two points found you may make many points at pleasure whereunto the said thread may also interpose which for more conveniency may be made at every angle or bending of the Wall or Cieling be they never so many So that if lines be drawn from point to point that said reflected hour-line will be also exactly drawn In like manner may the other hour-lines be drawn so that the Reflex or spot of the Sun from the said Horizontal Glasse scituated in the said window as before shining amongst the said reflected hour-lines drawn on the wall or Cieling will exactly shew the hour of the day desired Now if lines be drawn round about the said Room equal to the Horizon of the said Glasse it will shew when the Sun is in or neer the Horizon To draw the Aequator and Tropicks on any Wall or Cieling to any Horizontal reflecting Glasse 1 To draw the Reflected Aequator or equinoctial-Equinoctial-line on the Wall or Cieling which represents a great Circle TAke the thread fixed in the Center of the Glasse and move the end thereof to and fro in the meridian line drawn on the Cieling untill by help of a Quadrant the said thread be elevated equal to the complement of the Latitude which will be alwayes perpendicular to the reversed Axis marking in the Meridian where the end of that thread falls then on that point and the said meridian line on the Cieling erect a perpendicular line which line may be continued on any plane whatsoever and is the reflected Equinoctial line desired Note that all great Circles are right lines are alwayes drawn or projected from a right line 2. To draw the Tropicks Note that all Parallels of Declination are lesser Circles and are Conick Sections FIrst make or take out of some Book a Table of the Suns Altitude for each hour of the day calculated for the place or Latitude proposed when the Sun is in either of the Tropicks Then take the thread fixed in the center of the Glasse and by applying one side of a quadrant to the said thread and moving one end of it to and fro in the hour-hour-line proposed elevate the said thread answerable to the Suns height in that hour when he is in that Tropick you desire to draw and mark where the end of that thread so elevated toucheth in that hour-line proposed So may you in like manner finde a several point in each hour-line for the Suns height in that Tropick whereby a line may be drawn on the Wall or Cieling from point to point formerly made in the said hour-lines which the Tropick desired In like manner may any parallel of Declination be drawn If there be first calculated a Table of the Suns altitude at all hours of the day when the Sun hath any Declination proposed whereby may be drawn either the Parallels of the Suns place or the parallels of the length of the day To draw the parallels of Declination to any Reflected Glasse most easily by help of a Trigon first made on past board or other material FIx the Trigon to the reflected roversed Axis so that the center of the Trigon may be in the center of the Glasse then will the Equinoctial on the Trigon be perpendicular to the said Axis then take the thread fixed in the center of the Glasse and lay it along either of the Tropicks or other parallels of Declination required which is drawn on the said Trigon which thread must be continued so that the end thereof may touch any hour-line and on that hour-line mark the point of touch the thread being still laid on the same parallel of declination on the Trigon in the same manner finde a point in each hour-line Lastly draw a line by those points so found which will be the Tropick-line or other parallel of declination as the thread was laid on on the Trigon To draw the Azimuth-lines on any Wall or Cieling to any Horizontal reflecting Glasse Note that all Azimuths are great Circles FIrst find a vertical point either above to the Zenith or below to the Nadir of the Glasse by some called a perpendicular or plumb line and mark in what point it cuts the floor of the room which point I call the reflected vertical point wherein the end of a thread is to be fixed For by a point found in the reflected Axis of the Horizon the Azimuths may be drawn as by a point found in the reflected Axis of the Equinoctial the hour-lines may be drawn Then on pastboard or other material draw the points of the Compasse or other degrees and fix the center thereof in the center of the Glasse and the meridian thereof in the meridian of the world as was shewn in drawing the hour-lines being careful to place it horizontal Then take the thread fixed in the place of the glasse and draw it over any Azimuth which is desired to be drawn and at the further side of the Room fasten that thread with a small nail as it was in drawing the reflected hour-hour-lines Then take the thread whose end is fastened in the said reflect vertical point and bring that thread so as just to touch the said horizontal thread and augment it until the end thereof touch the wall or Cieling and there make a mark or point In like manner move the said thread whose end is fastened in the said vertical point higher or lower at pleasure till as formerly it touch the said horizontal thread and mark again whereabouts the end thereof toucheth the said Wall or Cieling Now by help of these two points found in the reflected Azimuth line the whole Azimuth line may be drawn for if as before in drawing the Hour-lines a thread be so scituated that it may interpose between the eye and the said two points you may make many points at pleasure to which the said thread so situated may also interpose which may be made at every angle or bending of the wall or Cieling as before whereby the reflected Azimuth-line desired may be drawn In like manner may the other reflected Azimuth lines be drawn Also there may be lines drawn parallel to the Horizon round about the room by help of the thread fixed in the center of the Glasse and a Quadrant for the elevation thereof which will shew the Suns altitude at any appearance thereof Thus have I shewed the drawing of a Reflected Dial from an Horizontal Glasse with all the usual furniture thereon though the wall or place on which it is to be drawn be never so gibous or irregular or in what shape soever Now the Glasse may be exactly situated Horizontal if you draw a reflected parallel for the present day and know also the true hour and so place the Glasse that the spot or reflex of the Sun may fall thereon on the Cieling for there is no way by an Instrument to do it the Glasse is so small Of Reclining Reflecting Glasses Reflected Dialling from any Reclining Glasse I shall now shew how to draw
will be reflected on By help of an ordinary Horizontal Dial for that Latitude FIrst extend several threads from the center of the Glasse to the extremity of the Reflected Horizon in the Room which for more conveniency and use may be the several hour-lines and may also serve as a bed to situate the Horizontal Diall on the Reflected Horizon having regard to situate the center of the Dial on the center of the Glasse and the Meridian of that Dial on the Reflected Meridian of the World Then to finde the point in the Reflected reversed Axis on the floor of the Room Take a thread one end thereof being fastened in the center of the Glasse and move the other end thereof to and fro in the reflected meridian under the Reflected Horizon until by help of a Quadrant the said thread is found to be depressed under the reflected Horizon equal to the latitude of the place and where the end of the said thread intersects or meets the Reflected Meridian either on the floor or wall that point is the reflected reversed Axis as was required In which point fasten one end of a thread which thread will be of great use in drawing the reflected hour-hour-lines on any wall or Cieling whatsoever Now if this thread whose end is fastened in a point on the reflected reversed Axis be taken and brought to touch any part of any one of the threads of the hour-hour-lines produced to and fastened in the reflected Horizon the said thread being continued so as the end thereof may touch the wall or Cieling and also any part of the said thread touch the hour-hour-line or thread proposed that point on the wall or Cieling is in the reflected hour-hour-line desired to be drawn Also the other point in the same reflected hour-hour-line may be found If the said thread whose end is fastened in the Reflected Axis be brought to touch some other part of the same hour-thread proposed so that when as before the end of the said thread toucheth the wall or Cieling some part of that thread may also touch the hour-hour-line desired which point of touch on the wall or Cieling is also another point in the said reflected hour-line desired By which two points so found as before the reflected hour-line may be drawn by a thread projecting by those points from the eye as it was formerly directed in drawing the reflected hour-hour-lines to an Horizontal Glasse To draw the Reflected Equinoctial line and also the Tropicks on any wall or Cieling to any Reclining Reflecting glasse 1 To draw the reflected Equinoctial line on the Wall or Cieling TAke that thread whose end is fastened in the center of the reclining glasse and move the other end thereof to and fro in the said Reflected meridian formerly drawn until by help of a quadrant the said thread is elevated above the reflected Horizon formerly drawn equal to the Complement of the Latitude which as before will be alwayes perpendicular to the reversed Axis and make a point in the said reflected meridian where the end of the said thread toucheth then on that point and the said reflected meridian on the Cieling raise a perpendicular line which is the Reflected Equinoctial line desired 2. To draw the reflected Tropicks or other Parallels of Declination FIrst as before make or take out of some Book a Table of the Suns Altitude for each hour of the day calculated for the place or Latitude proposed when the Sun is in either of the Tropicks or other parallel of Declination then take that thread whose end is fastened in the center of the Glasse move the other end thereof to and fro in the hour-line proposed until by applying one side of a quadrant to the said thread you find the said thread elevated above the reflected Horizon answerable to the Suns height in that hour proposed when he is in that Tropick or degree of Declination proposed Which altitude required will be found in the foresaid Table for that end calculated which said thread being of the elevation above the reflected Horizon as the said Table directeth then mark where the end of the thread so elevated toucheth the Wall or Cieling in that hour-hour-line so is one point found in the reflected parallel of Declination desired to be drawn In like manner find in the said Table in the same parallel or degree of declination what altitude the Sun hath at the next hour and elevate the said thread whose end is fastened in the center of the Glasse equal to the Suns altitude in that hour above the said reflected Horizon by help of the said Quadrant and where the other end of the said thread falleth in the hour-line proposed make another mark or point And so in like manner make the points belonging to that parallel of Declination in the remaining hour-lines according to the several Altitudes found in the said Table of Altitudes Then drawing by hand a line to passe through those several points so found as before which line is the reflected parallel of the Suns declination desired In like manner may be drawn all or any other parallel of Declination which may have respect to the Suns place or the length of the day as shall be desired Or To draw the said reflected Tropicks or other parallels of Declination without any Tables calculated only by help of a Trigon first made on pastboard or other material Note that all Parallels are lesser Circles FIrst as formerly is shewd in drawing the parallels of Delination to a Reflecting Horizontal Glasse fasten the Trigon on the reflected reversed Axis so that the center of the Trigon may be in the center of the Glasse then also will the Equinoctial on the Trigon be perpendicular to the said reflected reversed Axis then take the thread fixed in the center of the said Glasse which is also in the center of the Trigon and lay it upon that parallel of Declination drawn on the said Trigon whose reflected parallel is required to be drawn on the plane or Cieling then move the Trigon the thread lying on the said parallel until the end of the said thread touch any hour-hour-line on the said wall or Cieling in which point of touch on that hour-hour-line make a mark so will that point be in the reflected parallel of Declination desired In like manner move the said Trigon still keeping the thread on the same parallel until the end of that thread touch another hour-hour-line on the said plane or Cieling and there also make another mark And so in like manner find a point in each hour-line through which that reflected parallel must passe then drawing a line to passe through those several points on the said plane or Cieling which line is the reflected parallel of the Suns Declination desired In like manner may be drawn any other reflected parallel of Declination required To draw the reflected Azimuth-lines to any reclining Glasse on any plane whatsoever that the Sun-beams will be reflected on Here note that Azimuths
are great Circles FIrst know that the reflected vertical point in the Axis of the Reflected Horizon will alwayes be found in the reflected meridian And look how many degrees the reflected Horizon differs from the direct Horizon so many must the reflected Axis of the Horizon differ from the direct Axis of the Horizon Hence the reflected vertical point whereby the reflected azimuth-Azimuth-lines are drawn may be thus found Take that thread whose end is fixed in the center of the Glasse and move the other end thereof to fro in the reflected meridian until by applying one side of a quadrant thereto you find the said thread depressed just 90 degrees or perpendicular under the reflected Horizon then make a mark or point where the other end of the said thread toucheth the said reflected Meridian on the Wall Ground or Floor of the Room which point so found is the reflected vertical point desired in which point fasten one end of a thread Then on pastboard or other material draw the points of the Compasse or other degrees placing the center thereof in the center of the Glasse and the meridian thereof in the reflected meridian of the world which said pastboard must be also situated in the reflected Horizon just as the Horizontal Dial was formerly directed to be situated for drawing the reflected hour-lines And as the threads from the center fastened in the reflected Horizon were also the hour-lines on the Horizontal Diall whereby the reflected hour-hour-lines were drawn So now the threads from the center fastened in the Reflected Horizon may be the Horizontal Azimuth lines whereby the reflected azimuth-Azimuth-lines may be drawn Or if that thread which fastned in the center of the glass be drawn exactly over any azimuth-Azimuth-line the end whereof being fastened by a nail or other means in the reflected Horizon on the other side of the Room there may several points be found in the wall or Cieling through which the reflected Azimuth line must passe as followeth Take that thread one end of which is fastened in the said vertical point and bring it just to touch the Azimuth thread formerly fastened and continue it until the end thereof touch the wall or Cieling and also the thread it self touch the said Azimuth it self as before in which point of touch on the wall or Cieling make a mark through which point that reflected Azimuth-line must passe Then move the said string fastened in the said vertical point so that it may just touch the said thread again but in another place then as before continue that thread untill the end thereof touch the wall or Cieling again as before and there make another mark through which the said reflected Azimuth line must also passe In like manner may more points be found for your further guide in drawing that azimuth-Azimuth-line But two points being found will be sufficient To draw any reflected line by any two points given over any plane whatsoever without projecting by the eye FAsten two threads in the place of the center of the said reclining Glasse drawing the said threads straight fastening each of the other ends in the two reflected Azimuth-points formerly found on the wall or Cieling Then situate a thread cross or thwart the room so as it may crosse those other threads from the center neer at right angles and also just touch both of them in that situation By which said thread crosse the room may any number of points in the said reflected azimuth-Azimuth-line to be drawn be found at pleasure For if the end of another thread be also fastened in the center of the said Glasse making the other end thereof to touch the wall or Cieling but so that it may also just touch the said thread which is fastened crosse the room which point of touch on the said wall or Cieling is another point in the said reflected Azimuth line required to be drawn In like manner may more points be found at every angle or bending of the wall or Cieling for the exacter drawing the reflected Azimuth line required which doth find points whereby is drawn the same reflected Azimuth line or other lines as was formerly done by a thread so situated that it may interpose between the eye and any two points assigned on the wall or Cieling In like manner if the thread fastened on the further side of the room were removed on another Azimuth line on the said pastboard and then fasten it again on the further side of the room as before you may by help of the said thread fastened in the said vertical point find several points on the wall or Cieling through which that Azimuth-line will passe So may you either by this or the former way draw what Azimuth lines you please either in points of the Mariners Compasse or degrees as you please by drawing it first on pastboard as before is directed And note generally that such relation the point found on the floor or ground in the reflected reversed Axis hath to the hour-lines drawn on the Horizontal Dial in drawing the reflected hour-lines The same hath the Reflected vertical point found on the floor or ground to the Azimuths drawn on the pastboard in drawing the reflected Azimuth-lines To draw the reflected parallels of the Suns altitude or proportions of shadows to any reclining Glasse on any Plane whatsoever that the Sun-beams will be reflected on Here note that parallels of Altitude are lesser Circles therefore are not represented by a right line FIrst know generally that what respect the parallels of Declination have to the hour-hour-lines such have the parallels of Altitude to the Azimuths For if one end of a thread be fastened in the place of the center of the reclining Glasse and the other end moved to and fro in any reflected Azimuth line until the said thread be elevated any number of degrees proposed above the reflected Horizon the Elevation of which thread being found by applying a Quadrant thereto and making a mark or point where the end of the said thread toucheth the said reflected Azimuth drawn on the wall or Cieling that point so found is the point through which that Almicanâ—er or reflected parallel of the Suns altitude must passe In like manner remove the other end of the said thread fastned in the center of the Glasse to another reflected azimuth-Azimuth-line and as before move it higher or lower untill by applying the edge of a quadrant to that thread you find the said thread above the reflected Horizon the same number of degrees first proposed and at the end of the said thread in that Reflected azimuth-Azimuth-line drawn on the wall or Cieling I make another mark or point through which the same Reflected Almicanter or parallel of Altitude must also passe And so in like manner I find a point on each reflected Azimuth-line through which the same parallel of Altitude must passe Then drawing by hand a line to passe through these several points so found as before that line is the Reflected parallel
sine and enter the former extent between the Scale and the Thread and the foot of the Compasses will on the Line of equal parts shew the fourth Proportional The Proportion for finding the Altitude of a Tower at one Station by the measured distance may also be wrought in in equal parts and Sines For As the Cosine of the Ark at first Station To the measured distance thereof from the Tower So is the Sine of the said Ark To the Altitude of the Tower In that former Scheme the measured distance B H is 85 and the angle observed at H 48 d 29′ Wherefore I lay the Thread to the Sine of the said Ark in the Limb counted from the right edge and from the measured distance in the equal parts take the nearest extent to the Thread then laying the Thread to the Cosine of the said Ark in the Limb and entring the former extent between the Thread and the Scale I shall find the foot of the Compasses to fall upon 96 the Altitude sought So also in the Triangle A C B if there were given the side A C 194 the measured distance between two Stations on the Wall of a Town besieged and the observed angles at A 25 d 22′ at C 113 d 22′ if B were a Battery we might by this work find the distance of it from either A or C for having two angles given all the three are given it therefore holds As the Sine of the angle ot B 41d 16′ To its opposite side A C 194 So the Sine of the angle at C 66d 38′ the Complement To its Opposite side B A 270 the distance of the Battery from A Such Proportions as have the Radius in them will be more easily wrought we shall give some few Examples in Use in Navigation 1. To find how many Miles or Leagues in each Parralel of Latitude answer to one degree of Longitude As the Radius To the Cosine of the Latitude So the number of Miles in a degree in the Equinoctial To the Number of Miles in the Parralel So in 51 d 32′ of Latitude if 60 Miles answer to a degree in the Equinoctial 37 ‑ 3 Miles shall answer to one degree in this Parralel This is wrought by laying the Thread to 51 d 32′ in the Limb from the left edge towards the right then take the nearest distance to it from 60 in the equal parts which measured from the Center will be found to reach to 37 ‑ 3 as before The reason of this facil Operation is because the nearest distance from the end of the Line of equal parts to the Thread is equal to the Cosine of the Latitude the Scale it self being equal to the Radius and therefore needs not be taken out of a Scale of Sines and entred upon the first Tearm the Radius as in other Proportions in Sines of of the greater to the less when wrought upon a single Line only issuing from the Center where the second Tearm must be taken out of a Scale and entred upon the first Tearm 2. The Course and Distance given to find the difference of Latitude in Leagues or Miles As the Radius To the Cosine of the Rumb from the Meridian So the Distance sailed To the difference of Latitude in like parts Example A Ship sailed S W by W that is on a Rumb 56 d 15′ from the Meridian 60 Miles the difference of Latitude in Miles will be found to be 33 ‑ 3 the Operation being all one with the former Lay the Thread to the Rumb in the Limb and from 60 take the nearest distance to it which measured in the Scale of equal parts will be found as before 3. The Course and Distance given to find the Departure from the Meridian alias the Variation As the Radius To the Sine of the Rumb from the Meridian So the distance Sailed To the Departure from the Meridian In the former Example to find the Departure from the Meridian Lay the Thread to the Rumb counted from the right edge towards the left that is to 56d 15′ so counted and from 60 in the equal parts being the Miles Sailed take the nearest distance to it this extent measured in the said Scale will be found to be 49 ‑ 9 Miles and so if the converse of this were to be wrought it is evident that the Miles of Departure must be taken out of the Scale of equal parts and entred Parralelly between the Scale and the Thread lying over the Rumb Many more Examples and Propositions might be illustrated but these are sufficient those that would use a Quadrant for this purpose may have the Rumbs traced out or prickt upon the Limb Now we repair to the backside of the Quadrant Of the Line of on the right Edge of the Backside THe Uses of this Line are manifold in Dyalling in drawing Projections in working Proportions c. 1. To take of a Proportional Sine to any lesser Radius then the side of the Quadrant or which is all one to divide any Line shorter in length then the whole Line of Sines in such manner as the same is divided Enter the length of the Line proposed at 90 d the end of the Scale of Sines and to the other foot lay the Thread according to nearest Distance or measure the length of the Line proposed on the Line of Sines from the Center and observe to what Sine it is equal then lay the Thread over the like Arch in the Limb and the nearest distances to it from each degree of the Line of Sines shall be the Proportional parts sought And if the Thread be laid over 30 d of the Limbe the nearest distances to it will be Sines to half the Common Radius 2. From a Line of Sines to take off a Tangent the Proportion to do it is As the Cosine of an Arch To the Radius of the Line proposed So the Sine of the said Arch To the Tangent of the said Arch. Enter the Radius of the Tangent proposed at the Cosine of the given Arch and to the other foot lay the Thread then from the Sine of that Arch take the nearest distance to the Thread this extent is the length of the Tangent sought thus to get the Tangent of 20 d enter the Radius proposed at the Sine of 70 d then take the nearest distance to the Thread from the Sine of 20 d this extent is the Tangent of the said Arch in reference to the limited Radius Otherways by the Limb. Lay the Thread to the Sine of that Arch counted from the right edge whereto you would take out a Tangent and enter the Radius proposed down the Line of Sines from the Center and take the nearest distance to the Thread then lay the Thread to the like Arch from the left edge and enter the extent between the Scale and the Thread the distance of the Foot of the Compasses from the Center shall be the length of the
Sine of 13d and enter one foot of it on the Sine of 38d 28′ and to the other foot lay the Thread and in the Limb it shews the Amplitude sought to be 21d 12′ By changing the places of the two middle Tearms this Example will be turned into a Parralel entrance Lay the Thread to the Complement of the Latitude in the Limb and enter the Sine of the Declination between it and the Scale and you will find the same Ark in the Sines for the Amplitude sought as was before found in the Limb. Such Proportions of the greater to the less wherein the Radius is not ingredient that have two fixed or constant Tearms may be most readily performed by the single Line of Sines without the help of the Limb. An Example for finding the Suns Amplitude As the Cosine of the Latitude To the Sine of the Suns greatest declination So the Sine of the Suns distance from the next Equinoctial Point To the Sine of the Suns Amplitude Because the two first Tearms of this Proportion are fixed the Amplitude answerable to every degree of the Suns place may be found without removing the Thread To do it enter the Sine of the Suns greatest Declination 23d 31′ at the Sine of the Latitudes Complement and to the other foot lay the Thread where keep it without alteration then for every degree of the Suns place counted in the Sines take the nearest distance to the Thread and measure those extents down the Line of Sines from the Center and you will find the correspondent Amplitudes Example So when the Sun enters ♉ ♠♠♓ his Equinoctial distance being 30 d the Amplitude will be 18 d 41′ and when he enters ♊ ♌ ♠♒ Equinox distance 60 d the Amplitude will be 33 d 42′ and when he enters ♋ ♑ the greatest Amplitude will be 39d 50′ his distance from the nearest Equinoctial Point being 90 d. But for such Proportions in which there is not two fixed Tearms the best way to Operate them will be by the joint help of the Limbe and Line of Sines An Example for finding the Time of the day the Suns Azimuth Declination and Altitude being given By the Suns Azimuth is meant the angle thereof from the midnight part of the Meridian the Proportion is As the Cosine of the Declination To the Sine of the Azimuth So the Cosine of the Suns Altitude To the Sine of the hour from the Meridian Example So when the Sun hath 18 d 37′ North Declination if his Azimuth be 69 d from the Meridian and the Altitude 39 d the hour will be found to be 49 d 58′ from Noon So if there were given the Hour the Declination and Altitude by transposing the Order of the former Proportion it will hold to find the Azimuth As the Cosine of the Suns Altitude To the Sine of the hour from the Meridian So the Cosine of the Suns Declination To the Sine of the Azimuth from the Meridian Commonly in both these Cases the Latitude is also known and the Affection is to be determined according to Rules formerly given A Proportion wholly in Secants we have shewed before may be changed wholly into Sines but the like mutual conversion of the Sines into Tangents is not yet known however it may be done in 〈◊〉 of the 16 Cases wherein the Radius is ingredient for instance let the Proportion be to find the time of Sun rising As Radius To Tangent of Latitude So the Tangent of the Declination To the Sine of the hour from 6. Instead of the two first Tearms it may be As the Cosine of the Latitude To the Sine of the Latitude then instead of the Tangent of the Declination say So is the Sine hereof to a fourth Again As the Cosine of the Declination To that fourth So Radius To the Sine of the hour from six This being derived from the Analemm◠by resolving a Triangle one side whereof is the Arch of a lesser Circle If a Quadrant want Tangents or Secants in the Limb but may admit of a Sine from the Center the Tangent and Secant of the Latitude c may be taken out by what hath been asserted to half the common Radius and marked on the Limb and the Quadrant thereby fitted to perform most of the Propositions of the Sphoere in one Latitude and how to supply the Defect of a Line of Versed Sines in the Limb shall afterwards be shewne What hath been spoken concerning a Line of Sines graduated on a Quadrant from the Center may by help of the equal Limb be performed without it 1. A Proportional Sine may be taken off to any diminutive Radius By the Definition of Sines the right Sine of an Arch is a Line falling from the end of that Arch Perpendicularly to the Radius drawn to the other end of the said Arch So the Line H K falling Perpendicularly on the Radius F G shall be the Sine of the Arch H G and by the same Definition the Line G I falling perpendicularly on the Radius F H shall also be the Sine of the said Arch and whether the Radius be bigger or lesser this Definition is common but the Line G I on a Quadrant represents the nearest distance from the Radius to the Thread therefore a Sine may be taken off from the Limb to any Diminutive Radius to perform which Enter the length or Radius proposed down the streight Line that comes from the Center of the Quadrant and limits the Limb observe where the Compasses rests this I call the fixed Point because the Compasses must be set down at it at every taking off then to take off the Sine of any Arch to that Radius lay the Thread over the Arch counted in the Limb from the said edge of the Quadrant and take the nearest distance to it for the length of the Sine sought But to take out Sines to the Radius of the graduated Limb set down one foot at the Ark in the Limb and take the nearest distance to the two edge Lines of the Limb the one shall be the Sine the other Co-sine of the said Ark. 2. A Proportion in Sines alone may be wrought by help of the Limbe Take out one of the middle Tearms by the former Prop. and entring it down the right edge from the Center take the nearest distance to the Thread laid over the other middle Tearm in the Limbe counted from right edge then lay the Thread to the first Tearm in the Limb and enter that extent between the right edge Line and the Thread the distance of the foot of the Compasses from the Center is the length of the Sine sought to be measured in the Limb by entring one foot of that Extent in it So that the other turned about may but just touch one of the edge or side Lines of the Limb issuing from the Center or enter that Extent at the concurrence of the Limbe with the
equidistant one from another but having determined the distance between the two Extream Latitudes to which they are fitted for the the larger sine it will hold As the difference of the Secants of the two extream Latitudes It to the distance between the Lines fitted thereto So is the difference of the Secants of the lesser extream Latitude and any other intermediate Latitude To the distance thereof from the lesser extream And so for the lesser sine continued the other way having placed the two Extreams under the two former Extreams to place the imtermediate Lines the Canon would be As the difference of the sines of the two extream Latitudes Is to the distance between the Lines fitted thereto So is the difference of the sines of the lesser extream Latitude and of any other intermediate Latitude To the distance thereof from the lesser Extream Having fitted the distances of the greater sine streight Lines drawn through the two extream sines shall divide the intermediate Parralels also into Lines of sines proper to the Latitudes to which they are fitted Now for the lesser sines they are continued the other way at the ends of the former Parralells the Line proper to each Latitude should be divided into a Line of sines whose Radius should be equal to the sine of the Latitude of the other sine whereto it is fitted and so Lines traced through each degree to the Extreams but by reason of the small distance of these Lines the difference is so exceeding small that it may not be scrupled to draw Lines Diagonal wise from each degree of the two outward extream Sines for being drawn true they will not be perceived to be any other then streight Lines Whereas these Lines by reason of the latter Proportion should not fall absolutely to be drawn at the ends of the former Lines whereto they are fitted and then they would not be so fit for the purpose yet the difference being as we said so insensible that it cannot be scaled they are notwithstanding there placed and crossed with Diagonals drawn through each degree of the Extreams The Vses of the Diagonal Scale 1. To find the time of Sun rising or setting In the Parralel proper to the Latitude take out the Suns Declination out of the lesser continued sines and enter one foot of this extent at the Complement of the Declination in the Line of sines and in the equal Limb the Thread being laid to the other foot will shew the time sought In the Latitude of York namely 54d if the Sun have 20d of Declination Northward he rises at 4 and sets at 8 Southward he rises at 8 and sets at 4 2. To find the Hour of the Day or Night for South Declination In the Parralel proper to the Latitude account the Declination in the lesser continued sine and the Altitude in the greater sine and take their distance which extent apply as before to the Cosine of the Declination in the Line of sines on the Quadrant and laying the Thread to the other foot according to nearest distance it shews the time sought in the equal Limbe Thus in the Latitude of York when the Sun hath 20d of South declination his Altitude being 5d the hour from noon will be found 45 minutes past 8 in the morning or 15 minutes past 3 in the afternoon feré For North Declination The Declination must be taken out of the lesser sine in the proper Parralel and turned upward on the greater sine and there it shews the Altitude at six for the Sun or any Stars in the Northern Hemispere the distance between which Point and the given Altitude must be entred as before at the Cosine of the declination laying the thread to the other foot and it shews the hour in the Limb from six towards noon or midnight according as the Sun or Stars Altitude was greater or lesser then its Altitude at six So in the Latitude of York when the Sun hath 20d of North declination if his Altitude be 40d the hour will be 46 minutes past 8 in the morning or 14 minutes past 3 in the afternoon 4. The Converse of the former Proposition will be to find the Altitude of the Sun at any hour of the day or of any Star at any hour of the night I need not insist on this having shewn the manner of it on the small quadrant only for these Scales use the Limb instead of the lesser sines for Stars the time of the night must first be turned into the Stars hour and then the Work the same as for the Sun 5. To find the Amplitude of ehe Sun or Stars Take out the Declination out of the greater sine in the Parralel proper to the Latitude and measure it on the Line of sines on the lesser Quadrant and it shews the Amplitude sought So in the Latitude of York 54d when the Sun hath 20d of Declination his Amplitude will be 35d 35′ 6. To find the Azimuth for the Sun or any Stars in the Hemisphere For South Declination Account the Altitude in the lesser sine continued in the proper Parralel and the Declination in the greater sine and take their distance enter one foot of this extent at the Cosine of the Altitude on the Quadrant and lay the Thread to the other according to nearest distance and in the Limbe it shews the Azimuth from East or West Southwards So in the Latitude of York when the Sun hath 20d of South Declination his Altitude being 5d the Azimuth will be found to be 44d 47′ to the Southwards of the East or West For North Declination Account the Altitude in the lesser sine continued and apply it upward on the greater sine and it finds a Point thereon from whence take the distance to the declination in the said greater sine in the Parralel proper to the Latitude of the place and enter one foot of this Extent at the Cosine of the Altitude on the Line of sines and the Thread being laid to the other foot according to nearest distance shews the Azimuth in the Limbe from East or West So in the Latitude of York when the Sun hath 20d of North Declination and 40d of Altitude his Azimuth will be 23d 16′ to the Southwards of the East or West When the Hour or Azimuth falls near Noon for more certainty you may lay the Thread to the Complement of the Declination for the Hour or the Complement of the Altitude for the Azimuth in the Limbe and enter the respective extents Parralelly between the Thread and the Sines and find the answer in the sines We might have fitted one Scale on the quadrant to give both the houre and Azimuth in the Equall Limb by a Lateral entrance and have enlarged upon many more Propositions which shall be handled in the great Quadrants Mr Sutton was willing to add a Backside to this Scale and therefore hath put on particular Scales of his own for giving the requisites of an upright Decliner
six we may educe a single Proportion applyable to the Logarithms without natural Tables for Calculating the Hour of the day to all Altitudes By turning the third Tearm being a difference of Sines or Versed Sines into a Rectangle and freeing it from affection The two first Proportions to be wrought are fixed for one Declination The first will be to find the Suns Altitude or Depression at six The second will be to find half the difference of the Sines of the Suns Meridian Altitude and Altitude sought c. as before defined the Proportion to find it is As the Secant of 60d To the Cosine of the Declination So is the Cosine of the Latitude To the Sine of a fourth Arch. Lastly To find the Hour Get the sum and difference of half the Suns Zenith distance at the hour of six and of half his Zenith distance to any other proposed Altitude or Depression Then As the Sine of the fourth Arch Is to the Sine of the sum So is the Sine of the difference To the Sine of the hour from six towards Noon or Midnight according as the Altitude or Depression was greater or lesser then the Altitude or Depression at six Observing that the Sine of an Arch greater then a Quadrant is the Sine of that Arks Complement to a Semicircle Of the Stars placed upon the Quadrant below the Projection ALL the Stars placed upon the Projection are such as fall between the Tropicks and the Hour may be found by them with the Projection as in the Use of the small Quadrant Which may also be found by the fitted particular Scale not only for Stars within the Tropicks but for all others without when their Altitude is less then 62d and likewise their Azimuth may be thereby found when their Declination is not more then 62d. For other Stars without the Tropicks they may be put on below the Projection any where in such an angle that the Thread laid over the Star shall shew an Ark in the Limb at which in the Sines the Point of entrance will always fall And again the same Star is to be graved at its Altitude or Depression at six in the Sines and then to find the Stars hour in that Latitude whereto they are fitted will always for Northern Stars be to take the distance in the Line of Sines between the Star and its given Altitude and to enter that Extent at the Point of entrance laying the Thread to the other foot according to nearest distance and it gives the Stars hour in the equal Limb from six which may also be found in the Sines by a Parrallel entrance laying the Thread over the Star Example Let the Altitude of the last in the end of the great Bears Tail be 63d take the distance between it and the Star which is graved at 37d 30′ of the Sines the said Extent entred at the Sine of 23d the Ark of the Limb the Thread intersects when it lies over the said Star and by laying the Thread to the other foot you will find that Stars hour to be 46d 11′ from six towards Noon Meridian if the Altitude increase and in finding the true time of the night the Stars hour must be always reckoned from the Meridian it was last upon in this Example it will be 5 minutes past 9 feré Of the Quadrant of Ascensions on the backside This Quadrant is divided into 24 Hours with their quarters and subdivisions and serves to give the right Ascension of a Star as in the small Quadrant to be cast up by the Pen. It also serves to find the true Hour of the night with Compasses First having found the Stars hour take the distance on the Quadrant of Ascensions in the same 12 hours between the Star and the Suns Ascension given by the foreside of the Quadrant the said Extent shall reach from the Stars hour to the true hour of the night and the foot of the Compasses always fall upon the Quadrant Which Extent must be applyed the same way it was taken the Suns foot to the Stars hour Example If upon the 30th of December the last in the end of the Bears Tail were found to be 9 hours 05′ past the Meridian it was last upon the true time sought would be 16 minutes past 3 in the morning Another Example for the Bulls Eye Admit the Altitude of that Star be 39 d that Stars hour as we found it by the Line of Versed Sines was 3 ho 3′ from the Meridian if the Altitude increase then that Stars hour from the Meridian it was last upon was 57 minutes past 8 8 h 57′ If this Observation were upon the 23d of October the Complement of the Suns Ascension would be 9 30 The Ascension of that Star is 4 16 The true time of the night would be forty 10 43 three minutes past ten The distance between the Star and the Suns Ascension being applyed the same way by setting the Sun foot at the Stars hour will shew the true time sought When the Star is past the Meridian having the same Altitude the Stars hour will be 3′ past 3 and the true time sought will be 49′ past 4 in the next morning The Geometrical Construction of Mr Fosters Circle THe Circle on the Back side of the Quadrant whereof one quarter is only a void Line is derived from M. Foster's Treatise of a Quadrant by him published in Anâ—o 1638. the foundation and use whereof being concealed I shall therefore endeavour to explain it Upon the Center H describe a circle and draw the Diameter A C passing through the Center and perpendicularly thereto upon the point C erect a Line of Sines C I whose Radius shall be equal to the Diameter A C let 90d of the Sine end at I I say then if from the point A through each degree of that Line of Sines there be streight lines drawn intersecting the Quadrant of the circle C G as a line from the point D doth intersect it at B the Quadrant C G which the Author calls the upper Quadrant or Quadrant of Latitudes shall be constituted and if C I be continued as a Secant by the same reason the whole Semicircle C G A may be occupied hence it will be necessary to educe a ground of calculation for the accurate dividing of the said Quadrant and that will be easie for A C being Radius the Sine C D doth also represent the Tangent of the Angle at A therefore seek the natural Sine of the Ark C D in the Table of Natural Tangents and the Ark corresponding thereto will give the quantity of the Angle D A C then because the point A falls in the circumference of the Circle where an Angle is but half so much as it is at the Center by 31 Prop. 3. Euc. double the Angle found and from a Quadrant divided into 90 equal parts and their subdivisions by help of a Table so made may the Quadrant of Latitudes be accurately
it will be for there is given the Radius C D and the Tangent B E the two first Tearms of the Proportion with the Line C B the sum of the third and fourth Tearms to find out the said Tearms respectively and it will hold by compounding the Proportion As the sum of the first and second Tearm Is to the second Tearm So is the sum of the third and fourth Tearm To the fourth Tearm that is As C D + B E Is to B E So is C F + F B = C B To F B see 18 Prop. of 5 of Euclid or page 18 of the English Clavis Mathematicae of the famous and learned Mr Oughtred After the same manner is the Line Sol or Proportional Sines made that being also such a Line that any Ark being assumed in it to be a Sine the distance from that Ark to the other end of the Diameter shall be the Radius thereto A Demonstration to prove that the Line of Hours and Latitudes will jointly prick off the hour Diâ—tances in the same angles as if they were Calculated and prickt off by Chords Draw the two Lines A B and C B crossing one another at right angles at B and prick off B C the quantity of any Ark out of the line of Latitudes and then fit in the Scale of Hours so that one end of it meeting with the Point C the other may meet with the other Leg of the right angle at A from whence draw A E parralel to B C So A B being become Radius B C is the Sine of the Arch first prickt down from the line of Latitudes from the Point B through any Point in the line of Proportional Tangents at L draw the Line B L E and upon B with the Radius B A draw the Arch A D which measureth the Angle A B E to the same Radius I say there will then be a Proportion wrought and the said Arch measureth the quantity of the fourth Proportional the Proportion will be As the Radius To the Sine of the Ark prickt down from the Line of Latitudes So is any Tangent accounted in the Scale beginning at A To the Tangent of the fourth Proportional in the Schem it lies evident in the two opposite Triangles L C B and L A E by construction equiangled and consequently their sides Proportional Assuming A L to be the Tangent of any Ark L C becomes the Radius according to the prescribed construction of that Line it then lies evident As L C the Radius To C B the Sine of any Ark So is L A the Tangent of any Ark To A E the Tangent of the fourth Proportional Namely of the Angle A B E and therefore it pricks down the Hour-lines of a Dyal most readily and accurately the Proportion in pricking from the Substile being alwaies As the Radius To the Sine of the Stiles height So the Tangent of the Angle at the Pole To the Tangent of the Hour-line from the Substile Uses of the Graduated Circle To work Proportions in Tangents alone In any Proportion wherein the Radius is not ingredient it is supposed to be introduced by a double Operation and the Poportion will be As the first term To the second So the Radius to a fourth Again As the Radius is to that fourth So is the third Term given To the fourth Proportional sought In illustrating the matter I shall make use of that Theoremâ— for varying of Proportions that the Tangents of Arches and the Tangents of their Complements are in reciprocal Proportions As Tangent 23d to Tangent 35d So Tangent 55d to the Tangent of 67d. In working of this Proportion the last term may be found to the equal Semicircle or on the Diameter 1. In the Semicircle Extend the thred through 23d on the Diameter and through 3â— in the Semicircle and where it intersects the Circle on the opposite side there hold one end of it then extend the other part of it over 55 in the Diameter and in the Semicircle it will intersect 67d for the term sought 2. On the Diameter Extend the thred over 23d in the Semicircle and 35d on the Diameter and where it intersects the void circular line on the opposite side there hold it then laying the other end of it over 55 d in the Semicircle and it will cut 67 d on the Diameter If the Radius had been one of the terms in the Proportion the operation would have been the same if the Tangent of 45 d had been taken in stead of it To work Proportions in Sines and Tangents joyntly 1. If a Sine be sought the middle terms being of a different species Extend the thred through the first term on the Diameter being a Tangent and through the Sine being one of the middle terms counted in the unequal Quadrant and where it intersects the Opposite side of the Circle hold it then extend the thred over the Tangent being the other middle term counted on the Diameter and it will intersect the graduated Quadrant at the Sine sought Example If the Proportion were as the Tangent of 14d to the Sine of 29d So is the Tangent of 20d to a Sine the fourth Proportional would be found to be the Sine of 45d. 2. If a Tangent be sought the middle terms being of several kinds Extend the thred through the Sine in the upper Quadrant being the first term and through the Tangent on the Diameter being one of the other middle terms holding it at the Intersection of the Circle on the opposite side then lay the thred to the other middle term in the upper Quadrant and on the Diameter it shews the Tangent sought Example If the Suns Amplitude and Vertical Altitude were given the Proportion from the Analemma to find the Latitude would be As the Sine of the Amplitude to Radius So is the Sine of the Vertical Altitude To the Cotangent of the Latitude Let the Amplitude be 39d 54′ And the Suns Altitude being East or West 30 39′ Extend the thred through 39 54′ the Amplitude counted in the upper Quadrant and through 45d on the Diameter holding it at the intersection with the Circle on the Opposite side then lay the thred over 30d 39′ the Vertical Altitude and it will intersect the Diameter at 38d 28′ the Complement of the Latitude sought But Proportions derived from the 16 cases of right angled Spherical Triangles having the Radius ingredient will be wrought without any motion of the thred An Example for finding the Suns Azimuth at the Hour of 6. As the Radius to the Cosine of the Latitude So the Tangent of the Declination To the Tangent of the Azimuth from the Vertical towards Midnight Meridian Extend the thred over the Complement of the Latitude in the upper Quadrant and over the Declination in the Semicircle and on the Diameter it shews the Azimuth sought So when the Sun hath 15d of Declination his
Quadrant of Latitudes whereto belongs the two Parrallel Lines of Sines in the opposite Quadrants the upermost being extended cross the Quadrant of Latitudes The Proportion not having the Radius ingredient and being of the greater to the less Account the first Tearm in the line Sol and the second in the upper Sine extending the Thread through them and where it intersects the opposite Parrallel hold it then lay the Thread to the third Tearm in the line Sol and it will intersect the fourth Proportional on the upper Parrallel As the Sine of 30d To the sine of any Arch So is the Cosine of that Arch To the sine of the double Arch and the Converse By trying this Canon the use of these Lines will be suddenly attained Example As the sine of 30d To the sine of 20d So is the sine of 70d To the sine of 40d. But if it be of the less to the greater the answer must be found on the Line Sol. Account the first Tearm on the upper Sine and the second in the Line Sol and hold the Thread at the Intersection of the opposite Parrallel then lay the Thread to the third Tearm on the uper Parrallel and on the line Sol it will intersect the fourth Proportional if it be less then the Radius But Proportions having the Radius ingredient will be wrought without any Motion of the Thread As the Cosine of the Latitude To Radius So is the sine of the Declination To the sine of the Amplitude So in our Latitude of London when the Declination is 20d 12′ the Amplitude will be found to be 33d 42′ Extend the Thread through 38 d 28′ on the line Sol. and through the Declination in the upper Sine and it will intersect the opposite Parrallel Sine at 33 d 42′ the Amplitude sought The use of the Semi-Tangent and Chords are passed by at present The line Sol is of use in Dyalling as in Mr Fosters Posthuma page 70 and 71 where it is required to divide a Circle into 12 equal parts for the hours and each part into 4 subdivisions for the quarters and into such parts may the equal Semicircle be divided that if it were required to divide a Circle of like Radius into such parts it might be readily done by this Of the Line of Hours on the right edge of the foreside of the Quadrant This is the very same Scale that is in the Diameter on the Backside only there it was divided into degrees and here into time and placed on the outermost edge there needs no line of Latitudes be fitted thereto for those Extents may be taken off as Chords from the Quadrant of Latitudes by help of these Scales thus placed on the outward edges of the Quadrants may the hour-lines of Dyals be prickt down without Compasses To Draw a Horizontal Dyal FIrst draw the line C E for the hour-Hour-line of 12 and cross it with the Perpendicular A B then out of a Scale or Quadrant of Latitudes set of C B and C A each equal to the Stiles height or Latitude of the place then place the Scale of 6 hours on the edge of the Quadrant whereto the Line of Latitudes was fitted one extremity of it at A and move the Quadrant about till the other end or extremity of it will meet with the Meridian line C E then in regard the said Scale of Hours stands on the very brink or outward most edge of the Quadrant with a Pin Pen or the end of a black-lead pen make marks or points upon the Paper or Dyal against each hour and the like for the quarters and other lesser parts of the graduated Scale and from those marks draw lines into the Center and they shall be the hour-lines required without drawing any other lines on the Plain the Scale of Hours on the Quadrant is here represented by the lines A E and E B the hour lines above the Center are drawn by continuing them out through the Center And those that have Paper prints of this line may make them serve for this purpose without pricking down the hour points by Compasses by doubling the paper at the very edge or extremity of the Scale of Hours Otherwise to prick down the said Dial without the Line of Latitudes and Scale of hours in a right angled Parallellogram Having drawn C E the Meridian line and crossed it with the perpendicular C A B and determining C E to be the Radius of any length take out the Sine of the Latitude to the same Radius and prick it from C to A and B and setting one foot at E with the said Extent sweep the touch of an Arch at D and F then take the length of the Radius C E and setting down one foot at B sweep the touch of an Ark at D intersecting the former also setting down the Compasses at A make the like Arch at F and through the points of Intersection draw the streight lines A F B D and F E D and they will make a right angled Parallellogram the sides whereof will be Tangent lines To draw the Hour-lines Make E F or E D Radius and proportion out the Tangents of 15d and prick them down from E to 1 and 11 and draw lines 30 and prick them down from E to 2 and 10 and draw lines through the points thus found and through the points F and D and there will be 3 hours drawn on each side the Meridian line Again make A F or B D Radius and proportion out the Tangent of 15d and prick it down from A to 5 and from B to 7. Also proportion out the Tangent of 30d and prick it down from A to 4 and from B to 8 and draw lines into the Center and so the Hour-lines are finished and for those that fall above the 6 of clock line they are only the opposite hours continued after the like manner are the halfs and quarters to be prickt down Lastly By chords prick off the Stiles height equal to the Latitude of the place and let it be placed to its due elevation over the Meridian line Of Vpright Decliners DIvers Arks for such plains are to be calculated and may be found on the Circle before described 1. The Substiles distance from the Meridian By the Substilar line is meant a line over which the Stile or cock of the Dyal directly hangeth in its nearest distance from the Plain by some termed the line of deflexion and is the Ark of the plain between the Meridian of the Plain and the Meridian of the place The distance thereof from the Hour-line of 12 is to be found by this Proportion As the Radius To the Sine of the Plains Declination So the Cotangent of the Latitude To the Tangent of the Substile from the Meridian 2. For the Angle of 12 and 6. An Ark used when the Hour-lines are pricked down from the Meridian line in a Triangle or Parallellogram and not from
the Substile without collecting Angles at the Pole As the Radius Is to the Sine of the Plains Declination So is the Tangent of the Latitude To the Tangent of an Ark the Complement whereof is the Angle of 12 and 6. 3. Inclination of Meridians Is an Ark of the Equinoctial between the Meridian of the plain and the Meridian of the place or it is an Angle or space of time elapsed between the passage of the shaddow of the Stile from the Substilar line into the Meridian line by some termed the Plains difference of Longitude and not improperly for it shews in what Longitude from the Meridian where the Plain is the said Plain would become a Horizontal Dyal and the Stiles height shews the Latitude this Ark is used in calculating hour distances by the Tables and in pricking down Dyals by the Line of Latitudes and hours from the Substile As the Radius Is to the Sine of the Latitude So the Cotangent of the Plains Declination To the Cotangent of the Inclination of Meridians Or As the Sine of the Latitude to Radius So is the Tangent of the Plains Declination To the Tangent of Inclination of Meridians 4. The Stiles height above the Substile As the Radius Is to the Cosine of the Latitude So is the Cosine of the Plains Declination To the Sine of the Stiles height Or the Substiles distance being known As the Radius To the Sine of the Substiles distance from the Meridian So is the Cotangent of the Declination To the Tangent of the Stiles height Or The Inclination of Meridians being known As the Radius To the Cosine of the Inclination of Meridians So is the Cotangent of the Latitude To the Tangent of the Stiles height 5. Lastly For the distances of the Hour-lines from the Substilar Line As the Radius Is to the Sine of the Stiles height above the Plain So is the Tangent of the Angle at the Pole To the Tangent of the Hours distance from the Substilar Line By the Angle at the Pole is meant the Ark of difference between the Ark called the Inclination of Meridians and the distance of any hour from the Meridian for all hours on the same side the Substile falls and the sum of these two Arks for all hours on the other side the Substile These Proportions are sufficient for all Plains to find the like Arks without having any more if the manner of referring Declining Reclining Inclining Plains to a new Latitude and a new Declination in which they shall stand as upright Plains be but well explained for East or West Reclining Inclining Plains their new Latitude is the Complement of their old Latitude and their new Declination is the Complement of their Reclination Inclination which I count always from the Zenith and upon such a supposition taking their new Latitude and Declination those that will try shall find that these Proportions will calculate all the Arks necessary to such Dials So if an Upright Plain decline 25d in our Latitude of London from the Meridian The Substiles distance from the Meridian is 18d 34′ The Angle of 12 and 6 is 62 00 The Inclination of Meridians is 30 47 The Stiles height is 34 19 To Delineate the same Dial from the Substile by the Line of Latitudes and Scale of hours in an Equicrutal Triangle To Draw an Vpright Decliner An Vpright South Plain for the Latitude of London Declining 25d Eastwards TO prick down this Dial by the line of Latitudes and Scale of Hours in an Isoceles Triangle Draw C 12 the Meridian Line perpendicular to the Horizontal line of the Plain and with a line of Chords make the Angle F C 12 equal to the Substiles distance from the Meridian and draw the line F C for the Substile Draw the line B A perpendicular thereto and passing through the Center at C and out of the line of Latitudes on the other Quadrants or out of the Quadrant of Latitudes on this Quadrant set off B C and C A each equal to the Stiles height then fit in the Scale of 6 hours proper to those Latitudes so that one Extremity meeting at A the other may meet with the Substilar line at F. Then get the difference between 30d 47′ the inclination of Meridians and 30d the next hours distance lesser then the said Ark the difference is 47′ in time 3′ nearest then fitting in the Scale of hours as was prescribed Count upon the said Scale Hour Min.  0 3 from F to 10 1 3 11 2 3 12 3 3 1 4 3 2 5 3 3 And make points at the terminations with a pin or pen draw lines from those points into the Center at C they shall be the true hour-hour-lines required on this side the Substile Again Fitting in the Scale of Hours from B to F count from that end at B the former Arks of time Ho Min  00 03 from B to 4 1 3 5 2 3 6 3 3 7 4 3 8 5 3 9 And make Points at the Terminations through which draw Lines into the Center and they shall be the hour Lines required on the other side the Substile The like must be done for the halfs and quarters getting the difference between the half hour next lesser in this Example 22d 30′ under the Ark called the inclination of Meridians the difference is 1d 17′ in time 33′ nearest to be continually augmented an hour at a time and so prickt off as before was done for the whole hours By three facil Proportions may be found the Stiles height the Inclination of Meridians and the Substiles distance from the Plains perpendicular for all Plains Declining Reclining or Inclining which are sufficient to prick off the Dyal after the manner here described which must be referred to another place If the Scale of hours reach above the Plain as at B so that B C cannot be pricked down then may an Angle be prickt off with Chords on the upper side the Substile equal to the Angle F C A on the under side and thereby the Scale of hours laid in its true situation having first found the point F on the under side To prick down the former Dyal in a Rectangular ☉ blong or long square Figure from the Substile Having set off the Substilar F C assume any distance in it as at F to be the Radius and through the fame at right Angles draw the line E F D then having made F C any distance Radius take out the Sine of the Stiles height to the same Radius and entring it at the end of the Scale of three hours make it the Radius of a Tangent and proportion out Tangents to 3′ and set them off from F to 10 1 hour 3 and set them off from F to G 2 3 and set them off from F to H Again Take out the Tangents of the Complement of the first Ark increasing it each time by the augmentation of an hour namely 57′ and prick
them from F to I and from the points 1 ho. 57 and prick them from F to K and from the points 2 57 and prick them from F to E and from the points thus found draw lines into the Center Then for the other sides of the Square make C F the Radius of the Dyalling Tangent of 3 hours and proportion out Tangents to the former Arks namely 3′ and prick them from B to P Also to the latter Arks 57′ and prick them from A to N 1 ho. 3 and prick them from B to O Also to the latter Arks. 1 h. 57 and prick them from A to M 2 3 and prick them from B to L Also to the latter Arks. 2 57 and prick them from A to D and draw lines from these terminations into the Center and the Hour-lines are finished after the same manner must the halfs and quarters be finished And how this trouble in Proportioning out the Tangents may be shunned without drawing any lines on the Plain but the hour-lines may be spoke to hereafter whereby this way of Dyalling and those that follow will be rendred more commodious Lastly the Stile may be prickt off with Chords or take B C and setting one foot in F with that Extent sweep the touch of an occult Arch and from C draw a line just touching the outward extremity of the said Arch and it shall prick off the Angle of the Stiles height above the Substile To prick off the former Dyal in an Oblique Parallellogram or Scalenon alias unequal sided Triangle from the Meridian First In an Oblique Parallellogram DRaw CE the Meridian line and with 60d of a line of Chords draw the prickt Arch and therein from K contrary to the Coast of Declination prick off 62d the angle of 12 and 6 and draw the line C D for the said hour line continued on the other side the Center and out of a line of Sines make C E equall to 65d the Complement of the Declination then take out the sine of 38d 28′ the Complement of the Latitude and enter it in the line D C so that one foot resting at D the other turned about may but just touch the Meridian line the point D being thus found make C F equall to C D and with the sides C F and C E make the Parallellogram D G F H namely F H and G D equal to C E and E G and E H equal to D C. And where these distances sweeping occult arches therewith intersect will find the points H and G limiting the Angles of the Parallellogram Then making E H or C D Radius proportion out the Tangents of 15d and prick them down from E to 1 and 11 and 30 and prick them down from E to 2 and 10 and draw lines into the Center through those points and the angular points of the Parallellogram at H and G and there will be 6 hours drawn besides the Meridian line or hour line of 12. Then making D G Radius proportion out the Tangent of 15d and prick it down from D upwards to 5 and downward to 7 also proportion out the tangent of 30d and prick it from D to 8 and from F to 4 and draw lines into the Center and so the hour lines are finished after the same manner are the halfs and quarters to be proportioned out and pricked down and if this Work is to be done upon the Plain it selfe the Parallel F H will excur above the plain in that case because the Parallel distance of F H from the Meridian is equal to the parallel distance of D G the space G. 8. may be set from H to 4 and so all the hour lines prickt down To prick down this Dyal in a Scalenon or unequal sided triangle from the Meridian from E to D draw the streight line D E and from the same point draw another to F and each of them the former hour lines being first drawn shall thereby be divided into a line of double tangents or scale of 6 hours such a one as is in the Diameter of the Circle on this quadrant or on the right edge of the foreside and therefore by helpe of either of them lines if it were required to prick down the Dyal it might be done by Proportioning them out take the extent D E and prick it from one extremity of the Diameter in the Semicircle on the quadrant and from the point of Termination draw a line with black Lead to the other extremity which will easily rub out again either with bread or leather parings and take the nearest distance from 15 of the Diameter to the said line and the said extents 30 of the Diameter to the said line and the said extents 45 of the Diameter to the said line and the said extents shall reach from E to 11 and from D to 7 shall reach from E to 10 and from D to 8 shall reach from E to 9 and from D to 9 and the like must be done for the line E F entring that in the Semicirle as before or without drawing lines on the quadrant if a hole be drilled at one end of the Diameter and a thred fitted into it lay the thred over the point in the Diameter and take the nearest distances thereto Lastly from a line of Chords prick off the substilar line and the stiles height as we before found it This way of Dyalling in a Parallellogram was first invented by John Ferrereus a Spaniard long since and afterwards largely handled by Clavius who demonstrates it and shews how to fit it into all plains whatsoever albeit they decline recline or incline without referring them to a new Latitude the Triangular way is also built upon the same Demonstration and is already published by Mr Foster in his Posthuma for it is no other then Dyalling in a Parallellogram if the Meridian line C E be continued upwards and C E set off upwards and lines drawn from the point so found to D and E shall constitute a Parallellogram An Advertisement about observing of Altitudes IMagine a line drawn from the beginning of the line Sol to the end of the Diameter and therein suppose a pair of sights placed with a thred and bullet hanging from the begining of the said line as from a Center I say the line wherein the sights are placed makes a right angle with the line of sines on the other side of Sol and so may represent a quadrant the equal Limbe whereof is either represented by the 90d of the equal Semicirle or by the 90d of the Diameter and thereby an Altitude may be taken Now to make an Isoceles equicrural or equal legged triangle made of three streight Rulers the longest whereof will be the Base or Hipotenusal line thus to serve for a quadrant to take Altitudes withal will be much cheaper and more certain in Wood then the great Arched wooden framed quadrants Moreover the said Diameter line supplies all the uses of the
either under the window or any other convenient place in the Room Place the center of the said Horizontal Dial in the Center of the Hole or Nodus also scituate the said Dial exactly parallel to the Horizon and the meridian of the said Dial in the meridian of the world which as before may easily be done if at that instant you know the true hour of the day Then take the thread whose end is fixed in a point in the direct Axis and move it to and fro until the said thread doth interpose between your eye and the hour-line on the said Horizontal Dial which you intend to draw and then keeping your eye at that scituation make a point or mark in any place where you please or under the window so that the said thread or string may interpose between that point or mark so made and your eye as aforesaid which said point so sound will shew the true time of the day at that hour all the year long the Sun shining thereon so will that point together with the said thread serve to shew the hour instead of an hour-line In like manner the said thread fixed in the Axis may be again moved to and fro until the said thread doth interpose between the eye and any other hour-line desired on the said Horizontal Dial and then as before make another point or mark in any place at pleasure or under the said window by projecting a point from the eye so that the said thread also interpose between that point to be made and the eye so will that point so found shew the true time of the day for the same hour that did the hour line on the said Horizontal Dial which was shadowed by the said thread In like manner may be proceeded by help of that thread and the several hour-lines on the said Horizontal Dial to finde the other hour-points which must have the same numbers set to them as have the hour-lines on the said Horizontal Dial. Otherwise to make a Dial from a hole in any pane of glasse in a window and to graduate the hour-lines below on the Sell or Beam or on the ground that hole is supposed to be the center of the Horizontal Dial and being true placed the stile thereof if supposed continued will run into the point in the Meridian of the Cieling before found where a thread is to be fixed then let one extend a thread fastned in the center of the Horizontal Dial parallelly to the Horizon over each respective hour-line and holding it steady let another extend the thread fastened in the Meridian in the Cieling along by the edges of the former Horizontal thread and so this latter thread will finde divers points on the ground through which if hour-lines be drawn and the Sun shine through the hole in the pane of Glasse before made the spot of the Sun on the ground shall shew the time of the day For the points that will be thus found on the Beam or Transome the thread fixed in the Cieling or instead of it a piece of tape there fixed must be moved so up and down that the spot of the Sun may shine upon it and being extended to the Transome or Beam graduated with the hour-lines as before directed it there shews the time of the day Here note that it will be convenient to have that pane of Glasse darkened through which that spot is to shine In like manner may a Dial be made from a nail head a knot in a string tied any where a crosse or from any point driven into the bar of a window and the hour-lines graduated upon the Transome or board underneath To make a Reflected Dial on the Ceiling of the Room is onely the contrary of this by supposing the Horizontal Diall with its stile to be turned downwards and run into the true meridian on the ground where the thread is to be fixed and to be extended along by the former Horizontal thread held over the respective hours as before upward to find divers points in the Cieling as shall afterwards be shewed Of Dials to stand in the Weather These may be also made by help of an Horizontal Dial. DRive two nails or pins into the wall on which the edge of a Board of competent breadth may rest then to hold up the other side of the Board drive two hooks into the wall above whereto with cord or line the outside of the Board may be sustained and this Board being Horizontal place the Horizontal Dial its Meridian-line in the true Meridian of the world If a Plain look towards the South the stile of the Horizontal Dial continued by a thread from the center will run into the Plain which note to be the center of the new Dial as also that line is the new stile which must be supported with stayes when you fix it up By a thread from the center laid over every hour-line on the Horizontal Dial cross the Horizontal line of the Plain which note with the same hours the Horizontal Dial hath The hour-lines on the Plain are to be drawn from the center before found through those points and so cut off by the Dial or continued at pleasure If the Center of the Dial be assigned before you begin the work in such Cases you may remove the Horizontal Dial up and down keeping it still to the true position or hour till you finde the Axis or stile run into the Center But if the Plain look into the East or West then possibly the Axis of the Horizontal Dial will not meet with the Plain in such Cases you must fix a board so that it may receive the Axis the board being perpendicular to the Plain this stile or Axis is to be fastened to the Plain by two Rests the hour-lines may be drawn by the eye or shadowed out by a Light Bring the thread that represents the Axis or stile into any hour-point on the Horizontal Dial by your eye or shadow at the same time the thread or shadow making marks on the Plain shews where the hour-hour-lines are to passe After the same manner any hour-hour-line is to be drawn over any irregular or crooked Plain Further observe that any point in the middle or neer the end of the stile will as well shew the hour of the Day as the whole stile Of Refracted Dials IF you stick up a pin or stick or assign any point in any concave Boul or Dish to shew the hour and make that the center of the Horizontal Dial assigning the meridian-line on the edges of the Boul point out the rest of the hour-lines also on the edges of the Boul and taking away the Horizontal Dial elevate a string or thread from the end of the said pin fastned thereto over the Meridian-line equal to the Elevation of the Pole or the Latitude of the place then with a candle or if you bring the thread to shade upon any hour-point formerly marked out on the edges of
the Boul at the same time the shade in the Boul is the hour line And if the Boul be full of water or any other liquor you may draw the hour-lines which will never shew the true hour unlesse filled with the said Liquor again Reflected Dialling To draw a Reflected Dial on any Plain or Plains be they never so Gibous and Concave or Convex or any irregularity whatsoever the Glass being fixed at any Reclination at pleasure provided it may cast its Reflex upon the places proposed Together with all other necessary lines or furniture thereon viz. the Parallels of Declination the Azimuth lines the Parallels of Altitude or proportions of shadows the Planetary Hour-lines and the Cuspis of those Houses which are above the Horizon c. 1. If the Glasse be placed Horizontal upon the Transome of a window or other convenient place How upon the Wall or Cieling whereon that Glasse doth reflect to draw the Hour-lines thereon although it be never so irregular or in any form whatsoever CONSTRUCTIO FIrst draw on Pastboard or other Material an Horizontal Dial for the Latitude proposed Then by help of the Azimuth or at the time when the Sun is in the Meridian or by knowing the true hour of Day whereby may be drawn several lines on the Cieling Floor and Walls of the Room so as in respect of the center of the Glasse they may be in the true Meridian-circle of the World For if right lines were extended from the center of the said Glasse by any point though elevated in any of those lines so drawn it would be directly in the Meridian Circle of the World Now all Reflective Dialling is performed from that principle in Opticks which is That the angle of Incidence is equal to the angle of Reflection And as any direct Dial may be made by help of a point found in the direct Axis so may any Reflected Dial be also made by help of any point found in the Reflected Axis And in regard the reflected Axis for the most part will fall above the Horizon of the Glasse without the window so that no point there can be fixed therefore a point must be found in the said Reflected Axis continued below the Horizontal of the said Glasse until it touch the ground or floor of the Room in some part of the Meridian formerly drawn which point will be the point in the reversed Axis desired and may be found as followeth One end of the thread being fixed at or in the center of the said Glasse move the other end thereof in the meridian formerly drawn below the said Glasse until the said reversed Axis be depressed below the Horizon as the direct Axis was elevated above the Horizon which may be done by applying the side or edge of a Quadrant to the said thread and moving the end thereof to and fro in the said meridian until the thread with a plummet cut the same degree as the Pole is above the Horizontal Glasse and then that point where the end of the thread toucheth the Meridian either on the floor or wall of the room is the point in the reflected reversed Axis sought for Now if the Reversed Axis cannot be drawn from the Glasse by reason of the jetting of the window or other impediment that point in the reverse Axis may be found by a line parallel thereto by fixing one end of it on the Glasse and the other end in the meridian so as that it may be parallel to the floor or wall in which the reversed Axis-point will fall and finde the Axis point from that other end of the lath so if the same Distance be set from that point backward in the Meridian on the floor as is the Lath the point will be found in the Reversed Axis desired Thus having found a point in the reflected reversed Axis it is not hard by help whereof and the Horizontal Dial to draw the reflected hour-lines on any Cieling or Wall be it never so concave or convex To do which First note that all straight lines in any projection on any Plain do always represent great Circles in the Sphere such are all the hour-lines Place the center of this Horizontal Dial in the center of the Glasse the hour-lines of the said Dial being horizontal and the Meridian of the said Dial in the Meridian of the world which may be done by plumb lines let fall from the meridian on the Cieling Then fix the end of a thread or silk in the said center of the Dial or Glasse and draw it directly over any hour-line on the Dial which you intend to draw and at the further side of the room and there let one hold or fasten that thread with a small nail Then in the point formerly found on the reversed Axis on the â—oor fix another thread there as formerly was done in the center of the Diall then take that thread and make it just touch the thread on the hour-line of the Horizontal Dial extended in any point thereof it matters not whereabouts and mark where the end of that thread toucheth the Wall or Cieling and there make some mark or point Then again move the same thread higher or lower at pleasure till it as formerly touch the said same hour thread and mark again whereabouts on the wall or Cieling the end of the said thread also toucheth In like manner may be found more points at pleasure but any two will be sufficient for the projecting or drawing any hour-line on any plain how irregular soever For if you move a thread and also your eye to and fro until you bring the said thread directly between your eye and the points formerly found you may project thereby as many points as you please at every angle of the Wall or Cieling whereby the reflected hour-hour-line may be exactly drawn Again in like manner remove the said thread fastned in the center of the Horizontal Dial which also is the center of the Glasse on any other hour-hour-line desired to be drawn and as before fasten the other end of the thread by a small nail or otherwise at the further side of the room but so that the said thread may lie just on the hour-hour-line proposed to be drawn on the Horizontal Dial. Then as before take the thread fastened in the point on the reflected Axis and bring it to touch the thread of the hour-hour-line in any part thereof and mark where the end of that thread toucheth the said Wall or Cieling Then again as before move the said thread so as that it only touch the said thread of the hour-hour-line in any other part thereof and also mark where the end of that thread toucheth the said Wall or Cieling So is there found two points on the Wall or Cieling being in the reflected hour-line desired by help of which two points the whole hour-line may be drawn for if as before a thread be so scituated that it may interpose between the eye and the said
any Reflected Dial with all the Furniture that possible may be the Glass being set at any possible Reclination In the drawing of which there is principally to be considered 1 The Reflected Horizon 2 The Reflected Meridian Note the Horizon Meridian are two great circles 1 To draw the Reflected Horizon according to the situation of any reclining Glasse whatsoever FIrst let two pieces of nealed wire be fastened on the window on each side of the said Glasse the ends thereof being without the room in the air at whose ends let there be fastned a thread which may be pulled straight at pleasure by bending of the wire then bend those wires upward or downward until the thread fastened at the end of each wire be exactly horizontal with the center of the Glasse which may be tried by a quadrant Then I tie a string or thread cross the room in such sort that I may from most part of the thread see the reflecting glass and therein the said horizontal thread without the room Then on the said thread cross the room I tie a slipping knot to move to and fro at pleasure which knot I move to and fro on the said thread until by looking in the said Glasse I finde from my eye the said knot and part of the horizontal thread without all as it were in a right line the one interposing the sight of the other Then being careful to keep the knot in that position fasten one end of a thread in the place of the center of the reclining reflecting glasse and bring that thread so as just to touch the aforesaid knot augmenting that thread until the end thereof touch the wall or Cieling and there make a mark or point so is there one point found on the Wall or Cieling in the Reflected Horizon of the World Then I begin again and remove the position of that thread which went overthwart the Room either higher or lower at pleasure still having regard that I may from the most part of the said thread see the Reflecting Glasse and therein the same horizontal thread without the room Then as before I move the said knot on the said thread to fro until as before by looking in the said Glasse I find from my eye the said knot and part of the Horizontal thread both in one right line the one interposing the sight of the other and by the said knot I bring that thread whose end is fastened in the center of the said glasse and keeping it just to touch the said knot I continue it until the end thereof touch the Wall or Cieling as before and there I make another mark or point so is there two points found in the said reflected Horizon on the wall or Cieling By which said two points if a thread as before be so scituated that it may interpose between the eye and the said two points there may be many points made to be in the same interposition of the thread which as before may be made at every bending or angle of the Wall or Cieling whereby the reflected Horizon desired may be drawn by drawing a line from point to point round about the Room Which wil be the true reflected Horizon according to the situation of the glasse 2 To draw the Reflected Meridian according to the situation of any Reclining Glasse whatsoever FIrst take a lath or thin piece of wood of any convenient length at pleasure as some one and an half or two foot long and at each end thereof make a hole the one to hang a thread and plummet and the other is to put a small nail therin to fasten it in some part of the window over the center of the Glasse so that the thread and plummet may hang without the room then by help of the Suns Azimuth you may draw the meridian line as before as if the Glasse were horizontal and move the lath with the thread and plummet at the end of it to and fro until the thread and plummet be in the direct meridian of the world with the center of the Glasse Then as before tie a thread crosse the room in such sort that from or by some part of the said thread both the Reclining glasse and the thread to which the plummet is fastened may be seen at one time Then as before on the said thread which crosses the room I tie a slipping knot which I move to and fro on the said string until by looking in the said Glasse I find from my eye the said knot and some part of the perpendicular thread without all as it were in one right line the one shadowing or interposing the sight of the other being then very careful to keep that knot in the same position then take the thread whose end whereof being fastened in the said center of the Glasse and bringing it just to touch the said knot I augment that thread until the end thereof touch the said wall or Cieling and the said thread also touch the knot as before then in that place where the end of the said thread toucheth the wall or Cieling I make a mark which mark or point will be directly in the reflected meridian of the world according to the situation of that Glasse Then again I remove that thread overthwart the room on which the said knot is either higher or lower then it formerly was at pleasure still having regard that from some part of the said thread within you may see both the Reclining Glasse and the perpendicular thread without at one time and as before move the said slipping knot on the said thread until by looking in the said Reclining Glasse you see the said knot and some part of the perpendicular thread without in one right line so as the one shadows or hinders the sight of the other as before which knot then must not be removed from its situation then take that thread whose end is fastened in the Glasse and bring it to touch that knot the end of the said thread being continued to touch the wall or Cieling so is that point of touch on the Cieling another point found in the Reflected Meridian of the world So is there two points found in the said Reflected Meridian on the wall or Cieling by which if a thread as before be so situated that it may interpose between the eye and the said two points many points thereby in the said reflected Meridian may be made at every bending or angle of the wall or Cieling whereby the Reflected meridian desired may be drawn by drawing a line from point to point obliquely in the Room which will be the true Reflected Meridian of the world according to the situation of that Glasse Now this Reflected Horizon and Meridian being first drawn they will be of great use in drawing the Hour-lines together with all the furniture that possibly can be drawn on any Diall To draw the Reflected Hour-lines to any Reclining Glasse on any plane whatsoever that the Sun
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