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
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
the verticall circle passing through the zenith and nadir East and West and the line MAH crossing it at right angles shall represent the horizon These two being diuided in the same sort as the ecliptique and the equator the line drawne through each degree of the semidiameter AZ parallell to the horizon shall be the circles of altitude and the diuisions in the horizon and his parallels shall giue the azimuth Lastly if through 18 gr in AN be drawne a right line IK parallell to the horizon it shall shew the time when the day breaketh and the end of the twilight For example of this proiection let the place of the Sunne be the last degree of ♉ the parallell passing through this place is LD and therefore the meridian altitude ML and the depression below the horizon at midnight HD the semidiurnall arke LC the seminocturnall arke CD the declination AB the ascentionall difference BC the amplitude of ascenon AC The difference betweene the end of twilight and the day breake is very small for it seemes the parallell of the Sunne doth hardly crosse the line of twilight If the altitude of the Sunne be giuen let a line be drawne for it parallell to the horizon so it shall crosse the parallell of the Sunne and there shew both the azimuth and the houre of the day As if the place of the Sunne being giuen as before the altitude in the morning were found to be 20 gr the line FG drawne parallell to the horizon through 20 gr in AZ would crosse the parallell of the Sunne in ☉ Wherefore F ☉ sheweth the azimuth L ☉ the quantitie of houres from the meridian It seemes to be about halfe an houre past 6 in the morning and yet more then halfe a point short of the East The distance of two places may be also shewed by this proiection their latitudes being knowne and their difference of longitude For suppose a place in the East of Arabia hauing 20 gr of North latitude whose difference of longitude from London is found by an eclipse to be 5 ho. ½ Let Z be the zenith of London the parallell of latitude for that other place must be LD in which the difference of longitude is L ☉ Wherefore ☉ representing the site of that place I draw through ☉ a parallell to the horizon MH crossing the verticall AZ neare about 70 gr from the zenith which multiplied by 20 sheweth the distance of London and that place to be 1400 leagues Or multiplied by 60 to be 4200 miles 2 The Sphere may be proiected in plano by circular lines as in the generall astrolabe of Gemma Frisius by the help of the tangent on the side of the Sector For let the circle giuen represent the plane of the generall meridian as before let it be diuided into foure parts and crossed at right angles with EAE the equator and PS the circle of the houre of 6 wherein P stands for the North pole and S for the South pole Let each quarter of the meridian be diuided into 90 gr and so the whole into 360 beginning from P and setting to the numbers of 10 20 30. c. 90 at AE 180 at S 270 at E 360 at P. The semidiameters AP AAE AS A E may be diuided according to the tangents of halfe their arkes that is a tangent of 45 gr which is alwayes equall to the Radius shall giue the semidiameter of 90 gr a tangent of 40 gr shall giue 80 gr in the semidiameter a tangent of 35 gr shall giue 70. c. So that the semidiameters may be diuided in such sort as the tangent on the side of the Sector the difference being onely in their numbers Hauing diuided the circumference and the semidiameters we may easily draw the meridians and the parallels by the helpe of the Sector The meridians are to be drawne through both the poles P and S and the degrees before graduated in the equator The distance of the center of each meridian from A the center of the plane is equall to the tangent of the same meridian reckoned from the generall meridian PAESE and the semidiameter equall to the secant of the same degree As for example if I should draw the meridian PBS which is the tenth from PAES the tangent of 10 gr giueth me AC and the secant of 10 gr giueth me SC whereof C is the center of the meridian PBS and CS his semidiameter so AF a tangent of 20 gr sheweth F to be the center of PDS the twentith meridian from PAES and AG a tangent of 23 gr 30 M. sheweth G to be the center of P 69 S. c. The parallels are to be drawne through the degrees in AP AS and their correspondent degrees in the generall meridian The distance of the center of each parallell from A the center of the plane is equall to the secant of the same parallell from the pole and the semidiameter equall to the tangent of the same degree As if I should draw the parallell of 80 gr which is the tenth from the pole S first I open the compasses vnto AC the tangent of 10 gr and this giueth me the semidiameter of this parallel whose center is a little from S in such distance as the secant SC is longer then the radius SA The meridians and parallels being drawne if we number 23 gr 30 m. from E to ♋ Northward from AE to ♑ Southward the line drawne from ♋ to ♑ shall be the ecliptique which being diuided in such sort as the semidiameter AP the first 30 gr from A to ♋ shall stand for the sine of ♈ the 30 gr next following for ♉ the rest for ♊ ♋ ♌ c. in their order If farther we haue respect vnto the latitude we may number it from E Northward vnto Z and there place the zenith by which and the center the line drawne ZAN shall represent the verticall circle and the line MAH crossing it at right angles shall represent the horizon and these diuided in the same sort as AP the circles drawne through each degree of the semidiameter AZ parallell to the horizon shall be the circles of altitude and the circles drawne through the horizon and his poles shall giue the azimuths For example of this proiection let the place of the Sunne be in the beginning of ♒ the parallell passing through this place is ♒ ☉ L and therefore the meridian altitude ML and the depression below the horizon at midnight H ♒ the semidiurnall arke L ☉ the seminocturnall arke O ♒ the declination AR the ascensionall difference R ☉ the amplitude of ascension A ☉ Or if A be put to represent the pole of the world then shall PAESE stand for the equator and P ♋ S ♑ for the ecliptique and the rest which before stood for meridians may now serue for particular horizons according to their seuerall eleuations Then suppose the place of the Sunne giuen to be
seuerall mettalls and equall weight hauing the magnitude of the one to finde the magnitude of the rest Take the magnitude giuen out of the lines of Solids and to it open the Sector in the points belonging to the mettall giuen so the parallells taken from between the points of the other mettalls and measured in the lines of Solids shall giue the magnitude of their bodies Thus hauing cubes or spheres of equall weight but seuerall mettalls we shall finde that if those of tin containe 10000 D â—he others of iron wil contain 9250 those of copper 8222 those of siluer 7161 those of lead 6435 those full of quicksiluer 5453 and those of gold 3895. 2 In like bodies of seuerall mettalls and equall magnitude hauing the weight of one to finde the weight of the rest This proposition is the conuerse of the former the proportion not direct but reciprocall wherefore hauing two like bodies take the giuen weight of the one out of the lines of Solids and to it open the Sector in the points belonging to the mettall of the other body so the parallell taken from the points belonging to the body giuen and measured in the lines of Solids shall giue the weight of the body required As if a cube of gold weighed 38 l. and it were required to know the weight of a cube of lead hauing equal magnitude First I take 38 l. for the weight of the golden cube out of the lines of Solids put it ouer in the points of ♄ belonging to lead so the parallell taken from betweene the points of ☉ standing for gold and measured in the lines of Solids doth giue the weight of the leaden cube required to be 23 l. Thus if a sphere of gold shall weigh 10000 we shall finde that a sphere of the same diameter full of quicksiluer shall weigh 7143 a sphere of lead 6053 a sphere of siluer 5438 a sphere of copper 4737 a sphere of iron 4210 and a sphere of tin 3895. 3 A bodie being giuen of one mettall to make another like vnto it of another mettall and equall weight Take out one of the sides of the bodie giuen and put it ouer in the points belonging to his mettall so the parallell taken from between the points belonging to the other mettall shall giue the like side for the bodie required If it be an irregular bodie let the other like sides be found out in the same manner Let the bodie giuen be a sphere of lead containing in magnitude 16 D whose diameter is A to which I am to make a sphere of iron of equall waight If I take out the diameter A and put it ouer in the points of ♄ belonging to lead the parallell taken from betweene the points of ♂ standing for iron shall be B the diameter of the iron sphere required And this compared with the other diameter in the lines of solids will be found to be 23 d. in magnitude 4 A body being giuen of one mettall to make another like vnto it of another mettall according to a weight giuen First find the sides of a like bodie of equall weight then may we either augment or diminish them according to the proportion giuen by that which we shewed before in the second and third Prop. of Solids As if the bodie giuen were a sphere of lead whose diameter is A and it were required to find the diameter of a sphere of iron which shall weigh three times as much as the sphere of lead I take A and put it ouer in the points of ♄ his parallell taken from betweene the points of ♂ shall giue me B for the diameter of an equall sphere of iron if this be augmented in such proportion as 1 vnto 3 it giueth C for the diameter required CHAP. VI. Of the lines on the edges of the Sector HAuing shewed some vse of the lines on the flat sides of the Sector there remaine onely those on the edges And here one halfe of the outward edge is diuided into inches and numbred according to their distance from the ends of the Sector As in the Sector of fourteene inches long where we find 1 and 13 it sheweth that diuision to be 1 inch from the nearer end and 13 inches from the farther end of the Sector The other halfe containeth a line of lesser tangents to which the gnomon is Radius They are here continued to 75 gr And if there be need to produce them farther take 45 out of the number of degrees required and double the remainder so the tangent and secant of this double remainder being added shall make vp the tangent of the degrees required As if AB being the Radius and BC the tangent line it were required to find the tangent of 75 gr If we take 45 gr out of 75 gr the remainer is 30 gr and the double 60 gr whose tangent is BD and the secant is AD if then we adde AD to BD it maketh BC the tangent of 75 gr which was required In like sort the secant of 61 gr added to the tangent of 61 gr giueth the tangent of 75 gr 30 m. and the secant of 62 gr added to the tangent of 62 gr giueth the tangent of 76 gr and so in the rest The vse of this line may be To obserue the altitude of the Sunne Hold the Sector so as the tangent BC may be verticall and the gnomon BA parallell to the horizon then turne the gnomon toward the Sunne so that it may cast a shadow vpon the tangent and the end of the shadow shal shew the altitude of the Sunne So if the end of the gnomon at A do giue a shadow vnto H it sheweth that the altitude is 38 gr ½ if vnto D then 60 gr and so in the rest There is another vse of this tangent line for the drawing of the houre lines vpon any ordinary plane whereof I will set downe these propositions 1 To draw the houre lines vpon an horizontall plane 2 To draw the houre lines vpon a direct verticall plane First draw a right line AC for the horizon and the equator and crosse it at the point A about the middle of the line with AB another right line which may serue for the meridian and the houre of 12 then take out 15 gr out of the tangents and pricke them downe in the equator on both sides from 12 so the one point shall serue for the houre of 11 and the other for the houre of 1. Againe take out the tangent of 30 gr and pricke it downe in the equator on both sides from 12 so the one of these points shall serue for the houre of 10 and the other for the houre of 2. In like maner may you prick downe the tangent of 45 gr for the houres of 9 and 3 and the tangent of 60 gr for the houres of 8 and 4 and the tangent of 75 gr for the houres of 7 and 5.
of May if you obserue and find the lower sight to stay at 30 gr on the â—ledgeâ—acke of the Bow such is the altitude For the declination â—ledgeâ— 20 gr Northward the altitude of the Sunne betweene the â—ledgeâ—wo sights 40 gr the altitude of the equator 60 gr and thereâ—ledgeâ—re the latitude 30 gr as in the seuenth Prop. 12 To find the day of the moneth by knowing the latitude of the place and obseruing the meridian altitude of the Sunne Place your three sights according to your latitude the hoâ—ledgeâ—zontall sight to the center the lower sight to the latitude â—ledgeâ—d the vpper sight among the moneths Then when the â—ledgeâ—nne cometh to the meridian obserue the altitude looking â—ledgeâ— the lower sight through the horizontall and keeping the â—ledgeâ—wer sight still at the latitude but mouing the vpper sight â—ledgeâ—til it giue shadow vpon the middle of the horizontal sight â—ledgeâ— the vpper sight shall shew the day of the moneth requiâ—ledgeâ—d Thus in our latitude if you set the lower sight to 51 gr 30 â—ledgeâ— and obseruing finde the altitude of the Sunne betweene â—ledgeâ—at and the vpper sight to be 28 gr 30 m. this vpper sight â—ledgeâ—ll â—all vpon the ninth of October and the twelfth of Feâ—ledgeâ—uarâ—e And if yet you doubt which oâ— them two is the day â—ledgeâ—u may expect another meridian altitude and then if you â—ledgeâ—d the vpper sight vpon the tenth of October and the eleâ—ledgeâ—nth of Februarie the question will be soone resolued 13 To find the declination of any vnknowne starre and so to place it on the Bow by knowing the latitude of the place and obseruing the Meridian altitude of the Starre When you find a starre in the Meridian that is fit for obâ—ledgeâ—uation Set the center of the Bow to your eye the lower â—ledgeâ—ht to the latitude and moue the vpper sight vp or downe â—ledgeâ—till you see the horizon by the lower sight and the starre by the vpper sight then will the vpper sight stay at the declination and place of the starre Thus being in 20 gr of North latitude if you obserue anâ—redgeâ— find the meridian altitude of the head of the Crosier to bâ—redgeâ— 14 gr 50 m. The vpper sight will stay at 34 gr 50 m. and therâ—redgeâ— may you place this starre For by this obseruatioÌ„ the distance oâ—redgeâ— this starre from the South pole should be 34 gr 50 m. and thâ—redgeâ— declination from the equator 55 gr 10 m. And so for the resâ—redgeâ— The starres which I mâ—ntioneâ— beâ—â—re do come to the meridian in this order after the first point of Aries  Ho. Mi. The pole starre at 0 29 The rams head 1 46 The head of Medusa 2 44 The sidâ— of Perseus 2 58 The Buls eye 4 15 The goate 4 49 Orions left shoulder 5 5 Orions girdle the first the second the third 5 13 5 17 5 22 Orions right shoulder 5 35 The great dog 6 29 Castor 7 10 The little dog 7 20 Pollux 7 22 The Hydra's hart 9 9 The lions hart 9 48 The great beares backe 10 40 First in gr beares taile 12 37 The Virgins spike 13 5 Second in gr beaâ— taile 13 9 Third in gr beares taile 13 33 Arcturus 13 58 The formost gu rd 14 52 The North crowne 15 19 Thâ— hâ— dmost guard 15 25 Scorpions hart 16 7 The harpe 18 24 Vulturs hart 19 33 Swans taile 20 29 Fomahant 22 36 The end of the second Booke THE THIRD BOOKE Of the vse of the lines of Numbers Sines and Tangents for the drawing of houre-Houre-lines on all sorts of Planes THere are ten seuerall sorts of Planes which take their denomination froÌ„ those great circles to which they are parallels and may sufficiently for our vse be represented in this one fundamentall Diagram described before in the vse of the Sector and be knowne by their horizontall and perpendicular lines of such as know the latitude of the place and the circles of the sphere 1 An horizontall plane parallell to the horizon here represented by the outward circle ESWN 2 A verticall plane parallell to the prime verticall circle which passeth through the zenith and the points of East and West in the horizon and is right to the horizon and the meridian that is maketh right angles with them both This is represented by EZW 3 A polar plane parallell to the circle of the houre of 6 which passeth through the pole and the points of East and West being right to the Equinoctiall and the Meridian but inclining to the horizon with an angle equall to the latitude This is here represented by EPW 4 An equinoctiall plane parallell to the Equinoctiall which passeth through the points of East and West being right to the Meridian but inclining to the Horizon with an angle equall to the complement of the latitude This is here represented by EAW 6 A meridian plane parallell to the meridian the circle of the houre of 12 which passeth through the zenith the pole and the points of South and North being right to the horizon and the prime verticall This is here represented by SZN 7 A meridian plane inclining to the horizon parallell to any great circle which passeth through the points of South and North being right to the prime verticall but inclining to the horizon This is here represented by SGN 8 A verticall declining plane parallell to any great circle which passeth through the zenith being right to the horizon but inclining to the meridian This is represented by BZD. 9 A polar declining plane parallell to any great circle which passeth through the pole being right to the equinoctiall but inclining to the meridian This is here represented by HPQ 10 A declining inclining plane parallell to any great circle which is right to none of the former circles but declining from the prime verticall and inclining both to the horizon and the meridian and all the houre circles This may be here represented either by BMD or BFD or BKD or any such great circle which passeth neither through the South and North nor East and West points nor through the zenith nor the pole Each of these planes except the horizontall hath two faces whereon houre-lines may be drawne and so there are 19 planes in all The meridian plane hath one face to the East and another to the West the other verticall planes haue one to the South and another to the North and the rest one to the zenith and another to the nadir but what is said of the one may be vnderstood of the other To find the inclination of any Plane For the distinguishing of these Planes we may finde whethâ—r they be horizontall or verticall or inclining to the horizon and how much they incline either by the vsuall inclinatorie quadrant or by fitting a thread and plummet vnto the Sector For let the Sector be opened to a right angle the lines of Sines to an angle of 92 gr the inward edges of the Sector to 90
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
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
5 3  37 34 32 29 26 24 21 19 16 13 11 8 5 3  36 33 31 28 26 23 21 18 15 13 10 8 5 3  35 32 30 27 25 22 20 17 15 12 10 7 5 2 3 34 30 29 26 24 22 19 17 14 12 10 7 5 2  33 30 28 26 23 21 19 16 14 12 9 7 5 2  31 29 27 24 22 20 18 16 13 11 9 7 4 2  30 28 25 23 21 19 17 15 13 11 8 6 4 2 4 28 26 24 22 20 18 16 14 12 10 8 6 4 2  27 25 23 21 19 17 15 13 11 10 8 6 4 2  25 23 21 20 18 16 14 13 11 9 7 5 4 2  23 22 20 18 17 15 13 12 10 8 7 5 3 2 5 22 20 18 17 15 14 12 11 9 8 6 5 3 2  20 18 17 16 14 13 11 10 8 7 6 4 3 1  18 17 15 14 13 12 10 9 8 6 5 4 3 1  16 15 14 13 11 10 9 8 7 6 5 3 2 1 6 14 13 12 11 10 9 8 7 6 5 4 3 2 1  12 11 10 10 9 8 7 6 5 4 3 3 2 1  10 9 9 8 7 7 6 5 4 4 3 2 1 1  8 8 7 6 6 6 5 4 3 3 2 2 1 1 7 6 6 5 5 4 5 4 3 3 2 2 1 1 1  4 4 4 3 3 3 2 2 2 1 1 1 1 0  2 2 2 2 1 1 1 1 1 1 1 1 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 In the Chart let a meridian AB be drawne through A and in A with AB make an angle of the Rumb BAC Then open the compasses according to the latitude of the places to EF the quantitie of 6 gr in the meridian transferring them into the Rumb from A to C and through C draw the parallell BC crossing the meridian AB in B so the degrees in the meridian from A to B shall shew the difference of latitude to be 5 gr 2 By the Rumb and both latitudes to find the distance vpon the Rumb As the sine of the complement of the Rumb from the meridian is to the Radius So the difference of latitudes to the distance vpon the Rumb As if the places giuen were A in the latitude of 50 gr C in the latitude of 55 gr and the Rumb the third from the meridian Here I may take 5 gr for the difference of latitude out of the line of lines and put it ouer in the sine of 56 gr 15 m. for the complement of the third Rumb from the meridian Then keeping the Sector at this angle I take out the parallell Radius and measuring it in the line of lines I find it to be 6 gr and such is the distance vpon the Rumb which was required Or I may take the laterall Radius and make it a parallell sine of 56 gr 15 m. the complement of the Rumb from the meridian then keeping the Sector at this angle I take 5 gr for the difference of latitude either out of the line of lines or out of some other scale of equall parts and lay it on both sides of the Sector from the center either on the line of lines or of sines so the parallell taken from the termes of this difference and measured in the same scale with the difference shall shew the distance vpon the Rumb to be 6 gr or 120 leagues Or keeping the Sector at this angle I may take the difference betweene 50 gr and 55 gr out of the Meridian line and measuring it in the equator I shal find it to be equall to 8 gr 22 p. of the equator Wherefore I take the parallell betweene 822 and 822 out of the line of lines and measuring it in the line of lines I shall find it to be 989 which shewes that according to this proiection the distance vpon this third Rumb answerable to the former distance of latitudes will be equall to 9 gr 89 p. of the equator Or the Sector remaining at this angle I may take the difference betweene 50 gr and 55 gr out of the Meridian line and lay it from the center on both sides of the Sector either on the line of lines or of sines so the parallell taken from the termes of this difference shall be the very line of distance required the same with AC or EF vpon the chart which may serue for the better pricking downe of the distance vpon the Rumb without taking it forth of the Meridian line as in the former Prop. Or if the Rumb fall nearer to the equator that the laterall Radius cannot be fitted ouer in it this proposition may be wrought by parallell entrance For if I first take out the sine of 56 gr 15 m. and make it a parallell Radius by fitting it ouer in the sines of 90 and 90 or in the ends of the lines of lines and then take 5 gr for the difference of latitudes out of the line of lines and carrie it parallell to the former I shall find it to crosse both lines of lines in the points of 6 and so it giues the same distance as before Or if the distance be small it may be found by the former Table For the Rumb being found in the side of the Table and the difference of latitude in the same line the top of the columne wherein the difference of latitude was found shall giue the number of leagues in the distance required Or we may finde this distance in the Table of Rumbs in the fift Prop. following For according to the example looke into the Table of the third Rumb for 5 gr of latitude and there we shall finde 6 gr 01 parts vnder the title of distance So if the difference of latitude vpon the same Rumb were 50 gr the distance would be 60 gr 13 parts If the difference of latitude vpon the same Rumb were onely ½ of a degree the distance would be onely 60 parts such as 100 doe make a degree In the chart let a Meridian AB be drawne through A and parallels of latitude through A and C then in A with AB make an angle of the Rumb BAC so the distance taken from A to C and measured in the Meridian line according to the latitude of the places shall be found to be 6 gr or 120 leagues And such is the distance required 3 By the distance and both latitudes to find the Rumb As the distance vpon the Rumb to the difference of latitudes So is the Radius to the sine of the complement of the Rumb from the Meridian As if the places giuen were A in the latitude of 50 gr C in the latitude of 55 gr the distance betweene them being 6 gr vpon the Rumb First I take 6 gr for the distance vpon the Rumb and lay it on both sides of the Sector from the center then out of
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
whether the two stations be chosen at the one end of the bredth proposed or without it or within it if the line betweene the stations be perpendicular vnto the bredth as may appeare if in stead of the stations at A and H we make choise of the like stations at I and K. There might be other wayes proposed to work these Prop. by holding the Crosse euen with the distance and the Staffe parallell with the height but these would proue more troublesome and those which are deliuered are sufficient and the same with those which others haue set downe vnder the name of the Iacobs staffe CHAP. III. The vse of the Tangent lines in taking of Angles 1 To find an angle by the Tangent on the Staffe LEt the middle sight be alwayes set to the middle of the Crosse noted with 20 and 30 and then the Crosse drawne nearer the eye vntill the marks may be seene close within the sights For so if the eye at A that end of the Staffe which is noted with 90 and 180 beholding the marks K and N betweene the two first sights C and B or the marks K and P betweene the two outward sights the Crosse being drawne downe vnto H shall stand at 30 and 60 in the Tangent on the Staffe it sheweth that the angle KAN is 30 gr the angle KAP 60 gr the onâ— double to the other which is â—he râ—ason of the double numbers on this line of the Staffe and this way wil serue for any angle from 20 gr toward 90 gr or from 40 gr toward 180 gr But if the angle be lesse then 20 gr we must then make vse of the Tangent vpoÌ„ the Crosse 2 To find an angle by the Tangent of 20 vpon the Crosse Set 20 vnto 20 that is the middle sight to the middest of the Crosse at the end of the Staffe noted with 20 so the eye at A beholding the marks L and N close betweene the two first sights C and B shall seâ— them in an angle of 20 gr If the marks shâ—ll be nearer together as are M and N then draw in the Crosse from C vnto E if they be farther asunder as are K and N then draw out the Crosse from C vnto F so the quantitie of the angle shal be still found in the Crosse in the Tangent of 20 gr at the end of the Staffe and this will serue for any angle from 0 gr toward 35 gr 3 To find an angle by the Tangent of 30 vpon the Crosse This Tangent of 30 is here put the rather that the end of the Staffe resting at the eye the hand may more easily remoue th Crosse for it supposeth the Radius to be no longer then AH which is from the eye at the end of the Staffe vnto 30 gr about 22 inches and 7 parts Wherefore here set the middle sight vnto 30 gr on the Staffe and then either draw the Crosse in or out vntill the marks be seene between the two first sights so the quantitie of the angle will be found in the Tangent of 30 which is here represented by the line GH and this will serue for any angle from 0 gr toward 48 gr 4 To obserue the altitude of the Sunne backward Here it is fit to haue an horizontall sight set to the beginning of the Staffe and then may you turne your backe toward the Sun and your Crosse toward your eye If the altitude be vnder 45 gr set the middle sight to 30 on the Staffe and looke by the middle sight through the horizontall vnto the horizon mouing the Crosse vpward or downward vntill the vpper sight doe shadow the vpper halfe of the horizontall sight so the altitude will be found in the Tangent of 30. If the altitude shal be more then 45 gr set the middle sight vnto the middest of the Crosse and look by the inward edge of the lower sight through the horizontall to the horizon mouing the middle sight in or out vntill the vpper sight do shadow the vpper halfe of the horizontall sight so the altitude will be found in the degrees on the Staffe betweene 40 and 180. 5 To set the Staffe to any angle giuen This is the conuerse of the former Prop. For if the middle sight be set to his place and degree the eye looking close by the sights as before cannot but see his obiect in the angle giuen 6 To obserue the altitude of the Sunne another way Set the middle sight to the middle of the Crosse and hold the horizontall sight downward so as the Crosse may be parallell to the horizon then is the Staffe verticall and if the outward sight of the Crosse do shadow the horizontall sight the complement of the altitude wil be found in the tangent on the Staffe 7 To obserue an altitude by thread and plummet Let the middle sight be set to the middest of the Crosse and to that end of the Staffe which is noted with 90 and 180 then hauing a thread and a plummet at the beginning of the Crosse and turning the Crosse vpward and the Staffe toward the Sunne the thread will fall on the complement of the altitude aboue the horizon And this may be applied to other purposes 8 To apply the lines of inches to the taking of angles If the angles be obserued betweene the two first sights there wil be such proportion between the parts of the Staffe and the parts of the Crosse as betweene the Radius and the Tangent of the angle As if the parts intercepted on the Staffe were 20 inches the parts on the Crosse 9 inches Then by proportion as 20 vnto 9 so 100000 vnto 45000 the tangent of 24 gr 14 m. But if the angle shall be obserued betweene the two outward sights the parts being 20 and 9 as before the angle will be 48 gr 28 m. double vnto the former In all these there is a regard to be had to the parallax of the eye and his height aboue the Horizon in obseruations at Sea to the Semidiameter of the Sun his parallax and refraction as in the vse of other staues And so this will be as much or more then that which hath been heretofore performed by the Crosse-staffe CHAP. IIII. The vse of the lines of equall parts ioyned with the lines of Chords THe lines of equall parts do serue also for protraction as may appeare by the former Diagrams but being ioyned with the lines of Chords which I place vpon one side of the Crosse they will farther serue for the protraction and resolution of right line triangles whereof I will giue one example in finding of a distance at two stations otherwise then in the second Cap. Let the distance required be AB At A the first statioÌ„ I make choise of a station line toward C and obserue the angle BAC by the tangent lines which may be 43 gr 20 m then hauing gon an hundred paces toward C I make my second station at D
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.