ANIMADVERSIONS On the first part of the MACHINA COELESTIS Of the Honourable, Learned, and deservedly Famous Astronomer JOHANNES HEVELIUS CONSUL OF DANTZICK; Together with an Explication of some INSTRUMENTS MADE BY ROBERT HOOKE, Professor of Geometry in Gresham College, and Fellow of the Royal Society.  
LONDON, Printed by T. R. for John Martin Printer to the Royal Society, at the Bell in St. Paul's Churchyard. 1674.   

THE CONTENTS.  

THE Reason of the present Animadversions. page 1.  
How far Hevelius has proceeded. That his Instruments do not much exceed Tycho. The bigness, Sights and Divisions, not considerably differing. Tycho not ignorant of his new way of Division. p. 2.  
Proved by several passages out of his Works. p. 3, 4.  
That so great curiosity as Hevelius strives for is needless without the use of Telescopical Sights, the power of the naked eye being limited. That no one part of an Instrument should be more perfect than another. p. 4, 5.  
Hevelius his Letter of 1665. with his opinion of Telescopical Sights. p. 5, 6.  
That if Hevelius could have been prevailed on by the Author to have used Telescope Sights, his Observations might have been 40 times more exact than they are. p. 6, 7.  
That Hevelius his Objections against Telescope Sights are of no validity; but that Sights without Telescopes cannot distinguish a less Angle then half a Minute. p. 7.  
That an Instrument of 3 foot Radius with Telescopes, will do more than one of 3 score foot Raedius with common Sights, the eye being unable to distinguish. This is proved by the undiscernableness of spots in the Moon, and by an Experiment with Lines on a paper, by which a Standard is made of the power of the eye. p. 8.  
That it had been much to be wished that Tycho and Hevelius had, and that Observators for the future would, well consider this. p. 9  
That Altitudes of the Sun and some of the Moon may have been taken to greater exactness, but still short of what may be done with Telescopes. ibid.  
The Author's Engagement for describing an Instrument more perfect in 7 particulars. p. 10.  
A more particular Examination of Hevelius his Instruments, and first of his first Instrument, being a brass Quadrant. Hevelius having a very great aversion to Glass-Sights, used common. p. 11.  
Took great pains in the dividing it himself. p. 12.  
Of which he might have spared almost 80/90, if he had known either the first way of Diagonal Divisions, described in p. 12, 13. or the second way described and exemplified. p. 14, 15.  
Some inconveniences in the Contrivances about his first Instrument. p. 16.  
A Description of his second, third, fourth, fifth, and sixth Instruments, and some of their conveniences and inconveniences noted. p. 16, 17.  
Hevelius wholly rejecting all wooden Instruments, made better of Brass and Iron. p. 18.  
That notwithstanding there may be a good use made of Wood for the material of Instruments. Proved by the Experiment of an Instrument made long since by Sr. Ch. Wren. ibid.  
Hevelius his Reason for rejecting the use of wood-Instruments, not without some exceptions. p. 19  
Animadversions on the Description of three smaller metalline Instruments, one of 24, a second of 18, and a third of 12 inches, and particularly about the new way of Division, which he ascribes to Benedictus Hedreus. p. 20.  
That Hevelius was mistaken in supposing Hedreus his way more capable of Demonstration than Tichoes by Diagonals. p. 21.  
Tycho Brahe's Calculation of the quantity of Angles, made by Diagonals and equidistant parallel Circles. p 21, 22.  
'Tis strange that Tycho and Hevelius should not think of putting the parallel Circles at unequal Distances. ibid.  
How to calculate, and what those unequal Distances are. p. 23.  
Dr. Wallis his Letter to Hevelius about the same Subject, wherein that Doctrine is largely and fully handled. p. 23, 24, 25, 26  
The Diagonal Divisions more easy to be seen then those of Hedreus or Nonnius. Ticho's Description of Nonnius his way of Division. p. 27.  
Hevelius his Description of the way of Hedreus. p. 28.  
That this way of Hedreus is subject to great inequalities, proved by the Divisions on the Plate T. of Hevelius his Book. A practicable way of enlarging the small Divisions. p. 29.  
That Hedreus was not the first inventor of this way of Division, but Pierre Vernier was before him. p. 30.  
A second, third, and fourth Objection against this way of Divisions, drawn from a supposed unequal poise of the Plumb-Rule, caused by its unequal make or dust, or from the unpracticable way of hanging it either on a smaller or bigger Pin or Hole. ibid.  
Hevelius his invention for steadying the Quadrant ingenious, but the convertible Frame more easy for use. p. 31.  
Some Remarks in the Description of his large brass Quadrant, wherewith he took many Meridian Altitudes of the Sun. ibid.  
A new way hinted for making a Table of the fixed Stars, and regulating their places, by the help of a Mural Quadrant, some parts whereof are described as the way of Dividing, and of the Sights, and of poising the Tube and Observator, and of keeping the Tube from bending, etc. p. 32, 33.  
Some Difficulties therein now obviated, and some Objections answered. ibid.  
That this Subject deserves to be better enquired into, and to be promoted by some Prince. p. 34.  
Some Animadversions on the Description of Hevelius his large Quadrant of Brass. That the Instrument is good in its kind, but yet far short of what it might have been, if Glasses had been used for the Sights. ibid.  
How very small Seconds are even upon large Instruments, and how uncertain the Penumbra of the Sun's light is, contrary to the general Principles of Optical Writers, being sometimes bigger, sometimes less, according to the smallness and bigness of the hole through which it is trajected. p. 35.  
The curiosity of this Instrument further expressed by Hevelius, in the multitude of its parts and Contrivances, in the proper Turret for it, in the make and great use of Screws, for moving, fixing and dividing the Quadrant. p. 36.  
Some Objections and Emendations propounded, and a Conclusion on the whole Instrument and Apparatus. p. 36, 37.  
Some Animadversions on the large Sextant, and the way of using it, and on the difficulty acknowledged of taking Stars Distances from the Moon and Sun, and a way promised of doing them with more case. p. 38.  
The seeming difficulty and even impossibility of taking 8 several Distances in the Heavens, without failing one Second, and the reason why 'tis more likely that there could not be a greater certainty then of 4 Minutes in the whole. ibid.  
Hevelius his Letter concerning my Animadversions, and about Telescopical Sights. p. 39, 40, 41.  
An Answer to it. p. 41, 42, 43.  
A Conclusion of the Animadversions. That the learned World is obliged to Hevelius for what he hath done, but would have been more, if he had used other Instruments. p. 43, 44.  
That the Animadvertor hath contrived some hundreds of Instruments, each of very great accurateness for taking Angles, Levels, etc. and a particular Arithmetical Instrument for performing all Operations in Arithmetic, with the greatest ease, swiftness and certainty imaginable. p. 44,45.  
That the Reader may be the more certain of this, the Author describes an Instrument for taking Angles in the Heavens, whose perfection more than common consists, 1. In the manife sting of the Sights. 2. In the Divisions. 3. In the reflective construction of the Sights. 4. In its exact Perpendicularity. 5. In its fixation and motion fit for Observations. 6. In its facility for make; and 7. In its cheapness. p. 45, 46.  
An Explication of the make and singular conveniences of these new Sights. p. 46, 47, 48.  
An Explication of the new way of Dividing, and the great advantages of it above others. p. 48, 49, 50.  
Made more easy by the Explication of the Delineation in the 1, 2, 3, 4, 5, 6, 8, 9, 10, and 11th. Figures, expressing the Frame, hollow Centre, Movable arm, Screw-Frame, and Screw for the Divisions. The Obliquity of it to the Plain of the Quadrant, and the reason thereof. p. 51, 52.  
The way of certainly aetermining the Obliquity, and the resolving the whole Quadrant thereby into one grand Diagonal, and the magnifying thereof in a duple, triple, decuple, etc. Proportion. p. 53.  
Then follows a more particular Description of the Screw-Frame, its Collars, Centres, Screws, Handles, Indices, Opinion, Divisions, etc. p. 53, 54.  
How by these Indices is pointed out the Measure of the angle, in Degrees, Minutes, Seconds, etc. p. 55.  
The great advantage of these new ways of ordering Sights taken notice of. ibid.  
And the whole Contrivance more particularly described. p. 56.  
And explained by a Delineation, and the manner how they are applicable to a Quadrant or other Instrument. p. 57  
How they are made use of for taking an angle bigger than a Quadrant, is farther described, and made more intelligible by a Delineation. p. 58.  
The way of adjusting the two fixed Sights, so as to look forwards and backwards exactly in a right Line, and how to adjust and fix the Sight-Threads in the Tubes, with the reason thereof. p. 59, 60.  
A Description of the Water-Level, for setting the Instrument exactly Horizontal. Some Difficulties, and the way of preventing them proposed. p. 61, 62.  
This Instrument farther explained by a Delineation, and the reason of its accurateness manifested. p. 63.  
Some Difficulties about the make of the Glasses for these Levels, and some Expedients propounded, together with other ways and forms of Levels. p. 64, 65.  
After the Difficulties of Observations made the old ways are taken notice of, follows the Description of a new Method of moving and fixing Instruments for Observations, so as to prevent and obviate them. p. 66, 67, 68  
This is made more plain by a Delineation and Explication. p. 69.  
When the Circular Pendulum was first invented and published. p. 69, 70.  
Here by the way is published a Description of Wheel work, which may be called the perfection of Wheelwork, having the perfectest Idea that toothed Wheelwork is capable of, performing the same effect as if the Wheel and Opinion had an indefinite number of Teeth. p. 70, 71.  
A farther Explanation of the Pole or Conical hole of the axis. p. 72.  
A Description of the Frame for keeping the Instrument in its Perpendicularity, and yet always in the azimuth of the celestial Object, with a Digression of the great use of this Principle in Dialling, equalling Time, Clockwork, etc. p. 73.  
The way of finding an exact right angle or Quadrant, more particularly described and explained. An Objection about the inequality of the Divisions answered. p. 74, 75.  
Some Uses of this Instrument hinted: 1. For measuring the Refraction of the air. 2. For regulating the place of the sixth Stars. 3. Of the Planets. 4. For stating the Latitude of places. 5. For examining the influence of the Planets on the Earth. 6. For measuring a Degree, which was the cause of its Contrivance. 7. For measuring seen Distances. 8. For taking the Diameters of the Sun, Moon and Planets. p. 77.  
Where by the By are mentioned two other Instruments; one for taking Diameters to Seconds; and a second for looking on the Body of the Sun, without harming the eyes. p. 78.  
A ninth Use for Levelling, etc. with a short Conclusion. ibid.    

  

SOME ANIMADVERSIONS On the first Part of HEVELIUS His MACHINA COELESTIS, etc.  
HAVING lately perused a Discourse of Hevelius, newly published, entitled, Johannis Hevelii Machina Coelestis, pars prior Organographiam sive instrumentorum Astronomicorum omnium quibus Autor hactenus sidera rimatus & dimensus est accuratam delineationem & descriptionem, plurimis Iconibus aeri incisis illustratam & exornatam exhibens, etc. and finding it a Discourse about practical and mechanical Knowledge, and of that kind wherein Geometry seems to be more then ordinarily concerned; I thought it might not be ungrateful to my Auditory, (nor improper to the Subject of Sr. JOHN CUTLER ' s Lecture, which is partly Mechanical and partly Physical) to consider a little the Contents thereof: And somewhat the rather too, because having heretofore communicated to him somewhat of this Subject, which I had occasion to read in this place in one of my former CUTLERIAN Lectures, I find he hath made some Animadversions and reflections thereupon.  
I find then that this excellent Person hath been for the most part exceedingly circumspect, to find out the inconveniences and difficulties that do accrue to the best Observators, even with the best instruments, and has not been less industrious to find out ways to obviate and overcome them; In the doing of which, he seems not to have spared either for labour and vigilancy, or for any cost and charges that might effect his purpose, for which he hath highly merited the esteem of all such as are lovers of that Science: But yet if he had prosecuted that way of improving Astronomical instruments, which I long since communicated to him, I am of opinion he would have done himself and the learned World a much greater piece of service, by saving himself more than 1/10 of the charge and trouble, and by publishing a Catalogue ten times more accurate. For though I doubt not in the least but that he hath by his own extraordinary diligence, care and cost, corrected several mistakes and errors committed by the assistants of the Noble Tycho: yet I am not satisfied that his Instruments are capable of making Observations more accurately than those of Tycho, though 'tis possible they may do it with somewhat less trouble and inconvenience. For first, I find that those of Tycho were as large as those of Hevelius, and consequently were capable of as accurate and minute divisions, and of as long and convenient Sights. Secondly, I find that the Sights made use of by Hevelius are the very same, at least not at all materially differing from those of Tycho, being only naked Sights, made by a slit and edge, serving only to regulate the direction of the naked eye, but no ways capable of assisting the eye to distinguish more accurately the object. Thirdly, I find that though the way of Division made use of by Hevelius, be a very ingenious invention, and that which is Geometrically true and certain, yet if we consider the great difficulty there is in Mechanically performing it, we shall find it not much preferable, if altogether as good as that of Tycho. And 'tis plain enough that Tycho himself was not ignorant of it, though his particular reasons why he made no more use of it, we certainly know not: 'Tis very probable, because he thought it not altogether so accurate, as that he did make use of. For somewhat to this purpose he says himself, in the second Book of his Observations of the Comet of 1577. pag. 461. Hanc graduum in singula minuta, meaning the Division by Diagonal Lines; & etiam horum in dena scrupula secunda subdivisionem in omnibus meis machinis Astronomicis usurpo, eo quod illam multis ab hinc annis exquisitissimam expertus sum. Licet enim ejus demonstratio in Rectilineis parallelogrammis propriè conveniat, nihilominus arcualibus etiam in tam exili interstitio quod à recta linea insensibiliter differt, citra omne erroris vestigium convenienter applicatur. 'Tis true, Ticho's Objection against this way of Division by Diagonals is material, as to a Geometrical accurateness, but his Answer to it is altogether as material, that though it be not exactly true, yet it doth insensibiliter differre, and so long as the error is not discovered by sense, there can be no error committed in observation; and indeed the whole matter both one way and the other is insignificant, and but a vain curiosity to endeavour to divide an instrument into seconds, or parts smaller than a minute, for I shall by and by show that the eye can hardly distinguish minutes in the object: But were such niceness of Division of any use, 'tis easily enough to be done to Mathematical truth; for as I shall anon show, there is a certain distance of each of the parallel Circles, which being given, the strait Diagonal Lines will divide the degree, by the intersection with those parallel Circles, into exactly equal parts, which would have better answered Ticho's Objection, had he known it, which I wonder, I confess, how he could oversee, since he seems to have spent many thoughts on the matter; but this only by the By, because I shall speak more at large of it afterwards. But he proceeds to this other way of Divisions, which he, as well as Hevelius, ascribes to Nonnius, whereas the other that he approves of came first from England, as it appears by a passage in another Book of his, where he discourses somewhat of the same Subject.  
Altera Divisio ad clarissimi Mathematici Petri Nonnii in Libello de crepusculis propositione tertia imitationem per plures quadrantis arcus introrsum descriptos, & diversimodè subdivisos procedit; etsi autem in hac ipsa apprime ingeniosa Nonnii inventione aliquid Auctuarii loco expeditius à nobis additum est, ita ut exterior arcus in plurimas partiunculas dividatur, neque is ordo aut numerus arcuum sese introrsum concomitantium quem ille praefinivit sed multo expeditior & perfectior observetur; (I am apt to think he knew this very way, and here hints it:) Tamen quia haec subtilitas cum ad praxin deventum est plus habeat laboris quam fructûs, neque id in recessu praestet quod prima fronte pollicetur, ut alibi plenius ostendemus, idcirco apud nos dudum in usu esse desiit. From which words, and also from what he says in his first Book of the new Star in 1572. pag. 671. speaking of the comparison between these two ways of Divisions, to wit, Sit cujuscunque velit ingeniosa certe & apprime utilis est distributio, quam & ego postea arcualibus graduum divisionibus in quadrantibus sextantibus & armillis, non inconcinnè aut infrugiferè applicui. Licet enim demonstratio ejus in solis rectilineis superficiebus ad unguem se habeat; tamen cum quinorum vel denorum minutorum spatium in circumferentiis majusculis à rectilineo insensibiliter differat, hic quoque ejus usus satis commodus & ratus esse poterit, multoque Nonniana plurimorum arcuum intricata & difficili subdivisione expeditior aptiorque deprehenditur. From his Discourse I say in these two places, and from several others dispersed up and down his Works, which 'twould be too long now to quote, 'tis evident that Tycho was not ignorant of this way of Sub-division, so much applauded by Hevelius, invented by Petrus Nonnius, and promoted by Tycho himself; and yet we see he preferred that way of Diagonals, first made use of in England by the most skilful Mathematician Richard Cantzler, before it, rejecting the one and making use of the other in all his Instruments. But either of them will do well enough if the Divisions be done with great circumspection and care, and instruments of the size of those larger ones both of Hevelius and Tycho, are capable of Divisions ten times more accurate than are needful for common Sights, be they never so long, without making use of either Tichoes or Hevelius' way of Division, the eye being unable to distinguish a smaller Angle. To what purpose therefore is it to make the Divisions so fine, or any one part of the instrument or observation more accurate than another? since the power of distinguishing by the naked eye is that which bounds and limits all the other niceness, and whatever part is more curious than that can equalise, is of no significancy. For instance, in taking the altitude of a Star, it would be but labour lost to distinguish by the Diagonals, or otherwise to Seconds, whilst in the mean time you are not certain that the Plumb-line is true to a minute, or whilst you are not able to direct the Ruler, bearing the Sights to a greater certainty then to that of a minute. And the like might be said of the extraordinary curiosity in any two parts, and the failure in any third, that is essential to an observation; as fruitless it is to calculate to seconds, when the observations are not true to minutes, or to be certain by the Sights and Divisions to seconds, and uncertain in the Plumb-line to minutes.  
There is therefore one thing in Hevelius his Instruments, that though they be never so large, never so accurately divided, of never so choice and convenient materials, and never so tractable for use, and never so skilfully and industriously used, will notwithstanding make them all equal as to use, with one of about two or three foot radius of metal with Ticho's Sights and Diagonal Divisions, which is occasioned by the limited power of distinguishing by the naked eye.  
Something to this purpose I communicated to Hevelius in the year 65. and hoped that I might have thereby somewhat assisted him in his great and laborious Work, first by easing the eye, and next by making it capable of distinguishing more exactly, I having hinted to him the way how to reform and obviate that inconvenience by Telescopical or Perspective Sights, as also the way of making instruments of much less bulk, to do ten times more than 'twas possible to do with the largest instruments made the common way. In answer to which he returns me this Discourse, in a Letter to the Royal Society, in the year 65.  
MODUS ille observandi per Telescopia adminiculo Sextantis vel Quadrantis, videtur mihi vix adeò tutus, quam vulgaris, si pinnacidia rectè ac justè sint affixa. Haec enim sunt immobilia; Telescopia verò nullâ ratione adeò firmiter affigi possunt ut loco haud dimoveantur; etiamsi omni diligentiâ juxta methodum descriptum per totum Horizontem experiundo sint semel collocata. Adhuc quam arduum fit, eâ ratione verum eorum locum indagare, satis superque expertus sum; sic ut vix videam, an alicui circa restitutionem Fixarum Planetarumque adminiculo esse possint; in majoribus scilicet illis distantiis capiendis: In minoribus, largior, posse aliquid praestari; sed an Instrumenta, unius Spithamae radio instructa, elaborari possint multò exactius, quam optima quoevis, vulgares Dioptras habentia, licet 60 pedum radio elaborata, nollem adhuc asseverare. Multa namque in Theoriâ videntur certissima, quae in praxi satis longe nonnunquam à vero recedunt. Si quis mihi certas observationes quarundam distantiarum & quidem Fixarum, circa Eclipticam & Aequatorem existentium, illis ipsis Instrumentis, Dioptris Telescopicis-instructis habitas exhiberet: (utpote distantiam Lucidae ♈ à Palilicio; Palilicii à Polluce; Pollucis à Regulo; Reguli à Spicâ ♍; Spicae ♍ à Boreal. sinist. manus Serpentarii; Boreal. sinist. manus Serpentarii ab Aquilâ; Aquila à Marcab; & Marcab à Lucidâ Arietis) vellem protinus de rei illius certitudine & meum quale quale judicium far; sed antequam eas observationes obtineam, judicium suspendo. Interea utiq. fateor; si quis adminiculo minoris cujusdam Instrumenti observationes corporum Coelestium peragere potest, multò sane illum esse feliciorem, variis de causis, eo, qui per majora id praestare allaborat. Rationes dividendi Instrumenta, diversae quidem mihi probè cognitae sunt; easque etiam in usum transtuli; num autem sint eaedem quas Clarissimus Dominus Hookius novit, ac invenit, me prorsus latet: Si illi non adversum est, rogo, ut praecipuas communicet, ego ut meas intelligat rursùs studebo.  
Since which time I have not sent any other description of instruments, save that of the manner of making and using a Tube for a 60 foot Glass, which I am much pleased to find he makes use of, and should gladly have communicated any thing further, if I had not found they were esteemed insignificant. It did much trouble me, I confess, that I could not prevail with him to make use of Telescopical Sights at least, since with less trouble he would have afforded the World Observations, and a Catalogue of the Stars, ten times more exact. And I am the more sorry to find that he hath proceeded to finish his Machina Coelestis, by instruments not more accurate than those of Tycho, and that he still remains in the same opinion of Telescopical Sights, and other improvements of instruments. For pag. 293. of this first Part of his Machina Coelestis, speaking concerning Sights, he says, Possibly some may wonder that I do not make use of Telescopical Sights, since they are by some accounted better and more accurate, insomuch that there is one in the World hath proceeded so far, as to suppose Telescopical Sights to be ten, twenty, thirty, nay forty times more accurate than the common Sights; and that 'tis possible to make an instrument of a Span Radius to do more with Telescopical Sights, than an instrument of 60 foot with the common Sights. 'Twould be a thing of much moment could it be done, and not to be valued by money, but many things do seem true in the Theory, which do not answer upon Experience. You may perceive by comparing this slender Resutation with his Letter before, who he means by the Assertor of Telescopical Sights. But I am troubled he should think them so slight as not to deserve one trial in seven years' time, especially since by explaining the manner of making use of them much in the same sense with that which I sent him, he seems to have understood enough of the way to have made use of it if he would. As to his Objection, That the Glasses are apt to be broke, and the Pins or Threads are apt to be bend and broke, there is not the least colour for it, for they cannot without much labour and design be broken or put out of order, but if they were, it might as well be said, that the Plumb-line of any of his instruments may be broken, or his Sights bended, and the like, and therefore those instruments were not to be used. But these Objections I shall not urge against his instruments, nor a great many other I could produce of lesser moment, but only this one which is very fundamental, and cannot any ways be helped but by the help of Glasses, and that is, 'Tis impossible with Sights made after Tichoes or Hevelius his way, to distinguish any distance in the Heavens less than half a minute, or thirty Seconds, and hardly one of a hundred can distinguish a minute.  
And this being proved, what will become of all the machinations and contrivances for greater instruments, to show the Divisions of single or double Seconds? May not single minutes, nay half minutes, by the help of Diagonal Divisions, be sufficiently distinguished in an instrument of three foot Radius? What need is there then of all the other cumber? Certainly any one that will but try with the one and the other instrument, will find himself able to do as much with an instrument of three foot, as with one of threescore, since the eye cannot distinguish a less Angle, at lest none that I have yet met with hitherto. Who is there that by his bare eye can distinguish any of the Telescopical spots in the Moon, though some of them are above a minute in Diameter? As for instance, Who can see Mount Sinai, so called by Hevelius, which is a bright spot in a dark field, and consequently must appear near two minutes in Diameter to the naked eye? Or who can see the Palus Mareotis, or the Lacus niger, which are two dark spots in light fields, and each more than a minute in Diameter? Now if the eye cannot distinguish a smaller object than appears within the angle of half a minute, 'tis not possible to make any observation more accurate, be the instrument never so large.  
Now that any one may presently satisfy himself of the truth of what I assert, concerning the limited power of the naked eye, as to the distinguishing of Angles; Let him take a sheet of white Paper, and thereon draw two parallel Lines, as OO, and P P, in the 28th. Figure, at four or five inches distance, then draw as many other small lines between them at right angles to them, and parallel one with another, as he thinks convenient, as aa, bb, cc, dd, ee, ff, gg, hh, two, etc. and let them be drawn distant from each other an inch, then let him alternately blacken or shadow the spaces between them, as between aa and bb, between cc and dd, between ee and ff, between gg and hh, between two and kk, between ll and mm, etc. leaving the other alternately white, then let him expose this Paper against a Wall open to the light, and if it may be so that the Sun may shine on it, and removing himself backwards for the space of 287⅓ feet, let him try whether he can distinguish it, and number the dark and light spaces, and if his eyes be so good that he can, then let him still go further backwards and backwards from the same, till he finds his eyes unable any longer to distinguish those Divisions, there let him make a stand, and measure the distance from his eye to the aforesaid Paper, and try by calculation under what Angle each of those black and white spaces appears to his eye, for by that means it will be manifest how small an Angle his eye is capable of distinguishing, and beyond which it cannot reach: Which being once known, he hath a Standard, by which he is able to limit the bigness and exactness of his Instruments, if he make use of common Sights, beyond which all magnitude and curiosity is not only useless, but of much detriment upon many accounts.  
This is that Consideration which I could wish had occurred both to Tycho Brahe and to Hevelius, especially to the latter, who hath so earnestly endeavoured to outdo the former, and for the accomplishment thereof, seems to have spared no charge, labour, or endeavour he was able to expend. I hope at least that this public notice will for the future engage all such as shall attempt this Work, to be as solicitous about assisting the Eye in the discovery of the parts of the Object, as of distinguishing the Divisions of the Instrument, for the doing of the one without being able to reach the other, will avail nothing.  
Those therefore that desire or need Instruments to make Observations to Seconds, must take another course then any that I know yet described. 'Tis true indeed, That Altitudes of the Sun may be taken, with the Sights commonly used for that purpose, to what accurateness is desired, if the Instrument be large enough, because the Image of the Sun being transmitted by the upper Sight through a small round hole, is represented within a Circle upon the lower Sight, and by means of the eyes approaching near that Sight, 'tis possible by Instruments large enough, to arrive at the accurateness of a Second, in Observations made of that kind. And somewhat of this may be done also by the Moon, when very bright and clear, but in all the other celestial Bodies it has never yet been done.  
But then if we compare even this way with that of Telescopes, caeteris paribus, we shall find it much short, both as to clearness and distinctness, and therefore even here also Telescopical Sights are to be preferred, as I shall sufficiently manifest hereafter more at large, when I come to describe my own Instruments for this purpose; for I doubt not but to make it sufficiently plain, That by the help of an Instrument I have contrived, of three foot Radius, I will be able to make all Observations whatsoever, ten times more accurate, excepting those of the Sun, than any one can make with the largest Instrument, described either by Tycho or Hevelius, and to manage the same with a quarter the trouble, clutter, and Apparatus necessary to either of theirs, and to make the Divisions as accurate and sensible as can be desired.  
For the doing of which, I will show, First, How to make the Plain of the Instrument, that it shall not be subject to bending or warping, and yet be so light as to be easily manageable. Secondly, How to make the Divisions on that Instrument, so as to distinguish certainly and exactly to Seconds, without any trouble, or wearying the sight. Thirdly, I will show how to make the Sights of that Instrument, so as to distinguish the parts of the Object to Seconds, if need be, even by those who cannot distinguish to Minutes with common Sights, certainly, and without fallacy or error. Fourthly, How to make the Sights, so as to see two Objects, though never so far distant, with one glance of the eye. And Fifthly, I will show how to adjust the Perpendicular, so as to set it exactly upright and plain to a Second, so that if it meets with a diligent, accurate, and experienced Observator, it will serve to make as curious Observations as are hitherto desirable. Sixthly, I will show a way how to fix this Instrument, either for taking Altitudes or Azimiths, so as to be manageable with the least trouble imaginable, for Observations of that kind, and to be always steady and fixed in any Perpendicular posture, to whatever Azimith it be applied. Seventhly, I will explain an exact way for fixing the Instrument, so as to take the Distances of any two Stars, or celestial Object, and several other contrivances of the like nature. But of each of these hereafter, after I have examined over the several particulars mentioned by Hevelius, in his Descriptions of the Instruments and Contrivances made use of by himself.  
To pass by then his long Preface, and the Discourse of Instruments in general, which he hath premised in the first Chapter; I shall proceed to an examination of those Instruments of his own, which he doth more fully and particularly describe.  
The first of which kind I find to be a Quadrant of Brass, which he describes in the second Chapter, and begins with that first, as being an Instrument which he least esteemed, and which at length he made no use of, though for many Reasons I think of a quantity big enough, to be as good, nay better, than any he made use of. But of that anon.  
This Brass Quadrant was of three foot Radius, and so well fitted with cross Bars, and strengthened, that it was not subject to warp or bend; it had also a convenient Pedestal, and was made easy to be removed from place to place; it was suspended by a Cylinder placed on the backside, in the Centre of Gravity of the Quadrant, and could by this means more easily be moved to and fro to take any Altitude, than that way of Tichoes, who fixed his Cylinder at the upper corner: But it hath this of inconvenience that Tichoes hath not, namely, That the Plumb-Line or Perpendicular will be longer before it settle, and the Instrument somewhat more apt to warp. The Sights of it are the same with that of Tycho, and indeed the best of Common Sights, now commonly every where made use of in Instruments of that bigness, but far inferior to those which are made of Glasses, as I shall afterwards prove.  
The way of Sights which he describes, pag. 98. for taking the Altitude of the Sun, is very good, but yet far inferior to one fitted with the Object-Glass of a Telescope, though he had omitted the Tube, for he might thereby have enlarged the hole of the upper Sight to what bigness he pleased, and consequently have made the image of the Sun as bright as it should be thought convenient, and that without any manner of Penumbra, if the lower Sight were placed at the due distance of the Focus of that Object-Glass. And therefore I do wonder at his carefulness to inform his Reader aright, for fear he should understand a Telescope by the Tube he made use of, to keep off the adventitious light from the lower Sight, saying, pag. 99 Per Tubum autem mi Lector non intelligo Telescopium aliquod lentibus instructum, sed plane nudum ex charta constructum Tubulum, as if he had some dread of making use of Glasses in any of his Sights. Whether it were, that he supposed Glasses to have some hidden, un-intelligible, and mysterious way of representing the Object, or whether from their fragility, or from their uncertain refraction, or from a supposed impossibility of fixing them to the Sights, or whether from some other mysterious cause, which I am not able to think of or imagine, I cannot tell. Sure I am, that none of these I have named, are any thing at all considerable Objections against their use, and I have been so fully satisfied of the exceeding great use, nay absolute necessity of them in curious and exact Observations, that I do assure him there is not, nor can be any considerable Objection against them, which cannot easily be answered, nor any inconvenience, which cannot with ease be obviated and rectified; of which I shall say more hereafter.  
The Divisions of it were made wholly by himself, with extraordinary labour and curiosity, insomuch that he says, he could not only distinguish each minute of a Degree, but almost every quarter of a minute, sufficiently accurate for his Common Sights, if he could have only distinguished every half minute, and indeed much more than most men's eyes are able to reach. He seems to have been at infinite trouble and pains, to perform the Divisions made by the help of Diagonals, cutting parallel Circles, a way made use of by Tycho, and now so commonly known, that I think I need not spend time in the Explication thereof; only I must take notice, That whereas he supposes these Circles to be equally distant, he ought to have placed their Distances according to the Proportions of the differences of the Secants of some ten minutes, next successively following one another in some Degree of the Quadrant, which is easy to determine, from the Distance of the two extreme or bounding Circles; of which more hereafter.  
Now though the Circles ought not according to the strict Rules of Geometry, to be equally distant from each other, as Hevelius seems to suppose, yet I confess, unless the space wherein these Circles lie be very large, and the parts of a Degree that are to be distinguished, very small, there is no necessity of so curiously distinguishing those unequal Distances, but they may serve well enough for use, if they be taken equal, as Hevelius supposes, and indeed much more accurate, then 'tis possible to distinguish the Object by the bare eye; and therefore I shall not need to insist upon the further Explication thereof, especially because when I come to show a more accurate way of Sights, I shall also show a much more accurate way of Division, then either of those two of Tycho Brahe, or this set down by Hevelius, which is much the same with one of those which was 100 years since made use of by Tycho, and described, and is by him attributed to an English Mathematicians.  
But because this industrious and careful Person put himself to the trouble, of making and examining the Divisions himself, I could heartily have wished he had thought upon some such way as this, which I here describe, and call a Compendium of Diagonal Divisions, it being a way, whereby as ●9/90 of the trouble is saved, in performing the manual operation thereof, so I judge it to be much more certain, exact and plain, than the other way of Diagonals. My Reason for the first is plain, The Division of one Degree serving for the whole ninety: And my Reasons for the second are, First, Because it is much plainer to be distinguished, then by the help of the edge of a Ruler, lying over the Diagonals, one being able to see but one part of the Diagonal. And Secondly, I think it much better than a small fiducial Thread, which is very apt to be bended and broken, if it lies close to the Superficies of the Diagonal, and if it lies at a distance, a skew glance of the eye will much alter the seeming intersection of the Diagonals, which in this way are both prevented. The way then in short is nothing but this; Take a thin piece of clear Looking-glass Plate, well smoothed and polished on bothsides, and large enough one way to cover the whole breadth of the Rim of the Quadrant, on which the Diagonals were to be made, and the other way to cover two or three Degrees, (this I do the bigger, that the sides of the Arm may not shadow or darken the Divisions and numberings.) Suppose a a a a in the 29th. Figure, Plate 2. to represent such a Plate, upon this Plate describe with great care a Degree of the Quadrant you would have divided, and complete it with all its parallel Circles and Diagonals, as you would have done any one Degree upon the Quadrant, and if the Rim of the Quadrant be very broad in proportion to its Radius, you may by the Table of natural Secants or Tangents, set the parallels at their due Distances, but if the Rim be narrow, 'twill be sufficiently accurate to make their Distances equal. These Divisions must be done with Compasses, pointed with small Diamant Points, in the manner of those wherewith Glasiers cut their Glass. The Glass being thus divided and lined, number the Diagonals, and place it in the Frame of the Ruler, with the lined side next the Quadrant, so that moving it to and fro, the side of the Glass may immediately touch the Brass Rim of the Quadrant. This Brass Rim must be divided into 90 equal parts or Degrees▪ and at each Division strait Lines drawn from the Circumference towards the Centre, the whole breadth of the Limb, (at least as much as is made use of for the Glass-Plate, for the breadth of the Diagonals) the Frame to carry this Plate is a convenient Cavity, left in the movable Arm of the Quadrant, the whole manner of which will be better understood by the Delineation thereof, to which I shall therefore refer the Reader. The Distances of the parallel Circle; if unequal, may be easily set down true, according to the numbers of natural Tangents or Secants, with a pair of Compasses, contrived like Beam-Compasses, but having its Points to be set at any distance, desired by the help of a Screw, moving upon one side of the Beam, which I may have occasion to describe elsewhere more properly, and therefore will here omit it.  
Next, If this way had not pleased, I could have wished he had known this following, which is altogether as easy, and as Geometrically true, which I have contrived, and have made small Instruments thereby to show very minute Divisions, very easily and very plainly. I strike then upon the Limb of the Quadrant I would divide, being first made exceeding smooth and plain, a Circu●e very fine, and as lightly as possibly I can, so it be but discernible, and by the help of a very large Quadrantal Dividing Plate of ten foot Radius, I divide the said Quadrant in the faint Circle abovementioned, into 90 parts or Degrees, then by a peculiar contrivance of a very curious Point that strikes with a Spring, which I describe in another Discourse, the said Degrees are marked upon the Plate by curious, small, round and deep holes, these are by another Line without it, which is divided and figured the Common way, distinguished and numbered by Figures, according to the Common manner. Then for the sub-Divisions, I make a small Holdfast by a Screw, which is fixed on to the movable Arm of the Quadrant, this serves to hold the end of a Diagonal Hair, the other end of which is strained over the Supplementary Degree, till it lieth directly over some prickt-Hole of the curious Divisions, on the Limb of the Quadrant, this gives me the sub-Divisions of the Quadrant, to what accurateness I desire. The Supplementary Degree is a Degree of a very large Circle, put on upon a small Rule, fixed on to the side of the movable Arm, whose Magnitude and Distance is found by this Proportion, as the Distance between the end of the small Holdfast and the pointed Circle, is to the Radius of that Circle, so make the Distance between the said End and the Supplementary Circle to the Radius of that Circle. This will be more plain by a Scheme.  
Let aaa in the 30th. Figure represent a Quadrant, bbb a very fine Circle, struck on the Limb of the Quadrant, from the Centre l, which by a large Quadrant of 10 foot Radius, I divide into Degrees, and by a springing Point strike so many small Points, and number them to 90. beginning at m, and numbering towards i. Let dd represent the movable Arm, cc the holdfast, fixed upon the side of that Arm, which by a small Screw pincheth and holds fast a very fine Hair at k, ee the small Ruler fixed at right Angles, with the Line lkf, in this Line (through the Points l and k) I take a Point, as f, and through f I strike a part of a Circle fg, whose Centre is somewhere in the Line fkl produced, which I find by resolving this Proportion, as ki is to li, so will kf be to the Radius of the Supplementary Circle fg, which will fall somewhere in fkl produced, towards l, then take a Degree of that Circle, which will extend from f to g, and divide it into as minute Divisions as are necessary, and number them from f to g. Now to find what Angle the Sight dd maketh with the Sight mm, I strain the Hair hk, till I find it lie over the next Division Point towards the right hand, and observe in the Ruler ee, what part of a Degree is there marked, and on the Circle bbb, what Degree is marked, the sum of both which gives me the true Measure of the Angle ddlm. But this only by the By, and I will not now further enlarge on the Explication thereof, designing it for another Discourse, where I shall describe various, Mechanical and Practical ways, of accurately dividing Lines, into any assignable number of equal or proportional parts.  
To proceed then where I left off, to the examination of the Instruments of Hevelius, I find that together with the Brass Quadrant I was speaking of, he describes two Contrivances about it; The first is, How to set it presently to an upright, without the trouble of turning the Screws in the Pedestal, which is plain enough, and so much the better; but it hath this of inconvenience, that it must be altered for every Azimith, which is a very great one, and which by another way altogether as easy and plain, may be avoided; of which more hereafter.  
Another Contrivance about this Instrument, is a small Screw, for moving it and keeping it steady in any posture in the same Azimith, which is convenient enough, but will not perform what he afterwards supposes it capable of, as I shall afterwards show.  
The second Instrument, which in the third Chapter, pag. 102, 103, etc. 108. he describes, is a Sextant of Brass, of three foot Radius, carefully made, and divided with the same care and after the same way as the former. The Sights also are much the same, only whereas in the Quadrant he makes use of a Plate, with parallel edges for the Sight that is at the centre, and furthest from the eye; in this he makes use of a Cylinder, which way also Tycho made use of 100 years ago, and hath been ever since made use of. The other Sights next the eye are the same with the former: There is nothing singular in the Pedestal, nor in the Ball and Socket, only 'tis somewhat bigger than ordinary. His way of moving and fixing the Rule of it is convenient enough, and the same with his Instrument for moving and fixing his Quadrant, but 'tis not capable of performing what he promiseth for it.  
The third Instrument, which in the fourth Chapter he describes, is a Sextant of Iron, of four foot Radius, to be managed only by one Observator, by putting the Centre next the eye. The whole Instrument is little differing from the former, save only that the Cylinder at the Centre which is here next the eye, is covered with another hollow Cylinder, which is voluble and convertible about the former, and carries two small Slits for the Sights, which performeth the same as the other Sights, but nothing more, and as the Author himself affirms, is not so accurate for use as the other Sextant, where there are two Observators, and therefore was seldom made use of by him. But I shall anon show a way by which one Observator alone shall be able to take any Distance to a Semicircle with much more accurateness and conveniency than any two Observators can; and therefore will be an Instrument of the best use for Astronomical and Nautical affairs, for the perfecting both which I design it.  
The fourth Instrument, which in the fifth Chapter, from pag. 114, etc. to 123. he describes, is a Quadrant of six foot Radius, whose Frame was all made of dry Oak, but the Limb, Sights, Sockets, etc. were made of Brass, divided so as to see every quarter of a Minute distinctly, the Sights the same as in the first Quadrant, and the way of suspending it not much differing, save only, whereas in the former the Pedestal was movable, in this it is fixed, which is much better. And the Instrument is kept in an Aequilibrium, by the help of counterpoises hung at the end of a string, and cast over a Poultry, as is more visible by his Description. But this (as all other wooden Instruments do) he found to shrink and warp, and consequently to lose its exactness, and therefore he made little or no use thereof, but laid it aside, and made himself better of Brass.  
The fifth Instrument described in the sixth Chapter, from pag. 123. to 132. is a Sextant of Wood of six foot Radius, made in all particulars like the former Sextant of Brass of three foot; nor has it any other contrivance about it considerable, save only a rest made to slip up and down for the Observators to rest their Elbows upon. But this Instrument also he found to be vitiated by the shrinking and warping of the Wood, and therefore he laid that by also, and seldom made use of it.  
The sixth Instrument is a large Octant of Wood of eight foot Radius; this is made exactly according to the Form of Ticho's Octant, and serves for taking any Distance not exceeding 45 degrees. The Sights near the eye are made exactly as the former, but movable, so as to slip upon the Limbs of the Octant; the Divisions of it are performed by Diagonals as before, and gives a greater niceness of Division than the Eye is capable of distinguishing in the Object, and therefore of little use.  
And thus far the Author proceeded in Ticho's way. But finding these Instruments which were made for the most part of Wood to be subject to faileur, he aspired to get better Instruments made all of Brass or Iron, and wholly laid aside the rest as altogether useless. And I cannot but very much approve of his Judgement in so doing, for certainly caeteris paribus Instruments, well made of Brass or Iron, are much to be preferred before the best of Wood But yet neither are all manner of Wooden Instruments to be rejected; nor are all sorts of Metalline Instruments free from error, though 'tis confessed, if they be made and used with skill, they suffer not any considerable or sensible variation. First, I say, Wooden Instruments may be so contrived as very near to equalise those of Metal, the Joints and Plates for Divisions only being made of Metal, they being very easy to be rectified before, and examined after every time of using. Such a one was contrived by Sir Christopher Wren, being two square Wooden Tubes or Telescopes, joined together at the end next the Object by a Joint of Brass, and the Angle made by the opening of them, measured by a strait Rule equal to half the Radius, divided by Diagonals into 5000 equal parts, which will by the help of a Table of natural Signs or Subtenses, show the parts in Degrees, Minutes, and Seconds, of which I think I acquainted Hevelius some years since. Next Brass and Metalline Instruments, if they be not very carefully fortified against it, are more apt to bend then even those of Wood And the best way I have found to secure them true and plain in all postures, is to lay them on a Table or Frame of Wood, well fortified underneath against bending, and by the help of small Screws in several parts of the Instrument to adjust it upon that Frame; the whole Table and Quadrant being so counterpoised, as to be easily movable and fixed in any posture. But Hevelius is pleased, as I said before, wholly to lay aside all manner of Wooden Instruments as useless, and to endeavour the obtaining of Instruments of Brass or Iron. Name (says he pag. 136.) cum longâ experientiâ probe tandem didicerim, multo securius esse ex solido prorsus metal●o obtinere Instrumenta, tum quo majora & ampliora eo esse accuratiora & absolutiora, adhaec prioribus admodum Tichonicum constructis plurima deesse quibus ditari merito deberent, & quod iisdem de causis omnino necessum sit, ut parte corrigerentur & meliorentur, tam quà eorum materiam fructuram commotionem facilitandam divisionem quam alia diversa subsidia & adminicula, quo sic aptius, exquisitius, promptius, minorique labour, etc. ac temporis dispendio possent Astris exponi observationésque peragi. Idcirco omnem curam atque operam pro tenui ingenii mei facultatúmque mearum modulo à Deo concesso (reliqua sublimioribus ingeniis atque ampliori fortunâ Viris, sive posteritati nostrae relinquens) adhibui: quo minora, tam lignea universa ab Astris planè removerem, atque in ejus locum ex puro solidóque metallo, organa mihi compararem: & quidem ejusmodi, quae insigni amplitudine essent conspicua, simul commoditate regendi, simul aliquanto accuratioribus adhuc divisionibus, ad paulò subtiliores observationes obtinendas gauderent. His Reasoning indeed is very good, that since he had from much and long experience learned, that Instruments of Wood after Ticho's manner, were not to be trusted to by reason of their warping and shrinking, and consequently that Instruments of solid Metal were much to be preferred before them, and also that the larger the Instruments were, the more exactly they could be made and divided, and that the more easy they were to be moved, and the more steady and sure they were to be fixed in any position, the more convenient they were for use, he had therefore rejected all those Instruments which he had made after Ticho's way; and had endeavoured to procure for his own use such as were complete, both for their matter and form, having caused them to be made of Metal that which could not be subject to the inconvenience of warping, swelling, or shrinking, with the variety of Wether, or length of Time: And likewise of such a bigness as was capable of receiving more nice and curious Divisions; and in the dividing them had found such contrivances, and used such diligence, that they were more then ordinarily true and exact. As far as he has gone on with these Designs, he seems to have been even profuse in his expenses, and exceeding bountiful of his own care, labour, and diligence; but I could have wished heartily that it had been some other way employed. Those Instruments which he chiefly laboured to perfect, he professes to be Quadrants, Sectants, and Octants, after Ticho's manner, rejecting all other Instruments of whatsoever Figures, whether Radii, Astrolabs, Zodiacal or Aequinoctial Rings, Parallactical Instruments or Hoops, as more troublesome, and less accurate. But whether he hath in this his choice been rightly advised, I shall hereafter have more occa sion to examine when I come to describe an Apparatus of Instruments necessary for such a one as designs to promote and perfect the knowledge of the Celestial Bodies and their motions; wherein I shall show that of some Instruments rejected by him, there is a use absolutely necessary.  
The Instruments therefore that he begins with are three small Quadrants of Brass; the first of two foot, the second of eighteen inches, and the third of one foot Radius. Each of these Instruments, he says, were made somewhat larger than common Quadrants, to wit, of an arch of 110 degrees, which is to no other end, but only in order to show the subdivisions of each degree of the Quadrant, by the help of a new invented Perpendicular of Brass wherewith each of them was furnished. This Invention is by him highly extolled for most excellent and useful; and to that end is made use of for the division of all his other Instruments, both great and small. Hear what he says of it: Quiscunque hujus rei (to wit, the new way of subdividing the degrees of the Quadrant) primus fuerit repertor, sublimes profecto cogitationes exercuit, hoc ipso ad congruentem effectum dedu●endo, & inter praestantissima inventa meritissimo refertur, quod etiam minora Instrumenta remotis omnibus transversalibus Lineis, in singula minuta eorúmque particulas minimas subdividi liceat. He seems indeed both here, and elsewhere in many other places of his Book to be highly possessed with admiration of the sublimity, subtlety, and extreme usefulness of this invention, and seems very much concerned that the Author thereof should not certainly be known, but dares not father it upon any one positively. He says that one Benedictus Hedreus in a Work of his which he published Anno 1643. about the new and accurate Structure of the Geometrical Astrolab, describes it; but he gathers that he was not the Inventor himself, but rather that he got both this Invention and the whole Quadrant, which he describes out of the Observatory, or rather Repository of Tycho Brahes Instruments, for that it seems Tycho was the Inventor of this way of division; and yet, as I noted before, he preferred the way by Diagonals much before it, whatever Reason Hevelius had to be of a contrary Judgement. What this way is I shall by and by explain. But in the mean time I am sorry to find Hevelius joining with Hedreus in the Opinion or Demonstration, as Hevelius calls it, that the Sub-divisions by Diagonals is not capable of a Geometrical demonstration, especially in lesser Instruments, which have need of many Circles. I confess I understand not their meaning nor reasoning, nor why it should be less demonstrable in lesser than in greater Instruments; since 'tis very easily demonstrable both in greater and lesser Instruments, and as Geometrical as any other way of Division whatsoever: the Diagonal Line being always a piece of a Tangent Line, that is to say, the spaces between the Parallel Circles upon the Diagonals are always to be in proportion to the difference of some Tangent Lines, and the different distance of those Circles from the Centre are always in proportion of some Secants: And the way of finding what those Tangents or Secants are, and consequently what must be those Distances of the Parallel Circles I mentioned briefly before, and shall now more fully demonstrate. From which I will make it evident, that the Theory was not as Hedreus and Hevelius have supposed, uncapable of Calculation or Mechanical Demonstration.  
But first give me leave to show you what way Tycho Brahe made use of to demonstrate, or rather to find out the true Angle unto each equal Distance, which I find set down at the latter end of his Mechanics, as a Supplement to the rest. Divisionis puncta habentis transversalia modus talis est, ut 34 exprimit figura in qua singula denominata per Lincolas in decem interstitia aequalia discriminatum punctis notata sunt, sicque regula fiduciae quodcunque horum inter observandum transiens ipsum minutum gradus, quòd quaerebatur promit aut aliquotam ejus partem, prout ab hoc vel illo puncto removeri discernitur. Ut vero hoc etiam demonstratum hic addam ob sciolos fortè quosdam qui ea quae non satis capiunt carpunt sic habe.  
In Figura 34. Sat A centrum Instrumenti ejusque Semidiamiter AO, assumitur autem HEY, Particula in qua divisio ista per lineas transversas fit ea proportione quae est 1 ad 48. qualis in meis Instrumentis ut plurimum usurpatur. Cúmque AI ponatur partium 10000000000. integri canonis majoris Rhetici, erit earundem HEY 208333333 utpote pars quadragesima octava radii Arcus ●E sit 20″. & IV. 10″. horum sinus 29088779 YI. Sinus autem secundus eorundem 42308. VY. qui additus NV quod aequale est HEY facit NIGH 208375641. In triangulo igitur NYI ad Y rectangulo nota sunt duo Latera NIGH & YI. quare datur basis IN 210396208. una cum angulo NIY 82°. 3″. 10′. 47″′. cui additus YIA 89°. 50″. conficit NIA. 171°. 53″. 10′. 47″′. Basis verò NI in triangulo rectangulo NVI dividatur in decem partes aequales ut conveniant uni minuto 21039621 representatae per IB. Moxque in triangulo obliquengulo BIA. dantur duo latera IB & JA. radius, una cum angulo BIA, qui idem est cum NIA 171°. 53″. 10′. 47″′. prius reperto: quare innotescit angulus JAB 1″. 1′. 7″′. qui tantummodo 1″. esse deberet, ità ut major sit saltem 1′. 7″′. differentia sanè insensibili: similiter si FI assumatur, noven particularum erunt eae 189356587 habebimusque rursus triangulum FIA in quo dantur duo latera FI modo dictum una cum radio JA. & angulo FIAMETTA ab iisdem comprehenso velut antea exurgitque angulus JAF 9″. 1′▪ 6″′. qui debebat esse 9″ exact deficiente in ultimo minuto FN. 1′. 6″′. Porrò ut circa medium idem tentetur quod nunc apud extremitates fecimus inveniuntur eadem qua antea primo Angulus JAH 5″. 3′. 6″′. abundans 3′. 6″′. Secundo Angulus NAH 4″. 56′. 55″′. deficiens 3′. 5″′. Patet itaque quod maxima differentia, sive adjectiva, sive ablativa in hac pragmatia proveniat minimum quid ultra 3′. quam subtilitatem visus acumen discernere in quocunque tandem instrumento nullatenus sustinet, quae etiam per se otiosa est, quare frustra nodum in Scirpo quaerunt, si qui hanc nostram satis accuratam distributionis formam cavillari praesumant. By which 'tis evident that Tycho understood an inequality, and what it was, and that it was insensible, and so not to be regarded. Now 'tis to me very wonderful indeed that Tycho having thought of a way of calculating this inequality, should not think of an easy expedient of reforming it by putting the Parallel Circles at unequal, but their due proportionate distances. And 'tis much more strange that Hevelius should still affirm it to be a way not Geometrical: For to any one that considers this proportion, the inclination of a Diagonal Line being given to find the true distances of the Parallel Circles that shall divide any assignable part thereof in any proportion assigned: Nothing can be more easy: and for more expedition use may be made of the Table of Natural Tangents which is ready calculated to hand. For instance: Let BC represent a Diagonal Line subtending an angul of 10″. at the Centre A, produce the said Line BC to F, and let fall a Perpendicular, from the Centre A to E. Suppose then the Angle at B to be one Degree, then is BE the Tangent of 89°. to the Radius A and EC is the Tangent of 88 50″. and the differences between the Tangents of 88 50, 88, 51. 88, 52. 88, 53. 88, 54. 88, 55. 88, 56. 88, 57 88, 58. 88, 59 and 89. gives the Distances of the several Circles, C. 1 2 3 4 5 6 7 8 9 B. desired.  
Since the Reading of this Lecture, Dr. Wallis hath also described another way of finding these Distances, which he hath communicated in a Letter to Hevelius, and I have prevailed with the said Doctor to permit it to be here printed, being very ingenious and accurate, and proceeding by a differing method.  
Dr. Wallis his Letter to Hevelius.  
— SED & est cur communi omnium Literatorum nomine rebus praesertim caelicis addictorum reddam gratias, tum ob immensos in tanto apparatu sumptos erogatos, tam praetiosum conquirendo supellectilem Astronomicam, graphice hic descriptam, tum ob indefessos labores, insomnes noctes diesque occupatissimos coelestis acquirendis observationibus impensos; quarum vim ingentem, Thesauram supra aurum & margaritus praetiosum erudito orbi jam ante dederis, plura daturus indies, verum non est ut sperem me verbis aequare posse tua merita, qui ex privato penu sumptos plane Regios erogasti; onusque suscepisti non infeliciter, Herculeis Humeris (ne Atlanteis dicam) formidandum.  
Operis partem maximam jam evolvi, miratus inibi tanta molis Instrumentorum ingeniosissimum regimen, & subtilissimam divisionum administrationem, cum pari diligentia conjunctam in Regulis & Dioptris solicite curandis, & quidem si hoc deesset reliquus in cassum caederet labour; quip exiguus & vix evitabilis in Regulis aut Dioptris error, totum Instrumentum vitiaret, omnesque inficeret observationes, sed singulis immorari non licet, unum tamen est quod attingam breviter, nempe divisiones per Lineas Diagonales, circulos in limbo concentricos oblique secantes. Hanc dividendi methodum jam diu receptam, ipse retines & quidem merito, circulosque hos concentricos aequalibus intervallis disjunctos habes, quod quamvis in exiguorum aut etiam mediocrium Instrumentorum limbis latioribus aliquid erroris possit inducere in tuis tamen tantae amplitudinis Instrumentis cum limbis exiguae latitudinis (quod & tu recte moans) nihil quicquam erit discriminis quod in sensus occurrere possit. Hac tamen occasine libet hic subjicere, quod ea de re jam olim (circa A. 1650. aut 1651.) meditatus sum, atque apud adversaria mea nunc reperio: nempe si quis vellet minoris Instrumenti limbum latiorem Lineis Diagonalibus sic dividere, quibus intervallis oporteat concentricos illos circulos disponere ut angulos invicem aequales designarent illae cum tranversali intersectiones calculo Trigonometrico determinare.  
Divisio arcus in limbo quadrantis (aliusué ejusmodi Instrumenti) per circulos concentricos & rectam Diagonalem, sit latitudo limbi (RL =) L, Radius circuli intimi (ARE =) R, extimi (AZ = ALL =) LR = Z continentes angulum (RAZ =) A. dividendum in parts quotlibet aequales (quarum numerus n) rectis a, b, c, etc. (quarum longitudo quaeritur) facientibus ad RZ diagonalem, angulus α, β, γ, δ, etc. adeoque angulus etc. sitque ARZ = O & AZR = V. Datis ergo crucibus R, Z cum angulo contento A. (adeoque reliquorum summa O + V) inveniuntur reliqui O obtusus V accutus.) Nam Z + R. ZR:: Ita tangens tangentem & deinde cognitis angulis O & (adeoque reliquo α) cum trajecto▪ latere R habetur latus a. nempe sin a. R:: sinus O. a. & pari  
Praxis sit R = 1. L = 0, 2. Z = 1, 2. A = 10″. ergo 55″. tum Z + R = 2, 2. cui respondet angulus 89°. 5″. 0′. 17″′. proxime. Ergo fere cujus sinus 0, 0174511. nempe idem qui sinus 0°. 59″. 59′. 43″′.  
Deinde secandus sit A in 10 parts quarum quaelibet sit 1″. quaeruntur igitur a, b, c, d, e, f, g, h, i, ' nempe.  
Praxis altera sit R = 1. L = 0, 1. Z = 1, 1. A = 10′. ergo cujus tangens 687, 5488693, & 2, 1. 0, 1:: 687, 5488693.32, 7404223½ = tang. 18°. 15″. 1′. ergo cujus complementum ad semicirculum cujus sinus 0, 0319827. ergo  
Hactenus adversaria, ubi duos casus expendimus, nempe cum latitudo limbi ponetur pars quin a & pars decima Radii brevioris, & angulus dividendus 10 minutà prima tanta fere 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, quantum feret vulgaris canon Trigonometricus: & quidem ultima unitas in ambiguo est nunc justo major nunc justo minor. Radium autem (ut ego soleo) facio L (non ut plerumque fit 10000000.) quo omnes multiplicationes & Divisiones per Radium faciendae praecidantur: Adeoq▪ sinus habeo pro partibus decimalibus, quibus itaque cum opus est, ciphras praemitto quo de unius integri loco constet.  
Simili processu utendum erit mutatis mutandis si latitudo limbi sumatur in alia quavis proportione ad Radii longitudinem. Sed commodius erit (ad vitandam molestiam toties quaerendi partem proportionalem) ut sumatur angulus O commodae magnitudinis (justis minutis primis determinandae absque annexis secundis tertiisve) atque ita quaeratur Radii maximi Z longitudo, eodem modo quae Reliquorum a, b, c, etc. puta si in praxi posteriori sumpto ut prius R = 1▪ & angulo A = 10″ sumatur angulus O non qui illic prodit 178, 10″, 1′, sed potius 178. 10″. cujus complementum ad duos Rectos est 10. 50″. hujus sinus in ipso canone habetur 0, 0319922 & reliquorum item α, β, γ, δ, etc. sinus similiter ibidem habebuntur, ut una tantum divisione opus sit pro singulis exhibendis ipsaque Radii Z Longitudo habetur non quidem precise ut prius▪ 1, 1; sed proxima (quae itaque sumenda erit) 109996 nempe.  
similiter omnino res succedit si sumptis Radiis RL cum angulo A quaeramus V & Radios intermedios, aut sumpto Radio L cum angulis AV quaerantur R & Radii intermedii.  
Verum si limbi latitudo sit Radii non nisi pars trigessima quadragessima, quinquagessima aut adhuc minor, atque angulus dividendus non quidem 10 minuta prima sed totidem secunda, aut minor adhuc, subtilior res est quam ut canon vulgaris Trigonometrious hic adhibeatur; & quae omnem sensum fugit, ipsique circuli concentrici distantris aequalibus quantum sensu possumus distinguere invicem disjuncti: quip unius pollucis pars millesima nedum decies aut centies millesima minor est discrepantia quam ut sensu: percipi possit. Sed nimius sum in re levi felicem itaque exeuntem annum tibi comprecatus longa sequentium serie contrivandum, valere jubeo.  
But to proceed. In the next place I think it will be suffiicently plain, to any one that shall try both the ways, that the Divisions are by Diagonals much easier distinguished by the eye, then by this way so applauded by Hevelius, and therefore I cannot choose but conclude with Hevelius, (pag. 140.) though to a quite differing end and sense: Sunt igitur splendidissimae tantum speculationes mentisque ideae quaecunque de Nonianis vel Hedrianis Divisionibus proferuntur. But because perhaps there may be several persons that have not yet perused this Book of Hevelius, nor that of Benedictus Hedreus, printed in 1643. nor Ticho's Mechanics, of a much longer standing, and thence may perhaps not so well understand what this way of sub-dividing is; give me leave a little to explicate it, and show you plainly what it is.  
The way then as it is described by Tycho Brahe, and ascribed by him to Petrus Nonius, that excellent Spanish Mathematician, who published it in his learned Book, de Crepusculis, supposing it also to have been heretofore used by Ptolemy, but (as Tycho is of opinion) without much reason, is this; Ut ducantur intra extremum quadrantem alii minores numero 44. successive sese comitantes, quorum extimus in 89. sequens in 88 tertius in 87. & sic deinceps donec ad ultimum & intimum perventum fuerit qui 46. portiones habebit. To which Description published in his Mechanica, he adds in the second Book, de Mundi Aetherei recensioribus Phenomenis, pag. 461. Altera Divisio ad Clarissimi Mathematici Petri Nonii— imitationem per plures quadrantis arcus introrsum descriptos & diversimode subdivisos procedit. Etsi autem in hac ipsa imprimis ingeniosa Nonii inventione, aliquid auctuarii loco expeditius à nobis additum est, ita ut exterior arcus in plurimas portiunculas dividatur; neque is ordo aut numerus arcuum sese introrsum concomitantium, quem ille praefinivit sed multo expeditior & perfectior observetur, tamen quia haec subtilitas cum ad praxin deventum est plus habeat laboris quam fructus, neque id in recessu praestet quod prima fronte pollicetur, ut alibi plenius ostendemus, idcirco apud nos dudum in usu esse desiit. [See more of this, pag. 62. Epistolarum Astronomicarum.]  
From which way of Division, this of Hevelius (which he ascribes to Hedreus, but is more properly ascribable to Pierre Vernier, as I shall afterwards show) is somewhat different, and possibly might be the same that Tycho Brahe contrived to compendifie that of Nonius.  
The way than is this, described by Hevelius, pag. 141▪ Quadrantes contractiores ita à me sunt adornati, ut limbos eorum tantum in integros & semigradus distinxerim; quae ut haec distinctio non nemini admodum rudis videatur, sufficit tamen affatim commonest andis singulis minutis primis; dummodo perpendiculi ex centro appensi extremitas limbum stringens in certas particulas sit subdivisa, imo quod magis de quo non nemo sane mirabitur, non solum haec rudior limbi subdivisio sufficiens exhibendis singulis minutis primis sed etiam pro denis quinis quinetiam singulis secundis in majoribus organis si videlicet nostrum Instrumentum directorium adhibeas. Oportet ut inferior illius pars curiosissime & levissime sit limata & levigata, ut limbum totum aequabilissime quidem tangat, sed nullibi nimis adhaereat; tum quovis loco liberrime pendeat atque divisionis tam quadrantis quam perpendiculi observator rite discernere valeat. Dividitur autem istud perpendiculum hac ratione, si videlicet spatium 31 semigraduum in limbo perpendiculi accuratissime denotes; idque primum in tres aequales parts, rursum quamlibet trientem in decem dividas; atque ita obtinebis spatiola paulo admodum ampliora quam spatiola unius semigradûs, quia intercapedo 31 partium in 30 transmutata necessario fiunt modice ampliores. Attamen si divisiones perpendiculi ad limbum quadrantis accedant circa extremitates perpendiculi, discrepantiola illa divisionum ab invicem vix ac ne vix cognoscitur; circa medietatem vero perpendiculi satis evidenter. In medio limbo perpendiculi & divisionum parvulus index & quidem inter 15 & 16 spaciolum constituitur pro discernendis integris & semi gradibus; quos accurate dictus index indicat, quando totum spatium perpendiculi in 30 partibus divisum in ipso limbo quadrantis spatium 31 partium exquisite subtendit. Ea tamen expressa lege si totum Instrumentum absolute ab omni parte sit constructum; quando vero iste index pauxillum promotior existit integro aliquo vel semigradu certissimum est indicium, observation iminuta quidem adhaerere aut integro aut semigradui adnumeranda, si index huic vel illi vicinior est. Cognoscitur autem minutorum numerus ex eo, quando lineola aliqua divisionum in perpendiculo cum una aliqua in limbo quadrantis prorsus in unam eandemque coincidit rectam. Nunquam enim nisi unica lineola in perpendiculo cum altera in quadrante, si exquisite peracta sunt omnia omnino concurrit. In isto igitur utriusque lineolae concursu ubi una eademque videlicet constituitur linea est terminus ipsorum minutorum vel integro gradui vel semi gradui adhaerentium.  
This same way is also made use of by Hevelius, for the Division of all his larger Instruments, as well as for the Division of this smaller, by fixing it upon the Perpendicular, as he afterwards mentions, cap. 15. pag. 307▪ where he also gives a fuller description of it, to which I refer the Reader.  
The way indeed is exceeding ingenious, and very much improved by Hevelius, but yet at the very best it is very difficult, both to make the Divisions, and much more difficult to distinguish them, as may be plainly enough seen even by that very Specimen published by Hevelius, in the first and second Figure of the Plate T. especially if it be viewed with a magnifying Glass or Lens; and I do wonder that Hevelius did not all this while think of making use of a Lens, to make the Divisions and Distinctions appear more plain, without which Seconds are not to be distinguished, by those kinds of Divisions even in an Instrument of 10 foot Radius, and by the help of it they may be made and distinguished, in Instruments of a quarter that bulk, as he may find, if he please to make use of the shallowest Object-Glass of that Microscope which he had from London; he may, I say, by looking upon the Divisions of the first and second Figures of the Table T. with his Microscope, plainly detect how far those Divisions are short of accurateness, and how many faults and inequalities the naked eye and unmachined hand do commit.  
It is therefore one of my ways for dividing and distinguishing Divisions, to make use of one, two, or three Lenses, whereby not only the eye is very much eased, but the judgement is very much augmented, and the hand directed, as I shall afterwards explain, when I come to show some particular ways of making Divisions.  
But because this Benedictus Hedreus, from whom Hevelius affirms he received this invention of dividing the Limb of the Quadrant, was not so ingenuous as to confess that he received this invention from another, and because perhaps the Book being small, may have been long since lost and forgotten, having accidentally met with one, I shall acquaint Hevelius, that one Pierre Vernier (as he calls himself) Captain & Chastelain pour sa Majesté au Chasteau▪ Dornans, Conseiller, & General de ses Monnoies au Conté de Bourgongne, printed at Brussels, by Francis Vivien, 1631. (to wit 12 years before Hedreus) a Treatise in French, which he calls, La construction l'Usage & les Proprietes du quadrant nouveau Mathematique, comme aussi la construction de la table des sinus de minute en minutes successivement par un seul maxim. De plus un abregé desdicts table's en une petite demi page avec son usage: & finallement la methode de troúver les angles d'un triangle par la cognoissance des costez & les costes par les angles sans l'ayde d'aucune table. In which he hath at large and very plainly described this way of dividing the Quadrant, to what accurateness is desired, and pretends it to be, as possibly it was, an invention of his own.  
But to return where I left to Hevelius his Division on the Quadrant by the help of the Brass-arm, I say, against this way, besides what I have already mentioned, I have a second Objection, and that is, that it requires a most exceeding great curiosity and care to make that Metal Pendulum or Plumb of Brass, so as to be exactly of equal weight and make on both sides of the supposed middle Line, for if it be not so, it may easily vary not only some Seconds, but even some Minutes from its exact Perpendicularity, and if so, 'tis to little purpose all the former curiosity about Subdivisions.  
Thirdly, The Perpendicular aught always to be kept very clean from Dust, for if a little more Dust settle on the one side then on the other, the Perpendicularity will be vitiated, and all the curiosity else about the Observation will be lost.  
Fourthly, If the Pin on which this Brass Perpendicular hangs be not of some bigness, it may easily warp, or bend; and if it be of a considerable bigness, it will not move easily, and consequently the Plumb will not hang tender, but stiff; in both which cases it can be of no use in the World for Astronomical Observations. Further, if it hang loose upon the Centre, which it must do to hang tender, then there will lie as material an Objection against it, for it's not moving true upon the Centre of the Instrument; and therefore upon the whole matter I conclude it to be an Invention indeed of great sublimity and subtleness, but of little or no use for Astronomy, to which Hevelius applies it. He had much better therefore have been content to have followed Tycho Brahe; and made use of a common Plumb Line and Diagonal Divisions, where there is occasion for them, for that is true and practicably capable of exhibiting the Subdivisions of a Degree, as Minute, as are necessary to common Sights.  
In the next place, before he leaves the Descriptions of these three smaller Quadrants, he mentions an Invention of his whereby he fixes the Quadrant in any altitude, and easily moves it steadily into any posture desired by the help of Screws. This Invention of his own contrivance he doth indeed very highly applaud, insomuch that he believes no good Astronomical Observations can be made without it. But he must pardon me if I am not altogether of his mind; I grant indeed the thing is exceedingly convenient, in comparison with any yet used, if it be well made, and that the way of applying it to the Quadrant be very facile and easy. But 'tis not always so necessary, but that Observations may be as conveniently made without it, as I shall afterward show, in the Description of the movable Axis, for continuing the Instrument in the Plain of the Object, whether a Distance or an Altitude be to be taken.  
In the next place he proceeds to describe his large Quadrant of Brass adjusted so as to take Altitudes and Azimuths, of which he makes a full and particular description; but the most considerable thing that is new in it is, that instead of a Screw used by Tycho for lifting and moving the Arm with Sights, he makes use of two Lines poised with Plumbets, by the pulling of this or that of which he is able to raise or sink the Ruler with Sights, all the rest of the contrivance being to make it stand perpendicularly in any Azimuth, which I think may be done to greater certainty with less trouble, by a way I shall afterwards show: As an Essential part of this Instrument, he takes occasion to give the description of the Turret or Observatory which he built for it, and the several contrivances about it, which I now omit.  
The use he made of this Instrument was for the taking the Meridian Altitudes of the Sun, of which he affirms to have taken a very great number, especially such as were of principal use for the regulating the motion of the Sun: Such as the Solstitial and Aequinoctial Altitudes, of which I hope we may expect an account in the second and third Part of his Machina Caelestis. I know not to what exactness he hath proceeded in taking his Meridian Altitudes of the Sun; but had he proceeded in the way by Telescopes, he might have taken all his Altitudes of that kind to a single Second, with great ease and certainty.  
And upon this occasion I hope it will not be unacceptable to my Astronomical Reader to hint a very expeditious and exceeding accurate way of making a Catalogue of all the visible, as well as the most considerable Telescopical Stars of the Heaven. For the doing of which there will not need a tenth part so much time as for the other ways that have already been made use of, and yet will very much exceed them all in accurateness and certainty. The way then in short is nothing but this: Let there be made a very large mural Quadrant, or rather Semicircle, of 30 foot Radius, fixed exactly in the Meridian against a Wall made of squared Stones, well jointed and cramped together, and settled on a foundation very firm and solid, to prevent all manner of slaking and swarving. Let the rim of this be made of Brass Plates, stayed in their due posture by cramps or bars of Iron fixed in the Wall, by running them with Lead: then having divided this Semicircle into 180 Degrees, and subdivided each Degree by the help of Diagonals, on a flat and well polished Plate of Glass, according to the way I before described into Minutes and Seconds: adapt to it a 30 foot Telescope, so that the Tube shall not warp, nor the Glasses deviate out of their true posture; the Focus of the Object Glass make to be exactly upon the edge of the Brass Limb, so that by the help of the Eye-glass, which is a deep Convex, the punctual place or altitude of a Star to a quarter of a hairs breadth, even to Seconds of a Minute, may be discovered: the trouble of dividing this Quadrant will be no more then of one of an ordinary size, the subdivision of one Degree subdividing and examining all the rest. The way of making the Tube of the Telescope so as not considerably to bend, may be done somewhat after that way of stiffning the Tubes of very long Telescopes, which I communicated to Hevelius, and you will find at large described in this Treatise of Hevelius: Save only, that instead of Ropes which I first made use of, I rather commend so many Braces of Wood Now though notwithstanding all the Diligence that can be this way used, the Tube do somewhat bend in the middle, yet it can be of no manner of significancy as to the vitiating the Observation; since first, the Object Glass always standeth in the same posture as to the Centre, and secondly, the Focas thereof is exactly in the edge of the Limb.  
Further, to prevent the inconvenience of looking up or in any other uneasy posture by the help of a reflex Metal one may always look Horizontally, that is, perpendicularly to the plain of the Wall or Mural Quadrant. And to prevent the trouble and labour of moving or lifting the Tube by the help of a long yard poised upon Centres on a Frame before the said Instrument, both the Tube & Arm for the Sight, and the Seat on which the Observator sits, may be counterpoised, so that by turning a Windle, he may easily raise himself with the Tube to any posture desired. The Object Glass is just before the Centre, and the Eye Glass looketh directly on the Divisions of the Limb, and there is nothing to strain or stir the Instrument itself, nor can the warping of the Tube, if there should be any, have any effect on the Observation: Of this I may say more on another occasion. By this means (in one Night's Observation) the Declinations of some hundreds of Stars may be taken to a Second by one single Observator, having only one or two Assistants to write down the Observations as fast as made. And at the same time the right Ascension of every one of them may be taken by the help of a very accurate Compound-circular Pendulum Clock, which I shall elsewhere describe, denoting even to ●f a Second of time the appulse of the Star to the Meridian: There needs indeed great exactness in every part of this Apparatus, and 'twill not be done without a considerable charge, and much labour and diligence in the performance thereof; but if we compare it with the methods and ways that have been hitherto used, we shall certainly find that the Observations will be near 30 times more accurate, the charge not a quarter, and the labour not near a tenth part so much as in other ways made use of by Tycho and Hevelius. And though it may be objected against this way (which indeed may be much more so against any other) that the refraction of the Air will considerably vary the Declination of such Stars as are very far South, yet since the same Instrument affords a way beyond any in the World for the discovering the several Refractions of the Air at several Altitudes above the Horizon, to the accurateness of a Second, by taking the Altitude of such Stars as never set in the North, in the greatest and least Altitude above the Horizon; a Table of such Refractions will easily rectify the Declination of the other Stars to as great accurateness. This Subject doth deserve a much larger and more particular Description of every Branch thereof, and the Encouragement of some Prince, whose Name and Honour will thereby be Registered among those glorious Celestial Bodies to all Posterity, and the succeeding Learned World will be obliged to celebrate his memory. But I fear this Age will hardly yield another Alphonsus, another Tycho, or another Hevelius, who have not spared to expend their utmost Endeavours in performing this task, though by other methods. But leaving this for another time, I shall proceed.  
In the third place than he goes on to describe his great Horizontal voluble Brass Quadrant, of which he says, he does not believe that ever the like was made by any, if the splendid Apparatus and the whole Fabric thereof be considered. It is in Diameter six foot and an half, and serves, as he affirms, to take Altitudes to Seconds; but yet he is necessitated to allow, that it is short both of Ticho's large wooden Quadrant, and of his large mural Quadrant; nor do I see any reason why Ticho's mural Quadrant should not take Meridian Altitudes somewhat more accurately, since I believe his Sights every whit as good, and his Divisions altogether as exact; what he might fail in diligence, I cannot say. I do believe this Instrument to be an exceeding good one of the kind, and that he hath from much practice and experience found out many contrivances, in order to the making it convenient to make Observations, and he hath not spared for cost, pains, study and industry, for the completing thereof; but still whether he be arrived to the greatest perfection, or to so great as to take Altitudes to Seconds, seems to me very dubious, and if he made use of the Sights before-described, wholly impossible. For first, a Degree upon the Limb is but about 6/● of an inch, and consequently a Minute is but the 50th. part of an inch, and a Second but the 3000th. part of an inch, which he that can distinguish with his naked eye, hath better than I, or I fear, any man now living. unbiased men, I grant, can do much toward the distinguishing very minute Divisions, by being able to bring the Object very near the eye, but the most unbiased must be yet very much shortened by Glasses, before he will be able to distinguish the 3000th. part of an inch, and when he hath distinguished it, which he may possibly do with a Microscope, how will he distinguish of the Penumbra, which is not certain even to a Minute? And though it may be said, it is the same, round the Circle, and the Circle is the true bigness of the Sun, so that if a Circle of a bigness, answering to the Diameter of the Sun, and the Distance of the lower Sight from the upper be described on the lower Sight, it must bond the Limb of the Sun, and that consequently it will be easy to distinguish when that Circle is perfectly filled with the figure of the Sun, admitted through the hole in the upper Sight. I answer, That this seems very probable and easy, and is indeed believed and asserted so by Optical Writers: But yet 'tis quite otherwise; for not to mention that there is confessed by all, that the Penumbra of this Circle must be as big at least as the Diameter of the hole above, through which it is trajected, which cannot be less than a Minute; I say, that experience doth demonstrate that it is quite otherways, and that the Limb of this Image painted on the lower Sight is terminated with a Penumbra, which is sometimes five or six times bigger than the Diameter of the hole, and which is yet stranger, the smaller the hole be, the bigger is the Penumbra, and the bigger (to a certain Degree) the less, but there is no bigness which will take it off quite, and the Diameter of the Sun that way taken, is sometimes bigger and sometimes less than it ought, and that to a very considerable quantity: Of which, and several other very strange proprieties of Light, I shall hereafter say more on another Subject.  
But to proceed. That he hath made this Instrument his chiefest, you may perceive by his pathetical describing thereof; for he says of it, pag. 184. Ad commodiorem hujus quadrantis usum, tot ac tot adminicula recens excogitata atque huic organo applicata fuere, ut nesciam à quibus primum inchoare debeam. Imo etiamsi vel▪ maxim velim, nullo tamen modo omnia & singula adeo perspicue vel delineare vel describere potero, ut universi praeprimis qui similia haud ipsimet oculis usurparunt quaevis recte ac plane intelligant, quinetiam credas velim utut aliis sunt attentiores atque hujus rei bene gnaros, aliquoties sane hocce Instrumentum visuros antequam dimidiam tantam partem debite animadvertant ac plenissime comprehendant. Quip & verum fateor nec ipse ego, licet singula ex meo solo cerebro prodierint ac confecta fuerint, possem adeo distincte tibi eum sub aspectum ponere nisi mihi hocce organum sub oculis assidue versaretur. Nihilotamen minus dabo operam, ut quantum fieri poterit, dilucide omnia proponam, reliqua veri exercitatis caeli metaloribus ulterius rimanda & perquirenda commit am, etc.  
And so he proceeds with the Description of this Quadrant, and the Apparatus about it, and first, he tells us of the weight of this Instrument, that it was 80 l. Next, of the shape of the Turret in which it was fixed, which is indeed very convenient and ingenious, it being so contrived, as to be voluble or convertible upon Truckles, having one only side open, and enclosed on all sides else, so that neither the Observator nor the Quadrant was much exposed to the injury of the weather, which is indeed of no small use in Astronomical Observations. But this may be done many other ways also. He tells us further of the admirable and prodigious use of Screws, in order to the setting and fixing the Quadrant. Next, As to the giving a motion to it, in order to follow the Sun and fixed Stars in their diurnal motion. Thirdly, As to perform all the Subdivisions of a Degree, not only into Minutes but into single Seconds. To all which I say first, As to the use of the small Hand-screws, I do grant, that in some cases they may have their conveniency, as to the moving and staying the Instrument. But then since he is fain to make use of two Screws, whereby both the hands must be employed to manage these Screws, I judge them too troublesome for that use, and that there is a much better way, whereby the Quadrant being once set into the Azimuth of the Stars, it shall continue to be so, and to move along with it, without any trouble to the Observator, so long as the Observator hath occasion to have it remain so, which (that I may hint that only now by the By) is a small Automaton, which shall continue it for many hours exactly, in the Azimuth of the Star desired, of which more hereafter.  
Next, Whereas he affirms this way capable to show Seconds as well as Minutes, I grant it may be capable; but then I must further affirm, that he hath not at all showed how that can be done, nor is it indeed feasible in his way, for he shows us not any way how to set it, that is, fix it certainly to any Degree: Now if he be not sure in the fixing it exactly to a Second, upon that Degree where he would begin his Division, 'tis a vain thing to be so accurate in the other Dimension, for he cannot be more certain, (let him be never so curious in the Subdivision with his Screw) than he is certain in the first fixing of his Screw to the Degree, for whatever he varies from the Degree in the setting, he varies at least as much in the Subdivisions, and consequently unless that be some ways taken care of, which I do not find, 'tis a nicety without use.  
To conclude therefore, I say, the Frame of this Instrument is extraordinary good, and by the help of some additions, as to the Sights, Divisions, Perpendicular and Erection, might be made as good as need be desired for any use in Astronomy, and 40 times better than what it is now made and described by Hevelius, or then any I have yet heard of to be made in the World. But as it is, it is not more exact than the large Instruments of the Noble Tycho Brahe, which he used 100 years since, and much short of his mural Quadrant, for taking Meridional Heights.  
He proceeds to the Description of his new and large Brass Sextant of six foot Radius: The Sights and the Divisions thereof are in nothing differing from those of the Quadrant, nor do I find any thing very considerable in the Description thereof; it was made use of by two persons in the same manner as the former Sextant, and like that of Tycho; but what grand inconveniences do attend that way of Observation, I shall afterwards show, when I come to explain how one person alone may be able to do it with less trouble by half, and ten times more exactness.  
But by the way, I cannot but take notice of what Hevelius ingeniously confesses, of the great difficulty there is in taking the Distance of fixed Stars from the Moon, which is from nothing else but the imperfections of his Common Sights, and all that difficulty vanishes, if the Sights be made another way. Next, He seems to make it a much more difficult business, to take the Distance of the Sun from Venus, when she is seen in the daytime; but by a way I shall hereafter show, it will not only be easy to take the Distance of the Sun in the daytime from Venus, but from Mars, from Jupiter, nay, from several of the fixed Stars.  
I shall pass by therefore his Apparatus, which seems very great and chargeable, since I shall elsewhere show a single, plain way, without any trouble or perplexity, how the matter may be quite otherwise ordered, much to the advantage of the Observator.  
As to what he asserts of his extraordinary care, diligence and pains, in dividing and examining the truth of his Instrument, I do no ways doubt it, but that he hath proceeded as far as it was possible for one to do in that way he made use of, but might have saved much of it, if he had thought of the way by Diagonals on Glass, which I have already described. Yet I should have been very glad to have seen the Distances, which he mentions to have taken of eight fixed Stars near the Ecliptic, to wit, Lucidae Arietis & Palilicii, Palilicii & Pollucis, Pollucis & Reguli, Reguli & Spicae, Spicae & in manu Serpentarii, in manu Serpentarii & Aquilae, Aquilae & Marchab, Marchab & Lucidae Arietis, and that to so great exactness, as not to miss one single Second in the whole Circle of the Heavens, taken at eight Observations. For to me indeed it seems one of the greatest affirmations I ever met withal, and not less then humanely impossible, were there no Refraction in the Air▪ and did all the Objects stand still in the Horizon, but the Refraction of the Air, were it much less than it is granted by all, would necessarily cause a variety of a great number of Seconds. And I durst undertake to demonstrate it to any, as plainly as any Geometrical Proposition, that it was wholly impossible for him, with all or any of the Instruments he hath described, to make any one of these Observations, to the certainty of 30 Seconds, whence if that uncertainty be 8 times multiplied, it will follow, he cannot be certain in the whole Circle to 240 Seconds, or 4 Minutes, which how much it is differing from one single Second, any one may judge.  
I had many other things to have added, which have occurred to me in the perusing of Hevelius his Book, but I will say no more at present by way of Objection, having, I fear, wearied the Reader, with showing him my doubts and scruples, especially about the imperfection of that way of Sights and Divisions made use of by him: Only, to make my Reader some mends for his patience, I shall describe a short Apparatus, which I have contrived for this purpose, and in the doing thereof, shall be as plain and brief as possible the matter will bear.  
Since the reading these Lectures, the Author having been acquainted, that some considerable Objections had been made against the certainty and accurateness of his Instruments, and that I had affirmed it impossible to perform what he had promised in his Book, he returns his Sentiments thereof in a Letter to Mr. Oldenburg, to this effect:  
— Caeterum percipio vestrates non omnes mihi adstipulari in isto Dioptrarum negotio, de quibus in machinae meae coelestis Organographia tractavi, verum etiamsi Cla. Hookius & Cla. Flamstedius aliique plane aliter sentiant, experientia tamen quotidiana me edocuit atque etiamnum docet, rem longe aliter se habere in magnis illis organis, quadrantibus scilicet sextantibus & octantibus imprimis quadrantibus Azimuthalibus aliisq, quadrantibus regulis constructis, quae nempe adeo procliviter commoveri & inverti (dum Dioptrae Telescopicae examinantur) imo nullo modo possunt, ut quidem Instrumenta illa trium quatuorve pedum perpendiculo constructa. Rei cum primis in eo consistit, quod nullam plane observationem suscipere possint suis Dioptris Telescopicis nisi prius denuo eas examinent ac rectificent; in quo tamen examine variâ viâ, tum jugiter utut studiosissime illud suscipiatur hallucinari datur. Adhaec in quadrantibus Azimuthalibus, octantibus & sextantibus, qua ratione examen istud adeo accurate nunquam non haud magno negotio temporisque dispendio institui possit, profecto nondum capio, vix mihi persuadeo ullibi adhuc ullum aliquem magnum quoddam Instrumentum 6 vel 9 pedum utpote sixth. octant. vel quadrantem cum regula vel quadrant. Azim. cum pinnacidiis Dioptricis construisse, eumque ad coelum felici aliquo successu adhibuisse, & quicquam solid observasse; si tentasset ac per annos aliquot observationibus continuo invigilasset sine dubio aliter sentiret. Hoc negotium enim non solum in eo consistit quod stellae aliquanto distinctius conspiciantur (quanquam fixae ab eo qui visu pollet & exercitatus est aequè bene nudis oculis discernantur) sed an Instrumenta ab omni parte correcte commonstrent, an pinnacidia Telescopica Instrumentis toties ad quasvis observationes rite imponi & tuto conservari queant; de quibus quidem id omni tempore aequè praecise fieri posse valde dubito. Quare Clarissimos illos viros humanissime rogatos volo nisi jam possideant ejusmodi vastissima organa utpote sixth. octant & quadrant. Azim. Dioptris Telescopicis munita, eaque coelo continuo admoneant, suspendant judicium paululum, donec longa annorum serie experti fuerint haud fuisse multoties egregie elusos. Nam ex una alterave observatione quadrant. aliquo leviori perpendiculo gaudenti obtentâ, res haec non est decidenda, sed si quis per 10 & amplius annos assidue observaverint, tum ab ovo seriam stellarum restitutionem per distantias susciperit, poterit quaedam certiora in medium hac de re proffer. De reliquo satis mirari nequeo, eas omnes qui ejusmodi Dioptris Telescopicis gaudent, nondum locorum suorum, elevationem poli ubi degunt & observationes peragunt, quantum sciam recte & omnino praecise determinasse & stabilivisse. Hucusque enim ad aliquot minuta integra Parisiis elevatio poli nondum est definita, alii quippe eandem observationem 48. 49″ alii 50″. alii 51″, alii 52″, alii 53″, alii 54″, 55″ imo ampliorem adhuc statuerunt: sicuti legere est ex discertatione Petri Petiti de latitudine Lutetiae, sed nolo in his prolixius esse; ad observationes ipsas provoco, tempus aliquando docebit quoru● observationes universas accuratiores fuerint, si modo nonnulli censuram suam eo usque rejicere possent. Nam video aliquos inter quos etiam Cl. Flamstedius invenitur, prout ex Epistola ad Cassinum apparet, jam judicium de nostris qualibus observationibus tulisse▪ priusquam illas adhuc viderunt examinarunt vel quicquam de iis cognoverunt. Nolo quidem vanus esse rerum mearum jactator, nec unquam mihi imaginatus sum rem in omni isto negotio circa scilicet restitutionem stellarum fixarum acu omnino tetigisse vel tangere posse. Sed hocce penitus mihi imaginor si totum istud negotium Dioptris Telescopicis suscepissem, quod non solum plurimos annos examinibus trivissem, sed spe sine dubio varia via (de qua hic non est discerendi locus) cecidissem. Exinde gratulor mihi me ad eam sententiam nondum trans●sse, ac me mea methodo universa perfecisse se quicquid praestitum Dei beneficio erit: an nihil amplius (ut putat Claris. Flamstedius) quam hactenus & quousque progressum fuerit, liberum erit cuique cum deinde viderit judicium suum exponere quinetiam integrum erit alium novum integrum catalogum super additis tot ac tot centenis nonis fixis, hactenus neglectis alia ratione construere: Verum nondum video an cura haec molestissima, taediosissima ac laboriosissima, quae non nisi multorum annorum vigiliis suscipi & peragi potest, aliquem adhuc serio tangat. Unam aut alteram stellam ope Telescopii vel Dioptrarum Telescopicarum, dum praecipuas ac majores fixas earumque intercapedines supponimus correctas ad debitum locum deducere, tum nonnunquam distantias nonnullas stellarum capere haec ludicra sunt; sed omnes conjunctim secundum longum & latum restituere, tum ductu continuo singulis serenis diebus ac noctibus, tam altitudinum solarium quam reliquarum stellarum observationibus operam dare, easque orbi exponere ut pateat motuum harmonia atque Instrumentorum certitudo, hoc artis hoc laboris est. Quando observationes 20 vel 30 annorum spatio continuatas ab utraque parte aliquando habebimus, nimirum tam quae Dioptris Telescopicis quam quae solummodo nostris ex coelo deprompta sunt res omnino clarior erit. Interea quilibet fruatur suo ingenio, ac sua ratione pro libitu rem tentet. Honorificum nobis omnibus erit pro modulo nostro à Deo concesso, rei literariae incrementum varia via promovere.  
To this Letter of Hevelius I have this to answer, That the Author neither hath, had, nor can have any experience, to show Telescopical Sights not to be as good as the Common, or that they are less applicable to large Quadrants, Sextants, Octants, or Azimuth Quadrants, or to any other Quadrants furnished with Rules, and so fixed, that they cannot be easily inverted, or turned, than they are to Quadrants or Instruments of 3 or 4 foot Radius. Nor is his Reason against them of any validity, that no Observation can be made, without a repeated previous examination and rectification of the Sights, in which, says he, notwithstanding all the care and diligence, there is a Reason of failure and mistake. For first, I say, There is less need of rectifying the Instruments or Sights, after they have been once adjusted, then of Instruments with Common Sights, all things being perfectly fixed, and so strong as not easily to be stirred or removed. I now begin to fear, that he hath not a true notion of the manner of performing the same, otherwise he would never have propounded such an Objection; and indeed he seems to say as much in the following words, Qua ratione examen illud omni tempore commode & sine magno temporis dispendio institui possit profecto nondum capio. Though I am very sorry that he should be so: for first, I thought I had about 9 years since, explained to him the way, when I exhorted him by all means to the use thereof; at least if he had not understood it thereby, I should, upon his desire, have sent him a more ample and particular Description thereof, or have procured an Instrument of that kind made and fitted for him here. But I fear, he had been some ways or other prepossessed or prejudiced against them, before I writ first unto him concerning them, at least before he writ that Answer, which I have before printed in the 5 and 6 Pages, for thereby it appears, that he was then of the same opinion he seems now to continue of. And whereas he thinks, that no trial hath ever been made of Telescopical Sights, to a large Instrument of 6 or 9 foot, I do assure him, (and I mis-remember, if I did not then acquaint him with as much) that I had then by me several, and particularly one of Sr. Christopher Wren's invention, furnished with two Perspective Sights of 6 foot long each, which I made use of for examining the motions of the Comet, in the year 1665. And if the same thing can be better done with a Quadrant of 6 inches Radius, than he can perform with one of 6 foot the common way, I think he might have concluded at least, that the same thing would be 10 times better done in one of 6 foot Radius, made after the same manner; of this, I am sure, I gave him then an account. Now it is not with these kinds of Instruments, as it is with Common Instruments, where 'tis not possible to make any better than one may be made of 3 foot Radius, because that is capable of Divisions, accurate enough to reach the power of the naked eye; but Instruments with Telescopical Sights, are capable to be made to distinguish minutes, seconds, nay single thirds, if they be proportionably augmented. Nor is there any need that a man must make 7 years' trial of an Instrument, before he can be certain of the greater excellency thereof, for I can be as certain with 3 or 4 times viewing an Object through a Telescope, and with my naked eye, that I can see it better, and distinguish many more and much smaller parts in it through the Telescope, than I can with my naked eye, as I could be, supposing I had been viewing it 20 years together. But yet I must assure Hevelius, my experience hath not depended upon 3 or 4 trials only; I cannot choose but wonder why he should be of that opinion, who hath not been less exercised in the use of the Telescope, than any at present in Europe: Possibly indeed his Telescopes were not altogether so good as now they are made, yet sure I am, he saw more with them then any one can see without them, as will sufficiently appear by his Phases of the Moon, Jupiter and Saturn. But I hope he will not wonder at me, though I do now venture to affirm, without staying 10 years or more to make Observations, that I can do more with a Quadrant, Sextant or Octant, of 1 foot Radius, furnished with Telescopical Sights and Screws, then can possibly be done with any other Instrument, furnished only with Common Sights, though 10, 20, 30, nay threescore foot Radius; nor does it at all follow, that the Latitude of Paris is not yet exactly known, because Monsieur Petit was ignorant of it; but it rather shows, that Observations made with Common Sights, (such as I suppose Monsieur Petit's Instruments and others, before the publishing of his Book were) are no ways capable of certainty to a minute or two.  
But I have done, and am sorry I have been forced to say so much in vindication of Telescopical Sights; and that in the doing thereof, I have been necessitated to take notice of the imperfections, that are the inseparable concomitants of Instruments made with Common Sights. Nor should I have published these my thoughts, had I not found them so highy decried by a person of so great Authority, fearing that thereby other Observators might have been deterred from making any use of them, and so the further progress of Astronomy might have been hindered. Nor would I willingly be thought to depretiate or undervalue the Works and performances of a person, so highly meriting the thanks of all the learned World, both for his great and liberal expense, and for his vast pains, care and diligence, in the performing a Work so highly useful to Astronomy and Navigation, and of such infinite tedium, trouble, labour and cost, to the undertaker. I do not in the least doubt, but that it will be a Work worthy so excellent a person, of perpetual esteem and fame, and much preferable to any thing yet done of the like kind in the World, and that he hath gone as far as it was possible for humane industry to go with Instruments of that kind, and that his Instruments were as exact, and complete, and fit for use, as such Instruments with Common Sights could be made, and that he hath calculated them with all the skill and care imaginable, and delivered them with all the candour and integrity. But yet I would not have the World to look upon these as the bound or non ultra of humane industry, nor be persuaded from the use and improvement of Telescopical Sights, nor from contriving other ways of dividing, fixing, managing and using Instruments for celestial Observations, then what are here prescribed by Hevelius. For I can assure them, that I have myself thought of, and in small modules tried some scores of ways, for perfecting Instruments for taking of Angles, Distances, Altitudes, Levels, and the like, very convenient and manageable, all of which may be used at Land, and some at Sea, and could describe 2 or 3 hundred sorts, each of which should be every whit as accurate as the largest of Hevelius here described, and some of them 40, 50, nay 60 times more accurate, and yet everyone differing one from another in some or other circumstantial and essential part. And that this may not seem altogether so strange, I will assure them, that I have contrived above 20 ways for dividing the Instrument, each of them as much distinct from each other as this of Hevelius, and that of Diagonals, and yet everyone capable of as great certainty and exactness at least, and some of them 100 times more. I have above a dozen several ways of adjusting the Perpendicularity or Horizontality of Instruments, all as exact as the common Perpendicular, and some of them very much more, even to what accurateness shall be desired, and yet each of these very differing one from another. I have as many differing kinds of Sights, for improving, directing, adjusting and ascertaining the Sight, some of which are applicable to some particular uses, but some for all, by means of which that part also may be improved to what accurateness is desired. I have various ways of fixing those Instruments, and appropriating them for this, that, or the other particular use. I have various mechanical ways for making and working the several parts of them with great expedition and certainty, which is a knowledge not less useful than the knowledge of the theory and use of them when made, there being so very few to be found in the World that can or will perform it. I have a mechanical way of calculating and performing Arithmetical operations, much quicker and more certainly than can be done by the help of Logarithms, which completes the whole business of measuring Angles. These I mention, that I may excite the World to inquire a little farther into the improvement of Sciences, and not think that either they or their predecessors have attained the utmost perfections of any one part of knowledge, and to throw off that lazy and pernicious principle, of being contented to know as much as their Fathers, Grandfathers, or great Grandfathers ever did, and to think they know enough, because they know somewhat more than the generality of the World besides: Reptat humi quicunque vult, Caelo restat itur, Caelo tentabimus ire. Let us see what the improvement of Instruments can produce.  
And now to make my Reader some amends for his patience, I shall give a Specimen or two, of each of the several parts that belong to the perfecting of celestial Instruments: And this I shall do, in the Description of an Instrument for taking all manner of Angles and Distances in the Heavens, which if increased in bulk, is capable of as great accurateness, as the Air or Atmosphere will ever permit celestial Observations to be made. It's perfection consists in seven several particulars. 1. In the Sights, which are such as may be made to discover the minutest part discoverable in an Object, they do no ways strain the eye, and are fit for all Sights, whether unbiased or old, etc. 2. In the Divisions, which are such as will distinguish the Angle, as minutely as the Sights will distinguish the parts or Objects. 3. In the Sights, being so contrived, that with one glance of the eye, both the Objects though a Semicircle distant, are at once distinguished and seen together. 4. In the method of setting it exactly perpendicular to a Second, if need be. 5. In its fixation and motion, it being so fixed and moved, that if once set to the Objects, it continues to move along with them, so long as 'tis necessary to continue, or be very certain of any Observation. 6. In it's not being difficult to be made and adjusted, and it's not being without industry and design put out of order, and its being presently, and with all imaginable ease rectified and again adjusted. 7. In it's not being very chargeable. First, For the Sights, They are no other than plain Telescopes, made with two convex Glasses, an Object and an Eye-Glass, of what length and charge shall be thought most convenient, fixed into square Boxes or Tubes of Iron or Brass, and having cross Clews at the Focus, made with very fine Hair, or silkworms Clews. One of these is fixed upon the side of the movable Bar or Plate of the Quadrant, the Object-Glass of which is next the Rim, and the Eye-Glass is next the Centre. The other of these is fixed upon the side of the Quadrant by several Screws, and care is taken to keep it from bending or sagging. This Tube is made of twice the length of the former, and hath at each end an Object-Glass, each of them of the same length with the former, and hath two Eye-Glasses in the middle, the manner of ordering which I shall show by and by under the third head.  
But first I shall explain the manner of fitting a Telescope for a Sight. Let a a b b in the 12th. Figure represent a Tube, in which let p represent the part toward the Object-Glass, whose Focus is at o, and let n represent the Eye-Glass, whose Focus also is at o, let s represent the point, where the eye being placed, the whole Eye-Glass n will be enlightened and filled with the Object, then make a small Tube about an inch in length, and of such bigness as it will just slice within the hollow of the Tube a a b b, and cross the Cavity of that strain two very fine Hairs or silkworms Clews, which may cross each other in the Centre of the Cavity, by the means of which Box, the said crossing Clews or Hairs may be moved to and fro, till they are exactly placed in the very Focus both of the Object-Glass and Eye-Glass, for if they be not there, the moving of the eye to and fro over the hole at s, will make the Threads seem to move upon the Objects, but if they be exactly in both the aforesaid Focus', the moving of the eye will not at all make the said Threads seem to move upon the Object, but they will appear as steady and fixed to the Object, as if they were strained and fastened to it. And though they are exceeding small, even as small as the Web of a Spider or Silkworm, they will appear very big and distinct, and much plainer and bigger than a Thread in the Common Sights, at the further end thereof, will to the naked eye, though above 100, nay 1000 times the bigness, which at the first glance will sufficiently discover the vast advantage these kind of Sights have above the Common ones. Nor is this way of Sights at all confined, but may be made to distinguish the smallest part of the Object desirable, even the parts appearing to the naked eye, under the Angle of a single second or third of a Degree, which is some hundred of times more curious than the naked eye can distinguish, without the help of them, for the Telescope can be made longer, and the Eye-Glass can be made deeper, and according as the Telescope is longer, and the Eye-Glass deeper, so will the Object appear bigger, and more minute parts be distinguished, the power of the eye being increased proportionably to the length of the Object Glass, and the charge of the Eye-Glass, and the goodness of them both. Now as Sights this way made, are capable of the greatest accurateness desirable, so they are so appropriated to the eye, that they no ways strain it, for they may be so ordered, as to make all those parts that are to be distinguished, to appear to the eye under the Angle of 3 or 4 minutes, which most eyes are able well to distinguish, without using too much attention or straining to discover them. This is no small convenience, to one that is to make many Observations one after another, for the eye by too much attention is apt to be suddenly wearied, and it doth very much harm and weaken the Sight, to endeavour to distinguish parts so small, as appear to the eye under the Angle of a minute, very few eyes being able to reach it at all, and most others not without much difficulty and endeavour. 'Tis further considerable upon this account, that 'tis fitted for all kinds of Sights: For a unbiased person, the Eye-Glass may be made to slide a little nearer the Cross in the Focus; and for an old or decayed Sight, the Eye-Glass may be moved a little longer or further off from the said Cross or Focus; for a dim Eye, the aperture of the Object-Glass may be augmented, and the Eye-Glass made shallower, or of a less charge; and for a weak, tender and curious Eye, the charge of the Eye-Glass may be augmented, and the aperture of the Object-Glass made less. And according to the several constitution of the Observators eyes, the manner of Sights may be accommodated, which the other Common Sights without the help of Glasses, are no ways capable of.  
The second thing wherein the perfection of this Instrument consists, is the way of making the Divisions, which I think, is far beyond the Common way, both for the certainty and ease of making, and secondly, for the plainness and certainty of it, in being distinguished; nor is it capable of less accurateness for measuring, than the Sights are for distinguishing. And it excels all the Common ways of Division in these particulars: 1. That it is made certain and not by guess, we being not at all to depend upon the care, credit and diligence of the Instrument-maker, in dividing, graving or numbering his Divisions, for the same Screw makes it from end to end, as you will see by and by. 2. That the Divisions are not at all difficult to be distinguished, and there is no uncertainty in the Fabric, nor can there be any reason of mistake, there being nothing to be looked after, but the Numbers expressed in Figures at large, sufficiently plain to any one that can read the Print of a large Church-Bible. It excels the Common ways thirdly, upon the account of its Compendium; for whereas by Tichoes or Hevelius' way, the Instrument must be made of 150 foot Radius at least, easily and certainly to discover and distinguish Seconds, in this way it may be made to do it within the compass of 3 foot Radius. And whereas in either of their ways, even in an Instrument of 150 foot Radius, the Divisions are not easily distinguished and discovered without the help of Glasses, in this way they are made so easy and plain, that a man cannot mistake, that is able by his naked eye to distinguish Decimals of an inch. Now that this is so, as I affirm, the Reader will easily understand, if he considers, first, that the bigness of a minute is hardly half an inch, in an Instrument of 150 foot Radins, and consequently the bigness of a second is but ●/120 of an inch, which to a good eye is but barely a visible point at the best advantage, and to most eyes is not distinguishable without much difficulty, and to very many not at all without the help of Glasses. Now though Hevelius pretends to be able to do much by the help of the new way of Nonnius, Vernier, or Hedreus, yet if he considers what I have now said, he will be of much another mind, a Radius of 10 foot being but a 15th. part of one of 150, and consequently every 120th. part of an inch, being no less than 15 whole Seconds. At least, I am sure, he will be convinced that his own is not true, if he look upon that Specimen of it which he hath printed in his Machina Coelestis, in the Plate T. with a moderately magnifying Glass, as I hinted to him before. He will further understand the truth of my Assertion, if he considers in the next place, that by the help of the Screw, I am able to make the bigness of a Minute as much as I please; for since in an Instrument of 5 foot Radius, a Degree is somewhat better than an inch, 'tis easy enough to understand, that there may be 30 Threads of a Screw in the length of an inch, and consequently there will be but 2 Minutes to fill up the whole Circle of the Index-Plate, and consequently if the Circle be 7 inches Diameter, the Circumference will be almost 22 inches about, and consequently the bigness of a Minute not less than 11, and the bigness of a Second not much less than the 5th. part of an inch. Now the Index-Plate e in the first and 11th. Figures, shows exactly the number of Revolutions, and the Hand 8 in the same Figures, shows the parts of a Revolution, and both these in Characters large and distinct enough; and therefore the certainty and truth of this Assertion cannot be further doubted.  
The way then for these Divisions is this: Make a Frame of a Quadrant of hammered Iron, after the manner expressed in the first Figure, and in the Centre thereof fix or raise a hollow Cylinder, whose hollow may be about a 40th. part of its Radius, and whose convex part may be about a 30th; leave this standing above the Plain of the Quadrant about 1/●● part of the Radius, let the outside of this Cylinder be made as exactly round as 'tis possible to be turned or wrought, then make a Ruler or Plate, with a round hole in it at one end, turned, grouned and fitted exactly about the abovementioned Cylinder, and as long as you design the Telescope for the Sights of the Quadrant, this by a Screw on the top thereof must be kept close and steady upon the said Cylinder: Upon the end next the Limb is to be fitted a Socket or Frame with Screws, to carry the Screw-Frame steady and firm, according to the contrivance expressed in the first and 11 Figures; this Plate must be filled or bended at that part of it which touches the Limb of the Quadrant, so as to lie obliquely to the Plain of the Quadrant, and to be parallel to the Plain of the Frame which carries the Screw, and upon the part beyond the Limb must be fixed with a Screw k, the Frame h h h, which carries the Screw 9 9 9, and the Index-Plate t 't; the contrivance of this Frame h h, is to keep the Screw 9 9 9 close against, and very steady to the Limb of the Quadrant, and is moved to and fro upon the Limb of the Quadrant b b b, by the help of the Screw turning upon and against the edge of the Quadrant; and this Screw by reason of its distance from the centre and eye, (the reason of the placing of which in that place you will understand by and by) being too far off to be reached by the hand, is turned by a small Rod of Iron, 0 0 0 in the first and 11 Figures lying by the side of the Ruler or Plate, which hath a small Wheel q q, at the end next the Limb, by which the Screw is turned round with it, and hath a small Handle or Windle p p next the Centre, by which it is made convenient to be so turned round. Upon the end of the abovementioned Screw-Frame h h, is fixed a round Plate t t, which is divided into 1, 2, 3, 4, or 5 hundred equal parts, according as it is in bigness, and as it shall be thought convenient, which Divisions are numbered and marked accordingly, serving to show what part of a Revolution is made of the aforesaid Screw; for the end of the Screw 9 9 9 coming out through the middle thereof, and a Hand 8 being fastened upon the said end, every turn of the Screw doth make a Revolution of the Index upon the said Plate; and consequently the motion of the arm made by one turn of the Screw, is actually and sensibly divided into 1, 2, 3, 4, or 5 hundred equal parts, which is so exceeding exact, and withal so Mathematically and Mechanically true, that 'tis hardly to be equallized by any other way of proceeding. This Description will be much better understood by the Explication of the Figure, and the several parts thereof.  
Let a a a a a, etc. represent the Frame of the Quadrant, consisting of 5 Bars, radiating from the Centre, steadyed all of them by a Quadrantal Limb, and a strait subtending Chord Bar; this whole Frame is to be made of very good Iron, partly wielded and partly sodered together with Brass; the breadth of the Bars may keep the same Proportions expressed in the Figure, and the thickness may be about 180 part of the Radius in large Instruments. In the Centre of this, out of the solid Bar, is to be raised a Cylinder, as d d, expressed above more plainly in the 2d. Figure; the outside of this Cylinder is to be turned and wrought, as Founders do their Stopcocks, as exactly as possibly it can be, and the end of the Iron Plate or movable arm c c c c, shaped as is expressed in the 3d. and first Figure, must be bored and wrought upon it very well, so as they may turn exactly true, evenly and smoothly, without any manner of sticking or shaking, which a good Workman will easily perform. This arm being put on the Cylinder, is screwed down fast by the help of a Screw-Plate, expressed in the 4th. and first Figures by e e, which hath two notches in it f f, by means whereof a Handle g g in the 6th. Figure, doth readily screw and unscrew it, as there is occasion. Between this screwed Plate and the hole of the Plate c c c c, is a thin Brass Plate, let on upon an 8 sided part of the Cylinder, that so the turning of the Plate c c c c, may not have any power to unscrew the Plate e e, which otherwise it is very apt to do. Why this Centre is thus made, and a hole left in the middle thereof, you will shortly understand more plainly. Upon the Iron Limb of the Quadrant last mentioned, is screwed and riveted a Limb of fine Brass, first cast into that shape, and then very well hammer-hardned and filled, represented in the Figure by b b b b: This, as I said, by many holes drilled through the Iron and the Brass, is screwed and riveted upon the iron Limb, so as about half an inch in a Quadrant of 5 foot Radius doth over-hang the iron Limb, and the ends thereof extend a considerable deal longer than the Quadrant, the reason and use of which you will by and by understand, when I give the Description of the Screw-Frame. The edge of this Brass Limb must be, by the help of the Plate cccc, and a File or Plain, cut very exactly round, to answer the Centre of the Quadrant, and the upper side thereof must be plained exactly smooth and flat, upon which Plain-side the Loop-holed Plate cccc must move, as is visible in the Figure. This Plate at two must be wrenched or wreithed, so that the Plain thereof must stand parallel to the Plain of the Index-Frame, and by the wreithing of it at two, as aforesaid, there is room left for the Screw to lie obliquely, without the Screws touching the aforesaid Plate, or grating against it. The reason why I put the Screw obliquely to the Plain of the Quadrant is, that that part of the Thread which toucheth the edge of the Limb, may be exactly at right Angles, or perpendicular to that Plain, and consequently that the Teeth upon the said edge, may likewise be exactly cross or perpendicular also, and consequently that no bending of the Rule cccc, (to the end of which the Frame of the Screw is fastened) may at all vary the Angle, nor any unequal thickness in the Limb of the Quadrant, but that the turning only of the Screw shall produce a variation, and that exactly proportionate to the number of Revolutions, and the parts thereof, showed by the Index.  
The way to know exactly what the obliquity of the Screw aught to be, to make the Teeth upon the Limb perpendicular, is to number how many Threads of the Screw there are in a known length, and what the Compass of the said Screw, or the Cylinder out of which it is made is, and multiplying the said Compass by the number of Revolutions into a Product, the Proportions of that Product to the known length, will give the obliquity of the Screw, the Product being the Radius, and the known length the Tangent of obliquity, thus; Suppose in the length of 4 inches, there be 83 Threads of the Screw, and that the Compass of the Cylinder of the Screw be Centesms of an inch, I multiply the by 83, the number of Revolutions, and it giveth me , that is 76 inches, and 36 Centesms of an inch, making this Product the Radius, and the known length, viz. 4 inches, the Tangent of the obliquity of the Thread of the Screw to the Axis thereof, or of the Axis of the Screw to the Plain of the Quadrant. The demonstration of this is so plain, that I need not insist upon it, for the length of the Thread of the Screw is the Secant, the Compass of the Cylinder is the Radius, and the bigness of the Thread, or the Distance between two Threads, is the Tangent, in a right angled Triangle, and the Screw is such a right angled Triangle, wound about a Cylinder, putting the Tangent thereof parallel to the Axis of the Cylinder, and consequently in the Mechanical trial of these Proportions, the more Threads are taken to make that comparison or measurement, the more exact is the inclination found. The consideration of which doth plainly show, how exact a way of Division this by the help of the Screw is, for the whole Quadrant is thereby resolved into one grand Diagonal, the same with the Triangle, the length of the Thread upon the Compass of the Cylinder being the Diagonal, and the Distance of the two ends of those Threads, in a Line parallel to the Axis, being the space to be divided by it, and consequently by augmenting the bigness or Compass of the Cylinder, and diminishing the Thread, you may augment the Diagonal in any Proportion assigned. Or by making the Hand or Index upon the end thereof, of double, treble, quadruple, decuple, etc. of the semi-Diameter of the Cylinder, out of which the Screw is made, you may duplicate, triplicate, quadruplicate, decuplicate, etc. the said length of the Diagonal, in Proportion to the space to be divided.  
The next thing then to be described is the Screw-Frame, made of Iron, much of the shape represented by h h h, in the first and 11 Figures: This Frame, by the help of a Screw through the aforesaid Plate, whose head is expressed by the round head k, is fixed on to the long Plate from the centre, and by the help of the Screw l, is forced and kept down very close, upon the edge of the Limb of the Quadrant; the Frame hath 4 Collars for the Screw-Pin to run against, which are indeed but half Collars, serving only to keep the Screw steady; two of these are made with most care, marked with m m, in the 11th. Figure against m i, doth rest the Shoulder of the Screw-Pin 3, which is kept close home against it, by the Cylinder g g, in the 10 and 11 Figures; the sharp Conical Point of this Screw 99, goeth into the Conical hole, at the end of the said Cylinder g g g. The shape of this Cylinder, and the Screw by which it is forced against the end of the Screw 99, is represented in the 10th. Figure; 7 in the 9th▪ Figure represents the Conical Point; 3 the place lying against the Choler m i; 6 the Screw that moves upon the edge of the Limb of the Quadrant; 5 the Nut or Opinion by which the Screw is turned by a Rod from the Centre, expressed alone in the 8th. Figure, but the manner how it lies in the Frame, is expressed by ppooo in Fig. 1.000 representing the Rod; pp the Handle by which it is turned; qq the Nut or Opinion that turneth the Opinion 5 of the Screw; sir the Collars or Holes that hold it fast to the movable Plate or arm of the Quadrant; s● representeth two small pieces that clip the edge of the Limb, and serve to keep the Screw-Frame steady and true in its oblique posture, and move equally on the Limb, by a strong springing of one side of it; tt representeth the Index-Plate, which is divided into what number of parts are thought necessary, 1, 2, 3, 4, or 5 hundred parts, according to the bigness of the Thread of the Screw at 6, a greater Thread requiring a more minute Division, and a smaller Thread requiring a more gross. These Divisions are pointed at by the Index 8 at the end of the Screw, and the number of Revolutions or Threads are marked on the Limb of the Quadrant, and pointed at by the Tongue ee, upon the which is fastened a small Pin f, serving to carry a Lens over the Point of the Tongue, which maketh the number of Threads appear more plain and big: The manner of doing which upon the Frame of the Screw, is so easy, that I shall not spend more time in the Explication thereof, and the manner of making the whole Instrument, will be easy enough to any ingenious Workman; but if any person desire one of them to be made, without troubling himself to direct and oversee a Workman, he may employ Mr. Tompion, a Watchmaker in Water-Lane near Fleetstreet; this person I recommend, as having employed him to make that which I have; whereby he hath seen and experienced the Difficulties that do occur therein, and finding him to be very careful and curious to observe and follow Directions, and to complete and perfect his Work, so as to make it accurate and fit for use.  
By the help of these Indices, 'twill be easy and plain to see how many Revolutions of the Screw, and what parts of a Revolution make a Quadrant of a Circle, and consequently 'twill be easy to make a small Table, which shall show what parts of a Quadrant, divided into Degrees, Minutes and Seconds, will be designed by the Revolutions, and parts of the Revolutions of the Screw. As for instance, If I find that 1600 Revolutions and make a Quadrant, then Revolutions make a Degree, and Millesms of a Revolution make a Minute, and about 5 Millesms make a Second, thence 'twill be easy to find (if you observe) an Angle to contain , that is, 294 Revolutions, and 358 Millesms of a Revolution, that the Content of that Angle in Degrees, Minutes and Seconds, is 16 Degrees, 32 Minutes, and 47 Seconds, which is plain enough, and much less subject to mistake, than the common way made use of. I shall therefore proceed to  
The third particular, wherein this Instrument excels all others, and that, is, That one Observator with a single glance of his eye, at the same moment doth distinctly see, that both the Sights of the Instrument are exactly directed to the desired Points of the two Objects, and this, though they be removed by never so great an Angle, nay, though they are opposite to each other directly in a Line. This, I question not, will by all that know any thing of Instruments, or celestial Observations, be accounted one of the greatest helps to such Observations, that was ever found out. For whereas other Instruments require two Observators, for taking a Distance in the Heavens, and Tycho generally made use of four, amongst which there was necessary so unanimous a concurrence in their readiness and certainty, that the failure of any one spoiled all the rest, and made the Observation become uncertain and of no use; and such Instruments as were contrived for one Observator, were accompanied with so great difficulty, in the adjusting to both the Objects, being both in a continual and swift motion, and but one to be seen at once, that they were generally left off and dis-used, there being so vast a trouble and fatigue of looking now upon one, then upon another, by many repeated trials, and so many new settings of the Instrument to the Objects in motion, before the Sights could be adjusted, besides the great uncertainty at the best, of several Minutes of the truth In this way, the Observator has no farther trouble, than first, to set the Plain of the Quadrant in the Plain of the Objects, and by the Screw to move the arm of his Instrument, till he perceive both the Objects to touch each other, in those Points he would measure the Distance between. That this is so, he will easily perceive, when he understands the method of so adapting two Telescopes, that by looking in at one small hole in the side of one of them, he will be able to see both those Objects distinctly to which they are directed, how much soever separated. The way then of doing it is in short this.  
Join them together at one end, by a hollow Joint that has a hole through it, about 1/4 of the hollow of the Tubes, prepare two square Tubes of Wood, Brass, Iron, etc. of what length you please, and directly against the Centre of this hole in the Joint, make a small hole, about the bigness of the blackest part or pupil of the eye, so as the eye looking in at that hole, may see perpendicularly into the lower Tube, then obliquely place two pieces of reflecting Metal, very well and truly polished, so as to reflect the Axis of both those Tubes, perpendicular or at right Angles, which is by fixing the Plain of the Plates, inclined to the said Axis, in an Angle of 45 Degrees, let the upper reflex Plate reach from the upper side of the Tube, so low as to touch the Axis or middle of the Tube, and let the lower extend over the whole Tube, from the top to the bottom, and from one side to the other. These will be known to be duly placed, if looking in at the small hole against the Centre of the Joint, the two round holes of the Tube do appear to the eye to coallesce into one, and that the eye sees directly through the lengths of them both alike. Then into these Tubes fit two Telescopes, with convex Eye-Glasses, and cross Threads for Sights in their Foci, that they may be both of them at due distance from the eye, looking in at the side-hole, then opening those Tubes upon the said Joint to any Angle, and looking in at the side hole, you shall plainly distinguish at once both the Objects, that are brought into the Tubes directly, and reflected up to the eye.  
That this may be the plainer understood, I shall add a Delineation thereof in plano.  
Let a a b b in the 12th. Figure represent the upper Tube, and c c c c the lower Tube, and let d d represent that part of the Joint, which belongs to the lower Tube, at one end, by which they are joined together, and can be opened in the manner of a Sector. Let i represent the hollow or centre of this Joint, which communicates the Cavities of the two Tubes. Let e e represent that part of the said Joint which belongs to the upper Tube, being only a hole through the lower side, big enough to encompass the Cylinder d d of the lower Tube; and let r r represent a Plate screwed or pinned on, to keep the parts of the Joint together instead of rivetting. Let s represent the hole in the side, by which the Eye h is to look in, and f the reflex Metal in the upper Tube, reaching only half way the Tube, and g g the reflex Metal in the under Tube, reaching over the whole Cavity; then will n o and p represent the Eye-Glass, Sight-Threads, and Object-Glass of the upper Tube, and k l and m the same parts in the lower, and whatever Angle the Tubes make to each other, whilst they open upon the beforementioned Joint, the Eye h looking in at s, will see directly by the Axis of them both, and see the Sight-Threads distinctly crossing the Points of the Objects, whose Distances are to be measured.  
These being thus explained, I suppose, it will be no difficult matter for any man to conceive, how these may be applied to the above-described Quadrant; for 'tis but supposing c c, the upper side of the under Tube in this Figure, to represent a p a p, the fixed side arm of the Quadrant, and d d the Joint of this, to represent d d the Joint of the Quadrant, and b b the under side of the upper Tube, to represent c c c the movable arm of the Quadrant, and applying two Tubes to these parts, and fitting them with reflecting Plates, Eye-Glasses, Sight-Threads, and Object-Glasses, at due Distances, the whole will be performed.  
These Tubes thus fitted, will serve to take any Angle less than a Quadrant, to what exactness i desired, but for bigger Angles, the Contrivance must be somewhat varied, the Description of which I shall now add.  
Let either of the two Tubes for the Sights, be made of double the length of the other, that is, let it be as long behind the Centre as before it, and make the Reflex-Glass, that it may be turned round, and reflect the Ray exactly backwards, as before it did forward, then fix into this other half of the Tube a Telescope-Sight, in all things fitted, adjusted, and like the other two, then adjust them, that they may look forwards and backwards in the same like, which being done, the Reader will easily understand how any Angle may be taken, even to the extent of two right ones: For 'tis plain enough, that the two Tubes I first described, applied to the Quadrant, will measure any Angle to a Quadrant or right Angle; and 'twill be as easy to understand, how by the help of the Reverse-Tube, any Angle between a Quadrant and two right Angles may be measured.  
To make this a little plainer to the Reader, let c c c c c in the 12th. Figure represent the under Tube or fixed Sight, s the hole or Eye-cell, t r a round piece carrying the reflex Metal g g; this is made to turn round, and the reflecting Metal g g being fixed to it within the Tube, is carried round also with it. Let s i k l m x represent the Ray passing forwards by the Eye-Glass, Thread-Sight, and Object-Glass; then this round piece t r being turned and made r t, as in the 13th. Figure, is represented, and with it the reflecting Metal g g; here marked q q, being turned also: the Line s q k l m y will represent the Ray reflected, and passing backwards by the reflex-Mettal q q, Eye-Glass k, Thread-Sight l, and Object-Glass y.  
The measure of the Angle is found by the same Apparatus or Screw-Plate; for as much as the Screw-Plate would show the Angle less than a Quadrant, if the forepart of the Tube were used, by so much is the Angle more than a Quadrant, if the reverse or back part of the Tube be used; and the same reason of the accurateness and certainty for the one, is good for the other, without being liable to any manner of Objection or Inconvenience.  
It remains therefore now only to show, First, How these two Perspective or Telescope Sights, placed within the same Tube, may be made to look exactly forwards or backwards in the same Line. And secondly, How they shall be adjusted to the Telescope, fixed upon the movable arm of the Quadrant, so as to know when the Division-Angle begins, and when they are opened to a Quadrant, right Angle, or 90 Degrees; for unless these be ascertained, and fixed to as great a measure of accurateness, as the contrivance of the Screw is capable of dividing, or the Telescope-Sights are capable of distinguishing, or the Perpendicularity ascertained, all the pains, care, industry, and curiosity, bestowed about the other, are of no use.  
First then, For fixing the Thread-Sights of the two Telescopes within the same Tube, so as to look directly forward and backwards, care must be taken, that every one of the four Glasses, that is to say, the two Object-Glasses, and the two Eye-Glasses, must be so steadily and securely fixed into the Tube, that they cannot by any means be stirred or removed; the manner of doing which, I suppose, so exceeding easy, that I need not spend time in describing a way to do it. Next, Sufficient care must be taken of the stiffness of the Tubes, that they may not warp or bend. Thirdly, One of the Thread-Sights must be fixed as firmly and securely as the Glasses, and so, that the crossing of the Threads may be, as near as possible, in the Axis of the Object and Eye-Glass, the other Thread-Sight must be left free, till by several trials it be found to stand exactly in the same Line with the first; the manner of doing which, I shall now describe.  
There being two Threads which cross each other, the one Perpendicular and the other Horizontal, care must be taken, that both these lie exactly in the same Lines with the Horizontal and Perpendicular Threads in the other Sights; and in order thereunto, there must be two Frames of Brass, represented in the 29 and 30 Figures of the 2d. Plate, of the bigness of the hollow of the Tube; these must have groves made in the Tube fit to receive them, in which they may by the help of Screws be moved, and made to slide to and fro, as there is occasion, for their adjusting. Next, They must lie so close together, that the Hairs may touch each other. And thirdly, They must cross exactly in the Focus of the Object and Eye-Glass. One of these Frames must carry the Perpendicular Thread, and by a Screw in the side of the Tube, must be movable to the right or left side, as there is occasion; the other Frame must carry the Horizontal Thread, and by a Screw in the top of the Tube, must be made to rise or fall in the Tube, as there is need. The Mechanical Fabric of which is so easy, that, I hope, I need not spend time in the further Description thereof, but refer the Reader to the 29 and 30 Figures.  
These things being thus done, from the top of some Turret, or any other Station, where two opposite places at a considerable distance, as half a mile, or a mile or two, can be plainly seen, find out two Points, which, at the first looking through your Glasses, you find to be shown out by the Crosses of the Thread-Sights, then note those Points very diligently, that you may be sure to find them and know them again, when you have removed the Glasses; this done, turn the ends of the Tube, and (if you were looking Eastwards and Westwards) turn that part towards the East which before looked Westwards, and vice versâ, and find out, the two Points you saw in the former Observation, then directing that part that hath the fixed Threads, to the Point that was seen before by the movable Threads, find out the other Point, which you will be sure to see within the compass of your Eye-Glass, and observe how far the cross Threads are now removed from it, either Northwards or Southwards, upwards or downwards, then, as near as you can, by your judgement half that Difference, and by the Screws move the Frames, that the Threads may stand in the middle between the two Points, then take notice again of the Points shown by the Threads, and turn the Tube again: Do this so many times, till you find upon converting the Tubes, that you see the same Points to be marked by the Crosses of the Thread-Sights, with which end soever you look on them, and then the Tube will be exact and fit for use.  
The reason of this adjusting will be sufficiently plain, to any one that shall consider the 14th. Figure: Where let v represent the middle of the Tube tub, or the place of the Eye, and let w represent the Object seen Westwards, and e the Object Eastwards, at the first view; then keeping the middle of the Tube exactly upon the same Point u, turn the end of the Tube t▪ towards the East, and the end b▪ towards the West, and find out first the Eastern Object e, and finding the other Cross to direct now to the Point p, and not to w, divide the Distance between the Point w, and the Point p, as exactly as you can, in half, which if you chance to hit exactly at first, it will be the middle Point m, but if you do not, but you rectify it only to r, then by the next turning of your Tube you will find s, where you must again rectify to half the Difference between s and r; now the Difference being grown yet less, you will a 3d. or 4th. time set it so exactly, as to see the Points m and e, which lie in the strait Line with the Centre of the double Tube.  
The 4th. thing wherein this Quadrant exceeds the Common, is for its accurateness for taking Altitudes; and this is done by the help of a Water-Level, for adjusting the exact Perpendicularity thereof. This Level may be made and fixed so exactly, that any Observator may be sure of the Level of his Instrument to a Second or two. The Level itself is nothing but a short Tube of Glass, about 6 or 8 inches long, Hermetically sealed at each end, and filled with a Liquor that will not freeze nor grow foul with standing.  
The Glass, as near as can be gotten, should be Cylindrical and strait, it being the better the nearer it be to a strait, provided it have a sensible bending or swelling in the middle, the gibbous part of which should be set upwards, and a proper Cell and Box made for it of Brass.  
This Glass is to be filled almost full of distilled Water, to which about a 3d. part of good Aquafortis or spirit of Niter hath been put▪ to keep the same from freezing, and also from growing foul, then carefully sealed up Hermetically, and placed in its Box of Brass, and with hard Cement fixed into the same, which by Screws is fixed to that side of the Quadrant, that is to lie Horizontal.  
The Brass Box being thus fixed to the right side of the long fixed Tube ap ap ap, and underneath the Quadrant, so as not to hinder the free movement of the arm c c c, as at x x; the next thing to be done, is by it to set the Quadrant truly Horizontal, which is thus performed.  
Setting the side a p a p a p Horizontal, and the Limb of the Quadrant upwards, and looking in at the Centre, take notice of two Objects in the Horizon opposite to each other, observe the limits of the bubble of Air on the top of the Liquor▪ on each side of the middle of the Level, and make a mark, then turning the ends of the Quadrant, set it, till the ends of the bubble stand as in the former Observation; then look again at those Objects in the Horizon, and find what the difference is between these opposite Objects, and those in the former Observation; then halve the difference between them as near as you can, and by your eye set the Sights to the middle between them, by inclining the Quadrant, then by the Screw that rectifies the Level, set the Glass-Level so, that the ends of the bubble may be equally distant from the middle, and convert the Quadrant again, and see if the ends of the bubble standing at the same marks, the two opposite Telescope-Sights do see the same Objects, for if so, you are assured of the perfect Horizontality of the Sights, upon the fixed arm of a p a p; but if you do not find it to direct to the same Objects, continue examining and converting, till you find it perfect.  
Now this way of Perpendicular being subject to the inconvenience of heat and cold, which doth rarify and condense the Liquor, and consequently make the bubble of Air less or more, care must be taken, to mark all the varieties of those kinds of the bubble, that are caused by the degrees of heat and cold, which you may thus easily effect.  
Reduce the Liquor in the Tube of the 24th. Figure, by the help of Ice and Salt, to as great a degree of cold as you can, then by the method newly directed, set the Quadrant Horizontal, and mark the two ends of the bubble with 44, then by gently applying heat to the ambient Air, warm likewise the Water, and observe the expansion thereof at both its ends, and mark them on the Glass with the point of a Diamant, as 33. 22. 11. 00. which being done, it will be exceeding easy at any time, to adjust the Quadrant to any accurateness desired, by being careful to see, that the two ends of the bubble be proportionably extended, as to 00▪ 11▪ 22. 33. 44, etc. or to any intermediate space.  
The Contrivance of fastening and adjusting this Level to the Quadrant or other Instrument, will be very easily understood, by the Delineation thereof in the 24th▪ Figure.  
Let a a a a represent the Frame or Plate of Brass, which by four Screws d d d d, is fixed to the Tube, as before. This Plate hath 4 upright Cheeks, b b, c c, between which the Brass Box e e e e, (into which the Cylindrical Glass-Level f f, is fixed with hard Cement) is held steady, without any manner of shaking. This Brass Box, at the end of it near the right hand, hath a Pevots, which are fitted exactly into 2 small holes in the Cheeks c c, and at the other end next the left hand, hath a small Screw-Pin g, which holds it down fast to the bottom Plate, and keeps it from rising out from between the Cheeks b b, which a very strong Spring lying underneath it, between the Plate a a, and the Box e e, would otherwise force it to do. By this Screw the Level is to be adjusted to the Sights of the Quadrant, by the way I just now described, and being once thus adjusted and fixed, 'tis not easily put out of order, without moving or altering the▪ Screw g, which may easily be prevented by 100 Contrivances.  
The Reason of the accurateness of this kind of Level, will be easily discovered, if we consider, that the upper part of the Tube being very near to a strait Line, is consequently either a part of a Circle of a very great Radius, or of some irregular Curve, very near of the same nature with a Circle, as to this business of Levelling, and consequently a Degree of the same will be proportionably large, and the flexure of the Tube may be made of a Curve of so large a Radius, that every Second of Inclination may cause a change in the Level of a very sensible length.  
This can hardly be performed by the ordinary way of Plumbets, without hanging from a vast height, which is not practicably to be performed, without almost infinite trouble, expense and difficulty, and when done, can be of no use in the World, as any one will grant, that considers the vast Apparatus that is requisite to obviate the great unsteadiness of Buildings, the motion of the Air, and a multitude of other encumbrances.  
Now the Curvature this way made may be a portion of a Sphere of 1000 foot Radius, or more, if it be desired, and consequently a Minute of the same will not be less than 2●/100 of a foot, and every Second will be almost half a Centesm of a foot, which is sufficiently distinguishable to the naked eye. So if the Glass Cylinder be 9 inches long, it may contain two whole Minutes of such a Circle between f and f, and one between 4 and 4, and consequently the said Glass may be set Horizontal to the certainty of a Second, which is hardly to be ascertained any other way.  
But there remains yet one great Difficulty, how to be able to make such a Curviture, for though the thing be true in theory, yet is it not without some trouble, put in practice. Very few Glass Canes are so conveniently bend, as is desirable, and 'tis as difficult to find them true strait.  
To prevent this, If Glass Canes be used, there must be much care taken, and many trials made, for the finding what pieces, and what side of those pieces will be most fit for this purpose, for our Glass-House Workmen know not yet a way, certainly to draw them of this or that curviture or straightness, nor are they easily ground into a straightness or curviture by the Glassgrinder afterwards, though that can be done with some trouble. But diligence and trial will quickly find some piece or other, that will be sufficiently exact for any trial, among those which are only drawn at the Glass-House. I made use of one of another form, such as is described in the 25th. Figure, which I found to do exceeding well, the dark part representing the Water, and the lighter part the Air. This was made of two Glasses, drawn in distinct Pipes at the Glass-House, but joined together in the Lamp, and the upper part of the larger or under Tube, was incurvated with its convexity downwards, so that the Water touched the middle part, and the bubbles of Air at each end thereof, communicated together by the small Pipe above. I tried also another way, by which I was more certain of the truth of the Curvity, and could make the Curvity of a greater Circle: This was by a long piece of a Looking-Glass-Plate, ground very smooth and polished, which by the help of Screws I bent upon the circular edges of a brass prismatical Box, and cemented the same very tied, with hard and soft Cement; this Plate had a hollow Channel ground in it the length thereof; which served to keep the bubble in the middle. By this means, 'tis not difficult to bend such a Plate, into the Curviture of a Circle of 50, 60, 100, 1000 foot Radius, and the Brass Box can easily be made to fill or empty, as there shall be occasion for the use thereof, so that the Bubble may be at any time left, of what bigness shall be desired. It will be convenient also to varnish the inside of this Brass Box with Lacker-Varnish, very thick and close, both to keep it from rusting, and also to preserve it from being corroded by Aquafortis, whensoever there shall be occasion to put it in, for the cleansing the inward tarnish and foulness of the Glass-Plate. This Curvity of the upper side of the Level may be made, by grinding the under side of such a long Plate of Looking Glass, upon a Convex Glass-Tool of 50, 60, 100, 1000 foot Radius, and polishing the same accordingly of that Figure: The Curvity of the said Plate is expressed in the 26th. Figure. Now what by this way may be done with Water and Bubbles of Air, the same may be done with the same Glasses turned upsidedown, by the help of an exactly round and polished Cylinder or Globule of Glass, Crystal, Cornelian, Agate, or other exceedingly hard and close Stone, after the manner represented in the 27th. Figure, for the Ball or Cylinder will naturally roll to the lowest part of the Concavity, and there stand. But in the doing of this, great care must be taken, that the Globule be exactly round and polished, and that the Concavity of the Plate be as smooth and well polished, and that they be both very clean and free from dust, otherwise the Cylinder or Globule will be apt to stand in a place where it should not, and consequently produce considerable errors.  
And here I cannot omit to take notice of a very curious Level, invented by Sr. Chr. Wren, for the taking the Horizon every way in a Circle. Which is done by a large Concave▪ ground and polished on a very large Sphere, and the Limb of it ground and polished on a flat, for by placing the same Horizontal, and rectifying it by a small quantity of Quicksilver, poured into the Concavity thereof, 'twill be easy, by looking by the flat polished Limb, to discover the true Horizon. The only inconvenience I find in it is, that the ☿ hath some kind of sticking to the Glass, but a small Crystal Bowl, I suppose, may remedy that inconvenience, and make it fit for use.  
The 5th. thing wherein this Instrument is made to excel others, is in its easinesses to be adjusted to the Objects, and in this, that being once adjusted, the whole Instrument is so ordered, as that it will remain constant to those Objects, though they are moved. The want of this is so great an inconvenience, in all other Instruments hitherto made use of, that almost all Observations have been thereby vitiated. And Hevelius, to prevent and obviate this, hath found out many Contrivances, but they are such, as though they do it in part, yet 'tis but in part, and that with much trouble and inconvenience. I need not spend time to show, how many inconveniences his way by 4 several Hand-Screws, to be managed by 2 Observators at the least, is subject to; they are indeed so many and so great, that it was not without very good reason, that he so often appeals to experience, for the truth is, there was great need of long practice and much experience, to be able to make an Observation in that way well, the removal of every one of those Screws, having an influence upon every one of the other, so as no Screw could be turned, but the whole Instrument was put out of its due situation, and both the Objects being continually in motion, the whole Instrument was to be rectified every moment. There was therefore necessary so great a judgement and dexterity, to manage every one of those Screws, that without an acquired habitude and handiness by long practice and experience, nothing could be done to any certainty, nay, not even to that little accurateness that the common Sights are able to reach. But this, though it were a very great unhappiness to Hevelius, that he was not furnished with better Contrivances, yet it no ways tends to his dispraise, for his most extraordinary and indefatigable care, pains and industry, is so much the more to be admired, esteemed and honoured, and will be so much the more, by such as have by experience found the difficulty, of making any one Observation certain in that way.  
But that he or any other, that hath a mind to make further Trials and Observations, may be freed from this intolerable trouble and difficulty, I have thought of this following Instrument, by means whereof the Quadrant being once adjusted, and set to the Objects, will continue to be so, for as long a time as shall be desired, without at all requiring the help of any one hand of the Observator, though he be but one.  
My way then in short is this: I make an Axis of very dry and strong Dram-Fir, of a bigness thick enough for its length, to defend it from bending; at the lower end of this, I fix into the middle of it, (well bound and hooped about with Iron) a Centre or Point of Steel, very well turned, hardened and sharp, which is to move in a conical hole fit to receive it, of as good and well hardened Steel; at the other end of this Rod, I fix another piece of Steel into the middle thereof, that, immediately contiguous to the Wood, hath a Neck very well turned and hardened, a little tapering from the Wood outward, which is to be moved in a Collar fit for it, as I shall show by and by; and at a convenient Distance from the said Neck, as at somewhat more than half the Radius of the Instrument, is made a Cylindrical Neck, fitted with a Collar of Brass, with a Joint, and other Apparatus, large enough to carry the Table and Instrument firm and true, without sliding or yielding in its Socket, after it be once set. This Axis by the Collar and conical hole below, I place parallel to the Axis, which by some trials is easily enough adjusted; about the Cylindrical Neck, at the upper end of this Axis, is a Socket of Brass fastened with a Screw, which Socket claspeth in a Joint, a short Arm, which hath at one end a Ball that is fitted into a Socket, that is fixed under the Table and Frame of the Quadrant, and at the other end a Counterpoise of Lead, to balance the weight of the whole Apparatus, about the Quadrant, upon the middle Line of the long Axis, than the Table and Quadrant is rectified, so as to lie in the Plain of the two celestial Objects, whether Planets or fixed Stars, and by the small Screws in the Sockets it is fixed in that Plain. What further adjusting is requisite, is done by the help of small Screws in the Quadrant itself, which are easily enough conceived without Description. The Table being adjusted to the Plain of the Objects, with the Quadrant on it, and all counterpoised pretty near by the poises underneath the Table, and the fixed Sight directed to one of the said Objects, the said Table and Instrument continues to be in that Plain, so long as is desired, without any farther trouble to the Observer, though the Objects continually change their places, and the fixed Sight remains directed at one of the Object, till the other can be found by the movable Sight. To effect which motion of the Table and Instrument, a Watch▪ work is fitted to the Axis, so as to make it move round in the same time, with a diurnal revolution of the Earth, and consequently to keep even pace with the seeming motion of the fixed Stars; the manner of doing which is thus: About some part of the Axis, where 'tis most convenient for the Room in which 'tis to be used, six an Octant of a Wheel of 3 foot Radius, let the Rim of this be turned true to the Centres of the Axis, and cut the edge thereof into 360 Teeth, there being so many half minutes of an hour in the 8th. part of a whole Revolution, though these minutes and hours which respect the fixed Stars, will be considerably shorter than the solar hours; then fit a Worm or Screw to these Teeth, that one revolution of the Worm being made in 1/●, a minute may move one Tooth forward; the revolution of the Worm is adjusted by a circular Pendulum, which is carried round by a Fly, moved in the form of a one wheeled Jack, from a swash toothed Wheel, fastened upon the shank of the Worm or Screw abovementioned; the weight that carries round this Wheel must hang upon the shank of the Worm, and must be of about a 3d. or 4th. part of the weight of the Quadrant and Table, that it may carry it round steadily and strongly; and the circular Pendulum must be so ordered, that the Observator may at any time of his Observation either shorten or produce the length thereof, so as to make it move quicker or slower, as there shall be occasion, which is done, by sliding the hole upon which the Pendulum makes its conical motion, a little higher or lower, without lifting up or letting down the Pendulum, or else by winding up the Thread of the Pendulum a little shorter, or letting it down a little longer, by the help of a Cylinder, above the hole or apex of the Cone, in which the Pendulum is moved.  
This whole Contrivance will be somewhat better understood by a Delineation. Let ab then in the 15th. Figure represent the Axis of Fir or Iron, c the conical Point at the bottom, d the conical centre or hole in which it is to move, e the Collar above, in which the tapering Neck of the iron far f is to be moved. The Axis of this is to be placed as exactly as may be, parallel to the Axis of the Earth: at the end or head of the Iron fg, is fitted a Socket hh, with a Screw 4, which will fix it to the head in any posture. This Socket hh in the 15 and 16 Figures, hath a large Joint to be stiffened by a Screw 5, in which Joint is moved a strong Bar of Iron, about 4 foot in length, to wit, 2 foot on each side of the Joint, the one end 6 hath a large weight or counterpoise of Lead 8, which serveth to counterbalance the whole weight of the Frame and Instrument upon the other, and can be screwed either nearer to or farther from the Joint, as there shall be occasion for poising; at the other end of the Iron is a large Ball of Iron 7, to which is fitted also a Socket of Brass 9, with a Screw to fix it and move it, as there shall be occasion. This Socket is fastened under the middle of a Table ss, upon the plain side of which the Quadrant is to lie. Upon some convenient part of this Axis is fixed an Octant or Sextant of a Circle, represented in the 15th. Figure edge-ways, and in the 17th. Figure broad-ways, by 33 two, whose circular edge 33 is cut into Teeth, as before is directed; unto these is adjusted a Worm or Screw k, which is the Axis or Arbour of the Wheel lll; this Wheel is moved round by the weight x, whose Line is coiled round the Barrel vu, and with it it turneth round the Fly nn, by the help of a Screw m, fixed upon the Arbour oo, in the manner of the Fly of a one wheeled Jack; this Fly moveth circularly the Pendulum pp, in the 15th. and 29th. Figures, which is shortened or lengthened, by slipping up and down the Cylinder qq, the Thread of the Pendulum being fastened at r.  
I shall not now spend any more time in the Explication of the making or contriving the circular Pendulum, reserving it for another opportunity and Discourse, wherein I shall show several useful Contrivances and Inventions about the same, and particularly about this and some other Experiments of motion, which was the cause of the Invention thereof by me long since, in the year 65. Upon which occasion, I cannot but take notice of a Publication, made by Christianus Hugenius Zulichemius Const. F. in his Book called, Horologium Oscillatorium sive demotu Pendulorum ad Horologia aptato demonstrationes Geome tricae; containing a short Description of a circular Pendulum-with somewhat about the Explication of it, without naming, me at all, as concerned therein, though I invented it, and brought it into use in the year 1665, and in the year 1666, I communicated it to the Royal Society, at their public Meetings, both as to the Theory and Practic thereof, and did more particularly explain the Isocrone motion of the Ball of a Pendulum, in a parabolical Superficies, and the Geometrical and Mechanical way of making the same move in such a Superficies, by the help of a Paraboloeid, which I caused also to be made and showed before the same Society, upon several days of their public Meeting, where besides many of the Society, were divers strangers of foreign parts. This many of the Royal Society can bear me witness, and the public Registers thereof do testify and make appear, and I was told by Sr. Robert Moray, that he did then write to Monsieur Zulichem concerning the same. But of this more hereafter, when I examine some other things in that Book, about finding the descent of heavy Bodies, and of finding the Longitude of places, and publish some more certain and practicable ways of doing them.  
This puts me in mind of publishing an Invention, which I made and produced before the Royal Society, in the same year 1666, much about the same time that I produced the Theory and Experiment of the circular Pendulum complete, which I called the perfection of Wheelwork, as being indeed founded on a principle capable of the greatest perfection can be imagined. It is in short, First, To make a piece of Wheelwork so, that both the Wheel and Opinion, though of never so small a size, shall have as great a number of Teeth as shall be desired, and yet neither weaken the Work, nor make the Teeth so small, as not to be practicable by any ordinary Workman. Next, That the motion shall be so equally communicated from the Wheel to the Opinion, that the Work being well made, there can be no inequality of force or motion communicated. Thirdly, That the Point of touching and bearing, shall be always in the Line that joins the 2 Centres together. Fourthly, That it shall have no manner of rubbing, nor be more difficult to be made then the common way of Wheelwork, save only that Workmen have not been accustomed to make it.  
First then, If there be a certain number, and no more of Teeth required to be made in a small Wheel, then must the Wheel and Opinion consist of several Plates or Wheels, lying one besides the other, in the manner they appear in the 20th. Figure. Where suppose it be required, that the Wheel shall have 1000 Teeth, and the Opinion 100, and yet that the Teeth both of the Wheel and Opinion have sufficient strength; take 10 Plates all of equal bigness and thickness, and by 2 or more Screws fix them firmly together, as if one Wheel, cut this Wheel into 100 Teeth, and complete it, then fit the middle hole upon the round neck of an Arbour, then unscrew the Plates, and place them in such order, that the Teeth may gradually follow each other, much after the manner as is expressed in the 20th. Figure, (though it be there very ill expressed, by reason of the mistake and failure of the Graver) and with such steps, that the last Tooth of one Degree, may within one step answer to the first Tooth of the next Degree. I call the 10 Teeth comprehended within the lighter part, abcd, or efgh, or iklm, a Degree of Teeth in steps, and dcfe, or hgkis, are Degrees of Notches between the Teeth, and the Tooth bc, which is the last towards the right hand, should have been placed within one step as low as eh, the first of the next Degree on the left side, (though it be much otherwise here graven) whence all the inequality in the touching, bearing or rubbing, in a Wheelwork thus well made, would be no more than what could be between the 2 next Teeth in one of the Degrees, which would be much less than a 10th. part, of what must necessarily happen in a Wheel of one Plate of 100 Teeth only.  
Secondly, If it be desired, that the Wheel and Opinion should have infinite Teeth, all the ends of the Teeth in the Degrees of the 20th. Figure, must by a Diagonal slope be filled off, and reduced to a strait, as in the 21, which may indeed be best made by one Plate of a convenient thickness, which thickness must be more or less according to the bigness of the sloped Tooth. And this is to be always observed in the cutting thereof, (though it be otherwise and very falsely expressed in the 21 Figure) that the end of one slope Tooth on the one side, be full as forward as the beginning of the next Tooth on the other, that is, that the end bc of one Tooth on the right side, be full as low as eh, the beginning of the next Tooth on the left side, (though by the Gravers mistake it be here quite otherwise expressed.) I shall not spend more time in explicating the Pinnions, rstu, rstu, of the 20 and 21 Figures, which are to answer the Teeth of the Wheels, they being plain enough to any person a little versed in Mechanics, and because the further and more full Explication of the form and reason of this and other Wheelwork, is comprised in another Discourse, which I may afterwards publish.  
But to proceed where I left at this Digression, to the finishing of the Description of the Instrument for moving the Quadrant, so as always to respect the Object. The conical hole, in which the end of the Axis is to move, may be made after the form expressed in the 18th. Figure, where aaaa represents an iron Frame screwed fast to the Floor, bbbb the iron piece, containing the conical steel hole, cccc 4 long Screws, by which the piece is moved and fixed in any part of the space, included within the Frame aaaa; this by a strong springing Frame underneath, is kept down close to the Superficies of the Floor, and cannot in any wise totter or shake. There is no great difficulty in the Contrivance, and therefore I shall proceed.  
In the next place then, having showed the way how to keep the Instrument, in the Plain of two Objects that are to be observed, I shall show, by what means a Quadrant may be kept always Perpendicular, and in the Azimuth of the celestial Object. And this I do, by a small addition to the former Contrivance; that is, Let ab in the 22 Figure, represent the Axis described in the former Contrivance, accommodated with all the Contrivances of the movable Centre below, of the Clockwork of the circular Pendulum, to keep it moving equally round in the middle, and of the Collar e above. But unto the small Neck f must be joined a semicircular piece of Iron cd, with a Center-hole in each arm at c and d, to receive the Pevots two, of the circular piece of Iron x, in the 22 and 23 Figures; upon the second Floor oo, must be steadfastly fixed a Bow or Frame of Iron hh, which must have a hole through it, exactly over the middle of the Plate x, this is to be a Collar for the Neck k, of a perpendicular Axis lk, which by means of a movable Centre fixed in the ceiling, in which the Point l moveth, may be exactly adjusted to a Perpendicularity; to this Axis at right Angles is fixed a Frame mm, steadied by the Brakets or Braces nn; upon this Frame the fixed Sights of the Quadrant, are laid and adjusted to an exact Horizontality, and the Plain of the Quadrant being once adjusted to the Plain of the celestial Object, will by the circular Pendulum moving the Axis ab, in an equal motion with that of the Object about the Axis of the Earth, be always kept in the Plain of the Object, whose Azimuth and Altitude is to be observed. Now the motion of the under or inclining Axis ab, is communicated to the perpendicular Axis lk, by means of the circular Plate x, in the 22 and 23 Figures, for the semicircular Arms cd of the lower Axis, taking hold of the Points 11 of the Plate x, and the semicircular Arms of the upper Axis, taking hold of the Points 22 of the said Plate, the perpendicular Axis is moved in a proportionate motion with the inclining Axis ab, which Proportion is Geometrically and strictly such as it ought to be, to keep the Plain of the Quadrant exactly in the Azimuth of the celestial Object, as any one never so little versed in Geometry, will easily find; and I shall hereafter more at large demonstrate, when I come to show, what use I have made of this Joint, for a universal Instrument for Dialling, for equalling of Time, for making the Hand of a Clock move in the Shadow of a Style, and for performing a multitude of other Mechanical Operations.  
The next thing I have to explain, is the way of finding how many Revolutions of the Screw, and what parts of a Revolution go to make a right Angle, or 90 Degrees upon the Quadrant. For the doing of which, I must, in a place where I can have a good Prospect for a semicircle, first direct both the Sights of the Telescopes directly at the same Object, and the same Point thereof, and then rectify the Indices to o, or the beginning of the Divisions; then I turn the Screw, till as near as I can measure with Compasses, the movable Telescope hath moved a Quadrant, and through the three Telescopes take notice of three Points in the Horizon, that is to say, two Points exactly opposite one to another, in respect of the Centre of the Quadrant, and a third pretty near the middle between them, in the same respect, which I further adjust thus; I showed before how I rectified the fixed Sights, so as to look exactly forwards and backwards, which being accordingly done, I observe the supposed right Angle, with the movable Sight on the Quadrant, and with the Sight fixed on the Quadrant looking forwards, and note diligently the two Objects pointed at; then without moving the Screw, or movable Arm upon the Quadrant, I find those Objects through the movable Sight, and the fixed Sight, looking backwards, and directing one of the Sights exactly to one Point, I observe, how much the other doth vary from the other Object, either by being within it or without it; then I half that Difference, as near as I can judge by my Sight, and move the movable Sight by the help of the Screw, so as to respect the middle Point: Then I observe this second found Angle, by the fixed Sight looking forwards, and by the movable Sight, and see whether there be any Difference, and if I find any, as near as possible, I adjust it again, to half this last Difference, and so continue to examine and adjust, till I am certain, that the Angles on each side of the movable Tube, between the same and the Sights, looking forwards and backwards, are equal to each other, and consequently are both right Angles, or Quadrants of a Circle. Which when I have found, I observe, by the Indices on the Screw-Plate and Limb, how many Revolutions, and what part of a Revolution, the Screw hath been turned to open that Angle; this Number I set, as the Number answering to 90 Degrees, and dividing that Number into 90 equal parts, I have the Numbers that belong to every Degree, and dividing the common Difference between them into 60 parts, I find the Numbers answering to the Minutes of the Quadrant, and dividing the common Difference between the Minutes into 60 parts, I easily make the Numbers answering to the Seconds; but these will be needless, for subducting the next Number, less than it in the Table from the Number observed, you have the Degree and Minute, and some Number perhaps over, which may presently be found by one small Table of the common Differences of Seconds. See page 55.  
Here me thinks I hear some object possibly, That the Divisions on the Quadrant, do not exactly correspond to the Divisions made on the Plate. I answer, That in part they do, and in part they do not. First, They concur, in that all the Divisions made by whole Revolutions, show exactly the same by the Indices, that they do upon the Quadrant. Secondly, I say, in part they do not, that is, the parts of any single Revolution, are not exactly and Mathematically the same pointed out by the Index, upon a Ring equally divided, that are made upon the Limb of the Quadrant. But yet, I say, they are sensibly equal even to the sense, assisted by a 60 foot Telescope, and consequently need no manner of rectification; but yet if any one will be so curious and nice, he may make the Divisions on the Index-Ring, according to the proportion of the Differences of the Tangents, that are subtended within half the compass of the distance of the two next Threads. As suppose in the abovementioned instance, half the Distance of two Threads be the Tangent of three Minutes, or thereabout; if we examine any large Table of Natural Tangents, we shall find the Differences between the Minutes themselves, even till six Minutes, (which is much more than double three) doth not differ above one or two parts of a thousand thousand, which is 1000 times more nice, than our Sight, even with Glasses, can arrive to, much less than will be the difference between the Differences of the Seconds; and therefore it will be a niceness merely notional, and of no use, and as such, aught to be omitted, and the plain and equal Divisions made use of, they being as to all sense true and perfect, and proper Divisions, though as to curiosity of Theory and Calculation, unequal.  
Now I have done, possibly some may say, To what purpose all this curiosity? To which I answer, That though possibly in many common cases 'tis of but little value, yet I conceive in general, that it is of infinite value, to any that shall design to improve Geography, Astronomy, Navigation, Philosophy, Physics, etc. And to instance in some particulars, I conceive,  
First, That one use of this Instrument, may be for taking the exact Refraction of the Air, from the Horizon to the Zenith; by which we shall be able not only to rectify all Observations, and clear them from Refractions, which in some Observations, especially those of Parallax, is absolutely necessary, but it may give us a new means to judge of the qualities and constitutions of the Air, as to the seasons of the year, and the temperature of the weather, which are to succeed. For 'tis most certain, that there is as great a variety in the refractiveness of the Air, as there is in the heat and cold, gravity and levity, dryness and moisture, rarefaction and condensation thereof, and sometimes when none of those do seem at all to be sensibly altered, its refractiveness hath been very much varied, which change does seem to proceed from some alterations in the upper Regions thereof, far removed from the Superficies of the Earth, and is sometimes many days in descending and fermenting, as it were deeper and deeper, into the lower Regions of the Air, before it descend so low as the bottom thereof next the Earth. But of this much more in another place.  
A second use is for regulating the places of the fixed Stars, as to their Longitudes and Latitudes, and Distances from one another, especially those within the Zodiac, by which we shall in a short time be able to judge, whether those Bodies that we account so fixed and constant, do not vary their Positions one to another, which I have very good grounds to believe they do.  
A third use of this Instrument, is for regulating the places of the Planets, by their Appulses to those fixed Stars, so that not only Astronomy will be perfected, but the Longitude of places upon the Earth, (a thing so highly advantageous for Trade and Navigation) will of consequence follow, which without such an Instrument as this, is in vain expected from the Heavens.  
A fourth use of this may be for stating the exact Latitude of places to a Second, whereby we shall quickly know, whether those Latitudes do vary, as well as the variation of the Loadstone, which hath been conjectured, not without somewhat of probability, but is hardly to be determined, without some such accurate way of Trial, as this Instrument is capable of performing.  
A fifth use of it may be, for examining what influence the approach or recess of the other Planets have upon the Earth, as to its Periodical motion, and what influence the Earth hath upon them as to theirs; for I have good ground to believe, each of these to have influence upon one another, and to cause such motions, as have hitherto much confounded all Astronomical Hypotheses and Calculations: Of which I shall say more on another occasion.  
A sixth use may be for measuring the quantity of a Degree upon the Earth; the best Experiment of that kind, that is yet public to the World, is that of Mr. Norwood, made between London and York: But if we examine with what Instruments he made it, we shall find, that he was not certain in either of his Latitudes to a Minute, and consequently could not be certain of the quantity of the Earth, answering to his supposed mark to two miles, and consequently it could not be made the common standard of all measure. But by the means of this Quadrant, all Latitudes may be certainly taken to a Second, and consequently the error in 150 miles, cannot be more than the 30th. part of a mile, and consequently a foot, or yard, or red, this way stated, cannot vary above a 6000 part of its length, which is sufficiently accurate for a universal and common standard of all measure and quantity, to which all other measures in the World should be referred and proportioned. This was the occasion of the contriving and making thereof; His Sacred Majesty having commanded me to see that Experiment accurately performed, and to give Him a true Account thereof, which had been before this performed, had not my indisposition of health prevented.  
A seventh use may be for measuring the Distance between two places, exactly in a strait Line. This it will perform to admiration, by the exactness of taking the Angles, if some length be exactly measured at the place that is to be the Object, insomuch that 'tis hardly possible, by any other means in the World, to come to that exactness, nay, though there were a continued Plain extended between the two places, whose Distances are to be found, and the same were carefully measured with Chains, Rods, or Wheels. By this means the Distance of a Ship on the Sea, can be found more exactly, than any other way whatsoever, by one or two Stations, and a multitude of Philosophical Trials under this Head, which are not practicably to be done with any tolerable accurateness, by other ways.  
An eight use may be for taking the exact Diameters of the Sun, Moon, and Planets, even to a Second, and the Distance of the smaller appearing Planets from the sixth Stars, near adjoining. Now because for this Design, it may perhaps seem a little too cumbersome, and by reason of its short Tubes, somewhat too small, I have therefore contrived an Instrument of 6 times the length or radius, which will take in an Angle of about 5 Degrees, and yet take in the whole Angle by one glance of the eye, and determine the measure thereof to less than a Second. I have likewise invented and made a new Helioscope, by which the Body of the Sun may be looked on as inoffensively to the eye, as a sheet of white Paper; of great use for such, as will make Physical Observations of that glorious Body. These I will in some ensuing Papers describe.  
A ninth may be for exactly taking the Level, for the conveyance of a River or Water from place to place; and under that Head of performing infinite of Philosophical Experiments, which can hardly be tried by any other way in the World, about the Refractiveness of the Air near the Earth, whereby distant places sometimes appear, and sometimes disappear, under the Horizon. By this means also the Rotundity of the Earth may be truly found, vastly surpassing any thing performed by the best Levels yet known. To this we may add, the height of Hills, if their distance be known, or their distance, if their height be known.  
I could have enlarged upon these, and have named divers others; but designing it only as an Answer to such, as may captiously put such a Question, I shall rather leave the pleasure of finding them, to such as shall really seek them, to be assisted thereby in their own undertake.  
FINIS.   

Errata.  
PAg. 2. l. 13. r. 9/10 p. 6. l. 14. r. aquilae. p. 13. l. 3. r. Mathematician. p. 15. l. 11. r. Fig. 32. p. 13. l. 28. r. Fig. 31. p. 18. l. 39 r. structuram. p. 21. l. 26. r. dena minuta. p. 21. l. 27. r. discriminatim. p. 22. l. 3. r. Fig. 35. p. 28. l. 34. r. quaedam. p. 32. l. 21. r. shaking. p. 33. l. 8. r. focus. p. 39 l. 28. r. res. p. 40. l. 11. r. admoveant. p. 40. l. 39 deal se.