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end_n	draw_v	line_n	require_v	1,461	5	9.1279	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64223	The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.	Taylor, John, 1666 or 7-1687.	1667 (1667)	Wing T533B; ESTC R221720	52,688	153

is 13 deg 30 min. PROBL. 8. The declination given to find the beginning and end of twilight or day-break Lay the thread to the declination counted the contrary way as in the last Problem and take from your Scale of altitudes 18 deg for twilight and 13 deg for day-break or clear light with this run one point of the Compasses along the line of houres on that side next the end until the other will just touch the thread and then the former point gives the respective times required Ex. gr At 7 deg North declination day breaks 8 minutes before 4 but twilight is 3 houres 12 minutes in the morning or 8 hours 52 minutes afternoon PROBL. 9. The declination and altitude of the Sun or any Star given to find their Azimuth in Northern declination Lay the thread to the altitude numbred on the limb of the moveable piece from 60 0 toward the end and when occasion requires continue your numbring forward upon the loose piece and take the declination from your line of altitude with this distance run one point of your Compasses along the line of Azimuths on that side the thread next the head until the other just touch the thread then the former point gives the Azimuth from South Ex. gr at 10 deg declination North and 30 deg altitude the Azimuth from South is 64 deg 40 min. PROBL. 10. The Suns altitude given to find his Azimuth in the aequator Lay the thread to the altitude in the limb counted from 60 0 on the loose piece toward the end and on the line of Azimuths it cuts the Azimuth from South Ex. gr at 25 deg altitude the Azimuth is 53 deg At 30 deg altitude the Azimuth is 41 deg 30 min. fere PROBL. 11. The declination and altitude of the Sun or any Star given to find the Azimuth in Southern declination Lay the thread to the altitude numbred on the limb from 60 0 on the moveable piece toward the end and take the declination from the Scale of altitudes then carry one point of your Compasses on the line of Azimuths on that side the thread next the end until the other just touch the thread which done the former point gives the Azimuth from South Ex. gr at 15 deg altitude and 6 deg South declination the Azimuth is 58 deg 30 min. PROBL. 12. The declination given to find the Suns altitude at East or West in North declination and by consequent his depression in South declination Take the declination given from the Scale of altitudes and setting one point of your Compasses in 90 on the line of Azimuths lay the thread to the other point on that side 90 next the head on the limb it cuts the altitude counting from 60 0 on the moveable piece Ex. gr at 10 deg declination the altitude is 12 deg 40 min. PROBL. 13. The declination and Azimuth given to find the altitude of the Sun or any Star Take the declination from the Scale of altitudes set one point of your Compasses in the Azimuth given then in North declinanation turn the other point toward the head in South toward the end and thereto laying the thread on the limb you have the altitude numbring from 60 0 on the moveable piece toward the end Ex. gr At 7 deg North declination and 48 deg Azimuth from South the altitude is 35 deg but at 7 deg declination South and 50 deg Azimuth the altitude is onely 18 deg 30 min. PROBL. 14. The altitude declination and right ascension of any Star with the right ascension of the Sun given to find the hour of the night Take the Stars altitude from the Scale of altitudes and laying the thread to his declination in the limb find his hour from the last Meridian he was upon as you did for the Sun by Probl. 5. If the Star be past the South this is an afternoon hour if not come to the South a morning hour which keep Then setting one point of your Compasses in the Suns right ascension numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece extend the other point to the right ascension of the Star numbred upon the same line observing which way you turned the point of your Compasses viz. toward the head or end With this distance set one point of your Compasses in the Stars hour before found counted on the same line and turning the other point the same way as you did for the right ascensions it gives the true hour of the night Ex. gr The 22 of March I find the altitude of the Lions heart 45 deg his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres 5 min. The right ascension of the Sun is 46 m. of time or 11 d. 30 m. of the Circle the right ascension of the Lions heart is 9 hour 51 m. fere of time or 147 deg 43 m. of a circle then by a line of twice 12 you may find the true hour of the night 7 hour 13 min. PROBL. 15. The right ascension and declination of any Star with the right ascension of the Sun and time of night given to find the altitude of that Star with his Azimuth from South and by consequent to find the Star although before you knew it not This is no more than unravelling the last Problem 1 Therefore upon the line of twice 12 or 24 set one point of your Compasses in the right ascension of the Star extending the other to the right ascension of the Sun upon the same line that distance laid the same way upon the same line from the hour of the night gives the Stars hour from the last Meridian he was upon This found by Probl. 5. find his altitude as you did for the Sun Lastly having now his declination and altitude by Probl. 8. or 10. according to his declination you will soon get his Azimuth from South This needs not an example By help of this Problem the Instrument might be so contrived as to be one of the best Tutors for knowing of the Stars PROBL. 16. The altitude and Azimuth of any Star given to find his declination Lay the thread to the altitude counted on the limb from 60 0 on the moveable piece toward the end setting one point of your Compasses in the Azimuth take the nearest distance to the thread this applyed to the Scale of altitudes gives the declination If the Azimuth given be on that side the thread toward the end the declination is South when on that side toward the head its North. PROBL. 17. The altitude and declination of any Star with the right ascension of the Sun and hour of night given to find the Stars right ascension By Probl. 5. or 14. find the Stars hour from the Meridian Then on the line twice 12 or 24 set one point of your Compasses in the Stars hour thus found and extend the other to the hour of the night Upon the same line

with this distance set one point of your Compasses in the right ascension of the Sun and turning the other point the same way as you did for the hour it gives the Stars right ascension PROBL. 18. The Meridian altitude given to find the time of Sunrise and Sunset Take the Meridian altitude from your particular Scale and setting one point of your Compasses upon the point 12 on the line of hours that is the pin at the end lay the thread to the other point and on the line of hours the thread gives the time required PROBL. 19. To find any latitude your particular Scale is made for Take the distance from 90 on the line of Azimuth unto the pin at the end of that line or the point 12 this applyed to the particular Scale gives the complement of that latitude the Instrument was made for PROBL. 20. To find the angles of the substile stile inclination of Meridians and six and twelve for exact declining plains in that latitude your Scale of altitudes is made for Sect. 1. To find the distance of the substile from 12 or the plains perpendicular Lay the thread to the complement of declination counted on the line of Azimuths and on the limb it gives the substile counting from 60 0 on the moveable piece Sect. 2. To find the angle of the Stile 's height On the line of Azimuths take the distance from the Plains declination to 90. This applyed to the Scale of altitudes gives the angle of the stile Sect. 3. The angle of the Substile given to find the inclination of Meridians Take the angle of the substile from the Scale of altitudes and applying it from 90 on the Azimuth line toward the end the figures shew the complement of inclination of Meridians Sect. 4. To find the angle betwixt 6 and 12. Take the declination from the Scale of altitudes and setting one point of your Compasses in 90 on the line of Azimuths lay the thread to the other point and on the limb it gives the complement of the angle sought numbring from 60 0 on the moveable piece toward the end This last rule is not exact nor is it here worth the labour to rectifie it by another sine added sith you have an exact proportion for the Problem in the Treatise of Dialling Chap. 2 Sect. 5. Paragr 4. CHAP. III. Some uses of the Line of natural signs on the Quadrantal side of the fixed piece PROBL. 1. How to adde one sign to another on the Line of Natural Sines TO adde one sine to another is to augment the line of one sine by the line of the other sine to be added to it Ex. gr To adde the sine 15 to the sine 20 I take the distance from the beginning of the line of sines unto 15 and setting one point of the Compasses in 20 upon the same line turn the other toward 90 which I finde touch in 37. So that in this case for we regard not the Arithmetical but proportional aggregate 15 added to 20 upon the line of natural sines is the sine 37 upon that line and from the beginning of the line to 37 is the distance I am to take for the summe of 20 and 15 sines PROBL. 2. How to substract one sine from another upon the line of natural sines The substracting of one sine from another is no more than taking the distance from the lesser to the greater on the line of sines and that distance applyed to the line from the beginning gives the residue or remainer Ex. gr To substract 20 from 37 I take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining PROBL. 3. To work proportions in sines alone Here are four Cases that include all proportions in sines alone CASE 1. When the first term is Radius or the Sine 90. Lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end then numbring the third on the line of sines take the nearest distance from thence to the thread and that applyed to the Scale from the beginning gives the fourth term Ex. gr As the Radius 90 is to the sine 20 so is the sine 30 to the sine 10. CASE 2. When the Radius is the third term Take the sine of the second term in your Compasses and enter it in the first term upon the line of sines and laying the thread to the nearest distance on the limb the thread gives the fourth term Ex. gr As the sine 30 is to the sine 20 so is the Radius to the sine 43. 30. min. CASE 3. When the Radius is the second term Provided the third term be not greater than the first transpose the terms The method of transposition in this case is as the first term is to the third so is the second to the fourth and then the work will be the same as in the second case Ex. gr As the sine 30 is to the radius or sine 90 so is the sine 20 to what sine which transposed is As the sine 30 is to the sine 20 so is the radius to a fourth sine which will be found 43 30 min. as before CASE 4. When the Radius is none of the three terms given In this case when both the middle terms are less than the first enter the sine of the second term in the first and laying the thread to the nearest distance take the nearest extent from the third to the thread this distance applyed to the scale from the beginning gives the fourth Ex. gr As the sine 20 to the sine 10 so is the sine 30 to the sine 15. When only the second term is greater than the first transpose the terms and work as before But when both the middle tearms be greater than the first this proportion will not be performed by this line without a paralel entrance or double radius which inconveniency shall be remedied in its proper place when we shew how to work proportions by the lines of natural sines on the proportional or sector side These four cases comprizing the method of working all proportions by natural sines alone I shall propose some examples for the exercise of young practitioners and therewith conclude this Chapter PROBL. 4. To finde the Suns amplitude in any Latitude As the cosine of the Latitude is to the sine of the Suns declination so is the radius to the sine of amplitude PROBL. 5. To finde the hour in any Latitude in Northern Declination Proport 1. As the radius to the sine of the Suns declination so is the sine of the latitude to the sine of the Suns altitude at six By Probl. 2. substract this altitude at six from the present altitude and take the difference Then Proport 2. As the cosine of the latitude is to that difference so is the radius to a fourth sine Again Proport 3. As the

cosine of the declination to that fourth sine so is the radius to the sine of the hour from six PROBL. 6. To finde the hour in any Latitude when the Sun is in the Equinoctial As the cosine of the latitude is to the sine of altitude so is the radius to the sine of the hour from six PROBL. 7. To finde the hour in any latitude in Southern Declination Proport 1. As the radius to the sine of the Suns declination so is the sine of the latitude to the sine of the Suns depression at six adde the sine of depression to the present altitude by Probl. 1. Then Proport 2. As the cosine of the latitude is to that summe so is the radius to a fourth sine Again Proport 3. As the cosine of declination is to the fourth sine so is the radius to the sine of the hour from six PROBL. 8. To finde the Suns Azimuth in any latitude in Northern Declination Proport 1. As the sine of the latitude to the sine of declination so is the radius to the sine of altitude at East or West By Probl. 2. substract this from the present altitude then Proport 2. As the cosine of the latitude is to that residue so is the radius to a fourth sine Again Proport 3. As the cosine of the altitude is to that fourth sine so is the radius to the sine of the Azimuth from East or West PROBL. 9. To finde the Azimuth for any latitude when the Sun is in the Equator Proport 1. As the cosine of the latitude to the sine of altitude so is the sine of the latitude to a fourth sine Proport 2. As the cosine of altitude to that fourth sine so is the radius to the sine of the Azimuth from East or West PROBL. 10. To finde the Azimuth for any latitude in Southern Declination Proport 1. As the cosine of the latitude to the sine of altitude so is the sine of the latitude to a fourth Having by Probl. 4. found the Suns amplitude adde it to this fourth sine by Probl. 1. and say As the cosine of the altitude is to the sum so is the radius to the sine of the Azimuth from East or West The terms mentioned in the 5th 7th 8th 10th Problems are appropriated unto us that live on the North side the Equator In case they be applyed to such latitudes as lie on the South side the Equator Then what is now called Northern declination name Southern and what is here styled Southern declination term Northern and all the proportion with the operation is the same These proportions to finde the hour and Azimuth may be all readily wrought by the lines of artificial sines only the addition and substraction must alwayes be wrought upon the line of natural sines CHAP. IV. Some uses of the Lines on the proportional side of the Instrument viz. the Lines of natural Sines Tangents and Secants PROBL. 1. To lay down any Sine Tangent or Secant to a Radius given See Fig. 1. IF you be to lay down a Sine enter the Radius given in 90 and 90 upon the lines of Sines keeping the Sector at that gage set one point of your Compasses in the Sine required upon one line and extend the other point to the same Sine upon the other Line This distance is the length of the Sine required to the given Radius Ex. gr Suppose A. B. the Radius given and I require the Sine 40. proportional to that Radius Enter A. B. in 90 and 90 keeping the Sector at that gage I take the distance twixt 40 on one side to 40 on the other that is C. D. the Sine required The work is the same to lay down a Tangent to any Radius given provided you enter the given Radius in 45 and 45 on the line of Tangents Only observe if the Tangent required be less than 45. you must enter the Radius in 45. and 45 next the end of the Rule But when the Tangent required exceeds 45. enter the Radius given in 45 and 45 'twixt the center and end and keeping the Sector at that Gage take out the Tangent required This is so plain there needs no example To lay down a Secant to any Radius given is no more than to enter the Radius in the two pins at the beginning of the line of Secants and keeping the Sector at that Gage take the distance from the number of the Secant required on one side to the same number on the other side and that is the Secant sought at the Radius given The use of this Problem will be sufficiently seen in delineating Dyals and projecting the Sphere PROBL. 2. To lay down any Angle required by the Lines of Sines Tangents and Secants See Fig. 2. There are two wayes of protracting an Angle by the Line of Sines First if you use the Sines in manner of Chords Then having drawn the line A B at any distance of your Compass set one point in B and draw a mark to intersect the Line B A as E F. Enter this distance B F in 30 and 30 upon the Lines of Sines and keeping the Sector at that Gage take out the Sine of half the Angle required and setting one point where F intersects B A turn the other toward E and make the mark E with a ruler draw B E and the Angle E B F is the Angle required which here is 40. d. A second method by the lines of Sines is thus Enter B A Radius in the Lines of Sines and keeping the Sector at that Gage take out the Sine of your Angle required with that distance setting one point of your Compasses in A sweep the ark D a line drawn from B by the connexity of the Ark D makes the Angle A B C 40 d. as before To protract an Angle by the Lines of Tangents is easily done draw B A the Radius upon A erect a perpendicular A C enter B A in 45 and 45 on the Lines of Tangents and taking out the Tangent required as here 40 set it from A to C. Lastly draw B C and the Angle C B A is 40 d. as before In case you would protract an Angle by the Lines of Secants Draw B A and upon A erect the perpendicular A C enter A B in the beginning of the Lines of Secants and take out the Secant of the Angle with that distance setting one point of your Compasses in B with the other cross the perpendicular A C as in C. This done lay a Ruler to B and the point of intersection and draw the Line B C. So have you again the Angle C B A. 40. d. by another projection These varieties are here inserted only to satisfie a friend and recreate the young practitioner in trying the truth of his projection PROBL. 3. To work proportions in Sines alone by the Lines of natural Sines on the

next the outward limb of the moveable and loose piece both Yet because it is requisite to have pins to keep the loose piece close in its place You may have two sights more to supply their place which sometimes you may make use of and so the number of sights may be five viz. two sliding sights one turning sight and two pin sights to put into the holes at the end of the fixed and moveable piece to hold the tenons of the loose piece close joynted Every one of these sights hath a fiducial or perpendicular line drawn down the middle of them from the top to the bottom where this line toucheth the graduations on the limb is the point of observation The places of these sights have an oval proportion about the middle of them only leaving a small bar of brass to conduct the fiducial line down the oval cavity and support a little brass knot with a sight hole in it in the middle of that bar which is ever the point to be looked at There are two wayes of observing an altitude with help of these sights The one when we turn our face toward the object This is called a forward observation in which you must alwayes set the turning sight next your eye This way of observation will not exactly give an altitude above 45 degrees The other way of observing an altitude is peculiar to the Sun in a bright day when we turn our back toward the Sun This is termed a backward observation wherein You must have one of the sliding sights next Your eye and the turning sight toward the Horizon This serves to take the Suns altitude without thread or plummet when it is near the Zenith PROBL. 1. To finde the Suns altitude by a forward observation Serve the turning sight to the center of those graduations you please to make use of whether on the inward or outward limb and place the two sliding sights upon the respective limb to that center this done look by the knot of the turning sight moving the instrument upward or downward until you see the knot of one of the sliding sights directly against the Sun then move the other sliding sight until the knot of the turning sight and the knot of this other sliding sight be against the horizon then the degrees intercepted 'twixt the fiducial lines of the sliding sights on the limb shew the altitude required PROBL. 2. To finde the distance of any two Stars c. by a forward observation Serve the turning sight to either center and apply the two sliding sights to the respective limb holding the instrument with the proportional side downward and applying the turning sight to your eye so move the two sliding sights either nearer together or further asunder that you may by the knot of the turning sight see both objects even with the knots of their respective sliding sights then will the degrees intercepted 'twixt the fiducial lines of the object sights on the limb show the true distance By this means you may take any angle for surveying c. PROBL. 3. To finde the Suns Altitude by a backward Observation Serve the turning sight to the center at the beginning of the line of sines and apply one of the sliding sights to the outward limb of the loose piece and the other to the outward limb of the moveable piece and turning your back toward the Sun set the sliding sight upon the moveable piece next your eye and slide it upward or downward toward the end or head until you see the shadow of the little bar or edge of the sight on the loose piece fall directly on the little bar on the turning sight and at the same time the bar of the sight next your eye and the bar of the turning sight to be in a direct line with the Horizon Then will the degrees on the limb intercepted 'twixt the fiducial lines of the sliding sights if you took the shadow of the bar or 'twixt the fiducial line of the sliding sight next your eye and the edge of the other sliding sight when you took the shadow of the edge be the true altitude required ΣΚΙΟΓΡΑΦΙΑ OR The Art of Dyalling for any plain Superficies LIB II. CHAP. I. The distinction of Plains with Rules for knowing of them ALL plain Superficies are either horizontal or such as make Angles with the Horizon Horizontal plains are those that lie upon an exact level or flat Plains that make Angles with the Horizon are of three sorts 1. Such as make right angles with the Horizon generally known by the name of erect or upright plains 2. Such as make acute angles with the horizon or have their upper edge leaning toward you usually termed inclining plains 3. Such as make obtuse angles with the horizon or have their upper edge falling from you commonly called reclining plains All these three sorts are either direct viz. East West North South Or else Declining From South toward East or West From North toward East or West All plain Superficies whatsoever are comprized under one of these terms But before we treat of the affections or delineation of Dials for them it will be requisire to acquaint you with the nature of any plain which may be found by the following Problems PROBL. 1. To finde the reclination of any Plain Apply the outward ledge of the moveable piece to the Plain with the head upward and reckoning what number of degrees the thread cuts on the limb beginning your account at 30. on the loose piece and continuing it toward 60 0 on the moveable piece you have the angle of reclination If the thread falls directly on 60 0 upon the moveable piece it s an horizontal if on 30. on the loose piece it s an erect plain PROBL. 2. To finde the inclination of any Plain Apply the outward ledge of the fixed piece to the plain with the head upward and what number of degrees the thread cuts upon the limb of the loose piece is the complement of the plains inclination PROBL. 3. To draw an Horizontal Line upon any Plain Apply the proportional side of the Instrument to the plain and move the ends of the fixed piece upward or downward until the thread falls directly on 60 0. upon the loose piece then drawing a line by the outward ledge of the fixed piece its horizontal or paralel to the horizon PROBL. 4. To draw a perpendicular Line upon any Plain When the Sun shineth hold up a thread with a plummet against the plain and make two points at any distance in the shadow of the thread upon the plain lay a ruler to these points and the line you draw is a perpendicular PROBL. 5. To finde the declination of any Plain Apply the outward ledge of the fixed piece to the horizontal line of your plain holding your instrument paralel to the horizon This done lift up the thread and plummer until the shadow of the thread fall directly upon the pin hole on the fixed