Selected quad for the lemma: world_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
world_n	circle_n	north_n	pole_n	1,821	5	11.6207	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64224	Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.	Taylor, John, mathematician.	1687 (1687)	Wing T534; ESTC R23734	190,995	514

opposite points Aries and Libra and maketh an Angle therewith called its Obliquity of 23° 30 ' represented by ♋ ♎ ♑ This Circle is divided into 12 Sines each containing 30° 00 ' As Aries ♈ Taurus ♉ Gemini ♊ Cancer ♋ Leo ♌ Virgo ♍ which are called Northern Sines Libra ♎ Scorpio ♏ Sagitarius ♐ Capricornus ♑ Aquarius ♒ and Pisces ♓ these are called Southern Sines 7. The Zodiack is a Zone or Girdle having 8 deg of Latitude on either side the Ecliptick in which space the Planets make their revolution This Circle is a Circle which regulates the Years Months and Seasons and is distinguished in the Scheme by the 12 Sines 8. The Colures are two Meridians dividing the Ecliptick and the Equinoctial into four equal parts one of which passeth by the Equinoctial points Aries and Libra and is called the Equinoctial Colure as P ♎ S. The other by the beginning of Cancer and Capricorn and is called the Solstitial Colure as P ♋ S ♑ 9. The Poles of the Ecliptick are two points 23° 30 ' distant from the Poles of the World as I and K. 10. The Tropicks are two small Circles Parallel unto the Equinoctial and distant therefrom 23° 30 ' limiting the Sun's greatest declination The Northern Tropick passeth by the beginning of Cancer and is therefore called the Tropick of Cancer as ♋ a D. The Southern Tropick passeth by the beginning of Capricorn and is therefore called the Tropick of Capricorn as B b ♑ 11. The Polar Circles are two small Circles parrallel to the Equinoctial and distant therefrom 66° 30 ' and from the Poles of the World 23° 30 ' That which is adjacent unto the North Pole is called the Artick Circle as G d I. and the other the Antartick Circle as Kd M. 12. The Zenith and the Nadir are two points Diametrically opposite the one to the other the Zenith is the Vertical point or the point over our heads as Z The Nadir is opposite thereto as the point N. 13. The Azimuths or Vertical Circles are great Circles of the Sphere concurring and intersecting each other in the Zenith and Nadir as Z f N. 14 The Horizon is a great Circle 90 deg distant from the Zenith and Nadir cutting all the Azimuths at Rightangles and dividing the World into two equal parts the upper and visible Hemisphere and the lower and invisible Hemisphere represented by H ♎ R. 15. The Meridian of a Place is that Meridian which passeth by the Zenith and Nadir of the place as P Z S N. 16. The Alinicanthars or Parallels of Altitude are small Circles parrallel unto the Horizon imagined to ●pass through every degree and minute of the Meridian between the Zenith and Horizon B a F. 17. Parallels of Latitude or Declination are small Circles parallel unto the Equinoctial they are called Parallels of Latitude in respect to any place on the Earth and Parallels of Declination in respect of the Sun or Stars in the Heavens 18. The Latitude of a place is the height of the Pole above the Horizon or the distance between the Zenith and the Equinoctial 19. The Latitude of a Star is the Arch of a Circle contained betwixt the Center of a Star and the Ecliptick line this Circle making Right-angles with the Ecliptick is accounted either Northward or Southward according to the Scituation of the Star. 20. Longitude on Earth is measured by an Arch of the Equinoctial contained between the Primary Meridian or Meridian of that place where Longitude is assigned to begin and the Meridian of any other place counted always Easterly 21. The Longitude of a Star is that part of the Ecliptick which is contained between the Star's place in the Ecliptick and the beginning of Aries counting them according unto the succession of Sines 22. The Altitude of the Sun or Stars is the Arch of an Azimuth contained betwixt the Center of the Sun or Star and the Horizon 23. Ascension is the rising of any Star or part of the Equinoctial to any degree above the Horizon and Descension is the setting of it 24. Right Ascension is the number of Degrees and Minutes of the Equinoctial i. e. from the beginning of Aries which cometh unto the Meridian with the Sun or Stars or with any portion of the Ecliptick 25. Oblique-Ascension is an Arch of the Equinoctial between the beginning of Aries and that part of the Equinoctial which riseth with the Center of a Star or with any portion of the Ecliptick in an Oblique Sphere and Oblique Descention is that part of the Equinoctial tha● setteth therewith 26. The Ascentional difference is an Arch of the Equinoctial being the difference betwixt the Right and Oblique-Ascension 27. The Amplitude of the Sun or Stars is the distance of the rising or setting thereof from the East or West point of the Horizon 28. The Parallax is the difference between the true and apparent place of the Sun or Star so the true place in respect of Altitude is in the line ACE or ADG the Sun or Star being at C or D and the apparent place in the Line BCF and BDH so likewise the Angles of the Parallax are ACB or ECF and ADB or GDB also in the said Scheme ABK representeth a Quadrent of the Globe or Earth on the Earth's Superficies A the Center of the Earth and B any point of the Earth's Surface 29. The Refraction of a Star is caused by the Atmosphere or Vapourous thickness of the Air near the Earth's Superficies whereby the Sun and Stars seem always to rise sooner and and set later than really they do SECT II. Of Astronomical Propositions PROP. I. The Distance of the Sun from the next Equinoctial point either Aries or Libra being known to find his Declination THE Analogy or Proportion As Radius or S. 90° To S. of the Sun's distance from the next Equinoctial point So it S. of the Sun 's greatest Declination To the S. of the Sun 's present Declination sought PROP. II. The Sun's place given to find his Right-Ascension This is the Analogy or Proportion As Radius or S. 90° To T. of the Sun's Longitude from the next Equinoctial point So is the Sc. of his greatest Declination To T. of his Right-Ascension from the next Equinoctial point PROP. III. To find the Sun's place or longitude from Aries his Declination being given This is the Analogy or Proportion As S. of the Suns greatest Declination To Radius or S. 90° 00 ' So is S. of his present Declination To S. of the Suns Place or Longitude from Aries PROP. IV. By knowing the Suns Declination to find his Right Ascension This is the Analogy or Proportion As Radius or S. 90° To Tc. of the Suns greatest Declination So is T. of the Declination given To S. of the Suns right Ascension required PROP. V. By knowing the Latitude of a Place and the Suns Declination to find the Ascensional

A 49 47 30 To Tc. ½ V. at E 15° 47 ' Which doubled is the Angle at E 31° 34 ' as was required PROP. VIII Case 8. Two Angles and a Side opposite to one of them being known to find the Interjacent Side In the Triangle ADE there is given the Angle E 31° 34 ' the Angle D 130° 03 ' and his opposite Side AE 70° 00 ' Now the Side ED is required First by Prop. 2. Case 2. I find AD opposed to E to be 40° 00 ' and then work thus Take the Sum and Difference of the Angles then also find the Difference of the two Sides Now say As S. ½ X. VV D and E 49° 14 ' 30 To S. ½ Z VV D and E 80 48 30. So is T ½ X cr s. AD and AE 15 00 00 To T. ½ cr s. ED 19° 15 ' 00 which being doubled is the Side ED 38° 30 ' as required PROP. IX Case 9. Two Sides and their Included Angle being known to find the third Side In the Triangle APZ there is given the Side ZP 38° 30 ' the Side PA 70° and the Angle P let be 31° 34 and the Side AZ is required The Resolution of this Case depends on the Catholike proposition of the Lord of Marchiston by supposing the Oblique-Triangle to be divided by a supposed Perpendicular falling either within or without the Triangle into two Rectangulars Now in the Triangle AZP let fall the Perperpendicular ZR so is the Triangle AZP divided into two Rectangulars ARZ and ZRP Now the Side AZ may be found at two Operations thus say As the Radius or S. of 90° 00 ' To Sc. of the included V P. 31 34. So is T. of the lesser Side PZ 38 30 To T. of a fourth Arch. 34 08. If the contained Angle be less than 90° take this fourth Arch from the greater Side but if it be greater than 90° from its Complement unto 180° the Remainder is the Residual Arch Now again say As Sc. of the fourth Arch. 34° 08 ' To Sc. Residual Arch. 35 52 So Sc. of the lesser Side PZ 38 30 To Sc. AZ the Side required 40 00 But note that many times the Perpendicular will fall without the Triangle as it doth now within in such case the Sides of the Triangle must be continued so will there be two Rectangulars the one included within the other as in the Triangle HIK the Perpendicular let fall is KM falling on the Side HE and so the two Rectangulars found thereby will be IM K and KMH and so by the directions in the former proposition find out the Side IK if required to be found PROP. X. Case 10. Two Angles and their Interjacent Side known to find the third Angle In the Triangle AZP there is given the Side ZP 38° 30 ' the Angle P 31° 34 ' and the Angle Z 130° 03 ' and the Angle at A is required First the Oblique-Triangle AZP being reduced into two Rectangulars ARZ and ZRP by Case 9 aforegoing I find the Angle RZP to be 64° 19 ' in the Triangle ZRP which taken out of Angle AZP 130° 03 ' leaves the Angle AZR 65° 44 ' Now the Angle A is found by this Analogy or Proportion As S. V. PZR 64° 19 ' To S. V. AZR 65 44 So is Sc. V. at P 31 34 To Sc. V. at A 30 28 which was required to be found out and known PROP. XI Case 11. Three Sides given to find an Angle In the Triangle APZ the Side AZ is 40° 00 ' the Side ZP is 38° 30 ' the Side AP is 70° 00 ' and the Angle Z is required To find which do thus Add the three Sides together and from half their Sum deduct the Side opposite to the required Angle and then proceed as you see in the Operation following ½Sum is 65° 07 ' 30 the Sc. ½ V. at Z which doubled is 130° 03 ' 12 the Angle at Z required PROP. XII Case 12. Three Angles given to find a Side In the Triangle AZP the Angle A is 30° 28 ' 11 the Angle Z 130° 03 ' 12 the Angle P is 31° 34 ' 26 and the Side AZ opposite to P is required This Case is likewise performed as the former Case or Proposition the Angles being converted into Sides and the Sides into Angles by taking the Complement of the greatest Angle unto 180° see the work which being doubled gives the Side AZ 40° 00 required to be found out and known ☞ But if the greater Side AP were required the Operation would produce the Complem 〈…〉 thereof unto a Semicircle or 180° therfo 〈…〉 substract it from 180° it leaves the remaining required Side sought Thus I have laid down all the Cases of Triangles both Right-lined and Spherical either Right or Oblique-angled I might hereunto have annexed many Varieties unto each Case and some fundamental Axioms which somewhat more would have Illustrated and Demonstrated those Cases and Proportions but because of the smallness of this Treatise which is intended more for Practice than Theory I have for brevity sake omitted them and refer you for those things to larger Authors who have largely discoursed thereon to good purpose CHAP. VI. Of ASTRONOMY ASTRONOMY is an Art Mathematical which measureth the distinct course of Times Days Years c. It sheweth the Distance Magnitude Natural Motions Appearances and Passions proper unto the Planets and fixed Stars for any time past present and to come by this we are certified of the Distance of the starry Sky and of each Planet from the Center of the Earth and the Magnitude of any fixed Star or Planet in respect of the Earth's Magnitude SECT I. Of Astronomical Definitions 1. ASphere or Globe is a solid Body containing onely one Superficies in whose middle there is a point called the Center from which all right or streight lines drawn unto the Circumference or Superficies are Equal 2. The Poles of the World are two fixed points in the Heavens Diametrically opposite the one to the other the one called the Artick or north-North-Pole noted in the Scheme by P. The other is called the Antartick or South-Pole as S. and is not to be seen of us being in the lower Hemisphere 3. The Axis of the World is an imaginary line drawn from the north-North-Pole through the Center of the Earth unto the South-Pole about which the Diurnal motion is performed from the East to the West as the line PS 4. The Meridians are great Circles concurring and intersecting one another in the Poles of the World as PES and Pc S. 5. The Equinoctial or Equator is a great Circle 90 deg distant from the Poles of the World cutting the Meridians at Right-angles and divideth the World into two Equal parts called the Northern and Southern Hemispheres as E ♎ Q. in Scheme 42. 6. The Ecliptick is a great Circle crossing the Equinoctial in the two

edge shall shew the Hour of the day Now to draw the Hour-lines with the Radius of your line of Chords on M strike the Arch QN which divide into 5 equal parts in the points ● ● ● c. Then lay a Ruler from M unto each of those points and it will cut the line JK in the points * * * c. through which points by prop. 4. § 1. chap. 4. draw Parallels to 6 I 6 as the lines 77 88 c. which shall be the true Hour-lines of an East Plane from 6 in the Morning till 11 before Noon Then for the Hour-lines of 4 and 5 you must prick off 5 as far from 6 as 6 is from 7 and 4 as far as 6 is from 8 and draw the Hour-lines 55 and 44 as before Thus is your Dial compleated and in the forming of which you have made both an East and a West Dial which is the same in all respects only whereas the Arch H G through which the Equinoctial passed in the East Dial was described on the right hand of the Plane in the West it must be drawn on the left hand and the Hour-lines 4 5 6 7 8 9 10 and 11 in the Forenoon in the East Dial must be 8 7 6 5 4 3 2 and 1 in the West in the Afternoon as in the Figure plainly appeareth Now you may find the distance of the Hour-lines from the Substile by this Analogy or Proportion As the Radius To the Height of the Stile So is the Tangent of any Hours distance from 6 To the distance thereof from the Substile PROP. IV. How to draw the Hour-lines on a direct South and North Plane This Plane or Dial must stand upright having his face or Plane if it be a South Dial directly opposite unto the South but if a North Plane directly opposite unto the North now admit it be required to make a Direct South Dial for the Latitude of 51° 32 ' To make which first describe the Circle ABCD to represent an E●ect direct South Plane cross it with the Diameters CB and AD then out of your Line of Chords take 38° 28 ' the Complement of the Latitude and set it from A unto a and from B unto b Then lay a Ruler from C unto a and it will cut the Meridian ARD in P the Poles of Plate V Page 261 As Radius or S. 90° To Sc. of the Latitude So is T. of the Hour from Noon To T. of the Hour-line from the Meridian PROP. V. How to draw the Hour-lines on an Horizontal Plane This Horizontal Plane or Dial is one of the best and most usefull Dials in our Oblique Hemisphere Admit it be required to make an Horizontal Dial for the Latitude of 51° 32 ' To make which first describe the Circle AB CD which representeth your Horizontal Plane Then cross it with the two Diameters ARC and BRD Then take 51° 32 ' out of your Line of Chords and set it from B to a and from C to b Then lay a Ruler from A unto a and it will cut the Meridian BD in P the Pole of the World Then lay a Ruler from A unto b and it will cut ABD the Meridian in the point AE where the Equinoctial cutteth the Meridian then through the three points A AE and C draw the Equinoctial Circle whose Center is at H and found as in the former proposition Then divide the Semicircle ADC into 12 equal parts in the points ● ● ● c. Then lay a Ruler to R the Center of the Plane and on those points so shall the Equinoctial Circle AAeC be by it divided into 12 unequal parts in the points * * * * c. Then a Ruler laid unto P the Pole of the World and those Points shall cut the Semicircle CDA in those Points I I I c. Lastly from the Center R and through those Points let there be drawn right lines which shall be the true Hour-lines of such an Horizontal Plane from 6 in the Morning untill 6 at Night but for the Hours of 4 and 5 in the Morning and 7 and 8 in the Evening they are delineated by producing 4 and 5 in the Evening through the Center R and 7 and 8 in the Morning extending them out unto the other side of the Plane so shall you have those Hour-lines also on your Plane delineated as you see in the Figure The Stile of this Plane may be a thin Plate of Brass cut exactly unto the Quantity of an Angle of 51° 32 ' and set Perpendicular on the Meridian line for the forming of this Stile take out of your Line of Chords 51° 32 ' and set it from D unto e and draw Re which shall be the Axis of the Stile you may also prefix the Halves and Quarters of Hours in the very same manner as the Hours themselves were drawn Now to find out the distance of the Hour-lines from the Meridian say As the Radius or S. 90° To the S. of the Latitude So is the T. of the Hour from Noon To the T. of the Hour-line from the Meridian Line These kinds of Dials being so frequently used with us in this Oblique Sphere for the help of the speedy delineating of them I have annexed hereunto the Table of Longomontanus wherein the Hour-lines for many Latitudes are calculated A Table shewing the Distance of the Hour-lines from the Meridian in these Degrees of Latitude An Horizontal Dial Latitude The Hours from the Meridian A South Erect Dial Latitude xi i. x. ii ix iii viii iv vii v. vi D M D M D M D M D M D M 30 7 38 16 6 26 34 40 54 61 49 90 00 60 31 7 51 16 34 27 14 41 42 62 28 90 00 59 32 8 4 17 1 27 53 42 30 63 6 90 00 58 33 8 17 17 27 28 34 43 17 63 45 90 00 57 34 8 30 17 54 29 13 44 5 64 42 90 00 56 35 8 43 18 20 29 49 44 46 64 56 90 00 55 36 8 56 18 45 30 25 45 28 65 27 90 00 54 37 9 9 19 9 31 1 46 9 65 58 90 00 53 38 9 21 19 34 31 37 46 50 66 29 90 00 52 39 9 33 19 57 32 9 47 26 66 55 90 00 51 40 9 46 20 20 32 40 48 1 67 20 90 00 50 41 9 58 20 43 33 14 48 37 67 45 90 00 49 42 10 10 21 7 33 47 49 13 68 11 90 00 48 43 10 22 21 29 34 17 49 44 68 32 90 00 47 44 10 24 21 50 34 46 50 14 68 52 90 00 46 45 10 43 22 12 35 15 50 45 69 14 90 00 45 46 10 54 22 33 35 44 51 16 69 37 90 00 44 47 11 5 22 33 36 10 51 43 69 53 90 00 43 48 11 16 23 12 36 35 52 9 70 10 90 00 42 49 11 26 23 32 37 1 52 35 70 28

90 00 41 50 11 36 23 51 37 27 53 1 70 43 90 00 40 51 11 46 24 9 37 50 53 24 70 58 90 00 39 52 11 56 24 26 38 13 53 46 71 12 90 00 38 53 12 5 24 44 38 36 54 8 71 27 90 00 37 54 12 14 25 2 38 59 54 30 71 41 90 00 36 55 12 23 25 18 39 18 54 50 71 53 90 00 35 56 12 32 25 33 39 38 55 9 72 4 90 00 34 57 12 46 25 49 39 58 55 28 72 16 90 00 33 58 12 48 26 5 40 18 55 46 72 27 90 00 32 59 13 56 26 19 40 36 56 1 72 38 90 00 31 60 13 58 26 30 40 53 56 15 72 47 90 00 30 PROP. VI. How to draw the Hour-lines on an Erect declining Plane These Planes are made to set on the sides of Houses wherein the Meridian is always a Perpendicular drawn on the Plane in whose top is the Center where the Substile and the Hour-lines all meet Now before we can delineate the Hour-lines on any such Planes two things must be given As the Latitude of the Place and the Planes Declination by having which we must find these three things viz. The Poles height above the Plane The distance of the substile from the Meridian And the Plane's difference of Longitude For the finding of which Requisites by Geometrical Projection we describe on the Dial Plane these Circles of the Sphere viz. The Horizon Meridian and Equinoctial which being described in their true Position on the Plane we proceed thus Admit it be required to make a Direct South Dial on an Erect Direct South Plane Declining Westward 24° 20 ' in the Latitude of 51° 32 ' Now in order to find the requisites before mentioned describe the Circle ZHNO and cross it with the two Diameters ZQN and H QO now Z is the Zenith N the Nadir ZQN the Hour-line of 12 HQO the Horizon Now seeing the Plane declines S. W. 34° 20 ' make Na and Ob each equal to 34 20 Then a Ruler layed from Z to a will cut the Horizon in S the South point of the Horizon through which draw the Meridian ZSN whose Center is at Y found as in the fourth Proposition aforegoing Then a Ruler laid from Z to b will cut the Horizon in W the West point thereof Now the Horizon and the Meridian being projected on the Plane take out of your line of Chords 51° 32 ' which place from H unto c and from N unto d then lay a Ruler from W unto c and it cutteth the Meridian in P the Pole of the World. Then through P and Q draw the line PQD which representeth the Axis of the World and the Substilar line of the Dial then lay a Ruler from W to d it cutteth the Meridian in AE so is W AE two points through which the Equinoctial must pass whose Center is found as afore to be at M being always in the Axis of the World so have you on your Plane the Horizon HQO the Meridian ZPSAe N and the Equinoctial LAeKWG described on the Plane as required Now first to find the Poles height above the Plane which in this Scheme is represented by BP Lay a Ruler from G unto P and it shall cut the Plane in V then measure the distance BV on your line of Chords and you will find it to contain 34° 33 ' which is the Poles height above the Plane Secondly To find the distance of the Substile from the Meridian represented in the Scheme by the Arch ZB or ND which measured as afore will appear to be 18° 08 ' the distance of the Substile from the Meridian Thirdly To find the Plane's Difference of Longitude which in the Scheme is represented by the Angle AEPK lay a Ruler from P unto AE and it cutteth the Plane in X then measure the Arch DX as afore and so will you find the Planes Difference of Longitude to be 30° 00 ' Thus by Geometrical Projection have we found all the three Requisites Now to find them by Arithmetical Calculation observe these Analogies or Proportions 1. For the Poles height above the Plane say As Radius or S. 90° To Sc. of the Latitude 38° 28 ' So is Sc. of the Declination 65° 40 ' To S. of the Poles height above the Plane 34° 33 ' 2. For the Distance of the Substile from the Meridian say As the Radius or S. 90° 00 ' To the S. of the Plane's Declination 24° 20 ' So is Tc. of the Latitude 38° 28 ' To the T. of the Substilar Distance from the Meridian 18° 10 ' 3. For the Plane's Difference of Longitude say As the Sc. of the Latitude 38° 28 ' To the Radius or S. 90° 00 ' So is S. of the Substilar Distance 18° 10 ' To the S. of the Difference of Longitude 30 Deg. Or it may be found thus As the S. of the Latitude To the Radius So is the T. of the Declination To the T. of the Difference of Longitude required These things found we come now to shew how the Hour-lines may be projected To project which observe First to lay a Ruler from P the Pole of the World to AE the Intersection of the Equinoctial with the Meridian and it will cut the Plane in x where begin to divide the Semicircle L x G into 12 Equal parts in the Points ● ● ● ● c. Then lay a Ruler from Q to every of those parts and it shall cut the Equinoctial and divide it into 12 unequal parts in the points * * * * c. Then a Ruler laid from P the Pole of the World unto each of these points it will divide the Plane into 12 unequal parts in the Points I I I I c. Then by a Ruler laid from the Center Q to those points draw right lines which shall be the true Hour-lines proper unto such a Declining Plane as you see plainly demonstrated by the Scheme Now the Substilar line falleth in this Dial just on the Hour-line of 2 in the Afternoon because the Plane declineth Westerly The Angle of the Stile is DQR 34° 33 ' which may be either a Plate or Wyre brought into such an Angle which must be placed Perpendicular to the Plane and directly over the Substilar line QD 2. Now the distance of the Hour-lines from the Substilar line may also be found by this Analogy or Proportion As the Radius To the S. height of the Pole above the Plane So is the T. of the Hour-line from the Meridian of the Plane To the T. of the Hour-line from the Substile Thus have you compleated your Dial as you see in the Scheme and here you may take notice that having finished a West Decliner you have also made an East Decliner if you only convert the Hour-lines of the West Decliner in such manner as you see in Fig. 72. on the East