one dayes variation in 300 yeares as is observed by Mr. Philips The Vses of the Quadrant WIthout rectifying the Bead nothing can be performed by this Projection except finding the Suns Meridian Altitude being shewn upon the Index by the intersection of the Parallel of declination therewith Also the time when the Sun will be due East or West TRace the Parallel of Declination to the right edge of the Projection and the houre it there intersects in most cases to be duly estimated shewes the time sought thus when the Sun hath 21 deg of North declination we shall find that he will be due East or West about three quarters of an houre past 4 in the afternoon or a quarter past 7 in the morning The declination is to be found on the Back-side of the quadrant by laying the thread over the day of the moneth To rectifie the Bead. LAy the thread upon the graduated Index and set the Bead to the observed or given Altitude and when the Altitude is nothing or when the Sun is in the Horizon set the Bead to the Cypher on the graduated Index which afterwards being carried without stretching to the parallel of Declination the threed in the Limbe shewes the Amplitude or Azimuth and the Bead amongst the houres shewes the true time of the day Example Upon the 24th of April the Suns declination will be found to 16 deg North. Now to find his Amplitude and the time of his rising laying the threed over the graduated Index set the Bead to the beginning of the graduations of the Index and bring it without stretching to the parallel of declination above being 16 d and the threed in the limbe will lye over 26 deg 18′ for the Suns Amplitude or Coast of rising to the Northward of the East and the Bead amongst the houres sheweth 24 minutes past 4 for the time of Sun rising Which doubled gives the length of the night 8 houres 49 min. In like manner the time of setting doubled gives the length of the day The same to find the houre and Azimuth let the given Altitude be 45 degrees HAving rectified the Bead to the said Altitude on the Index and brought it to the intersect the parallel of declination the thread lyes over 50 degrees 48′ For the Suns Azimuth from the South And the Bead among the houres shewes the time of the day to be 41 minutes past 9 in the morning or 19 minutes past two in the afternoon Another Example wherein the operation will be upon the Reverted taile Let the altitude be 3 deg 30′ And the declination 16 deg North as before TO know when to rectify the Bead to the upper or neather Altitude will be no matter of difficulty for if the Bead being set to the neather Altitude will not meet with the parallel of declination then set it to the upper Altitude and it will meet with Winter parallel of like declination which in this case supplyes the turn So in this Example the Bead being set to the upper Altitude of 3 deg 30′ and carried to the Winter parallel of declination The thread in the Limbe will fall upon 68 deg 28′ for the Suns Azimuth from the North and the Bead among the houres shewes the time of the day to be either 5 in the morning or 7 at night Another Example Admit the Sun have 20 degr of North Declination as about the 9th of May and his observed altitude were 56 deg 20′ having rectified the Bead thereto and brought it to intersect the parallel of 20 deg among the houres it shewes the time of the day to be 11 in the morning or 1 in the afternoon and the Azimuth of the Sun to be 26 deg from the South The Vses of the Projection TO find the Suns Altitude on all houres or Azimuths will be but the converse of what is already said therefore one Example shall serve When the Sun hath 45 deg of Azimuth from the South And his Declination 13 deg Northwards Lay the threed over 45 deg in the Limbe and where the threed intersects the Parallel of Declination thereto remove the Bead which carried to the Index without stretching shewes 43 deg 50′ for the Altitude sought Likewise to the same Declination if it were required to find the Suns Altitude for the houres of 2 or 10. Lay the threed over the intersection of the houre proposed with the parallel of Declination and thereto set the bead which carried to the Index shewes the Altitude sought namely 44 deg 31′ The same Altitude also belongs to that Azimuth the threed in the former Position lay over in the Limbe This Projection is of worst performance early in the morning or late in the evening about which time Mr. Daries Curve is of best performance whereto we now addresse our selves Of the curved line and Scales thereto fitted This as we have said before was the ingenious invention of M. Michael Dary derived from the proportionalty of two like equiangled plain Triangles accommodated to the latitude of London for the ready working of these two Proportions 1. For the Houre As the Cosine of the Latitude is to the secant of the Declination So is the difference between the sine of the Suns proposed and Meridian Altitude To the versed sine of the houre from noone and the converse and so is the sine of the Suns Meridian Altitude to the versed sine of the semidiurnal Arke 2. For the Azimuth The Curve is fitted to find it from the South and not from the North and the Proportion wrought upon it will be As the cosine of the Latitude is to the Secant of the Altitude So is the difference of the versed sines of the Suns or Stars distance from the elevated Pole and of the summe of the Complements both of the Latitude and Altitude to the versed sine of the Azimuth from the noon Meridian Which will not hold backward to find the Altitude on all Azimuths because the altitude is a term involved both in the second and third termes of the former proportion If the third terme of the former Proportion had not been a difference of Sines or Versed sines the Curved line would have been a straight-line and the third term always counted from one point which though in the use it may seem to be so here yet in effect the third term for the houre is always counted from the Meridian altitude Here observe that the threed lying over 12 or the end of the Versed Scale and over the Suns meridian altitude in the line of altitudes it will also upon the curve shew the Suns declination which by construction is so framed that if the distance from that point to the meridian altitude be made the cosine of Latitude the distance of the said point from the end of the versed Scale numbred with 12 shall be the secant of the declination to the same Radius being both in one straight-line by the former constitution of the threed and instead
reference to the Suns abode above the Horizon the time of setting converted into degrees is also called the Semi-diurnal Ark the time of Sun rising so converted is called the Semi-nocturnal Ark doubled gives the whole length of the Night so upon the 27th day of April the Sun having 17d of Declination the length of the day is 15 hours and the length of the night 9 hours To find the Suns Altitude on all Hours or at any time proposed In Summer time if the hour proposed be before 6 in the morning or after it in the evening lay the Thread to the hour in the Limb the Bead being first rectified to the Winter Ecliptick and amongst the Paralels of Altitude above the upper Horizon it shews the Altitude sought Example So the Sun having 16d of declination Northwards as about the 24th of April laying the Thread over the Declination I set the Bead to the Winter Ecliptick and if it were required to find what Altitude the Sun shall have at 36 minutes past 6 in the afternoon lay the Thread over the same in the Limb and the Bead among the Parralels of Altitude will fall upon 7d At all other times the Operation is alike the Bead being rectified to that Ecliptick that is proper to the season of the year Lay the Thread over the proposed hour in the Limb and the Bead amongst the Parralels of Altitude sheweth the Altitude sought Example So if it were required the same day to find what Altitude the Sun should have at 19 m past 2 in the afternoon Lay the Thread in the Limb over the time given and the Bead among the Parralels of Altitude will fall upon 45d for the Altitude sought To finde the Suns Altitude on all Azimuths IN the Summer half year if the Azimuth propounded be more Northward then the Azimuth of the Sun shall have at the hour of 6 The Bead must be rectified to the Winter Ecliptick and brought to the Azimuth proposed above the upper Horizon and there among the Parralels of Altitude it sheweth the Altitude sought So about the 24th of April when the Suns Declination is 16d his Azimuth at 6 of the clock will be found to be 76d 54 m from the South Then if it were required to find the Suns Altitude upon an Azimuth more remote as upon 107d from the South laying the Thread over the Declination I set the Bead to the Winter Ecliptick and afterwards carrying it to the Azimuth proposed among the Parralels of Altitude above the upper Horizon it falleth upon 7d for the Suns Altitude sought In all other Cases bring the Bead rectified to the Ecliptick proper to the season of the year to the Azimuth proposed and among the Parralels of Altitude it sheweth the Altitude sought So far the same day I set the Bead to the Summer Ecliptick and if it were required to know what Altitude the Sun shall have when his Azimuth is 50d 48′ from the Meridian carry the Bead to the said Azimuth and among the Parralels of Altitude it will fall upon 45d for the Altitude sought The Hour of the night Proposed to find the Suns Depression under the Horizon IMagine the Sun to have as much Declination on the other side the Equinoctial as he hath on the side proposed and this Case will be co-incident with the former of finding the Suns Altitude for any time proposed the reason whereof is because the Sun is always so much below the Horizon at any hour of the night as his opposite Point in the Ecliptick is above the Horizon at the like hour of the Day Such Propositions as depend upon the knowledge of the Suns Altititude are to find the Hour of the Day and the Azimuth or true Coast of the Sun THe Suns Altitude is taken by holding the Quadrant steady and letting the Sun Beams to pass through both the Sights at once and the Thread hanging at liberty shews it in the equal Limb if this be thought unsteady the Quadrant may rest upon some Concave Dish or Pot into which the Plummet may have room to play but for greate Quadrants there are commonly Pedistalls made The Altitude supposed to find the Hour of the Day and the Azimuth of the Sun in Winter REctifie the Bead to the Winter Ecliptick and carry it along amongst the Parralels of Altitude till it cut or intersect that Parralel of Altitude on which the Sun was observed and the Thread in the Limb sheweth the hour of the Day and the Bead amongst the Azimuths sheweth the Azimuth of the Sun Example So about the 18 of October when the Suns Declination is 13d 20′ South if his observed Altitude were 18d the true time of the day would be found to be either 36 minutes after 9 or 24 minutes past 2 and his Azimuth would be 37 degrees from the South To finde the Hour of the Day and the Azimuth of the Sun at any time in the Summer half year IT was before intimated That if the question were put when the Sun hath less Altitude then he hath at the hour of 6 of the clock that then the Operation must be performed among those Parralels above the upper Horizon in the reverted Tail the Bead being rectified to the Winter Ecliptick and that it might be known what Altitude the Sun shall have at 6 of the clock by bringing the Bead rectified to the Summer Ecliptick to the left edge of the Quadrant So admitting the Sun to have 16d of North Declination which will be about the 24 April I might finde his Altitude at 6 of the Clock by bringing the Bead rectified to the Summer Ecliptick to the left edge of the Quadrant to be 12d 28 m whence I conclude if his Altitude be less the Bead must be rectified to the Winter Ecliptick and be brought to those Parralels above the upper Horizon and it may be noted that the Suns Altitude at 6 is always less then his declination Example Admit the 24th of April aforesaid the Suns observed Altitude were 7d laying the Thread over the Suns Declination or the day of the month I rectifie the Bead to the Winter Ecliptick and bring it to the said Parralel of Altitude above the upper Horizon and the Thread intersects the Limb at 9d 3 m shewing the hour of the day to be 24 minutes past 5 in the morning or 36′ past 6 in the evening and the Bead amongst the Azimuths shews the Azimuth or Coast of the Sun to be 107d from the South Another Example But admitting the Sun to have more Altitude then he hath at the hour of 6 the Operation notwithstanding differs not from the former but only in rectifying the Bead which must be set to the Summer Ecliptick and then carried to the Parralel of the Suns observed Altitude and the Thread will intersect the Limb at the true time of the day and the Bead amongst the Azimuths sheweth the true Coast of the Sun So upon the 24th of April aforesaid the Suns observed
difference of sines between the Scale and the Thread and then it will hold As the first Tearm To that fourth So Radius To the Sine of the Azimuth and may be either a Lateral or Parralel entrance according as it falls out and as the Radius is put either in the second or third place in all these Directions the introducement of the Radius is supposed according to to the general Advertisement The finding of the Amplitude this way presupposeth the Vertical Altitude known and then the Proportion derived from the Analemma not from the 16 Cases is As the Radius To the Tangent of the Latitude So the Sine of the Vertical Altitude To the Sine of the Amplitude So also to find the time of Sun rising As the Cosine of the Declination To Secant of the Latitude So the sine of the Suns Altitude at 6 To the Sine of the hour of rising from six To find the Suns Azimuth at six of the clock otherways then by the 16 Cases As the Cosine of the Suns Altitude at 6 To Tangent of the Latitude So is the difference of the sines of the Suns Altitude at 6 and of his Vertical Altitude To the sine of the Azimuth from the Vertical To find the time when the Sun shall be due East or West As Cosine of the Declination To Secant of the Latitude So the difference of the Sines of the Suns Altitude at 6 and of his Vertical Altitude To the Sine of the hour from 6 When the Sun shall be due East or West These Proportions derived from the Analemma are general both for the Sun and Stars in all Latitudes but when the Declination either of Sun or Stars exceed the Latitude of the place this Proportion for finding the Azimuth cannot be at some times conveniently performed on a Quadrant but must be supplyed from another Proportion whereof more hereafter Of the Hour and Azimuth Scales on the Edges of the Quadrant These Scales are fitted for the more ready finding the Hour and Azimuth in one Latitude being only to facilitate the former general Way The Labour saved hereby is twofold first the Suns declination is graved against the Suns Altitude of 6 in the Hour scale and the said Declinations continued at the other end of the said hour Scale to give the quantity of the Suns Depression in Winter equal to his Altitude in Summer and secondly they are of a fitted length as was shewed in the Description of the Quadrant and thereby half the trouble by introducing the Radius shunned The Vse of the Azimuth Scale The Altitude and Declination of the Sun given to find his Azimuth Take the distance between the Suns Altitude in the Scale and his Declination in Summer time in that Scale that stands adjoyning to the side in Winter in that Scale that is continued the other way beyond the beginning and laying the Thread to the Complement of the Suns Altitude in the lesser sines which is double numbred enter this extent between the Scale and the Thread parralelly and the foot of the Compasses sheweth in the Line of Sines the Azimuth accordingly Declination 23 d 31′ Altitude 47 d 27′ the Azimuth thereto would be 25 d from East or West in Summer and if the Altitude were 9 d 43′ in Winter the Azimuth thereto would be 30 d either way from the Meridian And so when the Sun hath no Altitude lay the Thread over 90 d in the lesser Sines and enter the extent from the beginning of the Azimuth Scale to the Declination and you will finde the Amplitude which to this Declination will be 39 d 50′ The Vses of the Hour Scale To find the Hour of the Day TAke the distance between the Suns Altitude in the hour Scale and his Declination proper to the season of the year then laying the Thread to the Complement of the Suns Declination in the lesser sines enter the former extent between the Scale and the Thread and the foot of the Compasses sheweth the sine of the hour Example If the Declination were 13 d North and the Altitude 37d 13′ take the distance between it in the Scale and 13 d in the prickt Line then laying the Thread to 77 d in the lesser sine enter that extent between the Scale and the Thread and the resting foot will shew 45 d for the hour from 6 that is either 9 in the forenoon or 3 in the afternoon The Converse of the former Proposition will be to find the Suns Altitude on all Hours The Thread lying over the Complement of the Suns Declination in the lesser sines from the sine of the hour take the nearest distance to it then set down one foot of that extent in the hour Scale at the Declination and the other will reach to the Altitude Example At London for these Scales are fitted thereto I would find the Suns Altitude at the hours of 5 and at 7 in the morning in Summer when the Sun hath 23 d 31′ of Declination Here laying the Thread to 23 d 31′ the Suns declination from the end of the lesser Sines being double numbred from the sine of 15 d taking the nearest distance to it set down one foot of this extent at 23 d 31′ the declination it reaches downwards to 9 d 30′ and upwards to 27 d 23′ the Suns Altitude at 5 and 7 a clock in the morning in Summer Another Example Let it be required to find the Suns Altitudes at the hours of 10 or 2 when his declination is 23d 31′ both North and South The Thread lying as before over the lesser sines take the nearest distance to it from 60d in the sines the said extent set down at 23d 31′ in the prickt Line reaches to 53 d 44′ for the Summer Altitude and being set down at 23d 31′ on the other or lower continued Line reaches to 10d 28′ for the Winter Altitude The Hour may be also sound in the Versed sines by help of this fitted hour Scale Take the distance between the Suns Altitude admit 36d 42′ and his Meridian Altitude to that Declination 61 d 59′ and enter one foot of this extent at the sine of 66 d 29′ and laying the Thread to the other foot according to nearest distance and it will lye over the hours of 8 in the morning or 4 in the afternoon in the Versed sines in the Limb and thereby also may the time of Suns rising be found by taking the distance from 0 to the Meridian Altitude and entring it at the Cosine of the Declination as before and the Converse will find the Suns Altitudes on all hours by taking the distance from the Co-sine of the Declination to the Thread laid over the Versed sine of the hour from Noon and the said Extent will reach from the Mridian Altitude in the fitted Scale to the Altitude sought To find the time of Sun rising or setting Lay the Thread over the Complement of the declination as before in the lesser sines
repaire to the Moneth you are in and those figures that stand against it shewes you what dayes of the said moneth the Weeke day shall be the same as it was the first day of March. Example For the yeare 1660 having found that the first day of March hapned upon a Thursday looke into the column against June and February you will find that the 7th 14th 21th and 28th dayes of those Moneths were Thursdayes whence it might be concluded if need were that the quarter day or 24th day of June that yeare hapneth on the Lords day Of the Epact THe Epact is a number carried on in account from yeare to yeare towards a new change and is 11 dayes and some odde time besides caused by reason of the Moons motion which changeth 12 times in a yeare Solar and runnes also this 11 dayes more towards a new change the use of it serves to find the Moones age and thereby the time of high Water To know the Moons age ADde to the day of the Moneth the Epact and so many days more as are moneths from March to the moneth you are in including both moneths the summe if lesse then 30 is the Moones age if more subtract 30 and the residue in the Moons age prope verum Example The Epact for the year 1658 is 6 and let it be required to know the Moons age the 28 of July being the fift moneth from March both inclusive 6 28 5 The summe of these three numbers is 39 Whence rejecting 30 the remainder is 9 for the Moons age sought The former Rule serves when the Moneth hath 31 dayes but if the Moneth hath but 30 Dayes or lesse take away but 29 and the residue is her ages To find the time of the Moones comming to South MUltiply the Moones age by 4 and divide by 5 the quotient shewes it every Unit that remaines is in value twelve minutes of time and because when the Moon is at the full or 15 dayes old shee comes to South at the houre of 12 at midnight for ease in multiplication and Division when her age exceedes 15 dayes reject 15 from it Example So when the Moon is 8 dayes old she comes to South at 24 minutes past six of the clock which being knowne her rising or setting may be rudely guessed at to be six houres more or lesse before her being South and her setting as much after but in regard of the varying of her declination no general certaine rule for the memory can be given Here it may be noted that the first 15 dayes of the Moones age she commeth to the Meridian after the Sun being to the Eastward of him and the later 15 dayes she comes to the Meridian before the Sun being to the Westward of him To find the time of high Water TO the time of the Moones comming to South adde the time of high water on the change day proper to the place to which the question is suited the summe shewes the time of high waters For Example There is added in a Table of the time of high Water at London which any one may cast up by memory according to these Rules it is to be noted that Spring Tides high winds and the Moon in her quarters causes some variation from the time here expressed Moones age Moon South Tide London Dayes Ho. mi. Ho. Mi 0 15 12 3 00 1 16 12 48 3 48 2 17 1 36 4 36 3 18 2 24 5 24 4 19 3 12 6 12 5 20 4  7 00 6 21 4 48 7 48 7 22 5 36 8 36 8 23 6 24 9 24 9 24 7 12 10 12 10 25 8 00 11 00 11 26 8 48 11 48 12 27 9 36 12 36 13 28 10 24 1 24 14 29 11 12 2 12 This Rule may in some measure satisfie and serve for vulgar use for such as have occasion to go by water and but that there was spare roome to grave on the Epacts nothing at all should have been said thereof A Table shewing the houres and Minutes to be added to the time of the Moons comming to South for the places following being the time of high Water on the change day  H. m. Quinborough Southampton Portsmouth Isle of Wight Beachie the Spits Kentish Knocke half tide at Dunkirke 00  00 Rochester Maulden Aberdeen Redban West end of the Nowre Black taile 00  45 Gravesend Downes Rumney Silly half tide Blackness Ramkins Semhead 1  30 Dundee St. Andrewes Lixborne St. Lucas Bel Isle Holy Isle 2  15 London Tinmouth Hartlepoole Whitby Amsterdam Gascoigne Brittaine Galizia 3  00 Barwick Flamborough head Bridlington bay Ostend Flushing Bourdeaux Fountnesse 3  45 Scarborough quarter tide Lawrenas Mountsbay Severne King sale Corke-haven Baltamoor Dungarvan Calice Creeke Bloy seven Isles 4  30 Falmouth Foy Humber Moonles New-castle Dartmouth Torbay Caldy Garnesey St. Mallowes Abrowrath Lizard 5  15 Plymouth Weymouth Hull Lin Lundy Antwerpe Holmes of Bristol St. Davids head Concalo Saint Malo 6  00 Bristol foulnes at the Start 6  45 Milford Bridg-water Exwater Lands end Waterford Cape cleer Abermorick Texel 7  20 Portland Peterperpont Harflew Hague St. Magnus Sound Dublin Lambay Mackuels Castle 8  15 Poole S. Helen Man Isle Catnes Orkney Faire Isles Dunbar Kildien Basse Islands the Casquers Deepe at halfe tide 9  Needles Oxford Laysto South and North Fore-lands 9  45 Yarmouth Dover Harwich in the frith Bullen Saint John de luce Calice road 10  30 Rye Winchelsea Gorend Rivers mouth of Thames Faire Isle Rhodes 11  15 To find the Epact for ever IN Order hereto first find out the Prime Number divide the yeare of the Lord by 19 the residue after the Division is finished being augmented by an Unit is the Prime sought and if nothing remaine the Prime is an Unit. To find the Epact MUltiply the Prime by 11 the product is the Epact sought if lesse then 30 but if it be more the residue of the Product divided by 30 is the Epact sought there note that the Prime changeth the first of January and the Epact the first of March Otherwise Having once obtained the Epact adde 11 so it the Summe if lesse then 30 is the Epact for the next yeare if more reject 30 and the residue is the Epact sought Caution When the Epact is found to be 29 for any yeare the next yeare following it will be 11 and not 10 as the Rule would suggest A Table of the Epacts belonging to the respective Primes Pr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Ep. 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 The Prime number called the Golden Number is the number of 19 years in which space the Moone makes all variety of her changes as if she change on a certain day of the month on a certain yeare she shall not change the same day of the moneth again till 19 yeares after and then it doth not happen upon the same houre of the day yet the difference doth not cause
concentrick to the Limb to keep the other at a certainty in stretching and the other to be rectified for use To rectifie the Bead. LAy the Thread over the day of the Month in its proper Circle and if the season wherein the Quadrant is to be used be in the Winter half year set the Bead by removing it to the Winter Ecliptick but in Summer let it be set to the lower or Summer Ecliptick and then it is fitted for use One Caution in rectifying the Bead is to be given and that is in Summer time if it be required to find the hour and Azimuth of the Sun by the Projection before the hour of 6 in the morning or after it in the evening or which is all one when the Sun hath less Altitude then he hath at 6 of the clock then must the Bead be rectified to the Winter Ecliptick and the Parralels above the Horizon in the Reverted Tail are those which will come in vse To find what Altitude the Sun shall have at 6 of the clock in the Summer half year This will be easily performed by bringing the Bead that is rectified to the Summer Ecliptick to the left edge of the Quadrant and-there among the Paralels of Altitude it shews what Altitude the Sun shall have at 6 of the clock It also among the Azimuths shews what Azimuth the Sun shall have at the hour of 6. Example So when the Sun hath 17 degrees of North Declination as about the 27 of April his Altitude at the hour of 6 will be found to be 13d 14 m and his Azimuth from the Meridian 79d 14 m whence I may conclude if his observed Altitude be less upon the same day and the Hour and Azimuth sought the Bead must be set to the Winter Ecliptick and the Operation performed in the reverted Tail Here it may be noted also that the exactest way of rectifying the Bead will be either from a Table of the Suns Declination laying the Thread over the same in the graduated Circle or from his true place laying it over the same in the proper Ecliptick or from his right Ascension counted in the Limb. Or Lastly from his Meridian Altitude on the right edge of the Quadrant for these do mutually give each other the Bead being rectified to the respective Ecliptick as before for Example To find the Suns Declination The Thread laid over the day of the Moneth intersects it upon that Circle whereon it is graduated which in the Summer half year is to be accounted on this side the Equinoctial North and in the Winter-half year South so laying the Thread over the 27th day of April it intersects the Circle of Declination at 17 degrees and so much was the Suns Declination To find the Suns true place The Thread lying as before shews it on the respective Ecliptick So the Thread lying over the 17 of April will cut the Summer Ecliptick in 17d 7 m of Taurus or in 12d 53 m of Leo which agrees to the 26 day of July or thereabouts the Thread intersecting both these days at once and the opposite points of the Ecliptick hereto are 17d◠m in Scorpio about the 20 of October and 12d 53 m of Aquarius about the 22d of January all shewed at once by the Threads position To find the Suns right Ascension Lay the Thread over the day of the Month as before and it intersects it in the equal Limb whence taking it in degrees and minutes of the Equator whilst the Sun is departing from the Equator towards the Tropicks it must be counted as the graduations of the Limb from the left edge towards the right but when the Sun is returning from the right edge towards the left the right Ascension thus found must be estimated according to the season of the year From June 11 to Sept. 13 It must have 90 degrees added to it Sep. 13 to Dec. 11 It must have 180 degrees added to it Dec. 11 to Mar. 10 It must have 270 degrees added to it But in finding the Hour of the night by the Quadrant we need no more then 12 hours of Ascension for either Sun or Star and the Limb is accordingly numbred from the left edge towards the right from 1 to 6 in a smaller figure and thence back again to 12 and the other figures are the Complements of these to 12. so that when the Sun is departing from the Equator towards the Tropicks his right Ascension is always less then 6 hours and the Complement of it more but when he is returning from the Tropicks towards the Equator it is always more then 6 hours and the Complement of it less the odd minutes are to be taken from the Limb where each degree being divided into 4 parts each part signifies a Minute of time and to know whether the Sun doth depart from or return towards the Equator is very visible by the progress and regress of the days of the month as they are denominated on the Quadrant Example So the Thread laid over 17d of Declination which will be about the 27 April The Suns right Ascension will be 44d 37 m In time 2 h 58′ 26 July The Suns right Ascension will be 135 23 In time 9 2 20 October The Suns right Ascension will be 224 37 In time 2 58 22 January The Suns right Ascension will be 315 23 In time 9 2 But here the latter 12 hours are omitted Such Propositions as require the use of the Bead are To find the Suns Amplitude or Coast of rising and setting from the true East or West Bring the Bead being rectified to either of the Eclipticks it matters not which to either of the Horizons and the Thread will intersect the Amplitude sought upon both alike Example The Suns Declination being 17 North or South the Suns Amplitude will be found to be 28◠2m. The Amplitude before found for the Summer half year is to be accounted from East or West Northwards and in the Winter half year from thence Southwards To find the time of the Suns rising or setting The Thread lying in the same Position as in the former Proposition intersects the Ascensional difference in the Limb which may there be counted either in degrees or Time Example So the Bead lying upon the Horizon being rectified to 17◠of Declination the Thread intersects the Limb at 22d 38 m which is 1 h 30 m of time and so it shews the time of Suns rising in Summer or setting in Winter to be at half an hour past 4 and his rising in Winter and setting in Summer to be at half an hour past 7. To find the length of the Day or Night The time of the Suns rising and setting are one of them the Complement of the other to 12 hours so that one of them being known the other will be found by Substraction the time of Suns setting is equal to half the length of the day and this doubled gives the whole length of the day in
is the Square of the Radius To the Square of the Tangent of half the angle sought and by changing the 2d Tearm into the place of the first As the Rectangle of the sines of the differences of the Leggs from the half sum of the 3 sides Is to the Rectangle of the sines of the half sum of the three sides and of the difference of the Base therefrom So is the Square of the Radius To the Square of the Cotangent of half the angle sought These Proportions are published in order to their Application to the Serpentine Line which will be accomodated for the sudden operating of any of them the Axioms to be remembred are not many the Reader will meet with their Demonstration and Application in Mr Newtons Trigonometry now in the Press and said to be near finished The four Proportions in plain Triangles when three sides are given to find an angle without the Cadence of Perpendiculars are demonstrated in the 27 Section of the late Miscellanies of Francis van Schooten The Construction of diverse Instruments will require a Table of the Suns Altitudes to the Hour and Azimuth assigned And for the Acurate bounding in of the Lines it may be a Table of Hours and Azimuths to any Altitude assigned for the easie Calculating whereof I am desired for the ease and benefit of the Trade to render this part of Calculation as facil as I can and therefore shall handle it the more largely To Calculate a Table of Hours to all Altitudes in all Latitudes The 1. Proportion shall be to find the Suns Altitude in Summer or Depression in Winter at the hour of 6. As the Radius To the sine of the Latitude So the sine of the Declination To the sine of the Altitude or Depression sought This remains fixed for all that day the Suns Declination supposed not to vary and then it holds As the Cosine of the Declination To the Secant of the Latitude So in Summer is the difference in Winter the sum of the sines of the Suns Altitude proposed and of his Altitude or Depression at 6 To the sine of the hour from 6 towards noon in Winter and in Summer also when the given Altitude is greater then the Altitude of 6 but when it is less towards midnight This Proportion also holds for Calculating the Horary distance of any Star from the Meridian In like manner to Calculate the Azimuth As the sine of the Latitude To sine of the Declination So is the Radius To the side of the Suns Altitude or Depression in the prime Vertical that is being East or West This remains fixed for one day Then As the Cosine of the Altitude To the Tangent of the Latitude So in Summer is the difference and in Winter the sum of the sines of the Suns Altitude proposed and of his Vertical Altitude or Depression To the sine of the Azimuth towards noon Meridian in Winter and in Summer also when the given Altitude is greater then the Vertical Altitude or Depression but when it is less towards Midnight Meridian This Proportion is general either for Sun or Stars when the Declination is less then the Latitude of the place But when it is more say as before As the sine of the Latitude To the sine of the Declination So is the Radius To a fourth we may call it a Secant Again As Cosine Altitude To the Tangent of the Latitude So in declinations towards the Depressed Pole is the sum but towards the Elevated Pole the difference of this Secant and of the sine of the Sun or Stars Altitude To the sine of the Azimuth from the Vertical towards the noon Meridian Before Application be made the latter part of these Proportions being of my own peculiar Invention and of very great use both for Calculation and Instrumentally it will be necessary to demonstrate the same For the Hour from the Analemma Having in the Scheme annexed drawn the Equator and Horizon the two prickt Lines passing through the Center as also the Prime Vertical and Axis the two streight Lines passing through the same Let I X and L M represent two Parralells of Declination on each side the Equator and O X a Parralel of the Suns Altitude in Summer and P Q of his Depression in Winter at the hour of 6 because these Parralells pass through the Intersection of the Parralells of Declination with the Axis Let R S be a Parralel of Altitude after 6 and T V a Parralel of Altitude before it from the Intersections of these Parralells of Altitude with the Parralels of Declination let fall Perpendiculars on the Parralells of the Suns Altitude or Depression at 6 and then we shall have divers right Lined right angled Triangles Constituted in which we shall make use of the Proportion of the sines of angles to their opposite sides an Axiom of common demonstration In the Triangle A F E As the sine of the angle at F the Radius To its Opposite side A E the sine of the Declination So the sine of the Latitude the angle at E to A F the sine of the Suns Altitude at 6. Again in the two Opposite Triangles A B C the smaller before the greater after 6. As the Cosine of the Latitude the sine of the angle at A To its Opposite side B C the difference of the sines of the Suns Altitude at 6 and of his proposed Altitude So is the Radius sine of the angle at B To C A the sine of the hour from 6 in the Parralel of Declination in the lower Triangle before in the upper after 6. So in the Winter or lower Triangle A B D C. As Cosine of the Latitude sine of the angle at A To B C the sum of the sines of the Suns Depression at 6 B D and of his given Altitude D C So is the Radius the sine of the angle at B To A C the sine of the hour from 6 towards noon in the Parralel of Declination The sine of the hour thus found in a Parralel is to be reduced by another Analogy to the common Radius and that will be As the Radius of the Parralel I A the Cosine of the Declination Is to the common Radius E AE So is any other sine in that Parralel To the sine of the said Arch to the common Radius Now it rests to be proved that both these Analogies may be reduced into one and that will be done by bringing the Rectangle of the two middle Tearms of the first Proportion with the first Tearm under them as an improper Fraction to be placed as a single Tearm in the second Proportion being in value the answer found in the Parralel and then we have the Rule of three to Operate as it were in whole Numbers and mixt The Proportion will run As the Cosine of the Declination To Radius So the said Improper Fraction To the Answer and so proceeding according to the
from six in the sines towards noone if the Altitude fell beyond the Star point otherwise towards midnight Example For the Goat Star let its Altitude be 40 deg and past the Meridian the houre of that Star will be 44′ from six for the Compasses fall upon the sine of 11 deg 4′ the houre is towards noon Meridian because the Altitude is greater then 34 deg the point where the Star is graved the thread lying over the Star intersects the Limbe at 25 deg 47′ if the distance between the Star and its Altitude be entred at the sine of that Arke and the thread laid to the other foot the houre will be found in the equal Limbe the same as before For Stars of Southwardly Declination BEcause the Star point cannot fall the other way beyond the Center of the quadrant therefore the distance between the Star point and the Center must be increasing by adding the sine of the Stars Altitude thereto which will fall more outwards towards the Limbe and then that whole extent is to be entred as before Example The Virgins Spike hath 9 deg 19′ of South Declination the Depression of that Star at six will be found by help of the particular sine to be 7 deg 17′ and at that Arke in the sines the Star is graved if the Altitude of that Star were 20 deg the sine thereof added to the Star will be equal to the sine of 29 deg 6′ this whole extent entred at the sine of 37 deg 52′ the Arke of the Limbe against which the Star is graved and the thread laid to the other foot the houre of that Star if the Altitude increase will be 19′ past 9. To find the true time of the right THis must be done by turning the Stars houre into the Suns houre or common time either by the Pen as hath been shewed before which may be also conveniently performed by the back of this quadrant for the thread lying over the day of the moneth sheweth the Complement of the Suns Ascension in the Limbe Or with Compasses on the said quadrant of Ascensions THe thread lying over the day of the moneth take the distance between it and the Star on the said quadrant the said extent being applyed the same way as it was taken the Suns foot to the Stars houre shall reach from the Stars houre to the true houre of the night and if one of the feet of the Compasses fall off the quadrant a double remedy is els-where prescribed Example If on the 12th of January the houre of the Goat Star was 16′ past 5 from the Meridian the true time sought would be 49′ past 1 in the morning Example If upon the third of January the houre of the Virgins Spike were observed to be 19′ past 9 the true time sought would be 45′ past 2 in the morning To find the time of a Stars rising and setting THe Ascentional difference is graved against the Star the Virgins Spike hath 48′ of Ascentional difference that is to say that Stars houre of rising is at 48′ past 6 and setting at 12′ past 5 And the true time of that Stars rising upon the third of January will be at 22′ past 10 at night and of its setting at 47′ past 8 in the morning found by the former directions Of the rest of the lines on the back of this quadrant THey are either such as relate to the motion of the Sun or Stars or to Dialling or such as are derived from Mr Gunters Sector The Tangent of 51 deg 32′ put through the whole Limbe is peculiarly fitted to the Latitude of London and will serve to find the time when the Sun will be East or West as also for any of the Stars that have lesse Declination then the place hath Latitude Lay the thread to the Declination counted in the said Tangent and in the Limbe it shewes the houre from 6 if reckoned from the right edge Example When the Sun hath 15 deg of North Declination the time of his being East or West will be 12 deg 17′ in time about 49′ before or after six ferè The Suns place is given in the Ecliptick line by laying the thread over the day of the moneth in the quadrant of Ascensions of which see page 16 17 of the small quadrant Of the lines relating to Dialling SUch are the Line of Latitudes and Scale of houres of which before and the line Sol in the Limbe of which I shall say nothing at present it is onely placed there in readinesse to take off any Arke from it according to the accustomed manner of taking off lines from the Limbe to any assigned Radius The requisite Arkes of an upright Decliner will be given by the particular lines on the Quadrant for the Latitude without the trouble of Proportionall worke 1 The substiles distance from the Meridian ACcount the Plaines declination as a sine in the fitted hour Scale on the right edge of the fore-side and just against it in the annexed Tangent stands the substiles distance from the meridian If an upright Plaine decline 30 deg the substiles distance will be 21 deg 41 minutes 2 The Stiles height Count the Complement of the Plaines Declination in the said fitted houre scale as a sine and apply it with Compasses to the line of sines issuing from the Center for the former Plaine the stiles height will be found 32 deg 37′ 3 The Inclination of Meridians Account the stilts height in the annexed tangent of the fitted hour Scale and just against it in the sine stands the Complement of the Inclination of meridians which for the former plaine will be found to be 36 deg 25′ 4 The Angle of 12 and 6. Account the Plaines Declination in the Limbe on the Backside from the right edge and lay the thread over it and in the particular Tangent it shewes the Angle between the Horizon and six 32 deg 9′ in this Example the Complement whereof is the Angle of 12 and 6 namely 57 deg 51 min. Also the requisite Arkes of a direct East or West reclining or inclining Dial may be found after the same manner for this Latit 1 The substiles distance ACcount the Plaines Reinclination in the Limbe on the Backside from the left edge and in there lay the thread and in the particular Tangent it shewes the Arke sought So if an East or West plain recline or incline 60 deg the substiles distance will be found to be 32 deg 12′ 2 The stiles height Account the Reinclination in the particular Sine on the foreside and in the Limbe it shewes the stiles height which for the former Example will be found to be 42 deg 41′ 3 The inclination of Meridians The Proportion is As the Sine of the Latitude to Radius So is the sine of the substiles distance To the sine of the inclination of Meridians when the substiles distance is lesse then the Latitude of the place it may be found in the
59 17 04 31 15 03 17 21 Dayes November ☉ R. A. ☉ Decl. H. M. D. M. 1 15 07 17 38 2 15 11 17 54 3 15 15 18 10 4 15 19 18 26 5 15 23 18 41 6 15 27 18 56 7 15 31 19 11 8 15 36 19 26 9 15 40 19 40 10 15 45 19 53 11 15 49 20 07 12 15 53 20 19 13 15 58 20 32 14 16 02 20 44 15 16 07 20 56 16 16 11 21 08 17 16 15 21 19 18 16 19 21 29 19 16 23 21 39 20 16 28 21 49 21 16 32 21 58 22 16 36 22 08 23 16 40 22 16 24 16 44 22 24 25 16 49 22 32 26 16 53 22 39 27 16 57 22 46 28 17 02 22 52 29 17 06 22 58 30 17 11 23 03 31     Dayes December ☉ R. A. ☉ Decl. H. M. D. M. 1 17 15 23 08 2 17 20 23 13 3 17 25 23 17 4 17 29 23 20 5 17 34 23 23 6 17 38 23 26 7 17 42 23 28 8 17 47 23 29 9 17 51 23 30 10 17 56 23 31 11 18 00 23 31½ 12 18 05 23 31 13 18 09 23 30 14 18 14 23 29 15 18 19 23 27 16 18 24 23 25 17 18 28 23 22 18 18 33 23 19 19 18 37 23 15 20 18 41 23 11 21 18 45 23 07 22 18 49 23 02 23 18 54 22 56 24 18 58 22 50 25 19 03 22 43 26 19 07 22 36 27 19 11 22 29 28 19 16 22 21 29 19 20 22 13 30 19 25 22 04 31 19 30 21 55 A Rectifying Table for the Suns Declination  Years Years Years  1657 1661 1665 1669 1673 1659 1663 1667 1671 1675 1660 1664 1668 1672 1676 Moneths min. min. min. January 3 s 2 a 5 a 4 s 3 a 7 a 5 s 4 a 9 a February 5 s 5 a 10 a 5 s 5 a 11 a 6 s 5 a 11 a March 6 s 5 a 13 s 5 a 5 s 12 a 5 a 5 s 12 a April 5 a 5 s 11 a 5 a 5 s 10 a 4 a 4 s 9 a May 4 a 4 s 8 a 3 a 3 s 6 a 2 a 2 s 4 a June 1 a 1 s 2 a 0 s 0 a 0 s 1 s 1 a 3 s July 2 s 2 a 5 s 3 s 3 a 7 s 4 s 4 a 9 s August 5 s 5 a 10 s 5 s 5 a 11 s 6 s 5 a 12 s Septēber 6 s 5 a 13 s 6 a 5 s 13 a 6 a 5 s 12 a October 6 a 5 s 12 a 5 a 5 s 11 a 4 a 5 s 9 a Novem. 3 a 4 s 7 a 2 a 3 s 5 a 1 a 2 s 3 a Decemb. 0 a 1 s 1 a 1 s 0 a 1 s 2 s 1 a 3 s The use of the Rectifying Table NOte that the minutes under the respective years is to be added or substracted to or from the Suns Declination in the former Table as is noted with the letter a or s and also note that the first figure in each moneth stands for the first 10 dayes of the moneth and the second for the second 10 days the third for the last 10 dayes except in March or September which in March will be the first 9 dayes only and in September the first 12 dayes Example I would know the Suns Declination the 15 day of May 1668. Now because this day of the moneth falls in the second 10 dayes I look in the Table under the year 1663 and right against May you shall finde that in the second place of the moneth stands 6 a which shews me that I must adde 6 minutes to the Suns Declination in the former Table 21 degrees 5 min. that stands against the 15 day of May and then I find that the Sun will have 21 deg 11 min. of North Declination and so for the rest which will never differ above two minutes from the truth but seldome so much and for the most part true Note that the former Table of the Suns Declination is fitted exactly for the year 1666. by the Rules Mr. Wright gives in his Correction of Errours and from his Tables and may indifferently serve for the years 1658 1662. 1670 1674 without any sensible errour and the Table of Right Ascensions will not vary a minute of time in many years FINIS Errours in the Horizontal Quadrant PAge 5 line 6 in an Italian letter should not have been distinct nor in another letter from the former line page 5. line 9. for quarter read half p. 5. l. 13. r. of a quadrant p. 11. l. 7. r. 63 d. 26′ p. 19. l. 7. r. the same day to p. 23. l. 17. r. and ends at 32′ past 9. p. 27. l. 7. for N R r. N Z. p. 28. l. 4. r. in the parallel p. 30. l. 9 l. 10. r. 23 d. 31′ p. 38. l. 4. r. Is to the sine p. 50. l. 5. r. whereof the Diameter AN APPENDIX Touching REFLECTIVE DIALLING By JOHN LYON Professor of this or any other part of the Mathematicks neer Sommerset House in the Strand LONDON Printed Anno Domini 1658. DIRECT DIALLING By a Hole or Nodus To draw a Dial under any window that the Sun shines upon by help of a thread fastened in any point of the direct Axis found in the Ceiling and a hole in any pane of glasse or a knob or Nodus upon any side of the window or window-post CONSTRUCTIO FIrst draw on pastboard or other material an Horizontal Dial for the Latitude proposed Then by help of the Suns Azimuth which may be found by help of a general Quadrant at any time or by knowing the true hour of the day with the help of the said Horizontal Dial and draw that true Meridian from the hole or Nodus proposed both above on the Cieling and below on the walls and floor of the Room so that if a right line were extended from the said hole or Nodus by any point in any of those lines it would be in the meridian Circle of the World To finde a point in the direct Axis of the world which will ever fall to be in the said Meridian in which point the end of a thread is to be fastened FIrst fix the end of a thread or small silk in the center of the Hole or Nodus and move the other end thereof up or down in the said meridian formerly drawn on the Cieling or wall untill by applying the side of a Quadrant to that thread it is found to be elevated equal to the Latitude of the place so is that thread directly scituated parallel to the Axis of the world and the point where the end of that thread toucheth the meridian either on the Cieling or wall is that point in the direct Axis sought for wherein fix one end of a thread which thread will be of present use in projecting of hour-points in any place proposed then To find the Hour-points