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Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	line_n	point_n	require_v	1,303	5	9.1027	5	false

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52257	Cosmographia, or, A view of the terrestrial and cœlestial globes in a brief explanation of the principles of plain and solid geometry applied to surveying and gauging of cask : the doctrine of primum mobile : with an account of the Juilan & Gregorian calendars, and the computation of the places of the sun, moon, and fixed stars ... : to which is added an introduction unto geography / by John Newton ...	Newton, John, 1622-1678.	1679 (1679)	Wing N1055; ESTC R17177	190,483	519

as that by the Sights on your Ruler you may look to the other Station this done turn your Ruler to that Object whose distance you desire to know and observe how many Degrees of the Circle are cut by the Ruler as suppose 36 Degrees as the Angle ACD in Fig. 30. Then removing your Instrument to D lay the Ruler on the Diameter thereof and then turn the whole Instrument about till through your Sights you can espy the mark set up at your first Station at C and there fix your Instrument and then upon the Centre of your Circle turn your Ruler till through the Sights you can espy the Object whose distance is inquired suppose at A and observe the Degrees in the Circle cut by the Ruler which let be 112 which is the Angle ADC and let the distance between your two Stations be DC 326 Foot so have you two Angles and the side between them in a plain Triangle given by which to find the other sides the which by protraction may be done as hath been shewed in the fifth Proposition of Chapter 8. but by the Table of Sines and Tangents the Proportion is As the Sine of DAC is to DC so is the Sine of ACD to the Side AD. Or as the Sine of DAC is to the given Side DC So is the Sine of ADC to the Side AC 6. There is another Instrument called the plain Table which is nothing else but a piece of Board in the fashion and bigness of an ordinary sheet of paper with a little frame to fasten a sheet of paper upon it which being also set upon a Staff you may by help of your Ruler take a distance therewith in this manner Having measured the distance between your two Stations at D and C draw upon your paper a Line on which having set off your distance place your Instrument at your first Station C and laying your Ruler upon the Line so drawn thereon turn your Instrument till through the Sights you can espy the Station at D then laying your Ruler upon the Point C turn the same about till through the Sights you can espy the Object at A and there draw a Line by the side of your Ruler and remove your Instrument to D and laying your Ruler upon the Line DC turn the Instrument about till through the Sight you can espy the Mark at C and then laying your Ruler upon the Point D turn the same till through the Sights you can espy the Object at A and by the side of your Ruler draw a Line which must be extended till it meet with the Line AC so shall the Line AD being measured upon your Scale of Equal Parts be the distance of the Object from D and the Line AC shall be the distance thereof from C. 7. And in this manner may the distance of two three or more Objects be taken from any two Stations from whence the several Objects may be seen and that either by the plain Table or Theodolite CHAP. XI How to take the Plot of a Field at one Station from whence the several Angles may be seen ALthough there are several Instruments by which the Plat of a Field may be taken yet do I think it sufficient to shew the use of these two the plain Table and Theodolite 2. In the use of either of which the same chain which is used in taking of heights and distances is not so proper I rather commend that which is known by the Name of Gunter's Chain which is four Pole divided into 100 Links being as I conceive much better for the casting up the Content of a Piece of Ground than any other Chain that I have yet heard of whose easie use shall be explained in its proper place 3. When you are therefore entered the Field with your Instrument whether plain Table or Theodolite having chosen out your Station let visible Marks be set up in all the Corners thereof and then if you use the plain Table make a mark upon your paper representing your Station and laying your Ruler to this Point direct your Sights to the several Corners of the Field where you have caused Marks to be set up and draw Lines by the side of the Ruler upon the paper to the point representing your station then measure the distance of every of these Marks from your Instrument and by your Scale set those distances upon the Lines drawn upon the paper making small marks at the end of every such distance Lines drawn from Point to Point shall give you upon your paper the Plot of the Field by which Plot so taken the content of the Field may easily be computed Example Let Fig. 31. represent a Field whose Plot is required your Table being placed with a sheet of paper thereupon make a Mark about the middle of your Table as at A. apply your Ruler from this Mark to B and draw the Line AB then with your Chain measure the distance thereof which suppose to be 11 Chains 36 Links then take 11 Chains 36 Links from your Scale and set that distance from A to B and at B make a mark Then directing the Sights to C draw a Line by the side of your Ruler as before and measure the distance AC which suppose to be 7 Chains and 44 Links this distance must be taken from your Scale and set from A to C upon your paper And in this manner you must direct your Sights from Mark to Mark until you have drawn the Lines and set down the distances between all the Angles in the Field and your station which being done you must draw the Lines from one Point to another till you conclude where you first began so will those Lines BC. CD DE. FG. and GB give you the exact Figure of the Field 4. To do this by the Theodolite in stead of drawing Lines upon your paper in the Field you must have a little Book in which the Pages must be divided into five Columns in the first Column whereof you must set several Letters to signifie the several Angles in the Field from which Lines are to be drawn to your place of standing in the second and third Columns the degrees and parts taken by your Instrument and the fourth and fifth to set down your distances Chains and Links this being in readiness and have placed your Instrument direct your Sights to the first mark at B and observe how many Degrees are comprehended between the Diameter of your Instrument and the Ruler and set them in the second and third Columns of your Book against the Letter B which stands for your first Mark then measure the distance AB as before and set that down in the fourth and fifth Columns and so proceed from Mark to Mark until you have taken all the Angles and Distances in the Field which suppose to be as they are expressed in the following Table   Degr. Part Chains Links B 39 75 11 56 C 40 75 7 44 D 96

Sun 's or Earth's Motion IN the first part of this Treatise we have spoken of the primary Motion of the Planets and Stars as they are wheeled about in their diurnal motion from East to West but here we are to shew their own proper motions in their several Orbs from West to East which we call their second motions 1. And these Orbs are supposed to be Elliptical as the ingenious Repler by the help of Tycho's accurate observations hath demonstrated in the Motions of Mars and Mercury and may therefore be conceived to be the Figure in which the rest do move 2. Here then we are to consider what an Ellipsis is how it may be drawn and by what Method the motions of the Planets according to that Figure may be computed 3. What an Ellipsis is Apollonius Pergaeus in Conicis Claudius Mydorgius and others have well defined and explained but here I think it sufficient to tell the Reader that it is a long Circle or a circular Line drawn within or without a long Square or a circular Line drawn between two Circles of different Diameters 4. The usual and Mechanical way of drawing this Ellipsis is thus first draw a line to that length which you would have the greatest Diameter to be as the Line AP in Figure 8 and from the middle of this Line at X set off with your compasses the Equal distance XM and XH 5. Then take a piece of thred of the same length with the Diameter AP and fasten one end thereof in the point M and the other in the point H and with your Pen extend the thred thus fastened to the point A and from thence towards P keeping the thread stiff upon your Pen draw a line from A by B to P the line so drawn shall be half an Ellipsis and in like manner you may draw the other half from P by D to A. In which because the whole thred is equal to the Diameter AP. therefore the two Lines made by thred in drawing of the Ellipsis must in every point of the said Ellipsis be also equal to the same Diameter AP. They that desire a demonstration thereof geometrically may consult Apollonius Pergaeus Claudius Mydorgius or others in their treatises of Conical Sections this is sufficient for our present purpose and from the equality of these two Lines with the Diameter a brief Method of calculation of the Planets place in an Ellipsis is thus Demonstrated by Dr. Ward now Bishop of Salisbury 6. In this Ellipsis H denotes the place of the Suns Center to which the true motion of the Planet is referred M the other Focus whereunto the equal or middle motion is numbred A the Aphelion where the Planet is farthest distant from the Sun and slowest in motion P the Perihelion where the Planet is nearest the Sun and slowest in motion In the points A and P the Line of the mean and true motion do convene and therefore in either of these places the Planet is from P in aequality but in all other points the mean and true motion differ and in D and C is the greatest elliptick AEquation 8. Now suppose the Planet in B the line of the middle motion according to this Figure is MB the line of the true motion HB The mean Anomaly AMB. The Eliptick aequation or Prosthaphaeresis MBH which in this Example subtracted from AMB the remainer AHB is the true Anomaly And here note that in the right lined Triangle MBH the side MH is always the same being the distance of the Foci the other two sides MB and HB are together equal to AP. Now then if you continue the side MB till BE be equal to BH and draw the line HE in the right lined Triangle MEH we have given ME = AD and MH with the Angle EMH to find the Angles MEH and MHE which in this case are equal because EB = BH by Contraction and therefore the double of BEH or BHE = MBH which is the Angle required And that which yet remaineth to be done is the finding the place of the Aphelion the true Excentricity or distance of the umbilique points and the stating of the Planets middle motion CHAP. X. Of the finding of the Suns Apogeon quantity of Excentricity aend middle motion THe place of the Suns Apogaeon and quantity of Excentricity may from the observations of our countrey man Mr. Edward Wright be obtained in this manner in the years 1596 and 1497 the Suns entrance into ♈ and ♎ and into the midst of ♉ ♌ ♍ and ♒ were as in the Table following expressed   1596 1597     D. H. M. D. H. M.   Ianuary 25. 00.07 24. 05.54 ♒ 15 March 9. 18.43 10. 00.37 ♈ 0 April 24. 21.47 25. 03.54 ♉ 15 Iuly 28. 01.43 28. 09.56 ♌ 15 September 12. 13.48 12. 19.15 ♎ 0 October 27. 15.23 27. 21.50 ♍ 15 And hence the Suns continuance in the Northern Semicircle from ♈ to ♎ in the year 1596 being Leap year was thus found   d. h. From the 1. of Ianuary to ☉ Entrance ♎ 256. 13. 48. From the 1. of Iun to ☉ Entrance ♈ 69. 18.43 Their difference 186. 19.05 In the year 1597 from the 1 of Ianuary to the time of the ☉ Entrance into ♎ 255. 19.15 To the ☉ entrance into ♈ 69. 09.37 Their difference is 186. 18.38 And the difference of the Suns continuance in these Arks in the year 1596 and 1597 is 27′ and therefore the mean time of his continuance in those Arks is days 186. hours 18. minutes 51. seconds 30. And by consequence his continuance in the Southern Semicircle that is from ♎ to ♈ is 178 days 11 hours 8 minutes and 30 seconds In like manner in the year 1596 between his entrance into ♉ 15. and ♍ 15 there are days 185. 17.36 And in the year 1597 there are days 185. 17.56 And to find the middle motion answering to days 186. hours 18. Minutes 51. seconds 30 I say As 365 days 6 hours the length of the Julian year is to 360 the degrees in a Circle So is 186 days 18 hours 51′ 30″ to 184 degrees 03′ 56″ In like manner the mean motion answering to 185 days 17 h. 46′ is 183 degrees 02′ 09 Apparent motion from ♈ to ♎ 180. 00.00 Middle motion 184. 03.56 Their Sum 364. 03.56 Half Sum is the Arch. SME 182. 01.58 In 1596 from 15 ♒ to 15 ♌ there are days 185 hours 01 minutes 36. In 1597. days 135. hours 4. 02′ And the mean motion answering thereunto is 182 d. 30′ 36″ Apparent motion from 15 ♉ to 15 ♍ 180. Middle motion 185. 17. 56. 181. 04.53 Half Sum is 183. 32. 26 From 15 ♒ to 15 ♌ Days 185. 04 h. 02′ Apparent motion 180. Middle motion 182. 30. 36 Half Sum 181. 15. 18 Now then in Fig. from PGC. 181. 32. 26 deduct NKD 180 the Remainer is DC+NP 1. 32. 26. Therefore DC or NP. 46. 13 whose Sine is HA. And from XPG. 181. 15. 18 deduct