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
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
greatest Declination to the sine of the greatest Ascensional difference which converted into Time gives the time of the Suns rising or setting before or after 6 by which Ark of the Limbe the Horizon is limitted Then to divide it say As the Radius to the Tangent of the assigned Amplitude So is the Sine of the Latitude To the Tangent of the Ascensional difference agreeing thereto which counted in the Limb from it the Amplitudes may be divided on both the Horizons and note if these Amplitudes be not coincident with those the Azimuths have designed then are the said Azimuths drawn false To inscribe the Stars on the Projection Such only and no other as fall between the two Tropicks may be there put on Set one foot of the Compasses in the Center of the Quadrant and extend the other to that place of either of the Eclipticks as corresponds to the given declination of the Star and therewith sweep an occult Ark I say then that a Thread from the Center of the Quadrant laid over the Limb to the Stars right Ascension where it intersects the former occult Ark is the place where the proposed Star must be graduated Of the Almanack There is also graved in a Rectangular Square or Oblong a perpetual Almanack which may stand either on the foreside or back of the Quadrant as room shall best permit On the Backside of the Quadrant there is 1. On the right edge a Line of Signs issuing from the Center the Radius whereof is in length 5 inches 2. On the left edge a Line of Chords issuing from the Center 3. On the edges of the Quadrant there are also two Scales for the more ready finding the Hour and Azimuths in one Latitude the Hour Scale is no other then 62d of a Line of Sines whose Radius is made equal to half the Secant of the Latitude being fitted for London to the common Radius of the Sines the prickt Line of Declination annexed to it and also continued beyond the other end of it to the Suns greatest Declination is also a portion of a Line of Sines the Radius whereof is equal to the Sine of the Latitude taken out of the other part of the Scale or which is all one the Sine of the Suns greatest declination is made equal to the Sine of the greatest Altitude at the hour of 6 taken out of the other part of the Scale which at London is 18d 12 m 4. The Azimuth Scale is also 62d of a Line of Sines whose Radius is made equal to half the Tangent of the Latitude to the common Radius of the Sines the Line of the Declination annexed to it and continued beyond it To the Suns greatest Declination is also a portion of a Line of Sines of such a length whereof the Sine of the Latitude is equal to the Radius of the Sines of the other part of this fitted Scale or which is all one the length of the Suns greatest Declination is made equal to the Suns greatest Vertical Altitude which in this Latitude is 30d 39′ of the other Sine or Line of Altitudes The Limbe is numbred both with degrees and time from the right edge towards the left Between the Limbe and the Center are put on in Circles the Scales following 1 A Line of Versed Sines to 180 degrees 2. A Line of Secants to 60d the graduations whereof begin against 30d of the Limbe to apply which Vacancy and for other good uses there is put on a Line of 90 Sines ending where the former graduations begin this is called the lesser Sines 3. A Line of Tangents graduated to 63d 26′ 4. A Line of Versed Sines to 60d through the whole Limbe called the Versed Sines quadrupled because the Radius hereof is quadruple to the Radius of the former Versed Sines 5. A Line of double Tangents or Scale of hours being the same Dyalling Scale as was described on the foreside 6. A Tangent of 45d or three hours through the whole Limbe for Dyalling which may also be numbred by the Ark doubled to serve for a Projection Tangent alias a Semi-tangent 7. In another Quadrant of a Circle may be inscribed a portion of a Versed Sine to eight times the Radius encreased of that of 180d called the Occupled Versed Sine and at the end of this from the other edge another portion of a Versed Sine to 12 times the Radius encreased may be put on 8. Lastly above all these is the Scale of Hours or Nocturnal with Stars names graved within it towards the Center this is divided into 12 equal hours and their parts and the Stars are put on from their right Ascensions only with their declination figured against them All the Lines put on in Quadrants of Circles must be inscribed from the Limbe by help of Tables carefully made for that purpose an instance shall be given how the Line of Versed Sines to 180d was inscribed and after the same manner that was put on must all the rest Imagine a Line of Versed Sines to 180d to stand upon the left edge of a Quadrant from the Center with the whole length thereof upon the Center sweep the Arch of a Circle and then suppose Lines drawn through each graduation or degree thereof continued parralel to the right edge till they intersect the Arch formerly swept which shall be divided in such manner as the Line of Versed Sines on this Quadrant is done But to do this by Calculation A Table of natural Versed Sines must first be made which for all Arks under 90d are found by substracting the Sine Complement from the Radius so the Sine of 20d is 34202 which substracted from the Radius rests 65798 which is the Versed Sine of 70d And for all Arks above 90d are got by adding the Sine of the Arks excess above 90d unto the Radius thus the Versed Sine of 110d is found by adding the Sine of 20d to the Radius which will make 134202 for the Versed Sine of the Said Ark. This Table or the like of another kind being thus prepared the proportion for inscribing of it will hold As the length of the Line supposed to be posited on the left edge Is to the Radius So is any part of that length To the Sine of an Arch which sought in the Tables gives the Arch of the Limbe against which the degree of the Line proposed must be graduated But in regard the Versed Sine of 180d is equal to the double of the Radius the Table for inscribing it will be easily made by halfing the Versed Sine proposed and seeking that half in the Table of natural Sines so the half of the Versed Sine of 70d is 32899 which sought in the Table of natural Sines gives 19d 13′ fore of the Limb against which the Versed Sine aforesaid is to be graduated and so the half of the Versed Sine of 110d is 67101 which answers to 42d 9′ of the sines or
serve for Mensurations Protractions and Proportional work The ground of working Proportions by single natural Lines is built upon the following grounds That Equiangled Plain Triangles have the sides about their equal angles Proportional and this work hath its whole dependance on the likeness of two equiangled Plain Right angled Triangles as in the figure annexed let A B represent a Line of equal parts Sines or natural Tangents issuing from the Center of the Quadrant supposed at A and let A C represent the Thread and the Lines B C E D making right angles with the Line A C or with the Thread the nearest distances to it from the Points B and E. I say then that this Scheme doth represent a Proportion of the greater to the less and the Converse of the less to the greater First of the greater to the less and then it lies As A B to B C So A E to A D whence observe that the length of the second Tearm B C must be taken out of the common Scale A B and one foot of that extent entred at B the first Tearm the Thread must be laid to the other foot at C according to the nearest distance then the nearest distance from the Point E to the Thread that is from the third Tearm called Lateral entrance being measured in the Scale A B gives the quantity of the 4 â—h Proportional Secondly of the less to the greater And then it lies As B C to A B So E D to A E Or As E D to A E So B C to A B by which it appears that the first Tearm B C must be taken out of the common Scale and entred one foot at the second Tearm at B and the Thread laid to the other at C according to nearest distance then the third Tearm E D must be taken out of the common Scale and entred between the Thread and the Scale so that one foot may rest upon the Line as at E and the other turned about may but just touch the Thread as at D so is the distance from the Center to E the quantity of the 4th Proportional and this is called Parralel entrance because the extent E D is entred Parralel to the extent B C To avoid Circumlocution it is here suggested that in the following Treatise we use these expressions to lay the Thread to the other foot whereby is meant to lay it so according to nearest distance that the said foot turned about may but just touch the Thread and so to enter an extent between the Thread and the Scale is to enter it so that one foot resting upon the Scale the other turned about may but just touch the Thread Another chief ground in order to working Proportions by help of Lines in the Limb is That in any Proportion wherin the Radius is not ingredient the Radius may be introduced by working of two Proportions in each of which the Radius shall be included and that is done by finding two such midle tearms one whereof shall always be the Radius as shall make a Rectangle or Product equal to the Rectangle or Product of the two middle Tearms proposed to find which the Proportion will be As the Radius To one of the middle Tearms So the other middle Tearm To a fourth I say then that the Radius and this fourth Tearm making a Product or Rectangle equal to the Product of the two middle Tearms these may be assumed into the Proportion instead of those and the answer or fourth Tearm will be the same without Variation and therefore holds As the first Tearm of the Proportion To the Radius So the fourth found as above To the Tearm sought Or As the first Tearm of the Proportion Is to the fourth found as above So is the Radius To the Tearm sought and here observe that by changing the places of the second and third Tearm many times a Parralel entrance may be changed into a Lateral which is more expedite and certain then the other having thus laid the foundation of working any Proportion I now come to Examples 1. To work Proportions in equal parts alone If the first Tearm be greater then the second take the second Tearm out of the Scale and enter one foot of that extent at the first Tearm laying the Thread to the other foot then the nearest distance from the third Tearm to the Thread gives the 4th Proportional sought to be measured in the Scale from the Center If the first Tearm be less then the second still as before keep the greatest Tearm on the Scale and enter the first Tearm upon it laying the Thread to the other foot then enter the third Tearm taken out of the Scale between the Thread and the Scale and it finds the 4th Proportional Example Admit the Sun shining I should measure the length of the Shaddow of a Perpendicular Staff and find it to be 5 yards the length of the Staff being 4 yards and at the same time the length of the Shaddow of a Chimny the Altitude whereof is demanded and find it to be 22½ yards the Proportion then to acquire the Altititude would be As the length of the shaddow of the Staff To the length of the Staff So the length of the shadow of the Chimney To the height thereof that is As 5 to 4 So 22 ‑ 5 to 18 yards the Altitude or height of the Chimney sought Enter 4 or the great divisions upon 5 laying the Thread to the other foot then the nearest distance from 22 ‑ 5 to the Thread measured will be 18 and in this latter part each greater division must be understood to be divided into ten parts And so if the Sun do not shine the Altitude might be obtained by removing till the Top of a Staff of known height above the eye upon a level ground be brought into the same Visual Line with the Top of the Chimney and then it holds As the distance between the Eye and the Staff To the height of the Staff above the eye So the distance between the Eye and the Chimney To the height of the Chimney above the Eye Some do this by a Looking Glass others by a Bowl of Water by going back till they can see the top of the object therein and then the former Proportion serves mutatis mutandis But Proportions in equal parts will be easily wrought by the Pen the chief use therefore of this Line will be for Protraction Mensuration and to divide a Line of lesser length then the Radius of the Quadrant Proportionally into the like parts the Scale is divided which may be readily done and so any Proportional part taken off to do it Enter the length of the Line proposed at the end of the Scale at 10 and to the other foot lay the Thread the nearest distances from the several parts of the graduated Scale to the Thread shall be the like Proportional parts to the length of the Line proposed the Proportion thus wrought is
how much it falls beyond it bring the said foot to the Center and let the other fall backward on the Line then will the distance between the said other foot where it now falleth and the place where it stood before be equal to the excess of the former foot beyond the Center which accordingly thence measured helps you to the Arch sought and its Complement both at once with due regard to the representation of the Line this should be well observed for it will be of use on other Instruments A difference of Versed sines thus taken out to the Common Radius must be entred but once down from the Center To take out a Difference of Versed sines to half the common Radius Count both the Arks proposed on the Versed sines in the Limbe and find what Arks of the equal Limbe answer thereto then out of the Line of sines take the distance between the said Arks and you have the extent required which being but half so large as it should be is to be entred twice down from the Center To measure a difference of Versed sines to half the common Radius The Versed sines are largest at that end numbred with 180d count the given Ark from thence and laying the Thread over the equal Limb find what Ark answers thereto then setting down the Compasses at the like Ark in the Line of sines from the end of it towards the Center mind upon what Ark it falls the Thread laid to the like Ark in the Limb sheweth on the Versed sines the Ark sought To save the labour of drawing a Triangle I shall deliver the Proportion for the Azimuth derived from the general Proportion As the Cosine of the Latitude To the Secant of the Altitude Or As the Cosine of the Altitude To the Secant of the Latitude So is the difference of the Versed sines of the Sun or Stars distance from the Elevated Pole and of the Ark of difference between the Latitude and Altititude To the Versed sine of the Azimuth sought as it falls in the Sphoere that is from the Midnight Meridian And So is the difference of the Versed sines of the Polar distance and of the Ark of difference between a Semicircle and the sum of the Latitude and Altitude To the Versed Sine of the Azimuth from Noon Meridian A Canon derived from the Inverse of the general Proportion to finde the Distance of places in the Ark of a great Circle As the Secant of one of the Latitudes To the Cosine of the other So the Versed sine of the difference of Longitude To the difference of the Versed sines of the Ark of distance sought and of the Ark of difference between both Latitudes when in the same Hemisphere or the Ark of the sum of both Latitudes when in both Hemispheres which difference added to the Versed sine of the said Ark gives the Versed sine of the Ark of distance sought And So is the Versed sine of the Complement of the difference of Longtitude to 180 d. To the difference of the Versed sines of the Ark of distance sought and of an Ark being the sum of the Complements of both Latitudes when in one Hemisphere Or the sum of the lesser Latitude encreased by 90d and of the Complement of the greater Latitude when in different Hemispheres which difference substracted from the Versed sine of the said Ark there will remain the Versed sine of the Ark of distance sought This Proportion is to be wrought after the same manner as we found the Suns Altitudes on all hours universally and the difference to be measured in the Line of sines as representing the former half of a Line of Versed sines according to the Directions given for measuring of a difference of Versed sines to the common Radius or Radius of the Quadrant By altring the two first Tearms of the Proportion above we may work this Proposition by positive entrance As the Radius To the Cosine of one of the Latitudes So the Cosine of the other Latitude To a fourth Again As the Radius To the Versed sine as above expressed in both parts So is that fourth To the difference as above expressed An Example for finding the distance between London and Bantum in the Arch of a great Circle the same that was proposed in Page 96. Bantam Longitude 140d Latitude 5d 40′ South London Longitude 25 50′ Latitude 51 32 North. difference of Longitude 114 10 Sum 57 12 Lay the Thread to 5d 40′ in the Limb counted from left edge and from 38d 28′ in the sines the Complement of our Latitude take the nearest distance to it then lay the Thread to 114d 10′ in the Versed sines and entring the former extent down the Line of sines from the Center take the nearest distance to it then laying the Thread over 57d 12′ in the Versed sine it cuts the Limbe at 13d 15′ from the right edge at the like Ark set down one foot of the former extent in the Line of sines and the other will reach to the Sine of 41d 42′ then laying the Thread over the like Ark in the Limb it will intersect the Versed sines at 109d 18′ the Ark of distance sought to be converted into Leagues or Miles according to the number of Leagues or Miles that answer to a degree in each several Country Thus when we have two sides with the angle comprehended to find the third side either to half or the whole common Radius without a Line of natural Versed sines from the Center or by the Proportions in page 93 or a third way which I pretermit to the great Quadrant and thus the Reader may perceive this small Quadrant to be many ways both Universall and particular which are of sudden performance though tedious in expression Three sides to find an angle Each of the Proportions in Rectangles and Squares before delivered for the Tables may as before suggested be reduced to single Tearms an instance shall be given in that which finds the Square of the sine of half the Angle sought Add the three sides of the Triangle together and from the half sum substract each of the sides including the angle sought then it will hold As the sine of one of the Comprehending sides rather the greater that the entrance may be Lateral Is to the sine of the difference of the same side from the half sum So is the sine of the difference of the other comprehending side To a fourth sine Again As the Sine of the other comprehending side Is to that fourth sine So is the Radius To half the Versed sine of the angle sought And So is the Diameter To the whole Versed sine To work this on the Quadrant Upon the first Tearm in the Line of sines being the greatest containing side enter the extent of the second and to the other foot lay the Thread then from the third Tearm in the sines take the nearest distance to it Which extent enter at
Altitude be well given These Scales in their Use presuppose the Hour and Azimuth of the Sun to be nearer the noon Meridian then 60d. Operation to find the Hour Take the distance between the Altitude and the Declination proper to the season of the year out of the Hour Scale and enter one foot of this Extent at the Cosine of the Declination in the Line of sines and laying the Thread to the other foot according to nearest distance it shews the hour from noon in the Versed sines Quadrupled Example When the Sun hath 23d 31′ of North Declination and 60d of Altitude the hour from noon will be 13d 58′ to be Converted into time When the hour is found to be less then 40d from Noon the former extent may be doubled and entred as before and it shews the hour in the Versed sines Octupled And when the hour is less then 30d from Noon the former extent may be tripled and entred as before and after this manner it is possible to make the whole Limb give the hour next Noon the Versed Sine Duodecupled lies on the other side of the Quadrant and in this case an Ark must first be found in the Limb and the Thread laid over the said Ark counted from the other edge will intersect the said Versed Sine at the Ark sought To find the Suns Azimuth TAke the distance in the Azimuth Scale between the Altitude and the Declination proper to the season of the year and entring it at the Cosine of the Altitude laying the Thread to the other foot according to nearest distance it will shew the Azimuth in the Versed Sines quadrupled or when the Azimuth is near Noon according to the former restrictions for the hour the extent may be doubled or tripled and the answer found in the Versed Sines Octupled or Duodecupled as was done for the hour Example So when the Sun hath 23d 31′ of North Declination his Altitude being 60d. The Azimuth will be found to be 26d 21′ from the South By the like-reason when we found the Hour and Azimuth in the equal Limb by the Diagonal Scale if those extents had been doubled the Hour and Azimuth near six or the Vertical might have been found in a line of Sines of 30d put thorow the whole Limb but that we thought needless FINIS THE DESCRPITION AND VSES Of a Great Universal Quadrant With a Quarter of Stofters particular Projection upon it Inverted Contrived and Written by John Collins Accomptant and Student in the MATHEMATIqUES LONDON Printed in the Year 1658. The DESCRIPTION Of the Great Quadrant IT hath been hinted before that though the former contrivance may serve for a small Quadrant yet there might be a better for a great one The Description of the Fore-side On the right edge from the Center is placed a line of Sines On the left edge from the Center a line of Versed Sines to 180d. The Limb the same as in the small Quadrant Between the Limb and the Center are placed in Circles a Line of Versed Sines to 180d another through the whole Limb to 90d. The Line of lesser Sines and Secants The line of Tangents The Quadrat and Shaddows Above them the Projection with the Declinations Days of the moneth and Almanack On the left edge is placed the fitted Hour and Azimuth Scale Within the Projection abutting against the Sines is placed a little Scale called The Scale of Entrance being graduated to 62d and is no other but a small line of Sines numbred by the Complements At the end of the Secant is put on the Versed Sines doubled that is to twice the Radius of the Quadrant and at the end of the Tangents tripled to some few degrees to give the Hour and Azimuth near Noon more exactly The Description of the Back-side On the right edge from the Center is placed a Line of equal part being 10 inches precise decimally subdivided On the out-side next the edge is placed a large Chord to 60â— equal in length to the Radius of the Line of Sines On the left edge is placed a Line of Tangents issuing from the Center continued to 63d 26′ and again continued apart from 60d to 75d The equal Limb. Within it a Quadrant of Ascensions divided into 24 equal hours and its parts with Stars affixed and Letters graved to refer to their Names Between it and the Center is placed a Circle whereof there is but three Quadrants graduated The Diameter of this Circle is no other then the Dyalling Scale of 6 hours or double Tangents divided into 90d. Two Quadrants or the half of this Circle beneath the Diameter is divided into 90 equal parts or degrees The upper divided Quadrant is called the Quadrant of Latitudes From the extremity of the said Quadrant and Perpendicular to the Diameter is graduated a Line of Proportional Sines M Foster call it the Line Sol. Diagonal-wise from one extremity of the Quadrant of Latitude to the other is graduated a line of Sines that end numbred with â—0 d that is next the Diameter being of the same Radius with the Tangents Opposite and parallel thereto from 45d of the Semicircle to the other extremity of the Diameter is placed a Line of Sines equal to the former Diagonal-wise from the beginning of the Line Sol to the end of the Diameter is graduated a Line of 60 Chords From the beginning of the Diameter but below it towards 45d of the Semicircle is graduated the Projection Tangent alias a Semi-tangent to 90d being of the same Radius with the Tangents The other Quadrant of this Circle being only a void Line there passeth through it from the Center a Tangent of 45d for Dyalling divided into 3 hours with its quarters and minutes Below the Diameter is void space left to graduate any Table at pleasure and a Line of Chords may be there placed Most of these Lines and the Projection have been already treated upon in the use of the small Quadrant those that are added shall here be spoke to Of the Line of Versed Sines on the left Edge issuing from the Center THis Line and the uses of it were invented by the learned Mathematician M. Samuel Foster of Gresham Colledge deceased from whom I received the uses of it applyed to a Sector I shall and have added the Proportions to be wrought upon it and in that and other respects diversifie from what I received wherein I shall not be tedious because there are other ways to follow since found out by my self The chief uses of it are to resolve the two cases of the fourth Axiom of Spherical Trigonometry as when three sides are given to find an Angle or two sides with the Angle comprehended to find the third side which are the cases that find the Hour and Azimuth generally and the Suns Altitudes on all hours For the Hour the learned Author thought meet to add a Zodiaque of the Suns-place annexed to it both in the use of his Sector as also
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-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-lines are to passe After the same manner any 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-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
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-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-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-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-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-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-line desired to be drawn Also the other point in the same reflected 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-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-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-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-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
As the length of the graduated Scale To any lesser length So the parts of the Scale To the Proportional like parts to that other length Of the Line of Tangents on the left edge of the Quadrant THe chief Uses of this Scale will be to operate Proportions either in Tangents alone or jointly either with Sines or equal parts to prick down Dyals and to proportion out a Tangent to any lesser Radius To work Proportions in Tangents alone 1. Of the greater to the less Enter the second Tearm taken out of the Scale upon the first laying the Thread to the other foot then the nearest distance from the third Tearm to the Thread being taken out and measured from the Center shews the 4th Proportional But if the Proportion be of the less to the greater Enter the first Tearm taken out of the Scale upon the second and lay the Thread to the other foot then enter the third Tearm taken out of the Scale between the Thread and the Scale and the foot of Compasses will shew the 4 Proportional Example Of the greater to the less As the Tangent of 50d To the Tangent of 20d So the Tangent of 30 To the Tangent of 10d. To work this take the Tangent of 20◠in the Compasses and entring one foot of that extent at 50d lay the Thread to the other according to the nearest distance then will the nearest distance from the Tangent of 30d to the Thread being measured on the Line of Tangents from the Center be the Tangent of 10d the fourth Proportional By inverting the Order of the Tearms it will be Of the less to the greater As the Tangent of 20d To the Tangent of 50d So the Tangent of 10d To the Tangent of 30d to be wrought by a Parralel entrance This Scale of Tangents is continued but to two Radii or 63d 26′ whereas in many Cases the Tearms given or sought may out-reach the length of the Scale in such Cases the Propprtion must be changed according to such Directions as are given for varying of Proportions at the end of the 16 Cases of right angled Sphoerical Triangles In two Cases all the Rules delivered for varying of Proportions will not so vary a Proportion as that it may be wrought on this Line of Tangents First when the first Tearm is greater then 63d 26′ the length of the Scale and the rwo middle Tearms each less then 26d 34′ the Complement of the Scale wanting In this Case if any two Tearms of the Proportion be varied according to the Rules for varying of Proportions there will be either in the given Tearms or Answer such a Tangent as shall exceed the length of the Scale but it may be remedied by a double Proportion by the reason before delivered for introducing the Radius into a Proportion wherein it is not ingredient As the Radius To the Tangent of one of the middle Tearms So the Tangent of the other middle Tearm To a fourth Tangent Again As the Radius To that fourth Tangent So is the Cotangent of the first Tearm To the Tangent of the fourth Ark sought The Radius may be otherways introduced into a Proportion then here is done but not conducing to this present purpose and therefore not mentioned till there be use of it which will be upon the backside of a great Quadrant of a different contrivance from this upon which this trouble with the Tangents will be shunned An Example for this Case As the Tangent of 65d To Tangent of 24d So the Tangent of 20d To what Tangent the Proportion will find 4d 10′ Divided into two Proportions will be As Radius To Tangent 24d So Tangent of 20d To a fourth the quantity whereof need not be measured Again As Radius To that fourth So the Tangent of 25d the Complement of the first Tearm To the Tangent of 4d 26′ the fourth Tangent sought Operation First enter the Tangent of 24d on the Radius or Tangent of 45d laying the Thread to the other foot then take the nearest distance to it from 20d and enter that extent at 45d laying the Thread to the other foot then will the nearest distance from 25d to the Thread if measured from the Center be the Tangent of 5d 26′ sought The second Case is when the first Tearm of the Proportion is less then the Complement of the Scale wanting and the two middle Tearms greater then the length of the Scale This ariseth from the former for if the Tearms given were the Complements of those in the former Example they would be agreeable to this Case and so no further direction is needful about them for the Tangent sought would be the Complement of that there found namely 84d 34′ Hence it may be observed that a Table of natural Tangents only to 45d or a Line of natural Tangents only to 45d may serve to operate any Proportion in Tangents whatsoever To Proportion on out a Tangent to any Radius Enter the length of the Radius proposed upon the Tangent of 45d and to the other foot of the Compasses lay the Thread according to the nearest distance then if the respective nearest distances from each degree of the Tangents to the Thread be taken out they shal be Tangents to the assigned Radius Because the Tangents run but to 63d 26′ whereas there may be occasion in some declining Dyalls to use them to 75d though seldom further to supply this defect they may be supposed to break off at 60 and be supplied in a Line by themselves not issuing from the Center or only pricks or full-points made at each quarter of an hour for the 5th hour that is from 60d to 75d and so these distances prickt again from the Center as here is done either one way or other the Proportion will hold As the common Radius of the Tangents Is to any other Assigned Radius So is the difference of any two Tangents to the common Radius To their Proportional difference in that Assigned Radius And so having Proportioned out the first four hours the 5th hour may be likewise Proportioned out and pricked forward in one continued streight Line from the end of the 4th hour To work Proportions in Sines and Tangents by help of the Limb and Line of Tangents issuing from the Center THough this work may be better done on the backside where the Tangents lye in the Limb and the Sines issue from the Center and where also there is a Secant meet for the varying of some Proportions that may excur yet they may be also performed here supposing the Radius introduced into any Proportion wherein it is not ingredient the two middle Tearms not being of the same kind as both Tangents or both Sines To find the 4th Proportional if it be a Sine Lay the Thread to the Sine in the Limb being one of the middle Tearms and from the Tangent being the other middle Tearm take the nearest distance
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
said Line and lay the Thread to the other foot according to the nearest distance and in the Limbe it shews the Ark sought Whence may be observed how to prick of an angle by Sines instead of Chords From this and some other following Propositions I assert the Hour and Azimuth may be found generally by the sole help of the Limb of a Quadrant without Protraction How from the Lines inscribed in the Limbe to take off a Sine Tangent Secant and Versed Sine to any Radius if less then half the common Radius of the Quadrant IT hath been asserted that a Sine may be taken off from the Limb and by consequence any other Line there put on for by being carried thither they are converted into Sines and put on in the same manner for by the Definition of Sines if Lines were carryed Parralel to the right edge of the Quadrant from the equal degrees of the Limb to the left edge they would there constitute a Line of Sines and the Converse To find the fixed Point enter the Radius proposed twice down the Line of Sines from the Center or which is all one Lay the Thread over 30 d of the Limb counted from the right edge towards the left and enter the limitted Radius between the Thread and the Scale so that one foot turned about may just touch the Thread and the other resting on the Line of Sines shews the fixed Point at which if the Compasses be always set down and the Thread laid over any Ark in the Tangent Secant or lesser Sines the nearest distances from the said Point to the Thread shall be the Sine Tangent Secant of the said Ark to the limitted Radius But for such Lines as are put on to the common Radius as the Tangent of 45 d c. the Radius is to be entred but once down from the Center to find the fixed Point Of the Line of Secants This Line singly considered is of small use but junctim with other Lines of great use for the general finding the Hour and Azimuth Mr Foster makes use of it in his Posthuma to graduate the Meridian Line of a Mercators Chart which is done by the perpetual addition of Secants and the like may be done from this Line lying in the Limb but a better way wil be to do it from a well graduated Meridian Line by doubling or folding the edge of the Chard thereto and so graduate it by the Pen. Of the Line of Tangents The joint use of this Line with the Line of Sines is to work Proportions in Sines and Tangents in any Proportion wrought by help of Lines in the Limb wherein the Radius is not ingredient the Radius must be introduced according to the general Direction If the two middle Tearms be Sines there must be one Proportion wrought wholly on the Line of Sines on the Backside and another on the Line of Tangents on the foreside but such Cases are not usual But if the two middle Tearms be Tangents the first Operation must be on the line of Tangents on the foreside and the latter on the line of Sines on this backside unless the Radius be ingredient A general Direction to work Proportions when the middle Tearms are of a different Species If a Sine be sought Lay the Thread to the Tangent in the Limb being one of the middle Terms and from the Sine being another of the middle Terms take the nearest distance to it then lay the Thread to the other Tangent in the Limb being the first Tearm and enter the former extent between the Scale and the Thread and the foot of the Compasses on the Line of Sines will shew the fourth Proportional Example If the Proportion were As the Tangent of 30 d To the Sine of 25d So is the Tangent of 20 d To the Sine of 15 d 27′ Lay the Thread over the Tangent of 20 d in the Limb and from the Sine of 25 d take the nearest distance to it then lay the Thread to the Tangent of 30 d and the former extent so entred that one foot resting on the Sines the other foot turned about may but just touch the Thread and the resting foot will shew 15 d 27′ for the Sine sought 2. If a Tangent be sought Lay the Thread to the Tangent being one of the middle Tearms and from the other middle Tearm being a Sine take the nearest distance thereto then Enter one foot of that extent at the first Tearm being a Sine and the Thread laid to the other foot shews the fourth Proportional in the Line of Tangents in the Limb. Example So if the Proportion were As the Sine of 25 d To the Tangent of 30 d So is the Sine of 32 d To a Tangent the fourth Proportional would be found to be the Tangent of 35 d 54′ If the answer fall near the end of the Scale of Tangents the latter entrance may be made by laying the Thread to the first Tearm in the Limb and by a Parralel entrance an Ark found on the Line of Sines then if the Thread be laid over the like Ark in the Limb it will intersect the Tangent sought These Directions presuppose the varying of the Proportion when the Tangens excur the length of the Scale according to the Directions in the Trigonometrical part but as before suggested those Directions are insufficient when one of the Tearms or Tangents are less then the Complement of the Scale wanting and the other greater then the length of the Scale for two such Arks cannot be changed into their Complements without still incurring the same inconvenience in this Case only change the greater Tearm which may be done by help of the Line of Secants for As the Tangent of an Arch To the Sine of another Arch So is the Cosecant of the latter Arch To the Cotangent of the former And by Transposing the Order of the Tearms As a Sine To a Tangent So the Cotangent of the latter Arch To the Cosecant of the former Example If the Proportion were As the Sine of 8d To the Tangent of 25d So is the Sine of 60d To the Tangent of 71d Here we might foreknow by the nature of the Tearms that the Tangent sought would be large or finde by tryal that it cannot be wrought upon the Quadrant We may therefore vary it thus As the Tangent of 25d To the Sine of 8d So the Secant of 30d To the Tangent of 19d the Complement of 71d the Arch sought Lay the Thread over 8d in the lesser Sines and set down one foot of the Compasses at the Sine of the same Arch the Thread lyes over in the Limb and take the nearest distance to the Thread laid over the Secant of 30 then lay the Thread to the Tangent of 25d and enter the former extent between the Thread and the Line of Sines and the distance of the foot of the
Compasses from the Center measured on the Tangents on the foreside sheweth 19d. But a more general Caution in this Case without the help of the Secants would be by altring the larger Tangent into its Complement by introducing the Radius and operating the Proportion on the greater Tangent of 45d. If the Proportion were As the Tangent of 70d To the Sine of 60d So the Tangent of 25d To the Sine of 8d 27′ By introducing the Radius at two Operations it would be easily wrought As Radius To Tangent 25d So Sine 60d To a fourth Again As the Radius To the Tangent of 20 d So that fourth To the Sine sought So the former Example wherein a Tangent is sought may be likewise varied As Radius To Tangent 25d So Sine 60d To a fourth Again As that fourth To the Radius So is the Sine of 8d To the Cotangent of the Arch sought namely to the Tangent of 19d as before Two Proportions with the Radius in each are as suddenly done as one without the Radius Operation Lay the Thread over the Tangent of 25d in the greater Tangents and from the Sine of 60d take the nearest distance to it enter that extent at 90 or the end of the Line of Sines laying the Thread to the other foot according to the nearest distance then enter the Sine of 8 parralelly between the Scale and the Thread and the distance of the foot of the Compasses from the Center is the Tangent of the Complement of the Ark sought to be measured in the greater Tangents by setting down one foot at 90d and the Thread laid to the other according the nearest-will lye over the Tangent of 19 d. An Example with the Radius ingredient and a Sine sought Data Latitude and Declination to find the time when the Sun shall be East or West As the Radius To the Cotangent of the Latitude So the Tangent of the Declination To the Sine of the hour from 6. To be wrought by the help of the lesser Tangents When the Radius comes first and two Tangents in the middle change the largest Ark into its Complement to bring it into the first place and the Radius into the second then take out the Tangent of the other middle Ark either from the foreside from the Scale or out of the Limb by setting one foot at the Sine of 90 d and taking the nearest distance to the Thread laid over the Tangent given then laying the Thread to the Tangent of the first Ark enter the former extent between the Scale and the Thread and the foot of the Compasses will shew the Sine sought Otherways the two middle Tearms being Tangents as also when the first Tearm and one of the middle Tearms is a Tangent change the Radius and one of those Tangents into Sines For As the Radius To the Tangent of any Ark So is the Cosine of the said Ark To the Sine thereof And As the Tangent of any Ark To Radius So is the Sine of that Ark To the Cosine thereof And so the former Proportion changed will be As the Sine of the Latitude To the Cosine of the Latitude So the Tangent of the Declination To the Sine of the hour from six When the Sun shall be East or West Example If the Declination were 23d 30′ North in our Latititude of London 51d 32′ to find the Sine sought Lay the Thread to the Tangent of the Declination in the Limb and from the Complement of the Latitude in the Sines take the nearest distance to it then lay the Thread to the Sine of the Latitude in the lesser Sines and enter the former extent between the Thread and the Scale and the foot of the Compasses sheweth the answer in degrees if the Thread be laid to the Ark found in the Limb it there sheweth it in Time So in this Example the time sought is 20d 14′ or in Time 1 h 17 h before 6 in the morning or after it in in the Evening If the Latitude and Declination were given To find the Suns Azimuth at the Hour of 6. As the Radius To Cosine of the Latitude So the Tangent of the Suns Declination To the Tangent of his Azimuth from the Vertical In this Case a Tangent being the 4th Tearm sought the Operation is very facil Lay the Thread to the Tangent of the Declination in the lesser Tangents and from the Cosine of the Latitude take the nearest distance to it and either measure that extent on the Tangents on the foreside or set one foot of that extent upon the Sine of 90d and to the other lay the Thread and it will intersect the Tangent sought in the Limb So in our Latitude when the Sun hath 23d 30′ of declination his Azimuth at the hour of 6 will be 15 d 9′ from the East or West Another Example So if the Suns distance from the nearest Equinoctial Point were 60 d his right Ascension would be found to be 57 d 48′ The Proportion to perform this Proposition is As the Radius To the Cosine of the Suns greatest Declination So the Tangent of the Suns distance from the next Equinoctial Point To the Tangent of the Suns right Ascension or when the Tangents are large As the Cosine of the Suns greatest declination To Radius So the Cotangent of the Suns distance from the Equinoctial Point To the Cotangent of his right Ascension By what hath been said it appears that the working Proportions by the natural Lines is more troublesome then by the Logarithmical however this trouble wil be shunned in the use of the great Quadrant by help of the Circle on the backside I now come to shew how the Hour of the Day and the Azimuth of the Sun may be found universally by the Lines on the Quadrant which is the principal thing intended The first Operation for the Hour will be to find what Altitude or Depression the Sun shall have at the hour of 6. The Proportion to find it is As the Radius To the Sine of the Latitude So the sine of the Suns Declination To the sine of the Altitude sought Example So in Latitude 51 d 32′ the Suns declination being 23 d 31′ To find his Altitude or Depression at 6 Lay the Thread to the Sine of the Latitude in the Limb and from the sine of the Suns Declination take the nearest distance to it which extent measured from the Center will be found to be 18 d 12′ This remains fixed for one Day and therefore must be recorded or have a mark set to it Afterwards the Proportion is As the Cosine of the Declination To the Secant of the Latitude Or As the Cosine of the Latitude To the Secant of the Declination So in Summer is the difference but in Winter the Sum of the sines of the Suns proposed or observed Altitude and of his Altitude or Depression at 6 To
the Sine of the hour from 6 towards Noon in Winter as also in Summer when the Altitude is more then the Altitude of 6 otherways towards Midnight To Operate this In Winter to the sine of the Suns Depression at 6 add the sine of the Altitude proposed by setting down the extent hereof outward at the end of the former extent in Summer take the distance between the sine of the Suns Altitude and the sine of his Altitude at 6 and enter either of these extents twice down the Line of sines from the Center then lay the Thread to the Secant being one of the middle Tearms and take the nearest distance to it Lastly enter one foot of this extent at the first Tearm being a Sine and to the other foot lay the Thread and in the equal Limb it shews the hour from 6 which is accordingly numbred with hours But when the Hour is neer Noon the answer may be found in the Line of Sines with more certainty by laying the Thread to the first Tearm in the Limb and entring the latter extent Parralelly between the Scale and the Thread Otherways Enter the aforesaid sum or difference of sines once down the Line of Sines from the Center and laying the Thread to the Secant being one of the middle Tearms take the nearest distance to it then lay the Thread to the first Ark in the lesser sines and enter the former extent between the Thread and the Scale and the foot on the Compasses on the Line sheweth the Sine of the Hour Example If the Altitude were 45d 42′ take the distance between it and the sine of 18 d 12′ before found enter this extent twice down the Line of sines from the Center and laying the Thread over the Secant of 51 d 32′ take the nearest to it then entring one foot of this extent at 66 d 29′ in the Line of Sines the Thread being laid to the other according to nearest distance will lye over 45 d in the Limb shewing the hour to be either 9 in the morning or 3 in the afternoon and so it will be found also in the latter Operation by entring the first extent once down the sines and taking the distance to the Thread lying over the Secant of the Latitude and then laying the Thread to 66 d 29′ in the 〈◊〉 and entring that extent between the Scale and the Thread To find the Suns Azimuth The first Operation will be to get the Suns Altitude in the Vertical Circle that is being East ar West As the sixe of the Latitude To Radius So is the Sine of the Declination To the sine of the Altitud So in our Latitude of London when the Sun hath 23d 31′ of declination his Vertical Altitude in Summer will be found to be 30d 39′ and so much is the Depression when he hath as much South declination This found either by a Parralel entrance on the Line of Sines by laying the Thread to the sine of the Latitude in the Limb and entring the sine of the Declination between the Scale and the Thread or by a Lateral entrance in the Limbe changing the Radius into the third place and then enter the sine of the Declination on the Sine of the Latitude laying the Thread to the other foot and in the Limb it shewes the Altitude sought having found this Ark let it be recorded or have a mark set to it because it remains fixed for one Day afterwards the Proportion to be wrought is As the Cosine of the Altitude To the Tangent of the Latitude So in Summer is the difference in Winter the sum of the Sines of the Suns Altitude and of his Vertical Altitude or Depression To the Sine of the Azimuth from the East or West towards noon Meridian in Winter as also in Summer when the given Altitude is more then the Vertical Altitude but if less towards the Midnight Meridian This Proportion may be wrought divers ways on the Quadrant after the same manner as the former I shall therefore illustrate it by some Examples Declination 13d Latititude 51d 32′ Vertical Altitude 16d 42′ Proposed Altitude in Summer 41d 53′ Proposed Altitude in Winter 12 13 Enter the aforesaid sum or difference of Sines twice down from the Center of the Quadrant and take the nearest distance to the Thread being laid over the Tangent of the Latitude this extent set down at the Cosine of the Altitude and lay the Thread to the other foot and in the Limbe it shews the Azimuth sought So in this Example the Azimuth will be found to be 40d both in Summer and Winter from East or West towards Noon Meridian Otherways Enter the aforesaid sum or difference of sines but down from the Center and take the nearest distance to the Thread laid over the Tangent of the Latitude then lay the Thread to the Complement of the Altitude in the lesser sines and enter the former extent between the Scale and the Thread and the answer will be given in the Line of Sines supposing the declination unchanged if the Altitude were 9d 21′ both for the Winter and the Summer Example the Azimuth at London would be 9d 22′ from the East or West Northwards in Summer and 35 d Southwards in Winter Hitherto we suppose the Latitude not to exceed the length of the Tangents whether it doth or not this Proportion may be otherways wrought by changing the two first Tearms of it Instead of the Co-sine of the Altitude to the Tangent of the Latitude we may say As the Cotangent of the Latitude To the Secant of the Altitude So when the Sun hath 23 d 31′ of North Declination in our Latitude and his Altitude 57 d 7′ take the distance between the sine thereof and the sine of 30 d 39′ the Altitude of East and enter it once down from the Center and take the nearest distance to the Thread laid over the Secant of the Altitude viz. 57 d 7′ then lay the Thread to 38 d 28′ in the Tangents and enter the former extent between the Scale and the Thread and the Compasses on the Line of Sines will rest at 50 d for the Azimuth from East or West Southwards because the Altitude was more then the Vertical Altitude Otherways without the Secant in all Cases by help of the greater Tangent of 45d Enter the aforesaid Sum or difference of the Sines once down from the Center and lay the Thread to the Tangent or Cotangent of the Latitude in the greater Tangents and take the nearest distance to it Then for Latitudes under 45d enter the former extent at the Complement of the Altitude in the Line of Sines and find the answer in the Limb by laying the Thread to the other foot or if it be more convenient make a Parralel entrance of it and find the answer in the Sines as before hinted But for Latitudes above 45 d first find a fourth by entring the sum or
and enter the extent between the ☉ Altitude which is nothing that is from the beginning of the Hour Scale to the Declination between the Scale and the Thread and the foot of the Compasses shews it in the Line of sines which may be converted into Time by help of the Limb. If these Scales be continued further in length as also the Declinations they will after the same manner find the Stars hour for any Star whatsoever to be converted into common Time as in the uses of the Projection as also the Azimuth of any Star that hath less declination then the place hath Latitude but of this more in the next Quadrant In Dyalling there will be often use of natural sines whereas these Scales are continued but to 62 d if therefore it be desired to take out any sine to the same Radius the rest of the Scale wanting may be easily supplyed for the difference of the sines of any two Arks equidistant from 60 d is equal to the sine of their distance Thus the sine of 20 d is equal to the difference of the sines of 40 d and 80 d Arks of like distance from 60 d on each side and so may be added either to 40 d forward or the other way from the end of the Scale In finding the Hour and Azimuth by these Scales not in the Versed sines the Directions altogether prescribe a Parralel entrance but if the Extent from the Altitude to the Declination be entred at the Cosine of the Altitude or of the Declination in the Line of sines according as the Case is and the Thread laid to the other foot the Hour and Azimuth may be found in the lesser sines by a Lateral entrance Or if the said Extent be doubled and entred as before hinted the answer will be found in the equal Limb. Example to find the Suns Azimuth Declination 23 d 31′ North. Altitude 41 34 Having taken the distance between these two Tearms in the Azimuth Scale and doubled it enter one foot in the Line of sines at 48 d 26′ the Complement of the Altitude and laying the Thread to the other according to nearest distance it will lye over 15 d of the equal Limb for the Suns Azimuth from the East or West Southwards The Vse of the Versed Sin 's in the Limbe It may be noted in the former general Proportion I have used the word Azimuth from Noon or Midnight Meridian though not so proper because they are more universal and common to both Hemispheres other expressions besides their Verbosity would be full of Caution for the following Proportion in our Northern Hemispere without the Tropick that finds it from the South between the Tropick of Cancer and the Equinoctial when the Sun comes to the Meridian between the Zenith and the Elevated Pole would find it from the North wherefore it is fit to be retained A general Proportion for finding the Hour As the Cosine of the Declination To the Secant of the Latitude Or As the Cosine of the Latitude To the Secant of the Declination So is the difference of the Sines of the Suns Altitude proposed and of his Meridian Altitude To the Versed Sine of the hour from Noon◠And So is the sum of the sines of the Suns proposed Altitude and of his Midnight Depression To the Versed sine of the hour from Midnight And So is the sine of the Suns Meridian Altitude To the Versed sine of the Semidiurnal Ark And So is the sine of the Suns Midnight Depression To the Versed sine of the Seminocturnal Ark. The Operation will be like the former I shall therefore onely illustrate it by one Example the Meridian Altitude is got in Winter by differencing in Summer by adding the Declination to the Complement of the Latitude if the sum exceed 90 d the Complement thereof to 180 d is the Meridian Altitude An Example for finding the Hour from Noon Declination 23d 31′ North the 11th June Comp. Latitude 38 28 London  61 59 Meridian Altitude Proposed Altitude 36 42 take the distance between the sines of these two Arks and enter it once down the Line of sines from the Center and take the distance to the Thread laid over the Secant then enter one foot of that extent at the sine being the first Tearm and to the other lay the Thread and in the Versed sines in the Limb it will lye over the Versed Sine of the hour from Noon In this Example if the Thread be laid over the Secant of 51d 32′ the extent must be entred at the sine of 66d 29′ 23 31 the extent must be entred at the sine of 38 28 either way the answer will fall upon 60 d of the Versed sine shewing the Hour to be either 8 in the forenoon or 4 in the afternoon If the hour fall near noon then the extent of the Compasses may be Quadrupled and entred as before and look for the answer in the Versed Sines Quadrupled Or before the distance be took to the Thread the extent of difference may be entred four times down from the Center The Converse of this Proposition will be to find the Suns Altitude on all Hours universally As the Secant of the Latitude To Cosine Declination Or As the Secant of the Declination To Cosine Latitude So the Versed sine of the hour from Noon To the difference of the sines of the Suns Meridian Altitude and of his Altitude sought to be substracted from the sine of the Meridian Altitude and there will remain the sine of the Altitude sought So in Latitude of London if the Suns Declination were 13 d 00′ and the hour from noon 75 d that is either 7 in the morning or 5 in the afternoon Lay the Thread over the Versed sine of the hour from noon namely 75 d and from the sine of 77 d the Complement of the Declination take the nearest distance to it then lay the Thread to the Secant of the Latitude and enter the former extent between the Scale and the Thread and you will find a sine equal to the difference sought which sine take between the Compasses and setting down one foot at the sine of 51 d 28′ the Meridian Altitude the other foot turned towards the Center will fall upon the sine of 19 d 27′ the Altiude sought A General Proportion for the Azimuth Get the Remainder or Difference between these two Arks the Suns Altitude and the Complement of the Latitude by Substracting the less from the greater and then the Proportion will hold As the Cosine of the Latitude Is to the Secant of the Altitude Or As the Cosine of the Altitude To the Secant of the Latitude So is the sum of the sines of the Suns Declination and of the aforesaid Remainder To the Versed Sine of the Azimuth from the Noon Meridian in Summer only when the Suns Altitude is less then the Complement of the Latitude In all other Cases So is the
difference of the said sines To the Versed sine of the Azimuth as before from Noon Meridian Example The 11th of June aforesaid the ☉ having 23 d 31′ of North declination his Altitude was observed to be 18 d 20′ which substracted from 38 d 28′ the remainder is 20 d 8′ take out the sine thereof and set down one foot at the sine of 23 d 31′ and set the other forwards towards 90 d then take the nearest to the Thread laid over the Secant of the Latitude 51 d 32′ enter one foot of this Extent at the Complement of the Altitude by reckoning the Altitude it self from 90 d towards the Center and the Thread laid to the other foot cuts the Line of Versed sines at 105 d the Azimuth from the South The same day when the Altitude was more then the Colatitude suppose 60 d 11′ the Remainder will be found to be 21 d 43′ take the distance between the sine thereof and of 23 d 31′ and because the Extent is but small enter it four times down the Line of sines from the Center and take the nearest distance to the Thread laid over the Secant of the Latitude which entred at the Cosine of the Altitude the Thread laid to the other foot shews 25 d in the Quadrupled Versed Sines for the Azimuth from the South The Proportion hence derived for the Amplitude As the Cosine of the Latitude To Secant of the Declination c. as before So in Summer is the sum in Winter the difference of the sines of the Suns Declination and of the Complement of the Latitude To the Versed Sine of the Amplitude from Noon Meridian The Proportion for the Azimuth will be better exprest by making the difference to be a difference of Versed sines How the Versed sines in the Limbe may be spared in Case a Quadrant want them If a Quadrant can only admit of a Line of sines from the Center the common Quadrant of Mr Gunters very well may on the right edge above the Margent for the Numbers of the Azimuths it may be easily fitted for any or many Latitudes by setting Marks or Pricks to the Tangent and Secant of the Latitudes in the Limbe which may be taken out by help of the Limb Line of Sines or by Protraction and either of these general Proportions wrought upon it or those which follow if it be observed that whensoever the Thread lyes over the Versed sine of any Ark in the Limbe it also at the same Time lyeth over a Sine equal to half that Versed sine to the common Radius Now because the sine of 30d doubled is equal to the Radius let it be observed whether the sine cut by the Thread be greater or less then 30 deg When it is less let the Line of Sines represent the former half of a Line of Versed Sines and take the sine of the Ark the Thread lay over and enter it twice forward from the end of the Scale towards the Center and you will obtain the Versed sine of the angle sought When it is more take the distance between the sine of 30 d and the said sine and letting the Line of sines represent the latter half of a Line of Versed sines enter the said distance twice from the Center and you will obtain the Versed sine of the Arch sought namely the sine of an Arch whereto 90 d must be added Three sides to find an Angle a general Proportion As the Sine of one of the Sides including an angle Is to the Secant of the Complement of the other including side So is the difference of the Versed Sines of the third side and of the Ark of difference betwen the two including sides To the Versed Sine of the Angle sought And So is the difference of the Versed sines of the third side and of the sum of the two including sides To the Versed sine of the sought angles Complement to 180d To repeat the Converse when two sides and the angle comprehended are given to find the third side were needless If one of the containing sides be greater then a Quadrant instead of it in referrence to the two first Tearms of the Proportion take the Complement thereof to 180d for the reputed side but in differencing or summing the two containing sides alter it not And further note that the same Versed sine is common to an Ark less then a Semicircle and to its Complement to 360 d. The Operation of this Proportion will be wholly like the former so that there needs no direction but only how to take out a difference of two Versed sines to the common Radius seeing this Quadrant of so small a Radius is not capable of such a Line from the Center And here note that the difference of two Versed sines less then a Quadrant is equal to the difference of the natural sines of the Complements of those Arks. And the difference of two Versed sines greater then a Quadrant is equal to the difference of the natural sines of the excess of those Arks above 90 d. And by consequence the difference of the Versed sines of two Arks the one less the other greater then a Quadrant is equal to the sum of the natural sines of the lesser Arks Complement to 90d and the greater Arks excess above it And so a difference of Versed sines may be taken out of the Line of natural sines considered as such Or the Line of sines may be considered sometimes to represent the former half of a Line of Versed Sines as it is numbred with the small figures by its Complements from the end of it to 90d at the Center and sometimes the latter half of it and then the graduations of it as Sines must be considered as numbred from 90 d to 180 d at the end of it and so a difference to be taken out of it by taking the distance between the two Tearms which if the two Arks fall the one to be greater the other less then 90 d will be a sum of two sines as before hinted and in this Case the sine of the greater Arks excess above 90 d to be set down outwards if it may be at the Versed sine of the lesser Ark or which is all one at the sine of that Arks Complement and the distance from the Exterior foot of the Compasses to the Center will be equal to the difference of the Versed sines of the Arks proposed To measure a difference of Versed sines to the common Radius In this Case also the Line of sines must sometimes represent the former sometimes the latter half of a Line of Versed sines and then one foot of the difference applyed to one Ark the other will fall in many Cases upon the Ark sought each Proportion variously exprest so that possibly either one or the other will serve in all Cases But if one of the feet of the Compasses falls beyond the Center of the Quadrant To find
in this Latitude which he hath often made upon Rulers for Carpenters and other Artificers and Diallists and whereof he was willing to afford them a Print whereto I have added other Scales for giving the Hour and Azimuth near Noon On the Backside are drawn these Lines A large Dyalling Scale of 6 hours or double Tangents with a Line of Latitudes fitted thereto A large Chord A Line for the Substiles distance from the Meridian A Line for the Stiles height A Line for the angle of 12 and 6. A Line for the inclination of Meridians All these Scales relate to Dyalling An Azimuth Scale being two Lines of natural sines of the same Radius set together at O and thence numbred with Declinations this Scale must be made of the same sine that the hour Scale following is made of continued from O one way to 38d 28′ and the other way to 23d 31′ or further at pleasure but numbred from the beginning which is at the end of that 38d 28′ the Complement of the Latitude with 10d 20′ c. up to 60d. The Hour Scale is no other then a Line of sines with the declinations set against the Meridian Altitudes in the Latitude of London the Radius of which sine is equal in length to the Dyalling Scale of hours Of the Vses of these Scales The Line of Hours and Latitudes is general for pricking down all Dialls with Centers as will afterwards be shewed in the Use of the great Quadrant and by help of the Scale of Hours may the Diameter of a such a Circle be graduated as is placed in on the back of the great Quadrant and the Line of Latitudes will serve as a Chord to divide the upper Quadrant and the Hour Scale or Line of Sines will serve as a Chord to divide a Semicircle whose Diameter is equal to the Scale of Hours into 90 equal parts and their Subdivisions and hereby may Proportions in sines and Tangents or Tangents alone be wrought by Protraction and so the necessary Arks in Dyalling found generally as is done by Mr Foster in the three last Schems of his Posthuma this will easily be understood if the use of the Circle on this Quadrant be well apprehended The particular Scales give the requisite Arks of upright Decliners in this Latitude by inspection for count the plaines Declination in the Line of Chords and a Square laid over it intersects all those Arks or to be found by applying the Declination taken out of the Chords with Compasses to every other Line Example So if an upright Plain decline 35d from the Meridian The Substiles distance from the Meridian will be 24d 30′ The Stiles height 30 38 The Inclinations of Meridians 41 49′ The angle of 12 and 6 54 10 These particular Scales also resolve some of the Cases of right angled Sphoerical Triangles relating to the Motion of the Sun or Stars thus Of the Line of the Stiles height Account the Declination in the Line for the Stiles height and against it in the Chord stands the Amplitude of the Sun or Stars from the Meridian Example for Amplitude So when the Sun hath 18d of Declination his Amplitude will be 67d 13′ from the Meridian and 29d 47′ from the Vertical The reason hereof is because the two first fixed Tearms of the Proportion that Calculate the Stiles height are the Radius and the Co-sine of the Latitude and the two first Tearms that Calculate the Amplitude are the Cosine of the Latitude and the Radius and therefore must as well serve in this Case as in that On this Stile Line may be found the Suns Altitudes on all hours when he is in the Equinoctial by applying the hour from six taken from the Chords to the other end of the Stile Line Of the Substiler Line Hereby we may find the time of Sun rising and setting take the Declination out of the Substilar Line and measure it on the Line of Chords Example So when the Sun hath 18 of North Declination the Ascensional difference is 24d 9′ in time 1 hour 36½ minutes and so much the Sun rises and sets from six Hereby may be also found the Equinoctial Altitudes to every Azimuth Of the Line for the Angle of 12 and 6. Hereby we may find the time when the Sun will be due East or West Account the Complement of the Declination in this Scale and against it in the Chords stands the hour from six Example So when the Sun hath 18d of North Declination he will be East or West at 7 in the morning or 5 in the afternon By these Scales the requisites of an East or West Reclining or Inclining Diall in this Latitude may be found 1. The Substiles distance from the Meridian Account the Complement of the Reclination Inclination in the Chords and against it in the Line for 12 and 6 stands the Complement of the angle sought 2. For the Stiles height Apply the Reclination in the lesser sines on the Diagonal Scale in the Parralel proper to the Latitude to the greater sine and it shewes the Ark sought 3. For the Inclination of Meridians This may be also found on the Diagonal Scale when the Substiles distance is not more then the Latitude By Accounting the Substiles distance on the greater sine and applying it to the lesser 4. For the Angle of 12 and six Account the Complement of the Reclination in the Chords and against it in the Substilar Line is the Complement of the angle sought So if an East or West Plain Recline or Incline 35d. The Substiles distance from the Meridian will be 45d 52′ The Stiles height 26 41 The Inclination of Meridians 66 27 And the angle of 12 and 6 56 55 Of the Hour and Azimuth Scales This Scale is fitted to find the Hour from Noon in the Versed sine augmented and the Proportion to be wrought by it the same as delivered in the use of the small Quadrant As the Cosine of the Declination Is to the Secant of the Latitude So is the difference of the sines of the Suns proposed and Meridian Altitude To the Versed sine of the hour from Noon And of this one Proportion we make two by introducing the Radius As the Radius is to the Secant of the Latitude So is the former distance To a fourth By fitting the Radius of the sines equal in length to the Secant of the Latiude this first Proportion is removed for the said difference of sines taken out of this fitted Scale is the 4th Proportional the Proportion that remains to be wrought upon the Quadrant is As the Cosine of the Declination Is to the difference of the sines taken out of this fitted Scale So is the Radius To the Versed sine of the hour from Noon By this means if in the same Proportion as we increase the length of the fitted Scale we also increase the versed sines lying in the Limb we may find the hour and Azimuth near noon with certainty if the
in the use of his Scale published since his death entituled Posthuma Fosteri that the Suns place being given which for Instrumental use might be obtained by knowing on what day of each moneth the Sun enters into any Signe and allowing a degree for every days motion come by it prope verum and being sought in the annexed Zodiaque which is no other then two lines of 90d. Sines each made equal to the Sine of 23d 31′ the Suns greatest Declination just against it stands the Suns Declination if accounted in the Versed Sine from 90d each way but this for want of room and because the Declination is more easily given by help of the day of the moneth I thought fit to omit the rather because it may also be taken from the Table of Declinations But from hence I first observed that if the two first terms of a Proportion were fixed if two natural Lines proper to those terms were fitted of an equal length and posited together if any third term be given to find a fourth in the same proportion it would be given by inspection as standing against the third but if the Lines stand asunder or a difference be the third term application must be made from one Line to the other with Compasses as in the same Scale there is also fitted a Line of 60 parts equal in length to the Radius of a small Sine serving to give the Miles in every several Latitude answerable to one degree of Longitude Three sides given to find an Angle the Proportion As the difference of the Versed Sines of the Sum and difference of any two Sides including an Angle Is to the Diameter So is the difference of the Versed Sines of the third side And of the Ark of difference between the two including Sides To the Versed Sine of the Angle sought And so is the difference of the Versed Sines of the third side And of the sum of the two including sides To the Versed Sine of the sought Angles Complement to a Semicircle Corollary And seeing there is such proportion between the latter terms of the fore-going Proportion as between the former omitting the two first terms it also holds As the difference of the Versed Sines of the third side and of the Ark of difference between the two including sides Is to the Versed Sine of the Angle sought So is the difference of the Versed Sines of the third side And of the sum of the two including sides To the Versed Sine of the sought Angles Complement to 180d. And this is the Proportion M. Foster makes use of in his Scale page 25 and 27. to find the Hour and Azimuth by Protraction as also in page 68. in Dyalling when three sides are given to find an Angle by constituting two right angled equi-angled plain Triangles the legs whereof consist of the 4 terms of this Proportion But in that Protraction work the first and third terms of the Proportion are given together with the sum of the second and fourth terms to find out the said terms respectively The Proportion for the Hour As the difference of the Versed Sines of the Sum and difference of the Complement of the Latitude and of the Sun or Stars distance from the Elevated Pole Is to the Diameter or Versed Sine of 180d So is the difference of the Versed Sines of the Complement of the Altitude and of the Ark of difference between the Complement of the Latitude and of the Polar distance To the Versed Sine of the Hour from Noon And if the latter clause of the third term be the Sum of the Co-latitude and Polar distance the Proportion will find the Versed Sine of the hour from midnight And if the sum of any two Arks exceed a Semicircle take its Complement to 360d for the same Versed Sine is common to both When the Declination is towards the Elevated Pole the Polar distance is the Complement of it to 90d and when towards the Depressed Pole the Polar distance is equal to the Sum of 90d and of the Declination added together Example Let the Suns Declination be 15d 46′ North Complement 74d 14′ The Complement of the Latitude 38 28 Sum 112 42 Difference 35 46 And let the Altitude be 20d Complement 70 00 Operation Take the distance between the Versed Sines of 35d 46′ and of 112d 42′ and entring one foot of that extent at the end of the Versed Scale at 180d lay the thred to the other foot according to nearest distance then take the distance between the Versed Sines of 35d 46′ and 70d and entring that extent parallelly between the Thred and the Scale and the other foot will rest upon the Versed Sine of 77d 32′ the quantity of the Hour from the Meridian being either 50′ past 6 in the morning or 10′ past 5 in the afternoon The Reader may observe in this work that the thred lies over a Star by entring the first extent as also that there is the same Star graduated at 35d 46′ of the Versed Sine and this no other then the Bulls eye having 15d 46′ of North Declination for which Star in this Latitude there needs be no summing or differencing of Arks in regard the Stars declination varies not So to find that Stars hour at any time having any other Altitude only lay the thred over that Star in the Quadrant and take the distance between the Star in the Scale and the Complement of its Altitude and enter that extent parallelly between the Thred and the Scale and it finds the Stars hour from the Meridian Thus when that Star hath 39d of Altitude its hour from the Meridian will be found to be 45d 54′ in time 3 hours 3½′ which to get the true time of the night must be turned into the Suns hour by help of the Nocturnal on the Back-side But admitting the Suns Declination and Altitude to have been the same with the Stars the true time of the day thus found would have been 56½′ past 9 in the morning or 3½′ past 3 in the afternoon and thus the Reader may have what Stars he pleases put on of any Declination and for any Latitude and they may be put on at such a distance from the Center that the distance from it to the Star may be a Chord to be measured in the Limb to give the Stars Ascensional difference or the like conclusion And thus the thred being once laid and the former point found for one example to the Suns Declination neither of them varies that day which is a ready general way for finding the time of the day for the Sun To find the Semidiurnal and Seminocturnal Arks. SUppose the Sun to have no Altitude and the Complement of it to be 90d and then work by the former precept and you will find the Semidiurnal Ark from the beginning of the Line and the Seminocturnal Ark from the end of the Line which doubled and turned into time shews the length of the Day and
Night and the difference between 90d and either of those Arks is the Ascensional difference or time of rising and setting from 6. To find the Azimuth generally The Proportions for this purpose have been delivered before from which it may be observed that there are no two terms fixed and therefore to every Altitude the containing sides of the Triangle namely the Complements both of the Altitude and Latitude must be summed and differenced when the Proposition is to be performed on this Line solely and the Operation will be after the same manner as for the hour namely with a Parallel entrance and this is all I shall say of the Authors general way and of any other that he used I never heard of those ways that follow being of my own supply By help of this Line to work a Proportion in Sines alone wherein the Radius leads As the Radius Is to the Sine of any Ark So is the Sine of any other Ark To the Sine of a fourth Ark. This fourth Sine as I have said before is demonstrated by M. Gellibrand to be equal to half the difference of the Versed Sines of the Sum and difference of the two middle terms of the Proportion Operation Let the Proportion be As the Radius Is to the Sine of 40d So is the Sine of 27 To a fourth Sine Sum 67 Difference 13 Take the distance between the Versed Sines of the said sum and difference and measure it on the Line of Sines from the Center and it will reach to 17d the fourth Sine sought By help of this Line may the Divisions of the line Sol or Proportional Sines be graduated to any Radius less then half the Radius of the Quadrant the Canon is As the Versed Sine of any Ark added to a Quadrant Is to the Radius or length of the Line Sol So is the Versed Sine of that Arks Complement to 90d To that length which pricked backward from the end of the Radius of the said Line shall graduate the Arch proposed Example Suppose you would graduate 20 of the Line Sol enter the Radius of the said Line upon the Versed Sine of 110d laying the thred to the other foot and from the Versed Sine of 70d take the nearest distance to the thred which prick from the end of the Line Sol towards the beginning and it shall graduate the said 20d. This Line Sol is made use of by M. Foster in his Scale for Dyalling The Line of Versed Sines was placed on the left edge of the foreside of the Quadrant for the ready taking out the difference of the Versed Sines of any two Arks and to measure a difference of two Versed Sines upon it which are the chief uses I shall make of it whereas to Operate singly upon it it would be more convenient for the hand to have it lie on the right edge of the Quadrant An example for finding the Azimuth generally by help of Versed Sines in the Limb and of other Lines on the Quadrant I shall rehearse the Proportion As the Cosine of the Latitude is to the Secant of the Altitude Or As the Cosine of the Altitude is to the Secant of the Latitude So is the difference of the Versed Sines of the Suns distance from the Elevated Pole and of the Ark of difference between the Latititude and Altitude To the Versed Sine of the Azimuth from the midnight meridian And making the latter clause of the third term the Complement of the Sum of the Latitude and Altitude to a Semicircle the Proportion will find the versed Sine of the Azimuth from the noon Meridian Example Altitude 51d 32′ Latitude 34 32 Complement 55d 2◠Difference 17 00 ☉ distance from elevated Pole 66 29 Operation in the first Terms of the Proportion On the Line of Versed Sines take the distance between 17d and 66d 29′ and entring it twice down the line of Sines from the Center take the nearest distance to the thread laid over the Secant of 51d 32′ the given Altitude and entring one foot of this Extent at the Sine of 55d 28′ the Complement of the Latitude lay the thred to the other foot according to nearest distance and in the line of Versed Sines in the Limb it will lie over 95d for the Suns Azimuth from the midnight meridian And the Suns declination supposed the same he shall have the like Azimuth from the North in our Latitude of London when his Altitude is 34d 32′ for the sides of the Triangle are the same Another Example To find it in the versed Sine of 90d Latitude 47d 27′ Altitude 51 32 Sum 98 59 Complement 81 1 Polar distance 66 29 Take the distance in the Line of Sines as representing the former half of a Line of Versed Sines between these two Arks counted towards the Center viz. 66d 29′ and 81d 01′ and enter this extent twice down the Line of Sines from the Center and take the nearest distance to the thred lying over the Secant of the Latitude 47d 27′ then enter one foot of this extent at 51d 32′ counted from the end of the Sines towards the Center laying the thred to the other foot according to nearest distance and in the Versed Sine of 90d it shews the Azimuth to be 65d from the South in this our Northern Hemispere Of the fitted Particular Scale and the Line of Entrance thereto belonging THis Scale serves to find both the Hour and Azimuth in the Latitude of London to which it is fitted in the equal Limb by a Lateral or positive Entrance it consists of two Lines of Sines The greater is 62d of a Sine as large as can stand upon the Quadrant the Radius of the lesser Sine is made equal to 51d 32′ of this greater being fitted to the Latitude The Scale of Entrance standing within the Projection and abutting on the Line of Sines is no other but a portion of a Line of Sines whose Radius is made equal to 38d 28′ of the greater Sine of the fitted Scale and this Scale of Entrance is numbred by its Complements up to 62d as much as is the Suns-greatest meridian Altitude in this Latitude The ground of this Scale is derived from the Diagonal Scale the length whereof bears such Proportion to the Line of Sines whereto it is fitted as the Secant of the particular Latitude doth to the Radius which is the same that the Radius bears to the Cosine of the Latitude and consequently making the Line of Sines to represent the fitted Scale the Radius of that Sine whereto it is fitted must be equal to the Cosine of the Latitude and so we needed no particular Scale but this would remove the particular Scale or Scale of Entrance nearer the Center and would not have been so ready as this fitted Scale however hence I might educe a general method for finding the hour and Azimuth in the Limb without Tangents or Secants The first Work would be
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
divided but the Author made his Table in page 5. without doubling to be graduated from a Quadrant divided into 45 equal parts Again If upon the Center C with a pair of Compasses each degree of the line of Sines be transferred into the Semicircle C G A it shall divide it into 90 equal parts the reason whereof is plain because the Sine of an Arch is half the chord of twice that Arch and therefore the Sines being made to twice the Radius of this circle shall being transferred into it become chords of the like Arch to divide a Semicircle into 90 equal parts Again upon the point A erect a line of Tangents of the same Radius with the former Sine which we may suppose to be infinitely continued here we use a portion of it A E. If from the point C the other extremity of the Diameter lines be drawn cutting the lower Semicircle as a line drawn from E intersects it at F through each degree of the said Tangent the said lower Semicircle shall be divided into 90 equal parts the reason is evident a line of Tangents from the Center shall divide a Quadrant into 90 equal parts and because an Angle in the circumference is but half so much as it is in the Center being transferred thither a whole Semicircle shall be filled with no more parts The chief use of this Circle is to operate Proportions in Tangents alone or in Sines and Tangents joyntly built upon this foundation that equiangled plain Triangles have their sides Proportional In streight lines it will be evident from the point D to E draw a streight line intersecting the Diameter at L and then it lies as C L to C D so is A L to A E it is also true in a Circle provided it be evinced that the points B L F fall in a streight line Hereof I have a Geometrical Demonstration which would require more Schemes which by reason of its length and difficulty I thought fit at present not to insert possibly an easier may be found hereafter As also an Algebraick Demonstration by the Right Honourable the Lord Brunkard whereby after many Algebraick inferences it is euinced that as L K is to K B ∷ so is L N to ◠F whence it will follow that the points B L F are in a right line If a Ruler be laid from 45d of the Semicircle to every degree of the Quadrant of Latitudes it will constitute upon the Diameter the graduations of the Line Sol whereby Proportions in Sines might be operated without the other supply From the same Scheme also follows the construction of the streight line of Latitudes from the point G at 90◠of the Quadrant of Latitudes draw a streight Line to C and transfer each degree of the Quadrant of Latitudes with Compasses one foot resting upon C into the said streight line and it shall be constituted To Calculate it The Line of Latitudes C G bears such Proportion to C A as the Chord of 90d doth to the Diameter which is the same that the Sine of 45d bears to the Radius or which is all one that the Radius bears to the Secant of 45 d which Secant is equal to the Chord of 90 d from the Diagram the nature of the Line of Latitudes may be discovered Any two Lines being drawn to make a right angle if any Ark of the Line of Latitudes be pricked off in one of those Lines retaining a constant Hipotenusal A C called the Line of Hours equal to the Diameter of that Circle from whence the Line of Latitudes is constituted if the said Hipotenusal from the Point formerly pricked off be made the Hipotenusal to the Legs of the right angle formerly pricked off the said Legs or sides including the right angle shall bear such Proportion one to another as the Radius doth to the sine of the Ark so prickt off and this is evident from the Schem for such Proportion as A C bears to C D doth A B bear to B C for the angle at A is Common to both Triangles and the angle at B in the circumference is a right angle and consequently the angle A C B will be equal to the angle A D C and the Legs A C to C D bears such Proportion by construction as the Radius doth to the Sine of an Ark and the same Proportion doth A B bear to B C in all cases retaining one and the same Hypotenusal A C the Proportion therefore lies evident As the Radius the sine of the angle at B To its opposite side A C the Secant of 45d So is the sine of the angle at A To its opposite side B C sought Now the quantity of the angle at A was found by seeking the natural Sine of the Ark proposed in the Table of natural Tangents and having found what Ark answers thereto the Sine of the said Ark is to become the third Tearm in the Proportion But the Cannon prescribed in the Description of the small Quadrant is more expedite then this which Mr Sutton had from Mr Dary long since for whom and by whose directions he made a Quadrant with the Line Sol and two Parrallel Lines of Sines upon it as is here added to the backside of this Quadrant Of the Line of Hours alias the Diameter or Proportional Tangent This Scale is no other then two Lines of natural Tangents to 45 d each set together at the Center and from thence beginning and continued to each end of the Diameter and from one end thereof numbred with 90 d to the other end This Line may fitly be called a Proportional Tangent for whersoever any Ark is assumed in it to be a Tangent the remaining part of the Diameter is the Radius to the said Tangent So in the former Schem if C L be the Tangent of any Ark the Radius thereto shall be A L. In the Schem annexed let A B be the Radius of a Line of Tangents equal to C D and also parralel thereto and from the Point B to C draw the Line B C and let it be required to divide the same into a Line of Proportional Tangents I say Lines drawn from the Point D to every degree of the Tangent A B shall divide one half of it as required from the similitude of two right angled equiangled plain Triangles which will have their sides Proportional it will therefore hold As C F To C D So F B To B E and the Converse As the second Tearm C D To the fourth B E So is the first C F To the third F B and therefore C F bears such Proportion to F B as C D doth to B E which is the same that the Radius bears to the Tangent of the Ark proposed If it be doubted whether the Diameter wil be a double Tangent or the Line here described such a Line a Proportion shall be given to find by Experience or Calculation what Line
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-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-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
distance between the Meridian Altitude and the given Altitude and enter that extent so upon the sines that one foot resting thereon the other turned about may just touch the thread the distance between the resting foot and the Center is equal to the versed sine of the ark sought and being measured from the end of the line of sines towards the Center shewes the ark sought Example When the Suns Declination is 15 d. North if his Altitude were 35 d. 21 m. the time of the day would be found 45 d. from noon that is 9 in the morning or 3 in the afternoon Of finding the Azimuth generally THough this may be found either by the sines alone in the equal limbe as before mentioned or by versed sines as was instanced for the hour see also page 239 240 241 yet where the Sun hath vertical Altitude or Depression as in places without the Tropicks towards either of the Poles it may be found most easily in the equal limb by the joynt help of sines and tangents by the proportions in page 175. First find the vertical Altitude as is shewed in page 174. Then for Latitudes under 45 d. Enter in Summer Declinations the difference but in Winter Declinations the sum of the sines of the vertical Altitude and of the proposed Altitude once done the line of sines from the Center and laying the thread over the Tangent of the Latitude take the nearest distance to it then enter that Extent at the complement of the Altitude in the Sines and lay the thread to the other foot and in the limb it shewes the Azimuth from the East or West Example For the Latitude of Rome to witt 42 d. If the Sun have 15 d. of North Declination his vertical Altitude is 22 d. 45 m. If his given Altitude â—e 40 d. the Azimuth of the Sun will be 17 d. 33 m. to the Southward of the West If his declination were as much South and his proposed Altitude 18 d. his Azimuth would be 41 d. 10 m. to the Southwards of the East or West For Latitudes above 45. If we assume the Rad. of the quadrant to be the tangent of the Latit the Rad. to that Tang. shall be the co-tangent of the Latit wherefore lay the thread to the complement of the Latitude in the line of Tangents in the limb and from the complement of the Altitude in the Sines take the nearest distance to it I say that extent shall be cosine of the Altitude to the lesser Radius which measure from the Center and it finds the Point of entrance whereon enter the former Sum or difference of sines as before directed and you will find the Azimuth in the equal limb Or if you would find the answer in the sines enter the first extent at 90 d. laying the thread to the other foot then enter the Sum or difference of the Sines of the vertical and given Altitude so between the scale and the thread that one foot turned about may but just touch the thread the other resting on the sines and you will find the sine of the Azimuth sought Example For the Latitude of Edinburg 55 d. 56 m. If the Sun have 15 d. of Declination his vertical Altitude or depression is 18 d. 14 m. the Declination being North if his proposed Altitude were 35 d. the Azimuth of the Sun would be 28 d. to the South-wards of the East or West But if the Declination were as much South and the Altitude 10 d. the Azimuth thereto would be 46 d. 58 m. to the South-wards of the East or West The first Operation also works a Proportion to witt As the Radius Is to the cotangent of the Latitude So is the cosine of the Altitude To a fourth sine I say this 4th sine beares such Proportion to the Radius as the cosine of the Altitude doth to the tangent of the Latitude for the 4th term of every direct Proportion beares such Proportion to the first terms thereof as the Rectangle of the two middle terms doth to the square of the first term But as the rectangle of the co-tangent of the Latitude and of the cosine of the Altitude is to the square of the Radius So is the cosine of the Altitude is to the tangent of the Latitude or which is all one So is the co-tangent of the Latitude To the secant of the Altitude as may be found by a common division of the rectangle and square of the Radius by either of the terms of the said rectangle by help of which notion I first found out the particular scales upon this quadrant All Proportions in sines and Tangents may be resolved by the sine of 90 d. and the Tangent of 45 d. on this quadrant if what hath been now wrote and the varying of Proportions be understood as in page 72 to 74 it is delivered Because the Projection is not fitted for finding the Azimuth there are added two particular scales to this quadrant namely the particular sine in the limb and the scale of entrance abutting on the sines fitted for the Latitude of London Lay the thread to the day of the moneth and it shewes the Suns Declination in the scales proper thereto Then count the Declination in the Limbe laying the thread thereto and in the particular sine it shewes the Suns Altitude or Depression being East or West To find the Suns Azimuth FOr North Declinations take the distance between the sines of the vertical Altitude and given Altitude but for South Declinations adde with your compass the sine of the given Altitude to the sine of the vertical Altitude enter the extent thus found at the Altitude in the scale of entrance laying the thread to the other foot according to nearest distance and in the equal limb it shewes the Azimuth sought from the East or West or it may be found in the sines by laying the thread to that arch in the limb that the Altitude in the scale of entrance stands against in the sines and entring the former extent paralelly between the thread and the sines Example So when the Sun hath 13 d. of Declination his vertical Altitude or Depression is 16 d. 42 m. If the Declination were North and his Altitude 8 d. 41 m. his Azimuth would be 10 d. to the North-wards of the East or West But if it were South and his Altitude 12 d. 13 m. the Azimuth would be 40d to the South-wards of the East or West By the same particular scales the hour may be also found To find the time of Sun rising or setting TAke the sine of the Declination and enter it at the Declination in the Scale of entrance and it shewes the time sought in the equal limâ—e from six Example When the hath 10 d. of Declination the Ascensional difference is 49 m. which added to or substracted from six shewes the time of rising and setting To find the hour of the day for South Declination IN
to graduate the same way The arke of 30 degr 30 degr Co-latitude 38 28 Summe 68 28 Difference 8 28 Count 68 deg 28′ from O towards D and from the point A draw a line to it Again in the said upper line count 8 deg 28′ upwards from O towards C from the point B lay a ruler over it where it intersects the line last drawn is the point where 30 d. of the curve is to be graduated To graduate the upper part of the curve requires no other directions the same arkes serve if the account be but made the other way and in accounting the summe the ruler laid over B. in the lower line instead of A and in counting the difference over A instead of B neither is there any Scheme given hereof the Practitioner need onely let the upper line be the line of altitudes on the left edge of the quadrant continued out to 90 deg at each end and to that end next the Center set C and to the other end D. So likewise let that end of the Versed Scale next the right edge of the quadrant be continued to 180 deg whereto set A and at the other end B and then if these directions be observed and the same distance and position of the lines retained it will not be difficult to constitute a Curve in all respects agreeing with that on the fore-side of the quadrant Of the houre and Azimuth Scale on the right edge of the Quadrant THis Scale stands outwardmost on the right edge of the quadrant and consists of two lines the one a line of 90 sines made equal to the cosine of the Latitude namely to the sine of 38 deg 28′ and continued the other way to 40 deg like a Versed sine The annexed line being the other part of this Scale is a line of natural Tangents beginning where the former sine began the Tangent of 38 deg 28′ being made equal to the sine of 90 deg this Tangent is continued each way with the sine towards the Limbe of the quadrant it should have been continued to 62 deg but that could not be without excursion wherefore it is broken off at 40 degrees and the residue of it graduated below and next under the Versed sine belonging to the Curve that runnes crosse the quadrant being continued but to halfe the former Radius Of the Almanack NExt below the former line stands the Almanack in a regular ob-long with moneths names graved on each side of it Below the Almanack stands the quadrat◠and shadowes in two Arkes of circles terminating against 45 deg of the Limbe below them a line of 90 sines in a Circle equal to 51 deg 32′ of the Limbe broken off below the streight line and the rest continued above it Below these are put on in Circles a line of Tangents to 60 degrees Also a line of Secants to 60 deg with a line of lesser sines ending against 30 deg of the Limbe counted from the right edge where the graduations of the Secant begins Last of all the equal Limbe Prickt with the pricks of the quadrat Abutting upon the line of sines and within the Projection stands a portion of a small sine numbred with its Complements beginning against 38 deg 28′ of the line of sines this Scale is called the Scale of entrance Upon the Projection are placed divers Stars how they are inscribed shall be afterwards shewne The description of the Back-side Put on in quarters or Quadrants of Circles 1 THe equal Limbe divided into degrees as also into houres and halves and the quarters prickt to serve for a Nocturnal 2 A line of Equal parts 3 A line of Superficies or Squares 4 A line of Solids or Cubes 5 A Tangent of 45 degrees double divided to serve for a Dyalling Tangent and a Semitangent for projections 6 The line Sol aliàs a line of Proportional Sines 7 A Tangent of 51 degrees 32′ through the whole Limbe 8 A line of Declinations for the Sun to 23 deg 31′ Foure quadrants with the days of the Moneth 9 10 11 12 13 The Suns true place with the Charecters of the 12 Seignes 14 The line of Segments with a Chord before they begin 15 The line of Metals and Equated bodies 16 The line of Quadrature 17 The line of Inscribed bodies 18 A line of 12 houres of Ascension with Stars names Declinations and Ascensional differences Above all these a Table to know the Epact and what day of the Weeke the first day of March hapned upon by Inspection continued to the yeare 1700. All these between the Limbe and the Center ON the right edge a line of equal parts from the Center decimally sub-divided being a line of 10 inches also a Dyalling Tangent or Scale of 6 houres the whole length of the quadrant not issuing from the Center On the left edge a Tangent of 63 deg 26′ from the Center Also a Scale of Latitudes fitted to the former Scale of houres not issuing from the Center and below it a small Chord The Vses of the Quadrant Lords-day 1657 63 68 74 ☉ 85 91 96 anno 25 1 26 3  4 11 6 epact Monday 58 ☽ 69 75 80 86 ☽ 97 anno 6  7 14 9 15  17 epact Tuesday 59 64 70 ♂ 81 87 92 98 anno 17 12 18  20 26 22 28 epact Wenesday ☿ 65 71 76 82 ☿ 93 99 anno  23 29 25 1  3 9 epact Thursday 60 66 ♃ 77 83 88 94 ♃ anno 28 4  6 12 7 14  epact Friday 61 67 72 78 ♀ 89 95 700 anno 9 15 11 17  18 25 20 epact Saturday 62 ♄ 73 79 84 90 ♄ 701 anno 20  22 28 23 29  1 epact Dayes the same as the first of March March 1 8 15 22 29 November August 2 9 16 23 30 August May 3 10 17 24 31 Jnuary October 4 11 18 25 0 October April 5 12 19 26 00 July Septem 6 13 20 27 00 December June 7 14 21 28 00 February Perpetual Almanack Of the Vses of the Projection BEfore this Projection can be used the Suns declination is required by consequence the day of the moneth for the ready finding thereof there is repeated the same table that stands on the Back-side of this quadrant in each ruled space the uppermost figure signifies the yeare of the Lord and the column it is placed in sheweth upon what day of the Weeke the first day of March hapned upon in that yeare and the undermost figure in the said ruled space sheweth what was the Epact for that yeare and this continued to the yeare 1701 inclusive Example Looking for the yeare 1660 I find the figure 60 standing in Thursday Column whence I may conclude that the first day of March that yeare will be Thursday and under it stands 28 for the Epact that yeare Of the Almanack HAving as before found what day of the Weeke the first day of March hapned upon
one dayes variation in 300 yeares as is observed by Mr. Philips The Vses of the Quadrant WIthout rectifying the Bead nothing can be performed by this Projection except finding the Suns Meridian Altitude being shewn upon the Index by the intersection of the Parallel of declination therewith Also the time when the Sun will be due East or West TRace the Parallel of Declination to the right edge of the Projection and the houre it there intersects in most cases to be duly estimated shewes the time sought thus when the Sun hath 21 deg of North declination we shall find that he will be due East or West about three quarters of an houre past 4 in the afternoon or a quarter past 7 in the morning The declination is to be found on the Back-side of the quadrant by laying the thread over the day of the moneth To rectifie the Bead. LAy the thread upon the graduated Index and set the Bead to the observed or given Altitude and when the Altitude is nothing or when the Sun is in the Horizon set the Bead to the Cypher on the graduated Index which afterwards being carried without stretching to the parallel of Declination the threed in the Limbe shewes the Amplitude or Azimuth and the Bead amongst the houres shewes the true time of the day Example Upon the 24th of April the Suns declination will be found to 16 deg North. Now to find his Amplitude and the time of his rising laying the threed over the graduated Index set the Bead to the beginning of the graduations of the Index and bring it without stretching to the parallel of declination above being 16 d and the threed in the limbe will lye over 26 deg 18′ for the Suns Amplitude or Coast of rising to the Northward of the East and the Bead amongst the houres sheweth 24 minutes past 4 for the time of Sun rising Which doubled gives the length of the night 8 houres 49 min. In like manner the time of setting doubled gives the length of the day The same to find the houre and Azimuth let the given Altitude be 45 degrees HAving rectified the Bead to the said Altitude on the Index and brought it to the intersect the parallel of declination the thread lyes over 50 degrees 48′ For the Suns Azimuth from the South And the Bead among the houres shewes the time of the day to be 41 minutes past 9 in the morning or 19 minutes past two in the afternoon Another Example wherein the operation will be upon the Reverted taile Let the altitude be 3 deg 30′ And the declination 16 deg North as before TO know when to rectify the Bead to the upper or neather Altitude will be no matter of difficulty for if the Bead being set to the neather Altitude will not meet with the parallel of declination then set it to the upper Altitude and it will meet with Winter parallel of like declination which in this case supplyes the turn So in this Example the Bead being set to the upper Altitude of 3 deg 30′ and carried to the Winter parallel of declination The thread in the Limbe will fall upon 68 deg 28′ for the Suns Azimuth from the North and the Bead among the houres shewes the time of the day to be either 5 in the morning or 7 at night Another Example Admit the Sun have 20 degr of North Declination as about the 9th of May and his observed altitude were 56 deg 20′ having rectified the Bead thereto and brought it to intersect the parallel of 20 deg among the houres it shewes the time of the day to be 11 in the morning or 1 in the afternoon and the Azimuth of the Sun to be 26 deg from the South The Vses of the Projection TO find the Suns Altitude on all houres or Azimuths will be but the converse of what is already said therefore one Example shall serve When the Sun hath 45 deg of Azimuth from the South And his Declination 13 deg Northwards Lay the threed over 45 deg in the Limbe and where the threed intersects the Parallel of Declination thereto remove the Bead which carried to the Index without stretching shewes 43 deg 50′ for the Altitude sought Likewise to the same Declination if it were required to find the Suns Altitude for the houres of 2 or 10. Lay the threed over the intersection of the houre proposed with the parallel of Declination and thereto set the bead which carried to the Index shewes the Altitude sought namely 44 deg 31′ The same Altitude also belongs to that Azimuth the threed in the former Position lay over in the Limbe This Projection is of worst performance early in the morning or late in the evening about which time Mr. Daries Curve is of best performance whereto we now addresse our selves Of the curved line and Scales thereto fitted This as we have said before was the ingenious invention of M. Michael Dary derived from the proportionalty of two like equiangled plain Triangles accommodated to the latitude of London for the ready working of these two Proportions 1. For the Houre As the Cosine of the Latitude is to the secant of the Declination So is the difference between the sine of the Suns proposed and Meridian Altitude To the versed sine of the houre from noone and the converse and so is the sine of the Suns Meridian Altitude to the versed sine of the semidiurnal Arke 2. For the Azimuth The Curve is fitted to find it from the South and not from the North and the Proportion wrought upon it will be As the cosine of the Latitude is to the Secant of the Altitude So is the difference of the versed sines of the Suns or Stars distance from the elevated Pole and of the summe of the Complements both of the Latitude and Altitude to the versed sine of the Azimuth from the noon Meridian Which will not hold backward to find the Altitude on all Azimuths because the altitude is a term involved both in the second and third termes of the former proportion If the third terme of the former Proportion had not been a difference of Sines or Versed sines the Curved line would have been a straight-line and the third term always counted from one point which though in the use it may seem to be so here yet in effect the third term for the houre is always counted from the Meridian altitude Here observe that the threed lying over 12 or the end of the Versed Scale and over the Suns meridian altitude in the line of altitudes it will also upon the curve shew the Suns declination which by construction is so framed that if the distance from that point to the meridian altitude be made the cosine of Latitude the distance of the said point from the end of the versed Scale numbred with 12 shall be the secant of the declination to the same Radius being both in one straight-straight-line by the former constitution of the threed and instead
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-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-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-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-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-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-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
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-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-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-lines may be drawn Or if that thread which fastned in the center of the glass be drawn exactly over any 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-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-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-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-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
Azimuth shall be 9d 28′ from the Vertical at the hour of 6 in our Latitude of London Another Example to find the time when the Sun will be due East or West Extend the thred over the Latitude in the Semicircle and over the Declination on the Diameter and in the Quadrant of Latitudes it shews the Ark sought The Proportion wrought is As the Radius to the Cotangent of the Latitude So is the Tangent of the Declination To the Sine of the Hour from 6. Example So when the Sun hath 15◠of North Declination in our Latitude of London the Hour will be found 12d 18′ from 6 in time 49⅕′ past 6 in the morning or before it in the afternoon Another Example to find the Time of Sun rising As the Cotangent of the Latitude to Radius So is the Tangent of the Declination To the Sine of the Hour from 6 before or after it Lay the thred to the Complement of the Latitude in the Semicircle and over the Declination on the Diameter and in the Quadrant of Latitudes it shews the time sought in degrees to be converted into common time by allowing 15◠to an hour and 4′ to a degree So in the Latitude of London 51d 32′ when the Sun hath 15◠of Declination the ascensional difference or time of rising from 6 will be 19d 42′ to be converted into common time as before By what hath been said it appears that the Hour and Azimuth may be found generally by help of this Circle and Diameter For the performance whereof we must have recourse to the Proportions delivered in page 123. whereby we may alwaies find the two Angle adjacent to the side on which the Perpendicular falleth which may be any side at pleasure for after the first Proportion wholly in Tangents is wrought to find either of those Angles will be agreeable to the second case of right angled Spherical Triangles wherein there will be given the Hypotenusal and one of the Legs to find the adjacent Angle only it must be suggested that when the two sides that subtend the Angle sought are together greater then a Semicircle recourse must be had to the Opposite Triangle if both those Angles are required to be found by this Trigonometry otherwise one of them and the third Angle may be found by those directions by letting fall the perpendicular on another side provided the sum of the sides subtending those Angles be not also greater then a Semicircle or having first found one Angle the rest may be found by Proportions in Sines only IN the Triangle ☉ Z P if it were required to find the angles at Z and ☉ because the sum of the sides ☉ P and Z P are less then a Semicircle they might be both found by making the half of the Base ☉ Z the first Tearm in the Proportion and then because the angles ☉ Z are of a different affection the Perpendicular would fal without on the side ☉ Z continued towards B as would be evinced by the Proportiod for the fourth Ark discovered would be found greater then the half of ☉ Z hence we derived the Cannon in page 124 for finding the Azimuth Whereby might also be found the angle of Position at ☉ so if it were required to find the angles at ☉ and P the sides ☉ Z and Z P being less then a Semicircle the Perpendicular would fall within from Z on the side ☉ P as would also be discovered by the Proportion for the fourth Ark would be found less then the half of ☉ P. But if it were required to find both the angles at Z and P in this Case we must resolve the Opposite Triangle Z B P because the sum of the sides ☉ Z and ☉ P are together greater then a Semicircle and this being the most difficult Case we shall make our present Example The Proportion will be As the Tangent of half Z P Is to the Tangent of the half sum of Z B and P B So is the Tangent of half their difference To a fourth Tangent That is As Tangent 19d 14′ Is to the Tangent of 86d 30′ So is the Tangent of 9d 30′ To a fourth Operation Extend the Thread through 19 d 14′ on the Semicircle and 9 d 30′ on the Diameter and hold it at the Intersection on the opposite side the Semicircle then lay the Thread to 86 d 30′ in the Semicircle and it shews 82 d 44′ on the Diameter for the fourth Ark sought Because this Ark is greater then the half of Z P we may conclude that the Perpendicular B A falls without on the side Z P continued to A. fourth Ark 82d 44′ half of Z P is 19 14 sum 101 58 is Z A difference 63 30 is P A Then in the right angled Triangle B P A right angled at A we have P A and B P the Hypotenusal to find the angle B P A equal to the angle ☉ Z P. The Proportion is As the Radius Is to the Tangent of 13d the Complement of B P So is the Tangent of P A 63d 30′ To the Cosine of the angle at P. Extend the Thread through 13 d on the Diameter and through 63 d 30′ in the Semicircle counted from the other end and in the upper Quadrant it shews 27 d 35′ for the Complement of the angle sought And letting this Example be to find the Hour and Azimuth in our Latitude of London so much is the hour from six in Winter when the Sun hath 13 d of South Declination and 6 d of Altitude in time 1 ho 50⅓ minutes past six in the morning or as much before it in the afternoon To find the Azimuth Again in the Triangle Z A B right angled at A there is given the Leg or Side Z A 101 d 58′ and the Hipotenusal Z B 96 d to find the angle B Z P here noting that the Cosine or Cotangent of an Ark greater then a Quadrant is the Sine or Tangent of that Arks excess above 90 d and the Sine or Tangent of an Ark greater then a Quadrant the Sine or Tangent of that Arks Complement to 180 d it will hold As the Radius To the Tangent of 6d So is the Tangent 78d 2′ To the Sine of 29 d 44′ found by extending the Thread through 78 d 2′ on the Semicircle counted from the other end alias in the small figures and in the Quadaant it will intersect 29 d 44′ now by the second Case of right angled Sphoerical Triangles the angle A Z B will be Acute wherefore the angle ☉ Z B is 119 d 44′ the Suns Azimuth from the North the Complement being 60 d 16′ is the angle A Z B and so much is the Azimuth from the South To work Proportions in Sines alone THat this Circle might be capacitated to try any Case of Sphoerical Triangles there are added Lines to it namely the Line Sol falling perpendicularly on the Diameter from the end of the