20 5 SbW 13 5 10 39 SSW 1 50 1 29 Â Style 0 0 SWbS 9 25 7 38 SW 20 40 16 58 SWbW 31 55 26 45 WSW 43 10 37 11 WbS 54 25 48 30 West 65 40 60 48 WbN 76 55 73 58 WNW 88 10 87 44 These angles being knowne if on the center V at the verticall point you describe an occult circle and therein inscribe the chords of these angles from the line VH and then draw right lines through the verticall point and the termes of those chords the lines so drawne shall be the azimuths required The like reason holdeth for the drawing of the azimuths vpon all other inclining planes wheof you haue another example in the Diagram belonging to the meridian incliner Pag. 126. Or for further satisfaction you may finde where each azimuth line shall crosse the equator As the sine of 90 gr to the sine of the latitude So the tangent of the azimuth from the meridian to the tangent of the equator from the meridian Extend the compasses from the sine of 90 gr vnto the line of our latitude 51 gr 30 m. the same extent will reach in the line of tangents from 10 gr vnto 7. gr 50. m. for the intersection of the equator with the azimuth of 10 gr from the meridian Againe the same extent will reach from 20 gr vnto 15 gr 54 m. for the azimuth of 20 gr And so the rest as in these tables Azim Equat. Gr. M. Gr. M. 10 0 7 50 20 0 15 54 30 0 24 20 40 0 43 18 50 0 13 0 60 0 53 35 70 0 65 3 80 0 77 18 90 0 90 0 Azim Equat. Gr. M. Gr. M. 11 15 8 51 22 30 17 58 33 45 27 36 45 0 38 2 56 15 49 30 67 30 62 6 78 45 75 44 90 0 90 0 By which you may see that the azimuth 90 gr distant from the meridian which is the line of East and West will crosse the equator at 90 gr from the meridian in the same point with the horizontall line and the houre of 6. And that the azimuth of 45 gr will crosse the equator at 38 gr 2 m. from the meridian that is the line of SE will crosse the equator at the houre of 9 and 28 m. in the morning and the line of SW at 2 ho. 32 min. in the afternoone and so for the rest whereby you may examine your former worke CHAP. XX. To describe the parallels of the horizon in the former planes THe parallels of the horizon commonly called Almicanters or parallels of altitude whereby we may know the altitude of the Sunne aboue the horizon haue such respect vnto the horizon as the parallels of declination vnto the equator and so may be described in like maner In an horizontall plane these parallels will be perfect circles wherefore knowing the length of the style in inches and parts and the distance of the parallell from the horizon in degrees and minutes As the tangent of 45 gr is to the length of the style So the cotangent of the parallell to the semidiameter of his circle Thus in the example of the horizontall plane Pag. 164. if AB the length of the style shall be 5 inches and that it were required to finde the semidiameter of the parallell of 62 gr extend the compasses from the tangent of 45 gr vnto 5.00 in the line of numbers the same extent will reach from the tangent of 28 gr the complement of the parallell vnto 2.65 and if you describe a circle on the center A to the semidiameter of 2 inches 65 cent it shall be the parallell required In all vpright planes whether they be direct verticals or declining or meridian planes these parallels will be conicall sections and may be drawne through their points of intersection with the azimuth lines in the same maner as the parallels of declination through their points of intersection with the houre-houre-lines To this end you may first finde the distance betweene the top of the style and the azimuth and then the distance betweene the horizon and the parallell both which may be represented in this maner On the center B and any semidiameter BH describe an occult arke of a circle and therein inscribe the chords of such parallels of altitude as you intend to draw on the plane I haue here put them for 15. 30. 45 and 60 gr then draw right lines through the center and the termes of those chords so the line BH shall be the horizon and the rest the lines of altitude according to their distance from the horizon That done consider your plane which here for example is the South face of our vertical plane p. 168 wherein hauing drawne both the horizontall verticall lines as I shewed before first take out AB the length of the style pricke that downe in this horizontall line from B vnto A then take out all the distances between B the top of the style and the seuerall points wherein the verticall lines do crosse the horizontal transfer them into this horizontal line BH from the center B and at the terms of these distances erect lines perpendicular to the horizon noting them with the number or letter of the azimuth from whence they were taken so these perpendiculars shall represent those azimuths and the seuerall distances betweene the horizon and the lines of altitude shall giue the like distances betweene the horizontall and the parallels of altitude vpon the azimuths in your plane Vpon this ground it followeth 1 To find the distance between the top of the style and the seuerall points wherein the azimuths do crosse the horizontall line Hauing drawne the horizontall and azimuth lines as before looke into the table by which you drew them and there you shall haue the angles at the zenith Then As the cosine of the angle at the zenith is to the sine of 90 gr So the length of the style to the distance required Azimuths Ang Ze Tangent Secant Par. 15. Par. 30. Gr. M Inch P. Inch P. Inch. P. Inch. P. South 0 0 0 0 10 00 2 68 5 77 SbE 11 15 1 99 10 20 2 73 5 90 SSE 22 30 4 14 10 82 2 90 6 24 SEbS 33 45 6 68 12 03 3 23 6 94 SE 45 0 10 00 14 14 3 80 8 16 SEbE 56 15 14 97 18 00 4 82 10 40 ESE 67 30 24 14 26 13 7 02 15 08 EbS 78 45 50 27 51 26 13 73 29 60 East 90 0 Infinit Infinit Infinit Infinit As in our example of the verticall plane where AB the length of the style was supposed to be 10 inches extend the compasses from the sine of 78 gr 45 m. the complement of 11 gr 15 m. the angle at the zenith belonging to SbE and SbW vnto the sine of 90 gr the same extent wil reach from 10.00 the length of the style vnto 10.20 for the distance betweene the top of the
DE SECTORE RADIO The description and vse of the Sector in three bookes The description and vse of the Crosse-Staffe in other three bookes For such as are studious of Mathematicall practise LONDON Printed by WILLIAM IONES and are to be sold by IOHN TAP at Saint Magnus corner 1623. THE DESCRIPTION AND VSE OF THE SECTOR For such as are studious of Mathematicall practise LONDON Printed by WILLIAM IONES 1623. THE FIRST BOOKE OF THE SECTOR CHAP. I. The Description the making and the generall vse of the Sector A Sector in Geometrie is a figure comprehended of two right lines containing an angle at the center and of the circumference assumed by them This Geometricall instrument hauing two legs containing all varietie of angles the distance of the feete representing the subtenses of the circumference is therefore called by the same name It containeth 12 seuerall lines or scales of which 7 are generall the other 5 more particular The first is the scale of Lines diuided into 100 equall parts and numbred by 1.2.3.4.5.6.7.8.9.10 The second the lines of Superficies diuided into 100 vnequall parts and numbred by 1.1.2.3.4.5.6.7.8.9.10 The third the lines of Solids diuided into 1000 vnequal parts numbred by 1. 1. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. The fourth the lines of Sines and Chords diuided into 90 degrees and numbred with 10. 20. 30. vnto 90. These foure lines of Lines of Superficies of Solids and of Sines are all drawne from the center of the Sector almost to the end of the legs They are drawne on both the legs that euery line may haue his fellow All of them are of one length that they may answere one to the other And euery one hath his parallels that the eye may the better distinguish the diuisions But of the parallels those onely which are inward most containe the true diuisions There are three other generall lines which because they are infinite are placed on the side of the Sector The first a line of Tangents numbred with 10.20.30.40.50.60 signifying so many degrees from the beginning of the line of which 45 are equall to the whole line of Sines the rest follow as the length of the Sector will beare The second a line of Secants diuided by pricks into 60 degrees whose beginning is the same with that of the line of Tangents to which it is ioyned The third is the Meridian line or line of Rumbs diuided vnequally into degrees of which the first 70 are almost equall to the whole line of Sines the rest follow vnto 84 according to the length of the Sector Of the particular lines inserted among the generall because there was voyd space the first are the lines of Quadrature placed betweene the lines of Sines and noted with 10.9.8.7 S. 6.5.90 Q. The second the lines of Segments placed betweene the lines of Sines and Superficies diuided into 50 parts and numbered with 5.6.7.8.9.10 The third the lines of Inscribed bodies in the same Sphere placed betweene the scales of Lines and noted with D. S. I. C. O. T. The fourth the lines of Equated bodies placed between the lines of Lines and Solids and noted with D. I. C. S. O. T. The fift are the lines of Mettalls inserted with the lines of Equated bodies there being roome sufficient and noted with these Characters ☉ ☿ ♄ ☽ ♀ ♂ ♃ There remaine the edges of the Sector and on the one I haue set a line of Inches which are the twelth parts of a foote English on the other a lesser line of Tangents to which the Gnomon is Radius 2 Of the making of the Sector LEt a Ruler be first made either of brasse or of wood like vnto the former figure which may open and shut vpon his center The head of it may be about the twelth part of the whole length that it may beare the moueable foote and yet the most part of the diuisions may fall without it Then let a moueable Gnomon be set at the end of the moueable foote and there turne vpon an Axis so as it may sometime stand at a right angle with the feete and sometimes be inclosed within the feet But this is well knowne to the workeman For drawing of the lines Vpon the center of the Sector and semidiameter somewhat shorter then one of the feet draw an occult arke of a circle crossing the closure of the inward edges of the Sector about the letter T. In this arke at one degree on either side from the edge draw right lines from the Center fitting them with Parallels and diuide them into an hundred equall parts with subdivisions into 2.5 or 10. as the line will beare but let the numbers set to them be onely 1.2.3.4 c. vnto 10. as in the example These lines so divided I call the lines or scales of Lines and they are the ground of all the rest In this Arke at 5 degrees on either side from the edge neere T drawe other right lines from the Center and fit them with Parallells These shall serue for the lines of Solids Then on the other side of the Sector in like manner vpon the Center equall Semidiameter drawe another like Arke of a circle heere againe at one neere degree on either side frō the edge neere the letter Q draw right lines from the Center and fit them with parallells These shall serue for the lines of Sines At 5 Degrees on either side from the edge neere Q drawe other right lines from the center and fit them with parallels these shall serue for the lines of Superficies These foure principall lines being drawne and fitted with parallels we may drawe other lines in the middle betweene the edges and the lines of Lines which shall serue for the lines of inscribed bodies and others betweene the edges and the Sines for the lines of quadrature And so the rest as in the example 3 To diuide the lines of Superficies SEeing like Superficies doe hold in the proportion of their homologall sides duplicated by the 29 Pro. 6 lib. Euclid If you shall find meane proportionals between the whole side and each hundred part of the like side by the 13 Pro. 6 lib. Euclid all of them cutting the same line that line so cut shal conteine the divisions required wherefore vpon the center A and Semidiamiter equall to the line of Lines describe a Semicircle ACBD with AB perpendicular to the diameter CD And let the Semidiameter AD he divided as the line of Lines into an hundred parts AE the one halfe of AC diuided also into an hundred parts so shall the diuisions in AE be the centers from whence you shall describe the Semicircles C 10. C 20. C 30. c. diuiding the lin AB into an hundred vnequall parts and this line AB so diuided shall be the line of Superficies and must be transferred into the Sector But let the numbers set to them be onely 1.1.2.3 vnto 10 as in the example 4 To
betweene the lines giuen as they are lines by the fifth Prop. of Lines then open the Sector in the lines of Superficies according to his number to the quantitie of the one and a parallell taken betweene the points of the number belonging to the other line shall be the meane proportionall Let the lines giuen be A and C. The proportion between them as they are lines wil be found by the fifth Prop. of Lines to be as 4 to 9. Wherefore I take the line C and put it ouer in the lines of Superficies betweene 9 and 9 and keeping the Sector at this angle his parallell betweene 4 and 4 doth giue me B for the meane proportionall Then for proofe of the operation I may take this line B and put it ouer between 9 and 9 so his parallel between 4 and 4 shall giue me the first line A. Whereby it is plaine that these three lines do hold in continuall proportion and therefore B is a meane proportionall betweene A and C the extremes giuen Vpon the finding out of this meane proportion depend many Corollaries as To make a Square equall to a Superficies giuen IF the Superficies giuen be a rectangle parallellogram a meane proportionall betweene the two vnequal sides shall be the side of his equall square If it shall be a triangle a meane proportion betweene the perpendicular and halfe the base shal be the side of his equal square If it shall be any other right-lined figure it may be resolued into triangles and so a side of a square found equall to euery triangle and these being reduced into one equall square it shall be equall to the whole right-lined figure giuen To finde a proportion betwene Superficies though they be vnlike one to the other IF to euery Superficies we find the side of his equall square the proportion betweene these squares shall be the proportion betweene the Superficies giuen Let the Superficies giuen be the oblonge A and the triangle B. First between the vnequall sides of A I find a meane proportionall and note it in C this is the side of a square equall vnto A. Then betweene the prependicular of B and halfe his base I finde a meane proportionall and note it in B this is the side of a Square equall to B but the proportion between the squares of C and B will be found by the first Prop. of Superficies to be as 5 to 4 and therefore this is the proportion betwene those giuen Superficies To make a Superficies like to one Superficies and equall to another First between the perpendicular and the base of B I find a meane proportionall and note it in B as the side of his equall square then betweene the perpendicular of the triangle A and halfe his base I find a meane proportionall and note it in A as the side of his equall square Wherefore now as the side B is to the side A so shall the sides of the Rhomboides giuen be to C and D the sides of the Romboides required his pendicular also to E the perpēdicular required Hauing the sides and the perpendicular I may frame the Rhomboides vp and it will be equall to the triangle A. If the Superficies giuen had been any other right-lined figures they might haue been resolued into triangles and then brought into squares as before Many such Corollaries might haue been annexed but the meanes of finding a meane proportionall being knowne they all follow of themselues 7 To finde a meane proportionall betweene two numbers giuen FIrst reckon the two numbers giuen on both sides of the Lines of Superficies from the center and mark the termes whereunto they extend then take a line out of the Line of Lines or any other scale of equall parts resembling one of those numbers giuen and put it ouer in the termes of his like number in the lines of Superficies for so keeping the Sector at this angle the parallell taken from the termes of the other number and measured in the same scale from which the other parallell was taken shall here shew the meane proportionall which was required Let the numbers giuen be 4 and 9. If I shall take the line A in the Diagram of the sixt Prop. resembling 4 in a scale of equall parts and to it open the Sector in the termes of 4 and 4 in the lines of Superficies his parallell betweene 9 and 9 doth giue me B for the meane proportionall And this measured in the scale of equall parts doth extend to 6 which is the meane proportionall number between 4 and 9. For as 4 to 6 so 6 to 9. In like maner if I take the line C resembling 9 in a scale of equall parts and to it open the Sector in the termes of 9 and 9 in the lines of Superficies his parallell between 4 and 4 doth giue me the same line B which will proue to be 6 as before if it be measured in the same scale whence D was taken 8 To find the square roote of a number 9 The roote being giuen to find the square number of that roote IN the extraction of a square roote it is vsuall to set pricks vnder the first figure the third the fifth the seuenth and so forward beginning from the right hand toward the left and as many pricks as fall to be vnder the square number giuen so many figures shall be in the roote so that if the number giuen be lesse then 100 the roote shall be only of one figure if lesse then 10000 it shall be but two figures if lesse then 1000000 it shall be three figures c. Thereupon the lines of Superficies are diuided first into an hundred parts and if the number giuen be greater then 100 the first diuision which before did signifie only one must signifie 100 and the whole line shall be 10000 parts if yet the number giuen be greater then 10000 the first diuision must now signifie 10000 and the whole line be esteemed at 1000000 parts and if this be too little to expresse the number giuen as oft as we haue recourse to the beginning the whole line shall increase it selfe an hundred times By this meanes if the last pricke to the left hand shall fall vnder the last figure which will be as oft as there be odde figures the number giuen shall fall out betweene the center of the Sector and the tenth diuision but if the last prick shall fall vnder the last figure but one which will be as oft as there be euen figures then the number giuen shall fall out betweene the tenth diuision and the end of the Sector This being considered when a number is giuen and the square roote is required take a paire of compasses and setting one foote in the center extend the other to the terme of the number giuen in one of the lines of Superficies for this distance applied to one of the Lines of Lines shall shew what the Square roote is without opening the Sector Thus
figures if aboue these and lesse then 1000000000 it shall be but three figures c. whereupon the lines of Solids are diuided first into 1000 parts and if the numbers giuen be greater thē 1000 the first diuision whcih before did signifie onely one must signifie 1000 and the whole line shall be 1000000 if yet the number giuen be greater then 1000000 the first diuision must now signifie 1000000 and the whole line be esteemed at 1000000000 parts and if these be to little to expresse the numbers giuen as oft as wee haue recourse to the begining the whole line shall encrease it selfe a thousand times By these meanes if the last pricke to the left hand shall fall vnder the last figure the number giuen shall be reckoned at the beginning of the lines of Solids from 1 to 10 and the first figure of the roote shall be alwayes either 1 or 2. If the last pricke shall fall vnder the last figure but one then the number giuen shall be reckoned in the middle of the line of Solids betweene 10 and 100 and the first figure of the roote shall be alwayes either 2 or 3 or 4. But if the last pricke shall fall vnder the last figure but two then the number giuen shall be reckoned at the end of the line of Solids betweene 100 and 1000. This being considered when a number is giuen and the cubique roote required Set one foote of the compasses in the center of the Sector extend the other in the line of Solids to the points of the number giuen for this distance applied to one of the line of Lines shall shew what the cubique roote is without opening the Sector So the nearest roote of 8490000 is about 204. The nearest roote of 84900000 is about 439. The nearest roote of 849000000 is about 947. On the contrary a number may be cubed if first we extend the compasses to the number giuen in the line of Lines and then apply the distance to the lines of Solids as may appeare by the former examples 10 Three numbers being giuen to finde a fourth in a triplicated proportion AS like Superficies do hold in a duplicated proportion so like solids in a triplicated proportion of their homologall sides and therefore the same worke is to be obserued here on the lines of Solids as before in the lines of Superficies as may appeare by these two examples If a cube whose side is 4 inches shall be 7 pound weight and it be required to know the weight of a cube whose side is 7 inches here the proportion would be As 4 are to a cube of 7 so 7 to a cube of 37 ½ And if I tooke 7 out of the lines of Solids and put it ouer in 4 and 4 in the lines of Lines his parallell between 7 and 7 measured in the lines of Solids would be 37 ½ and such is the weight required If a bullet of 27 pound weight haue a diameter of 6 inches and it be required to know the diameter of the like bullet whose weight is 125 pounds here the proportion would be As the cubique root of 27 is vnto 6 so the cubique root of 125 is vnto 10. And if I tooke 6 out of the line of Lines and put it ouer in 27 and 27 of the lines of Solids his parallell betweene 125 and 125 measured in the line of Lines would be 10 and such is the length of the diameter required The end of the first booke THE SECOND BOOKE OF THE SECTOR Containing the vse of the Circular Lines CHAP. I. Of the nature of Sines Chords Tangents and Secants fit to be knowne before hand in reference to right-line Triangles IN the Canon of Triangles a circle is commonly diuided into 360 degrees each degree into 60 minutes each minute into 60 seconds A semicircle therefore is an arke of 180 gr A quadrant is an arke of 90 gr The measure of an angle is the arke of a circle described out of the angular point intercepted betweene the sides sufficiently produced So the measure of a right angle is alwayes an arke of 90 gr and in this example the measure of the angle BAD is the arke BC of 40 gr the measure of the angle BAG is the arke BF of 50 gr The complement of an arke or of an angle doth commōly signifie that arke which the giuen arke doth want of 90 gr and so the arke BF is the cōplement of the arke BC the angle BAF whose measure is BF is the complement of the angle BAC and on the contrary The complement of an arke or angle in regard of a semicircle is that arke which the giuen arke wanteth to make vp 180 gr and so the angle EAH is the complement of the angle EAF as the arke EH is the complement of the arke FE in which the arke CE is the excesse aboue the quadrant The proportions which these arkes being the measures of angles haue to the sides of a triangle cannot be certaine vnlesse that which is crooked be brought to a straight line and that may be done by the application of Chords Right Sines versed Sines Tangents and Secants to the semidiameter of a circle A Chorde is a right line subtending an arke so BE is the chorde of the arke BCE and BF a chorde of the arke BF A right Sine is halfe the chorde of the double arke viz. the right line which falleth perpendicularly from the one extreme of the giuen arke vpon the diameter drawne to the other extreme of the said arke So if the giuen arke be BC or the giuen angle be BAC let the diameter be drawne through the center A vnto C and a perpendicular BD be let downe from the extreme B vpon AC this perpendicular BD shall be the right sine both of the arke BC and also of the angle BAC and it is also the halfe of the chord BE subtending the arke BCE which is double to the giuen arke BC. In like maner the semidiameter FA is the right sine of the arke FC and of the right angle FAC for it falleth perpendicularly vpon AC and it is the halfe of the chord FH This whole Sine of 90 gr is hereafter called Radius but the other Sines take their denomination from the degrees and minutes of their arks Sinus versus the versed sine is a segment of the diameter intercepted betweene the right sine of the same arke and the circumference of the circle So DC is the versed sine of the arke CB and GF the versed sine of the arke BF and GH the versed sine of the arke BH A Tangent is a right line perpendicular to the diameter drawne by the one extreme of the giuen arke and terminated by the secant drawne from the center through the other extreme of the said arke A Secant is a right line drawne from the center through one extreme of the giuen arke till it meete with the tangent raised from the diameter
be described out of the angular point at any other distance let the semidiameter be turned into a parallell chord of 60 Gr. then take the chord of this arke and carrie it parallell till it crosse in like chords so the place where it stayeth shall giue the quantitie of the angle As in the former example if I make the semidiameter AB a parallell chord of 60 Gr. and then keeping the Sector at that angle carrie the chord BC parallell till it stay in like chords I shall finde it to stay in no other but 11 Gr. 15 M and such is the angle BAC 10 Vpon a right line and a point giuen in it to make an angle equall to any angle giuen FIrst out of the point giuen describe an arke cutting the same line then by the 5. Prop afore find the chord of the angle giuen agreeable to the semidiameter and inscribe it into this arke so a right line drawne through the point giuen and the end of this chord shall be the side that makes vp the angle Let the right line giuen be AB and the point giuen in it be A and let the angle giuen be 11 gr 15 m. Here I open the compasses to any semidiameter AB but as oft as I may conueniently to the laterall semiradius and setting one foot in A I describe an occult arke BC then I seeke out the chord of 11 gr 15 m. and taking it with the compasses I set one foote in B the other crosseth the arke in C by which I draw the line AC and it makes vp the angle required 11 To diuide the circumference of a circle into any parts required IF 360 the measure of the whole circumference be diuided by the number of parts required the quotient giueth the chord which being found will diuide the circumference So a chord of 120 gr will diuide the circumference into 3 equall parts a chord of 90 gr into 4 parts a chord of 72 gr into 5 parts a chord of 60 gr into 6 parts a chord of 51 gr 26. into 7 parts a chord of 45 gr into 8 parts a chord of 40 gr into 9 parts a chord of 36 gr into 10 parts a chord of 32 gr 44 m. into 11 parts a chord of 30 gr into 12 parts In like maner if it be required to diuide the circumference of the circle whose semidiameter is AB into 32 first I take the semidiameter AB and make it a parallell chord of 60 gr then because 360 gr being diuided by 32 the quotient will be 11 gr 15 m. I find the parallell chord of 11 gr 15 m. and this will diuide the circumference into 32. But here the parts being many it were better to diuide it first into fewer and after to come ouer it againe As first to diuide the circumference into 4 and then each 4 parts into 8 or otherwise as the parts may be diuided 12 To diuide a right line by extreme and meane proportion THe line to be diuided by extreme and meane proportion hath the same proportion to his greater segment as in figures inscribed in the same circle the side of an hexagon a figure of six angles hath to a side of a decagon a figure of ten angles but the side of a hexagon is a chord of 60 gr and the side of a decagon is a chord of 36 gr Let AB be the line to be diuided if I make AB a parallell chord of 60 gr and to this semidiameter find AC a chord of 36 gr this AC shall be the greater segment diuiding the whole line in C by extreme and meane proportion So that As AB the whole line is vnto AC the greater segment so AC the greater segment vnto CB the lesser segment Or let AC be the greater segment giuen if I make this a parallell chord of 36 gr the correspondent semidiameter shall be the whole line AC and the difference CB the lesser segment Or let CB be the lesser segment giuen if I make this a parallell chord of 36 gr the correspondent semidiameter shall be greater segment AC which added to CB giueth the whole line AB To auoid doubling of lines or numbers you may put ouer the whole line in the Sines of 72 gr and the parallell sine of 36 gr shall be the greater segment Or if you put ouer the whole line in the sines of 54 gr the parallell sine of 30 gr shall be the greater segment and the parallell sine of 18 gr shall be the lesser segment CHAP. III. Of the proiection of the Sphere in Plano 1 THe Sphere may be proiected in Plano in streight lines as in the Analemma if the semidiameter of the circle giuen be diuided in such sort as the line of Sines on the Sector As if the Radius of the circle giuen were AE the circle thereon described may represent the plane of the generall meridian which diuided into foure equal parts in E P AE S and crossed at right angles with EAE and PS the diameter EAE shall represent the equator and PS the circle of the houre of 6. And it is also the axis of the world wherein P stands for the North pole and S for the South pole Then may each quarter of the meridian be diuided into 90 gr from the equator towards the poles In which if we number 23 gr 30 m. the greatest declination of the Sun from E to 69 Northwards from AE to ♑ Southward the line drawne from 69 to ♑ shal be the ecliptique and the lines drawne parallell to the equator through ♋ and ♑ shall be the tropiques Hauing these common sections with the plane of the meridian if we shall diuide each diameter of the Ecliptique into 90 gr in such sort as the Sines are diuided on the Sector The first 30 gr from A toward 69 shall stand for the sine of ♈ The 30 gr next following for ♉ The rest for ♊ ♋ ♌ c. in their order So that by these meanes we haue the place of the Sun for all times of the yeare If againe we diuide AP AS in the like sort and set to the numbers 10. 20. 30. c. vnto 90 gr the lines drawne through each of these degrees parallell to the equator shall shew the declination of the Sunne and represent the parallels of latitude If farther we diuide AE AAE and his parallels in the like sort and then carefully draw a line through each 15 gr so as it makes no angles the lines so drawne shall be ellipticall and represent the houre-circles The meridian PES the houre of 12 at noone that next vnto it drawne through 75 gr from the center the houres of 11 and 1 that which is drawne through 60 gr from the center the houres of 10 and 2. c. Then hauing respect vnto the latitude we may number it from E Northward vnto Z and there place the zenith by which and the center the line drawne ZAN shall represent
the verticall circle passing through the zenith and nadir East and West and the line MAH crossing it at right angles shall represent the horizon These two being diuided in the same sort as the ecliptique and the equator the line drawne through each degree of the semidiameter AZ parallell to the horizon shall be the circles of altitude and the diuisions in the horizon and his parallels shall giue the azimuth Lastly if through 18 gr in AN be drawne a right line IK parallell to the horizon it shall shew the time when the day breaketh and the end of the twilight For example of this proiection let the place of the Sunne be the last degree of ♉ the parallell passing through this place is LD and therefore the meridian altitude ML and the depression below the horizon at midnight HD the semidiurnall arke LC the seminocturnall arke CD the declination AB the ascentionall difference BC the amplitude of ascenon AC The difference betweene the end of twilight and the day breake is very small for it seemes the parallell of the Sunne doth hardly crosse the line of twilight If the altitude of the Sunne be giuen let a line be drawne for it parallell to the horizon so it shall crosse the parallell of the Sunne and there shew both the azimuth and the houre of the day As if the place of the Sunne being giuen as before the altitude in the morning were found to be 20 gr the line FG drawne parallell to the horizon through 20 gr in AZ would crosse the parallell of the Sunne in ☉ Wherefore F ☉ sheweth the azimuth L ☉ the quantitie of houres from the meridian It seemes to be about halfe an houre past 6 in the morning and yet more then halfe a point short of the East The distance of two places may be also shewed by this proiection their latitudes being knowne and their difference of longitude For suppose a place in the East of Arabia hauing 20 gr of North latitude whose difference of longitude from London is found by an eclipse to be 5 ho. ½ Let Z be the zenith of London the parallell of latitude for that other place must be LD in which the difference of longitude is L ☉ Wherefore ☉ representing the site of that place I draw through ☉ a parallell to the horizon MH crossing the verticall AZ neare about 70 gr from the zenith which multiplied by 20 sheweth the distance of London and that place to be 1400 leagues Or multiplied by 60 to be 4200 miles 2 The Sphere may be proiected in plano by circular lines as in the generall astrolabe of Gemma Frisius by the help of the tangent on the side of the Sector For let the circle giuen represent the plane of the generall meridian as before let it be diuided into foure parts and crossed at right angles with EAE the equator and PS the circle of the houre of 6 wherein P stands for the North pole and S for the South pole Let each quarter of the meridian be diuided into 90 gr and so the whole into 360 beginning from P and setting to the numbers of 10 20 30. c. 90 at AE 180 at S 270 at E 360 at P. The semidiameters AP AAE AS A E may be diuided according to the tangents of halfe their arkes that is a tangent of 45 gr which is alwayes equall to the Radius shall giue the semidiameter of 90 gr a tangent of 40 gr shall giue 80 gr in the semidiameter a tangent of 35 gr shall giue 70. c. So that the semidiameters may be diuided in such sort as the tangent on the side of the Sector the difference being onely in their numbers Hauing diuided the circumference and the semidiameters we may easily draw the meridians and the parallels by the helpe of the Sector The meridians are to be drawne through both the poles P and S and the degrees before graduated in the equator The distance of the center of each meridian from A the center of the plane is equall to the tangent of the same meridian reckoned from the generall meridian PAESE and the semidiameter equall to the secant of the same degree As for example if I should draw the meridian PBS which is the tenth from PAES the tangent of 10 gr giueth me AC and the secant of 10 gr giueth me SC whereof C is the center of the meridian PBS and CS his semidiameter so AF a tangent of 20 gr sheweth F to be the center of PDS the twentith meridian from PAES and AG a tangent of 23 gr 30 M. sheweth G to be the center of P 69 S. c. The parallels are to be drawne through the degrees in AP AS and their correspondent degrees in the generall meridian The distance of the center of each parallell from A the center of the plane is equall to the secant of the same parallell from the pole and the semidiameter equall to the tangent of the same degree As if I should draw the parallell of 80 gr which is the tenth from the pole S first I open the compasses vnto AC the tangent of 10 gr and this giueth me the semidiameter of this parallel whose center is a little from S in such distance as the secant SC is longer then the radius SA The meridians and parallels being drawne if we number 23 gr 30 m. from E to ♋ Northward from AE to ♑ Southward the line drawne from ♋ to ♑ shall be the ecliptique which being diuided in such sort as the semidiameter AP the first 30 gr from A to ♋ shall stand for the sine of ♈ the 30 gr next following for ♉ the rest for ♊ ♋ ♌ c. in their order If farther we haue respect vnto the latitude we may number it from E Northward vnto Z and there place the zenith by which and the center the line drawne ZAN shall represent the verticall circle and the line MAH crossing it at right angles shall represent the horizon and these diuided in the same sort as AP the circles drawne through each degree of the semidiameter AZ parallell to the horizon shall be the circles of altitude and the circles drawne through the horizon and his poles shall giue the azimuths For example of this proiection let the place of the Sunne be in the beginning of ♒ the parallell passing through this place is ♒ ☉ L and therefore the meridian altitude ML and the depression below the horizon at midnight H ♒ the semidiurnall arke L ☉ the seminocturnall arke O ♒ the declination AR the ascensionall difference R ☉ the amplitude of ascension A ☉ Or if A be put to represent the pole of the world then shall PAESE stand for the equator and P ♋ S ♑ for the ecliptique and the rest which before stood for meridians may now serue for particular horizons according to their seuerall eleuations Then suppose the place of the Sunne giuen to be
seuerall mettalls and equall weight hauing the magnitude of the one to finde the magnitude of the rest Take the magnitude giuen out of the lines of Solids and to it open the Sector in the points belonging to the mettall giuen so the parallells taken from between the points of the other mettalls and measured in the lines of Solids shall giue the magnitude of their bodies Thus hauing cubes or spheres of equall weight but seuerall mettalls we shall finde that if those of tin containe 10000 D â—he others of iron wil contain 9250 those of copper 8222 those of siluer 7161 those of lead 6435 those full of quicksiluer 5453 and those of gold 3895. 2 In like bodies of seuerall mettalls and equall magnitude hauing the weight of one to finde the weight of the rest This proposition is the conuerse of the former the proportion not direct but reciprocall wherefore hauing two like bodies take the giuen weight of the one out of the lines of Solids and to it open the Sector in the points belonging to the mettall of the other body so the parallell taken from the points belonging to the body giuen and measured in the lines of Solids shall giue the weight of the body required As if a cube of gold weighed 38 l. and it were required to know the weight of a cube of lead hauing equal magnitude First I take 38 l. for the weight of the golden cube out of the lines of Solids put it ouer in the points of ♄ belonging to lead so the parallell taken from betweene the points of ☉ standing for gold and measured in the lines of Solids doth giue the weight of the leaden cube required to be 23 l. Thus if a sphere of gold shall weigh 10000 we shall finde that a sphere of the same diameter full of quicksiluer shall weigh 7143 a sphere of lead 6053 a sphere of siluer 5438 a sphere of copper 4737 a sphere of iron 4210 and a sphere of tin 3895. 3 A bodie being giuen of one mettall to make another like vnto it of another mettall and equall weight Take out one of the sides of the bodie giuen and put it ouer in the points belonging to his mettall so the parallell taken from between the points belonging to the other mettall shall giue the like side for the bodie required If it be an irregular bodie let the other like sides be found out in the same manner Let the bodie giuen be a sphere of lead containing in magnitude 16 D whose diameter is A to which I am to make a sphere of iron of equall waight If I take out the diameter A and put it ouer in the points of ♄ belonging to lead the parallell taken from betweene the points of ♂ standing for iron shall be B the diameter of the iron sphere required And this compared with the other diameter in the lines of solids will be found to be 23 d. in magnitude 4 A body being giuen of one mettall to make another like vnto it of another mettall according to a weight giuen First find the sides of a like bodie of equall weight then may we either augment or diminish them according to the proportion giuen by that which we shewed before in the second and third Prop. of Solids As if the bodie giuen were a sphere of lead whose diameter is A and it were required to find the diameter of a sphere of iron which shall weigh three times as much as the sphere of lead I take A and put it ouer in the points of ♄ his parallell taken from betweene the points of ♂ shall giue me B for the diameter of an equall sphere of iron if this be augmented in such proportion as 1 vnto 3 it giueth C for the diameter required CHAP. VI. Of the lines on the edges of the Sector HAuing shewed some vse of the lines on the flat sides of the Sector there remaine onely those on the edges And here one halfe of the outward edge is diuided into inches and numbred according to their distance from the ends of the Sector As in the Sector of fourteene inches long where we find 1 and 13 it sheweth that diuision to be 1 inch from the nearer end and 13 inches from the farther end of the Sector The other halfe containeth a line of lesser tangents to which the gnomon is Radius They are here continued to 75 gr And if there be need to produce them farther take 45 out of the number of degrees required and double the remainder so the tangent and secant of this double remainder being added shall make vp the tangent of the degrees required As if AB being the Radius and BC the tangent line it were required to find the tangent of 75 gr If we take 45 gr out of 75 gr the remainer is 30 gr and the double 60 gr whose tangent is BD and the secant is AD if then we adde AD to BD it maketh BC the tangent of 75 gr which was required In like sort the secant of 61 gr added to the tangent of 61 gr giueth the tangent of 75 gr 30 m. and the secant of 62 gr added to the tangent of 62 gr giueth the tangent of 76 gr and so in the rest The vse of this line may be To obserue the altitude of the Sunne Hold the Sector so as the tangent BC may be verticall and the gnomon BA parallell to the horizon then turne the gnomon toward the Sunne so that it may cast a shadow vpon the tangent and the end of the shadow shal shew the altitude of the Sunne So if the end of the gnomon at A do giue a shadow vnto H it sheweth that the altitude is 38 gr ½ if vnto D then 60 gr and so in the rest There is another vse of this tangent line for the drawing of the houre lines vpon any ordinary plane whereof I will set downe these propositions 1 To draw the houre lines vpon an horizontall plane 2 To draw the houre lines vpon a direct verticall plane First draw a right line AC for the horizon and the equator and crosse it at the point A about the middle of the line with AB another right line which may serue for the meridian and the houre of 12 then take out 15 gr out of the tangents and pricke them downe in the equator on both sides from 12 so the one point shall serue for the houre of 11 and the other for the houre of 1. Againe take out the tangent of 30 gr and pricke it downe in the equator on both sides from 12 so the one of these points shall serue for the houre of 10 and the other for the houre of 2. In like maner may you prick downe the tangent of 45 gr for the houres of 9 and 3 and the tangent of 60 gr for the houres of 8 and 4 and the tangent of 75 gr for the houres of 7 and 5.
and Azimuth CHAP. I. Of the description of the Quadrant HAuing described these standing planes I will now shew the most of these conclusions by a small Quadrant This might be done generally for all latitudes by a quarter of the generall Astrolabe described before in the vse of the Sector pag. 58 and particularly for any one latitude by a quarter of the particular Astrolabe there also described pag. 63. which if it be a foote semidiameter may shew the azimuth vnto a degree and the time of the day vnto a minute but for ordinary vse this smaller Quadrant may suffice which may be made portable in this maner 1 Vpon the center A and semidiameter AB describe the arke BC the same semidiameter will set of 60 gr and the halfe of that will be 30 gr which being added to the former 60 gr will make the arke BC to be 90 gr the fourth part of the whole circle and thence comes the name of a Quadrant 2 Leauing some little space for the inscription of the moneths and dayes on the same center A and semidiameter AT describe the arke TD which shall serue for either tropique 3 Diuide the line AT in the point E in such proportion as that AT being 10000 AE may be 6556 and there draw another arke EF which shall serue for the Equator 5 This part of the ecliptique may be diuided into three Signes and each Signe into 30 A Table of right Ascensions Gr. ♈ ♉ ♊ Gr. M. Gr. M. Gr. M. 0 0 0 27 54 57 48 5 4 35 32 42 63 3 10 9 11 37 35 68 21 15 13 48 42 31 73 43 20 18 27 47 33 79 7 25 23 9 52 38 84 32 30 27 54 57 48 90 0 gr by a table of right ascensions made as before pag. 60. As the right ascension of the first point of ♉ being 27 gr 54 m. you may lay a ruler to the center A 27 gr 54 m. in the Quadrant BC the point where the ruler crosseth the Ecliptique shall be the first point of ♉ In like maner the right ascension of the first point of ♊ being 57 gr 48 m. if you lay a ruler to the Gr. Parts 1 176 2 355 3 537 4 723 5 913 6 1106 7 1302 8 1503 9 1708 10 1917 11 2130 12 2348 13 2571 14 2799 15 3032 16 3270 17 3514 18 3763 19 4019 20 4281 21 4550 22 4825 23 5108 Tro 5252 center A and 57 gr 48 m. in the quadrant the point where the ruler crosseth the ecliptique shal be the first point of ♊ And so for the rest but the lines of distinction between Signe Signe may be best drawne from the center G. 6 The line ET betweene the equator and the tropique which I call the line of declination may be diuided into 23 gr ½ out of this Table For let AE the semidiameter of the equator be 10000 the distance betweene the equator and 10 gr of declination may be 1917 more between the equator and 20 gr 4281 the distance of the tropique from the equator 5252. 7 You may put in the most of the principall starres betweene the equator and the tropique of ♋ by their declination from the equator and right ascention from the next equinoctial point As the declination of the wing of Pegasus being 13 gr 7 m. the right ascension 358 gr 34 m. from the first point of ♈ or 1 gr 26 m. short of it If you draw an occult parallell through 13 gr 7 m. of declination and then lay the ruler to the center A and 1 gr 26 m. in the quadrant BC the point where the ruler crosseth the parallell shall be the place for the wing of Pegasus to which you may set the name and the time when he cometh to the South in this maner W. Peg. * 23 Ho. 54 M. and so for the rest of these fiue or any other starres  Ho. M. R. Ascen Decl. M Pegasus wing * 23 54 1 26 13 7 Arcturus * 13 58 29 37 21 10 Lions heart * 9 48 32 58 13 45 Buls eye * 4 15 63 33 15 42 Vultures heart * 19 33 66 56 7 58 8 There being space sufficient between the equator and the center you may there describe the quadrat and diuide each of the two sides farthest frō the center A into 100 parts so shall the Quadrant be prepared generally for any latitude But before you draw the particular lines you are to fit foure tables vnto your latitude First a table of meridian altitudes for diuision of the circle of dayes and moneths which may be thus made Consider the latitude of the place and the declination of the Sun for each day of the yeare If the latitude and declination be alike both North or both South ad the declinatiō to the complement of the latitude if they be vnlike one North and the other South substract the declination from the complement of the latitude the remainder will be the meridian altitude belonging vnto the day Thus in our latitude of 51 gr 30 m. Northward whose complement is 38 gr 30 m. the declination vpon the tenth day of Iune will be 23 gr 30 m. Northward wherefore I adde 23 gr 30 m. vnto 38 gr 30 m. the summe of both is 62 gr for the meridian altitude at the tenth of Iune The declination vpon the tenth of December will be 23 gr 30 m. Southward wherefore I take these 23 gr 30 m. out of 38 gr 30 m. there wil remain 15 gr for the meridian altitude at the tenth of December and in this maner you may find the meridian altitude for each day of the yeare and set them downe in a table Dies 0 5 10 15 20 25 30 Mo Gr. M Gr. M. Gr. M Gr. M. Gr. M. Gr. M Gr. M. Ianuary 16 31 17 24 18 26 19 37 20 57 22 24 23 58 February 24 17 25 59 27 45 29 35 31 29 33 25  March 34 35 36 33 38 32 40 30 42 27 44 22 46 15 April 46 37 48 26 50 11 51 50 53 25 54 53 56 15 May 56 15 57 29 58 35 59 33 60 22 61 2 61 31 Iune 61 36 61 54 62 0 61 58 61 45 61 22 60 49 Iuly 60 49 60 6 59 14 58 13 57 4 55 48 54 24 August 54 7 52 36 50 59 49 17 47 31 45 41 43 49 September 43 26 41 30 39 33 37 36 35 38 33 41 31 46 October 31 46 29 53 28 3 26 16 24 35 22 59 21 29 Nouember 21 12 19 51 18 39 17 36 16 43 16 0 15 28 December 15 28 15 7 15 0 15 2 15 17 15 44 16 22 The Table being made you may inscribe the moneths and dayes of each moneth into your quadrant in the space left below the tropique For lay the ruler vnto the center A and 16 gr 31 m. in the quadrant BC there may you draw a line for the end of December and beginning of
whether the two stations be chosen at the one end of the bredth proposed or without it or within it if the line betweene the stations be perpendicular vnto the bredth as may appeare if in stead of the stations at A and H we make choise of the like stations at I and K. There might be other wayes proposed to work these Prop. by holding the Crosse euen with the distance and the Staffe parallell with the height but these would proue more troublesome and those which are deliuered are sufficient and the same with those which others haue set downe vnder the name of the Iacobs staffe CHAP. III. The vse of the Tangent lines in taking of Angles 1 To find an angle by the Tangent on the Staffe LEt the middle sight be alwayes set to the middle of the Crosse noted with 20 and 30 and then the Crosse drawne nearer the eye vntill the marks may be seene close within the sights For so if the eye at A that end of the Staffe which is noted with 90 and 180 beholding the marks K and N betweene the two first sights C and B or the marks K and P betweene the two outward sights the Crosse being drawne downe vnto H shall stand at 30 and 60 in the Tangent on the Staffe it sheweth that the angle KAN is 30 gr the angle KAP 60 gr the onâ— double to the other which is â—he râ—ason of the double numbers on this line of the Staffe and this way wil serue for any angle from 20 gr toward 90 gr or from 40 gr toward 180 gr But if the angle be lesse then 20 gr we must then make vse of the Tangent vpoÌ„ the Crosse 2 To find an angle by the Tangent of 20 vpon the Crosse Set 20 vnto 20 that is the middle sight to the middest of the Crosse at the end of the Staffe noted with 20 so the eye at A beholding the marks L and N close betweene the two first sights C and B shall seâ— them in an angle of 20 gr If the marks shâ—ll be nearer together as are M and N then draw in the Crosse from C vnto E if they be farther asunder as are K and N then draw out the Crosse from C vnto F so the quantitie of the angle shal be still found in the Crosse in the Tangent of 20 gr at the end of the Staffe and this will serue for any angle from 0 gr toward 35 gr 3 To find an angle by the Tangent of 30 vpon the Crosse This Tangent of 30 is here put the rather that the end of the Staffe resting at the eye the hand may more easily remoue th Crosse for it supposeth the Radius to be no longer then AH which is from the eye at the end of the Staffe vnto 30 gr about 22 inches and 7 parts Wherefore here set the middle sight vnto 30 gr on the Staffe and then either draw the Crosse in or out vntill the marks be seene between the two first sights so the quantitie of the angle will be found in the Tangent of 30 which is here represented by the line GH and this will serue for any angle from 0 gr toward 48 gr 4 To obserue the altitude of the Sunne backward Here it is fit to haue an horizontall sight set to the beginning of the Staffe and then may you turne your backe toward the Sun and your Crosse toward your eye If the altitude be vnder 45 gr set the middle sight to 30 on the Staffe and looke by the middle sight through the horizontall vnto the horizon mouing the Crosse vpward or downward vntill the vpper sight doe shadow the vpper halfe of the horizontall sight so the altitude will be found in the Tangent of 30. If the altitude shal be more then 45 gr set the middle sight vnto the middest of the Crosse and look by the inward edge of the lower sight through the horizontall to the horizon mouing the middle sight in or out vntill the vpper sight do shadow the vpper halfe of the horizontall sight so the altitude will be found in the degrees on the Staffe betweene 40 and 180. 5 To set the Staffe to any angle giuen This is the conuerse of the former Prop. For if the middle sight be set to his place and degree the eye looking close by the sights as before cannot but see his obiect in the angle giuen 6 To obserue the altitude of the Sunne another way Set the middle sight to the middle of the Crosse and hold the horizontall sight downward so as the Crosse may be parallell to the horizon then is the Staffe verticall and if the outward sight of the Crosse do shadow the horizontall sight the complement of the altitude wil be found in the tangent on the Staffe 7 To obserue an altitude by thread and plummet Let the middle sight be set to the middest of the Crosse and to that end of the Staffe which is noted with 90 and 180 then hauing a thread and a plummet at the beginning of the Crosse and turning the Crosse vpward and the Staffe toward the Sunne the thread will fall on the complement of the altitude aboue the horizon And this may be applied to other purposes 8 To apply the lines of inches to the taking of angles If the angles be obserued betweene the two first sights there wil be such proportion between the parts of the Staffe and the parts of the Crosse as betweene the Radius and the Tangent of the angle As if the parts intercepted on the Staffe were 20 inches the parts on the Crosse 9 inches Then by proportion as 20 vnto 9 so 100000 vnto 45000 the tangent of 24 gr 14 m. But if the angle shall be obserued betweene the two outward sights the parts being 20 and 9 as before the angle will be 48 gr 28 m. double vnto the former In all these there is a regard to be had to the parallax of the eye and his height aboue the Horizon in obseruations at Sea to the Semidiameter of the Sun his parallax and refraction as in the vse of other staues And so this will be as much or more then that which hath been heretofore performed by the Crosse-staffe CHAP. IIII. The vse of the lines of equall parts ioyned with the lines of Chords THe lines of equall parts do serue also for protraction as may appeare by the former Diagrams but being ioyned with the lines of Chords which I place vpon one side of the Crosse they will farther serue for the protraction and resolution of right line triangles whereof I will giue one example in finding of a distance at two stations otherwise then in the second Cap. Let the distance required be AB At A the first statioÌ„ I make choise of a station line toward C and obserue the angle BAC by the tangent lines which may be 43 gr 20 m then hauing gon an hundred paces toward C I make my second station at D
Ianuary then laying your ruler to the center A and 24 gr 17 m. in the quadrant there draw the end of Ianuary and beginning of February and so the rest which may be noted with I F M A M I c. the first letters of each moneth and wil here fall betweene 15 gr and 62 gr The second Table which you are to fit may serue for the drawing and diuiding of the horizon For drawing of the horizon As the cotangent of the latitude to the tangent of the greatest declination So the sine of 90 gr to the sine of intersection where the horizon shall crosse the tropiques So in our latitude of 51 gr 30 m. we shall find the horizon to cut the tropique in 33 gr 9 m wherefore if you lay the ruler to the center A and 33 gr 9 m. in the quadrant the point where the ruler crosseth the tropique shall be the point where the horizon crosseth the tropique And if you finde a point at H in the line AC whereon setting the compasses you may bring the point at E and this point in the tropique both into a circle the point H shall be the center and the ark so drawne shall be the horizon Then for the diuision of this horizon As the sine of 90 gr to the sine of the latitude So the tangent of the horizon to the tangent of the arke in the quadrant which shall diuide the horizon So in our latitude of 51 gr 30 m. we shall finde 7 gr 52 m. belonging to 10 gr in the horizon and 15 gr 54 m. belonging to 20 gr And so the rest as in this Table Ho Gr. M Ho Gr. M Ho Gr. M. Ho Gr. M. Ho Gr. M. Ho Gr. M. 0 0 0 15 11 51 30 24 19 45 38 2 60 53 35 75 71 5  0 47  12 39  25 11  39 1  54 41  72 19  1 34  13 27  26 4  40 0  55 48  73 33  2 21  14 16  26 57  41 0  56 56  74 48  3 8  15 4  27 50  42 0  58 4  76 3 5 3 55 20 15 54 35 28 43 50 43 0 65 59 13 80 77 18  4 42  16 43  29 37  44 1  60 22  78 33  5 29  17 33  30 32  45 3  61 31  79 49  6 17  18 22  31 27  46 5  62 41  81 5  7 4  19 12  32 22  47 8  62 52  82 21 10 7 52 25 20 2 40 33 18 55 48 11 70 65 3 85 83 37  8 39  20 53  34 14  49 14  66 15  84 53  9 27  21 44  35 10  50 19  67 27  86 10  10 14  22 36  36 7  51 24  68 39  87 26  11 2  23 27  37 4  52 29  69 52  88 43 15 11 51 30 24 19 45 38 2 60 53 35 75 71 5 90 90 0 Wherefore you may lay the ruler to the center A and 7 gr 52 m. in the quadrant BC the point where the ruler crosseth the horizon shall be 10 gr in the horizon and so for the rest but the lines of distinction between each fift degree will be best drawne from the center H. The third Table for drawing of the houre-lines must be a Table of the altitude of the Sunne aboue the horizon at euery houre especially when he cometh to the equator the tropiques and some other intermediate declinations If the Sunne be in the equator and so haue no declination As the sine of 90 gr to the cosine of the latitude So the cosine of the houre from the meridian to the sine of the altitude Thus in our latitude of 51 gr 30 m. at six houres from the meridian the Sunne wil haue no altitude at fiue the altitude will be 9 gr 17 m at foure 18 gr 8 m at three 26 gr 7 m at two 32 gr 37 m. at one 36 gr 58 m at noone it will be 38 gr 30 m. equall to the complement of the latitude If the Sunne haue declination the meridian altitude wil be found as before for the Table of dayes and moneths If the houre proposed be six in the morning or six at night As the sine of 90 gr to the sine of the latitude So the sine of the declination to the sine of the altitude Thus in our latitude the declination of the Sunne being 23 gr 30 m. the altitude will be found to be 18 gr 11 m the declination being 11 gr 30 m. the altitude will be 9 gr If the houre proposed be neither twelue nor six As the cosine of the houre from the meridian to the sine of 90 gr So the tangent of the latitude to the tangent of a fourth arke So in our latitude and one houre from the meridian this fourth ark will be found to be 52 gr 28 m. at two 55 gr 26 m. at three 60 gr 39 m. at foure 68 gr 22 m. and at fiue houres from the meridian 78 gr 22 m. Then consider the declination of the Sunne and the houre proposed if the latitude and declination be both alike as with vs in North latitude North declination and the houre fall between noone and six take the declination out of the fourth arke the remainer shall be your fift arke But if either the houre fall between six and midnight or the latitude and declination shall be vnlike adde the declination vnto the fourth arke and the summe of both shall be your fifth arke or if the summe shall exceed 90 gr you may take the complement vnto 180 gr This fifth arke being knowne As the sine of the fourth arke to the sine of the latitude So the cosine of the fift arke to the sine of the altitude Thus in our latitude of 51 gr 30 m. Northward the Sunne hauing 23 gr 30 m. of North declination if it shall be required to finde the altitude of the Sunne for seuen in the morning here because the latitude and declination are both alike to the Northward and the houre proposed falleth betweene noone and six you may take 23 gr 30 m. the arke of the declination out of 78 gr 22 m. the fourth arke belonging to the fift houre from the meridian so there will remaine 54 gr 52 m. for your fift arke Then working according to the Canon you shall find As the sine of 78 gr 22 m. your fourth arke to the sine of 51 gr 30 m. for the latitude So the sine of 35 gr 8 m. the cōplement of your fift ark to the sine of 27 gr 17 m. the altitude required If in the same latitude and declination it were required to finde the altitude for fiue in the morning
houre past 2 in the afternoone and which of these is the true time of the day may be soone knowne by a second obseruation for if the Sunne rise higher it is the forenoone if it become lower it is the afternoone 5 The houre of the night being giuen to find how much the Sunne is below the horizon The Sunne is alwayes so much below the horizon at any houre of the night as his opposite point is aboue the horizon at the like houre of the day and therefore the beade being set if the question be made of any houre of the night in the Summer then moue it to the like houre of the day in the Winter if of any houre of the night in Winter then moue it to the like houre of the day in Summer so the degrees which the thread cutteth in the Quadrant shall shew how much the Sunne is below the horizon at that time As if it be required to know how much the Sunne is below the horizon the 10 of April at 4 of the clock in the morning the bead being set to his place according to the time in the Summer houres bring it to 4 of the clocke in the afternoone in the Winter houres and so shall you finde the thread to cut 8 gr and about 30 m. in the quadrant and so much is the Sunne below the horizon at that time 6 The depression of the Sunne supposed to giue the houre of the night with vs or the houre of the day to our Antipodes Here also because the Sun is so much aboue the horizon at all houres of the day as his opposite point is below the horizon at the like houre of the night therefore first set the bead according to the time then bring the thread to the degree of the Suns depression below the horizon so shall the bead fall on the contrary houre-lines and there shew the houre of the night in regard of vs which is the like houre of the day to our Antipodes As if the 10 of April the Sunne being then in the beginning of ♉ and by supposition 8 gr 30 m. below the horizon in the East it be required to know what time of the night it is first set the bead according to the day in the Summer houres then bring the thread to 8 gr 30 m. in the quadrant so shall the bead fall among the Winter houres on the line of 4 of the clocke in the afternoone wherefore to our Antipodes it is 4 of the clocke in their afternoone and to vs it is then 4 of the clocke in the morning 7 The time of the yeare or the place of the Sunne being giuen to find the beginning of day-breake and end of twi-light This proposition differeth little from the former for the day is said to begin to breake when the Sun cometh to be but 18 gr below our horizon in the East and twi-light to end when it is gotten 18 gr below the horizon in the West wherefore let the bead be set for the time and then bring the thread to 18 gr in the quadrant so shall the bead fall on the contrary houre-lines and there shew the houre of twilight as before So if it be required to know at what time the day begins to breake on the tenth of April the Sun being then in the beginning of ♉ first set the bead according to the time in the Summer houres and then bring the thread to 18 gr in the quadrant so shall the bead fall among the Winter houres a little more then a quarter before 3 in the morning and that is the time when the day begins to break vpon the tenth of April CHAP. VII Of the Horizon THe Horizon is here represented by the arke drawne from the beginning of declination towards the end of February diuided vnequally and numbred by 10. 20. 30. 40. 1 The day of the moneth or the place of the Sunne being knowne to finde the amplitude of the Sunnes rising and setting Let the bead rectified for the time be brought to the horizon and there it shall shew the amplitude required As if the day giuen be the 15 of May the Sunne being in the fourth degree of ♊ the bead rectified and brought to the horizon shall there fall on 35 gr 8 m. such is the amplitude of the Sunnes rising from the East and of his setting from the West which amplitude is alwayes Northward when the Sunne is in the Northerne Signes and when he is in the Southward Signes alwayes Southward 2 The day of the moneth or the place of the Sunne being giuen to finde the ascensionall difference Let the bead rectified for the time be brought to the horizon so the degrees cut by the thread in the quadrant shall shew the difference of ascensions As if the day giuen be the 15 of May the Sunne being in the fourth degree of ♊ let the bead be rectified and brought to the horizon so shall the thread in the quadrant shew the ascensionall difference to be 28 gr and about 50 m. Vpon the ascensionall difference depends this Corollarie To find the houre of the rising and setting of the Sun and thereby the length of the day and night The time of the Sunnes rising may be guessed at by the 3 of the last Cap. but here by the ascensionall difference it may be better found and that to a minute of time For if the ascensionall difference be conuerted into time allowing an houre for 15 gr and 4 minutes of an houre for each degree it sheweth how long the Sunne riseth before six of the clock in the Summer and after six in the Winter As if the day giuen be the 15 of May the Sunne being in the fourth of ♊ and his ascensionall difference found as before 28 gr 50 m this conuerted into time maketh 1 ho. and somewhat more then 55 m. of an houre wherefore the Sun at that time in regard it was Summer rose 1 ho. and full 55 m. before 6 of the clocke and so hauing the quantitie of the semidiurnall arke the length of the day and night need not be vnknowne CHAP. VIII Of the fiue Starres I Might haue put in more Starres but these may suffice for the finding of the houre of the night at all times of the yeare and first I make choice of Ala Pegasi a starre in the extremitie of the wing of Pegasus in regard it wants but 6 minutes of time of the beginning of ♈ but because it is but of the second magnitude and not alwayes to be seene I made choice of foure more one for each quarter of the Ecliptique as of Oculus ♉ the Buls eye whose right ascension conuerted into time is 4 ho. 15 m then of Cor ♌ the Lions heart whose right ascension is 9 ho. 48 m next of Arcturus whose right ascension is 13 H. 58 m and lastly of Aquila or the Vultures heart whose right ascension is 19 H. 33 m. These fiue starres haue all of