20 36 11 26 0 13 11 53 20 01 23 51 20 80 12 60 1 20 10 31 19 75 23 51 10 20 15 10 90 N26 11 88 20 21 23 52 20 61 12 26 0 81 10 68 19 98 23 52 11 19 93 10 53 0 66 12 21 20 41 23 53 20 41 11 93 0 41 11 03 20 20 23 53 12 19 70 10 16 1 05 12 55 20 61 23 51 20 21 11 60 0 03 11 38 20 41 23 51 13 â—9 46 09 80 1 45 12 88 20 80 23 50 20 00 11 25 0â—36 11 73 20 61 23 50 14 19 23 09 43 1 83 13 20 20 98 23 48 19 80 10 90 0 75 12 08 20 81 23 48 15 18 98 09 06 2 23 13 53 21 16 23 45 19 58 10 56 1 15 12 43 21 01 23 45 16 18 73 08 70 2 63 13 85 21 33 23 41 â—9 35 10 21 1 55 12 78 21 20 23 40 17 18 48 08 31 3 03 14 16 21 50 23 38 â—9 13 9 85 1 93 13 11 21 38 23 35 18 18 21 07 93 3 41 14 48 21 66 23 33 18 90 9 50 2 31 13 45 21 55 23 30 19 17 95 07 55 3 80 14 80 21 81 23 26 18 65 9 15 2 71 13 78 21 7â— 23 25 20 17 66 07 16 4 18 15 10 21 96 23 21 18 41 8 78 3 10 14 11 21 88 23 18 21 17 40 06 78 4 56 15 40 22 10 23 15 18 16 8 41 3 50 14 43 22 03 23 10 22 17 11 06 40 4 95 15 70 22 23 23 06 17 91 8 05 3 88 14 76 22 18 23 01 23 16 81 06 01 5 33 15 98 22 36 22 98 17 66 7 68 4 28 15 08 22 31 22 91 24 16 53 05 63 5 71 16 28 22 48 22 90 17 40 7 31 4 66 15 40 22 45 22 81 25 16 23 05 25 6 10 16 56 22 60 22 80 17 11 6 95 5 05 15 70 22 58 22 70 26 15 91 04 86 6 48 16 83 22 71 22 70 16 85 6 56 5 43 16 00 22 70 22 58 27 15 61 04 48 6 85 17 11 22 81 22 60 16 56 6 20 5 81 16 30 22 80 22 46 28 15 30 04 06 7 23 17 38 22 90 22 48 16 28 5 83 6 20 16 60 22 90 22 33 29 14 98 03 68 7 60 17 65 22 98 22 35 16 00 5 45 6 58 16 90 23 00 22 20 30 14 66  7 96 17 90 23 06 22 21 15 71 5 06 6 96 17 â—8 23 08 22 05 31 14 35  8 33  23 15  15 41 4 68  17 46  21 90 A Table of the Suns Declination for the years 1657 1661 1665 1669.  Ianu. Febr. Mar Apr. May. June July Aug. Sep. Octo Nov Dece Dayes south south sout north north north north north nort south south south 1 21 73 13 76 3 40 08 60 18 08 23 20 22 13 15 20 4 40 7 25 17 66 23 15 2 21 56 13 43 3 00 08 96 18 33 23 26 22 00 14 90 4 01 7 63 17 93 23 21 3 21 38 13 08 2 61 09 33 18 58 23 31 21 85 14 60 3 63 8 00 18 20 23 28 4 21 21 12 75 2 21 09 70 18 83 23 36 21 70 14 28 3 25 8 36 18 46 23 33 5 21 03 12 41 1 81 10 05 19 06 23 41 21 53 13 96 2 86 8 75 18 71 23 38 6 20 83 12 06 1 41 10 40 19 30 23 45 21 36 13 65 2 48 9 11 18 96 23 43 7 20 63 11 71 1 01 10 75 19 51 23 48 21 20 13 33 2 08 9 48 19 21 23 46 8 20 43 11 35 0 63 11 10 19 73 23 50 21 03 13 01 1 70 9 85 19 45 23 50 9 20 21 11 00 0 23 11 45 11 95 23 51 20 85 12 68 1 31 10 21 19 68 23 51 10 20 00 10 63 N16 11 78 20 16 23 52 20 66 12 35 0 91 10 58 19 91 23 52 11 19 76 10 26 0 55 12 11 20 36 23 53 20 46 12 01 0 53 10 93 20 13 23 53 12 19 53 09 90 0 95 12 46 20 56 23 51 20 26 11 68 0 13 11 30 20 35 23 51 13 19 30 09 53 1 35 12 80 20 75 23 50 20 06 11 35 0â—26 11 65 20 56 23 50 14 19 05 09 16 1 73 13 11 20 93 23 48 19 85 11 15 0 65 12 00 20 76 23 48 15 18 80 08 80 2 13 13 45 21 11 23 46 19 63 10 65 1 05 12 35 20 96 23 45 16 18 55 08 41 2 51 13 76 21 28 23 43 19 41 10 30 1 43 12 68 21 15 23 41 17 18 28 08 05 2 90 14 08 21 45 23 38 19 20 9 95 1 83 13 20 21 33 23 36 18 18 03 07 66 3 30 14 40 21 61 23 33 18 96 9 60 2 21 13 36 21 51 23 31 19 17 75 07 28 3 68 14 70 21 76 23 28 18 71 9 25 2 61 13 70 21 68 123 26 20 17 46 06 90 3 08 15 01 21 91 23 23 18 48 8 88 3 00 14 03 21 83 23 20 21 17 18 06 51 4 46 15 31 22 06 23 16 18 23 8 51 3 38 14 35 22 00 23 11 22 16 90 06 13 4 85 15 61 22 20 23 10 17 98 8 15 3 78 14 68 22 15 23 03 23 16 60 05 75 5 23 15 90 22 33 23 01 17 73 7 78 4 16 15 00 22 28 22 95 24 16 30 05 35 5 61 16 20 22 45 22 91 17 46 7 41 4 55 15 31 22 41 22 85 25 16 00 04 96 6 00 16 48 22 56 22 83 17 20 7 05 4 95 15 61 22 55 22 73 26 15 70 04 56 6 36 16 76 22 68 22 73 16 93 6 68 5 33 15 91 22 66 22 61 27 15 38 04 18 6 75 17 03 22 78 22 61 16 65 6 30 5 71 16 21 22 76 22 50 28 15 06 03 78 7 11 17 30 22 88 22 51 16 36 5 93 6 10 16 51 22 86 22 36 29 14 75  7 50 17 56 22 96 22 38 16 08 5 55 6 48 16 81 22 96 22 â—â— 30 14 43  7 86 17 83 23 05 22 26 15 80 5 16 6 86 17 10 23 06 22 08 31 14 10  8 23  23 13  15 50 4 78  17 38  21 93 A Table of the Suns right Ascension in hours and minutes  Janu Febr. Mar. Apr. May Iune Jul Aug. Sept. Octo. Nov. Dece Dayes H. M. H. M. H. M. H M H. M H. M H M H. M H. M H. M H. M H. M 1 19 53 21 68 23 45 1 33 3 21 5 30 7 36 9 40 11 30 13 10 15 10 17 23 2 19 61 21 75 23 51 1 38 3 28 5 36 7 43 9 46 11 35 13 16 15
Artificiall SINES and TANGENTS as also of the LOGARITHMS IN Astronomie Dialling and Navigation By JOHN NEWTON LONDON Printed Anno Domini 1654. A Mathematicall Institution The second Part. CHAP. I. Of the Tables of the Suns motion and of the equation of time for the difference of Meridians WHereas it is requisite that the Reader should be acquainted with the Sphere before he enter upon the practise of Spherical Trigonometri the which is fully explained in Blundeviles Exercises or Châ—lmades translation of Hues on the Globes to whom I refer those that are not yet acquainted therewith that which I here intend is to shew the use of Trigonometrie in the actuall resolution of so me known Triangles of the Sphere And because the Suns place or distance from the next Equinoctial point is usually one of the three terms given in Astronomical Questions I will first shew how to compute that by Tables calculated in Decimal numbers according to the Hypothesis of Bullialdus and for the Meridian of London whose Longitude reckoned from the Canarie or Fortunate Islands is 21 deg and the Latitude North 51 deg 57 parts min. or centesms of a degree Nor are these Tables so confined to this Meridian but that they may be reduced to any other If the place be East of London adde to the time given but if it be West make substraction according to the difference of Longitude allowing 15 deg for an houre and 6 minutes or centesms of an houre to one degree so will the sum or difference be the time aequated to the Meridian of London and for the more speedy effecting of the said Reduction I have added a Catalogue of many of the chiefest Towns and Cities in diverse Regions with their Latitudes and difference of Meridians from London in time together with the notes of Addition and Substraction the use whereof is thus Suppose the time of the Suns enterance into Taurus were at London Aprill the 10th 1654 at 11 of the clock and 16 centesms before noon and it be required to reduce the same to the Meridian of Vraniburge I therefore seeke Vraniburge in the Catalogue of Cities and Places against which I finde 83 with the letter A annexed therefore I conclude that the Sun did that day at Uraniburge enter into Taurus at 11 of the clock and 99 min. or centesms before noon and so of any other Problem 1. To calculate the Suns true place THe form of these our Tables of the Suns motion is this In the first page is had his motion in Julian years compleat the Epochaes or roots of motions being prefixed which sheweth the place of the Sun at that time where the Epocha adscribed hath its beginning the Tables in the following pages serve for Julian Years Moneths Dayes Houres and Parts as by their Titles it doth appear The Years Moneths and Dayes are taken compleat the Houres and Scruples current After these Tables followeth another which contains the Aequations of the Eccentrick to every degree of a Semicircle by which you may thus compute the Suns place First Write out the Epocha next going before the given time then severally set under those the motions belonging to the years moneths and dayes compleat and to the hours and scruples current every one under his like onely remember that in the Bissextile year after the end of February the dayes must be increased by an unit then adding them all together the summe shall be the Suns mean motion for the time given Example Let the given time be 1654 May 13 11 hours 25 scruples before noon at London and the Suns place to be sought The numbers are thus  Longit. ☉ Aphel ☉ The Epocha 1640 291.2536 96.2297 Years compl 13 359.8508 2052 Moneth co April 118.2775 53 Dayes compl 12 11.8278 6 Hours 23 9444  Scruples 25 102     SuÌ„ or mean motioÌ„ 782.1643 96.4308 2. Substract the Aphelium from the mean Longitude there rests the mean Anomalie if it exceed not 360 degrees but if it exceed 360 degr 360 being taken from their difference as oft as it can the rest is the mean Anomalie sought Example The ☉ mean Longitude 782.1643 The Aphelium substracted 96.4308 There rests 685.7335 From whence deduct 360. There rests the mean Anomalie 325.7335 3. With the mean Anomalie enter the Table of the Suns Eccentrick Equation with the degree descending on the left side if the number thereof be lesse then 180 and ascending on the right side if it exceed 180 and in a straight line you have the Equation answering thereunto using the part proportional if need require Lastly according to the title Add or Substract this Equation found to or from the mean longitude so have you the Suns true place Example The Suns mean longitude 782.1643 Or deducting two circles 720. The Suns mean longitude is 62.1643 The Suns mean Anomalie 325.7335 In this Table the Equation answering to 325 degrees is 1.1525 The Equation answering to 326 degrees is 1.1236 And their difference 289. Now then if one degree or 10000 Give 289 What shall 7335 Give the product of the second and third term is 2119815 and this divided by 10000 the first term given the quotient or term required will be 212 fere which being deducted from 1.1525 the Equation answering to 325 degr because the Equation decreased their difference 1.1313 is the true Equation of this mean Anomalie which being added to the Suns mean longitude their aggregate is the Suns place required Example The Suns mean longitude 62.1643 Equation corrected Add 1.1313 The Suns true place or Longitude 63.2956 That is 2 Signes 3 degrees 29 minutes 56 parts The Suns Equation in this example corrected by Multiplication and Division may more readily be performed by Addition and Substraction with the help of the Table of Logarithmes for As one degree or 10000 4.000000   Is to 289 2.460898 So is 7335 3.865400   To 212 fere 2.326298 The Suns mean Motions Epochae Longitud ☉ Aphelium ☉  ° ′ ″ ° ′ ″ Per. Jul. 242 99 61 355 85 44 Mâ—â—di 248 71 08 007 92 42 Christi 278 98 69 010 31 36 An. Do. 1600 290 95 44 095 58 78 An. Do. 1620 291 10 41 095 90 39 An. Do. 1640 291 25 36 096 21 97 An. Do. 1660 291 40 33 096 53 56 1 356 76 11 0 01 58 2 359 52 22 0 18 17 3 359 28 30 0 04 74 B 4 000 03 00 0 06 30 5 359 79 11 0 07 89 6 359 55 19 0 09 47 7 359 31 30 0 11 05 B 8 000 05 97 0 12 64 9 359 82 08 0 14 22 10 359 58 19 0 15 78 11 359 34 30 0 17 36 B 12 000 08 97 0 18 94 13 359 85 08 0 20 52 14 359 00 19 0 22 11 15 359 37 30 0 23 69 B 16 000 11 97 0 25 25 17 359 88 08 0 26 83 18 359 64 19 0 28 41 19 359 40 28 0 30 00 B 20 000 14 97 0 31
25 degr PC is given by substracting 25 degr from PZ 38 degr 47 min. the complement of the poles height the angle CP 1 is 15 degrees one hours distance and the angle at C right we may finde C 1 by the first case of right angled spherical triangles for As the Radius 90 10.000000 Is to the sine of PC 13.47 9.367237 So is the tangent of CP 1. 15. 9.428052 To the tangent of C 1 3.57 8.795289 And this being all the varieties save onely increasing the angle at P I need not reiterate the work 3. Of South reclining more then the pole This plane in the fundamental Scheme is represented by the prickt circle EAW of which in the same latitude let the reclination be 55 degrees from which if you deduct PZ 38 deg 47 min. the complement of the poles height there will remain PA 16 deg 53 min. the height of the north pole above the plane and instead of the triangle PC 1 in the former plane we have the triangle PA 1 in which there is given as before the angle at P 15 deg the height of the pole PA 16 deg 53 min. and therefore the same proportion holds for As the Radius 90 10.000000 Is to the sine of PA 16.53 9.454108 So is the tangent of A 15. 9.428052 To the tangent of A 1. 4.36 8.882160 The rest of the hours as in the former are thus computed varying onely the angle at P. The Geometricall Projection These arches being thus found to draw the Dials true consider the Scheme wherein so oft as the plane falleth between Z and P the Zenith and the North pole the South pole is elevated in all the rest the North the substile is in them all the meridian as in the direct North and South Dials in which the stile and hours are to be placed as was for them directed which being done let the plane reclining lesse then the pole be raised above the horizon to an angle equal to the complement of reclination which in our example is to 65 degr and the axis of the plane point downwards and let all planes reclining more then the pole have the hour of 12 elevated above the horizon to an angle equal to the complement of the reclination also that is in our example to 35 deg then shall the axis point up to the North pole and the Diall-fitted to the plane Probl. 10. To draw the hour-lines upon any direct North reclining or inclining plane THe direct north reclining planes have the same variety that the South had for either the plane may recline from the Zenith just to the Equinoctial and then it is a Polar plane as I called it before because the poles of the plane lie in the poles of the world or else the plane may recline more or lesse then the Equinoctial and consequently their poles do fall above or under the poles of the world and the houre lines do likewise differ from the former Of the Polar plain This place is well known to be a Circle divided into 24 equall parts which may be done by drawing a circle with the line of Chords and then taking the distance of 15 degrees from the same Chord drawing streight lines from the center through those equall divisions you have the houre-lines desired The houre-lines being drawn erect a streight pin of wier upon the center of whâ—â— length you please and the Diall is finished yet seeing our Latitude is capable of no more then 16 houres and a halfe the six houres next the South part of the Meridian 11 10 9 1 2 and 3 may be left out as uselesse Nor can the reclining face serve any longer then during the Suns aboad in the North part of the Zodiac and the inclining face the rest of the year because this plain is parallel to the Equinoctial which the Sun crosseth twice in a year These things performed to your liking let the houre of 12 be placed upon the Meridian and the whole plain raised to an angle equall to the complement of your Latitude the which in this example is 38 deg 47 min. so is this Polar plain and Diall rectified to shew the true houre of the day 2. Of North reclining less then the Equator The next sort is of such reclining plains as fall between the Zenith and the Equator and in the Scheme is represented by the pricked circle EFW supposed to recline 25 degrees from the Zenith which being added to PZ 38 deg 47 min. the complement of the poles elevation the aggregate is PF 63 deg 47 min. the height of the Pole or stile above the plane And therefore in the triangle PF1 we have given PF and the angle at P to finde F1 the first houres distance from the Meridian upon the plain for which the proportion is As the Radius 90 10.000000 Is to the sine of PF 63.47 9.951677 So is the tangent of FP1 15 9.428052 To the Tangent of F1 13.48 9.379729 In computing the other houre distances there is no other variety but increasing the angle at P as before we shewed 3. Of North reclining more then the Equator The last sort is of such reclining plain as fall between the Horizon and Equator represented in the fundamental Scheme by the prickt circle EBW supposed to recline 70 deg And because the Equator cutteth the Axis of the world at right angles all planes that are parallel thereunto have the height of their stiles full 90 deg above the plane and by how much any plane reclineth from the Zenith more then the Equator by so much less then 90 is the height of the stile proper to it and therefore if you adde PZ 38 deg 47 min. the height of the Equator unto ZB 70 deg the reclination of the plain the totall is PB 108 deg 47 mi. whose complemenc to 180 is the arch BS 71 deg 53 min. the height of the pole above the plain To calculate the houre lines thereof we must suppose the Meridian PFB and the houre circles P1 P2 P3 c. to be continued till they meet in the South pole then will the proportion be the same as before As the Radius 90 10.000000 To the sine of PB 71.53 9.977033 So is the tangent of 1PB 15 9.428052 To the tangent of B1 14.27 9.405085 And so are the other houre distances to be computed as in all the other planes The Geometricall Projection The projection of these planes is but little differing from those in the last Probl. for the placing the hours and erecting the stile they are the same and must be elevated to an angle above the horizon equall to the complement of their reclinations which in the North reclining lesse then the Equator is in our example 65 degrees and in this plane the houres about the meridian that is from 10 in the morning till 2 in the afternoon can never receive any shadow by reason of the planes small reclination from the
Zenith and therefore needlesse to put them on In the North reclining more then the Equator the plane in our example must be elevated 120 degr above the horizon and the stiles of both must point to the North pole Lastly as all other planes have two faces respecting the contrary parts of the heavens so these recliners have opposite sides look downwards the Nadir as those do towards the Zenith and may be therefore made by the same rules or if you will spare that labour and make the same Dials serve for the opposite sides turn the centers of the incliners downwards which were upwards in the recliners and those upwards in the incliners which were downwards in the recliners and after this conversion let the hours on the right hand of the meridian in the recliner become on the left hand in the incliner and contrarily so have you done what you desired and this is a general rule for the opposite sides of all planes Probl. 11. To draw the hour-lines upon a declining reclining or declining inclining plane DEclining reclining planes have the same varieties that were in the former reclining North and South for either the declination may be such that the reclining plane will fall just upon the pole and then it is called a declining Equinoctial or it may fall above or under the pole and then it is called a South declining cast and west recliner on the other side the declination may be such that the reclining plane shall fall just upon the intersection of the Meridian and Equator and then it is called a declining polar or it may fall above or under the said intersection and then it is called a North declining East and West recliner The three varieties of South recliners are represented by the three circles AHB falling between the pole of the world and the Zenith AGB just upon the pole and AEB between the pole and the horizon and the particular pole of each plane is so much elevated above the horizon upon the azimuth DZC crossing the base at right angles as the plane it self reclines from the Zenith noted in the Scheme with I K and L. 1. Of the Equinoctiall declining and reclining plane This plane represented by the circle AGB hath his base AZB declining 30 degrees from the East and West line EZW equal to the declination of the South pole thereof 30 degrees from S the South part of the Meridian Easterly unto D reclining from the Zenith upon the azimuth CZD the quantity ZG 34 degrees 53 min. and Passeth through the pole at P. Set off the reclination ZG from D to K and K shall represent the pole of the reclining plane so much elevated above the horizon at D as the circle AGB representing the plane declineth from the Zenith Z from P the pole of the world to K the pole of the plane draw an arch of a great circle PK thereby the better to informe the fancie in the rest of the work And if any be desirous to any declination given to fit a plane reclining just to the pole or any reclination being given to finde the declination proper to it this Diagram will satisfie them therein for in the Triangle ZGP we have limited First the hypothenusal PZ 38 degrees 47 min. Secondly the angle at the base PZG the planes declination 30 degrees Hence to finde the base GZ by the seventh case of right angled spherical triangles the proportion is As the Radius 90 10.000000 To the co-sine of GZP 30 9.937531 So the tangent of PZ 38.47 9.900138 To the tangent of GZ 34.53 9.837669 the reclination required If the declination be required to a reclination given then by the 13 case of right angled spherical triangles the proportion is As the Radius 90 10.000000 To the tangent of ZG 34.53 9.837669 So the co-tangent of PZ 38.47 10.099861 o the co-sine of GZP 39. 9.937530 And now to calculate the hour-lines of this Diall you are to finde two things first the arch of the plane or distance of the meridian and substile from the horizontal line which in this Scheme is PB the intersection of the reclining plane with the horizon being at B. And secondly the distance of the meridian of the place SZPN from the meridian of the plane PK which being had the Diall is easily made Wherefore in the triangle ZGP right angled at G you have the angle GZP given 30 degrees the declination and ZP 38 degr 47 min. the complement of the Pole to finde GP and therefore by the eighth case of right angled spherical triangles the proportion is As the Radius 90 10.000000 To the sine of ZP 38.47 9.793863 So is the sine of GZP 30 9.698970 To the sine of GP 18.12 9.492833 Whos 's complement 71 deg 88 min. is the arch PB desired The second thing to be found is the distance of the Meridian of the place which is the houre of 12 from the substile or meridian of the plane represented by the angle ZPG which may be found by the 11 Case of right angled sphericall Triangles for As the Radius 90 10.000000 Is to the sine of GP 18.12 9.492833 So is the co-tang of GZ 34.53 10.162379 To the co-tang of GPZ 65.68 9.655212 Whos 's complement is ZPK 24 deg 32 min. the arch desired Now because 24 deg 32 min. is more then 15 deg one houres distance from the Meridian and lesse then 30 deg two houres distance I conclude that the stile shall fall between 10 and 11 of the clock on the West side of the Meridian because the plain declineth East if then you take 15 deg from 24 deg 32 min. there shall remain 9 deg 32 min. for the Equinoctiall distance of the 11 a clock houre line from the substile and taking 24 deg 32 min. out of 30 deg there shall remain 5 deg 68 min. for the distance of the houre of 10 from the substile the rest of the houre distances are easily found by continual addition of 15 deg Unto these houre distances joyn the naturall tangents as in the East and West Dials which will give you the true distāces of each houre from the substile the plane being projected as in the 5 Pro. for the east west dials or as in the 8 Prob. for the Equinoctial according to which rules you may proportion the length of the stile also which being erected over the substile and the Diall placed according to the declination 30 deg easterly and the whole plain raised to an angle of 55 deg 47 min. the complement of the reclination the shadow of the stile shall give the houre of the day desired 2. To draw the houre lines upon a South reclining plain declining East or West which passeth between the Zenith and the Pole In these kinde of declining reclining plains the South pole is elevated above the plane as is clear by the circle AHB representing the same which falleth between the Zenith and the North pole and therefore hideth