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 leasure, 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.

[Page 2]Secondly, A way of shortning 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 Dials, by the tangent projection.

• 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 Dials, in Mr. Foster's way, &c.
• 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 Rayes from the surfaces of black Glasses, which are grownd 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 [Page 3] is lost by every reflection, and that every reflection doth duplicate, triplicate, quadruplicate, quintuplicate, &c. 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 easie 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, blew, 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 Rayes; 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 waies 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 Rayes 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's way, of covering the Glass with the soot of a [Page 4] 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 Optick 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 Opticks 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 waies of doing which I have represented in the adjoyning Table, wherein I have expressed ten several waies 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 shewed before the Royal Society, at divers of their publick Meetings at Arundel house, in the year 1668, and it remains upon their Register.

This (as I then shewed) 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 [Page 5] and full. And that would not be subject to lose its Figure, which all our specular Mettals 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 easie use and ma­nage.

This I attempted with several sorts of Mettal, made with ♃, ♀, ♂, Antimony and Arsenick, but most of these compound Mettals 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 glase 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 foil'd 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 unfoil'd superficies of the Glass. The first from the surface of the Glass before it entred; this, as it weakned 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 weakned much more from the second reflection it suffered at the unfoil'd superficies of the Glass, from the reflection of the Air, or [...]ther, which is much stronger than that of Glass, at its re-entring into the Air. Besides this, I find that the substance of most Glass is so imperfectly mixt, that there is in the very best much of veinyness and inequa­lity of Refraction in the parts thereof, and thence, though [Page 6] 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 leasure and opportunity I design to ex­perience farther▪ But this only by the by, in relation to the shortning 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 Raies 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 errour 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 o, and b g h i k o, represent the two side Rayes 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 [Page 7] 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 shortned 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 Fogg, thin Clouds, or the like.

If it be thought more convenient to have this long Tube to lie alwaies 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 above-mentioned, 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 fixt to their due lengths and positions.

I explained also at the same time to the Royal Society, at their publick Meeting at Arundel-house, several other waies 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 hindred 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 [Page 8] 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 waies contrived, which I shall not now spend more time in describing, it being so easie 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, &c. 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 alwaies directed at the body of the Sun, which will be no small [...]ase to an Observer, that is to delineate the figures on Paper. [Page 9] 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 superiour 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 eaven 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 straight Limb, or Ruler; the end of which must move upon Centres or Rowlers, the centres or axes of which Rowlers 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 straight Screw, in the manner of a pair of dividing Com­passes, where the same care must also be had, that the axes of the Rowlers 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 thred, otherwise it will be apt to be moved out of a straight by screwing a large thred; and the thred, whether greater or less, must be made by degrees with a pair of cutting-stocks, that may be set closer every time of screwing.

[Page 10]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 incompassing 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 center 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 vvill it be difficult, after it is known by Observa­tion, how many Revolutions, and vvhat part of a Revolution answers to five whole degrees, to calculate a Table of Sub­tenses, which shall shew vvhat 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, rowling upon the limb of the Quadrant, by the help of two Wires vvhich vvere coyled about those Rowlers, and the ends thereof were fastned 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, vvith great ease and certainty.

I also at the same time made another Frame with a straight Screw, vvhich opened to five degrees only, vvith Tumbrels or Rowlers like a pair of dividing Compasses (after the same manner vvith 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 vvhich it vvas fixt, and the help of a very deep Plano convex lens, vvhose plain side vvas turned downwards towards the Plate, and the convex side towards the eye, the said Frame vvas moveable from five degrees to five degrees, upon the whole limb of the Quadrant or Instru­ment, by vvhich▪ Instrument I could vvith great ease actu­ally and accurate divide an angle into every five Seconds, [Page 11] 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 vvay vvill 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 vvith vveights, the vvay of which I shall shortly describe: Beginning from the very centre of the In­strument, and [...] the end thereof, to set up so many Deal­boards joyned to the end of each other in a streight line, or else to strain a pretty big Line, vvhich shall cut the measured line of distance from the center of the Instrument at Right­angles, 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 Boards 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 fixt to the same place, so as still to respect the same point or beginning of the Divisions upon the Boards 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 vvhole degrees, or vvhatever Angle else you design it to take in; then (for the first and third way) reckon how many vvhole Revolutions, and vvhat part of a Revolu­tion goeth to make up that vvhole Angle, and subdivide the same by a small Table into Minutes and Seconds, and you vvill presently find by the Trial, that you vvill be able to di­vide to a strange accurateness upon those Boards, by the help of your Tubes and Screw, even at the distance of 1000, 1500, or 2000 foot, and even almost to equalize the Divisions by your Compasses, when at the very Boards. And by this you [Page 12] 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 streight 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 Rowlers 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, vvhich I find also is somewhat troublesome in the Instrument of three foo [...] Radius, vvhen 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 moveable Arm, is made, by unscrew­ing, to draw off the long Screw from touching the threds 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 Threds 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 Threds of the Limb, into which it must be steadily guided by hand, the Index e e will shew upon the Limb the number of Threds or Revolutions from the beginning, and the Index 8 will shew what part of a Revolution there is to be joyned 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 fixt, nor need I much to shew, how the Tubes may be stiffned to keep them from warping very [Page 13] much; A small matter of warping not creating any sensible errour, 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 straight 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 Joynt, mentioned in page 73, as a Universal Instrument for describing all manner of Dials. 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 Ecliptick, 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 Dials in Mr. Foster' s way. For communicating a round motion through any irregularly bent way, without shaking or variation, and the like.

First, The Instrument for describing all manner of Dials 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 joyned together by a Joynt, which from the applicability of it to, and fitness for all kinds of motions and flexures, I call a Universal Joynt. 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 Stile above the Plain, and to put the Axis in its due scituation accordingly, that is, parallel to the Axis of the Earth, and [Page 14] then by the Plumbet at the end thereof to rectifie 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 it self. But in such Dials as are in or near a Polar-plain, it will be convenient to make use of a small Thred to extend from the Cross, till it touch the Plain in the several hours, halfs, quarters, minutes, &c. The Arms of the Joynt 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 Joynt 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 fastned, which are to be joyned 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 fastned 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.

[Page 15]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 Collers, 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 Joynt 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 Collers e f g and h, and these Collers 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 collers be wrought true and exact, the Axis or middle lines of them will alwaies cut each other in the same point, howsoever they be turned round within their Collers; 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 shew the motion of the Sun in the Ecliptick, 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 [Page 16] 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 legg'd 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 fastned. At q must be made a Joynt 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 Joynt, as they are apart and distinct, you may see in the 9th. Figure, and as they are joyned all together [...]it for motion you may see in the tenth Figure, to which also the description of every part is ad­joyned in words referred to by the help of Literal marks, [Page 17] 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 Center, 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 Joynt be so contrived and wrought that the Axis of the two Rods may alwaies cut each other in the centre of the medium Cross or Plate, and that the said Centre, whatever change happens to the Joynt, may alwaies 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 Dials, 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 Tropick, or of any other pa­rallel Circle, you must rectifie 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.

[Page 18]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, &c. are of very great use, as I shall afterwards shew.

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 Sea-mans 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 Joynt, being made to joyn the arbor of the Wheel that goeth round in twenty four hours, with the arbor of the hand, performeth it without any other Wheel or Pinion in the Dial or Face part of the Clock; if the Arbor 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 Arbor 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 alwaies 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 alwaies moved in a plain which passeth through the Arbor of the Clock, and maketh equal pro­gressions in equal spaces about the said Arbor, but unequal progression about the Centre of the Dial, according to the differing Inclinations: And those Inclinations being both in the Sun-Dial and Clock-Dial the same, it will follow, that [...]he [Page 19] Hand of the Clock must alwaies move in the shadow of the Style, if the Hand be once rectified to the true Plain, and the Axis or Arbor 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 Coelestial Object, should, by the aforesaid Joynt or Instrument, be kept alwaies 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 Joynted into either of the Arms 11, of the cir­cular Plate X, in the 22 and 23d. Figure of my Animadver­sions, and the upper end joynted 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 Clock-work it self 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 alwaies moved in the superficies of the Cone described, by the radiation from the Coelestial 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 alwaies directed to the Coelestial Object, if the Arm of the Quadrant be moved per­pendicular over it, and parallel to it, that also must be alwaies directed to it. And hence it may very easily be conceived, how the aforesaid Semicircular Arms may be readily and cer­tainly rectified to any Coelestial 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 loosing the Screw h to rectifie it to the Object by the sight, and then immediately to fix it in the said posture by the afore­said Screw; the Clock-work 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 [Page 20] 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 moves 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 alwaies 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 Joynt to the Axis or Arbor must vary also, and that variation must al­waies be the same with the variation of the declination of the Sun, which is twenty waies mechanically performable in Clock-work, so that the motion shall be performed by the Clock-work 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 Arbor of the Clock-work to have the same inclination to the plain of the Dial, that the Axis of the Earth, or a line paralel to it hath; and rectifying the Hand into the true plain of the Axis, or Inclined arbor, 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 streight Style, howsoever inclined to the plain of the Dial, then must there be another Joynt 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 scituate 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 [Page 21] 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 exprest 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 it self may perform it. This may be done twenty waies easily enough, which I suppose will be suffi­ciently obvious to any knowing Mechanick, 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 easie, 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 [Page 22] 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 alwaies to move over that point of the Ecliptick 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 Suns motion in the excentrick of the Ecliptick. 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, culminatings, 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 shew 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 Joynt, 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, &c. every of which prin­cipal parts hath a vast variety, and the compound and decompound principles have a variety almost infinite.