Horometria: or the compleat diallist

THE FOURTH BOOK. Shewing how to resolve all such Astro∣nomicall Propositions (as are of ordinary use in this Art of Dialling) by help of a Quadrant fitted for the same purpose. (Book 4)

CHAP. I. The description of the Quadrant.

HAving in the second and third Books shewed Geometrically the working of most of the ordinary Propositions Astro∣nomicall, with the delineation of all kinde of plain wall Di∣als howsoever, or in what lati∣itude soever scituated, still keep∣ng within the limits of our plane, and yet not tyed to the use of any Instrument.

I will now shew how you may performe the former

work exactly, easily and speedily, by a plain Quadrant fit∣ted for that purpose; the description whereof is after this manner.

Having prepared a piece of Box or Brasse in manner of a Quadrant, draw thereon the two Semidiameters A B and A C, equally distant or parallel to the edges, cutting one the other at right angles in the center A, upon which cen∣ter A, with the Semidiameter A B or A C, describe the arch B C, this arch is called the limbe, and is divided into 90 equall parts or degrees; and subdivided into as many parts as quantity will give leave, being numbered from the left hand towards the right after the usuall manner.

Then let the Semidiameter A B be divided into 90 un∣equall parts, (called right Sines) either from the Table of naturall Sines by help of a decimall Scale, equall to the Semidiameter A B, or else by taking the neerest extents from each degree of your Quadrant, unto the side A B, and placing them upon the side A B each after other, from the center A towards B, you shall exactly divide the Se∣midiameter A B into 90 unequall divisions called right Sines.

This being done, draw the line D E from the Sine of 45 degrees counted in the line of Sines unto 45 degrees counted in the Quadrant, then from the point E draw the line E F parallel to A B, making the square A D E F, the side D E whereof (for distinction) may be called a Tangent line, and the side E F a Co-tangent line, then draw the Diagonall line A E, which you may call the line of Latitudes.

Then upon the center A, with the distance A D or A F describe the arch D F, which you may divide into six equall parts, by laying your Rule upon each 15th. degree in the

Quadrant, and the center A as at g h I k l F, from which points draw slope lines to each 15th. degree in the Qua∣drant, numbered backward, as F P, l O, k E, I n, h m, g B; these lines so drawn are to be accounted as hours, then dividing each space into two equall parts, draw other slope lines standing for half hours, which may be distin∣guished from the other, as they are in the figure.

[illustration] Dialling diagram

Now because in the latter part of this Book there is often required to use a line of Chords to severall Radiuses,

therefore upon the edge of the Quadrant A C, you may have a line of Chords, divided as in the figure, and so the Quadrant being at hand will supply the uses of the Scale mentioned in the preceding Book, and also a Chord of any Circle, whose Radius is lesse then the line A C may be taken off and in that case supply the use of a Sector.

To this Quadrant, as to all others of this kind in their use is added Sights, with a threed, bead, and plummet ac∣cording to the usuall manner.

CAAP. II. Of the use of the line of Sines. Any Radius not exceeding the line of Sines being known, to finde the right Sine of any arch or angle thereunto belonging.

IF the Radius of the Circle given be equall to the line of Sines, there needs no farther work, but to take the other Sines also out of the line of Sines.

But if it be lesser, then take it betwixt your compasses, and set one foot in the Sine of 90 degrees, and with the o∣ther lay the threed to the neerest distance, which you may doe by turning the compasses about, till the moving point thereof doe onely touch the threed and no more: the threrd lying still in this position, take the neerest extent thereunto, from any Sine you think good, and it shall be the like Sine agreeable to the Radius given.

As for example, let the circle B C D E in the following chapter represent the meridian circle, let B D be the Ho∣rizon, and C E the verticall circle; and let F G be the dia∣meter

of an almicanter, and so F H the Semidiameter there∣of; which being given it is required to finde the Sines both of 30 and 50 degrees, agreeable to that Radius, first there∣fore, take the given Radius betwixt your compasses, and with one foot set in the Sine of 90 degrees, with the o∣ther lay the threed to the neerest distance, the threed lying still in this position, take the neerest extents thereunto, from the Sine of 30 and likewise of 50, these distances place upon the Radius F H from H to N, and from H to R, so shall H N be the Sine of 30 degrees, and H R the Sine of 50 degrees, agreeable to the Radius F H the thing desired.

CHAP. III. The Right Sine of any arch being given to finde the Radius.

TAke the Sine given betwixt your compasses, and set∣ting one foot in the like Sine in the line of Sines, with the other lay the threed to the neerest distance, the threed lying still in this position, take the shortest extent thereunto from the Sine of 90 degrees, which distance shall be the Radius required.

As for example, let H R be the given Sine of 50 de∣grees, & it is required to find the Radius answering there∣unto, take H R with your compasses and set one foot in the Sine of 50 deg. and with the other lay the threed to the neerest distance, which being kept in this position, if you take the shortest extent thereunto, from the Sine of 90 you shall have the line H F for the Radius required.

[illustration] Dialling diagram

CHAP. IV. The right Sine, or the Radius of any Circle being given, and a streight line resembling a Sine, to finde the quantity of that unknown Sine.

FIrst, take the Radius, or the right sine given, and set∣ting one foot of your Compasses either in the like sine or in the Radius of the line of Sines, and with the other, lay the threed to the neerest distance, then take the right line given, and six one foot in the line of sines, moving it till the moveable foot touch the threed at the neerest ex∣tent, so shall the fixed foot stay at the degree of the sine required.

As for example, let F H be the Radius given, and H N the streight line given resembling a Sine, first with the di∣stance F H from the Sine of 90 lay the threed to the neerest distance; the threed lying still in this position, take the line H N and fixing one foote of your compasses in the line of Sines, still moving it to and fro till the moveable foote thereof, doth onely touch the threed, so shall the fixed foote rest at the Sine of 30 degrees in the line of Sines; this 30 degrees is the arch, of which H N is the Sine, F H in the last chapter being the Radius.

CHAP. V. The Radius of a circle not exceeding the line of Sines being given, to finde the chords of every arch.

IF the Radius given, shall be equall to the line of Sines, then double the Sine of halfe the arch, and you shall have the chord of the whole arch, that is, a Sine of 10 deg. dou∣bled giveth a chord of 20 deg. & a Sine of 15 deg. doubled giveth a chord of 30 deg. and so of the rest, as in the third chapter, the line I O the Sine of I C an arch of 30 deg. be∣ing doubled giveth I L the chord of ICL which is an arch of 60 deg.

And if the Radius of the circle given, be equall to the Semi-radius (the sine of 30 deg.) of the line of sines; then you neede not to double the lines of sines as before, but onely double the numbers: so shall a sine of 10 deg. be a chord of 20 deg. and a sine of 15 deg. be a chord of 30 deg. and so of the rest, but if the Radius of the circle given, be lesse then the semi-radius of your line of sines, then take it

betwixt your compasses and setting one foot in the sine of 30 deg. with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the neerest extent in what Sine you think good, onely doubling the number, and you shall have the Chord desired.

As for example, let A C be the diameter of the circle in the third Chapter, and it is required to find a Chorde of 30 degrees, therefore first, I take A C betwixt my compas∣ses and setting one foot in the Sine of 30 deg. with the other I lay the threed to the neerest distance: which being kept at this angle, I take it over from the sine of 15 deg. which doth give me I C the Chord of 30 deg. which was desired.

And if the Radius given, be greater then the Sine of 30. and yet lesse then the Radius of the line of Sines; then with the Radius given, and from the Sine of the comple∣ment of halfe the arch required, lay the threed to the nee∣rest distance; then taking it over at the neerest extent from the sine of the whole arch, you shall have your desire.

As for example, let the Radius A C of the circle in the third Chapter be given; and a Chord of 30 deg. required: the halfe of 30 deg. is 15 deg. the complement whereof is 75 deg. therefore I take the Radius with my compasses, and setting one foot in the Sine of 75 deg. with the other I lay the threed to the neerest distance: the threed lying still in this position, I take the shortest extent thereunto from the Sine of 30 deg. which giveth I C the Chord of 30 deg. which was desired.

Now by the converse of this Chapter, if you have the Chord of any arch given, you may thereby find out the Radius.

CHAP. VI. To divide a line by extream and mean proportion.

A Right line is said to be divided by extream and meane proportion, when the lesser Segment thereof, is to the greater, as the greater is to the whole line.

Let A B be the line to be so divided, this line I take with my compasses, and setting one foot in the sine of 54 deg. and with the other I lay the threed to the neerest distance: which lying still in this position, I take it over from the sine of 30 deg. which distance shall be the greater segment A C dividing the whole line in the point C; or the threed lying in the former position, if you shall take the shortest extent thereunto from 18 deg. you shall have B C for the lesser segment, which will divide the whole line by extream and mean proportion in the point C from the end B, so that as B C the lesser segment, is to A C the greater segment; so is A C the greater segment, to A B the whole line, as was required

[illustration] Dialling diagram

CHAP. VII. To find a mean proportionall line between two right lines given.

A Mean proportionall line is that, whose square is equall to the long square, contained under his two extreams.

First, ioyn the two given lines together, so as they may make both one right line; the which divide into two equall parts; and with the one halfe thereof, setting one foot in the Sine of 90 deg. with the other lay the threed to the nee∣rest extent, which lying still in this position, take the di∣stance betwixt the middle point, & the point of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but onely touch the threed; now from the complement of the sine where the fixed foot so resteth, take the shortest extent unto the threed, which shall be the mean proprtional line required.

[illustration] Dialling diagram

As for example, let A and B be two lines given, between which it is required to find a mean proportionall line, first joyne the two lines together in F, so as they both make the right line C D, which divide into two equall parts in the point E, then with either halfe of which, setting one foot in the sine of 90 deg. with the other lay the threed to the neerest distance: then keeping the threed in this positiō, take the distance betweene the middle point E and F, the place of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but onely touch the threed, and the fixed foot will stay about 22 deg. 30 min. the complement whereof is 67 deg. 30 min. from which take the shottest extent unto the threed lying as before, which shall be the line H, the meane proportional line be∣twixt the two extreames A and B, which was required.

CHAP. VIII. Having the distance of the Sun from the next equinoctiall point, to find his declination.

FIrst, lay the threed upon 23 deg. 30 min. the suns great∣est declination, counted on the limbe of the quadrant, the threed lying still open at this angle, take the shortest ex∣tent thereunto from the sine of the distance of the Sunne from the next equinoctiall point, this distance being ap∣plied to the line of sines from the centre A, shall give you the sine of the suns declination.

So in the figure of the 13 chapter, the sun being in the 29 deg. of Taurus at K, which is 59 deg. from C the equi∣noctiall point Aries; the declination of the sun will be found about 20 deg. the line C M which was required.

CHAP. IX. The declination of the sun, and the quarter of the ecliptick which he posseseth being given, to find his place.

TAke the sine of the suns declination from the line of sines, & setting one foot in the sine of the suns greatest declination, with the other lay the threed to the neerest di∣stance so shall it shew upon the limb, the distance of the sun from the next equinoctiall point.

So in the figure of the 13 chapter C M the declination of the sun being 20 deg. and K the angle of the suns great∣est declination, the line C K will be found to be 59 deg. for the distance of the sun from the next equinoctiall point which was required.

CHAP. X. Having the latitude of the place, and the distance of the sun from the next equinoctiall point, to find his amplitude.

TAke the sine of the suns greatest declination betwixt your compasses, and setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest di∣stance; which lying still in this position, set one foot in the sine of the suns distance from the next equinoctiall point, and with the other take the neerest extent unto the threed, so shall you have betwixt your compasses the Sine of the Amplitude.

As in the figure of the 13 chapter, the angle at N being 37 deg. 30 min. the complement of the latitude, and K the angle of the Suns greatest declination, and C K 59 deg. the distance of the Sun from the equinoctiall point Aries, the line C N will be found to be the Sine of 34 deg. 9 min. the amplitude required

CHAP. XI. Having the declination and amplitude to finde the height of the pole.

FIrst, take the sine of the Suns declination, and set one foot in the sine of the Amplitude, and with the other lay the threed to the neerest distance, so shall the threed upon the limbe, shew the complement of the latitude.

So in the figure of the 13 Chapter, the declination C M being 20 deg. and the amplitude C N being 34 deg. 9 min. & the angle at M being Right, we shall find the angle at N

to be 37 deg. 30 min. the complement whereof is 52 deg. 30 min. which was required for the latitude of the place.

CHAP. XII. Having the latitude of the place, and the declination of the Sun, to find his amplitude.

WIth the Sine of the declination set one foot in the co∣sine of the latitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the amplitude required: so in the figure of the next Chapter, the angle CNM being 37 deg. 30 min. the cosine of the la∣titude, and C M the declination here 20 deg. and the angle at M being right, we shall finde the base C N to be the Sine of 34. which was required for the Suns amplitude.

CHAP. XIII. Having the elevation of the pole, and amplitude of the Sun, to find his declination.

FIrst, lay the threed to the amplitude counted in the limb, then take it over at the shortest extent, from the cosine of the latitude, so shall you have the Sine of the suns decli∣nation betwixt your compasses.

So in this figure, the amplitude C N being. 34 deg. 9 min. and the angle at N being cosine to the latitude, the an∣gle at M being a right angle, we shal find CM to be 20 deg. for the declination of the sun which was required.

[illustration] Dialling diagram

CHAP. XIIII. Having the latitude of the place, and the declination of the Sun, to find his height in the Vertical circle.

FIrst, take the sine of the declination of the Sun, and setting one foot in the Sine of the latitude, with the o∣ther lay the threed to the neerest distance; so shall it shew upon the limb the height of the Sun in the Verticall circle.

So in the figure of the last Chapter, the angle C I O being 52 deg. 30 min. the latitude of the place, and C O the Suns declination 20 degrees, and the angle C O I

being a right angle we may find C I to be a sine of 25 deg. 32 min. the height of the sun in the Vertical circle which was required.

CHAP. XV. Having the latitude of the place, and the distance of the Sun from the next Equinoctiall Point, to find his height in the verticall circle.

FIrst, take the sine of the suns greatest declination, & set∣ting one foot in the sine of the latitude, with the other lay the threed to the neerest distance: the threed lying still in this position; from the sine of the suns place take the nee∣rest extent thereunto, which shall be the sine of the suns height in the Vertical circle.

So in the figure of the 13 chapter, the angle at I being 52 deg. 30 min. which is the latitude of the place, and the angle at K the suns greatest declination, and K C being 59 deg. the suns distance from the next equinoctiall point, we shall find C I to be 25 deg. 32 min. for the height of the sun in the Vertical circle.

CHAP. XVI. Having the latitude of the place and the declination of the Sun, to finde the time when the Sun commeth to be due east or west.

WIth the sine of the declination, set one foot in the sine of the latitude, and with the other lay the threed to the neerest distance: then take it over at the neerest extent

from the co-sine of the latitude; which distance keep; and setting one foot in the co-sine of the declination, with the other lay the threed to the neerest distance: so shall it shew upon the limbe, the quantitie of degrees betwixt the houre of six and the East or West points.

So in the figure in the 13 Chapter, the declination C O being 20 deg. and the angle O I C being 52 deg 30 min. the complement whereof is the angle O C I, we may find the sine O I which distance keep; now seeing O I is a sine of the Radius O F & not of AEC, therefore by the 4 Chapter, you may find the quantity of that unknown sine; for seeing the Radius O F is the cosine of the declination, therefore set one foot therein, and with the other distance kept, lay the threed to the neerest distance: so shall it shew upon the limbe 16 deg. 30 min. which converted into time maketh 1 houre, and 6 min. for the quantitie of time betweene the hour of six and the suns being in the East or West points.

CHAP. XVII. Having the latitude of the Place, and the declination of the sun, to find his altitude at the houre of six.

FIrst, take the threed, and lay it upon the declination counted in the limbe; then from the sine of the latitude, take it over at the shortest extent; which distance shall be the sine of the height of the sun at the houre of six.

So in the figure of the 13 Chapter, the angle at L being a right angle, and L O C being 52 deg. 30 min. the latitude of the place, and C O the declination of the sun being 20 deg. we shall find C L to be the sine of 15 deg. 44 min for the height of the sun at the houre of six, which was equired.

CHAP. XVIII. Having the latitude of the place, and the height of the sun at the hour of six, to find what azimuth he shall have at the houre of six.

FIrst, with the sine of the suns height at the houre of six, set one foot in the sine of the latitude, and with the other lay the threed to the neerest distance: then take the least distance thereunto from the cosine of the latitude, now with this distance setting one foot in the cosine of the alti∣tude, with the other lay the threed to the neerest distance as before: so shall it shew upon the limbe, the azimuth of the sun from the East or West points.

So in the figure of the 13 chapter, the angle CLO being a right angle, and the angle L C O being 37 deg. 30 min. the cosine of the latitude, the angle L O C must be 52 deg. 30 min. the latitude of the place being the complement of the angle L C O, and C L being 15 deg 44 min. (as by the last chapter it did appeare) we shall find L O to be the sine of 12 deg. 30 min. for the azimuth of the sun from the East or West, at the hour of six as was required.

CHAP. XIX. Having the declination of the sun, to finde his Right Ascension.

FIrst, with your compasses take the sine of the suns de∣clination given, and setting one foot in the sine of the suns greatest declination, with the other lay the threed to

the neerest distance: then at the least distance from the co∣sine of the suns greatest declination take it over: now again, with this distance lay the threed to the neerest distance from the cosine of the declination given, so shall it shew upon the limbe the right ascension of the sun.

So in the figure of the 13 chapter, C O the suns declinati∣on being 20 deg. and the angle O K C being 23 deg. 30 min. the suns greatest declination, and the angle K C O being the complement of the angle O K C we shall find K O to be the sine of 56 deg. 50 min. for the right ascen∣sion of the sun required.

CHAP. XX. Having the latitude of the place, and the declination of the Sun, to finde the Ascensionall difference.

FIrst, take the sine of the Suns declination, and setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance: then at the least di∣stance take it over from the sine of the Latitude: with which, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance, so shall it shew upon the limbe the Suns Ascensionall dif∣ference.

So in the figure of the 13 Chapter, the angle M C N being 52 deg. 30 min. and the angle C N M being the complement thereof, the one being the latitude, and the other the co-latitude, and C M being 20 deg. the sine of the Suns declination, we shall finde M N 28 deg. 19 min. for the difference of Ascensions, which being converted

into time, maketh 1 houre, and somthing better then 53 min.

Now when the Sun hath North declination, if you take this difference of Ascension (which is 1 houre 53 min.) out of 6 houres, there will be left 4 hours 7 min. for the time of Sun rising, and if you adde it unto 6 hours, the same will be 7 houres 53 min. for the time of Sun setting.

And so contrarily, when the Sun hath South declinati∣on, if you adde this ascensionall difference to 6 hours, you shall have the time of his rising, and if you take it away from 6 hours, that which is left shall be the time of Sun setting.

CHAP. XXI. The Latitude of the place, the Almicanter, and declination of the Sun being given, to finde the Azimuth.

IF the Suns declination be Northward, then by the 14 or 15 Chapters get his height in the Verticall Circle for the day proposed: from the sine of which take the distance unto the sine of the Suns altitude observed: with this di∣stance, setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance; unto which (being kept still in this position) take the least di∣stance from the sine of the Latitude, with this distance, set∣ting one foot in the co-sine of the Suns altitude, with the other lay the threed again to the neerest distance, so shall it shew upon the limb the Suns Azimuth from the East or West, either Northward or Southward.

So in this figure, having N M the distance betwixt the

sine of 14 deg. 33 min. (the height of the Sun in the Verticall circle) and the sine of 30 deg. 45 min. the height of the Sun at the time of observation, and 52 deg. 30 min.

[illustration] Dialling diagram

the angle N O M the Latitude of the place, the comple∣ment whereof is 37 deg. 30 min. the angle M N O, we shall finde M O to be the sine of 23 deg. 17 min. the Azi∣muth from the East or West points Southward.

And here note, when the declination is Northward, that as when the latitude of the Sun given, and his height in the

Verticall circle is equall, he is directly in the East or West, so when his altitude given is greatest, then is the Azimuth toward the South, and when his altitude given is least, then is the Azimuth towards the North.

But if the declination of the Sun be Southward, then by the 10 or 12 Chapters, finde the Amplitude for the day proposed.

Now first, take the sine of the Suns altitude, and setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance, which threed lying still in this position, take it over at the shortest extent from the sine of the Latitude, this distance adde to the sine of the Amplitude, by setting one foot in the sine of the Ampli∣tude, and extending the other upon the line of sines, these two being thus joyned, take them betwixt your Compas∣ses, setting one foot in the co-sine of the Suns altitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the Suns Azimuth from the East or West, towards the South.

So in this figure, having V C or T N, 19 deg. 7 min. the Amplitude for the day proposed, and T V the sine of the Suns altitude being 13 deg. 20 min. and 52 deg. 30 min. the angle V X T, the latitude of the place; and the angle T V X, the complement thereof; we shall finde X N to be the sine of 40 deg. 11 min. the Azimuth of the Sun from the East or West points Southward, which was required.

CHAP. XXII. The latitude of the place, the declination and altitude of the Sun being given, to finde the houre of the day.

IF the declination of the Sun be Northward, finde the height of the Sun at the houre of six by the 17 Chapter, betwixt which sine, and the sine of the Suns altitude given, take the distance upon the line of sines, with which di∣stance, setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the shortest ex∣tent from the sine of 90 deg. with this distance, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the houre of six.

So in this figure, having M N the distance betwixt the sine of 9 deg. 5 min (the height of the Sun at the houre of six) and the sine of 42 deg. 33 min. the height of the Sun given, and the angle M O N 52 deg. 30 min. the La∣titude of the place, and his complement M N O, we shall finde N O to be the sine of 60 deg. the quantity of time from the houre of six, which 60 deg. is four hours of time. And here also note, that if the altitude given be greater then the altitude of the Sun at the houre of six, then is the time found to the Southward of the houre of six; but if it be lesser then is it to the Northward.

But if the declination of the Sun be Southward, finde his depression at the houre of six, by the 17 Chapter, for

the day proposed, which will be equall to his height at six, if the quantity of declination be alike.

[illustration] Dialling diagram

Now take the sine of this depression, and adde it to the sine of his altitude observed, by setting one foot in the sine of his altitude, and extending the other upon the line of sines: These two being thus joyned together in one, take them betwixt your compasses, and setting one foot in the co-sine of the latitude as before, and with the other, lay the threed to the neerest distance: which lying still in this po∣sition, take it over at the shortest extent from the sine of

90 deg. with this distance, setting one foot in the co-sine of the declination as before, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the houre of six.

So in this figure, having the sine of 15 deg. 24 min. the altitude of the Sun given, and the sine of 9 deg. 5 min. his depression at the houre of six, joyned both together in one streight line, as T V, and having the angle T X V 52 deg. 20 min. the Latitude given, and the angle T V X the co-latitude, we shall find T X to be the sine of 45 deg. the quantity of time from the houre of six, which con∣verted into time will make three houres.

The end of the fourth Book.