The making of an Instrument to finde out the situation of any plat or Diall, and to place them already prepared. CAP. I.

FOr asmuch as it is necessary before you can drawe my Dial, to know how your plat is already or ought afterward to be placed, it shalbe ex­pedient to shew the making and vse of an Instrument, whereby you may examine and try all plain plats and place all dials being ready made and prepared.

Prouide therefore an halfe circle of Peare-trée, Walnut­trée, Bor, or any other close grained and solide wood, being well seasoned, so that the alteration of the weather at any time may not make it change from the first proportion ther of. Let it be perfectly tryed on both sides of an euen thicke­nesse, halfe or thrée quarters of an inch thick or more if you will, and sir or eight inches broad as you thinke good. The edge or side A. B. must be very right. Thē draw the Line C. D. thrée quarters of an inch equally distant from the side A. B. place one foot of your compasses in the centre E. which is the midst of the line C. D. and with the other draw halfe a circle from C. to A. deuide it into two equall partes or qua­drants, and laying your ruler vpon the centre E. and vpon this diuision drawe the line E. F. this done, deuide eche of [Page] these quadrants into 90. equall degrées or partes, which you may thus doe: First, deuide eche of them into thrée e­quall partes, and euery of these into thrée other partes, so shall you haue 9. diuisions in either quadrant, whereof eue­ry one shal represent ten degrées. Againe, part euery one of these into two, and ech of those into fiue (if you can) and so is eche quadrant diuided into 90. degrées.

Moreouer, it were necessary if your Instrument were large inough, to parte ech degrée into 60. minutes, or at the least into 3. whereof ech parte may conteine 20. minutes. This done, fasten a thread wel wared in the centre E. with a plommet of leade on the end, so that it may moue at frée libertie.

You may also (if you will) vpon this side of the quadrant between E. D. draw the diall whose delineation is taught in the 28. Chapter.

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The [...]omer [...] of [...] Instrument thus finished, let the other be altogether li [...]e vnto it, except that in stead of the the thread with the pl [...]t, you must haue a plombrule [Page] made in this maner.

Prepare a péece of very good wood, try it perfectly on both sides to an equal thicknes, let it be about half an inch thick, and twise so broad as the space betwéene the line C. D. of your instrument and the edge A. B. in the midst therof draw the right line G. H. and placing one foote of your compasses in the point G. with the other make a halfe circle according to the breadth of the ruler, whereby you may cut the end round, make the other end with a sharp point right in the end of the line G. H. as you sée in this figure.

Let the distance betwéen the point G. and the end H. be so much as is the semidiameter of the quadrant, make by G. an hole so great that a plōmet of lead hanging by a thread from the sharpe end, may moue therein at libertie: Make also an other round or square hole, wherein you may place a néedle touched with the Loadstone, as in the compasses or dyals you sée. Finally, make a little hole in the point or cen­tre G. and with a broad headed na [...]e of brasse fasten it vpon the centre E. of your Instrument, so as it may moue & turne which way you wil, and then your Instrument is finished.

Note that you may make your Instrument in forme of a quadrangle rightangled, as Witekindꝰ teacheth, cut­ting a round furrow without the circle and degrées made, for the plōmet to hang at libertie in: so shall you néede but one side of your instrument to vse, with a compasse or needle touched with the Loadstone for declinations. But the form of the other, whose making we haue taught, seemeth more commnodious, beholde the Figure.

The vse of this instruement. CAP. 2.

The vse of this instrument is in examining and placing the plat or dial. In immoueble plats, to know how they [Page] stand, as if you would make a Diall vpon a wall, first you must vnderstand whether the wall be erect or els recline or incline: then whether it be direct East, West, North, or South, or els decline, which you cannot know without the helpe of this Instrument. Againe, in moueable plats for the placing of them, as if you haue made a South diall or any o­ther kinde, by this Instrument you shall easily and perfect­ly place them.

The making of a Horizontall, or plaine lying Sunne Dial. CAP. 3.

YOur plat being prepared smoothe and plaine, drawe vpon it two lines, as in the figure fol­lowing, the one A. B. the other C. D. cutting themselues squirewise, that is, making right angles in the point E. vpon which make the quadrant of any circle from the line E. C. to the line E. A. or E B and write at C. the North, at D. the South, at A. the East, at B. the West. And the line C. A. which here is the quadrant being deuided into 90. degrées or parts, the eleua­tion [Page 5] of the Pole shalbe accounted in it (which in our exam­ple is 52. d. from C. to A. and at the ende of this number drawe a line from the centre E. which shalbe E F represen­ting the stile and artrée of the world. Then draw an other line K. L. by C. or by some other point of the line D. C. squire wise, so long as you can which shalbe called the touch line, or line of Contingence. Then measuring with your compasses the least distance of the point O. and the line E. F. or the Stile, the one foot placed in O. which is the point of intersection, and the other extended toward E. where it shal chaunce to diuide or be placed in the line E. C. marke that point or centre with the letter G. and draw with your cō ­passes a halfe circle vpon this centre for the equinoctial cir­cle, from H. by C. to I. whose diameter must be equally di­stant to the line L. K. Then diuide this halfe circle into 12. equall parts: this done, lay a ruler vpon the centre G. and vpon euery marke or diuision made in the halfe Equator, and where the ruler shal touch y^• line of contingence, there make marks or pricks, by which pricks draw lines from E. for the houres. E. C. is the 12. houre, E. B. the 6. in the mor­ning, E. A. the 6. at euening, the rest you may sée in y^• figure.

And whereas in Sommer the 4. and 5. in the morning, and also the 7. and 8. at euening, shalbe necessary in this kinde of Diall: prolong or draw the lines of 4. and 5. at euening, beyond the centre E. which shall shew the houres of 4. and 5. in the morning. And likewise the 7. and 8. in the morning, for the 7. and 8. at euening.

You may obserue an order both in these and in all other erect direct dials, by diuiding the one halfe of the Equator, drawing houre lines for the forenone, and obseruing the same distance from the meridian line, on the other side for the afternoone: for the line of the 11. houre in the forenoone is of like distance from the Meridian, that the 1. is in the afternoone, and the 10. as 2. and so of the rest.

When you would draw or make the halfe houres, you must diuide euery parte of the Equator into 2. equall parts, [...] the ruler and the line of contingence as you find in the [Page] drawing of the houre lines.

And this remember for the drawing of the halfe houre­lines, not onely in this kinde, but also in all other kindes of dials, which afterward shall follow.

The Stile must be fixed in the centre E. hanging directly ouer the Meridian line E. C. with so great an angle, as the lines C. E. F. make, declining from that on neither sides.

The Equinoctiall circle, the Quadrant, the line of the Stile and of Contingence, must be lightly drawen, because they ought to be put out againe, in that they serue to no vse but for the drawing of the Diall. And this likewise remē ­ber in all other kindes of Dials, that the preparatiue or pricked lines must after the making of the Dial be omitted and extinguished, as altogither vnprofitable.

This and all other kindes of Dials may most fitly be drawen vpon a clean paper, and then with the help of your compasses placed on the plat.

The making of a South Erect, direct Diall. CAP. 4.

THe making of this, is almost altogether like vnto other before. Draw here also a line A. B. parallel to the Horizon, out of the middest wherof, let the line C. D. perpendicu­ler to that be extended: and let the Quadrant be drawen from A. to D. or B. to D. parted into 90. degrées, as before. Aboue C. let Zenith be written, beneath D. Na­dir; nigh vnto B. the East, and by A. the West. In y^• Qua­drant number the Eleuation of the Pole (which in our ex­ample is 52. degrées) from A. toward D. And at the end hereof draw from the centre C. the line C. E. representing the Stile: and by D. draw the line of Contingence, I. K. squirewise, as before. Then the least distance of the point or intersection D. and the Stile being taken with your Com­passes, extend th [...]n in the line D. C. the one foote placed in D. set the other in F. toward C. and draw by F. the Diame­ter of the halfe circle of the Equator equally distant to the line K. L. which must be made vpon the centre F. from G. by D. to H. and diuide it into 12. equall partes, vpon euery one of which and the centre F. the ruler being placed, wher­soeuer it shal happen to touch the line of contingence, there make markes. Then from the centre C. by these marks the houre lines must be drawen. The line C. A. shall shew the 6. in the morning, C. B. the 6. at euening, C. D. the 12. &c. The Stile must be placed or fixed in the centre C. han­ging precisely ouer the line of the 12. houre, with so great a distance, as the angle D. C. E, is. This kinde of Dial doth receiue and shew onely but 12. houres at the most.

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The making of a North Erect, Direct Diall. CAP. 5.

HEre as in the South make a line Paral­lele to y^• Horizon, A. B. cut it squire wise with the Perpendicular C. D. let C. be the centre. At C. write Zenith, at D. Nadir, [...]c. From C. draw the Quadrant of a circle to D. or B. deuide this into 90. Degrées, account the Eleuation of the pole (which in our example is 52. d) from A. toward C. Draw at the end of of this number the line E. F for y^• Stile. Afterward draw the line of Contingence by C. squire-wise, and take the shortest distance, with your compasses be­twéene the poynt C. and the Stile, placing the one foote in the point C. extend the other toward E. in the line C. D: ma­king a poynt or pricke G: wherupon (as a centre) the same [Page 7] widenesse of the compasses remaining, describe the halfe Equator by C. ended with the Diameter H. I. equidistant to the line of Contingence. When you haue deuided thus halfe circle of the Equator into twelue equall partes, lay the ruler vpon the centre G. and vpon each diuision of the Equator, and where it shall touch the Contin­gent line, make markes. This done, draw the lines for the houres, by those markes from the centre E. but those very fewe, that is, two nigh vnto A. and two by B. but prolong and extend them beyond the centre E. so that their contraries may bée made: for in this kinde of Dials there bée but onely ten houres profitable, that is, 4, 5, 6, 7, 8, before noone, and 4, 5, 6, 7, 8, after noone; which shew but only in Sommer, from the entring of the Sunne into ♈, vntill such t [...]ne as it entreth ♎.

The Stile must bée fixed in the centre E. placed vp­ward, directly ouer the lint E. C. with so great an angle as I. E. C. is. Let the line C. D. bée placed vpward per­pendicularly, but so that it may not bée almost séene, as afterward seruing to no vse, because it is onely a line preparatiue.

Beholde the Figure.

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The making of the East and West Erect Dials. CHAP. 6.

IF you know the making of one of these, you may easily make them both, for they be very like, differing onely in the na­ming of the houres, for the one contai­neth houres for the forenoone, and the o­ther for the afternoone.

You must therefore on your plat make the quadrant of a circle A. B. C. which may afterward easilie bée put out, as all the other lines must bée, except the [Page 8] houre lines: let the side A. B. be Perpendiculare: B. C. Paral­lele to the horizon, and let the arke beholde the South, which being diuided into 90. degrées, nomber therin the ele­nation of the Pole dounward from A. toward C. by the end of this number, and by the Centre B, draw a line so long as your plat will giue you leaue, whose South ende shall be­hold precisely the Equinoctiall circle. At which end draw a circle whose diameter shall bee almost the third part of the line. Then draw another diameter or line in the centre, squire wise to the other, which shall shew the axtrée of the world, and be the line for the 6. hour. After ward at the cut­ward sides of the circle, draw two contingent lines, one be­neath, the other aboue, so that they may be Parallele to the middle line. Deuide each quarter of the circle into sixe equal partes. Then place the ruler vpon the centre, and eache of those markes or partes, and where it toucheth the lines of contingence there make markes in them. Afterward draw a line by those two markes which be next to the 6. houre, in the lines of contingence, which may be equallie distant from the line of the 6. houre.

In like maner doe with the rest; so that you may haue in the East Diall, two aboue the 6. houre, the 4. and 5. in the morning, and vnder it 7, 8, 9, 10, 11. In the West Diall like wise 7. and 8. in the euening aboue the 6. houre, and vnder it 6, 5, 4, 3, 2, 1. Neither of them doe shew the 12. houre, because at that time the Sunne beames be Parallele to the plat.

Fixe the Stile in the centre of the circle right vp from the Plat, so long as the Semidiameter of the circle is: only shewing the houre with the very top or end thereof. Yet it were more conuenient to haue it placed along ouer the line of the 6. houre, being a plate of yron or some other metall, being so broad as the Semidiamer of the circle is.

Behold the figures folowing.

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Note that these fiue kinds of Dials before taught may be made vpon a stone cut square in forme of a die.

The making of a South Erect declining Diall, which may be placed on any vpright wall what­soeuer. CAP. 7.

IN all Declining Dials because the Stile doth not hang directly ouer the meridianline, therefore you must first finde out and place the Substile (which is the line ouer which the Stile directly hangeth) and lik wise the line of the Stile: which may bée both eastly and spéedily performed in this maner.

First, by your instrument séeke out the Declination of [Page] the wall or plat, whereupon you would make a diall, which for example suppose I had found to be 50. degrées.

[...] Therefore drawe the Horizontall line A. B. let the Meri­dian C. D. cut it squire wise in the point E. which may be in any place of the line A. B. and let the line F. G. cut the meridi­an squire-wise in the point C. which you may like wise take in what parte of the Meridian you thinke most conuenient. Upon the center E. describe the quadrant of a circle from C. to A. if the declinatiō of your plat be West-ward, or from C. toward B. if East-ward (as in this example it is) diuide it into 90. degrées and number in it from C. toward B. the E­leuation of the Equator 38. degrées, and laying your ru­lerThis is al­waies so much as the Eleua­tion of the Pole, wan­teth of 90. degrees. vpon the center E. and vpon the end of this number draw the line E. H.

Againe account in the quadrant from C. toward B. the declination of the plat being 50. degrées, and draw the line of Declination E. I. from E. to the end of this number.

Then take your compasses, set one foote in the point C. extend the other vnto H. (which is the intersection point of the line E. H. with the line C. G.) and that same widnesse re­mayning, place one foote in the centre E. and with the other marke the point 1. in the line E I. and from the point I. draw the line I. K. squire-wise to the Meridian E. C. placing K. at the intersection it maketh with E. C.

Moreouer take with your compasses the iust length of the line I. K. and placing one foote in C. turne the other to­ward F. and make the point L. in the line C. F. drawe the Substile from E. by L. Let the line of Contingence bée drawne squire-wise to the Substile in the point L.

This done, take the distance betwéene the centre E. and the point K. and place it in this line of Contingence from the point L. vnto M. and make the point M. that the line E. M. may be made from E. vnto M. for the stile.

Measure with your compasses the least distance be­twéene the point L. and the Stile, and with the same wide­nes, one foote remaining in L. turne the other toward E. and make the centre O. in the Substile: vpon which de­scribe [Page 10] the Equinoctiall circle. Then placing your ruler vp­on O. being the centre of the Equinoctiall circle, and N. (which is the point of intersection of the Meridian and Contingent) marking where it cutteth the circumference, for there you must begin to diuide it into 24. equall partes, not withstanding those 12. are onely in vse which are next the contingent.

Finally, place your ruler vpon the centre O. and vp­on the seuerall diuision points of the Equator, and where it toucheth the line of Contingence make markes, by which from the center E. draw the houre lines.

Place the Meridian perpendicularly vpon the wall, the centre E. vpward, the Stile pointing downeward. Let the Stile hang directly ouer the Substile making an angle e­quall to E. M. L.

The making of a North Erect Declining Diall. CAP. 8.

THe North Erect declining Dial differeth from the South onely herein, that the centre is to be placed downeward, the Stile pointing vpward to the North Pole, and that the Meridian representeth the 12. houre at midnight and not at noone. Therefore if the de­clination bée toward the East, you must account the houre lines from that which is as it were the Meridian forth­ward 1, 2, 3, 4, 5, 6 &c.

But if it decline toward the West, number them backe­ward, 11, 10, 9, 8, &c. omitting the first 3. which are before Sunne rising and after Sunne setting in our eleuation.

Let the line E. D. bée placed perpendicularly vpon the plat, and the Stile point vpward to the North Pole. Com­pare this figure with the former.

Another way to make the South Erect Declining Diall. CAP. 9.

ALbeit we haue plainely and perfectly shewed the making of the South and North erect Declining Dials, in the two former Chapters; yet to satis­fie them that delight in variety; here is also declared another way, whereby you may make them, namely, by the helpe of Arithmetick, and the Table of Sines which isRead the [...]se of the Table of Sines. placed in the end of this booke for this intent.

Therefore the Eleuation of the Pole being knowen, and the Declination of your plat, by your Instrument found out:

Multiply the Sine of the complement of the Eleuati­on, by the Sine of the complement of the Declination, di­uide the product by the whole Sine (which is 100000) and you shall haue a quotient Sine, whose arke is the di­stance of the Stile from the Substile, which distance kéep.

Then take the complement of this distance, and the Ele­nation of the Pole, and multiply the Sine of the lesser by the whole Sine, parting the product by the Sine of the greater: the quotient Sine which shall come of this diuisi­on, shall giue you an Arke, whose complement is the di­stance of the line of the Substile from the Meridian, which distance you shall likewise kéepe. For better instruction herein, consider the example.

The making of a North Erect declining Diall another way. CAP. 10.

THe distance of the Stile from the Substile, and of the Substile from the Meridian, is foūd out altogether like to the South Erect decli­ning. Therefore you may resort thither for the working hereof, I will onely drawe the Figure.

[Page 13]Let the line A. B. being parallele to the Horizon, cut C. D. squire wise, placing E. at the intersection Draw the qua­drant from C. to B. diuide it into 90. ^d. accounting therein from C. toward B. the distance of the Substile from y^e line, which is (as it were) the Meridian, which distance is 28. ^d. 55^m. At the end of this number draw the line E. F. for the Substile. Then number from that line the distance of the Substile and the Stile, which is 25. degrées, 48 minutes toward B. Draw likewise at the end of this number the line for the Stile E. G. This done, let the line of Con­tingence be drawen squire wise by the point F. and then ta­king the least distance betwéene the point F. and the Stile, extend the compasses in the line F. E. the one foote being pla­ced in F. where the other shall deuide the line, place the let­ter H. With the same widenes of the compasses draw vpon the centre H. the Equator, and where the ruler shall touch the same, being laid vpon the centre H. and the intersection of the Contingent line, and y^t which is (as it were) the Meri­dian, begin to deuide it into 24. equall partes. Finish all things remaining, as in y^e South erect declining, onely this excepted, the Stile being fixed in the centre E. must be pla­ced vpward, beholding the substile with so great a distance or angle, as the letters F. E. G. do shew. The line C. D. being applyed to the plat perpendicularly, sheweth as it were the 12. houre at midnight: therefore account from that the houre lines, as they follow in order, 1, 2, 3. which houres haue no vse in this kinde of Diall for our Eleuation, but from 4. &c.

Let the line A. B. be parallel to the Horizon, being light­ly drawen, as the other preparatiue lines be, seruing after­ward to no vse.

Note, if the plat decline towards the East, as this figure doth, it is for the forenoone: and you must account the houre lines from that which is (as it were) the Meridian, forward 4, 5, 6. &c. If it decline toward the West, account them backward 11, 10, 9. as we shewed before.

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The making of a Meridionall or South Reclining direct Diall. CAP. 11.

IF the Reclination of the plat be lesse then the complement of the Eleuation of the Pole, adde this complement, and the Re­clination of the plat together, and with this number as if it were the Eleuation of the Pole make a South erect direct di­all, and it will be perfect and fit for your plat.

The making of a North Diall Reclining Direct. CAP. 12.

IN these kindes, if the Reclination bée lesse then the Eleuation of the Pole, sub­tract that from this, & kéep the remain­der for the Eleuation of the Pole: ac­cording to which Eleuation, make a North Diall Erect Direct, and it will be fit for your plat. In which the houres about Noone are placed, being profitable onely in sommer: [Page 16] and not then vnlesse the Reclination be greater then the di­stance of y^• Trop [...]k of S, from y^• Zenith. And the more your Diall Reclineth, the longer time it will shew in Sommer.

The making of an East and West Diall reclining. CAP. 13.

FIrst multiplie the Sine of the Eleua­tion of the Pole, by the Sine of the Reclination of the plat, and diuide the product by the whole Sine: whereof shall come a quotient, whose arke is the distance of the Stile, from the Substile.

Compare the Complement of this distance with the Complement of the Eleuation of the Pole, and which you shall find least, multiply the Sine thereof by the whole Sine, parting the Product by the Sine of the greater. The quotient shall yéelde an arke, whose Complement shall be the distance of the Substile from the Meridian.

The making of a South reclining declining Diall. CAP. 14.

FIrst, in this kinde, as in other which follow, you must finde out and place the Meridian, the Substile, and the Stile, which being done, that which remai­neth is finished as in those before.

Therefore the Declination and the Reclination of the plat being known by your instrument, multiplie the Sine of the Declination by [Page 19] the Sine of the complement of the Reclination, diuiding the product by the whole Sine. The quotient Sine shall yéeld an Arke, whose complement shall be named the Com­plement to be repeated.

Afterward augment the Sine of the Complement of the Declination, by the whole Sine, diuide the Product by the Sine of the Complement to be repeated: where of shall come a Sine, whose arke shalbe the distance of the Meridi­an from the Horizon.

Againe, multiply the Sine of this distance, by the Sine of the Complement of the Reclination, parte the Product by the whole Sine, the arke of whose quotient shalbe cal­led the Eleuation of the Meridian.

Then compare this Eleuation of the Meridian, with the Eleuation of the Pole, and which you shall finde least, sub­tract that from the greater, and that which remaineth kéep, (for it shall be called the difference kept) being mindefull which of them was the greatest.

This done, multiply the Sine of the Complement to be repeated, by the Sine of the difference kept: diuiding the Product by the whole Sine, whereof shall come a Sine, whose arke shalbe the distance of the Stile from y^• substile.

To conclude, compare the Complement of this distance, with the complement of the difference kept, and which you shall finde least, multiply the Sine thereof by the whole Sine: part the Product by the Sine of y^• greater, and there­of shall come a Sine, the Complement of whose arke shalbe the distance of the Substile from the Meridian.

• Declining 18. ^d.
• Reclining 25. ^d.

First looke out in the Table the Sine of the Declinati­on, which is 30901. Then the Complement of the Recli­nation being 65. ^d. the Sine hereof is 90630. Afterward multiply 90630: by 30901. and the product 2800557630 [Page] diuide by the whole Sine, which is 100900. the quotient shalbe 28005. whose arke is 16. ^d. 16. ^m. The complement whereof being 73. ^d. 44. ^m. is the complement to be re­peated.

This done, multiply 95105 the Sine of the Comple­ment of the Declination 72. ^d. by the whole Sine 100000. the product shalbe 9510500000. which being diuided by 95996 the Sine of the complement to be repeated, the quo­tient shalbe 99071, whose arke is 82. ^d. 11. ^m. which is the distance of the Meridian from the Horizon.

Afterward augment the Sine of this distance 99071. by the Sine of the complement of the Reclination, which is 90630. and the product 8978804730 part by the whole Sine: and the quotient 89788 shall yéeld an arke 63. ^d. 53. ^m. which is the Eleuation of the Meridian.

Then compare the Eleuation of the Meridian, with the Eleuation of the Pole, which in this example is 52. ^d. and finding the Eleuation of the Pole to be least, substract this namely 52. degrées) from 63. ^d. 53. ^m. the Eleuation of the Meridian, and there remaineth 11. ^d. 53. ^m. which is the difference kept.

Now multiply 95996 being the Sine of the Comple­ment to be repeated, which is 20591. the product is 1976653636. which diuided by the whole Sine, the quoti­ent shalbe 19766, whose arke 11. ^d. 24. ^m. is the distance of the Stile from the Substile.

Then to conclude. y^• Complement of this distance, which is 78. ^d. 36. ^m. being compared with the Complement of the difference kept, which is 78. ^d. 7. ^m. you shall finde the Complement of this difference kept to be least, wherefore multiply 97856 the Sine thereof, by the whole Sine, and the product 9785600000. part by the Sine of the greater, to wit, of the Complement of the distance of the Stile from the Substile, which is 98027. the quotient shalbe 99825. whose arke is 86. ^d. 37. ^m. The Complement whereof is [Page 20] 3. ^d. 23 ^m. which is the distance of the Substile from the Meridian.

There be thrée sundry kindes of these Dials, the one dif­fering from the other.

The making of a North reclining decli­ning Diall. CAP. 15.

The making of a South Inclining direct Diall. CAP. 16.

IF the inclination of the plat be lesse then the Eleuation of the pole, take that from this, accoūting the remainder for the Eleuation of the Pole. And with this number as if it [Page 29] were the Eleuafion of the Pole make a Diall like to the South erect direct, and it shall be fit for your plat.

But if the Inclination of the plat be equal with the Ele­uation of the Pole, make your Diall altogether like to the North reclining direct, whose reclination likewise is equall to the Eleuation of the Pole. They differ onely herein, that this is placed downward to the earth, and that vpward.

But if the Inclination of the plat be greater then the Eleuation of the Pole, substract this from that, and with the number remaining, as if it were the Eleuation of the Pole, make your Dial like to the North erect direct.

The making of a North inclining direct Diall. CAP. 17.

IF the Inclination of the plat be lesse then the complement of the Eleuation of the pole, adde the inclination to the Eleuation: and with that number, as if it were the Eleuation of the Pole, make a North crect direct diall, for your plat.

But if the inclination be equall with the complement of the Eleuation of the Pole, then make a Diall as you were taught in the second kinde of South reclining direct: for as that plat so reclining in the South is placed vpward: so is this placed down ward. Note that this North diall shew­eth but onely 4. howers, the two first in the morning, and the two last in the euening, and that only in the middest of Sommer.

But if the inclination be greater then the Complement of the Eleuation of the Pole, substract this lesser out of the other being greater: and with the number remaining, as if it were the Eleuation of the Pole, make your diall like to an Horizontall. This kinde likewise sheweth very few howers.

The making of the East and West inclining Dials. CAP. 18.

MUltiply the Sine of the Eleuation of the Pole by the Sine of the Inclination of the plat, di­uiding the product by the whole Sine: the quotient arke shall be the distance of the Stile from the Substile.

Then compare the complement of this distance with the complement of the Eleuation of the Pole, increasing the Sine of the lesser by the whole Sine, part the product by the Sine of the greater, where of shall come a Sine, the Complement of whose arke shalbe the distance of the Sub­stile from the Meridian.

[Page]It shall not be necessarie to shew any example hereof, be­cause the East and West reclining bée altogether like to this. Wherefore resort thither, for the working of your example, for those distances in drawing this figure shall bée here obserued.

The making of a South inclining declining Diall. CAP. 19.

FIrst multiplie the Sine of the Decli­nation, by the Sine of the complement of the inclination, parting the product by the whole Sine. The quotient Sine shall yéeld an arke, whose complement shall bée named the complement re­peated.

Then increase the Sine of the Complement of the De­clination by the whole Sine, and the product diuide by the Sine of the Complement repeated: whereof shall come a Sine, whose arke is the distance of the Meridian from the Horizon.

Afterward, this Sine being multiplied by the Sine of the Complement of the inclination, and the product parted by the whole Sine: the arke of the quotient Sine shall bée the Eleuation of the Meridian. Which arke you must adde to the Eleuation of the Pole. And if the totall number bée lesse then 90. ^d. it shall bée named the Doubtfull arke. But if it be greater then 90. ^d. take it from 180, and let the re­mainder be called the Doutfull arke.

This done, augment the Sine of the Complement re­peated, by the Sine of the Doutfull arke: & the product ari­sing thereof being diuided by the whole Sine, the quotient arke shall be the distance of the Stile from the Substile.

Now compare the Complement of this distance, with the Complement of the Doutful arke, multiplying the Sine of the lesser, by the whole Sine, diuiding the product by the Sine of the greater: the arke of the quotient Sine com­ming thereof shall bée the distance of the Substile from the Meridian.

[Page]But note, if the doubtfull arke be found without subtrac­tion from 180, (which is if it bée lesse then 90 ^d.) then you must subtract the distance of the Substile from the Meridi­an out of 180, and the number remaining shall be the true distance of the Meridian from the Substile.

Marke this likewise, if the doubtfull arke be equall 90 degrées, let the Complement repeated be the distance of the Stile from the Substile: then shall there bée iust 90 ^d. for the space betwéene the Substile and the Meridian, as be­fore is taught in the North declining reclining, whither you may resort for the working hereof. Here it shall bée sufficient to shew two examples, with the delineation of their figures: the one, where the doubtfull arke is subtrac­ted from 180 ^d. and the other, where there is no subtraction hereof, because it is lesse then 90 degrées.

• Inclining 45 degrees.
• Declining 45 degrees.

Forasmuch as this is altogether like to the North recli­ning 45 ^d. declining 45 ^d. whither you may resort: I wil omit the working of the former part of this exāple, vntil I come to the finding out of the distance of the Substile from the Meridian. The former part of this example you may finde to bée thus wrought in the North reclining declining:

• 1 First the complement repeated 60 ^d.
• 2 Then the distance of the Meridian from the Horizon 54 ^d. 44 ^d.
• 3 Next the Eleuation of the Meridian 35 ^d. 16 ^m.
• 4 The compounded arke there, which we call the doubt­full arke here 87 ^d. 16 ^m.
• 5 The distance of the Stile frō the Substile 59 ^d. 54 ^m.
• 6 Last, the distance of the Substile from the Meridian 84 ^d. 33 ^m.

But here (as before is taught) you must subtract this [Page 32] distance from 180 ^d. because the doubtfull arke was lesse then 90 ^d. and then the remainder 95 ^d. 27 ^m. shall bée the true distance of the Substile from the Meridian.

In the delineation of this diall, first, as before, make a line paralele to the Horizon A. B. Draw another perpen­dicular C. D. cutting the former squire-wise. Make two quadrants from A. by D▪ to B. If the plat decline westward, number the distance of the Meridian from the Horizon from A. toward D. &c. But if the declination be Eastward (as in this example it is) then account this distance of the Meridian from the Horizon, which is 54 ^d. 44 ^m. from B. to­ward D. drawing at the ende hereof from the centre C. the Meridian line which shall shew the 12 houre. Account likewise from that toward A. 95 ^d. 27 ^m. the distance of the Substile from the Meridian: making in like maner the line E. C. for the Substile. From this line on which side you will, number the distance of the Stile from the Substile, be­ing 59 ^d. 54 ^m. drawe by this the line G. representing the Stile. Let the line of Contingence cut the Substile squire­wise in any point where you will. Make the Equator, be­gin the diuision thereof, and draw the houre lines in all re­spects as in the North reclining 21 degrées 30 minutes, de­clining 30 degrées. Cap. 15.

[Page]

• Inclining 33. ^d. 40. ^m. Where the dout­full arke is sub­tracted frō 180.
• Declining 31. ^d. Where the dout­full arke is sub­tracted frō 180.

First 51503 the Sine of the Declination being multipli­ed, by 83227 the Sine of the Complement of the inclinati­on: and the product 4286440181 diuided by the whole Sine: the arke of the quotient is 25 ^d. 23. ^m. The com­plement whereof being 64 ^d. 37 ^m. is the complement re­peated.

Then I increase the Sine of the Complement of the declination which is 85716, by the whole Sine, and I part the product 8571600000, by 90346 the Sine of the Com­plement repeated. The quotient 94875 yeeldeth an arke [Page 33] 71 ^d. 35 ^m. which is the distance of y^• Meridian frō y^• Horizō.

Agayne, I multiply this Sine 94875, by the Sine of the Complement of the Inclination being 83227, and the product 7895961425 arising thereof, I diuide by the whole Sine. The arke of the quotient is 52 ^d. 9 ^m. which is the Eleuation of the Meridian.

Now this being added to the Eleuation of the Pole 52 ^d. and the total number 104 ^d. 9 ^m. taken from 180, there re­mayneth 75 ^d. 51 ^m. which is the doubtful arke.

Therefore I augment 90346 the Sine of the Comple­ment repeated, by 96930 the Sine of the doubtful arke, and the product 8760399890 I parte by the whole Sine. The arke of the quotient 61 ^d. 10 ^m. is the distance of the Sub­stile from the Stile.

This done, I compare 28. ^d. 50. ^m. the Complement of this distance, with 14. ^d. 9. ^m. the Complement of the dout­full arke, multiplying y^• Sine of the lesser, which is 24446, by the whole Sine, diuiding the product 2444600000, by 48226 the Sine of the greater. The quotient Sine 50741 yéeldeth an arke 30 ^d. 30. ^m. whose Complement 59 ^d. 30 ^m. is the distance of the Substile from the Meridian.

First draw a line A. B. Parallele to the Horizon: drawe another C. D. squire-wise to the line A. B. make the two qua­drantsThe deline­ation of the figure. 71. from A. by D. to B. And because this Declination is on the left hand, number from A. towards D. 7 ^d. 35 ^m. the distance of the Meridian from the Horizon, drawing by this from C. the Meridian line. From that toward B. account 59 ^d. 30 ^m. the distance of the Substile from the Meridian, making in like manner at the ende of this number from C. the Substile line E. Againe from this forward (if you can, or else backward) number the distance of the Stile from the Substile, which is 61 ^d. 10 ^m. Draw likewise by this frō C. the line G. for the Stile. Let the Contingent cut the Substile squire-wise in any place, where you will, &c. The making of the reclining declining Dials before, shall teach you the finishing of this likewise.

[Page]

The making of a North inclining declining Diall. CAP. 20.

THe working of this kinde is altogether like the South reclining declining diall.

First therefore multiplie the Sine of the Decli­nation, by the Sine of the Complement of the inclination, diuiding the product by the whole Sine. The quotient shalt giue an arke: whose Complement shall be named the Com­plement [Page 34] repeated.

Then augment the Sine of the Complement of the De­clination by the whole Sine, & the product part by the Sine of the Complement repeated. Whereof will come a quoti­ent Sine: whose arke shall be the distance of that which is as it were the Meridian from the Horizon.

For the finding out of the Eleuation of the Meridian, be­cause it may bée done two sundry waies, I will set them downe both, referring the choyse to your selfe.

Now comparing the Eluation of the Pole with the E­leuation of the Meridian, subtract the lesser from the grea­ter, retaining the number remaining, which shall be called the difference kept.

Afterward increase the Sine of the difference kept, by the Sine of the Complement repeated, and diuide the pro­duct by the whole Sine. The quotient arising of this diui­sion shall giue an arke, which shall be the distance betwéene the Stile and the Substile.

To conclude, the Complement of this distance being compared with the Complement of the difference kept, multiplie the Sine of the lesser by the whole Sine, and part the product by the Sine of the greater. The Complement of the arke of the quotient Sine, shall be the distance of the Substile from the Meridian.

But note, if the Eleuation of the Meridian, & the Eleua­tion of the Pole be found equal, make an Equinoctial dial as before you were taught in the Meridionall reclining decli­ning: [Page] for there is no difference, but only that this is placed downward, and the South vpward.

There be of this north, as of the South reclining decli­ning, thrée sundrie kinds of Dials, as by the examples and figures following you may perceiue.

• Inclining 20 ^d.
• Declining 30 ^d.

First I multiplie 50000 the Sine of the declination, by 93969 the Sine of the Complement of the inclination, and the product thereof 4698450000, I diuide by the whole Sine 100000, & the quotient Sine 46984 yéeldeth an arke 28 ^d. 2 ^m. whose Complement 61 ^d. 58 ^m. is the Comple­ment repeated.

Then I augment the Sine of the Complement of the de­clination, which is 86602 by the whole Sine, parting the product 8660200000, by 88267 the Sine of the Comple­ment repeated: the quotient is 98113. The arke therof be­ing 78 ^d. 51 ^m. is y^• distance of the Meridian frō the horizon.

This done, I compare the Eleuation of the Meridian 67 ^d. 14 ^m. with the Eleuation of the Pole 52 ^d. subtracting the lesser from the greater, and there remaineth 15 ^d. 14 ^m. which is the difference kept.

[Page 35]Then I multiply 26275 the Sine of the difference kept, by 88267 the Sine of the Complement repeated: parting the product 2318455682, by the whole Sine. The arke 11. ^d. 47. ^m. of the quotient Sine 23184, is the distance of the Stile from the Substile.

Now the Complement of this distance being 76. ^d. 36. ^m. compared with 74. ^d. 47. ^m. the Complement of the diffe­rence kept, I encrease 96494 the Sine of the lesser, by the whole sine, and y^• product therof 9649400000, I distribute by 97277 the Sine of the greater. And the quotient is 99191. whose arke is 82. ^d. 43. ^m. The complement wher­of being 7. ^d. 17. ^m. is the distance of the Substile from the Meridian.

The delineation of those Dials, whose plat lyeth nigh to the Eleuation of the Pole. CAP. 21.

WHen as the declination and reclination, or declination and inclination, shall cause the plat of your Diall to lye nigh to the Eleuation of the Pole, then the hower lines must be drawen very long, to make the distances somwhat large, so that they may be discerned one from another. And [Page] commonly these folowing be such as here we meane.

• Reclining direct.
• Declining erect.
• Declining reclining.

• Reclining.
• Inclining.

• Inclining direct.
• Declining erect.
• Declining inclining.

In all these kindes oftentimes the distance betwéen the Stile and the Substile is so small that it is not possible to draw your Diall, except the plat be very large: which at al times you shall not haue.

Therefore the best way is this. First drawe your Diall very true (as before hath béen taught) vpon a large paper, making your quadrant so great as may be. Draw likewise the howre lines very long, which will cause the greater di­stance betwéen the Stile and the Substile, and in like ma­ner betwéen the howre lines. It will also procure y^• greater Equator, which wil be a great help in the drawing of your Dial, as by the figure following you may perceiue.

When you haue thus drawen your Diall, then at y^• end of the paper wher the lines be of greatest distance, draw the line I. K. parallel to the Horizontall line, make likewise the other line L M. (according to the greatnes of your plat) of e­quall distance from y^• other. This done, finde out (or rather draw a line vpon your plat which may represent) the true Horizon of the plat: cutting the paper in sunder by the line I. K. and the line L. M. placing it vpon your plat very plain, so that one of the edges may be parallel to y^• Horizon. Then make markes vpon the plat at both ends of all the howre lines drawing by those marks, y^• lines for y^• howers. Draw likewise light lines vpon the plat for the Stile & the Sub­stile, easily to be extinguished, as you did vpon the paper.

Let the Stile N. be fastened ouer y^• Substile as you haue [Page 38] béen taught in the East, & West, or Equinoctiall Dialls.

But here note, that the Stile must not be of equal height at both ends. But let it be so high, as the distance betwéene the Stile & the Substile is in the place where it standeth: as in the examples folowing you may plainely perceiue.

How to draw the 12. Signes in all kindes of Dials before mentioned. CAP. 22.

FOr somuch as sometime the 12 signes of the Zodiake are placed in Sunne Dialls, know in what signe y^• Sunne is at any time (which albeit any kinde will receiue, yet most commonly the verticall directly opposite to y^• South are garnished therewith.) Therefore in drawing the 12. Signes, in these South, and all other kindes of Dialls before mentioned, doe thus.

Prepare a little Table of Iron, Brasse, or close grained wood, in breadth 3. or 4. in length 5. or 6. inches. Chose for the Diameter one of the shortest sides, wherein draw vpon the centre A. halfe a circle to be deuided into two quadrants distinguished with a line drawen from the centre A. to the arke B. parte ech quadrant into 90. degrées. The line A. B. is héere in place of the Equator, shewing in the arke the be­ginning of Aries and Libra. Moreouer how to finde the o­ther Signes on both sides, séeke in the Table of the declina­tion of the Sunne from the Equinoctial circle, which is af­terward expressed. First finde out the degrées of the di­stance (in this Table) of the beginning of Taurus from the Equator being 11. degrées 30. minutes. Account this di­stance in the quadrant from the letter B. towards the left hand, and at the end of this number draw a line from the centre A. which shall shew the beginning of Taurus and Virgo. Account like wise the same distance on the other side from the line A. B. and make a line for the beginning of Pisces and Scorpius. Againe in the 20. ^d. 12. ^m. from the line A. B. you shal haue the beginning of Gemeni & Leo: and so [Page 39] many degrées and minutes on the other side, the beginning of Aquarius and Sagittarius. To conclude on both sides in the 23 degrées 30 minutes there must bée on the one part Cancer, and on the other Capricorne.

Afterward draw lines from the centre A by euery marke so long as your Table will receiue▪ and at the end of these lines let the caracters of the 12 signes of the Zodiake bée fixed: as in the figure following you may sée.

How to place the vnequall houres in a Horizontall Diall. CAP. 23.

FIrst you shall vnderstand, that by an vnequall hower is ment the 12 part of the day whether it be short or long. For when the Sunne is in the Equi­noctiall circle, the day hath 12 equall, and like wise 12 vnequall houres. But in winter solstice, when the dayes be at the shortest, and containeth with vs (where the Pole is eleuated aboue the Horizon 52 de­grées) onely 7 houres, 36 minutes: then the vnequall houres be lesse. For if you diuide these 7 houres 36 minutes by fractions into 12: you shall finde onely 38 minutes of an equall houre, to make one vnequall houre. But in the sommer solstice, when the dayes bée at the longest, and hath 16 houres, 24 minutes: then one houre with 24 minutes maketh one houre vnequall. These bée like wise called the houres of the Planets, and are placed in Dialls on this maner.

Your Diall being made and prepared, draw vpon it the 12 signes of the Zodiake (as you were taught before) so large as your plat will giue leaue, &c.

Note that alwaies the Meridian line, or 12 equall houre, [Page 41] is the 6 vnequall houre. Marke likewise, that when the Sunne entreth into the beginning of ♈ and ♎, both the e­quall and the vnequall houres be of like quantitie. For the 7 equall houre in the morning is the first vnequall, and the 8 equall the 2 planetare houre, &c.

But in the Tropike of ♋ it is otherwise: for then the vn­equall houres be greater then the equal. Therefore account how many houres and minutes is in the longest day for your Eleuation, changing all the minutes of those houres into one totall summe. Then diuide this number by 12: and the quotient shall shew how many minutes of an e­quall maketh one vnequall houre. Example hereof: with vs the longest day is 16 houres 24 minutes. Where­fore I multiplie 60 which is the minutes of an equal houre, by 16, which is the summe of houres of the longest day, and the product ariseth to 960, whereunto I adde 24 the mi­nutes remayning, then the number shall be 984, which be­ing diuided by 12, the quotient is 82. Wherefore I conclude that 82 minutes of an equall, maketh one vnequall or pla­netare houre. Then to draw them in Dialls worke thus.

First in the Tropike of ♋ diuide the space betwéene each houre into 60 equall partes, but because those distances bée so small, for the most part, that this cannot be done: there­fore it shall suffice to diuide euery one of them into thrée e­quall partes. Then shall euery parte conteine 20 ^m. and thrée of them 60 ^m. which is one equall houre.

When as therefore you will place the 7 vnequall houre, account from the Meridian line, or 12 houre, 4 parts and 2 minutes (which is 82 minutes) and make there a marke. Then place your ruler by this point or marke, and the in­tersection of the line of ♈ and ♎, and the first houre after noone: drawe a line from one Tropike to another, that is from ♋ to ♑.

Then for the placing of the 8 vnequall houre, account from the 7 last made, 4 of the foresaid parts and two mi­nutes make there a marke, by which and the intersection of [Page] the line of ♈ and ♎ and the line of the second houre after noone, the ruler being placed, draw a line, as before.

For the 9, account from the 8 last made, likewise 4 parts and two minutes, drawing a line as before. Doe in like maner for the drawing of all the other remaining: that is, for the 10, and 11. The 12 being the Sunne sitting shall néede no line.

As you haue finished the vnequall houres on this side for the afternoone, doe in like maner for those in the fore­noone: accounting from the Meridian 4 parts and 2 ^m. of an equall houre, to each vnequall, drawing, as before, lines from one Tropike to another.

You may place the figures for the vnequall or planetare houres, at the ende of the lines vnder the Tropike of Capri­corne, or aboue the line of Cancer as you will.

Note that the same part of the Stile which sheweth the 12 signes, shall like wise shew the vnequall houres. For better vnderstanding hereof:

Behold the Figure following.

[Page 42]

How to place the houres vnequall in a South erect direct Diall. CAP. 24.

THere is almost the same maner of pla­cing the vnequall houres in this kind, as in the Horizontall, differing onely herein, that whereas you diuided the spaces betwéene the houre lines, in the Tropike of ♋, (which is next to the centre of your Diall) and accoun­ted 4 partes 2 minutes of an equall, to one vnequall houre: in this you must diuide the Tropike of ♑, (which is like­wise next to the centre of the Diall) taking onely 2 partes wanting 2 minutes, (that is 38^m.) of an equall houre to ech vnequall.

Therefore your Diall being prepared, and the 12 signes of the Zodiake placed therein, &c. account how many houres and minutes be in the shortest day in your country, changing all the houres into minutes (giuing to each houre 60 minutes.) Diuide the product by 12, and the quotient shall shew how many minutes of an equall, will be answe­rable to each vnequall houre. Example, the shortest day with vs is 7 houres, 36 minutes, which being multiplied into one grosse number of minutes, the product is 456, which I diuide by 12, and the quotient is 38. Therefore I affirme that 38 minutes of a vulgar or equall, maketh one planetare or vnequall houre.

Then diuide the Tropike of ♑ here, as in the Horizon­tall you did the Tropike of ♋. Afterward account from the Meridian line 38^m. (or 2 parts wanting, 2^m.) of an e­quall houre to each vnequall, &c.

Finish all thinges else in this Meridionall erect direct, as you were taught in the Horizontal. The figure ensueth.

[Page 43]

The placing of the howres vnequall in East and West Dials. CAP. 25.

FIrst draw your Diall true, then place the 12 Signes of the Zodiake in it: which you may doe on this wise. The Stile of your Dial being fixed therin, as before hath béen taught, take your Trigonall (before prescribed) and place it vpon the edge of the Stile: then according to the direction of the thread (as in y^• South Diall before) make marks vpon the plat, as in the Figure following you may sée. Afterward by these markes draw lines as before you did, placing at the ends of them the Caracters of the 12 Signes.

Moreouer note that where the centre A. of your Trigo­nall doth touch the edge of the Stile, there fasten a small péece of Iron or such like thing, which may shew she place of the Sunne in euery Signe, and likewise the planetare or vnequall howres.

[Page 44]

When as therfore you would place the howres vnequal in those kindes of Dialls, account the m [...]mber of howres and minutes, which the halfe of the longest day in your Country doth conteine.

The making of an Horizontall Sphericall or hollow Diall. CAP. 26.

FIrst prepare your Sphere or plat perfectly hollow, of what quantity you wil. Then with your compasses diuide the vpper brimme thereof, into 4 equall partes, marking it with the letters A. B. C. D. Afterward open your Compasses to the widenes of one Quadrant, either A. C. or A. D. and with that widenesse, one foote being placed in the point D. draw an obscure or light line from A. to B. This done, set one foote of your Compasses in the point C and with the other you shall try whether the plat be perfectly Sphericall or not: for if it be, your Compasses will fall in the same line which you made before, but if it be not perfect you may a­mend it. And then draw this line manifest, so that it may continue. For it shall be the Meridian line, shewing the 12 howre. Againe, place one foote in the point A. drawing with the other a light line from C. to D. then one foote being set in B. you may try with the other, as before, the truth of the line last made. Now where these two lines cut eche o­ther in the bottome of the plat, place the letter E. Then di­uiding the quadrant A. E. into 90. ^d. and accounting therein the Eleuation of the Pole from E. toward A. make y^• point F. by which the arke line of ♈ and ♎, representing the equi­noctiall circle, shall be drawen. Afterward séeke out the greatest declination of the Sun from the Equator, (which you shall finde in the table of the declination of the Sunne) which is 23. ^d. 30. ^m. Then account 23. ^d. 30. ^m. from F. to­ward E. making there a mark for the arke of Cancer. Like­wise from F. toward A. number the same distance, for the arke line of Capricorne. This done, account from F. toward E. 20. ^d. 12. ^m. make there also a marke for the line of ♊ and [Page 46] ♌. And the like space from F. towards A. for the arke of ♒ and ♐. To conclude, number from F. towards E. 11. ^d. 30. ^m. making there also a marke for the [...]ine of ♉ and ♍: and so much from F. toward A. for ♓ and ♏.

When you haue thus made marks for al y^• arkes of the 12 Signes, open your compasses to the Quadrant of the Spheare, that is, from A. to C. which widenes of the com­passes remaining, place one foote in the point F. in the arke of ♈ and ♎ in the Meridian: and where the other foot shall touch the same Meridian towards B make the point G. which shall represent the Pole antartic [...], by which as it were from a centre, draw a line from the point D. by F. to C. which shall be the line of Aries and Libra. Then one foote of your compasses remaining in the point G. with the other draw lines from one side of the plat to the other, by euery marke before made for the 12. Signes of the Zodiake.

The 12. Signes being thus finished, procéede to the di­uision of the equall howres on this manner. Diuide y^• arke of the Equator into 12. equall partes, beginning at D. by F. ending in C. Then open your Compasses to the quadrāt of the plat, and the same widenesse of them remaining, place one foote on the first point of the diuision next C. in the Equator, and if you haue diuided the Equator equally, the other foote will touch the first part beyond F. towards D. by which from the centre G to the edge of the plat, draw a line, which shall shew the first howre afternoone. This done, remoue your compasses (that widenesse remaining) placing one foote in the second part from C. towards F. and the other foote touching the second part from F towards D. make a line as before, from the centre G. to the brimme of the plat, to shew the 2. howre afternoone. In like manner, finish all the other howre lines, namely the 3, 4, 5, 6 7, 8, for the afternoone. Then doe likewise on the other side for the howres in the forenoone, namely the 11, 10, 9, 8, 7, 6, 5, and 4 drawing lines from the centre G. by euery diuision, to the brimme of the plat, &c.

The making of a South Spherical erect direct Diall. CAP. 27.

YOu shall vnderstand that the making of this kind differeth almost nothing from the Horizontall: except onely in the accounting the beginning of the Eleuation of the Pole, & drawing the houres vnequall.

First your Sphear or plat being prepared, and parted in­to foure Quadrants, as before in the Horizontall you were taught, diuide the quadrant E. A. into 90 degrees. Then number in it the Eleuation of the Pole from A. toward E. marking the ende of the number with F. Afterward open your compasses to the quadrant of the plat, and the one foote placed in F. extend the other in the Meridian line towards B. making the point G. where it shall touch the Meridian line: from which point as a centre you shall draw the lines for the 12 signes, as you were taught in the Horizontall. Diuide likewise, and draw the lines for the equall houres, as before in the Horizontall.

The placing of the houres vnequall in this kinde is done, as in the South Erect Direct vpon a plaine superficies, ex­cept onely that whereas there you did draw them with a ruler, here you shall doe it with compasses on this manner. Open your compasses to the Quadrant of the plat, that is, from A. to C. accounsing from the Meridian in the Tro­pike of Capricorne, two parts wanting two minutes of an equall houre, making there a marke: And that widenesse of your compasses remaining, draw a line from that marke made in the Tropike ♑, by the intersection of the line of A­ries and Libra, & the line of the first houre in the afternoone, vnto the Tropike of Cancer. Draw all the other lines for the vnequall houres in like maner, as in the figure you may plainely sée,

Fixe the Stile in the centre G. as you did in the Horizon­tall, placing the figures for the equall and vnequall houres, [Page 48] and the caracters to the 12 signes, as in the example follow­ing, or otherwise as you will.

The making of a Diall vpon a Quadrant, or the Table described in the beginning of the Booke. CAP. 28.

IF you will make this Diall vppon a Quadrant alone, prepare and diuide it altogether like one of the Quadrants of the table, whose making is prescribed in the first Chapter: or if you will make it vpon that instrument alreadie prepared, worke thus.

Diuide the diameter E. D. into foure equall partes; then leauing out the fourth part next E. describe vpon the middle point betwéene C. and S. an obscure halfe circle: which di­uide into sixe equall parts. From each point of which diui­sion draw light lines vnto the Semidiameter D. E. touching it squire wise. Then draw nigh to the line D. E. two other lines parallele thereto: which with the crosse lines drawne from the halfe circle may make 12 vnequall quadrangles, wherein the 12 signes of the Zodiake may be figured: the North towards D. the South toward S. And to the intent you may proportionally part these Quadrangles into fiue or ten degrées: diuide those sixe parts of the halfe circle e­qually into so many degrées. Then drawe againe from those diuisions right liues to the Dimetient D. E. squire­wise, as before.

This done vpon the point E. from S. that is to say, from the end of ♐ and beginning of ♑, drawe an arke to the Se­midiamer D. E. which shall be for the winter Tropike. A­gaine vpon the same centre, draw another from the begin­ning of ♈ and ♎ for the Equator. To conclude, make an­other from the angle D. that is, from the end of ♊ and begin­ning of ♋, which shall bée for the Summer Tropike. By the helpe of these arkes, you shall draw the houre lines, so [Page 49] that you first know the height of the Sunne aboue the Ho­rizon, at euery houre, when it occupieth the beginning of ♑, ♈ and ♋. Whose altitude is thus found out.

Take the Eleuation of the Pole, and the Complement thereof: also the declination of the Sunne from the Equi­noctiall, and the distance of the Sunne from the Meridian, accounting 15 degrées for euery houre, with the Comple­ment of this distance. Then if you desire to knowe the al­titude of the Sunne at sixe of the clock either in the mor­ning or at euening (at which houre onely in Sommer it is aboue the Horizon) multiply the Sine of the Eleuation of the Pole, by the Sine of the declaration of the Sunne, diui­ding the product by the whole Sine, and you shall haue your desire.

But if the Sunne shall bée distant from the Meridian fewer howers then sixe, multiply the Sine of this distance (giuing to euery houre 15 degrées) by the Sine of the Com­plement of the altitude of the Pole, and the product hereof diuide by the whole Sine. Then take the arke of the quo­tient from 90 degrées, and the first number found out shall remaine, which must be kept. Then compare the Sine of this number found out, with the Sine of the Eleuation of the Pole, augment the lesser by the whole Sine, diuiding the product by the greater: Whereof shall come a quotient Sine, to the complement of whose arke adde the declination of the Sunne, if it shall be in the North signes, or subtract it from that, if it occupieth the South signes. And if the num­ber which commeth hereof shall bée greater then 90, take it from 180, and you shall haue the second number found out. The Sine of this number found out being multiplied, by the Sine of the first number found out, shall yéelde a product, which part by the whole Sine, and the arke of the quotient Sine shall shew the desired altitude of the Sunne for the houre propounded.

But if the distance of the Sunne from the Meridian shall excéede sixe houres (that is 90 degrées) take the de­grées [Page] of that distance from 180, multiplying the Sine of the remainder by the Sine of the Complemēt of the altitude of the Pole, diuiding the product by the whole Sine, and subtract the arke of the quotient from 90, the remainder shal be called the First number found out. The Sine of which number found out compared with the Sine of the Eleuati­on of the Pole, increase the lesser by the whole Sine, and di­stribute the product by the greater. Then take the Com­plement of the Declination of the Sunne, from the arke of the quotient, and you shall haue the second number found out.

Finally the Sine of the first number found out, and the Sine of the second being multiplied by themselues, and the product parted by the whole Sine; the quotient Sine shall yéeld an arke, which shall bée the altitude of the Sunne.

But when the Sunne occupieth the beginning of ♈ or ♎ you shall finde the altitude thereof euery houre, onely by multiplying the Sine of the Complement of the distance of the Sunne from the Meridian, by the Sine of the Com­plement of the Eleuation of the Pole, diuiding the product by the whole Sine, the quotient which commeth hereof shall yéeld the arke of your desire. At both the sixe houres, because the one sixe is the rising, and the other the sitting, there is then no altitude of the Sunne aboue the Ho­rizon.

Moreouer, to know how much the Meridian altitude is, of the Sunne entring into ♋, adde the greatest Declinati­on thereof, to the Complement of the Eleuation of the Pole: and by subtracting the greatest Declination of the Sunne, from the complement of the Eleuation of the Pole, you shall likewise haue the Meridian altitude of the Sunne entring into ♑.

You shall likewise séeke the height of the Sunne, being in the 10 degrée of ♉ aboue the Horizon, at 8 of the clock be­fore noone, and at 4 after noone: at 7 in the morning, and 5 [Page 51] in the euening: at both 6. and the 5. houre in the morning, and 7. at night. Also at 5. in the morning and 7. in the eue­ning, the Sunne being in the beginning of ♊.

But because the working of this, to finde out these alti­tudes, requireth much time and labour, we will set downe these distances ready found out, calculated for the Eleua­tion of the Pole 50. ^d. and 52 ^d. which you may likewise vse without any notable difference, where the Pole is Eleua­ted 49. ^d. 51 ^d. and 53 ^d.

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If therefore you will make your Quadrant Horologi­call for the Eleuation of the Pole 50. ^d. extend a thread, or lay a ruler, from the centre E. by the 63. ^d. 30. ^m. of the limbe of the quadrant, beginning at F and where the thread so placed shall touch the Tropike of ♋, there make a point or marke. Againe, let the thread be placed vpon the 40. ^d. of the limbe, and where it shall touch the Equator, there also make a marke. Thirdly let y^• thread be drawen by the 16. ^d. 30. ^m. of the quadrant: and where it shall touch the Tropike of ♑, there make likewise a marke. This done, search out the centre (by the 5. Proposstion 4. Euclid) and ioyne these thrée markes into one arke, which shall be the line for the 12 hower. Afterward place the thread vpon the 61. ^d. 2. ^m. of the limbe, and where it shall cut the Tropike of ♋ note it. Draw likewise the thread vpon the 38. ^d. 23. ^m. of the [Page 51] limbe, marke where it cutteth the Equator. Then by the 15▪ ^d. 19. ^m. of the limbe extend the thread, making a marke in the section therof and the Tropike of ♑. Drawe into one arke these thrée markes (as you did before) finding out the common centre: so shall you haue the line of the 11. hower before noone, and 1. after noone. In like maner are the rest of the hower lines drawen by 3. points accoun­ted and found out by the altitude of the sunne. But because before 9. and after 3. in this Eleuation of the Pole it will not serue for this delineation, there must another be made by the table of the 12. Signes or Quadrangles. Therfore draw that from the 10. ^d. of ♉, so shall you haue a line or place wherein the third points may be marked. Then to prepare the points of the arke for the howers of 8. and 4. of 7. and 5. and both the 6. besides the two markes made in the Tropike of ♋ and the Equator, let the third be noted in the arke drawen from the 10. degrée of ♉. [...] draw the line of the 5. and 7. howers, vse besides the Tropike of Cancer and this drawen from the 10. degrée of ♉, the third which shall come from the beginning of ♊. The line of 4. in the mor­ning, and 8. in the euening is very short, included in y^• an­gle next vnto I. Thus you haue finished 9 arkes or lines for the houres, of which each one of them doth shew two houres: except one which is onely for the [...]2 houre.

And to the intent that there should be no space left voyd and vnprofitable betwéene the Tropike of ♑ and the cen­tre E. you may drawe in that place the vnequall houres, which you shall very easily doe on this maner. Describe vpon the centre E. an arke very little distant from the Tro­pike of ♑, to be diuided into 6 equall parts, and take the middle point betwéene the beginning of that arke at S. and betwéene E. Then vpon this point▪ as it were a centre, make a halfe circle from S. to E. which shall be the line of the 6. vnequall hower, that is of midday. Afterward one foote of the compasses being placed in the same Semidiame­ter E. D. and remoued eche way as occasion shall require: [Page] and the other extended in the meane time so far that it may touch the second point of the arke diuided into 6. partes, and the centre E. draw an arke or line from the centre to that point, which shal shew the 5. and 7. howers. Inlike maner are the other 4. drawen, one foote of the compasses being re­moued, as necessity shal require, in the Semidiameter E. D (which may be prolonged if it shall be too short) & the com­passes so opened, that the other foot may touch the point of the diuided arke and the centre E. and ioyne them all into one arke line: then fixe a thread in the centre E. well wax­ed, hauing two small beades vpon it to moue with a plum­met of iron or lead. Last of all prepare two sights of brasse or other metall with ioynts, which you must place in one right line vpon the side or edge A. B. so that you may direct them, and againe turne them downe vpon that side A. B. at your pleasure: that (if you make this Dial vpon the Table or instrument prescribed in the beginning of this book) they be no hinderance to you in trying or examining of your plats. Thus is your diall prepared: but for further instruc­tion beholde the Figure.

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How to make an instrument, whereby you may know the iust howre of the night by the Starres. CAP. 27.

FIrst prepare a plat or Table of brasse or firme and solide wood, which will not change or bend, somwhat thinne, let the forme of it be round, thrée in­ches broad, or more or lesse as you will. Draw a circle nigh vnto the edg, and diuide it into 12. equal parts: wherein shal be placed the 12. Signes of the Zodiake, then diuide ech of these parts into 30. equal partes, which shall shew the number of the daies that the Sunne moueth in euery Sign. Make a second circle, wher­in you may number the daies, and a third circle wherein write the Caracters of the 12. Signes, as you sée in the Fi­gure following.

Draw likewise another circle, wherein you may write the daies of euery moneth in the yéere, which you may doe by any Calender, but for more easines beholde the Table following, whereby you may likewise doe it.

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Enter this Table, and you shal finde that the first day of January must be placed against the 20. ^d. 13. ^m of Capri­corne, and the 15. day against the 5. ^d. 33. ^m. of Aquarius, &c.

Therfore lay your ruler vpon the centre A. of your plat, and vpon the 20. ^d. 13. ^m. of ♑, and where it shall touch the circle which you made for the daies of euery moneth, there make a marke which shall shew the first day of January. Then lay it vpon the 5. ^d. 33. ^m. of ♒ and the centre A. and where it shall touch the circle, make a marke for the 15. day of January. Then lay your ruler vpon A. and the 21. ^d. 44. ^m. of ♒, and make likewise a marke for the 31. day of January. Again place the ruler vpon the centre A. and the [Page 54] 5. ^d. 52. ^m. of ♓, for the 14. day of Februarie: worke thus with all the rest, vntil you haue set down the beginning and middest of euery month, as the Table doth direct you, then hauing found out the beginning & middest of euery month, you may at ease diuide euery space into so many partes, as there be daies in the month, which it serueth for, according as you sée in this figure following. It shall be also necessary to make one circle to write the number of the daies of the moneth, and another for the names of the monthes: you must make also a hole in the centre of this plat, of such big­nesse, as you may sée a Starre thorow it: as in the figure you may perceiue.

Hauing prepared this plat with the circles vpon it, for the 12. Signes and months, you must prepare an other whéele full of téeth, which you shall make on this maner.

Take a thime Table or platte of the same matter your [Page] other was, draw a circle vpon it so great as the inward cir­cle of the first plat, and diuide it into 24. equall parts. Then draw lines from the centre to euery one of those parts, vn­till you haue so many as there be howers in the longest night in your Country. Then cut téeth by these lines for the howres, and write the number of them vpon the téeth as you sée in this figure. Let the tooth for the 12. howre be so long from the centre to the end, as is betwéen the centre A. of the other plat and the circle of the 12. Signes. You must likewise make a hole in the centre of this whéele, of the same bignes of that, in the other plat.

This done, you must prepare a ruler, which shalbe pla­ced vpon this whéele. Take therefore a thin péece of wood or brasse, and draw vpon it a light line: at y^• one end therof in the middest of the line make a hole of the same bignesse which that is in the great plat and tothed whéele. Then draw a circle halfe an inch wide or more if you will, after­ward cut the ruler round at the end, cutting off likewise the one halfe of the breadth of the ruler by the line.

[Page 55]Let the length of the ruler from th [...] [...]ntre to the end be of such quantitie, as is from the centre of the great plat to the outward edge, and an inch or more if y [...].

The foreside of your instrument being finished: the back­side must haue two small holes, and a handle, which may turne and moue about. Prepare them thus.

Vpon your great plat on the backside, right against the fourth degrée of ♍, nigh the edge make a small hole, where­in the little pinne of the handle may bée pla [...]ed, when néede requireth: and write by this the greater Beare. Againe, right against the 28 ^d. 1 ^m. of ♎ on the backside make ano­ther hole likewise, so that the handle being: remoued to it, the pinne may enter, as in the other. And by this, write the lesser Beare. &c.

[Page]Prepare for your handle a thin plate of yron, brasse, or woode, about sixe inches in length or more if you wil. Draw in the middest of it a line: at the one ende thereof make a hole in the middest of the line, of such bignesse as the hole in the greater plat is. Vpon this handle you must fasten a litle short pinne right in the line, so that the handle being faste­ned with the pinne toward the backside of the great plat, and being moued the pinne may fall directly into the hole.

Beholde the Figure following.

Prepare a hollow pinne of yron or brasse, according to this figure aboue, so great as it may easilie enter into the hole of the handle, so that it may turne vpon it: but let it goe close into the great plat, that it may remaine immoueable. Let the Tothed whéele and the ruler moue likewise vpon it. Then place the Tothed whéele and the ruler on the for­mer side of the plat vpon the pinne, and on the backside the handle with the little pinne towards the back of the plat: fa­sten them so together, that the handle, the Tothed whéele, and the ruler may turne vpon the plat seuerally, or each by themselues at your pleasure, and as néede shall require. Notwithstanding you must bée carefull, that the hole in your pinne may bée so great, as you may perfectly sée the North Starre in the night thorow it. Then your instru­ment shall bée finished, whose vse followeth afterward.

For better instruction beholde the figure.

The making of a Diall, to knowe the houre by the Moone. CAP. 30.

PRepare a square péece of wood or metall 3 or 4 inches ouer, draw thereon a cir­cle so great as you can, draw also an­other within that. Now because the Moone finisheth her course in 29 dayes, 12 houres, and 44 minutes, part the inward into 30 parts in this maner; let 29 be equally diuided, and the 30 which is the last must not be so great by a third part: wherefore diuide one of those 29 [Page] equall parts into thrée parts, and take two of them for the 30 and last part.

Cut out that which is within the inward limb, and pre­pare another of the same or like wood or metall somewhat thicker, and worke it so that it may goe into the former so much of the thicknes as it may be equal on the backside, and that which remaineth of the thicknes, let it hang ouer the foreside of the first plate to kéepe it from falling through: draw a circle vpon it, and diuide it into 24 equall parts: draw from these parts houre lines so many as shall suffice for the longest night: cut that which ouerhangeth with téeth at euery houre, but especially at the 12 houre make a long tooth, fixe a wier in the centre for the Stile equally di­stant from the circle on each side. Let the Stile hang so [Page 58] much beneath the plat, as it is aboue, because you shall haue as much vse on the backside as before.

Cut out all that which is within the inward circle, except a little portion to support the Stile, so that this shall bée but as a hoope. Drawe the houre lines on the backside of this plat, as on the foreside, so that the one may be right against another: then drawe lines on the inside from euery one of these lines on the foreside to the lines on the backside. This plat or whéele must be moueable and turne within the former.

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