Horlogiographia optica. Dialling universall and particular: speculative and practicall. In a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. With diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. Illustrated by diverse opticall conceits, taken out of Augilonius, Kercherius, Clavius, and others. Lastly, topothesia, or, a feigned description of the court of art. Full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. Together with many usefull instruments and dials in brasse, made by Walter Hayes, at the Crosse Daggers in More Fields. / Written by Silvanus Morgan.

THE PRAECOGNITA ASTRONOMICAL.

CHAP I.

Of the division of the World, by accidentall scituation of the Circles.

COSMUS, the World, is divided by Microcosmus the little World, into substantiall and imaginary parts: Now the substantiall are those materiall parts or substance of which the World is compacted and made a Body, by the inter-folding of one Sphear within another, as is the Sphear of Saturn, Jupi∣ter, Mars, Sol, &c. And these of themselves have a gentle and proper motion, but by violence of the first mover,

have a racked motion contrary to their own proper moti∣on: whence it appears, that the motion of the heavens are two, one proper to the Sphears as they are different in themselves, the other common to all.

By Phebus motion plainly doth appear, How many dayes doe constitute one yeare.

Will you know how many days doe constitute a year, he telleth you who saith,

Ter centum ter viginti cum quinque diebus Sex horas, neque plus integer Annus habet.

Three hundred sixty five dayes, as appear, With six houres added, make a compleat Year.

The just period of the Suns proper revolution.

Perpetuus Solis distinguit tempora motus.

The Imaginary part traced out by mans imagination, are Circles, such is the Horizon, the Equator, the Meridi∣an, these Circles have of themselves no proper motion, but by alteration of place have an accidentall division, di∣viding the World into a right Sphear, cutting the paral∣lels of the Sun equally or oblique, making unequall dayes and nights: whence two observations arise:

First, Where the parallels of the Sun are cut equally, there is also the dayes and nights equall.

Secondly, Where they are cut oblique, there also the dayes and nights are unequall.

The variety of the heavens are diversly divided into Sphears, or severall Orbs, and as the Poets have found

out a Galazia, the milkie way of Juno her brests, or the way by which the gods goe to their Palaces, so they will assigne to each Sphear his severall god.

Goddesse of

Heralts.

Caliope in the highest Sphears doth dwell,

Astrologie.

Amongst the Stars Urania doth excell,

Philosophie.

Polimnia, the Sphear of Saturn guides,

Gladnesse,

Sterpsicore with Jupiter abides.

Historie,

And Clio raigneth in mans fixed Sphear.

Tragedic.

Melpomine guides him that gvids the year

Solace.

Yea, and Erata doth fair Venus sway.

Loud Instruments.

Mercury his Orbe Euturpe doth obey.

Ditty.

And horned Cynthia is become the Court Of Thalia to sing and laugh at sport.

Where they take their places as they come in order.

The Sphear is said to be right where the Poles have no elevation, but lie in the Horizon, so that to them the E∣quinoctiall is in the Zenith, that is, the point just over their heads.

The Sphear is oblique in regard of its accidentall divi∣sion, accidentally divided in regard of its orbicular form; orbicular in regard of its accidentall, equall variation orbi∣cular, it appears before in the Praecognita Philosophicall, his equall variation is seen by the equall proportion of the earth answering to a Coelestiall degree, for Circles are in proportion one to another, and parallel one to another are cut equally, so is the earth to the heavens; having considered them as before, we will now consider another sort of Sphear, which is called parallel.

This parallel Sphear is so that the parallels of the Sun are parallel to the Horizon, having the Poles in their Ze∣nith, being the extream intemperate, colde, and frozen Zone: Ovid in his banishment complaines thus thereof.

Hard is the fright in Scythia I sustain, Over my head heavens Axis doth remain.

CHAP II.

Of the Circles of the Horizon, the Equator, and the Meridian.

THe greatest Circle of a Sphear is that which divides it in two equall parts, and that be∣cause it crosseth diametrically, and the dia∣meter is the longest line as can be struck in a Circle, and therefore the greatest, which great Circles are represented in the follow∣ing figure, representing the Circles of a Sphear in an ob∣lique Latitude, according to the Latitude or elevation of the Pole here at London, which is 51 deg. 32 min. being North Latitude, because the North Pole is elevated.

The Horizon is a great Circle dividing the part of hea∣uen seen, from where we imagine an Antipodes, the inha∣bitants being to us an Antipheristasin, our direct opposites, so that while the Sun continues visible to us, it is above our Horizon, and so continues day with us, while it is night with our opposites; and when the Sun goes down with us it appears to them, making day with them while it remain∣eth night with us, and according to the demonstration, is expressed by the greot Circle marked NSEW, signifying

the East, West, North, and South parts of the Horizon. So now if you imagine a Circle to be drawn from the Suns leaving our sight, through those Azimuth points of heaven, then that Circle there imagined is the Horizon, and is accidentally divided as a man changes his place, and divides the World in a right or oblique Sphear.

[illustration]

The Meridian is a great Circle scituated at right angles to the Horizon, equally passing between the East and West points, and consequently running due North and South, and passeth through the Poles of the World, being sted∣fastly fixed, it is represented by the great Circle marked NDSC, and is accidentally divided, if we travell East

or West, but in travailing North or South altereth not, & when the Sun touches this Circle, it is then mid-day or Noon: Now if you imagine a Circle to passe from the North to the South parts of the Horizon, through your Zenith, that Circle so imagined is your Meridian, from which Meridian we account the distance of houres.

The Aequinoctiall likewise divides the World in two equall parts, crossing at right angles between the two Poles, and is therefore distant from each Pole 90 degrees, and is elevated from the Horizon on the contrary side of the Poles elevation, so much as the Pole wants of 90 deg. elevation, demonstrated in the Scene by the Circle passing from A to B, and is accidentally elevated with the Poles as we change our Horizon, and when the Sun touches this Circle, the dayes and nights are then equall, and to those that live under this Citcle the dayes and nights hang in equilibra continually, and the Sun doth move every houre 15 degrees of this Circle, making the houre lines equall, passing 15 degrees in one houre, 30 degrees in two houres, 45 degrees in three houres, 60 degrees for four, and so in∣creasing 15 degrees as you increase in houres. This I note to the intent you may know my meaning at such time as I shall have occasion ro mention the Aequinoctiall distances.

The Axis of the World is that which the Stile in every Diall represents, being a line imaginary, supposed to passe through the center of the World, from the South to the North part of the Meridian, whose outmost ends are the Poles of the World, this becomes the Diameter, about which the World is imagined to be turned in a right Sphear having no elevation, in an oblique to be elevated above the Horizon and the angle at the center, numbred on the arch of the Meridian between the apparent Pole and the Hori∣zon,

is the elevation thereof, represented by the streight line passing from E to F, the arch EN being accounted the elevation thereof, which according to our demonstration is the Latitude of London.

[illustration]

The Stars that doe attend the Artick or North Pole, are the greater and lesser Beare, the last star in the lesser Bears tale is called the Pole Star, by reason of its neerness to it: this is the guide of Mariners, as appeareth by Ovid in his exile, thus

You great and lesser Bear whose Stars doe guide Sydonian and Graecian ships that glide

Even you whose Poles doe view this lesser Ball, Under the Western Sea neere set at all.

The stars that attend the Southern Pole is the Cross, as is seen in the Globes.

Lord be my Pole, make me thy Style, Lord then Thy name shall be my terminus ad quem.

Video Coelos opera manuum tuarum, lunam & stellas que tu fundasti.

CHAP III.

Of the severall sorts of Planes, and how they are known.

DYals are the dayes limiters, and the bounders of time, whereof there are three sorts: Hori∣zontall, Erect, Inclining: Horizontall are al∣wayes parallel to the Horizon: Erect, some are erect direct, others erect declining: Incli∣ning also are direct or declining: for more explanation the figure following shall give you better satisfaction, where the Horizon marked with diverse points of the Compasse shall explain the demonstration: Now if you imagine Cir∣cles to passe through the Zenith A, crossing the Horizon in his opposite points, as from SW through the verticall point A, passing to the opposite point of South-west to North-East, those, or the like circles, are called Azimuthes, parallel to which Azimuthes all erect Sciothericals doe stand.

Those Planes that lie parallel to the Horizontall Circle are called Horizontall planes, and his Style makes an angle with the Pole equall to the elevation thereof; then the ele∣vation of the Pole is the elevation of the Style.

[illustration]

Erect Verticals are such which make right angles with the Horizon, and lie parallel to the Verticall point, and these, as I told you before, were either direct or declining.

Direct are those that stand in a direct Azimuth, behold∣ing one of the four Cardinall Quarters of the World, as either direct East, West, North, or South, marked with these letters NEWS, or declining from them to some other indirect Azimuth or side-lying points.

Erect North and South are such as behold those Quar∣ters, and cuts the Meridian at right angles, so that the planes crosse the Meridian due East and West, and the Poles are their Styles, equally elevated according to the aequino∣ctiall altitude, being the complement of the Poles ele∣vation. For in all North Faces, Planes, or Dials, the Style beholds the North Pole, and in all South faces, the Style beholds the South Pole: therefore, where the North Pole is elevated, there the North Pole must be pointed out by the Style, and where the South Pole is elevated vice versa.

The second sort of Verticals are declining, which ate such that make an acute angle with the Quarter from which they decline; for an acute angle is lesse then a right angle, and a right angle is 90 degrees: these declining Planes ly∣ing in some accidentall Azimuthe.

For supposing a Diall to turn from the South or North towards the East or West, the Meridian line of the South declines Eastward, happening in these Azimuthes or be∣tween them.

• South declining East
• South declining West
• ...

  S by E	11	15	Or to these points of the West decli∣ners, or be∣tween them.	S by W	11	15
  S S E	22	30	S S W	22	30
  S E by S	33	45	S W by S	33	45
  South-East	45	00	South West	45	00
  S. E by E	56	15	S W by W	56	15
  E S E	67	30	W S W	67	30
  E by S	78	45	W by S	78	45
  East	90	00	West.	90	00

Again, North decliners, declining toward the East and West, doe happen in these Azimuthes or between them.

• North declining East
• North declining West
• ...

  N by E	11	15	Or to these points of the West decli∣ners, or be∣tween them.	N by W	11	15
  N N E	22	30	N N W	22	30
  N E by N	33	45	N W by N	33	45
  North-East	45	00	North West	45	00
  N E by E	56	15	N W by W	56	15
  E N E	67	30	W N W	67	30
  E by N	78	45	W by N	78	45
  East.	90	00	West.	90	00

By which it appeareth that every point of the Compasse is distant from the Meridian 11 degrees 15 minutes.

The third sort of planes are inclining, or rather reclining, whose upper face beholds the Zenith, and in that respect is called Reclining, but if a Diall be made on the nether side, and thereby respect the Horizon, it is then called an incliner, so that the one is the opposite to the other.

These planes are likewise accidentally divided, for they are either direct recliners, reclining from the direct points of East, West, North; and South, and in this sort happens the direct Polar and Aequinoctiall planes, as infinite more according to the inclination or reclination of the plane, or they are as erect planes doe become declining recliners, which looke oblique to the Cardinall parts of the World, and obtusely to the parts they respect.

Suppose a plane to fall backward from the Zenith, and by consequence it falls towards the Horizon; then that represents a Reclining plane, such you shall you suppose the Aequinoctiall Circle in the figure to represent, reclining from the North Southwards 51 degrees from the Zenith, or suppose the Axis to represent a plane lying parallel to it, which falls from the Zenith Northward reclining 38 de∣grees, one being Aequinoctiall, the other a Polar plane.

But for the inclining decliners you shall know them thus, forasmuch as the Horizon is the limiter of our sight, and being cut at right angles representeth the East, West, North, and South points, it may happen so that a plane may lie between two of these quarters in an accidentall Azimuth, and so not beholding one of the Cardinall Quar∣ters is said to decline: Again, the said plain may happen not to stand Verticall, which is either Inclining or Recli∣ning, and so are said to be Inclining Decliners: First, be∣cause they make no right angle with the Cardinal Quarters: Secondly, because they are not Verticall or upright.

There are other Polar planes, which lie parallel to the Poles under the Meridian, which may justly be called Me∣ridian plains, and these are erect direct East and West Dials, where the poles of the plane remain, which planes if they recline, are called Position planes, cutting the Horizon in the North and South points, for Circles of position are nothing but Circles crossing the Horizon in those points.

CHAP IV.

Shewing the finding out of a Meridian Line after many wayes, and the Declination of a Plane.

A Meridian Line is nothing else but a line whose outmost ends point due North and South, and consequently lying under the Meridian Circle, and the Sun comming to the Meridian doth then cast the shadow of all things Northward in our Latitude;

so that a line drawn through the shadow of any thing per∣pendicularly eraised, the Sun being in the Meridian, that line so drawn is a Meridian line, the use whereof is to place planes in a due scituation to their points respective, as in the definition of this Circle I shewed there was acci∣dentall Meridians as many as can be imagined between place and place, which difference of Meridians is the Lon∣gitude, or rather difference of Longitude, which is the space of two Meridians, which shews why noon is sooner to some then others.

The Meridian may be found divers wayes, as most com∣monly by the Mariners compasse, but by reason the needle hath a point attractive subject to errour, and so overthrow∣eth the labour, I cease to speake any further.

It may be found in the night, for when the starre called Aliot, seems to be over the Pole-starre, they are then true North, the manner of finding it, Mr. Foster▪ hath plainly laid down in his book of Dyalling, performed by a Qua∣drant, which is the fourth part of a circle, being parted into 90 degrees.

It may also be fouhd as Master Blundevile in his Booke for the Sea teacheth, being indeed a thing very necessary for the Sea, which way is thus: Strike a Circle on a plain Superficies, and raise a wire, or such like, in the center to cast a shadow, then observe in the forenoon when the shadow is so that it just touches the circumference or edge of the Circle, and there make a mark; doe so again in the afternoon, and at the edge where the shadow goes out make another mark, between which two marks draw a line; which part in halfe, then from that middle point to the center draw a line which is a true Meridian.

Or thus, Draw a great many Circles concentricall one

within another, then observe by the Circles about noone when the Sun casts the shortest shadow, and that then shall represent a true Meridian, the reason why you must observe the length of the shadow by circles & not by lines is, because if the Sun have not attained to the true Meridian it wil cast its shadow from a line, and so my eye may deceive me, when as by Circles the Sun casting shadow round about, still meetes with one circumference or other, and so we may observe diligently. Secondly, it is proved that the shadow in the Meridian is the shortest, because the Sun is neerest the Verticall point. Thirdly, it is proved that it is a true Meridian for this cause, the Sun, as all other Lumi∣nous bodies, casts his shadow diametrically, and so being in the South part casts his shadow northward, and is there∣fore a true Meridian.

But now to finde the declination of a wall, if it be an erect wall draw a perpendicular line, but if it be a declining reclining plane, draw first an horizontall line, and then draw a perpendicular to that, and in the perpendicular line strike a Style or small Wyre to make right angles with the plane, then note when the shadow of the Style falleth in one line with the perpendicular, and at that instant take the altitude of the Sun, and so get the Azimuthe reckoned from the South, for that is the true declination of the wall from the South. The distance of the Azimuthes from the South, or other points, are mentioned in degrees and minutes in the third Chapter, in the definition of the seve∣rall sorts of planes: or by holding the streight side of any thing against the wall, as is the long Square ABCD, whose edge AB suppose to be held to a wall, and suppose again that you hold a thrid and plummet in your hand at E, the Sun shining, and it cast shadow the line EF, and at the

same instant take the alti∣tude of the Sun, thereby getting the Azimuthe as is taught following, then from the point F, as the center of the Horizon., and from the line FE, reckon the distance of the South, which suppose I finde the Azimuthe to be 60 degrees from the East or West, by the propositions that are delivered in the end of this Booke, and because there is a Quadrant of a Circle between the South, and the East or West points, I substract the distance of the Azimuthe from 90 degrees, and it shall leave 30, which is the declination of the wall, equall to the angle EFG: but to finde the inclination or reclination, I shall shew when I come to the use of the Universall Qua∣drant, or having first found the Meridian line, you may prick down the Azimuthe.

[illustration]

CHAP V.

Shewing what houre-lines may be drawn upon any Plane.

LIght being the cause primary of shadows, sha∣dows being but the imitation of the secondary cause, that is substance, doth delineate unto us the passing away of time, by receiving light on the substance casting shadow.

Neverthelesse, substance receives not light, if either they want the immediate beame or reflecting light, which is the reason that some Dials are vacant of diverse houres, or else are vacant for a certain Season of the year, wherefore we will shew some reasons why the Sun beams cannot be re∣ceived

[illustration]

on diverse planes, which is caused by the acciden∣tall scituation thereof, which we will consider by this fi∣gurative demonstration, by the Analemma, described in Mr. Gunters Book.

The Sun, though he never moves from the line Eclip∣tique wherein he hath his annuall or yearly motion, yet

have a declination from the Aequinoctiall North or South, making his diurnall or daily motion, altering the dayes and nights according to all the diversities thereof: for the Sun being in the Aequinoctiall hath no declination, but in his diurnall motion still declyning from the Aequinocti∣all makes his progresse towards the North or South, descri∣beth many parallel Circles, being parallel to the Aequino∣ctiall, whose farthest distance from either side is 23 deg. 30 minutes, so that so many degrees that the Sun is distant from the Aequinoctiall, so much is its declination.

Now if you imagine the Circle before described to re∣present the Meridian Circle which crossed diametrically, which diameter shall represent the Aequinoctiall, then lay∣ing down the greatest declination, on either side of it, drawing two lines at that distance, on either side of the Aequinoctiall, parallel to it, represent the Tropicks, the upper representing the Tropick of Cancer, marked with GE, the other the Tropick of Capricorn, marked with HI: and if from each severall degree you draw parallels too, they doe represent the parallels of the Sun, which shall shew the diurnall motion of the Sun: now if you crosse these parallels with a line from E to H, that then represents the Ecliptique; now if you crosse the Aequino-Ctiall at right angles with another line, that line represents the Axis of the World: then if you lay down from the Poles the elevation thereof, to wit, the North and South Poles, according to the elevation of the North Pole down∣ward, where the number of degrees end make a mark; then account the same elevation from the South Pole up∣ward, and there also make a mark, from which two marks draw a right line, which shall represent your Horizon, and cuts the parallels of the Sun according to the time of his abiding above the Horizon.

As in example, they that live under the Aequinoctiall have their dayes and nights equall, for under the Aequi∣noctiall the Poles lie in the Horizon, and have no elevation, so that you see the Axis AB cuts the Aequinoctiall at right

[illustration]

angles, and then must needes cut the parallels of the Sun equall, so that the continuing of the Sun above the Poles or Horizon, is equall to his continuance under the Poles or Horizon; so that there is represented a right Sphear where the dayes and nights are equall. But if the Pole hath ele∣vation,

as here at London, 51 degrees 30 minutes, then the Horizon is represented by CD, where you see that then the horizon cuts the parallel of the Sun oblique, represent∣ing an oblique Sphear, so that now the lines grow longer while the Sun declines in them towards the North pole A, then the day is represented by the parallel EF, and the night by FG, when the Sun is in his greatest North decl∣nation, so that you see that then in the night the Sun is no lower under the Horizon then from C to G, and then 'tis twi-light all night. Again, the Sun having his greatest de∣clination towards B the South Pole, then he continues but the arch ID above the Horizon, then the day is represent∣ed by KI: and the night by HK: Thus you see the reason of the dayes and nights inequality in an oblique Sphear, and equality in a right, you may likewise perceive by those parallels, why the Sun cannot shine on all Diall Planes, as we will now shew.

First, An East and West Diall lies parallel to the Me∣ridian, therefore the Sun in the Meridian cannot shine on them; neverthelesse, though an East and West Diall cannot have the houre of 12 on it, yet an East or West position may, because it crosseth the Horizon in the North and South.

Secondly, A direct North Diall can have but morning and evening houres on it, and then of no use but when the Sun hath North declination, for then his Amplitude or distance from the East and West is Northward, and so at morning or night shines on the face thereof.

Thirdly, A North reclining may shew all the houres all the year, if it recline from the North Southward, the quantity of the complement of the least Meridian altitude, but if but the complement of the elevation of the Aequi∣noctiall,

and so become a Polar Plane, it can then but shew while the Sun is in the North Signes, for the Dyall lying parallel to the Aequinoctiall while the Sun is in South de∣clination cannot shine on the plane because it lies under.

All upright planes declining from the South may have the houre line of 12, so also may all North decliners, but not in the Temperate Zone, which is contained between the degrees. South incliners also may have the line of 12, whose upper face is not below the least Meridian altitude, as also if greater then the greatest Meridian altitude, then doth the upper face want it.

Fifthly, all North recliners reclining more then the great∣est meridian altitudes complement, may have all the houres but will shew but one part of the yeare.

Sixthly, All South declinets or recliners may have the line of 12 on them. And now having proceeded thus far in some theoricall demonstration or grounds of Dials for the Geometricall projection, we will in the next Chapter lay down the theoricall demonstration for the Arithmeticall Calculation, and so proceed to our practicall way of opera∣tion as ensueth.

CHAP VI.

Being the definition of the severall lines of Sines, Tangents, and Secants, to be understood before we can come to Arithmeticall Calculation.

THe Arithmeticall part for calculation is thus to be understood, there are certain right lines from e∣very degree of a Quadrant, named by certain words of art, which for illustration we will con∣sider in their nomination and definition, whose names are

these, Sines, Tangents, and Secants, which have certain arches of degrees and minutes of a Quadrant answering thereto, and howsoever the Question is propounded, are resolved by these numbers as in the Golden Rule, by still adding the second and third number together in stead of Multiplication, and substracting the first instead of Divisi∣on, doth leave the arch of the Question as it was propoun∣ded, which well considered, nothing shall seeme difficult.

[illustration]

A right Sine is halfe the subtense of the double arch, which subtenses are represented by the lines passing from D to D▪ and from E to E, and from F to F, the halfe of which lines subtending the arches are the right Sines.

A Tangent is a right line without the peripherie to the extremity of the Secant to the Radius being perpendicu∣lar eraised, such is represented by the line BC.

A Secant is a right line drawn from the center through the circumference to the Tangent, such is represented by the line AB, the Semidiameter of the same Circle is called the Radius.

You may furthermore for very convenient uses have those lines placed on a Ruler, for if from one degree of one Quadrant of a Semicircle you draw lines to the same degree of the other Quadrant, cutting the line GA, that line so cut shall be a line of Sines, and if from the centre you draw lines to the Tangent line through every degree of the Quadrant, that line so cut is a Tangent line, whose use is most exquisite and infinite for the solution of many excel∣lent propositions.

CHAP VII.

Being the fundamentall Diagram for the Geometricall projection of Dials.

THe Style being the representation of the Axis of the World, doth become the Gnomon or substance casting shadow on all Planes lying parallel to some Circle or other, as to circles of Azimuthes in all Verticall Dials.

So that the figure following is a representation of divers semidiameters, doth plainly shew the theoricall ground of the practick part hereof.

Where the line in the demonstration, noted the semidi∣ameter of the Horizon, signifies the Horizon, for so sup∣posing it to represent an Horizontall Diall, the style or Axis must be elevated above it, according to the Poles ele∣vation above the Horizon, and then the semidiameter or Axis of the World represents the style or Axis casting sha∣dow being the line AC.

The Geometricall projection of Dials.

[illustration]

Where note by the way, that if you set one foot of the Compasses in B, and with the Semidiameter of the Equa∣tor, fix the other foot in the line BC, keeping that last foot fast, and at that center draw a Quadrant divided into six parts, & a ruler from the center of the Equator through each division, shall divide the line AB as a contingent line, and if from C to these marks on the line AB you draw lines, it shall be the houre lines of a Verticall Diall.

But supposing a Diall to stand verticall, or upright to the Horizon AB, as the line BC, then that is represented by the semidiameter of the Verticall, and his style again represented by the semidiameter or Axis AC, being di∣stant

from the Verticall equall to the complement of the Poles elevation, and again, the Aequinoctiall crossing the Axis at right angles, the semidiameter thereof is represent∣ed by the line BD, the reason why the angle at A hath to his opposite angle at C, the complement of the angle at A, is grounded on this, the three angles of any right lined triangle are equall to two right angles, and a right angle consists of 90 degrees: now the angle at B is 90 degrees, being one right angle, and the angle at A being an angle of 51 degrees, which wants of 90 39 degrees, which is the angle at C, all which being added together doe make 180 degrees, being two right angles: here you see that having two angles, the third is the complement of 180 degrees.

CHAP VIII.

Of the proportion of shadows to their Bodies.

SEeing the Zenith makes right angles with the Horizon, and a right angle consisteth of 90 degrees, the middle point betwixt both is 45 degrees, the Sun being at that height, the shadow of all things perpen∣dicularly raised, are equal to their bodies, so also is the Radius of a Circle equall to the Tangent of 45 degrees: and if the Sunne be lower then 45 degrees it must necessary follow the shadow must exceed the substance, because the Sun is nigh the Horizon, and this is called the adverse or contrary shadow.

Contrarily, if the Sun exceed this middle point, the substance then exceeds the shadow, because the Sun is neerer

the verticall point. Mr. Diggs in his Pantometria laying down the manifold uses of his Quadrant Geometricall, doth there shew, that having received the Sun beams through the Pinacides or Sights, that when the Suns alti∣tude cuts the parts of right shadow, then the shadow ex∣ceeds the substance erected casting shadow as 12 exceeds the parts cut: But in contrary shadow contrary effects.

CHAP IX.

To finde the Declination of the Sun.

TO give you Orontius his words, it is con∣venient to take the beginning from the greatest obliquation of the Sun, because on that almost the whole harmony of all Astronomicall matters seeme to depend, as shall be manifest from the discourse of the succeeding Canons.

Wherefore prepare of commodious and elect substance, a Quadrant of a Circle parted into 90 equall parts, on whose right angled Radius let be placed two pinacides or sights to receive the beams of the Sun.

Then erect it toward the South in the time of the Solsti∣cials, either in Cancer the highest annuall Almicanther, or in Capricorn the lowest annuall▪ meridian altitude, also ob∣serve the equilibra, or equality of day and night in the time of the Aequinoctials, from the Meridian altitude thereof substract the least Meridian altitude, which is, when the Sun enters in the first minute of Capricorn, the remainer is the Declination, or substract the Aequinoctiall altitude from

the greatest Meridian altitude, the remainer is the Decli∣nation of the greatest obliquity of the Sun in the Zodiaque.

The height of the Sun is observed by the Quadrant when the beames are received through the sights by a plum∣met proceeding from the center, noting the degree of alti∣tude by the thrid falling thereon.

You may also take notice that for the continuall variati∣on of the Suns greatest declination it ought to be observed by faithful Instruments: for as Orontius notes that Claudius, Ptolomie found it to be 23 degrees 51 minutes and 20 se∣conds, but in the time of Albatigine the same number of degrees yet but 35 minutes, Alcmeon found it of little lesse, to wit 33 minutes, Purbachi and some of his Disciples doe affirme the same to be 23 degrees only 28 minutes, yet Jo∣hanes Regiomontan. in the tables of Directions, hath alotted the minutes to be 30, but since Dominick Maria an Italian, and Johannes Varner of Norimburg testifie to have found it to be 29 minutes, to which observation our works doe ex∣actly agree. Albeit all did observe the same well neere by like Instruments, neverthelesse, not justly by exact constru∣ction, or by insufficient dexterity of observation some small difference might happen, but not so much as from Ptolo∣mie to our time.

Having this greatest Declination, to finde the present Declination is thus, by calculation: As the Radius, is to the Sine of the greatest Declination; so is the Sine of the Suns distance from the next Aequinoctiall point, that is Aries or Libra, to the declination required: wherefore in the Naturall Sines, as in the Rule of Proportion, multiply the second by the third, divide by the first, the Quotient is the Sine of the Declination. Or by the naturall Sines, adde the second and third, and substract the first, the remainer is the Sine of the present Declination.

Degre.	♈	♎	♉	♏	♊	♐	Degre.
D	m	D	m	D	m
0	0	0	11	29	20	10	30
1	0	24	11	50	20	23	29
2	0	47	12	11	20	35	28
3	1	11	12	31	20	47	27
4	1	35	12	52	20	58	26
5	1	59	13	12	21	9	25
6	2	23	13	32	21	20	24
7	2	47	13	52	21	30	23
8	3	10	14	11	21	40	22
9	3	34	14	30	21	49	21
10	3	58	14	50	21	58	20
11	4	21	15	8	22	7	19
12	4	45	15	27	22	15	18
13	5	8	15	45	22	23	17
14	5	31	16	3	22	30	16
15	5	55	16	21	22	37	15
16	6	18	16	38	22	43	14
17	6	41	16	56	22	50	13
18	7	4	17	12	22	55	12
19	7	27	17	29	23	0	11
20	7	49	17	45	23	5	10
21	8	12	18	1	23	9	9
22	8	34	18	17	23	13	8
23	8	57	18	32	23	17	7
24	9	19	18	47	23	20	6
25	9	41	19	2	23	22	5
26	10	3	19	16	23	24	4
27	10	25	19	30	23	26	3
28	10	46	19	44	23	27	2
29	11	8	19	57	23	27	1
30	11	29	20	10	23	28	0
De	♓	♍	♒	♌	♑	♋	De

But I have here ad∣ded a Table of Decli∣nation of the part of the Ecliptique from the Aequinoctiall, the use whereof you may dis∣cern is very plain, for if you finde the Signe on the top, and the de∣grees downward, the common angle shall be the Declination of the Sun that day. As if the Sun being in the 10 degree of Taurus or Scorpio, the declina∣tion shall bee 14 de∣grees 50 minutes, and if you finde the Signe in the bottome, you shall seeke the degrees on the right hand up∣ward, so the 20 de∣greee of Leo or Aqua∣rius hath the same de∣clination with the for∣mer.

The ende of the Praecog∣nita Astronomicall.