Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	side_n	2,221	5	9.5367	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A68509	The newe attractiue Containyng a short discourse of the magnes or lodestone, and amongest other his vertues, of a newe discouered secret and subtill propertie, concerning the declinyng of the needle, touched therewith under the plaine of the horizon. Now first founde out by Robert Norman hydrographer. Heerevnto are annexed certaine necessarie rules for the art of nauigation by the same R.N.	Norman, Robert, fl. 1590.; Borough, William, 1536-1599.	1581 (1581)	STC 18647; ESTC S112416	31,726	60

latitude The second the ark of the verticall circle contayned betweene your Zenith and the Sunne which is the complement of the Sunnes eleuation at the tyme of the obseruation The third side is an arke of the circle of declination comprehended betweene the Sunne and the eleuated Pole this arke is found by addyng or subtractyng the declination of the Sunne to or from the quadrant or 90. d. whiche must be done with this consideration that if you be on the same side of the Equinoctiall that the Sunne is you are to subtract the declination from the quadrant If on the other side to add it to the same so haue you the three sides of the sphericall triangle giuen Then the substance of the work consisteth in findyng the quātitie of the angle of the same triangle at the Zenith for the complement thereof to the Semicircle or two right angles is the Horizontall distance of the Sunnes Azimuth from the meridian which beyng compared with the variation of the Sunnes shadowe vppon the Instrument giueth the thyng required LEt FACE be the meridian wherein A. the Zenith C. the Pole AD. the verticall circle or Azimuth of the Sunne passyng by B. the place of the Sunne at the tyme of the obsetuation BD. the eleuatiō of the Sunne BA the complement of the eleuation AC the complement of the latitude BC. the ark of the circle of declination or the chord of the same ark FGE the plaine of the Horizō Now from the three angles of the triangle ABC let fall 3. perpendicular lines to the plaine of the Horizō AG. CH. and BK and by the 6. of the 11. of Euclide these three lines shall be parallelles Then let fall a perpendicular line from C. vppon AG. in the poinct L. from B. an other perpendicular vppon the same line AG. at the poinct M. And from the same point M. erect a perpendicular line to N. which shalbe parallell and equall to LC Then ioyne B. and N. together So haue you a right-lined triangle BMN. whose angle at M. is equall to the angle A. of the sphericall triangle ABC By the 4. definition of the 11. of Euclide for the like reason is of obtuse angles as of acute or sharp And the sides therof BM and MN are giuen BM the sine of BA and MN equall to LC the sine of CA. And the third side BN is found by subtracting the square of NC frō the square of the chord BC. as in the 47. of the first of Euclide And in rightlined triangles the three sides beyng giuen the angles are also giuen by the 44. 45. c. of the first of Regiomontanus and by the 7. proposition of the 13. Chapter of Copernicus his first booke For example I take the former obseruation of the 16. October 1580. and work as followeth The eleuation of the Pole CE. 51. d. 32′ the sine thereof CH. 78297. The eleuation of the Sunne BD. 17. d 0′ the sine thereof BK 29237. The arke BC. 102 d. 30′ the chord thereof BC. 155976. The complement of the eleuation of the Sunne BA 73. d. 0′ the sine thereof BM 95630. The complement of the latitude AC 38. d. 28′ the sine therof LC 62205. equall to MN Now out of CH. 78297. subtract NH equall to BK 29237. Rest NC 49060. Then out of the chord BC. squared 24328512576. Take the square of NC 2406883600. Rest the square of BN 21921628976. The roote thereof is 148059. the side BN So are the three sides of the triangle giuen BN 148059. MN 62205. BM 95630. Now to finde the angle M. I subtract from the square of BM the bigger side whiche is 9145096900. the square of MN the lesser side which is 3869462025. Rest 5275634875. which diuided by the base BN 148059. giueth 35631. which number I take out of the said base rest 112428. the half therof 56214. is IN. the lesser case or shorter parte of the base diuided by the perpendicular line MI. fallyng vpon the same frō the obtuse angle M. whiche subtracted from the whole base BN 148059. leaueth IB 91845. the greater case or longer part thereof Now it is manifest that these two cases or parts of the base BI and IN. are the sines of the two sharpe angles IMB. and NMI made of the obtuse angle M. by the perpendicular fallyng from the same angle to the base and the arks of them ioyned together are the quantitie of the obtuse angle NMB. Therefore to reduce them to the nombers of the sines first for the greater case BI making BM the whole sine say BM BM BI BI If. 95630. giue 100000. then shall 91845. giue 96042. The ark thereof is 73. d. 49′ 38″ Againe for the lesser case making MN the whole sine say MN MN IN. IN. If. 62205. giue 100000. then 56214. giueth 90376. Whose ark is 64. d. 38′ 45″ And addyng these two arks together they giue 138 d. 28′ 23″ the ark or quantitie of the obtuse angle NMB. equall to the sphericall angle BAC And deductyng it frō the Semicircle 180. d. there resteth 41● d. 31′ 37″ the angle FAD the Horizontall distance of the Sunnes Azimuth from the meridian subtractyng that from 52. d. 35′ the variation found vppon the Instrument from North to West in the fornoone resteth 11. d. 3′ 23. ″ the variation of the Needle from the meridian the thyng that was proposed to be found And comparyng the same with the afternoones obseruation you shall finde it 11. d. 31′ 37″ the cause of this difference I haue declared in the former Chapter If the Reader be delighted with varietie of demonstration of this matter let him peruse the 34. proposition of the 4. of Regiomontanus and the 13. proposition of the 14. Chapter of the first booke of Copernicus But whereas you see this calculation to differ from the former in some odde seconds the reason thereof is not as it might bee taken the different nature of the rules but in workyng thereof omitting the fractions in the diuisions and neglectyng the proportionall parts of the sines and arks In these exāples I haue vsed y e abridged table of 100000. the whole sine which though it giue some ease in the working yet it is not so exact as that of 10000000. of Erasmus Reinholdus Vnto the which with his Canon foecundus aunswerable to the same if the third Canon of the Hypothenusaes were annexed we should haue an entire table for the doctrine of triangles that might worthely bee called The table of tables Whiche thyng though Georgius Ioachimus Rheticus haue well begun and framed it orderly frō ten minuts to ten yet is it left very rawly for suche as desire the exact truth of thynges I haue therefore for myne owne ease and vse calculated the complement of this table and almost ended it for the whole quadrant from minut to minut whiche if in the meane tyme before I haue finished I shall not finde it extant by any other I will publishe it

of the tyme of the day by any one obseruation in any place besides that it is of all other that hitherto haue been vsed at Sea the most tedious and vnfit for that purpose it is also by reason of the variation not considered ●iere false and erronious For the true meridian which is the grounde of his purpose is as farre to seeke as the thyng he promiseth to giue by the same The like may be said of al other Instruments made vpon the same ground whether they serue for the Sea or Land The same Author in the 4. Chapter of his booke entreting of saylyng vpon the poincts of the Cumpas saieth that in sailing South or North he shall passe by the Poles of the world and keepe vnder one meridian till he come to the place from whence he first departed And vppon the poincts of East and West out of the Equinoctiall he shall saile vnder a parallell till he returne to the place from whence he went But in saylyng vppon the poinct of Northeast he shall describe a spirall line inclyning by little and little to wardes the Pole as in his demonstration thereof in the same Chapter appeareth But for want of due consideration of the variation his rules reasons and demonstrations and suche others hitherto giuen for like purposes are friuolous and false For if he direct his saylyng by the Cumpas as of necessitie he must beyng the onely Instrument for that purpose it is manifest that whether he sayle North or South East or West or by what other poinct so euer the Cumpas not respectyng alwaies the Pole of the world as he supposeth but some other poinct or poincts distant from the same shall leade hym accordyngly whereby he shall neither keepe vnder one meridian nor vnder one parallell of latitude neither make such a spirall line to the Pole of the world as he demōstrateth His fault in settyng downe those rules is so muche the greater in that he acknowledgeth in the Chapter next before the variation at Antwerp to bee about 9. d. from North to East accordyng to Mercators position of the Magneticall Pole which he also confirmeth by his owne experience But it seemeth he hath followed that excellent Mathematician Petrus Nonius especially concernyng the saylyng vppon the poincts of East and West For he in his first booke of the rules and Instruments of Nauigation enforceth hym self to proue and demonstrate that in saylyng East or West out of the Equinoctiall the course is performed by peeces of great circles and yet describeth a parallell But how that may stande with the principles of Geometry I referre the iudgement to the expert Mathematicians for it is like as a circle should be made of straight lines which is impossible It appeareth in the discourse that he hath made of those matters that he had not a right iudgement of the nature of the Cumpas in saylyng admitting the same to shew the Pole without Variation for if he had he would neuer haue entred into suche a Labyrinth as he did But he thought it a great absurditie that the Cumpas in euery Horizon should shewe the meridian and Poles of the world by the poincts of South and North and by the poincts of East and West to shewe in the Horizon the verticall and Equinoctiall East and West being a great circle and yet in saylyng East or West except in the Equinoctiall it should performe but a parallell But it is to be vnderstood that albeit the poincts or lynes of the Cumpas doe alwaies in euery Horizon represent great circles in the Heauens the points of South and North the meridian and the poincts of East and West the verticall circle of East and West eche crossyng other at right angles and likewise of the other poincts The reason whereof is because the Cumpas lieth euery where leuell with the Horizon so as a perpendicular line descendyng from the centre thereof at right angles with the plaine of the same will alwaies fall vpon the centre of the earth and consequently be the Semidiameter of a great circle So that where so euer the Cumpas be caried these circles are supposed to be caried about with it and the view of euery thyng in the Horizon represented by the poincts thereof is likewise in great circles Yet in saylyng by the Cumpas the poincts of South and North onely describe great circles generally which are the meridians the points of East and West describe a great circle in the Equinoctiall onely in all other places out of the Equinoctiall they describe but parallells And the saylyng vppon any other poinct of the Cumpas from any place describeth a spirall line according to the angle it maketh with the meridian And hereby in saylyng vpō the poincts of East or West out of the Equinoctiall the North poinct alwaies respectyng the Pole the course performeth a parallell accordyng to the distance of the centre of the Cumpas from the Pole The maner thereof you may perceiue by fastnyng a small threed or Virginall wier at the Pole of a Globe or centre of a circle which shall represent a moueable meridian to bee caried about the Globe or circle and fixe vppon the same a small Flye of a Cumpas so as the line of South and North be answerable to the thréed or wier and the North poinct thereby alwaies respect the North Pole then in turnyng the threed about the Globe or circle vpon the Pole or center if the center of the Flye bee out of the Equinoctiall between it and the Pole albeit the points of East and West crossyng the same line and moneable meridian at right angles doe shewe the verticall East and West vpon the Globe which is a great circle yet in cariyng the same Flye vpon the threed or moueable meridian about the Pole or center you shall by the center of the same Flye describe but a parallell accordyng to the distance thereof from the Pole of the Globe or center of y e circle not vnlike the circular motiō of a Horse drawyng in a mill who though he looke forth straight in a right line yet beyng fastned to the beame of the mill is forced to make his course in a circle whose Semidiameter is the lēgth of the beame contayned betweene the Horse and the centre of the mill or mispost And as in the Equinoctiall the line of South and North in the Cumpas by supposition representing the meridian is parallell to the Axis of the earth which is the common section of all the meridian plaines and the line of East and West crossyng the same Axis at right angles representeth the verticall East and West which is the Equinoctiall imaginyng to descende from the centre of the Cumpas a line to fall perpendicularly and at right angles with the Axis of the worlde whiche shall bee at the centre of the earth and in saylyng East or West by the Cumpas the imagined perpendicular line being caried about with the same makyng alwaies right angles