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end_n	draw_v	equal_a	line_n	2,859	5	9.5591	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A40031	Elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by Samuel Foster ...	Foster, Samuel, d. 1652.; Twysden, John, 1607-1688.; Wingate, Edmund, 1596-1656.	1654 (1654)	Wing F1632; ESTC R7034	96,404	198

another sliding plate of the same breadth that the former was na●●ly M O here equall to M O in the former and M N must in both be justly equall that it may both slip in the former cavity so as just to fill it and that the same limbs centers and threeds may in both agree A lesser Ellipsis I say must be described which though it be lesse yet it must be like and proportionall to the former Therefore assuming any length as M P for the longer Radius you must to that Radius make M R and M S equall● to the Sine of your Latitude and so describe and divide it as formerly was ordered Then again to this new Ellipsis there must a new Scale of Tangents be made whose length must be limited as the former was in this manner To M P as Radius finde the Co-sine of your Latitude and make that Co-sine a Radius or Tangent of 45 gr and according thereunto continue the Scale of Tangents as far as it will go It must begin upon the same line that the other Scale of Tangents began and must go the same way with it And in this new way of ordering the Scales and the Ellipses it must be considered what the other parts of the Instrument do signifie The end of the lower plate upon which the Scales begin must be accounted for the North the other end C D is to be taken for South The side lines are the Meridian The degrees of the limbs are to be used as the degrees of the Equinoctiall The threed applyed to those degrees are the Meridians comming from the Centers which are the North Pole The Tangents are the degrees of Altitude The Ellipsis notes out that Almicantar which the threed Index stands unto upon the Tangent Scale And further It is to be here noted that if the Meridians or threeds comming from the center were rightly divided those divisions or parts of it should signifie the degrees of Declination from the Equinoctiall But the inconvenience of it is that because one Ellipsis is to represent every Almicantar therefore the Radius cannot possibly be at all times the same in length but must varie according to the removing or severall positions of the said Ellipsis That is when the Ellipsis signifies the Horizon or stands at the beginning of the Scales without any altitude at all then the Radius M O in the first moving plate or M P in the second moving plate is the Scale of Declinations but at other times when the Ellipsis signifies any Almicantar then must the Secant of that Almicantar of Altitude be taken as the Radius for that particular work But alwayes the Radius of what length soever it be must be divided as a whole line of Sines the greatest parts beginning towards the center the least parts ending at the remotest end from the center and yet then again the numeration of the parts or declinations of it must begin at the remotest end and must end in the center at the Pole It might be applyed to use in the motions of the Stars but that would be troublesome therefore it shall suffice to make it usefull only for the Sun The best way that for the present I know whereby to divide this Radius thus of severall lengths is this Divide the length from the center T to C into 90 Sines the greatest parts beginning at the center but numbered the contrary way This line may be drawn from the center T into some part of the limbe where it may have room to receive divisions Then with your Compasses take the Radius T V equall to M O and from the end of the former Scale open the threed to the least distance and where the threed stayes there make a mark upon the limbe V D for there is a new Scale to begin for the Secants that were mentioned before Then lay the threed from T to 80 in the opposite limb and move the Ellipsis till the point O fall just under the threed and take T O in your Compasses and put in that distance as before so shall you put in 10 gr into this new Scale In like manner lay the threed to 70 60 c. and bring the Ellipsis till the point O lies under it so shall T O in these severall positions give the lengths to be inscribed in this new Scale for 20 gr and 30 gr c. all the rest must be done in like manner till you have put in as many as will come within the reach of the limb which will be upon 60 gr or thereabout The like should also be done for the lesser Ellipsis working in the same manner by the threed and the point P. These are called the new inscribed Scales The former line of Sines serves to them both indifferently Now when things are thus fitted the uses of them will be such as in particular Astrolabes is vulgarly known Some of the Uses are here mentioned 1. Having the Suns Declination and Altitude to finde the Houre and Azimuth First For North Declinations Set the Ellipsis to the Altitude counted in the Tangent Scale Then count the same Altitude in the new inscribed Scale of Secants and thereto lay the threed Afterwards take the least distance from the Suns declination counted in the fore-mentioned inscribed Scale of Sines to the threed Set one foot of that distance in the center and extend the other till it crosse the Ellipsis and where it crosseth thither apply the threed which in the limb will shew the Houre and scruple in degrees required The same point of the Compasses doth also immediately shew upon the Ellipsis the Azimuth sought for If the Compasses do crosse the Ellipsis twice as somtimes it will take that crossing that is furthest from the Meridian Secondly for South declinations The work is in a manner the same only you must note that the threed will crosse the Ellipsis twice and you are in this case to take that crossing which is neerest to the Meridian By supposing the Altitude to be 00 that is by laying the Ellipsis upon the line T V and using the Declination as before you may finde both the Amplitude and Ascensionall difference as before and what else doth thereon depend But because these things being done neer the Horizon will not prove good especially when the Declination is little from the Equinoctiall and because the work at the best is but troublesome I shall here break off supposing that I have already written too much A DEMONSTRATION of the Ellipticall Diall upon an Horizontall Plain shewing the reason why the same Diall by an upright Index should shew the true Houre THE reason is principally deduced from the Sphere it self and secondly from the Orthographicall Projection of the Sphere upon the plain of the Horizon which as it doth represent the Sphere it selfe so it doth perform the same conclusions with the like certainty that the Sphere doth and that upon this generall ground that

  44 26 29 36     48 15 27 54   IX 51 57 26 7 III.   55 32 24 14     59 1 22 16     62 24 20 14   VIII 65 41 18 8 IV.   68 54 15 59     72 2 13 47     75 7 11 33   VII 78 9 9 17 V.   81 9 6 59     84 7 4 39     87 4 2 20   VI. 90 0 0 0 VI. FOr this purpose here are two Tables joyned together both of them made for the Latitude of London 51 gr 30 min. the like to which every man may calculate for his own place The first of them is a Table of such angles as are made by the hour lines coming through the center of the Ellipsis with the Meridian line or line of 12. And it was made by that rule which was given in this case at the beginning of this Treatise Namely As the Sine of your Poles elevation which is here 51 gr 30 m. is to the Radius So is the Tangent of each houre and their quarters counted from 12 a clock To the Tangent of the angle required The second Table is of the Altitudes of each houre and quarter in the Equinoctial Circle above the Horizon and it is calculated by this Rule As the Radius To the Co-sine of your Latitude which is here the Sine of 38 gr 30 min. So is the Sine of each houre and their quarters counted from 6 a clock To the Sine of the altitude which is here sought Then having computed these two Tables you may by help of them both draw and divide the Ellipsis into its true houres and quarters by these following directions First draw the two lines 6 6 and D E crossing each other at right angles in A and let A D be the Meridian and 6 A 6 the two six a clock lines Then upon A as a center describe the Circle 6 D 6 of any convenient bignesse and upon the same Circle from D on both sides of it set on the houres and their quarters as 11 and 1 10 and 2 c. by help of the degrees and minutes of the angles set to every houre in the former Table And through every of these points inscribed into the Circle draw lines from the center A as A 11 A 10 A 1 A 2 with the rest of the houres and their quarters if you will Fourthly looke in the second Table for the altitudes of the Equinoctiall at every houre Then count those Altitudes in the line of Sines and take with your Compasses the several distances of them from A transfer the said distances from the center A to every houre respectively so shall those intersections give you the points through which the Ellipsis must be drawn Thus the altitude of 7 and 5 a clock which is 9 gr 17 m. being taken upon the line of Sines from A towards S is inserted into 4 houres at the note m. And the altitude at 8 and 4 which in the Table is 18 gr 8 min. being taken and transferred to the four houres at n do give four points more The rest of the altitudes give each of them two houres only as at o i and r is done And the last of all at 12 gives one only point at t. The like may be done for the quarters And so through these points thus found the Ellipsis may easily be drawn and the lines formerly drawn give the divisions that are due unto it Note 1. This is propounded only for Horizontal plains but it may without difficulty be applyed to any other direct plain The trouble that is comes by reason that the foresaid plains have a different elevation of the Pole from that elevation that belongs to the Horizon or place where they are to stand and consequently there will be required two new Tables for that elevation which is proper to the plain the calculation of which will easily be done by the two former proportions set down for that end Note further that by these two Tables you may prick down an Ellipsis upon any leaning not upright plain whatsoever But the Index must then lie in the Zenith line of the place not of the plain and the Ellipsis or Index must move in the Meridian of the place not of the plain and the Zodiac must be like or proportional to the Zodiac of the Horizon but augmented for the most part in this proportion As the Radius To the Secant of that angle that the Meridian line upon the plain makes with the Horizon which angle must be gott●n by a Clinatorie So the Radius of the Horizontal Zodiac which is the Co-sine of the Latitude To the Radius of the Zodiac proper to the plain which Zodiac must be set according to the line of 12 upon the plain 2. This way of delineation may likewise be applyed to all other plains which decline and are not direct But in these will be found more difficulty then in the former plains which declined not unlesse it be required to put in the houres that are proper to the plain for in that case the work is the same which was in them But if the houres of the place are to be inscribed as most usually they are then there will be some trouble by reason that the difference of the plains Longitude seldome falls upon any just houre I purpose not here to shew the way it being such as will prove un-pleasant to the unskilfull and such as have knowledge will finde it out quickly Perhaps the pleasure of the thing done will recompense their labour SECT IV. Here follow some Vses and Varieties of this Ellipticall Diall USES FIrst If the Index stand still and the Ellipsis be made moveable you may upon that plate which is immoveable and on which the Index standeth describe a Circle whose center must be the foot of the Index And drawing a Meridian line upon the said immoveable plate just under the Meridian of the Ellipsis and from the foot of the Index you may divide that Circle beginning from the Meridian line into 360 gr or so many of them as shall be needfull By these degrees you shall finde the Suns Azimuth by the shadow of the upright Index and by the divisions of the Ellipsis you may know the houre of the day So both Azimuth and Houre are shewed upon the two plates by one upright Index This kinde of Diall must be set in its true position for it will not set it selfe and must there be fixed If it be upon an Horizontall or any direct plain you may draw a Meridian line thereon and either draw the Azimuths true upon it and so make the Ellipsis to move to and fro according to that Meridian Or else if both your plates be to be fixed upon any Base prepared for them you may first upon that Base draw a true Meridian line and to that line make the Meridian of the Diall to agree The like must

an houre from 12 in the Ellipsis the remaining breadth N O or E F or G H may be just fit to A B the breadth of the hollow Area in which this lesser rectangled Parallelogram E F G H may be moved to and fro as there shall be occasion And let the thicknesse or depth of this moving plate be the same with the thicknesse of the limbe of the lower plate The division of the Ellipsis may be made according to the rules before given 3. In the midst of this lesser plate let a hole be pierced quite through it and in that hole let a threed be extended as I K which is to serve for an Index by which the said Semiellipsis is to be rectified And therefore it is best that the threed I K be laid just in the longer diameter which is the 6 a clock line 4. Take the lesser Parallelogram and put it into its place as close to the end A B as it will go and when it is there laid observe where the threed lies upon the lower Area and draw the line upon which it lies which must be the line 90 90 upon which the Scales that are inscribed must take their beginning And likewise you are to take the longer Radius O M and as the plate there lies set one foot of that Radius upon O and let the other reach as far as it will which must be upon the rising limbe beyond the inner edge of it so much as M N comes to in that point you are to make one center so again turning your lesser plate about you must lay it in the contrary way close to A B as you did before and from the point O you may finde the second center both these centers in each of which must a threed be fastned will happen to fall in the former line 90 and 90 neer the points T and V. Being thus far prepared you may take away the lesser plate and divide the Scales upon the greater plate in this manner 5. First For the two limbs of it they are nothing else but the degrees of a Circle projected thereon from the two centers which may therefore without difficulty be done either from the degrees of a Circle or else by Tangent lines as any man will easily perceive 6. Secondly For the other Scales you must make them parallel to the edge of the limbs There are three of them in number The longest is a Scale of Tangents The middlemost is a Scale of houres made in the form of a line of Sines The third is a Zodiac or Scale of the year The Tangents are made thus Taking M O as Radius finde the Co-sine of your Latitude agreeable thereto and making that Co-sine as a Radius or Tangent of 45 degrees all the other Tangents are to be fitted accordingly Then to know how far of this line will be usefull put the lesser plate again into its place as close to the end C D as it will go so shall the threed I K shew the furthest point of it that will be usefull The middle Scale of Sines is made thus Count the complement of your Latitude upon the new made Tangent line and from the beginning of the Scale to that point make a line of Sines for six houres and their quarters so shall this Scale be fitted The Zodiac or Annual Scale is equall in length to the Tangent of 23 gr 30 min. and the manner of inscription with Tables whereby to do it are set down before and need not be here repeated The reason why this Instrument is thus contrived The intent of the contrivance is that it should shew both Azimuth and houre and that therefore there should be a Ci●●● and an Ellipsis sliding over it both of them made to one center so that the center should be in the midst and the Circle should be a whole entire Circle and the Ellipsis also whole But because the Ellipsis sliding upon the nether plain doth somtimes of necessity cover the center so as that there can be no threed fixed for use I was therefore compelled to order it in this manner that is before exprest that the motion of the upper plate and the centers migh● stand cleere without hin●●ing each other Then further it must be noted that the Instrument is not generall to all but particular to one La●●tude The uses of it do here follow which by reason of the Semiellipsis will not be so easie because many changes are required In generall note thus much 1. In using this Instrument you must alwayes suppose that you looke upon the North. 2. If the Declination of the Sun or Star be North hold A B towards you If South then hold C D towards you 3. If the Sun or Star be Eastward of the Meridian let the morning houres be towards you If they be Westward let the evening houres be towards you 4. Alwayes use that threed which is on the same side with the center of the Ellipsis And that side lin● of the larger plate is ever to be taken for the Meridian and consequently the other side must be esteemed for one of the two Coasts of East or West 5. Becau●e North is furthest from you therefo●● the furthest quarter of the limbe must be either North-east or North-west the neerest South-east or South-west These cautions being observed the conclusions that are to be wrought will be easie 1. By having the Azimuth to finde the Houre First Order your Instrument by the former cautions according to the present case of your observation And rectifie the upper ●late by help of the threed Index either to the Declination of the Star counted in the Tangent line or to the Suns place in the Zodiac Then count the Azimuth given upon the right quarter of the limbe and apply the threed thereto So shall the threed shew upon the Ellipsis the houre in which the Sun or Star is If the work be for the Sun then the houre thus shewed is the Suns houre or houre of the day If the work be for a Star it then shews the Stars houre only which must afterwards be reduced into the Suns houre that it may be the true houre of the night This change may most easilie be done by a common Nocturnall 2. By having the Houre in which the Sun or any Star is the Azimuth thereof may be found The Instrument being rectified lay the threed upon the houre counted in the Ellipsis the same threed will shew the Azimuth in the graduations upon the limbe 3. To finde the Ascensionall difference the Semidiurnall and Seminocturnall arkes with the time of their risings and settings In generall note this If the Sun or Star decline Northward then the Semidiurnall arke is greater than six houres if Southward then lesse The upper plate therefore being rectified either to the Suns place or to the Stars Declination the threed will shew upon the middle Scale the Ascensionall difference if you count the houres

or 3 houres R i the Sine of 60 gr or 4 hours R h the Sine of 5 houres R D or R 12 the whole Sine or Sine of 90 gr Then through each of these points at h i k l m R o p q draw a line parallel to the other conjugate Diameter namely to 6 R 6. This done make R 6 a Radius and to that Radius make m7 m5 07 05 each of them a Sine of 5 hours or 75 degrees Make l 8 l 4 p 4 p 8 a Sine of 4 hours or 60 gr Make k9 k3 q3 q 9 each of them a Sine of 3 houres or 45 gr So i 10 i● must be the Sine of 2 houres and h 11 h 1 the Sine of 1 houre or 15 gr By this work you shall upon the former parallel lines drawn through h i k l m o p q finde the houre points of 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 In which you are to signe out the houre points and through which you are to draw the Ellipsis as you see the figure for a pattern You might also have divided each of the Radiusses noted with R 6 into parts correspondent to a line of Sines as you see partly done at the letters s t u in the last of the two former Scheams as at s t u is expressed and through those points you must draw paralle●s to the other Diameter D R G. Then to the Radius D R make s 7 s 5 the Sine of 15 gr or 1 hour t 8 t 4 the Sine of 30 gr or ●wo hours u 9 u 3 the Sine of 45 gr or 3 houres and so forward by which work you shall finde the same houre points 3 4 5 6 7 8 9 c. that you did before and so you may finde out and draw the Ellipsis by them The same manner of work is to be performed for halves and quarters of the severall houres How the Zodiac is to be placed and limited IT is the same work with that which went before Pag. 185. yet I will here insert it again First work thus As the Radius is to the Tangent of 23½ gr So let E F which was formerly found to be the Sine of the Equinoctials altitude above the Horizon 38 gr 30 mn in respect of the Radius R 6 and must now again be here used be to a fourth quantity which suppose to be F W. Take therefore that length F W and set it from R to X on the North side in this example and through X draw X γ parallel to the Subindicall line R S. Then say again As the Radius is to the Tangent of 23 gr and an half So let R C which was before found to be the Sine of the Latitude of the place 51 gr and an half in respect of the Radius R 6 and comes now to be here used again be to a fourth quantity which suppose to be F z. Take therefore F z and set it upon the line formerly drawn through X from X contrary to the going of the Subindical line namely Southward in this example I say from X to γ Then from γ through R draw γ R π and make R π equall to R γ So shall γ π be the Zodiac both placed rightly and justly limited For division of it you must conceive that R γ or R π is a Tangent of 23 and an half gr the Suns greatest declination and consequently you may by it set on either 12 moneths or 12 signes as you will the manner whereof hath been often shewed in this Book and especially Pag. 10 11. and Pag. 14 15 c. so that now all this work is come to an end The Index must slide to and fro upon the line γ π or the Index standing the Ellipsis must move after the same rate γ π is the proper Meridian of the Index And if your work when you make this Diall according to this case here before put be exactly true you shall finde the Zodiacall line π γ to fall just into the houre lines of 5 and 7 as they ought to do because the difference of the Indexes Longitude was assumed to be 75 gr from the South Meridian of the place which is the same with those houres The signe ♋ must be placed at γ and ♈ at π in this Diall The Index must lie when it is in the Center R just over the line R S and must alwayes move in a parallel position to it self And it must in this example recline 45 gr or rise above the Horizon or above R S making an angle above that line 45 gr also But though it lie thus yet it must move in the line γ π and it must be so set on that when you project the fiduciall edge of it into a point as workmen use to do when with their eye they tr●e the straightnesse of a line all that edge may at once justly appear to run into the line γ π wherever the Index be set in the Zodiac This is to be done carefully I have here set two of them whose proper Meridians I mean proper to the Index which are the lines of their Zodiacs viz. γ π do not lie in one and the same right line or parallelism which is the samenesse of po●ition but make angles one to the other This is done because they should not lie in one Meridian for by that meanes they will alwayes set one another which the Horizontal Diall with the single Ellipsis to an upright Index will not do at all times These two I say will set themselves and if you adde a common Horizontall Diall more to stand between them as you see done there they will be the more specious and usefull for setting each other I have set that whose Index looks North-east upon the left hand you may set it on the right hand if you will and the other on the left For this makes no materiall change The Index is best to be a threed Other circumstances I remember none but what the Workman will be well able to go through Another way for the description of an Ellipticall Diall upon the Horizontall plain of 51 gr 30 min. Latitude whose Zodiac and the motion of the Index is performed upon the ●oures of 7 in the morning and of 5 in the afternoon as the same houres are drawn upon the common Horizontall Diall THis way was formerly written and though it need not to have been here placed in respect the way last given is the best yet because it doth somwhat differ from what was set down before I thought best not to omit it because it is delightfull to see differ●●t way ●● meet both in one effect 1. First you are to get this way the altitudes of each houre point in the Equino●tiall above the Horizon by this R●le As the Radius to the Sine of 38 gr 30 min. So ●he Sines of 75 60 45 30 15 gr to