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
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
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
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.
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
diuide the lines of Solids SEing like Solids do hold in the proportion of their homologall sides triplicated if you shall finde two meane proportionalls between the whole side each thousand part of the like side all of them cutting the same two right lines the former of those lines so cut shall containe the diuisions required Wherefore vpon the center A Semidiameter equall to the line of Lines describe a circle and diuide it into 4 equall parts CEBD drawing the crosse diameters CB ED. Then diuide the semidiameter AC first into 10 equall parts and betweene the whole line AD AF the tenth part of AC seeke out two meane proportionall lines AI and AH againe betweene AD and AG being two tenth parts of AC seeke out two meane proportionals AL and AK and so forward in the rest So shall the line AB be diuided into 10 vnequall parts Secondly diuide each tenth part of the line AC into 10 more and betweene the whole line AD and each of them seeke out two meane proportionalls as before So shall the line AB be diuided now into an hundred vnequall parts Thirdly If the length will beare it subdiuide the line AC once againe each part into ten more and betweene the whole line AD and each subdiuision seeke two meane proportionalls as before So should the line AB be now diuided into 1000 parts But the ruler being short it shall suffice if those 10 which are nearest the center be expressed the rest be vnderstood to be so diuided though actually they be diuided into no more then 5 or 2. and this line AB so diuided shall be the line of Solids and must be transferred into the Sector But let the numbers set to them be onely 1.1.1.2.3 c. vnto 10. as in the example 5 To diuide the lines of Sines and Tangents on the side of the Sector VPon the center A and semidiameter equall to the line of Lines describe a semicircle ABCD with AB perpendicular to the diameter CD Then diuide the quadrants CB BD each of them into 90. and subdiuide each degree into 2 parts For so if streight lines be drawne parallell to the diameter CD through these 90 and their subdiuisions they shall diuide the perpendicular AB vnequally into 90. And this line AB so diuided shall be the line of Sines and must be transferred into the Sector The numbers set to them are to be 10.20.30 c. vnto 90 as in the example If now in the poynt D vnto the diameter CD we shall raise a perpendicular DE and to it drawe streight lines from the center A through each degree of the quadrant DB. This perpendicular so diuided by them shall be the line of Tangents must be transferred vnto the side of the Sector The numbers set to them are to be 10.20.30 c. as in the example If betweene A and D another streight line GF be drawne parallell to DE it will be diuided by those lines from the center in like sort as DE is diuided and it may serue for a lesser line of Tangents to be set on the edge of the Sector These lines of Sines and Tangents may yet otherwise be transferred into the Sector out of the line of Lines or rather out of a diagonall Scale equall to the line of Lines by tables of Sines and Tangents In like manner may the lines of Superficies be transferred by tables of square rootes and the line of Solids by tables of cubique rootes which I leaue to others to extract at leasure 6 To shew the ground of the Sector LEt AB AC represent the leggs of the Sector then seuering these two AB AC are equall and their sections AD AE also equall they shall be cut proportionally and if we draw the lines BC DE they will be parallell by the second Pro. 6 lib. of Euclid and so the Triangles ABC ADE shal be equiangle by reason of the common angle at A and the equall angles at the base and therefore shall haue the sides proportionall about those equall angles by the 4 Pro. 6 lib. of Euclid The side AD shal be to the side AB as the basis DE vnto the parallell basis BC and by conuersion AB shall be vnto AD as BC vnto DE and by permutation AD shall be vnto DE as AB to BC. c. So that if AD be the fourth part of the side AB then DE shall also be the fourth part of his parallell basis BC The like reason holdeth in all other sections 7 To shew the generall vse of the Sector THere may some cōclusions be wrought by the Sector euen then when it is shut by reason that the lines are all of one length but generally the vse hereof consists in the solution of the Golden rule where three lines being giuen of a known denominaton a fourth proportionall is to be found And this solution is diuerse in regard both of the lines and of the entrance into the worke The solution in regard of the lines is sometimes simple as when the worke is begun and ended vpon the same lines Sometimes it is compound as when it is begun on one kind of lines and ended on another It may be begun vpon the lines of Lines finished vpon the lines of Superficies It may begin on the Sines and end on the Tangents The solution in regard of the entrance into the worke may be either with a parallell or else laterall on the side of the Sector I cal it parallell entrance or entring with a parallell when the two lines of the first denomination are applied in the parallells and the third line and that which is sought for are on the side of the Sector I call it laterall entrance or entring on the side of the Sector when the two lines of the first denomination are one the side of the Sector and the third line and that which is to be found out doe stand in the parallells As for example let there be giuen three lines A B C to which I am to find a fourth proportionall let A measured in the line of lines be 40 B 50 and C 60 and suppose the question be this If 40 Monthes giue 50 pounds what shal 60 Here are lines of two denominatiōs one of months another of pounds and the first with which I am to enter must be that of 40 monthes If then I would enter with a parallell first I take A the line of 40 and put it ouer as a parallell in 50 reckoned in the line of lines on either side of the Sector from the center so as it may be the Base of an Isoscheles triangle BAC whose sides AB AC are equal to B the line of the second denomination Then the Sector being thus opened I take C the line of 60 betweene the feete of the compasses and carrying them parallell to BC I finde them to crosse the lines AB AC on the side of the
Sector in D and E numbred with 75 wherefore I conclude the line AD or AE is the fourth proportionall and the correspondent number 75 which was required But if I would enter on the side of the Sector then would I dispose the lines of the first denomination A and C in the line of Lines on both sides of the Sector in AB AC in AD AE so as they should all meete in the center A and then taking B the line of the second denomination put it ouer as a parallell in BC that it may be the Basis of the Isocheles triangle BAC whose sides AB AC are equall to A the first line of the first denomination for so the Sector being thus opened the other parallell from D to E shall be the fourth proportionall which was required and if it be measured with the other lines it shal be 75 as before In both this manner of operations the two first lines do serue to opē the Sector to his due angle the difference betweene them is especially this that in parallell entrance the two lines of the first denomination are placed in the parallells B C D E in latterall entrance they are placed on both sides of the Sector in AB AD and in AC AE Now in simple solution which is begun and ended vpon the same kinde of lines it is all one which of the two latter lines be put in the secōd or third places As in our exāple we may say as 40 are to 50 so 60 vnto 75 or else as 40 are to 60 so 50 vnto 75. And hence it cōmeth that we may enter both with a parrallell on the sides two manner of wayes at either entrance and so the most part of questions may be wrought 4 seuerall wayes though in the propositions following I mention onely that which is most conuenient Thus much for the generall vse of the Sector which being considered and well vnderstood there is nothing hard in that which followeth CHAP. II. The vse of the Scale of Lines 1. To set downe a Line resembling any giuen parts or fraction of parts THe lines of Lines are diuided actually into 100 parts but we haue put onely 10 numbers to them These we would haue to signifie either themselues alone or ten times themselues or an hundred times themselues or a thousand times themselues as the matter shall require As if the numbers giuen be no more then 10 then we may thinke the lines onely diuided into 10 parts according to the numbers set to them If they be more then 10 and not more then 100 then either line shall containe 100 parts and the numbers set by them shall be in value 10.20.30 c. as they are diuided actually If yet they be more then 100 then euery part must be thought to be diuided into 10 and either line shall be 1000 parts and the numbers set to them shall be in value 100.200.300 and so forward still increasing themselues by 10. This being presupposed we may number the parts and fraction of parts giuen in the line of lives and taking out the distance with a paire of compasses set it by for the line so taken shall resemble the number giuen In this manner may we set downe a line resembling 75 if either we take 75 out of the hundred parts into which one of the line of lines is actually diuided and note it in A or 7 ½ of the first 10 parts and note it in B or onely ¾ of one of those hundred parts and note it in C. Or if this be either to great or to small we may run a Scale at pleasure by opening the compasse to some small distance and running it ten times ouer then opening the compasse to these ten run them ouer nine times more set figures to them as in this example and out of this we may take what parts we will as before To this end I haue diuided the line of inches on the edge of the Sector so as one inch containeth 8 parts another 9 another 10 c. according as they are figured and as they are distant from the other end of the Sector that so we might haue the better estimate 2 To encrease a line in a giuen proportion 3 To diminish a line in a giuen proportion TAke the line giuen with a paire of compasses and open the Sector so as the feete of the compasses may stand in the points of the number giuen then keeping the Sector at this angle the parallell distance of the points of the number required shall giue the line required Let A be a line giuen to be increased in the proportion of 3 to 5. First I take the line A with the compasses and open the Sector till I may put it ouer in the poynts of 3 and 3 so the parallell betweene the poynts of 5 5 doth giue me the line B which was required In like manner if B be a line giuen to be diminished in the proportiō of 5 to 3 I take the line B to it open the Sector in the poynts of 5 so the parallell betweene the points of 3 doth giue me the line A which was required If this manner of worke doth not suffice we may multiplie or diuide the numbers giuen by 1 or 2 or 3 c. And so worke by their numbers equimultiplices as for 3 and 5 wee may open the Sector in 6 and 10 or else in 9 and 15 or else in 12 and 20 or in 15 and 25 or in 18. and 30. c. 4 To diuide a line into parts giuen TAke the line giuen and open the Sector according to the length of the said line in the points of the parts wherevnto the line should be diuided then keeping the Sector at this angle the parallell distance betweene the points of 1 and 1 shall diuide the line giuen into the parts required Let AB be the line giuen to be diuided into fiue parts first I take this line AB and to it open the Sector in the points of 5 and 5 so the parallell betweene the points of 1 and 1 doth giue me the line AC which doth diuide it into the parts required Or let the like line AB be to be diuided into twenty three parts First I take out the line and put it vpon the Sector in the points of 23 then may I by the former proposition diminish it in AC CD in the proportion of 23 to 10 and after that diuide the line AC into 10 c. As before 5 To finde a proportion betweene two or more right lines giuen TAke the greater line giuen and according to it open the Sector in the points of 100 and 100 then take the lesser lines seuerally carry them parallell to the greater till they stay in like points so the number of points wherein they stay shall shew their proportion vnto 100. Let the lines giuen be AB CD first I take the line CD 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
at the other extreme of the said arke So if the giuen arke be CE or the giuen angle be CAE let the diameter be drawne through the center A to C and in C to AC be raised a perpendicular CI. Then let another line be drawne from the center A through E till it meet with the perpendicular CI in I the line CI is a Tangent and AI is the Secant both of the arke CE and of the angle CAE CHAP. II. Of the generall vse of Sines and Tangents 1 The Radius being knowne to find the right sine of any arke or angle IF the Radius of the circle giuen be equall to the laterall Radius that is to the whole line of Sines on the Sector there needs no farther worke but to take the other sines also out of the side of the Sector But if it be either greater or lesser then let it be made a parallell Radius by applying it ouer in the lines of Sines betweene 90 and 90 so the parallell taken from the like laterall sines shall be the sine required As if the giuen Radius be AC and it were required to find the sine of 50 Gr. his complement agreeable to that radius Let AB AB represent the lines of sines on the Sector and let BB the distance betweene 90 and 90 be equall to the giuen radius AC Here the lines A 40 A 50 A 90 may be called the laterall sines of 40 50 90 in regard of their place on the sides of the Sector The lines betweene 40 and 40 betweene 50 and 50 betweene 90 and 90 may be called the parallell sines of 40 50 and 90 in regard they are parallell one to the other The whole sine of 90 Gr. here standing for the semidiameter of the circle may be called the Radius And therefore if AC be put ouer in the line of Sines in 90 and 90 and so made a parallell radius his parallell sine betweene 50 and 50 shall be BD the sine of 50 required And because 50 taken out of 90 the complement is 40 his parallell sine betweene 40 and 40 shall be BG the sine of the complement which was required 2 The right sine of any arke being giuen to finde the Radius TVrne the sine giuen into a parallell sine and his parallell Radius shall be the Radius required As if BD were the giuen sine of 50 Gr. and it were required to finde the Radius let BD be made a parallell sine of 50 Gr. by applying it ouer in the lines of Sines betweene 50 and 50 so his parallell Radius betweene 90 and 90 shall be AC the Radius required 3 The Radius of a circle or the right Sine of any arke being giuen and a streight line resembling a Sine to find the quantitie of that vnknowne Sine LEt the Radius or right sine giuen be turned into his parallell then take the right line giuen and carrie it parallell to the former till it stay in like Sines so the number of degrees and minutes where it stayeth shall giue the quantitie of the Sine required As if BD were the giuen sine of 50 Gr. and BG the streight line giuen first I make BD a parallell sine of 50 Gr then keeping the Sector at this angle I carie the line BG parallell and find it to stay in no other but 40 and 40 and therefore 40 gr is his quantitie required 4 The Radius or any right Sine being giuen to finde the versed sine of any arke IF the arke whose versed sine is required be lesse then the quadrant take the sine of the complement out of the radius and the remainder shall be the sinus versus the versed sine of that arke As if AB being the laterall Radius it were required to find to find the versed line of 40 gr here the sine of the complement is A 50 and therefore B 50 is the versed sine required Or if I reckon from B at the end of the Sector toward the center the distance from 90 to 80 is the versed sine of 10 gr from 90 to 70 the versed sine of 20 gr from 90 to 60 is the versed sine of 30 gr and so in the rest If AD be the giuen sine of 50 gr and it be required to find the versed sine of 50 gr here because AD is vnequall to the laterall sine of 50 gr I make it a parallell And first I find the radius AC then the sine of the complement A 40 which being taken out of AC leaueth C 40 for the versed sine of 50 gr which was required But if the arke whose versed sine is required be greater then the quadrant his versed sine also is greater then the Radius by the right sine of his excesse aboue 90 gr As if AC being the Radius giuen it were required to find the versed sine of 130 gr here the excesse aboue 90 gr is 40 gr and therefore the versed sine required is equall to the Radius AC and A 40 both being set together 5 The Diameter or Radius being giuen to finde the Chords of euery arke The sines may be fitted many wayes to serue for chords 1 A sine being the halfe of the chord of the double arke if the sine be doubled it giueth the chord of the double ark a Sine of 10 gr doubled giueth a Chord of 20 gr and a Sine of 15 gr being doubled giueth a Chord of 30 gr and so in the rest As here BD the sine of BC an arke of 40 gr being doubled giueth BE the chord of BCE which is an arke of 80 gr Wherefore if the Radius of the circle giuen be equall to the laterall Radius let the Sector be opened neare vnto his length so that both the lines of Sines may make but one direct line so the distance on the sines betweene 10 and 10 shall be a chord of 20 the distance betweene 20 and 20 shall be a chord of 40 and the distance betweene 30 and 30 shall be a chord of 60 and so in the rest 2 Because a sine is the halfe of the chord of the double arke the proportion holdeth As the diameter FH vnto the radius AH so the chord BE vnto the sine DE or the chord GL vnto the sine AL and then if the radius AH be put for the diameter which is a chord of 180 gr the sine DE or AL shall serue for a chord of 80 gr and the semiradius which is the sine of 30 gr shall serue for a chord of 60 gr and go for the semidiameter of a circle and so in the rest So that by these meanes we shall not need to double the lines of Sines as before but onely to double the numbers And to this purpose I haue subdiuided each degree of the sines into two that so they might shew how far the halfe degrees do reach in the
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
9 or one of them in the point C then take out the semidiameter AV and prick it downe in those parallells from C vnto D and draw right lines from A vnto C and from V vnto D the line VD shall be the houre of 6 and if you diuide these lines AC and DC in such sort as you diuided the like line DC in the horizontall plane you shall haue all the houre points required Or you may find the point D in the houre of 6 without knowledge either of H or C. For hauing prickt downe AV in the meridian line and AE in the horizontall line and drawne parallels to the meridian through the points at E you may take the tangent of the latitude out of the Sector and fit it ouer in the sines of 90 and 90 so the parallell sine of the declination measured in the same tangent line shall there shew the complement of the angle DVA which the houre line of 6 maketh with the meridian then hauing the point D take out the semidiameter VA and pricke it downe in those parallels from D vnto C so shall you haue the lines DC and AC to be diuided as before The like might be vsed for the houre lines vpon all other planes But I must not write all that may be done by the Sector It may suffice that I haue wrote something of the vse of each line and thereby giuen the ingenuous Reader occasion to thinke of more The conclusion to the Reader IT is well knowne to many of you that this Sector was thus contriued the most part of this booke written in latin many copies transcribed and dispersed more then sixteene yeares since I am at the last contented to giue way that it come forth in English Not that I thinke it worthy either of my labour or the publique view but partly to satisfy their importunity who not vnderstanding the Latin yet were at the charge to buy the instrument and partly for my owne ease For as it is painefull for others to transcribe my copie so it is troublesome for me to giue satisfaction herein to all that desire it If I finde this to giue you content it shall incourage me to do the like for my Crosse-staffe and some other Instruments In the meane time beare with the Printers faults and so I rest Gresham Coll. 1. Maij. 1623. E. G. FINIS THE FIRST BOOKE OF THE CROSSE-STAFFE CHAP. I. Of the description of the Staffe THe Crosse-Staffe is an instrument wel knowne to our Sea-men and much vsed by the ancient Astronomers and others seruing Astronomically for obseruation of altitude and angles of distance in the heauens Geometrically for perpendicular heights and distances on land and sea The description and seuerall vses of it are extant in print by Gemma Frisius in Latin in English by Dr. Hood I differ something from them both in the proiection of this Staffe but so as their rules may be applied vnto it and all their propositions be wrought by it and therefore referring the Reader to their bookes I shal be briefe in the explanation of that which may be applied from theirs vnto mine and so come to the vse of those lines which are of my addition not extant heretofore The necessary parts of this Instrument are fiue the Staffe the Crosse and the three sights The Staffe which I made for my owne vse is a full yard in length that so it may serue for measure The Crosse belonging to it is 26 inches â…• betweene the two outward sights If any would haue it in a greater forme the proportion betweene the Staffe and the Crosse may be such as 360 vnto 262. The lines inscribed on the Staffe are of foure sorts One of them serues for measure and protraction one for obseruation of angles one for the Sea-chart and the foure other for working of proportions in seuerall kinds The line of measure is an inch line and may be knowne by his equall parts The whole yard being diuided equally into 36 inches and each inch subdiuided first into ten parts and then each tenth part into halfes The line for obseruation of angles may be knowne by the double numbers set on both sides of the line beginning at the one side at 20 and ending at 90 on the other side at 40 and ending at 180 and this being diuided according to the degrees of a quadrant I call it the tangent line on the Staffe The next line is the meridian of a Sea-chart according to Mercators proiection from the Equinoctiall to 58 gr of latitude and may be knowne by the letter M and the numbers 1. 2. 3. 4. vnto 58. The lines for working of proportions may be knowne by their vnequall diuisions and the numbers at the end of each line 1 The line of numbers noted with the letter N diuided vnequally into 1000 parts and numbred with 1. 2. 3. 4. vnto 10. 2 The line of artificiall tangents is noted with the letter T diuided vnequally into 45 degrees and numbred both ways for the Tangent and the complement 3 The line of artificiall sines noted with the letter S diuided vnequally into 90 degrees and numbred with 1. 2. 3. 4. vnto 90. 4 The line of versed sines for more easie finding the houre and azimuth noted with V diuided vnequally into about 164 gr 50 m. numbred backward with 10. 20. 30. vnto 164. Thus there are seuen lines inscribed on the Staffe there are fiue lines more inscribed on the Crosse 1 A Tangent line of 36 gr 3 m. numbred by 5. 10. 15. â—ledgeâ—nto 35 the midst whereof is at 20 gr and therefore I call it â—ledgeâ—he tangent of 20 and this hath respect vnto 20 gr in the Tangent on the Staffe 2 A Tangent line of 49 gr 6 m. numbred by 5. 10. 15. vnâ—ledgeâ—o 45 the midst whereof is at 30 gr and hath respect vnto â—ledgeâ—0 gr in the Tangent on the Staffe whereupon I call it the â—ledgeâ—angent of 30. 3 A line of inches numbred with 1. 2. 3. vnto 26 each inch â—ledgeâ—qually subdiuided into ten parts answerable to the inch line â—ledgeâ—pon the Staffe 4 A line of seuerall chords one answerable to a circle of â—ledgeâ—welue inches semidiameter numbred with 10. 20. 30. vnto â—ledgeâ—0 another to a semidiameter of a circle of six inches and â—ledgeâ—he third to a semidiameter of a circle of three inches both â—ledgeâ—umbred with 10. 20. 30. vnto 90. 5 A continuation of the meridian line from 57 gr of laâ—ledgeâ—tude vnto 76 gr and from 76 gr to 84 gr For the inscription of these lines The first for measure is â—ledgeâ—qually diuided into inches and tenth parts of inches The tangent on the Staffe for obseruation of angles with â—ledgeâ—e tangent of 20 and the tangent of 30 on the Crosse may â—ledgeâ—l three be inscribed out of the ordinary table of tangents The â—ledgeâ—affe being 36 inches in length the Radius for the tangent â—ledgeâ— the Staffe will be 13
point I would fall in the latitude of 51 gr 0 m. and the point K in the latitude of 51 gr 30 m. But the longitude of I would be onely 1 gr 30 m. and the longitude of K onely 3 gr 57 m. which is 33 m Westward from the meridian of the place to which the ship was bound Such is the difference betweene both these charts CHAP. VI. The vse of the line of Numbers 1 Hauing two numbers giuen to find a third in continuall proportion a fourth a fift and so forward EXtend the compasses from the first number vnto the second then may you turne them from the second to the third and from the third to the fourth and so forward Let the two numbers giuen be 2 and 4. Extend the compasses from 2 to 4 then may you turne them from 4 to 8 and from 8 to 16 and from 16 to 32 and from 32 to 64 and from 64 to 128. Or if the one foote of the compasses being set to 64 the other fall out of the line you may set it to another 64 nearer the beginning of the line and there the other foot will reach to 128 and from 128 you may turne them to 256 and so forward Or if the two first numbers giuen were 10 and 9 extend the compasses from 10 at the end of the line backe vnto 9 then may you turne them from 9 vnto 8.1 and from 8.1 vnto 7.29 And so if the two first numbers giuen were 1 and 9 the third would be found to be 81 the fourth 729 with the same extent of the compasses In the same maner if the two first numbers were 10 and 12 you may finde the third proportionall to be 14.4 the fourth 17.28 And with the same extent of the compasses if the two first numbers were 1 and 12 the third would be found to be 144 and the fourth to be 1728. 2 Hauing two extreme numbers giuen to find a meane proportionall between them Diuide the space betweene the extreme numbers into two equall parts and the foote of the compasses will stay at the meane proportionall So the extreme numbers giuen being 8 and 32 the meane betweene them will be found to be 16 which may be proued by the former Prop. where it was shewed that as 8 to 16 so are 16 to 32. 3 To find the square roote of any number giuen The square roote is alwayes the meane proportionall betweene 1 and the number giuen and therefore to be found by diuiding the space betweene them into two equall parts So the roote of 9 is 3 and the roote of 81 is 9 and the roote of 144 is 12. 4 Hauing two extreme numbers giuen to find two meane proportionals between them Diuide the space betweene the two extreme numbers giuen into three equall parts As if the extreme numbers giuen were 8 and 27 diuide the space betweene them into three equall parts the feet of the compasses will stand in 12 and 18. 5 To find the cubique roote of a number giuen The cubique roote is alwayes the first of two meane proportionals betweene 1 and the number giuen and therefore to be found by diuiding the space betweene them into three equall parts So the roote of 1728 will be found to be 12. The roote of 17280 is almost 26 and the roote of 172800 is almost 56. 6 To multiply one number by another Extend the compasses from 1 to the multiplicator the same extent applied the same way shall reach from the multiplicand to the product As if the numbers to be multiplied were 25 and 30 either extend the compasses from 1 to 25 and the same extent will giue the distance from 30 to 750 or extend them from 1 to 30 and the same extent shall reach from 25 to 750. 7 To diuide one number by another Extend the compasses from the diuisor to 1 the same extent shall reach from the diuidend to the quotient So if 750 were to be diuided by 25 the quotient would be found to be 30. 8 Three numbers being giuen to find a fourth proportionall This golden rule the most vsefull of all others is performed with like ease For extend the compasses from the first number to the second the same extent shall giue the distance from the third to the fourth As for example the proportion between the diameter and the circumference is said to be such as 7 to 22 if the diameter be 14 how much is the circumference Extend the compasses from 7 to 22 the same extent shall giue the distance from 14 to 44 or extend them from 7 to 14 and the same extent shall reach from 22 to 44. Either of these wayes may be tried on seuerall places of this line but that place is best where the seete of the compasses may stand nearest together 9 Three numbers being giuen to finde a fourth in a duplicated proportion This proposition concernes questions of proportion betweene lines and superficies where if the denomination be of lines extend the compasses from the first to the second number of the same denomination so the same extent being doubled shall giue the distance from the third number vnto the fourth The diameter being 14 the content of the circle is 154 the diameter being 28 what may the content be Extend the compasses from 14 to 28 the same extent doubled will reach from 154 to 616. For first it reacheth from 154 vnto 308 and turning the compasses once more it reacheth from 308 vnto 616 and this is the content required But if the first denomination be of the superficiall content extend the compasses vnto the halfe of the distance betweene the first number and the second of the same denomination so the same extent shall giue the distance from the third to the fouth The content of a circle being 154 the diameter is 14 the content being 616 what may the diameter be Diuide the distance betweene 154 and 616 into two equall parts then set one foote in 14 the other will reach to 28 the diameter required 10 Three numbers being giuen to find a fourth in a triplicated proportion This proposition concerneth questions of proportion betweene lines and solids where if the first denomination be of lines extend the compasses from the first number to the second of the same denomination so the extent being tripled shall giue the distance frō the third number vnto the fourth Suppose the diameter of an iron bullet being 4 inches the weight of it was 9 l the diameter being 8 inches what may the weight be Extend the compasses from 4 to 8 the same extent being tripled will reach from 9 vnto 72. For first it reacheth from 9 vnto 18 then from 18 to 36 thirdly from 36 to 72. And this is the weight required But if the first denomination shall be of the Solid content or of the weight extend the compasses to a third part of the distance betweene the first number and the second of the same denomination so
obserued and went to Limehouse with some of my friends and tooke with vs a quadrant of 3 foote semidiameter and two needles the one aboue 6 inches and the other 10 inches long where I made the semidiameter of my horizontall plane AZ 12 inches and toward night the 13 of Iune 1622 I made obseruation in seuerall parts of the ground and found as followeth Alt. ☉ AZM AZN Variat Gr. M. Gr. M. Gr. M. Gr. M. 19 0 82 2 75 52 6 10 18 5 80 50 74 44 6 6 17 34 80 0 74 6 5 54 17 0 79 15 73 20 5 55 16 18 78 12 72 32 5 40 16 0 77 50 72 10 5 40 20 10 71 2 64 49 6 13 9 52 70 12 64 25 5 47 CHAP. VI. Containing such nauticall questions as are of ordinary vse concerning longitude latitude Rumb and distance 1 To keep an account of the ships way THe way that the ship maketh may be knowne to an old sea-man by experience by others it may be found for some small portion of time either by the log line or by the distance of two knowne markes on the ships side The time in which it maketh this way may be measured by a watch or by a glasse Then as long as the wind continueth at the same stay it followeth by proportion As the time giuen is to an houre So the way made to an houres way Suppose the time to be 15 seconds which make a quarter of a minute and the way of the ship 88 feet then because there are 3600 seconds in an houre I may extend the compasses in the line of numbers from 15 vnto 3600 and the same extent will reach from 88 vnto 21120. Or I may extend them from 15 vnto 88 and this extent will reach from 3600 vnto 21120 which shewes that an houres way came to 21120 feete But this were an vnnecessary businesse to hearken after feet or fadoms It sufficeth our sea-men to find the way of their ship in leagues or miles And they say that there are 5 feet in a pace 1000 paces in a mile and 60 miles in a degree and therefore 300000 feete in a degree Yet comparing seuerall obseruations and their measures with our feete vsuall about London I find that we may allow 352000 feete to a degree and then if I extend the compasses in the line of numbers from 352000 vnto 21120 I shall find the same extent to reach from 20 leagues the measure of one degree to 1.2 and from 60 miles to 3.6 which shewes the houres way to be 1 league and 2 tenths of a league or 3 miles and 6 tenths of a mile But to auoid these fractions and other tedious reductions I suppose it would be more easie to keep this account of the ships way as also of the difference of latitude and the difference of longitude by degrees and parts of degrees allowing 100 parts to each degrâ—e which we may therefore call by the name of centosmes Neither would this be hard to conceiue For if 100 such parts do make a degree then shall 50 parts be equall to 30 minutes as 30 minutes are equall to 10 leagues And 5 parts shall be equall to 3 minutes as 3 minutes are equall to 1 league And so the same extent as before will reach from 100 parts vnto 6 which shewes that the houres way required is 6 cent such as 100 do make a degree and 5 do make an ordinary league This might also be done at one operation For vpon these suppositions diuide 44 feet into 45 lengths and set as many of them as you may conueniently betweene two markes on the ships side and note the seconds of time in which the ship goeth these lengths so the lengths diuided by the time shall giue the cent which the ship goeth in an houre Suppose the distance betweene the two markes to be 60 lengths which are 58 feet and 8 inches and let the time be 12 seconds extend the compasses from 12 to 1 in the line of numbers so the same extent will reach from 60 vnto 5. Or extend them from 12 vnto 60 and the same extent will reach from 1 vnto 5. This shewes that the ships way is according to 5 cent in an houre This may be found yet more easily if the log line shall befitted to the time As if the time be 45 seconds the log line may haue a knot at the end of euery 44 feete then doth the ship run so many cent in an houre as there are knots vered out in the space of 45 seconds If 30 seconds do seeme to be a more conuenient time the log line may haue a knot at the end of euery 29 feet and 4 inches and then also the centesmes will be as many as the knots Or if the knots be made to any set number of feet the time may be fitted vnto the distance As if the knots be made at the end of euery 24 feet the glasse may be made 24 seconds and somewhat more then an halfe of a second and so these knots will shew the cent If there be 5 knots vered out in a glasse then 5 cent if 6 knots then the ship goeth 6 cent in the space of an houre and so in the rest For vpon this supposition the proportion between the time and the feet will be as 45 vnto 44. But according to the common supposition it should seeme to be as 45 vnto 37 ½ or in lesser termes as 6 vnto 5. Those which are vpon the place may make proofe of both and follow that which agrees best with their experience 2 By the latitude and difference of longitude to find the distance vpon a course of East and West Extend the compasses from the sine of 90 gr vnto the sine of the complement of the latitude the same extent shal reach in the line of numbers from the difference of longitude to the distance So the measure of one degree in the equator being 100 cent the distance belonging to one degree of longitude in the latitude of 51 gr 30 m. will be found about 62 cent and ¼ Or if the measure of a degree be 60 miles the distance will be found about 37 miles and â…“ If the measure be 20 leagues then almost 12 leagues and ½ If the measure be 17 ½ as in the Spanish charts then somewhat lesse then 11 leagues sailing vpon this parallell will giue an alteration of one degree of longitude 3 By the latitude and distance vpon a course of East or West to find the difference of longitude Extend the compasses from the sine of the complement of the latitude to the sine of 90 gr the same extent wil reach in the line of numbers from the distance to the difference of longitude So the distance vpon a course of East or West in the latitude of 51 gr 30 m. being 100 cent the difference of longitude will be found 1.60 which make one degree and 60 centesmes or 1 gr 36 m.
gr and let a thread and plummet be hanged vpon a line parallell to the edges of one of the legs so that leg shall be verticall and the other leg parallell to the horizon If the plane seeme to be leuell with the horizon you may trie it by setting the horizontall leg of the Sector to the plane and holding the other leg vpright for then if the thread shall fall on his plummet line which way soeuer you turne the Sector it is an horizontall plane If the one end of the plane be higher then the other and yet not verticall it is an inclining plane and you may find the inclination in this maner First hold the verticall leg of the Sector vpright and turne the horizontall leâ— about vntill it lie close with the plane and the thread fall on his pluÌ„met line so the line drawne by the edge of that horizontall leg shall be an horizontall line Suppose the plane to be BGED and that BD were thus found to be the horizontall line vpon the plane then may you crosse the horizontall line at right angles with a perpendicular CF that done if you set one of the legs of the Sector vpon the perpendicular line CF and make the other leg with a thread and plummet to become verticall you shal haue the angle betweene the verticall line and the perpendicular on the Plane as before in the vse of the Sector pag. 50. and the complement of this angle is the inclination of the plane to the horizon To find the declination of a Plane The declination of a Plane is alwayes reckoned in the horizon betweene the line of East and West and the horizontall line vpon the Plane As in the fundamentall Diagram the prime verticall line which is the line of East and West is ECW if the horizontall line of the plane proposed shal be BCD the angle of declination is ECB But because a Plane may decline diuers wayes that we may the better distinguish them we consider three lines belonging to euery Plane the first is the horizontall line the second the perpendicular line crossing the horizontall at right angles the third the axis of the plane crossing both the horizontall line and his perpendicular and the plane it selfe at right angles The perpendicular line doth help to find the inclination of the plane as before the horizontall to finde the declination the axis to giue denomination vnto the plane For example in a verticall plane here represented by EZW the horizontall line is ECW the same with the line of East and West and therefore no declination the perpendicular crossing it CZ the same with the verticall line drawne from the center to the zenith right vnto the horizon and therefore no inclination The axis of the plane is SCN the same with the meridian line drawne from the South to the North and accordingly giues the denomination to the plane For the plane hauing two faces and the axis two poles S and N the pole S falling directly into the South doth cause that face to which it is next to be called the South face and the other pole at N pointing into the North doth giue the denomination to the other face and make it to be called the North face of this plane In like maner in the declining inclining plane here represented by BFD the horizontall line is BCD which crosseth the prime verticall line ECW and therefore it is called a declining plane according to the angle of declination ECB or WCD The perpendicular to this horizontall line is CF where the point F falleth in the plane QZH perpendicular to the plane proposed betweene the zenith and the North part of the horizon and therefore it is called a plane inclining to the Northward according to the arke FQ or the angle FCQ The axis of the plane is here represented by the line CK where the pole K is 90 gr distant from the plane and so is as much aboue the horizon at H and the other pole as much below the horizon at Q as the plane at F is distant from the zenith and this pole K here falling betweene the meridian and the prime verticall circle into the Southwest part of the world this vpper face of the plane is therefore called the Southwest face and the lower the Northeast face of the plane The declination from the prime verticall may be found by the needle in the vsuall inclinatorie Quadrant or rather by comparing the horizontall line drawne vpon the plane with the azimuth of the Sunne and the meridian line in such sort as before we found the variation of the magneticall needle For take any boord that hath one side strait and draw the line HO parallell to that side and the line ZM perpendicular vnto it and on the center Z make a semicircle HMO this done hold the boord to the plane so as HO may be parallel to BD the horizontall line on the plane and the boord parallell to the horizon then the Sunne shining vpon it hold out a thread and plummet so as the thread being verticall the shadow of the Sunne may fall on the center Z and draw the line of shadow AZ representing the common section which the azimuth of the Sunne makes with the plane of the horizon and let another take the altitude of the Sunne at the same instant so by resoluing a triangle as I shewed before pag. 65. you may find what azimuth the Sun was in when he gaue shadow vpon AZ Suppose the azimuth to be as before pag. 64. 72 gr 52 m. from the North to the Westward and therefore 17 gr 8 m. from the West we may allow these 17 gr 8 m. from A vnto V and draw the line ZV and so we haue the true West point of the prime verticall line then allowing 90 gr from V vnto S we haue the South point of the meridian line ZS and the angle HZV shall giue the declination of the plane from the verticall and the angle OZS the declination of the plane from the meridian Or we may take out onely the angle AZH which the line of shadow makes with the horizontall line of the plane and compare it with the angle AZV which the line of shadow makes with the prime verticall And so here if AZV the Sunnes azimuth shall be 17 gr 8 m. past the West and yet the line of shadow AZ 7 gr 12 m. short of the plane the declination of the plane shall be 24 gr 20 m. as may appeare by the site of the plane and the circles If the altitude of the Sunne be taken at such time as the shadow of the thread falleth on BD or HO and then a triangle resolued the declinatâ—on of the plane will be such as the azimuth of the Sunne from the prime verticall If at such a time as the shadow falleth on MZ the declination will be such as the azimuth of the Sunne from the meridian If it be a faire Summers day
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