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end_n	divide_v	equal_a	line_n	1,364	5	9.1106	5	false

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A40034	Posthuma Fosteri the description of a ruler, upon which is inscribed divers scales: and the uses thereof: invented and written by Mr. Samuel Foster, late professor of astronomie in Gresham-Colledg. By which the most usual propositions in astronomy, navigation, and dialling, are facily performed. Also, a further use of the said scales in deliniating of far declining dials; and of those that decline and recline, three severall wayes. With the deliniating of all horizontall dials, between 30 and 60 gr. of latitude, without drawing any lines but the houres themselves.	Foster, Samuel, d. 1652.	1654 (1654)	Wing F1635A; ESTC R218296	23,210	71

a skin of smooth Velame well pasted on a board you may work upon it many conclusions very exactly The Vses of the Sea-Chart Are set down in 12 Propositions by Master Gunter beginning page 121. In each of which Propositions is shewed how to resolve the Question upon the Chart it selfe which will be direction enough how the work must be performed without any more words here used The working of these propositions also may be applyed to the Scales of Sines and Tangents on the Ruler and wrought by protraction according to the rules given in the first Chapter if the proportions as he layes them down in the forecited pages be so applyed If a Scale of Rumbs be thought more expedient for these operations then is a Scale of Chords it may be put into some spare place of the Ruler His two Propositions page 114. 116 may be done upon the Chart as is there shewed but his second Proposition which is SECT. 2. To finde how many Leagues doe answer to one degree of Longitude in every severall Latitude THis I say may be done upon the Scales of Sines and equall parts And for this purpose the two last inches of the same Scale of equall parts being equall in length to the Radius or Sine of 90 are divided into 20 at one end and into 60 at the other end Take therefore upon the line of Sines the complement of the parallels distance from the Equator or the complement of the given Latitude and measuring it upon the Scale of 20 parts it will shew you what number of Leagues make one degree of Longitude in that parallel of Latitude And being measured upon the Scale of 60 parts it gives so many of our miles or so many minutes of the Equinoctiall or any other great circle as are answerable to one degree of Longitude in that Latitude Example Let it be required to finde how many Leagues doe answer to one degree of Longitude in the Latitude of 18 gr. 12 ' Take out of the line of Sines the complement of the given Latitude namely 71 gr. 48 ' Then applying this distance to the Scale of 20 equall parts you shall finde it to reach 19 and so many Leagues doe answer to one degree of Longitude in the Latitude of 18 gr. 12 ' And the same distance being measured upon the Scale of 60 equall parts will give you 57 parts and so many minutes of the Equator are answerable to one degree of Longitude in that parallel of Latitude So likewise in the Latitude of 25 gr. 15 ' if you take the complement thereof 64 gr. 45 ' out of the Scale of Sines and apply it to the former line of 20 you shall finde it to reach 18 parts and so many Leagues doe answer to one degree of Longitude in the Latitude of 25 gr. 15 ' In the Appendix to Master Norwoods Doctrine of Triangles there is by him laid down 15 Questions of sailing by the plain Sea-chart and others by Mercators Chart all which the line of Chords and equall parts will sufficiently perform if the work of the third Chapter of this Booke be rightly understood SECT. 3. How to set any place upon your Chart according to its Longitude and Latitude IF the two places lie under one parallel and so differ only in Longitude then the course leading from one to the other is East or West As A and E being two places under the parallel of 50 gr. and differing 5½ gr. in Longitude But if the two places differ only in Latitude and lie under one Meridian as A and B then the course is North or South CHAP. VI Of Projections of the Sphere FOr this purpose chiefely is the lesser line of Chords added being made to the same Radius that belongs to the Sines Secants and Tangents For when any Projection is to be made the fundamentall Circle must be of that common Radius and then the angles to be inserted upon it may be taken out of this line of Chords which is fitted to it See the second Book of the Sector Chap. 3. For these Tangents and Secants will performe the same things in those Stereographicke projections that there are done and in all other irregular projections likewise By this kinde of work may any Sphericall conclusions be performed by protraction in plano Also true Schemes of the Sphere may be drawn sutable to any question which will not a little direct in Sphericall calculations As suppose it were required to project the Sphere sutable to this Question Having the Latitude of the place the declination of the Sun and the Altitude of the Sun to finde either the Azimuth or the houre of the day First With the Radius of the line of Chords upon the center C describe the fundamentall Circle ZHNO representing the Meridian and draw the diameter HO for the Horizon and ZCN at right angles thereto ZN being the Zenith and Nadir points Then by your line of Chords set the Latitude of your place which let be 51 gr. 32 ' from Z to E and from N to Q drawing the line ECQ for the Equinoctiall and at right angles thereto the line MP for the axis of the World P representing the North and M the South Pole Secondly Supposing the Sun to have 20 gr. of North Declination take 10 gr. the Semitangent of the Declination out of the line of Tangents and set it from C to G. Likewise take 20 gr. the Declination from your line of Chords and set that distance upon the Meridian from E unto D and from Q unto K then describe the arke of a Circle which shall passe directly through the points DGK the center whereof will alwayes fall in the line CP if it were extended and this arke DGK shall be the line of the Suns course when his Declination is 20 gr. from the Equinoctiall Northward 1. The centers of the parallels of declination and of the parallels of altitude may readily be fouud by the Scale it selfe as in this projection having found the point F upon the Line ZC extend the line ZC without the circle and because the Suns altitude is 50 gr. take therefore out of your Scale the Secant of 40 gr. the complement of the Altitude and set that distance from C to I so shall the point I be the center of the parallel of Altitude Or take the Tangent of 40 gr. out of your Scale and place it from F to I either of which will fall in the point I the center of the parallel of 50 gr. In the same manner may the center of the parallel of Declination be found by taking out of your Scale the Secant complement of the Declination and setting it from C upon the line CP being extended and where that distance ends that is the center of the parallel 2. For the finding of the centers of the Hours and Azimuths the Scales of Secants and Tangents will much help you So the Azimuth from the South being 49 gr. 52

to draw the Horizontall and then the Verticall line perpendicular thereto 2. Next is the placing of the Meridian upon the plain in a true position In direct plains that re in-cline and in upright decliners the Meridian is the same with the plains Verticall line In East and West re in-cliners it is the same with the horizontall line In the rest it ascendeth or descendeth from the horizontall line and must be placed according to the rules hereafter given I. ZB iein-clination BG â™’ ZO OC co-sine of declination RC â™’ FO. ORD Radius DS â™’ BO Out of this Structure will follow DAE new Latitude lesse than 90. OR co-sine of new Declination DS Sine of Meridians ascension or descension II. ZB rein-clination BG â™’ ZO OA Sine of declination HAI â™’ FO. IO Radius AM â™’ IO Out of this Structure will follow OK is the sine of ZD or ND in the former figure where the new Latitude DAE will be found HO Sine of new declination AM co-sine of Meridians ascension or descension How all re/in-clining plains being counted as upright in their new Latitude are to be taken whether as North or South decliners As also How the Meridian line is to be placed whether ascending above or descending below the horizontall line and from which end of that line whether that which looks the same way with the declination of the plain or that which looks the contrary way In North re in-cliners If D fall below P the Recliners are North plains and the Meridian ascends above the horizontall line from that end of it which looks to the same Coast of declination Incliners are South plains and the Meridian descends below that end of the horizontall line which looks to the contrary Coast of declination If D fall above P the Recliners are South plains and the Meridian goes below contrary Incliners are North plains and the Meridian goes above the end looking the same way with declination In South re in-cliners If D fall above AE the Recliners are North plains and the Meridian goes above the horizontall line from the same end with the Coast of declination Incliners are South plains and the Meridian goes below the horizontall line from that end which is contrary to the Coast of declination If D fall below AE the Recliners are North plains and the Meridian goes below the horizontall line contrary Incliners are South plains and the Meridian goes above the horizontall line from that end which looks to the Coast of declination If D fall into P both re in-cliners are called Polar plains and the Meridian in both ascends from the Same end in Recliners contrary end in Incliners If D fall into E the Recliners are North plains and the Meridian ascends from the same descends from the contrary end to that which looks upon the Coast of declination Incliners are South plains and the Meridian ascends from the contrary descends from the same end that looks upon the Coast of declination East West Recliners are North plains declining from North So much as the complement of their re in-clination comes to This is their new declination their new Latitude is the complement of the Latitude of your place Incliners are South plains declining from South 4. For that which follows take notice of these four things First That from D to the neerest AE measured by the line of Chords gives the new Latitude in which the re in-clining plain is an upright declining plain Secondly That OR measured upon the line of Sines gives the complement of the plains new declination in that new Latitude this New declination is to the same Coast with the Old but alwayes lesse in quantity than it Thirdly That DS measured upon the Sines gives the quantity of the Meridians ascension or descension This gives the quantity the former rules gave the Coast Fourthly That in the description of the Diall you must only make use of the new Latitude and new Declination having nothing to doe with the other 5. Having the former things known you must by the Tangent and co-tangent of the new Latitude describe your Rectangled Parallelogram as in the 10 Chap. and according as the plain was discovered to be a decliner from the North or South you must make choice of your center place the substylar style and six a clock line by help of the Sine and co-sine of the new declination and new Latitude and then prick down and draw the houres all in the same form that was before shewed in the 10 Chap. for upright decliners This for the Dials description 6. Lastly for placing your Diall First Consider which way and how much your Meridian ascended or descended from the horizontall line Then goe to your plain and there draw the same Meridian line answerably setting off so many degrees by your Scale of Chords When this is done take your paper description and lay the Meridian of it either upon or else parallel to the Meridian drawn upon the plain and take care to place it the right way namely so as that the imaginary style of your paper or a reall pattern of the style cut fit and set upon the paper Diall may point into the North or South Pole according as the plain is esteemed to be a North or South plain After this is performed you may transfer each houre from the paper to the plain and so finish all the work CHAP. XV Concerning full East and West re-in-clining plains HEre in this sort of plain you are only to take notice that the new Latitude wherein they stand as erect plains is ever the complement of your own Latitude And the new declination in that Latitude is the complement of their re in-clination By knowing these you may describe the diall according to the 10 Chap. The Meridian line in all these lyeth in or parallel to the horizontall line All which things will appear also out of the former figures if according to them you should make a draught and suppose your plain to decline 90 degrees as all these East and West plains do All other things will follow of themselves agreeable to other plains CHAP. XVI Concerning re-in-cliners that are direct or have no declination IF the line CB be placed as is prescribed in the former figure and drawn quite through it will represent your plain that is re in-clining towards the North and without any declination So also BL if it be drawn quite through will represent such plains as re in-cline towards the South and have no declination For which lines so drawn or imagined only you may gather according to the former rules which of the Poles A or X is elevated and how much it is elevated which is shewed by the arke CA or LX You may also see which end of the Meridian is to be taken for the substilar line over which in these direct plains the stile is ever to be erected and must stand Then for drawing the houres you have no more to

shall you have S e the whole Tangent of 67 gr. 8 ' as is required Thus doe for any other which shall goe beyond the Scale CHAP. XXI A second way for re-in-cliners TAke notice of these terms 1. Verticall distance is the distance of the plains pole from the Vertex or Zenith of the place 2. Polar distance is the distance of the plains pole from the North pole Preparatory works 1. Draw the horizontall line upon the plain and crosse it at right angles with a Verticall line 2. Get the plains re in-clination and consequently the distance of the plains Pole from the Zenith of the place which is here called the Verticall distance 3. Get the plains declination and alwayes account how much it is from the North For that is here called the angle of Declination SECT. 1. By the Scale of Versed Sines how to finde the elevation of the Pole above the plain and which Pole it is whether North or South that is elevated First finde the summe and difference of The complement of your Latitude The plains Verticall distance And observe likewise that If the point of your Compasses applyed to the Scale doe fall just upon 90 then is your plain a Meridionall or Polar plain and hath no pole elevated above it If it fall short of 90 then is the North Pole elevated and the elevation is so much as the point fals short of 90. If it fall beyond 90 so much as it falls beyond so much is the South Pole elevated SECT. 2. To finde the plains difference of Longitude from the South part of your Meridian and which way the said difference of Longitude is to be taken First finde the sum and difference of The complement of your Latitude The fore-mentioned polar distance Then make AB equall to your whole line of Versed Sines And upon your Scale count your difference now found and the fore-mentioned verticall distance taking the distance of these two as they are numbered upon the Scale With which length upon A describe the arch CD Take also upon the Scale from the verticall distance to the fore-mentioned sum and with that length upon B describe the arke EF. Then draw the line FC so as to touch both these arks cutting the line AB in G so shall AG being measured upon the Scale give the plains difference of Longitude from the South which is here required This difference of Longitude is to be taken to the same Coast in the heavens unto which the plain declineth and may afterwards in the description of the Diall be easily accounted either from the South or North part of the Meridian viz. so as that the said difference may be alwayes lesso than 90 gr. SECT. 3. To finde how much the Substilar or plains proper Meridian must lie from the Verticall line of the plain and which way Frist Finde the summe and difference of The Polar distance The Verticall distance Then make AB equall to your whole Versed Scale And on the same Scale take the extent from the complement of your Latitude to the difference now before found with which length upon A as the center describe the arke CD Also upon the Scale take from the complement of your Latitude to the summe here before found and with that length upon the center B describe the arke EF. then draw the line FC justly touching both these arks and cutting the line AB in G so shall AG being applyed to the Scale give the quantity of the angle here required According to this angle the substylar line must alwayes stand off from the verticall line of the plain Which way must the Substilar lie from the Verticall line If the plain hath the North Pole elevated upon it then must the substilar alwayes lie from the upper end of the Verticall line towards the North Pole so much as the angle was in the last Section found to be If the South Pole be elevated then the substilar lyeth alwayes from the lower end of the verticall line towards the same South Pole according to the forenamed angle If the plain be Meridionall upon which neither of the Poles is elevated then the substilar must doe either or both these two according to the angle before found According to these Rules you may place the substilar line upon the plain in its true position requisite SECT. 4. To draw the houre-lines upon the plain FIrst consider by the first of these Sections whether it is the North or South pole that is elevated upon your plain If it be the North pole then must the center of your Diall stand downward and the style must point upward to the said North pole But if the South pole be elevated then the center of the Diall is to be set upward and the style comming from thence must point downwards into the South Pole Lastly From the center of the Diall A or B and through the said unequall parts draw right lines These last lines shall give you 12 of your houres required And if you draw each of them quite through the center you shall have the whole number of 24 of which you may take such as are sutable and necessary for your plain When your paper Diall is thus finished you may transfer it to your plain by laying the substilar upon or parallel to the substilar before placed upon the plain and so insert all the houres from the paper to the plain After all this you may make the style to the angle of the Poles elevation and fit it in according to its requisite place and position Note that because some of the houre points found in the Circle will happen so neere to the center of the Diall that you cannot well draw the houre-lines true you may therefore help your selfe by that direction which I have given in my Geometricall way This Geometricall way shall shortly be published by the Authors own copie with his own Demonstrations of the whole work For drawing houres upon plains that have small elevations and upon Polar plains use the former directions CHAP. XXII A third way for all re-in-clining Dials SECT. 1. To finde a re-in-clining plains difference of Longitude from the South part of your Meridian and how much the plains Meridian or substyle must lie from the Verticall line of the plain   I. II. III. Complement of your Latit 38 30 K 38 30 K 38 30 K Plains verticall distance 100 00 Z 60 00 Z 30 00 Z Their Summe 138 30 98 30 68 30 Their Difference 61 30 21 30 8 30 Their halfe Summe 69. 15 R compl. 20 45 V 49. 15 R comp. 40. 45 V 34. 15 R compl. 55. 45 V Their halfe difference 30. 45 S. compl. 59. 15 X 10. 45 S compl. 79. 15 X 4. 15 S compl. 85. 45. X Plains declinat from Sou. 50. 00 Y 140 00 Y 160 00 Y Describe a Circle with your common or lesser Scale of Chords And out of the same Scale make A Y plains declination from South

Out of the line ⊙ make A R R and A S S draw Y R M and M S B and make A D A B. Out of the same line ⊙ make A V V and A X X and draw Y V N and N X C. Then if K be lesse than Z C A D is the differ of Longitude required and C B is the angle between the substile the verticall line But if K be greater than Z C A D is the forementioned angle and C B is the difference of Longitude These two arks C D and C B must be measured from 90 in the line of Versed Sines and looke what number of degrees they there cut the same must be accounted for their quantities SECT. 2. To finde the elevation of the Pole above the plain and which of the Poles it is whether North or South that is elevated MEasure A B upon the Versed Sines from 90 as before the complement of that is E B. Measure also E C upon the same Scale in the same manner Count these quantities E B and E C so found upon the line ⊙ and set them from E to F and G and make Er R taking E r out of the Scale of Versed Sines from 90 Draw r F O and O G P. Measure E P upon your Scale of Chords it will there give you the polar distance If E P fall to be 90 it is a Meridionall plain and hath no Pole elevated If it be lesse than 90 the complement of it is the elevation of the North Pole If it be greater than 90. the excesse is the elevation of the South Pole Note that the three figures following have relation to the three Columns of the foregoing Table and to these rules last delivered Figure I. In this first Figure C A D 62 49 C B 34 13 E P 122 00 Figure II. In this second Figure C A D 110 29 C B 42 19 E P 36 28 Figure III. In this third Figure C A D 118 26 C B 44 56 E P 14 00 SECT. 3. Which way must the Substilar lie from the Verticall line THe Rules are the same with those before in the second way of Dialling where the same Question is propounded You may therefore have recourse to them Or thus Upon all plains whereon the North pole South pole is elevated the substilar must lie from the upper end lower end of the Verticall line towards the full North South For drawing the houres and finishing the Diall you must doe as is prescribed in the 4th Sect. of the former second way For having placed the Substilar and knowing the plains difference of Longitude you are to use the same course here that was there given It will be easie to doe these things in plains that are upright and have no re-in-clination Note in all these three wayes of Dialling All directions here given suppose you to be in the Northern Hemisphere of the world If therefore you should be in the Southern Hemisphere you may easily make these precepts serve there too by only altering the name of North Northen c. and South Southern c. one into the other FINIS This Scheme hath relation to the 16th Chapter page 54. Errata Page 32 in the Diagram for H P G read H F G. Page 51 for If D fall into E read If D fall into Ae Page 54 line 16 for For read From In the Diall Page 57 at the other end of the line M place the letter K upon the Substilar Page 62 line 19 for L P read S P. In the Diagram page 76 the letter R is wanting at the intersection of the line Y M with the line A E. The wayes how to effect these are shewed other-where and are here taken as known