THE Letter and Postscript to which this refers, are to be see ● in Mr. Oldenburg's Philosophical Transactions for the Month 〈…〉 July, 1670. which if the Reader please to peruse, he will see ho ● little it defer 〈…〉 to be so complained of by Dr. Holder.   

ERRATA.  
REad page 2. l. ult. not being so, p. 10. l. 32. impose on, p. 11. l. an't ●p. precedent to th' ● p. 15▪ ●●●. ingeniously, p. 16. l. 13. Labra, p. 17. l. 2. in different, l. 3. Dalgar ● l. 37. 〈…〉 me t ● p, ●, p. 19 l. penult. or a ●, p. 22. l. 3. prevailed with, p. 23. l. ●▪ knew, p. 25. l. 〈…〉 ght not be, p. 26. l. 25. renew, p. 27. l. 36. years or more, p. 3 ● l. 23. so too▪ (at lest of his supervising) p. 31. l. 39 owe no more, p. 32. l. 26. hath sin ● Points and Parenthesis omitted or misplaced, and some literal faults, the Read ●● is desired to correct or excuse.   

A DESCRIPTION OF HELIOSCOPES, And some other INSTRUMENTS MADE BY ROBERT HOOKE, Fellow of the Royal Society.  
Hos ego, etc. Sic vos non vobis——.  
LONDON, Printed by T. R. for John Martin Printer to the Royal Society, at the Bell in St. Paul's Churchyard, 1676.   

A DESCRIPTION OF HELIOSCOPES, And some other INSTRUMENTS.  
THE necessary avocations of business, and the urgent importunity of some, for the speedy publication of my Animadversions, made me conclude them in the Eleventh sheet, without staying to Explicate several things which I designed to go along with them. But having now retrieved a little more of leisure, both for Delineation and Description▪ for a further elucidation of what I have said, I shall make it my third Attempt, to explain;  
First, A Helloscope to look upon the body of the Sun, without any offence to the Observers eye.  
Secondly, A way of shortening reflective and refractive Telescopes.  
Thirdly, A way for using a Glass of any length, without moving the Tube.  
Fourthly, An Instrument for taking the Diameters of the Sun, Moon and Planets, or for taking any other Distances, to five or ten Degrees, to the certainty of a Second. Two of these I promised in the 78th. or last page of my Animadversions, and the other fall in as analogous to them.  
Fifthly, An Instrument for describing all manner of Dial's, by the tangent projection.  

Sixthly, The uses thereof;  
1. For adjusting the Hand of a Clock, so as to make it move in the shadow of a Dial, whose style is parallel to the Axis: Or,  
2. In the Azimuth of any Celestial Body, that is, in the shadow of an upright, or any other way inclining Style, upon any plain.  
3. For making a Hand move according to the true aquation of Time.  
4. For making all manner of Elliptical Dial's, in Mr. Foster's way, etc.  
5. For communicating a circular motion in a Curve Line, without any shaking: And for divers other excellent purposes.   
And first, For a HELIOSCOPE which shall so take off the brightness of the Sun, as that the weakest eye may look upon it, at any time, without the least offence. My contrivance is, By often reflecting the Rays from the surfaces of black Glasses, which are ground very exactly, fla ●, and very well polished, so to diminish the Radiations, that at length they become as weak and faint as those of the Moon in the twilight, so that one may with ease, and very much pleasure, view, examine and describe the phase of the Sun, and the maculae and faculae thereof, if any such happen to appear when the Observation is made, and it gives a good opportunity of discovering them, before we have any advertisement thereof from others. The reason of which will be sufficiently plain to such as consider, how great a quantity of the rays of Light is lost by every reflection, and that every reflection doth duplicate, triplicate, quadruplicate, quintuplicate, etc. the first proportion of loss. For Instance:  
Suppose I have a Helioscope made of an Object Glass, an Eye Glass, and four Reflecting Glasses, and that, by the first reflection, I lose ¼ of the Direct light, I affirm there will remain but 1/256 part of the Direct rays of the Sun, which can fall upon the eye at the last, for if every reflection doth lose ¼ of its Rays, and reflect but ¼, and that quarter loseth ¾, and reflects only ¼ of its received Light, there will remain but 1/16 part of the whole, and if this sixteenth part loseth three quarters of its Rays, and reflects only a fourth, it will follow, the remainder will only be 1/64 part of the whole, and if that be once more reflected, the Ray will return but with 1/256 part of its first light.  
This, although it be obvious, and easy enough now it is known, yet I do not find that any Person hath yet had thoughts of applying it to this use. The generality of Observers have hitherto made use of, either some very opacous and thick Glasses next the Eye, whether of red, green, blue, or purple Glass; others have diminished the Radiation, by covering the Glasses with a very thick and close coat of the soot of a Lamp; others, by casting the figure upon a piece of white Paper, whence 'tis reflected to the eye; Others have contracted the Aperture into a less circle, and thereby let in less Light, and so make use of one single Ray instead of a pencil of Rays; Others have expanded the figure of the Sun, by the help of Eye Glasses, into a circle of ten, twenty, or an hundred times its Diameter. But none of all these ways do come near this which I now describe by the help of three, four, or more Reflections, as any one upon trial will very plainly discover.  
First, As to the coloured Glasses, I cannot at all approve of them, because they tinge the Rays into the same colour, and consequently take off the truth of the appearance, as to Colour; besides, it superinduces a haziness and dimness upon the Figure, so that it doth not appear sharp and distinct. The same inconvenience is also produced by Monsieur Hugenius' way, of covering the Glass with the soot of a Lamp, though not to so great a degree. The Figure on paper, or a smooth white surface is not magnified enough, nor the difference of shadows so very distinct, though that doth very well, if the surface be very smooth, and the Object be magnified by a Hand Glass. That by the contracted Aperture is the worst of all, by reason of a certain propriety of Light not taken notice of yet by Optic Writers, the edges of Objects seeming ragged, of which I have hinted somewhat in my Animadversions, pag. 35, and shall shortly say much more, the whole ground of Optics depending thereon.  
The way of expanding the figure of the Sun by the Eye Glass, to me seems the best of all the rest, but that is apt to vitiate the Figure, to super-induce somewhat of Colour, and doth not give the smallest distinctions of lights and shadows, without somewhat of colour, and somewhat of haziness and dimness.  
The Glasses of this HELIOSCOPE may be made either by refracting or reflecting Spherical Glasses. The best way for taking in a large Angle, is, the using refracting Glasses, both for the Object and Eye Glasses; but the best way for taking in a small part, and for avoiding haziness, dimness, and colours, is, by Reflection, either in part, or in whole; that is, either to make the Object Glass only by way of Reflection, and the Eye Glass by that of Refraction, or, both the Object-glass and Eye Glass also by reflection, and to have no refraction at all. The several ways of doing which I have represented in the adjoining Table, wherein I have expressed ten several ways of placing the several Glasses, so as to be fit for the use designed.  
The first way represented in the first Figure, is, a sixty foot Object-Glass, contracted into a twelve foot Tube, by the help of four several Reflecting-plates placed between the Object-Glass and Eye-Glass. The Experiment of doing which, I produced and showed before the Royal Society, at divers of their public Meetings at Arundel house, in the year 1668, and it remains upon their Register.  
This (as I then showed) would be of exceeding great use in all manner of Perspectives and Telescopes, if we could find a good material that would make the Reflections very strong and full. And that would not be subject to lose its Figure, which all our specular Metals are very apt to do; for; by it, 'twould be possible to contract the Tubes for long Glasses into very short lengths, and so make them of easy use and manage.  
This I attempted with several sorts of Metal, made with ♃, ♀, ♂, Antimony and Arsenic, but most of these compound Metals I found to be very spongy, and consequently in the last polish to receive, though a very glaring polish, yet such as did much confound the Object by a kind of haziness, especially if Putty be used to glass it, and; for this purpose, Putty must not in any wise, that I yet know of, be used, it being so very apt to round off the edges of pores or scratches, which does much contribute to the haziness and confusion of the Object.  
If I made use of Glasses foiled with Quicksilver, which I found to give much the best reflection, yet I found this inconvenience, that a considerable part of the Ray was lost, by the double reflection at the unfoiled superficies of the Glass. The first from the surface of the Glass before it entered; this, as it weakened the Ray, so mingling with the other reflection that came from the bottom, it created some kind of haziness and confusion, if the two superficies of the Glass were parallel, but if they▪ were not parallel, it superinduced somewhat of Colour, unless it were helped by a contrary refraction in a second Reflecting-glass, after the manner of that which is delineated in the Fifth, Figure, where let a b represent the Object-Glass, ●g the first Reflecting-plate, whose thinnest side is to c and d ● the second Reflecting-plate, whose thinnest part is towards θ, which doth thereby take off the first Refraction of c g, and destroy the Colours superinduced by the first. The Ray also was weakened much more from the second reflection it suffered at the unfoiled superficies of the Glass, from the reflection of the Air, or ●ther, which is much stronger than that of Glass, at its re-entering into the Air. Besides this, I find that the substance of most Glass is so imperfectly mixed, that there is in the very best much of veinyness and inequality of Refraction in the parts thereof, and thence, though there were no visible vein appearing in the body of the Glass, and though both the surfaces thereof were very truly figured and polished, yet there was some kind of dimness superinduced upon the Objects, by the rays passing through those Glasses. But this was not in all, for I found some that did very well answer my expectation, and I am very apt to believe, that if a pot of Glass were made on purpose, by a way I know, the body thereof might be made perfectly clear, uniform, and transparent, without blebs, veins, or sands, which, when I have leisure and opportunity I design to experience farther▪ But this only by the by, in relation to the shortening the Tubes of Telescopes for the Moon, Planets, and other Objects, because it is not at all to our present purpose of making a Helioscope, where we make use only of the reflection of the first superficies of the Glass, and where our main aim and design, is, the loss of the strength and brightness of the Rays, and not for preserving the strength and briskness of the Rays, or augmenting them. And therefore for this use, the best material I have yet met with, is, black Glass, black Marble, and Glass of Antimony. For these substances being very dark and opaque, do reflect but a very small part of the Rays that fall upon it, and none of those that penetrate into it, especially if they be thick; and being of a very hard and permanent substance, are capable of receiving a very curious and exact polish, and qualified sufficiently to retain and keep it, without receiving injury from the Air, or ordinary wiping.  
But in the making of these Glasses for Long Telescopes, very great care and diligence must be used to make them of a true flat, and so much the more, by how much the nearer they are placed to the Object-Glass, and the further from the Eye-Glass; a little error at a great distance from the eye being vastly magnified to the eye at that distance, whereas a greater becomes insensible, if it be near the eye. Let a b, in the first, represent a sixty foot Glass, whose focus is at o; let a c d e f oh, and b g h i k oh, represent the two side Rays of the pencil of light, this Pencil, by the four Reflecting surfaces (γ , δ θ, ●●, η ) is broken into five shorter lengths (δ answering to c d, γ θ to g h, δ ● to d e, θ ● to h ●, ● η to e f, and ●● to i k, and lastly, η ● and ●, to f ● and k o) as will be sufficiently plain to any one that will but consider the Scheme.  
By this way four fifths of the length of the Tube is taken away, which is the most that can be taken away by four Reflections, every reflection running the whole length of the Tube, a lesser part of the length may be taken away in any proportion assigned, as in the second contrivance, described in the second Figure, two thirds are taken off, when the same Letters answer to the Object-Glass, Eye-Glass, the flexures of the side Rays of the Pencil, and the Reflecting-plates that make those flexures. The third and fourteenth Figures represent the Tube shortened by two or three reflections, and so serves to shorten the Tube by two thirds only. These are of use for a very strong Eye and with a small aperture of the Object-Glass, and when the Sun is near the Horizon, or its light is a little diminished, by a Fog, thin Clouds, or the like.  
If it be thought more convenient to have this long Tube to lie always Horizontal, and consequently, that there should be no need of having a Pole or Engine to raise the Tube: It may be framed somewhat like that in the fourth Figure, where the same Letters answer to all the parts abovementioned, or else like that in the sixth Figure, the Letters of both which being the same with the former, will easily explain them.  
Now in all these, and 20 other contrivances of this nature, with one, two, three, or four Reflecting-plates which may be presently thought of, the sight is directed exactly at the Sun, so that there will be little difficulty of finding it after the Glasses are fixed to their due lengths and positions.  
I explained also at the same time to the Royal Society, at their public Meeting at Arundel-house, several other ways of facilitating the use of very long Glasses, for other Objects in the heaven, by the help of one Reflecting plate only, and that was by a Tube fixed, either perpendicularly, horizontally, or obliquely, for it mattered not whether as to the seeing the Object in any part of the Heaven, supposing other circumstances hindered not, and the object could be as easily found as by the common Telescopes of the same length. But of these elsewhere.  
These contrivances with four Reflections, may be made use of by such whose sight is weak, but such as can endure it somewhat brighter, and would see the parts more strong, may make use of one of three Reflections only, like that of Fig. 14. which doth best suit my eye.  
Next, this Helioscope may be made by Reflection only, without any Refraction, and that may be done either in the manner of that in the seventh Figure, when a b represents a concave surface of a black Glass, whose focus is o, which, for Instance, we will suppose at the distance of forty foot, c d represents a clear plate of Glass of two flat surfaces, which are made not parallel but a little inclining, so as the reflection from that side which is furthest from the concave may be cast another way, and not fall at all upon the third Reflecting-plate ● η, and because the wedg-like form of this transparent plate of Glass, c d, will cause a refraction, and consequently a coloration of the Ray; therefore there must be another wedg-like Plate exactly as may be like the former, which at some distance, as at m p, where the reflection will not come to fall upon the Plate, ● η must be so fixed that the thinnest part of this may lie just upon the thickest part of c d, and the thickest of this over the thinnest of that, by which means both the false reflections and refractions will be removed. From ● η the Rays are reflected to γ θ, and from γ θ to o the focus, and so through the lens, z, to the eye x. This I take to be the best by Reflection; but it may be twenty other ways contrived, which I shall not now spend more time in describing, it being so easy a matter from the consideration of these I have mentioned, to make an hundred other variations of the principle.  
To this Helioscope may be fitted Instruments for measuring the Maculae, faculae, and Nebulae, visible in the body of the Sun, as also the spaces passed by them in a day, two, three, ten, etc. together with the variation of their Figures and Magnitudes; but the diameter of the body of the Sun will be better taken by the following Instrument. And by reason that it will be often necessary to draw their figures more exactly, the Engine that I have described in my Animadversions, in the 67, 68, and 69 pages, may be made use of to keep the Helioscope always directed at the body of the Sun, which will be no small ●ase to an Observer, that is to delineate the figures on Paper. When the brightness and radiation of the Moon, Venus or Jupiter, do somewhat offend the eye, they will presently lose their beards and look very distinct, if one reflection from glass be made use of in the Telescope.  
Another Instrument I promised to describe, is, for taking any such Diameters transits, or distance to the certainty of a second Minute, by which more may be done for the finding the Parallax of the superior Planets, and the Longitude on the Earth, then hath been ever yet done by all the Instruments that have been used in the World.  
1. This is made exactly, in all particulars like the Quadrant, as to its hollow centre, Screwd-limb, Screw-frame, and long Rod to turn the Screw from the Centre; and that the Screw-frame may be kept down the truer, upon the edge of the Limb, there should be made a small Arm to clasp behind the inward limb of the Instrument, after the manner represented in the 8th. Figure by w, by which means the Screw will be kept close, steady, and even to the outward edge of the Limb. The Letters in this 8th. Figure being the same with those of the 1 and 11th. Figures of the Animadversions, and representing the same parts, need no further explanation.  
2. Instead of this Screw upon a circular Limb, a Screw may be made to move upon a strait Limb, or Ruler; the end of which must move upon Centres or Rulers, the centres or axes of which Rulers must be exactly in the same line, when both the Perspective-sights are adjusted to the same Object, and the divisions began. The same thing may be done by a strait Screw, in the manner of a pair of dividing Compasses, where the same care must also be had, that the axes of the Rulers must be exactly in the same line, and the sides of the Incompassing-screw, being made of steel, must be made to spring about the long Straight-screw; this long Screw must be made of steel of half an inch of diameter at least, if it be made 18 inches long, and 'twill be best to screw it with a small thread, otherwise it will be apt to be moved out of a strait by screwing a large thread; and the thread, whether greater or less, must be made by degrees with a pair of cutting-stocks, that may be set closer every time of screwing.  
The manner of contriving the Centres and Sockets may be seen in the 12 and 13 Figures, where the 13 represents it in an end-way Prospect, and the 12 in a lateral or side-Prospect; 1 is the Rowler of the upper Tube, and 2 of the under, 33 the Screws to fasten them in the holes, 44 the encompassing or Socket-screw which springeth close to the Cylinder 5, 6 the Cylinderical smooth Socket which guides the Cylindricalscrew, so as to make its Axis pass exactly over the centre of the Rowler 22, and which, by means of a Ring 7 on the screw, keepeth the pointed-end thereof 8 against the stay or portance 9; 'tis not difficult how to make a Dividing-plate, and an Hand or Index thereunto, nor how it may be turned from the centre of the two Tubes by a long Rod, as in the 8th. Figure; nor will it be difficult, after it is known by Observation, how many Revolutions, and what part of a Revolution answers to five whole degrees, to calculate a Table of Subtenses, which shall show what part thereof goeth to make the subtense of every Minute and Second of the said angle.  
3. The same thing in the year 1665, I performed by a Rowler, rolling upon the limb of the Quadrant, by the help of two Wires which were coiled about those Rulers, and the ends thereof were fastened upon the limb of the Quadrant; for, by a large index on the end of this Rowler, I was able to move the arm of the Instrument to any fifth Second of the Quadrant, with great ease and certainty.  
I also at the same time made another Frame with a strait Screw, which opened to five degrees only, with Tumbrels or Rulers like a pair of dividing Compasses (after the same manner with this I have newly described, for taking Diameters or Distances to five degrees) and by the help of very curious Lines drawn upon a smooth Glass-plate, and Points very curiously made at every five degrees on the limb of the Quadrant, or Instrument on which it was fixed, and the help of a very deep Plano convex lens, whose plain side was turned downwards towards the Plate, and the convex side towards the eye, the said Frame was movable from five degrees to five degrees, upon the whole limb of the Quadrant or Instrument, by which▪ Instrument I could with great ease actually and accurate divide an angle into every five Seconds, and consequently take any angle to the accurateness of five Seconds; for, removing the Frame to the next division, less than the Angle desired, and then by the Glass, fixing one of the Arms that had the plate, exactly over the hole or point of division, by the Screw the remaining part of the Angle could be exactly measured.  
As to the method of dividing any of these, the best way will be to measure upon some Plain 1000, 1500, or 2000 foot in length, by two Rods of twenty foot long a piece, or else by Wires strained with weights, the way of which I shall shortly describe: Beginning from the very centre of the Instrument, and ● the end thereof, to set up so many Deal-boards joined to the end of each other in a straight line, or else to strain a pretty big Line, which shall cut the measured line of distance from the centre of the Instrument at Right-angles, and then by a Table of natural tangents, according to the distance from the centre of the Quadrant, put as Radius, to set and mark off upon those Board's or Lines the divisions of Degrees and Minutes, by Compasses or Rules, as exactly as may be, and mark them accordingly, that the Degrees may be distinguished very plainly from the Minutes: Then having adjusted the Instrument, so as to see the beginning of those Divisions through both the Tubes at once, to set both the Indices to o, or the beginning of the divisions, then keeping the undermost of the two Tubes fixed to the same place, so as still to respect the same point or beginning of the Divisions upon the Board's or Line, by the help of the Rod to turn the Screw or Rowl, till you find the upper Tube to respect the first minute, and then the first degree, and so till you see the last minute of the five whole degrees, or whatever Angle else you design it to take in; then (for the first and third way) reckon how many whole Revolutions, and what part of a Revolution goeth to make up that whole Angle, and subdivide the same by a small Table into Minutes and Seconds, and you will presently find by the Trial, that you will be able to divide to a strange accurateness upon those Board's, by the help of your Tubes and Screw, even at the distance of 1000, 1500, or 2000 foot, and even almost to equalise the Divisions by your Compasses, when at the very Board's. And by this you may easily examine, whether your Instrument doth make the sub-divisions exactly or not, which will be a great confirmation of the certainty and truth of your Instrument. But for the second way, by straight Screws, the Table of Sub-division into degrees, minutes, and seconds, must be proportioned according to the length of Subtenses answering to the Radius, which is the distance of the centre of the Rulers from the centre of the Instrument.  
Now, because in an Instrument of this bigness it will be somewhat troublesome to turn the whole Angle by the help of the Screw upon the Limb, which I find also is somewhat troublesome in the Instrument of three foo ● Radius, when the Angle is large, therefore for preventing of that trouble, and to be able immediately to open the Instrument to the Angle desired, or very near it. The Screw l (in the first Figure of my Animadv.) at the end of the movable Arm, is made, by unscrewing, to draw off the long Screw from touching the threads on the Limb, which being done, the Arm is at liberty to be moved to any part of the Quadrant, when by returning the Screw l, the Screw-frame and Screw is brought down again to take hold of the Threads of the Limb of the Instrument. The only care to be taken in this action, is, that neither the Index e e be at all moved out of its posture to the Index-frame h h, nor the Index 8 be moved at all about the rod of the Screw 999. It matters not at all though the Screw-rod 999 be turned round or moved, so as it be done by the Rod 000, and the handle thereof p p, or by the small handle x at the end of the Screw-rod, and that the Index 8 being very stiffly fix ● to the said Rod, be moved round with it by the same motion, without varying its position to the Rod; for being again brought down by the return of the Screw l, to take hold of the Threads of the Limb, into which it must be steadily guided by hand, the Index e e will show upon the Limb the number of Threads or Revolutions from the beginning, and the Index 8 will show what part of a Revolution there is to be joined to it.  
I hope I shall not need to spend time to explicate, how the Centre of these Tubes are to be made, nor how the Glasses and Thred-sights are to be fixed, nor need I much to show, how the Tubes may be stiffened to keep them from warping very much; A small matter of warping not creating any sensible error, I am not much conderned to prevent.  
If it be desired to make the Screw less, and only long enough to subtend one whole degree, which is enough in Instruments of fifty or sixty foot Radius, it may be done by a strait Screw very well, if care be used, which will very exactly take Diameters and Transits to a single Second.  
Another thing I promised further to explain, was, the contrivance of the Arms and Joint, mentioned in page 73, as a Universal Instrument for describing all manner of Dial's. For adjusting the Hand of a Clock, so as to make it move in the shadow of the Style of a Dial, that is, in the Plain of the right ascension of any Point, of the Ecliptic, or of the Heaven; or secondly, in the shadow of a perpendicular, or inclined Style: For dividing and describing all manner of Ellipses in any Analematical projection; and also, For making all manner of Elliptical Dial's in Mr. Foster' s way. For communicating a round motion through any irregularly bend way, without shaking or variation, and the like.  
First, The Instrument for describing all manner of Dial's by the Tangent projection, must be made in this manner, described in the 11th. Figure, in which there are two Axes or rods of Wire that are joined together by a Joint, which from the applicability of it to, and fitness for all kinds of motions and flexures, I call a Universal Joint. One of these Rods b b, is, by the help of a Frame a a, placed perpendicularly over the centre of the Dial, the sharp or pointed end thereof ● being sunk into the Centre, about which it is to be moved according as it shall be guided by the motion of the second Rod or Axis d d. This second Rod or Axis, is, by its Frame, to be moved and set so as to be parallel to the Axis of the World; then the Hand e e of this last being turned to the hour of Twelve on the Plate f f, the Hand of the first g g will point out upon the Dial-plain, the Meridian or Twelve of Clock Line.  
And so for describing any manner▪ of Dial, you have nothing to do but to find the Substile, and the altitude of the Style above the Plain, and to put the Axis in its due situation accordingly, that is, parallel to the Axis of the Earth, and then by the Plumbet at the end thereof to rectify the Meridian or Twelve of clock point: For then, by turning round the Axis or Rod d d by the handle, till you see the Index e e on the Axis to point at those Hours, halfs, quarters, or minutes you have a mind to take notice of in your Dial; by the second Index g g, you are directed to the true corresponding point in the Plain of the Dial itself. But in such Dial's as are in or near a Polar-plain, it will be convenient to make use of a small Thread to extend from the Cross, till it touch the Plain in the several hours, halfs, quarters, minutes, etc. The Arms of the Joint in this Operation are to be so fixed, that the axis of the Plate may cross the axis of the Rod at right Angles.  
The Universal Joint for all these manner of Operations, having not had time to describe the last Exercise, I shall now more particularly explain. It consisteth then of five several parts, each of which I shall describe in the 9 and 10 Fig.  
The two first parts are, the Rods and Axes A and B, on which the Semicircular Arms are fastened, which are to be joined together so, as that the motion of the one may communicate a motion to the other according to a proportion, which, for distinctions sake, I call Elliptical or Oblique.  
The two next parts are, the two Semicircular Arms C C and D D, which are fastened to the ends of those Rods, which serve to take hold of the four Points of the Ball, Circle, Medium, or Cross in the middle, X; each of these pair of Arms have two Centre-holes into which the sharp ends of the Medium are put, and by which the Elliptical or oblique proportion of Motion, is steadily, exactly, and most easily communicated from the one Rod or Axis to the other. These Centre-holes I call the Hands.  
The fifth and last thing, is, the Ball, Round-plate, Cross, or Medium X in the middle, taken hold of by the hands both of one and the other pair of Semicircular Arms, which, for distinctions sake, I henceforth call the Medium, and the two Points 11, taken hold of by the Hands of the Axis, I call the Points, and the other two Points 22, taken hold of by the second pair of Arms, I call the Pivots.  
First, for the Rods, they may be made of what bigness you think fit, according to the use for which you design the Instrument. The only care to be taken in the making of them, is, first that they may be exactly Cylindrical in those parts that move in Collars, and secondly, that the Axis or middle line of them do cut each other exactly in one point, which point must not vary upon any alteration or change of the Joint by bending the angle they make with each other, more or less, nor with the inclination of the Semicircular-arms to any desired obliquity, nor with the rotation or turning round of the whole Instrument. They require therefore a very dexterous, and a very knowing Artist, to make them as they ought to be, to perform their motion with exactness. Let a b then represent one of those Rods, and c d a second, which are turned exactly cylindrical within the Collars e f g and h, and these Collars are so disposed and fixed on some frame, that the middle line or axis of both these Cylinders may cut each other in the point e; if then both their necks and collars be wrought true and exact, the Axis or middle lines of them will always cut each other in the same point, howsoever they be turned round within their Collars; nor must this point i be varied, howsoever those two Axes are inclined to each other, so that though c d be inflected to l m, or n o, and so make either an obtuser or acuter Angle, yet the point i must be the centre of the Medium, where both the Axes concur and cut each other.  
Secondly, The Semicircular-arms may be made of what bigness, thickness, or strength, the occasion for which they are designed shall require; that is, if they are only to carry the Hand of a Clock in the shadow of a Common Dial, whether made after the Orthographical, Stereographical, or Horological projection; or if they are by an Annual motion to show the motion of the Sun in the Ecliptic, or the aequation of Time, a very small strength is sufficient; but if they are for carrying round a great Quadrant, such as that I have heretofore described, there they must be made stronger and more substantial. Care also must be had, that the inclining the Arms to any angle may not vary the centre of the Ball or Cross out of the point, where the two Axes cut each other. Both these Arms are to be made so as to be inclined to any angle; that is, that the Axis of the Medium, taken hold of by the Arms of Iron, may be made to incline to the axis of the Rod, on which they are in any angle desired, and being set to that Angle, to be steadily fixed, which may be done by a pin, screw or wedge; the way I make use of for the Azimuth-Instrument, described in the 73 p. of my Animadversions, is this which is delineated and explained in the 9th. Fig., where G represents a socket of Brass, movable cylindrically round about the end or neck B, of the Axis or Rod B B, the same with a b, in the 22 Fig. of my Animadversions, and fixable in any posture desired, by help of a side Screw h, such as is very commonly made use of for most Instruments that are fixed upon the end of a three legged Staff, and is commonly called a Cylinder and Socket; this Socket of Brass hath a small Rod of Iron, k, fixed into it at k, which is near the middle of its concave part, through this Rod there is made a small eye or hole, and through that hole a wedge-like pin m being thrust, serves to keep the Semicircular Iron-arms C C, steady and fixed in any posture they shall be rectified to. The Semicircular-arms C C, are to be made of very good Iron, or rather Steel, and to have a channel or grove quite through the middle of one of them, and extending the whole length of a quadrant of a Circle, namely from n to o, because, according to the variety of occasions, it may be varied to any point between n and o; and 'tis to be observed, that the Iron-rod k must be so far fixed out of the axis of the Socket g, as n is distant from i, or o from p the middle of the Iron-arms between i and i, that so when there is occasion, the Centre-hole or hands i may be moved to p and fastened. At q must be made a Joint in the Semicircular-arms, so that when the end n of the Arms is fixed in or near k, the other arm C may fall back from the point i, otherwise the circular motion, in many cases, cannot be continued quite round, and communicated from one Rod to the other, by help of the Medium or Plate x. The several pieces of this Joint, as they are apart and distinct, you may see in the 9th. Figure, and as they are joined all together ●it for motion you may see in the tenth Figure, to which also the description of every part is adjoined in words referred to by the help of Literal marks, which, I hope, will make it sufficiently plain to any Artist to understand.  
Thirdly, The medium Ball or Cross X, must be made of a bigness suitable to the Arms and Cylinders, and great care must be had that all the ends, points, or handles, lie exactly in the same plain, and that they be all equally distant from their Centre, at least, that any two opposite ones be so made, because it is not absolutely necessary that they should be so all four, though in most cases it be best; and farther, the Handles or Pivots ought to be exactly round, conical, or cylindrical, and the middle lines of them to cut each other at right angles, or upon a square; and in general, that all things about the said Joint be so contrived and wrought that the Axis of the two Rods may always cut each other in the centre of the medium Cross or Plate, and that the said Centre, whatever change happens to the Joint, may always keep exactly in the same very point, without any alteration.  
The shape of this Medium may be either, a Cross whose four ends hath each of them a Cylinder, which is the weakest way, 'tis described in the 9 and 10th. Figures by the Cross X; or secondly, it may be made of a thick plate of Brass, upon the edge of which are fixed four Pivots, which serve for the handles of the Iron-arms to take hold of; this is much better than the former, but hath not that strength and steadiness that a large Ball hath, which is the way I most approve of, as being strong, steady, and handsome; these are delineated in the aforesaid Figures, by X x, and X x x.  
If it be an Elliptical Dial to be described by the Orthographical projection, the former way for describing Tangent Dial's, gives the lines that divide the Ellipsis of the Equinox in its true proportions: and if you would have the Lines that divide the Ellipsis of either Tropic, or of any other parallel Circle, you must rectify the Semicircular Arms C C of the Axis B B, to the degree of the declination of that Parallel, and them proceeding as before, you have the Lines which from the aforesaid Circle divide the Ellipsis of that Parallel accordingly. Perpendiculars also, let fall from the ends of the Cross 11, give the true Ellipsis in the Orthographical projection answering to that Parallel.  
These Lines thus found, are the true azimuth Lines of the points or divisions of that Parallel, and are this way traced out exactly, without any trouble of Calculation, which for some purposes, in Surveying, Navigation, etc. are of very great use, as I shall afterwards show.  
The Universality of this Contrivance, for resolving almost all Spherical Questions, makes it of very great use in Navigation, if it be adapted as it ought to be, especially for the Common Seaman's use, who, with a very few Rules, will be able immediately to find the hour, and azimuth of any point in the Heaven, sufficiently accurate for most Observations that can be made at Sea; of which more hereafter.  
For making the Hand or Index of a Clock move in the shadow of the Style, made upon the Face of the Dial, and exposed to the Sun, this Joint, being made to join the arbour of the Wheel that goeth round in twenty four hours, with the arbour of the hand, performeth it without any other Wheel or Pinion in the Dial or Face part of the Clock; if the Arbour of the Clock that should have carried the Hand round in twenty four hours, be made to have the same inclination to the plain of the Dial that the Axis hath, whether parallel to the Axis or not, it matters not at all, so that the Hand be rectified accordingly as it ought to be, and that the Style of the Dial ariseth from the centre of the Dial, out-through which the Arbour is produced for carrying the Hand, and placed in its Parallel respect to the Axis, as it ought to be for a Tangent Dial. For the shadow-Line of the Axis upon the plain of the Dial, being always carried round the centre of a Dial in a plain, which passeth through the Axis or Style, and maketh equal progressions about it in equal spaces of Time, and unequal progressions upon the Dial-plain, according to the proportion of Inclination, and the whole Revolution being performed in twenty four hours, and the Hand of the Clock upon the Face of the Dial being always moved in a plain which passeth through the Arbour of the Clock, and maketh equal progressions in equal spaces about the said Arbour, but unequal progression about the Centre of the Dial, according to the differing Inclinations: And those Inclinations being both in the Sundial and Clock-Dial the same, it will follow, that ●he Hand of the Clock must always move in the shadow of the Style, if the Hand be once rectified to the true Plain, and the Axis or Arbour make its Revolution as it ought to do in twenty four hours.  
If it be further desired, for the ease of taking Azimuths and Altitudes, that the Arm of the Azimuth quadrant that is once adjusted to the Celestial Object, should, by the aforesaid Joint or Instrument, be kept always respecting and following the said Object in its Diurnal motion, it may be very easily performed by the help of a small perpendicular Ruler, whose lower end is Jointed into either of the Arms 11, of the circular Plate X, in the 22 and 23d. Figure of my Animadversions, and the upper end jointed into the movable Arm, at the same distance from the Centre of the Quadrant that the lower end is from the centre of the Plate X, and that the centre of the Quadrant be set exactly perpendicular over the centre of X; but then the divisions by the help of the Screw cannot be made use of, because the Clockwork itself is to turn and move the Arm: But it may be done by any Quadrant, where the minute Divisions are performed by the help of Diagonals. For the Arms of the Circular-plate 11 being always moved in the superficies of the Cone described, by the radiation from the Celestial Object to the centre of the Plate X, that is to say, the Line that passes through the Centre of the said Plate, and through the two Points 1 1, being always directed to the Celestial Object, if the Arm of the Quadrant be moved perpendicular over it, and parallel to it, that also must be always directed to it. And hence it may very easily be conceived, how the aforesaid Semicircular Arms may be readily and certainly rectified to any Celestial Object; that is, by fixing Telescopes or Common-sights upon the Circular-plate, so as the Axis of them may be parallel to the Line through 1 1, and losing the Screw h to rectify it to the Object by the sight, and then immediately to fix it in the said posture by the aforesaid Screw; the Clockwork of the said Instrument having been before that put into motion. The reason of all which will easily appear to any one that throughly considers, that all Celestial Objects seem, by the diurnal motion of the Earth, to move equally from East to West about the Axis of it, and would all do exactly so, were they not somewhat varied by their own proper periodical revolutions, which though it doth indeed make a real difference between their velocities about the Axis of the Earth, yet that difference is but small; and the same circular Pendulum will serve both for the Sun, Moon, Planets, and Stars, if at least the Pendulum p, in the fifteenth Figure, be a little lengthened or shortened, by lifting up or letting down the Rod q q, in proportion as the Body k moveth swifter or slower. And 'twill not be difficult to mark upon the Rod q q, the appropriated length of the Pendulum for the Sun, Moon, or Stars; but this only by the by.  
If in the next place it be desired, that the Hand of the Clock should be always carried round upon the face of the Clock, in the shadow of a Style perpendicular to that plain, by reason that the declination of the Sun daily varieth, the angles of the shadow about that Style varieth also, and consequently the inclination of the plate of the Joint to the Axis or Arbour must vary also, and that variation must always be the same with the variation of the declination of the Sun, which is twenty ways mechanically performable in Clockwork, so that the motion shall be performed by the Clockwork alone, without touching it with the hand. All the other directions that are requisite to adjust the Clockwork to such a Dial, is, only to make the Arbour of the Clockwork to have the same inclination to the plain of the Dial, that the Axis of the Earth, or a line parallel to it hath; and rectifying the Hand into the true plain of the Axis, or Inclined arbour, the equality of the motion of the Clockwork, according to the diurnal and annual motion of the Sun, we suppose also to be provided for.  
If the Hand of the Clock be desired to be moved in the shadow of any other straight Style, howsoever inclined to the plain of the Dial, then must there be another Joint like the former, added to the end of that Axis which was perpendicular to the plain of the Dial, and all the three Axes must be situate in respect of the Plain, in which the Hand on the end of the last is to move, that the inclination of the said Axes to each other, may represent the inclination of the Axis to the perpendicular axis of the Plain, and of that perpendicular Axis to the axis of the Style. Or, which is somewhat shorter, and may be made handsome enough, Let the two ends of the Hand represent the two points of the second circular Plate or Globe, extended long enough to reach to the hour Circle, then let the axis of this second Arm be placed in the axis of the inclined Style, and let the axis of equal motion, representing the axis of the diurnal motion of the Earth, be placed with such inclination to it, as the axis of the Earth hath to the oblique Axis or Style of the Dial, and the motion will be most exactly performed mechanically, and according to the truth of Geometry and Calculation.  
Now, in all these motions, care must be taken, to provide that the inclination of the declination of the Sun from the Equinoctial, be expressed by the ends 11, in the 22 and 23 Figures of the second Plate of my Animadversions, of the Cross, taken hold of by the semicircular arms c d, upon the end of the first Axis; that is, that the said arms may, by their revolution, make the line of the Cross describe such a cone about the first Axis, as the motion of the Sun doth about the axis of the Earth, making the centre of the Earth the apex of that Cone; which will be done, if the said semicircular Arms be moved, and set to the declination of the Sun for that day. Or, that an additional motion be added to the first Axis, that the Clock itself may perform it. This may be done twenty ways easily enough, which I suppose will be sufficiently obvious to any knowing Mechanic, and that without the help of Tooth-wheels or Pinions, which in works of this nature are in no wise to be made use of, by reason of their shaking and uncertainty, which I shall elsewhere describe.  
There is one only difficulty in this motion, and that is only in such Objects as pass over, or very near the Zenith or Nadir of the place, for in those cases, when the Object comes very near the Zenith, the obliquity of the motion of the one to the other is so very great, that the first Axis doth not move the second without some difficulty: But to remedy this, the expedient is as easy, and that is, by having a little barrel about the perpendicular Arm, to carry it forward as far and as fast as the first Inclined axis will permit it; which weight may be removed as soon as the Object is a little way past the Zenith.  
The next use that may be made of this, is, for carrying the Hand of a Clock so, as always to move over that point of the Ecliptic in which the Sun is, in a Stereographical projection of the Sphere upon the Plain of the Equinoctial, or in an Orthographical projection of the said Sphere upon the same Plain, so as to express thereby not only the differing right ascensions, but the anomaly also of the Sun's motion in the eccentric of the Ecliptic. And by this means the Face of the Clock may be made by a Planispherical projection, to represent the motion of all the Stars appearing in any Horizon that is not too near the Equinoctial, their Risings, settings, culminating, azimuths, and almicauters; Risings and settings of the Sun, the lengths of the Days and Nights, and of the Twilights and Dawnings, and many other Problems of the Sphere. And, which is a consequent of this, it may be made to show the equation of Time, which is necessary to be made use of for setting a pendulum Clock by the Sun, the manner of doing which I must refer to another opportunity, as I must also the use of this Joint, for drawing Ellipses, drilling and boring of bending Holes, for turning Elliptical and Swash-work, till I publish my description of a Turning Engine, capable to turn all manner of Conical Lines, and Conoeidical; all manner of Foliage and Flower-work, all variety of Basket or Breaded-work, all variety of Spiral and Helical-work, serving for the imitation of the various forms and carvings of all sorts of Shells▪ for cylindrical and conical Screws; all variety of Embossments and Statues; all variety of edged and Wheel-like work; all variety of Regularly shaped Bodies, whether the five Regular bodies of Plato, or produced from those by various sections or additions, of which the variety is infinite; all variety of bended Cylinders or Cones, and those whether round, in the manner of an Oxes-horn, or compressed and angular, like those of a Ram or Goat; for all manner of Swasht-work, Comprest-work, etc. every of which principal parts hath a vast variety, and the compound and decompound principles have a variety almost infinite.   

Appendix. Concerning the Eclipse of the Moon, observed in London.  
JAnuary the first, 1674/5;, being at Sr. Ionas Mores in the Tower of London, and making use of a Telescope of eight foot, and my pocket-Watch, whose balance was regulated with springs, I observed the Eclipse of the Moon, which began at about twenty minutes after five, the penumbra very much cheating the naked eye; for the Penumbra had darkened that side of the Moon, next the spot Grimaldi, about half an hour before, and grew darker and darker towards the edge where the Umbra entered, so that if the light of the Moon were diminished either by reflection upon dark Glass, or looking through a small hole, between a quarter and a third part of the Moon seemed eclipsed before the Umbra entered; but the Telescope discovered it plainly to be no true umbra, but penumbra.  
This I note, because such Persons as do not make use of a Telescope, but only of their naked eye, are very apt to be much deceived in their estimation of the beginning and end of the Eclipse.  
At 5. 48. we judged by the Telescope that the Moon was eclipsed six digits, or half; at 6. 19 the total Eclipse began, when the Moon appeared of a very red colour, especially towards that part of the Limb where the direct Rays left it, which was at the Mare Crisium, which is opposite to Grimaldi. Now the Sky being somewhat clearer, it being before hazy, with the Telescope I began to discover a great number of small Stars about the Moon, which appeared yet much more conspicuous, after I had taken off the apperture from the Object-glass, and amongst the rest, one seemed very conspicuous, and lay in the way of the Moon, which I diligently watched and observed, that it was just covered by the Moon at 6h. 47′. 30″. the Moon first covering it with that part of it which was almost perpendicularly under the centre of the Moon.  
About three quarters of an hour after the total immersion, the body of the Moon was exceeding dark, and almost unperceivable, being then near the centre of the Umbra, and afterwards the Eastermost or foremost part of the Limb of the Moon began to be enlightened, whereas before the West ●most Limb had been the brightest. This was also very notable, that that part of the Moon that was towards the North-Pole, a pretty while before the emersion of the Moon out of the total Eclipse, and even till the very emersion, and somewhat after too, appeared enlightened with a much brisker light than any other part of the body, except that which was next the Limb where the light again entered. From what cause this should happen, I know not; possibly it might be caused by a greater refraction of the Air near the North-Pole of the Earth, and I am much troubled, that I had not taken notice whether the like phenomenon had not happened to the body of the Moon before it had passed the centre of the Umbra. It was very manifest, that there was a considerable quantity of light that kept that Limb of the Moon which was next the light, conspicuous by the Telescope all the time of the total Eclipse; and 'tis very rational to ascribe it to the Rays of the Sun, refracted by the Air, or atmosphere of the Earth.  
I was very well pleased to observe the Moon to cover several small Stars that lay in its way, but I kept no account of them, but only watched diligently when the Star that entered behind the Moon at 6. 47. 30. would come out again, which I found it to do at 7. 30′. seeing it at the very moment of time that it began to appear again. And it was also at the same instant discovered by Sr. Ionas More, who was expecting it with another Tube.  
At 7. 58. the body of the Moon first emerged out of the Umbra at the spot Grimaldi, and soon after all those small Stars that were conspicuous before about the body of the Moon, vanished. However I had, before its first emersion out of the shadow, taken a little draught of the small Stars, according to their several positures and magnitudes, only by guess, that I might a week after, when the Moon was gone farther off, inquire what that Star was that had suffered so conspicuous an Eclipse, and that thereby I might the more certainly determine the true place of the Sun and Moon at that instant, which I found to be that in Bayer, touching the Ecliptic, in about 21°. 40′. of Cancer. The Umbra ceased wholly at eight of the Clock and five minutes, though the Penumbra then possessed almost a third of the Moon's Diameter, and lasted near half an hour after, before that side of the Moon was perfectly enlightened like the other.  
There was one Phenomenon very remarkable, which I took more especial notice of, as seeming to me very considerable for the determining that controversy, whether the Moon have an atmosphere or not, like that of the Earth? And that was, that after the Moon was entered wholly into the Umbra of the Earth, that part of the Limb of the Moon which was last enlightened, continued for a considerable while to have a very great brightness upon it, which extended on each side that part of the Limb, both northwards and southwards, to about a quadrant of the Moon's Limb, making a representation almost of a New Moon about a day or two old, and as the body of the Moon was immerged deeper into the shadow, so this brightness or light grew fainter and fainter, but still seemed to spread itself very far upon the Limb of the Moon only, and not upon the body thereof. That which was spread into the body being much fainter and weaker, and seeming (as I before noted) to proceed from the refraction of the Atmospheres of the Earth. Nor was this only conspicuous at the Moons entering into the total darkness, but as remarkable also at the exitus thereof out of the same, insomuch that some of those Persons, who at the same time viewed the same with me, verily believed the Moon was not wholly eclipsed so soon as really it was, nor continued so long in that obscurity, as very visibly it did by the space of two or three minutes. For I took especial notice when this enlightening of the Limb began again to appear, and I observed its increase, and spreading about the Limb, till the very instant that the immediate light of the Sun touched the very extremity of the Limb it self, which was indeed so very briskly bright and strong, that it did not only soon make the other light disappear, but also all the Telescopical Stars that were near to it, and towards the end also many of the more conspicuous Stars, especially such as were not far from the body of the Moon.   

Postscript.  
I Should have here taken leave of my Reader for this time, but that finding in the Transactions a passage inserted out of the French Journal de Scavans, about the invention of applying a Spring to the Balance of a Watch, for the regulating the motion thereof, without at all taking notice that this Invention was first found out by an Englishman, and long since published to the World: I must beg the Readers patience, whilst I, in vindication of my own right against some unhandsome proceedings, do acquaint him with the state of this matter.  
About seventeen years since, being very inquisitive about the regulating the measure of Time, in order to find the Longitude, I did from an Art of Invention, or mechanical Algebra (which I was then Master of) find out and perfect this contrivance, both as to the Theory and Experimental verification thereof, of which I then discoursed to divers of my Friends, but concealed the modus.  
About fifteen years since, to wit, in the year 1660, presently after his Majesty's happy Restauration, I was in treaty with several Persons of Honour (some of which are yet living, though one of them is since dead, but I have sufficient evidence to produce in his own writing that he was one) for the discovery thereof, upon proposed Articles of encouragement. This I can prove by undeniable Witnesses yet living, and I have still all the Papers, Articles, and Transactions of this matter by me, in their own hand-writing.  
In order to bring this Treaty to pass, I was necessitated to discover something of Invention about measuring Time, which was, this way of applying Springs to the arbour of the Balance of a Watch, for the regulating the vibrations thereof in all postures. And this I did, to the end that I might gain somewhat of belief in those Noble Persons (with whom I was to treat) That I had somewhat more than ordinary, and was not one of the heard-of Pretenders to that Invention: which effect it had, and their Treaty with me had finally been concluded for several Thousand pounds, had not the inserting one Clause broke it off, which was, That if after I had discovered my Inventions about the finding the Longitude by Watches, or otherwise (though in themselves sufficient) They, or any other Person should find a way of improving my Principles, he or they should have the benefit thereof, during the term of the Patent, and not I. To which Clause I could no ways agree, knowing 'twas easy to vary my Principles an hundred ways, and 'twas not improbable but that there might be made some addition of conveniency to what I should at first discover, it being facile Inventis addere. And judging it most unreasonable to be deprived of the benefit of my Inventions, in themselves sufficient, because others might vary them, or any other ways improve them, of which it was very probable they would have no thought, if they had not the advantage of being instructed by my discovery, it having lain hid some thousands of years already, as indeed the effect hath made evident and certain, there having been nothing done by any body else upon that matter for these fifteen years.  
Upon this point our Treaty was broken off, and I concealed the farther discovery of any of the other more considerable parts of my Inventions, for the regulating of Time-keepers, as hoping I might find some better opportunity of publishing them together with my way of finding the Longitude of Places, for which I hoped to have had some benefit for all the labour, study, and charge I had been at for the perfecting thereof. Upon this I was told, That I had better have then discovered all, since there were others that would find it out within six months; to which I answered, that I would try them one seven years; and it is now above twice seven, and I do not find it yet found out. Indeed Mr. Hugens hath made use of that part I discovered, and somewhat Mr. Leibnitz hath hit upon, but both of them are imperfect as I shall hereafter show.  
'Tis true, I was alarmed by one of those Persons about two years after that, who told me, That he had news that the Longitude was found out by a Person of Honour, by a way of carrying Mr. Hugens' s Pendulum-Clock, at Sea, by the help of a Ball and Socket, hung to the underside of the Deck of a Ship. But having a description of it, I presently told that Person, That that Invention would do mine no harm; and indeed we experimentally found it useless to that effect not long after, upon a trial made of carrying the said Clocks off to Sea in one of His Majesty's Pleasure-Boats, in the year 1662.  
The Invention indeed in itself was ingenious, and did much more than what Mr. Hugens did expect, as I was then informed by the Right Honourable the Earl of Kincardine, the Author and perfecter of that part of the Invention. But wanting a little addition (which I concealed, and Mr. Hugens hath not got yet that I hear of) it failed of the effect that was expected. Notwithstanding this, it was not long after published in Low Dutch, and presently after in English; wherein what made for it was related, but what made against it was concealed, though they were both equally known.  
But on the otherside, all that I could obtain was a Catalogue of Difficulties, first, in the doing of it, secondly, in the bringing it into public use, thirdly, in making advantage of it. Difficulties were propounded from the alteration of Climates, Airs, heats and colds, temperature of Springs, the nature of Vibrations, the wearing of Materials, the motion of the Ship, and divers others. Next, it would be difficult to bring it to use, for Seamen knew their way already to any Port, and Men would not be at the unnecessary charge of the Apparatus, and observations of the Time could not be well made at Sea, and they would no where be of use but in East and West India Voyages, which were so perfectly understood that every Common Seaman almost knew how to Pilot a Ship thither. And as for making benefit, all People lost by such undertake; much had been talked about the Praemiums for the Longitude, but there was never any such thing, no King or State would ever give a farthing for it, and the like; All which I let pass.  
At the earnest importunity of a Dear Friend of mine, since deceased, I did, in the year 1664, read several of my first Cutlerian Lectures upon that Subject, in the open Hall at Gresham College, at which were present, besides a great number of the Royal Society, many Strangers unknown to me. I there showed the ground and reason of that application of Springs to the Balance of a Watch, for regulating its motion, and explained briefly the true nature and principle of Springs, to show the Physical and Geometrical ground of them. And I explained above twenty several ways by which Springs might be applied to do the same thing, and how the Vibrations might be so regulated, as to make their Durations either all equal, or the greater slower or quicker than the less, and that in any proportion assigned. Some of these ways were applicable to lesser Vibrations, others to greater, as of 2, 3, 4, 5, 6. or what number of Revolutions were desired; the models of which I there produced, and I did at the same time show wherein the aforesaid Sea-Clocks were defective.  
All these particulars also were at several other times, at the Public meetings of the Royal Society, discoursed, experimented, and several Models produced. I did also, at the earnest desire of some Friends, in the year 1664 and 1665, cause some of the said Watches to be made, though I was unwilling to add any of the better applications of the Spring to them, as waiting a better opportunity for my advantage.  
Of all these things the Publisher of the Transactions was not ignorant, and I doubt not but Mr. Hugens hath had an account, at least he might have read so much of it in the History of the Royal Society as was enough to have given him notice of it, for page 247 of that History, amongst other Experimented Inventions, there are recounted several new ways of Pendulum Watches for the Pocket, wherein the motion is regulated by Springs, etc. The account of the several ways was given somewhat larger to the Learned Author of that excellent History, though he, as judging it more proper to his design, was pleased to give only this summary account. Mr. Hugens might therefore, if he had pleased, have mentioned the first Inventor, Nam ingenuum est fateri; as he might also that of the Circular Pendulum, which is mentioned in the same page of the aforesaid History.  
But though he would not please to confess he knew my published Invention, yet I am sure he hath manifested, that he knows no more than what I had formerly discovered, he having not in least mentioned the oath Contrivance, which is the principal, and without which the first part of the Invention is but lame and imperfect, and doth but limp on one leg, and will some time hobble, and stumble, and stand still. And the said Watches will not be tres Just, nor show the Longitude at Sea or Land, but, on the contrary, they will be subject to most Inequalities of motion and carriage, and with many of those motions will be apt to stand still, whatever to the contrary is affirmed in the French Journal, or in the English Transactions.  
I forbear now to mention any further the carriage of the Writer of the Transactions in this Affair, and begging my Readers excuse for this digression, I shall conclude this Tract with a short communication of the general ground of my Invention for Pocket-Watches, the number of particular ways being very great, which (that the true Lovers of Art, and they only may have the benefit of) I have set down in the Universal and Real Character of the late Reverend Prelate, my Honoured Friend Dr. John Wilkins, Lord Bishop of Chester, deceased. In which I could wish, that all things of this nature were communicated, it being a Character and Language so truly Philosophical, and so perfectly and thoroughly Methodical, that there seemeth to be nothing wanting to make it have the utmost perfection, and highest Idea of any Character or Language imaginable, as well for Philosophical as for common and constant use. And I have this further to desire of my Reader, who will be at the pains to decipher and understand this description, that he would only make use of it for his own information, and not communicate the explication thereof to any that hath not had the same curiosity with himself.  
This I do, not so much to hinder the spreading of this Description here delivered, as to revive, and, if possible, bring into use and practice that excellent Design: It being a Character and Language perfectly free from all manner of ambiguity, and yet the most copions, expressive and significative of any thing or Notion imaginable, and, which recommends it most to common use, the most easy to be understood and learned in the World. See Table the third.  
To fill the vacancy of the ensuing page, I have here added a decimate of the centesme of the Inventions I intent to publish, though possibly not in the same order, but as I can get opportunity and leisure; most of which, I hope, will be as useful to Mankind, as they are yet unknown and new.  

1. A way of Regulating all sorts of Watches or Time-keepers, so as to make any way to equalise, if not exceed the Pendulum-Clocks now used.  
2. The true Mathematical and Mechanichal form of all manner of Arches for Building, with the true butment necessary to each of them. A Problem which no Architectonick Writer hath ever yet attempted, much les ● performed. abccc ddeeeee f gg iiiiiiii llmmmmnnnnnooprr ssstttttuwuwuux.  
3. The true Theory of Elasticity or Springiness, and a particular Explication thereof in several Subjects in which it is to be found: And the way of computing the velocity of Bodies moved by them. c e i i i n o s s s t t u u.  
4. A very plain and practical way of counterpoising Liquors, of great use in H ●d ●aulicks Discovered.  
5. A new sort of Object-Glasses for Telescopes and Microscopes, much outdoing any yet used. Discovered.  
6. A new Selenoscope, easy enough to be made and used, whereby the smallest inequality of the Moon's surface and limb may be most plainly distinguished. Discovered.  
7. A new sort of Horizontal Sails for a Mill, performing the most that any Horizontal Sails of that bigness are capable of▪ and the various use of that principle on divers other occasions. Discovered.  
8. A new way of Post-Charriot for travelling far, without much wearying Horse or Rider. Discovered.  
9 A new sort of Philosophical-Scales, of great use in Experimental Philosophy. c d e i i n n o o p s s s t t u u.  

10. A new invention in Mechanics of prodigious use, exceeding the chimaeras of perpetual motions for several uses. a a a ae b c c d d e e e e e e g i i i l m m m n n o o p p q r r r r s t t t u u u u u.  
a a e f f h i i i i l l n r r s s t u u.    
FINIS.