals of Time between the successive Applies of a fixed Star to the same vertical Circle, are equal to the Intervals between it’s Applies to the Meridian. But to avoid the Uncertainty of the Air’s Refraction, as much as may be, the Face of the Building should be as near the Plane of the Meridian as possible. Note also, The Stars made Choice of should be those near the Equinoctial, by reason of their quick Passages.

The Motion of a Clock may be examined another way by the Sun. But we must here have regard to the Inequality of the Natural Days; for as has been said, these are not all equal between themselves, and though the Difference be but small, yet it often happens, that in the Space of several Days it becomes so considerable, as it must not be disregarded; for although we should have the most perfect Description of the solar Days, and the Motion of the Clock most exactly agreed with the true Measure of those Days; yet it would necessarily happen that at certain Times of the Year they would differ from one another a Quarter of an Hour, or even Half an Hour, and then again at stated Times return to their Agreement; as is to be seen in the Tables of the Equation of Days.

Now, to compare the Motion of a Clock by that of the Sun on a Sun-Dial, take the Equation of any Day, when the Clock is set to go with the Sun-Dial, and the Equation of that Day, you want to know how true the Clock goes. If the former Equation be greater than the latter, the Time by the Clock will exceed that of the Sun-Dial, by the Difference of those Equations; but if the Equation of the latter Day be found to be greater, the Time by the Sun Dial will be the greatest. For Example, If on the Fifth of March the Clock and Sun going together, the Equation of that Day be found in the Table to be 3′. 11″. and I want to know, whether or not, on the 20th Day of the same Month, the Clock truly measures equal Time, the Equation for this Day in the Table will be 7′. 27″. which because it is greater than that for the 5th, by 4′. 16″. the Time shewn by the Sun-Dial will be slower than that given by the Clock by the said Difference; wherefore if it be found different, it is easy to gather from thence how much every Day the Sun Dial goes faster or slower than the Clock.

Note, The Construction of an Equation Table of this Kind, is founded upon a twofold Cause, both known to Astronomers, viz. upon the Obliquity of the Ecliptick, and the Irregularity of the Sun’s Motion, because it is deduced as well by Reason as Experiments founded upon these very Clocks, without which it could not be inferred; for Observations of the Time of the Sun’s coming to the Meridian every Day for many Months have most evidently been found to agree.

Now, if upon Trial by both Methods, but rather by the first, the daily Error of the Clocks going should be so great as to amount to 3′. or 4′. this must be remedied by lengthening or shortening it’s Pendulum, in doing of which the following Rule should be observed, viz. the daily Motion of the Clock will be accelerated or retarded by so many Minutes, as the Pendulum is shortened or lengthened about $$\frac{7}{120}$$ Parts of an Inch taken so many times, as there are Minutes. The going of the Clock being thus almost corrected, the rest of the Correction is performed by moving upwards or downwards upon the Rod VV a small Weight D, as I have said before. (See the Figure in Mathematical Instruments.)

Mr Huygens says, that two such Clocks as these were carried to Sea in the Year 1664, in order to find the Longitude by them, being moved by a Spring instead of a Weight; and to avoid their being affected by the Motion of the Ship, the Clocks inclosed in a brass Cylinder, were suspended from a steel Ball, and the Stern, which continued the Motion of the Pendulum, (which Pendulum was half a Foot long) produced downwards, and governed it, was in Shape of the Letter F inverted; for fear least the Motion of the Pendulum should become circular, and thereby the Cessation of it’s Motion might arise. The Captain of the Ship that had these two Clocks being in Company at Sea with three others, on the Coast of Guinea, and failing from thence near the Equinoctial, had a Consultation with the other Captains to know where they were, they produced their Journals, the Captain with the Clocks using them to correct his Journal, by comparing their going with the Motion of the Sun. The Event was, that they all differed very much in their Reckoning from this Captain’s, viz. one said he was 80 Leagues, another 100, and another still more, from the Land, but this Captain said he was but about 30 Leagues, which was found to be true, and the next Day they arrived at one of the African Islands, called Del Fuego.

Mr Huygens, at page 17. of his Horologium Oscil. gives a more particular Account of this, as well as other Observations of other People finding the Longitude at Sea by other of these Clocks, whose Pendulums were half a Pound weight, and about nine Inches long, the Motion of whose Wheels was by a Weight, the whole Clock being put into a Case four Foot long, and at the Bottom of the Case there was hung an additional Weight of more than 100 Pounds, thereby to keep the Instrument better suspended in the perpendicular Posture.

This great Man Mr Huygens, in his Horologium Oscil. gives two other Contrivances for the equable going of his Clock, and for the Preservation of it’s Motion at Sea, in blowing Weather, both of which, not being long, are as follow: To the Crown Wheel, in which the Pallets of the Pendulum play, is hung a small Weight from a slender Chain, by which it is only moved, all the rest of the Machinery acting no otherwise than that in every half Second, this little leaden Weight will be restored to it’s first Altitude, almost after the same Manner as is to be seen in the Construction of our Clock already given; where one Weight is raised up by the Line, while the other (viz. that of the Pendulum) by it’s Gravity preserves the Motion of the Clock. Now by this Contrivance, every Thing being reduced, as it were, to one Wheel only, the Clock will appear to have a greater Equality in it’s Motion than it had before, viz. according to the former Construction. And something happened here very remarkable. When two such Clocks were suspended from the same Beam, supported upon two Props, the Motion of the Pendulum of each agreed so well by reciprocal Beats, as not in the least to vary, the Noise of each being always heard at the same Moment. And if by some means this Harmony was disturbed, it would be restored again of itself in a short time; I at first wondered at this uncommon Event, but afterwards, upon strict Enquiry, I found the Cause thereof to exist in the Motion of the Beam itself, although this was scarcely sensible. For the alternate Vibrations of the Pendulums, by any the least Weight communicate some Motion to the Clocks, and this Motion is impressed upon the Beam itself, and by that means if each Pendulum did not most exquisitely move with reciprocal Vibrations, it would at length happen, that the Motion of the Beam would altogether cease, which Cause nevertheless will not take place but when the Motion of the Clocks from thence are most equable, and consentaneous between themselves.

The other Contrivance of Mr Huygens’s, to preserve the Motion of the Clock at Sea in great Agitations of the Ship, is as follows. (See Fig. 22.) The Pendulum is of a triangular Form as ABC, at the Bottom of which the leaden Weight B is hung, and at the Angles A and C of it the same is also suspended from the cycloidal Plates or Cheeks ED, FG. The middle of the Base AC receives the perpendicular Axis HK of the horizontal Crown Wheel N, which turns at K by means of the Teeth of that Wheel, during which Time the Weight B moves the Base AC of the triangular Pendulum ACB. The Motion of all the Wheels, is not by a Weight, but by a steel Spring enclosed in a Barrel, and the Crown Wheel is below the other Wheels of the Clock. LL are the small lenticular Weights to regulate the Motion of the Pendulum.

The Manner of the Clock’s Suspension is expressed in the 23d Figure. where the Case AB is set upon two Axes, (one of which, as C, can only be seen) fitted to an iron Rectangle DE, which Rectangle likewise moves about the Axes FG, going into the Ends FG of the iron Gnomon FHKG, which Gnomon is firmly fixed to the Top of that part of the Ship wherein the Clock is, and to the Bottom of the Case is hung a Weight of fifty Pounds. By this Contrivance the Clock will preserve it’s upright Posture in all Inclinations of the Ship. The Axis C, together with that opposite to it, is so placed, as to be in a right Line with the two Points of Suspension of the Pendulum, that we have already spoken off, by which means it’s oscillatory Motion cannot move the Clock, than which there is nothing more liable to destroy the Motion of the Pendulum. Moreover, the Thickness of the Axes CC, of about an Inch, and the Bigness of the Gravity of the Weight hung on at the Bottom, are a Check upon the too great Motion of the Clock, and make it continually return to it’s upright Posture, when the Ship should happen to be in a great deal of Agitation.

Now one of these Clocks, thus managed, being tried at Sea, seems more promising than the others that have been already tried, to avoid being affected with the different Motions of a Ship. Thus far from Mr Huygens.

One Mr Harrison, several Years ago, made a Clock to go at Sea of a different Construction from those of Mr Huygens’s, whose Pendulum consisted of several horizontal Rods that see-sawed, or vibrated up and down. The Rods were partly of Wood, and partly of Brass, in order to check one another’s lengthening or shortening by heat or cold, and thereby keeping the Motion of the Clock equable. I have not seen the Clock, and have been only told this, by a Person who saw it many Years ago. The Inventor himself now lives in London. One of them was shewn to the Lords of the Admiralty, and approved of by them.

That great Mechanick, the late Mr George Graham, has made some of the bed Clocks in the World for the true measuring Time; they are partly described in the Philosophical Transactions at Numb. 432. There is also another curious Clock of his, described in Dr Desaguliers’s, Course of Experiments to measure Time to quarter Seconds.

Gesner, in his Pandecticks, says, that Richard Wallingford, an Abbot of St Albans, who lived in the Year 1326, made a Clock with wonderful Art, that had not it’s Fellow in all the World. Some Clocks have been made to shew apparent Time, viz. to go with the Sun at all Times of the Year. These are Curiosities indeed, but little else. Some Clocks shew the Sun’s annual Motion by the Addition of a Wheel going once round in 365 Days 5 Hours and 49 Minutes; and others shew the Moon’s synodical Revolution of 29 Days 12 Hours, and 44 Minutes, being the mean Time from one Conjunction to the next. There are other Clocks that shew the Motions of the Planets particularly; and that Instrument shewing the Motions and Phases of them all, of which there are many amongst us, made by our Mathematical Instrument-Makers, is called a Planetarium or Orrery. And indeed that described by me, is not so perfect, as some that have been made since, the superiors Planets not being in it.

Gasper Scottus the Jesuit, in his Technica Curiosa, seu Mirabilia Artis, printed in the Year 1664, in Quarto, says, that Father Schirley de Reita, in a Book called, The Eye of Enoch and Elias, gives the Construction of a Planetarium, representing all the Motions of the Planets, both true and mean, their Stations, Retrogradations, and Directions, without Epicycles or Equations, and that with a few Wheels, endless Screws and Pullies. The first Wheel of this Instrument that gives Motion to all the rest, is moved round by the Fall of Water. The Sun’s Motion in it is only 365 Days, and the Disks of the Planets are too great. See a particular Account of all the Parts of this Instrument in the aforesaid Book of Scottus’s; wherein likewise he treats of the different Machines that were in use to measure Time. The Book is curious, full of Figures; and although he sometimes fails in the Exactness to what we are now arrived, yet in my Opinion the Book is valuable enough.

At the End of Mr Huygens’s, Opuscula posthuma, printed in Quarto at Leyden, in the Year 1703, there is a masterly Description of the Motion of the Planets by Clock-Work, at least upon the Astronomical Principles assumed by him. He indeed herein makes the annual Motion of the Earth to be but 365 Days, when it should have been near 6 Hours more, but this he did to avoid increasing the Number of Wheels. He gives the best way of finding the Number of Teeth, that will make all the Motions the nearest the Truth possible, by the Method of finding the nearest Ratio in lesser Numbers, that shall approach to a given Ratio in greater Numbers. The Number of the Teeth of one of his Wheels is 300, of another 206, of another 219, of another 166, of another 158, of another 137, &c. He gives two very large Figures of his Machine, which is 2 Feet over.

At the End of the first Volume of Dr Desaguliers’s Course of Experimental Philosophy, printed in the Year 1745, is a Description of a Planetariam of the Doctor’s, with Figures, which I take to be one of the best of these Instruments that has yet appeared, and is 3 Feet over. The Doctor says, these sort of Machines had the Name of Orrery given to them by Sir Richard Steele, and the first that was made in England was by the late Mr George Graham about the Year 1712, which was sent over to Prince Eugene. But that it only shewed the Motion of the Moon round the Earth, and that of the Earth and Moon round the Sun. That this Instrument being in the Hands of Mr Rowley, a Mathematical Instrument-Maker, he copied it, adding Improvements of his own, and thereby got all the Praise due to Mr Graham, though Sir Richard Steele did not know Mr Graham to be the first Inventor.

In the Philosophical Transactions, at Numb. 479. for the Year 1746. the ingenious Mr Ferguson describes the Phænomena of Venus, as represented in an Orrery of his, agreeable to the Observations of S. Bianchini, who will have her Axis to be inclined 75°. from a Line supposed to be drawn perpendicular to the Plane of the Ecliptick, and that her diurnal Motion is performed in 24 Days and 8 Hours. Whereas in other Orrerys her Axis is perpendicular to the Plane of the Ecliptick, and her diurnal Motion about it is only 23 Hours. This Gentleman’s Improvement and Discoveries are ingenious enough, but it's Usefulness will be best: perceived by the Inhabitants of Venus.

I shall here say no more of these very elaborate, expensive and artificial Clock-Work Representations of the Motions, and Appearances of the heavenly Bodies, which at best are only amusive, and apt to afford more Honour to the Inventors, and Interest to the Makers of them, than any ways really promote the most useful Parts of Astronomy. The next Things I shall mention are some more concerning the useful Instruments Clocks.

Mr Huygens’s second Pendulum Clock, described by Mr Bion, has five Wheels of 15, 24, 48, 48, and 80 Teeth, and 2 Pinions of 8 Teeth each. Which are too many. For three Wheels of 25, 72, and 80 Teeth, and 2 Pinions of 8 and 10 Teeth, or else 3 Wheels of 25, 64 and 90 Teeth, and those 2 Pinions, or 3 Wheels of 30, 60, and 80 Teeth, and those two Pinions, or three Wheels of 21, 75, and 80 Teeth, and 2 Pinions of 7 and 10 Teeth, will cause one Revolution of the Wheel, with the greatest Number of Teeth, to be performed in one Hour, or 3600 Seconds; and thence to get a Revolution of 12 Hours, two Wheels of an equal Number of Teeth, with a Pinion of 6, and another Wheel of 72 Teeth, or a Pinion of 7, and a Wheel of 84, or a Pinion of 8, and a Wheel of 96, or a Pinion of 9, and a Wheel of 108 Teeth, will cause one Revolution in 12 Hours. Note, The way of finding the Number of Teeth of the principal Wheels and Pinions of a Clock is founded upon this Theorem, viz. that the Product of the Teeth of the Pinions multiplied by half the Number of Vibrations, is equal to the Product of the Teeth of all the Wheels; and hence, by the Method of finding all the Divisors of a Number, may be obtained all the several different Numbers of Teeth of the Wheels and Pinions that will produce one Revolution in the same given Time.

The Length of a Pendulum is the Distance from it’s Point of Suspension to it’s Centre of Oscillation. A Pendulum, whose Length is 39.2 Inches will vibrate one of it’s smallest Arches in one Second of Time. The Distance of the Centre of Oscillation of a simple Pendulum from it’s Point of Suspension, may be found by saying, As the Sum of the Weight of half the Rod, and the Weight of the Bob, is to the Sum of $$\frac{1}{3}$$ of the Weight of the Rod, and the Weight of the Bob, So is the Distance from the Point of Suspension, to the Centre of Gravity of the Bob, to the Distance of the Centre of Oscillation from the Point of Suspension. Hence the Centre of Oscillation lies above the Centre of the Bob. But when the Weight of the Rod is exceeding small, the Centre of Oscillation will be so near the Centre of Gravity of the Bob, or its Centre of Magnitude (supposing it to consist of uniform Matter of the same Density) that all three may be taken for one another, without any sensible Error. Twice the Time of one Vibration of a Pendulum, whose Centre of Oscillation describes the shortest circular Arch possible, is equal to the Time of a Body’s perpendicular fall by the Force of Gravity, through a Space twice the Length of the Pendulum. The Times of the Vibrations of two Pendulums of different Lengths, when their Centres of Oscillation describe the smallest circular Arches possible, are in the subduplicate Ratio of the Lengths of the Pendulums. Hence it is easy to find the Length of a Pendulum that shall perform one of it’s least Vibrations in a given Time: Suppose any given Number of Seconds. For it is but saying, As the Square of 60 Seconds, is to the Square of the given Number of Seconds, So is 39.2 Inches to the Length of the Pendulum wanted. The Time of the Vibration of a Pendulum, whose Centre of Oscillation describes any Arch of a Circle, will be to the Time of the Vibration of the least Arch possible, nearly as the vibrating Arch is to twice it’s Chord. Hence because the Times of the perpendicular Fall of all Bodies, through any Chords of the same Circle, are all equal between themselves; the Times of the Vibration of a Pendulum, describing small Arches of a Circle, not much different from their Chords, may be taken as equal between themselves, viz. equal to the Time of one Vibration of it’s smallest Arch. But since these Times of the Vibrations of the small Arches are not exactly and geometrically equal to one another, the longer Arches being really a longer Time describing than the shorter; and as many small Errors, separately considered, do not deserve Notice, yet a great many put together, must be taken Notice of and avoided; and for this Mr Huygens, in his Horologhim Oscill. proposes a Remedy, viz. by making the Centre of Oscillation of the Pendulum describe the Curve of the Cycloid, instead of the Arch of a Circle by means of two other equal Cycloids, by which Contrivance all the Vibrations, both great and shall, will be performed in the same Time; and the Time of any Vibration in this Cycloid is to the Time of the perpendicular Fall through the Length of the Pendulum, as the Circumference of a Circle is to twice the Side of the Square inscribed in that Circle. Note, The Demonstrations of the major Part of what is here said about Pendulums are given by Mr Huygens and others; and the Pendulum is supposed to move in a Vacuum free from the external Resistance of the Air, or any other Fluid in it.

Sir Isaac Newton, in the sixth Station of the second Book of his Mathematical Principles of Natural Philosophy, says, that all the Oscillations in the Cycloid of a Pendulum, moving in a Fluid which resists it’s Motion in the Ratio of the Velocity will be isochronal. And if the Medium a Pendulum moves in resists it in the duplicate Ratio of the Velocity, the very short Oscillations are the more isochronal, and the shortest of all are performed nearly in the same Times as in Vacuo, and the Times of them that are performed in greater Arches, are somewhat greater. And also the Times of all Oscillations, both great and small, seem to be prolonged by the Motion of the Medium the Pendulum swings in. This is only Theory. But in the general Scholium Sir Isaac deduces from some Experiment he made with Pendulums, that the Resistance of the Air to Globes moving in it, when they move swiftly, is nearly in the duplicate Ratio of the Velocity, but when slowly, a little greater than in that Ratio. He also’ says, that the Resistance of Globes moving in the Air is nearly in the duplicate Ratio of their Diameters, and makes the Resistance of the Rod of small Pendulums to be so considerable, as to be more than one third Part of the Resistance of the whole Pendulum. He makes the Resistance of a Pendulum vibrating in Water, to it’s Resistance when vibrating in Air, to be as 535 is to 1$$\frac{1}{2}$$, &c. from all which, joined to other Considerations, we may certainly conclude, that the Resistance of a Pendulum with a globular Bob, when swinging in the Air, causes the Times of the Vibrations to be longer than they would be in vacuo, and that the denser the Air is, the longer will be their Times of description; and the same is true of lenticular Bobs, although in a less Degree, because these Bobs meet with less Resistance from the Air than spherical ones of the same Breadth.

Again, the late Dr Derham, at Numb. 480. of the Philosophical Transactions, for the Year 1736, recites some Experiments he made on Pendulums vibrating in vacuo, and says, that the Arches of Vibrations in Vacuo were larger than in the open Air, or in the Receiver before it was exhausted. That the Enlargement or Diminution of the Arches of Vibrations were constantly proportional to the Quantity of Air, or Rariety or Density of it, which was left in the Receiver of the Air-Pump; and as the Vibrations were larger or shorter, so the Times were accordingly, viz. 2″ in an Hour slower, when the Vibrations were largest, and less and less as the Air was re-admitted, and the Vibrations shortened. But notwithstanding (says he) the Times were slower, as the Vibrations were, he had great Reason to conclude, that the Pendulum moved really quicker in Vacuo than in the Air; because the same Difference or Enlargement of the Vibrations (as $$\frac{1}{5}$$ of an Inch on each side) would cause the Movement instead of 2″ in an Hour to go 6″ or 7″ slower in the same Time, as he found by nice Experiments. The Doctor also says, that the Length of the Rod of a Pendulum, he found by trial was increased by the Summer Heat $$\frac{1}{10}$$ part of an Inch. All this from Dr Derham.

Hence the Resistance of the Air must certainly be a considerable Obstacle to the equable going of a Clock, although this Cause has been mostly disregarded by our most able Clock-Makers; and accordingly, the Irregularities in the going of a Clock are caused principally

1. By the Lengthening of the Pendulum, or its shortening by heat and cold.
2. By the Lengthening the Arch described by it’s Centre of Oscillation, or it’s shortening, caused by the greater or less Cleanness of the Clock, and the greater or less Weight, or Force of the Springs giving Motion to the Clock.
3. By the greater or less Density of the Air, (or other fluid mixed with it) causing the Times of the Vibrations to be increased or lessened, in proportion to that Density.
4. The Pendulum Clocks greater or less Distance from the Centre of the Earth.

It has been commonly said, that Clocks go faster in cold Weather than they do in warm Weather, because the Pendulum in cold Weather describes less Arches, and is shorter than in warm Weather. But this can only be true, when the Pendulum vibrates in Vacuo, viz. free from any external Resistance, or when this Resistance is always unalterable. But in dry hard frosty Weather, when the Air is very dense, and mixed with innumerable small icy Particles, (the Mercury in the Barometer being very high) the Pendulum may be so much resisted as the Times of the Vibrations caused thereby, may exceed the Times arising from the lessening of the small Arches described, and the shortening the Pendulum, both taken together. And then a Clock will go slower in such cold Weather rather than faster. The Dutch, who formerly wintered at Nova Zembla, could not make any of their Clocks go; and Captain Middleton (see the Philosophical Transactions, Numb. 465.) who wintered in the Year 1741, at Prince of Wales’s Fort on Churchill River in Hudson’s Bay, says, the Cold hindered the going of all Watches but one of Mr Graham’s, which went always too slow by 15″.

As the swinging of a Pendulum between the two cycloidal Cheeks removes one of the Obstacles to the equable going of a Clock, so does it’s swinging in Vacuo another; and the other Obstacle arising from the lengthening or shortening of the Pendulum by heat and cold, has been endeavoured to be overcome by making the Pendulum partly of Wood, and partly of Metal, so that the one may be as much contracted by heat, as the other is lengthened, and thereby the Pendulum preserve the same Length in all Vicissitudes of heat and cold. Mr George Graham (in the Philosophical Transactions, Numb. 392.) endeavours to do this by a glass or brass Tube of Mercury for a Pendulum, instead of a Rod and Bob. But I think he fails here, as well as in his Method to know the going of a Clock in an Air of the same Temperature by means of a Thermometer. For a Barometer, and perhaps Hygrometer, should be added in order to obtain the principal End of bell knowing and avoiding the Irregularities of the going of a Clock.

I shall conclude this Article upon Clocks with only mentioning two uncommon Treatises upon them, the one in Latin, and the other in French; in the former of which Treatises is a Contrivance to measure Time equably, by a Motion caused by three Weights, two of which form two perpetual Levers, by the means of which the third Weight is balanced, and these two Levers rest upon the Wheel of the Axis, which Axis shews the Minutes, Seconds, and Hour, by means only of an Index. And what is peculiar in this Invention is, that the Time it measures is not really in itself the same as that of common Clocks, nor is it a Pendulum, but another Thing, whose Motions are equable and regulated, as is that of a Pendulum; and besides, it will move in all Situations, either horizontal, perpendicular, and oblique. And therefore such a Contrivance as this has a Prospect of going well at Sea, and thereby being helps towards finding the Longitude. In the French Treatise you have seventeen Contrivances of Clocks, some of which are diverting enough, having small brass Balls perpetually dropping through Holes, and out again; others descend upon an inclined Plane (though these are not new Contrivances); others are Hour-Glasses, that turn up of themselves as soon as they are run out; another is a celestial Globe, turning about upon the Shoulders of an Atlas; another has the Hours placed horizontally, and another upright, &c. The two Treatises are,

• Mat. Campani de alemenis Horologium solo naturæ motu atque ingenio dimetiens & numerans momenta temperis constantissime æqualia. Romæ, 1677. Quarto.
• Recuil d’outrages curieuse ou Description du cabinet de M. de Servire; a Lyons, 1719. Quarto.

XIII. Of concave Mirrors or Speculums.

1. The Focus of parallel Rays is contained between the 4th and 5th Part of the Diameter of the great Circle of the Sphere of which the Speculum is a Segment; and so,
2. The Focus of one of these spherical Speculums is not a Point, but a small round Solid of such a Breadth.
3. The Diameter of the Aperture of one of these concave spherical Speculums, should not be a Chord of more than 18 Degrees of the Arch of the great Circle whose Segment that Speculum is.
4. Metalline Speculums are not so easy to polish as Glass ones quicksilvered over on the back-side, nor do they reflect so much Light.
5. These Instruments burn best when they are cold.
6. So that when they are exposed to the Meridian Sun in clear frosty Weather their Effects are greatest.
7. In the Focus of any of them directed to the Sun at Noon-Day there is not the least Appearance of a lucid Image, unless it falls upon an opake Body, and yet there is in that Place, and in some of the best of them, a Fire so intensely hot, that Stones are instantly melted by it and turned into Glass.
8. If the Back, of a concave Glass Speculum be covered over with a very white Composition of Tin and Mercury, the Reflection of the Image of the Sun from the Focus will be so strong, that the Eye will not be able to bear it's Brightness.
9. If a Piece of white Paper be put in the Focus of a large Concave of this kind, so as to receive the contracted Image of the Moon, when mining at full on the Meridian in a clear Winter’s Night; you will have so refulgent a Light that the strongest Eyes will not be able to bear it; and yet in the Focus there will be no Heat at all, instead thereof there will be found a very piercing cold.
10. The Heat of the Focus of a Concave-Speculum will be lessened, when acting upon any Thing laid upon a black Body in that Focus.
11. Whether, if the concave Surface of a Speculum were covered with some black polished Substance, the Effect of it’s focal Heat would be lessened.
12. The Rays reflected from the yellow Colour of Gold are vastly refulgent, as has been found by a wooden Concave polished, and nicely covered over with Leaf-Gold, which burned with an incredible Power; as did another covered over with Pieces of yellow Straw very accurately fitted together. Hence the different Colours of a Speculum causes different focal Heat.

The most eminent burning Concaves that we are certain have ever as yet been made, are those of Manfred Septala at Milan; who is said by Scottus, to have made a parabolical Speculum, that would burn almost at the Distance of 15 or 16 Paces; those of the Villet’s at Lyons, whereof one is of Metal, weighing about 400 Pounds; the concave and convex Sides Sides are spherical, the Diameter of the Aperture 43 Inches; that of the Sphere whereof it is the Segment 14 Feet, the focal Diftance 3$$\frac{1}{2}$$ Feet, and the focal Depth is half an Inch. By the focal Heat of this Instrument, Metals, Stone, Bricks, Allies, &c. are melted and turned into Glass. (See our Philosophical Transactions, Numb. 6. and the Paris Diary of the Learned for the Month of December, Anno 1675.) And, Lastly, Those of Mr Tschirnhaus, whose burning Effects are described in the Acla Eruditorum, published at Leipsick for Jan. 1687, The Diameter of the Aperture of this Speculum was almost three Leipsick Yards; it was made of Copper Plates not much exceeding the Back of a common Knife in thickness, and the Focus was two Yards distance from the Speculum. In imitation of these Speculums of Mr Tschirnhaus a certain celebrated Artist at Dresden, made larger burning Concaves of Wood, which produced Effects no less wonderful. Even some have made large Concaves of this sort, by properly placing 30, 40, or more square Pieces of concave or plain Speculums, on the under Surface of a wooden Concave, whose Effects were not much less than if the Surface had been covered all over with them; and after this manner may polyhedrous burning Concaves, either spherical or parabolical, of a vast Size be made.

In the Philosophical Transactions, at Numb. 483. in the Year 1747, there is an Account of a Mirror of one Mr Buffon, a Frenchman, consisting of a great Number of small plain Mirrors, each of about 4 by 3 Inches square, fixed at about 4 of an Inch from one another, upon a large wooden Frame about six Feet square, strengthened with many cross Bars of Wood for the mounting these Mirrors; each of them has three moveable Screws, which the Operator commands from behind, so contrived that the Mirror can be inclined to any Angle in any Direction that meets the Sun; and by this means the solar Image of each Mirror is made to coincide with all the rest. Twenty-four of these Mirrors thus placed, in a few Seconds of Time set fire to a Composition of Pitch and Tow at the Distance of 66 French Feet. Also a sort of Polyhedron, consisting of 168 small Mirrors, or flat Pieces of Glass of six Inches square each, set fire to some Beach Boards at the Distance of 150 Feet. This was done by the Marquis Nicolini. And in another Transaction of the Royal Society, Numb. 489, for the Year 1748, the same Mr Buffon says, he has made a polyhedron Speculum six Feet broad, and as many high, which burns Wood at the Distance of 200 Feet; melts Tin and Lead at the Distance of about 120 Feet, and Silver at 50 Feet; and besides, says, that Heat is not proportional to Light, nor do the Rays come from the Sun in parallel right Lines.

Whether the burning Speculums of Archimedes and Proclus, by which they are said to have burned the Enemy’s Ships at a Distance, (see Zonara’s Annal. Tzetza’s variarum Historiarum, Chiliad. 2. Galen’s Book de Temperamentis; and others of the Antients) were contrived after some such manner, or whether they be not rather fabulous, I leave others to judge. As to myself, I cannot assert, whether it be true or false, that Archimedes and Proclus could have made Speculums to produce such great Effects.

If a Light be set in the Focus of a concave spherical Speculum, the Rays are parallel after Reflection; so that the Light of a Candle placed in the Focus, will be strongly projected to a considerable Distance, whereby one may be enabled even to see to read at the Distance of 30 or 40 Yards.