this Instrument which at this time I omit CHAP. XLVI Things belonging to the vse of the Plaine Table Things belonging to the plaine Table TO this instrument as to all other appertaines a chaine or wire line of foure pearches long according to 16. foote and ½ or of thrée pearches long which is 16. yards and ½ let the pearches be noted with brasse rings at the ends thereof and then diuided into halfes quarters The Line with lesser rings fixed at each quarter and halfe that you may distinguish the same A Scale Compasse You must also prouide a scale of brasse or wood whether you please with a paire of brasse compasses pointed with stéele very neate and sharpe for it is rude to draw your lines Geometricall with Painters kéelers or blacke lead as M. Lucar would Also you must haue such sights for this Table as bee described in the vse of the Circumferentor whose vse are set downe in the 16. Chapter of the same booke or else you may haue such a quadrant as is spoken of in the first part of my Art of Geodetia as in the 26. Chapter These things had you may fall to worke CHAP. XLVII To take any Horizontall distance by the Plaine Table To take any distance by the Plaine Table IT were vaine to make many demonstrations of this worke since a few may as well suffice for this Instrument is but only fit to take longitudes and latitudes as for altitudes I hold him very troublesome and vnapt to performe the same though M. Lucar haue taken paines to illustrate him in that point howbeit finding by experience the cumbersome and vncertaine working thereby I thinke it better omitted then remembred You shall then vnderstand that you may performe any distance vpon this Table in the same order as you doe with my Staffe onely heere you must drawe lines vpon the paper and measure the same by your scale whereas the legges of the Staffe represent the lines and the diuisions your scale Therefore at the place whence the distance is required to any marke proposed place your Table which place call your first station then your Table lying parallel with your Compasses make a point in the paper to represent that first station wherevnto bring the fiduciall edge of your rule kéeping the one end of the said ruler vpon the point mouing the other vntill through the groue or sights you espye the marke whose distance is required the rule so resting drawe a line by the fiduciall edge thereof the Table resting espye out a second station let it make as néere as you may a right angle with the marke whose distance is required This marke so appointed out for your second station kéep the fidutiall edge of the rule vpon the foresaid point and so draw a line to point to your second station then let one measure the distance betwixt your first and second station which were best to be 1 10 part of the distance required So haue you finished all at your first station with this Prouiso that you haue regard to the degrées cut by the South end of the Néedle in the Card in the bottome of the boxe before you any wise alter the table and that you lay downe your stationary line by your scale and compasses limiting the same according to the line measured at the end thereof marke another pricke which call the pricke of your second station Then take vp your Table leauing a marke at your first station vnder the pricke made vpon the table representing the same Now must you beare your instrument to your second station where hauing placed the same in such sort that the pricke of your second station may directly stand ouer the marke representing your second station lay then the edge of your ruler vpon the stationary line kéeping the pricke of your second station next to your body turning about the table the ruler resting as before til through the sight you espy the marke left at your first station which done make fast the table with your screw A proofe of the worke Now for proofe of the exactnesse of your worke and to know if you haue truly taken your backe sight haue respect to the south end of the néedle for if it cut the like dgrées at this second station aâ— at the first you haue done well Hauing so done place againe the fiduciall edge of the rule vpon this point of your second station the one end being there fixed moue the other end vntill through the sights you sée the marke whose distance is required then draw a line by the fiduciall edge of the rule which will intercept with the line drawne from your first station thereunto therefore note the point of intersection and by your scale measure the distance from any one point to the other I meane by the same scale you laid downe your stationary line so haue you your desire Example The distance a b is required first thererfore I plant my Table at b then working as before I finde c my second station and so draw a line to point from b to a and another from b to c. Next I measure the line b c and finde it 7â— yaâ—dâ— which I lay downe vpon my paper with my Scale and Compasses Lastly I note the degrees cut by the South end of the needle which let be 40. This done I go to c and there againe plant the Table as before So do I make the stationary line protracted point iust to b and then noting the degrees cut a gaine by the needle I finde theÌ„ 40. as before which argues I haue well planted my Table To concluÌ„de I place the fiduciall edge of my rule vpon c mouing â—he other end vntill it intersect with the line representing a b therefore by my Scale I measure the line representing b a so haue I the distance of b a 135. yards by the same Scale might you haue expressed c a. CHAP. XLVIII The part of the distance of any thing being giuen to finde the rest VNderstanding the last Chapter so wee may thereby auoid many words and may most easily be performed by the Geodeticall Staffe as may appeare in the Propositions of the 18.19 or 32. Chapters of the second booke of the Geodeticall Staffe But to procéed a b is a distance required the part of that distance giuen is a c 50. pearches Then do I plant my instrument at a as I did in the last chapter at my first station drawing a line to represent a b infinitely then laying downe my scale vpon the same line the part giuen representing a c 50. pearches the instrument vnremoued I séeke a second station as in the last chapter which is d but the stationary line shall not be measured Lastly I note the degrée cut by the South end of my néedle then leauing one at a I cary my Instrument to d where I plant him in an respects as at a now must I finde the
point vpon the paper which represented c and thereupon lay the fiduciall edge of the rule mouing the other end vntill through the sights you seâ— c so wilt the edge of the ruler in the line vpon the paper representing â— d thâ—â— kâ—â—ping the â—uler â—pon that point d I moue the other 〈◊〉 vntill it pâ—â—oâ— to 〈◊〉 shall the fiduciall edge of the rule intersectâ— witâ— the â—â—â—e 〈◊〉 the papâ—â— represenâ—â—ng a b from the point of which intersection to the point a is b the termes of the line a b which being measured by your scale and compasses is found 133 pearches CHAP. XLIX To take the distance of any two townes or such like COnsider well the premises and this labour is already effected therefore plant your Instrument at a Latitudes as you were directed in the 29 chapter and let the latitude required be d c noâ—â—raw lines from a to point to d and c and also to b your second station now obseruing the former directions I â—emoue my instrument to b and so draw lines from b to point againe to d and c then doe I note the concurse or intersection of the said lines which I measure by the scale and compasses as before so will the stationary line a b bee 316 perches and the distance required d c 131 perches A note for many distances And here note if you had sought more distances as the distance of f e e d d c c. the labour is no more but to draw lines at euery station to point â—nto the distances required and then to note the intersection of matchy lines vpon the paper which after measure by your scale and compasses so shall you haue your stationary line g h 10 score the distance f e 12 score c. whereby you may plat any field and come not within the same as in the 8 chapter CHAP. L. To finde the Horizontall distance of any place from you standing by a new way vpon the Plaine-Table To finde any Horizontall distance after a new way IN the 44 chap. at twice I told you of 4 certaine rulers or ribs that were belonging vnto the Plaine-Table euery one being diuided into a 100 equall parts or more by these rulers ordered in their due place vpon the Plaine-Table shall I teach you to séeke the Horozintall distance of any place thus Lay the ruler with the sights vpon the very edge of one of the sides of the Plaine-table turning the Table about vntill through the sights you espy the marke whose distance is required but with this prouiso that the corner of the Table where the diuision take begining be neerest vnto you this done take the ruler with the sights the Table vnmoned and place the same vpon the right side the Table as before and then looking through the sights espy your second station in a knowne distance from your first station Next shall you beare your Instrument to your second station situating the Instrument by helpe of the Needle and backe sights here in all respects as it was at the first which being done lay the ruler ouer the corner or both sides of the Instrument remoouing the same vntill through the sights you espy the marke whose distance is required lastly note the equall parts vpon the ribs cut by each end of the ruler or sight hauing regarde to those parts that doe responde to the statinary line and also to the distance required for as the parts respondent to the stationary line are to the line it selfe being measured and knowne so are the parts respondent to the distance vnto the distance required therefore worke by the golden rule in this worke the line of distance and stationary line alwiaes cut at right angles this needeth no example for as it is most exact so it is most plaine easy The premises béeing considered and the doctrine before well vnderstood you may produce infinite wates to performe many rare conclusions but we cannot stand to set downe a demonstration to suite to euery proposition that may happen in the field chiefly for that let the demand stand howit will you may resolue the same by due regarding the prescript Now I will briefly touch the order of taking a plat of a field mannor c. by the plain Table according as we haue dealt with the Geodeticall staffe and other instruments before ayming to performe some such propositions here that were omitted in the other bookes for it would increase the volume ouer much to set downe euery kind in the vse of euery instrument since wee understanding what is said of the one may also be performed in the other and that much after one kind of method as I haue said before but indeed I haue here set downe such propositions that will best agrée with the Plaine Table and are aptest to be wrought thereon setting aside all impertinent demonstrations And you shall note for diuers good respects that I shall omit one thing that standeth firme and is ordinarily vsed in demonstrations of this nature that is lines to represent the Instrument the lines also drawne thereupon my reason is because I will not confound the worke with multitude of lines as also to saue the cutting of many figures whereby such that serued in the Glasse likewise serue in the Plaine Table CHAP. LI. To draw the plat of a peece of ground at one station where all the angles of the field may be seene from that place of standing At one station to get a plat FIrst goe round about the field and in euery angle set vp some marke then plant your table couered with paper in such a place as from thence you may sée all the angles of the field that done in a place conuenient of your table make a pricke or point to represent the place of standing from the point to each marke draw a visuall line by the edge of your Ruler then from your place of standing measure exactly with your wire line the iust distance in pearches to each seuerall mark and set those distances by the scale each vpon his own line which was drawne to those markes noting these seuerall points where these measures end Lastly from point to point by the edge of your Ruler drawe lines which shall include a figure proportionall to the field to be measured and the lines so drawne shall represent the hedges of the field as in this demonstration Your station is i the lines drawne from i to point to euery angle are i a i b i c i d i k i e i f i g and i h which are measured as is noted vpon each line as i a 27 pearches i b 9 ¾ pearches c. then from a to b I drawe a line and so go round so haue I made a figure proportionall which was required CHAP. LII To drawe the plat of any field by the rule taught in the last Chapter where you cannot from one place of the field see
line from that point along the edge of the instrument then kéeping the edge still at that point moue the instrument vntill the South end of the néedle cut the degrées noted at your second obseruation then draw another line by the edge of your instrument whereupon lay the line measured betwixt both your stations counted from the point first made towards the end of the said line and where that number ends there make a point which let represent your second station where place the edge of your instrument turning him about vntill the south end of the néedle cut the degrées you noted at the second station then by the fiduciall edge of the instrument draw a line note where it intersecteth with the first line for that is the place of the marke whose distance is required the distance of which from either of your stations may you measure by the Scale that you expressed the length of your stationary line by CHAP. LXXI To take an altitude onely by the old Circumferentor To take an altitude YOu must first get the horizontall distance vnto the thing whose length is required then plant your instrument perpendicular and moue the vane vntill through the hole therein you sée the top or the summitie of the altitude note then the equall parts cut by the side of the vane for such proportion as they beare vnto 100. the like doth the altitude vnto the distance multiply therefore the distance by the parts cut and diuide by 100. the quotient she weth the height which is correspondent to the leuell of your eye The ground of this worke is borrowed from the Iacobs Staffe as may appeare in the ninth Chapter of the fifth booke of the Geodeticall Staffe An inconuenience like to that in the Theodelitus But in taking of altitudes you shall haue it oftentimes so fall out that the altitude will be so high that you cannot bring the vane so low as to sée the top of the altitude by the hole and pins head When it so happens you must place the center of the Index vpon the wire in the shorter sight looking through the sight hole in the Index vntill by the wire and through the said sight you sée the summitie of the altitude then note the equall parts cut by the fiduciall edge of the Index vpon the right edge of the instrument for as those parts are in proportion to 60. the like proportion hath the distance vnto the height And so that proportion as those parts cut haue to the parts cut in the Index the very proportion hath the distance to the visuall line Therefore multiply the horizontall distance by 60. and diuide by the parts cut on the right edge of the instrument the quotient will shew the height Againe multiply the horizontall distance by the parts cut in the Index and diuide the same by the parts cut in the edge of the instrument the quotient sheweth the visual or hipothenusall line As you séeke altitudes so must you sinde profundities as I haue said often in the Geodeticall Staffe but the errour is great if the instrument be not exact paralell CHAP. LXXIII To take the plat of a peece of ground by the old or new Circumferentor DIuers wayes may bee set downe to fetch the plat of a péece of ground by this instrument To measure woodland or any other ground but I hold that most easie which is to be protracted by the Instrument it selfe because you shall not bee troubled to séeke the quantitie of angles which in this Instrument is ouer tedious Hauing therefore a péece of ground giuen you shall begin at some one corner and there plant your Instrument looking vnto the next corner and note what degrée the south end of your néedle cuts then with a chaine measure from the first corner to the second and note downe the degrées cut by the south end of the néedle and the length of the line measured Next go to the second angle and there conuey your sight to the third angle paralell to the hedge then measure the distance from the second corner vnto the third noting downe the degrées cut by the south end of the néedle the length of the line at your second obseruation Then go vnto the third angle and note the degrées cut and the length of them and so procéede from angle to angle noting the degrées cut and the length of euery line answering thereunto vntill you haue gone round about And if you being at any one angle and from thence can sée two or three angles more you shall not néed to remooue your instrument to any of them but onely from that angle obserue all the rest onely measuring the hedges With these notes you shall resort vnto a faire shéete of paper and there protract it downe thus In some place of the paper make a point and there place the fiduciall edge of your Instrument turning it about vntill the south end of the néedle cut like degrées as he did at your first obseruation then drawe a line by the fiduciall edge of the instrument whereupon from the said point to wards the other end lay downe the length of the first measured line which you must take with your compasse from your scale where that number ends in the said line there make a point where place the edge of your instrument mouing him about vntill the South end of your néedle cut like degrées hee did at your second obseruation then drawe a line by the fiduciall edge thereof whereupon lay the length of your second line and where that number ends make a point where as before place the edge of your instrument mouing him vntill the South end of the néedle cut like parts hee did at the third obseruation then drawe a line by the edge thereof whereupon lay the third line and where that number ends make a point as before placing there the edge of your Instrument turning him vntill the South end of the néedle cut like parts as at the fourth obseruation and so procéed laying downe the parts cut and the length of the lines vntill you haue gone round about by which meanes you shall lay downe the plat of the péece of ground in true forme then for the casting vp thereof resort vnto my booke of the art of measuring ground CHAP. LXXIIII To take a plat at one station from whence you you may see all the angles in the field by the old or new Circumferentor To take a plat at one station THis kinde of working is performed with as much ease as the former You shall therefore repaire into the field and finde some such place from whence you may behold all the corners in the said field where plant your instrument and then begin at some one angle whereunto direct your sight noting the degrées cut by the South end of the néedle then direct your sight vnto the second corner vpon the right hand and there againe note the
your Table made fast from the point of your standing draw a line into the next angle and measure the distance thither which measure set on the line drawne and then plant your Table againe in your third angle and in this order worke till you haue compassed the whole ground and if it fall out in the conclusion of your worke that the line and angle of your figure agrée with the line angle of the field then is your plat perfect if not you haue some errour And herein if I may aduise you begin your worke againe to finde your fault trust not to any helpe for the closing thereof wherein you shall but deceiue your selfe and happly of a small errour increase a greater for that you know not whether your saulte be in the lines or in the angles wherefore if your figures misse of closing aboue one pearche neuer trust vpon your worke And be sure when you plant your Instrument in one angle looke to the next that you so direct the sights that the visuall lines lye paralell to the hedge measured neither obseruing this paralelly is it materiall how far off the perimeter you place the Instrument alwaies prouided that you take your measure in the true direct place where the very hedge or bounds go for if you measure much within the hedge your lines fall too short if without too long therefore obserue the meane for in all things obseruare decorum is best This kind of measuring is principally and commonly vsed for woods for that in them a man cannot see the angles by any other me anes and may serue for all kind of other grounds and indéed commonly vsed of land meaters who hauing but this one Chapter and that rawly presume of the full knowledge of the vse of this instrument but how they performe it I leaue to those that shall try them which is but had as others before me haue reported CHAP. LVIII To draw the Plat of a field by placing the Instrument in euery angle thereof as in the last Proposition and yet measuring but one line in the whole Perimeter To draw the plâ—t of any field by going round about yet me asuring but one line FIrst place your instrument in that angle where you will begin which let be in such a place that the first line you go vpon may he of reasenable length then set vp a marke in the field in such a place as from thence you may see as many angles of the field as possible may bee which marke call your principall point and to that marke get the distance by your first and second station which let be in the first second angle in the field in order as before is set downe so shall you haue a point in your Table to represent that principall point in the field This done draw a line into the third angle and thereto remoue your instrument and hauing there placed it get the distance betwixt your two stations as you got your distance in the like case before which you shall performe thus Hauing first placed your Instrument by looking backe or by the néedle at his due situation lay your ruler either by the pricke on the paper that representeth the principall point in the field or by any point on the paper that you know representeth some marke in the field then turne your ruler about till you sée that marke in the field which the pricke by which your ruler lyeth doth represent and draw a line till it cut your stationary line and that point of interception sheweth the point on the paper where you stand in the field So in this order by placing your instrument in euery angle you may get the length of euery hedge seuerally with measuring but one in the whole and the conclusion will bee that in the end you shall make a figure with equall angles and lines proportionall to the plat of the field The premises bâ—â—ig well vnderstood and all things else well considered I will â—raue pardon and so cease further prosecution hereof presuming that there is sufficient said to open the whose scope of this chapter neither would I go about to fill the booke with many curious demonstrations and difficult questions to beguile the aspiring wit of the yong practitioner but onely set downe some such fewe things that were most requisite to bee knowne lest otherwise I should be held rather tedious then compendious and therefore I will hast to an end CHAP. LIX To take the plat of any Champion Field containing 2000. or 3000. Acres of ground by the plaine Table and yet neuer bee forced to change your paper YEt againe before I conclude I will giue you another way to séeke the plat of great Champion fields that containe 3000. or 2000. Acres by the Plaine Table which is not much differing from your worke by the Geodeticall Staffe You shall therefere place your Instrument in euery angle and so get euery angle and his sides To vse the plain Table and neuer change the paper not regarding the length of the conteining sides as you be wont then must you measure euery hedge and as you were wont to lay the same downe by your scale and compasse heere you shall but write the length of euery hedge vpon the lines drawne vpon your paper and responding thereunto so haue you finished and you shall neuer be forced to shift your paper nor haue the lines to runne off the same for that you may draw them as long or as short as you please Now when you come home vpon some shéete of paper protract all the angles one after another See the 3. chap. pro. 5. as you found them in the field allowing by your Scale and Compasse euery line his due length according as you finde the same note those figures vpon the said respondent lines and your conclusion will be to produce a figure like and proportionall to the field proposed This chapter is most excellent for the purpose before said and therefore worthy of note as they shall finde it that worke by the Plaine Table in countries that consist of great Champion fields CHAP. LX. What Chapter is most fit to vse in platting of ground as well such whose superficies is subiect to sight as others that be rough and full of wood as also to make choyce of the best instrument to performe the same as also to make a new kinde of particular YOu bee taught before to measure and plat any péece of ground whatsoeuer it resteth then for you to make choyce of such Propositions that are best as well in respect of the fashion of the field as in respect of the aptnesse of the Proposition Therefore for all grounds whose bounds and angles may all bée séene from one place vse the 51. Chapter and if you cannot trauers the same to measure it with your chaine by reason of pooles marshes or such like then is the 53. Chapter excellent as for the
the setting vpon the West side and the rising vpon the East side of the Card. So shall you find the 11 day of May the Sunne to rise néere 4 and set néere 8 then if you would knowe the length of the day and night you may repaire vnto the second Booke Chap. 10. of the Geodeticall Staffe CHAP. LXVII To find the amplitude of the rising of the Sunne and Starres To find the amplitude of the Sunne or stars IT is not vnknowne to any man tho meanely traueld in Astronomie that euery Horizon hath foure principall points viz. East West North and South then you must vnderstand that there is no starre nor the sunne that riseth iust East or setteth iust West vnlesse they be in the Equinoctiall which happeneth vnto the sunne but twice in the whole yeare but for starres if they rise once East or set West so doe they alwaies whereof there be but a few the starre in the pinion of the left wing of the Virgin the starre in Antinous left arme c. come néere thereunto but as the amplitude of a starre obserued one day is certaine and all one in any other day for that latitude so in the sunne doth it differ euery day and is called Amplitudo ortus This had Obserue the sunne or starre when they seeme as it were to touch the earth as béeing at point of rising or setting wherevnto turne the Instrument vntill through the slit in the longer sight and by the pinnes head you sée the sunne or starre then note the degr cut If you sought the setting multiply the degr cut by the West end in 3 which take from 90 so haue you your desire so the degr were vnder 30 but if the degr cut be aboue 30 multiply the degr cut by the East end in 3 then from the totall take 90 so haue you your desire and the setting shall be North from the Equinoctiall But if you séeke a rising you must consider whether the degrées cut by the East end be vnder 30 or aboue if they be vnder 30 multiply them by 3 so haue you your demand and it is North if they be aboue 30 sée what degrées the South end cuts which multiply by 3 substract from 90 to haue you your desire and the rising is South from the Equinoctiall Or thus with more ease hauing made your obseruation sée how many degrées are contained betwixt the West point of the Card and the South end of the Néedle for arising but for a setting sée how many degrées be included betwixt the East point of the Card and the South end of the Néedle which treble so haue you your desire But this Chapter is performed with farre more ease truth by my Topographicall Glasse CHAP. LXVIII Of the opposite degrees and how to find them Opposite degrees BY an opposite degrée is meant the opposite point of a Diameter or the point opposite vnto the degr cut by the South end of the Néedle that is the degr which the North end should fall vpon which is alwaies the halfe of a circle distant from the South end in this instrument 60 degrées so that if the degrées be lesse then 60 adde 60 thereunto but if more then 60 substract 60 from it and the total of the â—ne or the remainder of the other is your desire This néedeth no example CHAP. LXIX To find the quantitie of an Angle To find the quantitie of an angle THe quantitie of an angle is the portion of a circle included betwixt the 2 sides of any angle which is found vpon this instrument by the cutting of the Néedle at two obseruations in one place the lesser of which must be taken from the greater and the degrées which remain after substraction is the quantitie thereof But if the remainder after substraction excéed 60 then must the said remainder be taken from 120 so haue you the quantitie if your degrées be not direct then must you worke by the opposite degrees as in the 9 Chapter taking the lesser of those degrées from the greater And you must here note that all degrées cut at diuers obseruations in one or more places be called direct And such degrées as be opposite vnto direct degrées be called indirect and here note the tediousnesse of taking an angle by this instrument inrespect of my Staffe CHAP. LXX To take the distance of any marke by the old Circumferentor To take a distance AS I haue often times said in the 2 Booke of the Geodeticall staffe that there must be 3 things giuen as 2 lines and one angle or 2 angles and one line by which all dimensions are performed so in this kind of working you must alwaies haue two angles and one line giuen by helpe of which you may séeke any distance proposed thus Plant your instrument at the place appointed whence you desire the distance and there looking towards the said marke note the deg cut by the South end of the Néedle then appoint another place for your second station to which bring the sight as before noting the degr cut that done measure the distance betwixt the place you then stand at and the place appointed for your second station there againe plant your instrument looking through the fights vnto the marke whose distance is required then note the degr cut and so get the quantitie of both the angles as in the last Chapter When you haue gotten these two angles adde them both together which take from 60 so haue you the quantity of the angle at your marke then must you resort vnto the table of signes placed in the Instrument and there â—ind the signe of euery angle and note it downe and if the quantity of the angle excéed 30 substract the excesse or ouerplus from 30 and take the signe of the remainder These signes had and noted downe worke by the golden rule wherein the signe of the angle at the marke must be the first number the measure betwixt the two stations the second number and the signe of the other angles seuerally the third number according to the side which is sought and this worke is grounded vpon this Chapter In all right-lined Triangles the proportion of the one side vnto the other is such as the signe representing the angles be Or more briefe See the 7 Booke Axioma 2. of the Geo. St. The sides of opposite angles bee direct proportionall to their signes CHAP. LXXI To performe the last Chap. by protracting with the old or new Circumferentor To take a distance HAuing made your obseruations at each station note downe the degr cut by the South end of the needle and then protract thus Take a faire sheet of paper and fasten the same vpon a Plaine Table or such like with mouth glewe then shall you make a point vpon your paper to represent your first station there keepe the side of your instrument turning him vntill the needle cut the degree first noted then draw a
degrées cut by the South end of the néedle which note downe and so procéed rightwards from angle to angle noting the degrées cut by the South end You are taught this chapter with a demonstration lib. 6. cap. 3. of the Geodeticall staffe vntill you haue gone round about the field of which degrées cut you shall make a little Table to the end you may remember how many degrées were cut at the first second third c. corner Next shall you cause one to mete with a chaine the true distance of the first corner from your staffe which note downe against the the first degrée cut in your Table then mete the distance of the second corner from your instrument which note downe in your Table against the number of degrées cut at the second corner and thus procéed vntill you haue gone round about the field laying downe the distance of euery angle from your instrument against his proper degrée cut which done fall to protracting thus Hauing prepared a faire shéet of paper as you be taught before about the middest thereof make a point which call your station then apply the edge of your instrument thereunto mouing him about vntill the South end of the néedle cut the degrées you noted at the first corner which done draw a line by the edge of the instrument from the point made in the paper out at length then moue him rightwards vntill the South end of the néedle cut the degrées noted at the second corner and then by the edge of the instrument draw another line as before so go forward vntill you haue finished all the degrées cut by the south end of your néedle noted in your Table then with your compasse take from your scale the distance of the first angle from your instrument which lay in the line first drawne from the point made in the paper towards the other end of the line then take the distance of the second corner from your instrument which apply to the second line drawne in the paper and so procéed from line to line according as you be taught in the third chapter of the Art of measuring ground The length of euery line laid downe in such order as is said then must you draw lines from point to point in each line so shall you drawe the limits and proportion of the ground according as in the foresaid third chapter of the art of measuring ground by the Staffe And by this meanes may you measure ground at two stations measuring but one line in the whole plat in such order as I set downe in the fourth chapter of the sixth booke of the Geodetical Staffe And since what is said before may giue sufficient light to performe both this way and many other I will omit further spéech least I rather seeme tedious to the wise then facile to the vnlearned And you shall heere note that by taking perfect notes in the field where one closse boundeth vpon another you may take the plat of many flelds lying together and so saue a great labour CHAP. LXXIIII The degrees of a field being taken to finde whether the plat will close the lines being truly taken To know if your plat will close NOte downe the quantity of euery angle at each seuerall station as well as you doe the degrées cut then adde vp all the quantities together then multiply 60. by a number lesse by 2 then the number of the angles and if your worke be right the product thereof shall be equall to the totall of the quantities Example Let the number of angles be 8. froÌ„ which take 6. which is a lesse then multiply 60. by 6. and the product will be 360. which agrâ—â—ing with the totall of all your quantities of angles added together is one argument that the plat will close CHAP. LXXV To reduce Hipothenusall lines vnto Horizontall after another way then in the 6 booke 8 Chapter of the Geodeticall Staffe onely by the sights in the old Circumferentor To reduce Hypothenusall lines to Horizontal lines PRepare a marke to bee carried before you the just height of your line of leuel from the ground when the iustrument is planted vpon his rest this marke must be placed in the angle whereunto you looke hee must stand perpendicular and when you take the degrée looke your instrument stand perpendicular and then moue the vane vpon the sight vntill you sée the top of the marke before planted through the hole in the vane and by the pins head then in the Hypothenusall diuisions cut by the vane vpon the sight for they will shew you how much that line you shall measure will differ vpon the 100 from that line you should measure if the ground were leuel therefore when you haue measured that line proportion him according to the parts cut Example Suppose the parts of the Hipothenusall diuisions cut to be 4 and the line measured to be 30 pearches now you are to finde a number to beare like proportion to 30 as 100 beareth to 104 which you shall find to be 28 1 5 1 1 so that the line measured by the cheine to be 30 pearches must be laid downe 28 1 1 1 3 pearches in your protracting But for asmuch as these calculations be tedious in the field your best way is to note the Hipothenusall parts cut and then reduce them when you come home CHAP. LXXVII To performe the same by a Quadrant made of purpose A new Quadrant to proportion lines YOu shal prepare a Quadrant and then diuide the limbe thereof in 30 equall degrees setting number therupon as the coÌ„mon order is then shall you diuide the lower side of the Quadrant the is betwixt the first degrée the center into 30 equall parts raising perpendicular lines vpon each diuision which will be paralell vnto the other side this done prepare an Index of the length of the semidiameter of the Quadrant with a center hole therein this Index is to be fastened to the center of the Quadrant with a brasse pin or such like which also must be diuided into 30 such equall parts as the semidiameter was the Quadrant thus prepared you shall fore shorten the lines thus First for the taking of your notes in the field you must work as in the last Chapter onely here you must note the degrée of a circle cut by the vane in stéed of the Hypothenusall diuisions and then procéed thus Put the Index to the different angles in the limbe then number the line measured vpon the Index and note the perpendicular there cut by the edge of the Index for that shall shew you the length of the Horizontall line which must be protracted Example Let the different angles from the Horizon be taken 18 degr and the line measured 20 perches first count 18 degrees in the limbe then thereunto bring the edge of the Index next count the line measured viz. 20. pearches vpon the Index fromwards the center so shall you there
CHAP. V. To find the true square of a squared peece of timber consisting of two vnequall sides and 4 right angles the one side being onely knowne YOu must take the length of the broader of the two sides the which fit ouer in 60 and 60 amongst the coard diuisions the legges of the Staffe so resting the distance taken from 36 to 36 yeelds the true square of a peece of timber that béeing of equall longitude is also of equall quantitie But if both the sides c d and d b be knowne then worke by the next Chapter for this takes no notice of the thicknesse CHAP. VI. To find the square of any broad or flat peece of timber that consists of 4 right angles and two equall sides SVch a péece of timber as this the end thereof doth represent the iust forme of an Oblong and is thus squared take the longer and shorter side and ioyne them together in one right line the which right line made of the length of both these lines so ioyned make the diameter of a circle lastly vpon the point where the two lines were ioyned raise a perpendicular forthe length of that perpendicular to the circumference is the true side of the square Example A b c d is the end of a peece of timber c d the longer side d b the shorter therefore I take the length of c d and d b and ioynt them together in v making one right line thereof as r s next I part r s into two equall parts at w then placing the one foote of my compasse in w extending the other to s or r t describe the semicircle r t s lastly vpon v where the two lines were ioyned together raise a perpendicular v t which is equal vnto p q I conclude the square made of the line t v is equall to c d b a and thus of any such other Or get the square thus multiply the breadth in the thickenesse so is the square roote of the product the true square which you may easily find in the Geodeticall Staffe fol. 142. CHAP. VII To find the true square of any peece of timber whose ends are formed like a Diamond THe end of such a peece of timber as this doth represent the iust forme of a Rombus therefore doth consist of equall sides and Oblique angles the square whereof find thus Drawe a right line betwixt any of the two opposite angles noting the length of that line vpon which line let fall a plumb line from one of the subtended angles so hauing those two lines find the square as in the last Chapter Example Or the length of the perpendicular b m is the square falling vpon b c at right angles CHAP. VIII To find the square of any peece of timber consisting of three sides THe true square of all kind of Triangles whatsoeuer are found out by the 44 Chap. Metamorphosis 7. of the 6. Booke of the Geodeticall Staffe and therefore it were vaine to repeat it here againe As if you ioyne the perpendicular b d and halfe the base a c in one right line according to the 6 Chap. you shal find the square of the triangle a b c to be h i. CHAP. IX To find the square of any peece of timber containing 5 6 7 or 8 sides c. YOu must imagine in this Chapter as also in all the other sauing for round timber that I goe not about to tell you how much square that peece of timber would beare if it were reduced into a 4 square but I doe deliuer you the side of a péece of timber béeing iust foure square and of equall height with the peece proposed shall also be of equall quantitie which is right necessarie for the attaining of the number of square feet or solide content of any péece of timber Thus are you taught to find the square of any peece of timber of what fashion soeuer and if it beare none of these regular formes or that there be wood wanting take from one place adding the same vnto another thereby making it perfect regular and in such cases you must alwaies so doe CHAP. X. The square of any peece of tymber being found to tell how much of the same in length will make a square foot of tymber and consequently how many foote is in the whole peece A Solid or Cubicall foote of tymber doth containe 1728 cubicall inches for so many foure square inches may be taken out of one cubicall foot I meane such Inches that are square euery way like vnto a dye now hauing the square of any péece of tymber giuen square the same diuiding 1728 by the product so doth the quotient shew you how much of the length of the tymber must be taken to make a square foote by the which diuide the whole length or altitude of the tymber so doth the quotient acquaint you how many foote of tymber is in the péece Example The square of the tree a b is found by the third Chapter to be 27 inches whose square is 719 by which diuide 1728 so haue you 2 2 7 9 2 0 8 inches which is 2 ⅜ inches or two inches a quarter halfe a quarter whereby I conclude that as often as I can finde 2 ⅜ inches in the length or height of the tree or tymber so many square foote of tymber is in the same the tree a b is 8 foote high which diuide by 2 ⅜ inches or lay somuch of your rule out measuring one from a towards b calling euery 2 ⅜ inches a foote so by either of the waies shall you finde 40 foote of tymber in the said tree being squared some small quantity being ouer more then the same In the like manner must you deale with all other peeces of timber of what fashion soeuer first finding their square as before next the solide capacity euen as you be taught in this chapter By this Chapter may you measure out as many foote of tymber stone or such like as you please thereby cut off any number of feet from any peece of tymber as you shall be occasioned CHAP. XI To measure all kind of Tymber c. after another sort without regarde of the square THis kind of measure taketh no regard to the square feete in the tymber but vnto the solid capacity thereof but for that it is not much pertinent to the Geodeticall Staffe requiring rather numerall then instrumentall operation I will be the more briefe When you haue any péece of tymber stone pillar or such like whose solid content is required by the rules taught in the sixt book part 2 of my Geodeticall Staffe seeke the superficiall contents of the end of the tymber the which augment in the altitude or length of the same so is the product your desire Example The like must you do with any other peece of tymber of what fashion soeuer but if the tymber trapeze much vse the middest or difference of the ends as in the 4 Chapter
Diameter thereof standing paralel with the fiduciall edge of the Index g and f. 9 In the middest of this Demicircle vnder 12 in the Hysometricall Scale may you fixe a plumbe and ouer the Bore with the Needle a certaine point iust vnder 12 which will serue to keepe the Instrument paralel and vpright which the crosse Needle will as wel doe but both are not amisse the old song is Two strings are good to one bowe Some reasonable distance as an inch and better drawe a line b m paralel to h g whereunto drawe 11 paralels at such distance as they be in the figure wherein must be placed the degr figures and characters of the 12 Signes 〈◊〉 as in the demonstration they be euery paralel is diuided as g k is by placing the one foote of your compasse in f and so fetching each degree from the line g f to the other paralel and at the ending of euery third degree the line is strooke quite through so that there be two lines paralel to g m and k l strooke quite through and these lines doe limit the beginning and ending of euery signe You must also note the South and North signes at the head as here they be In the very point h there is the ordinary fight placed such as be in Quadrants so is the graduating of this sight finished This fore peece of the moueable Sight so finished there must be another peece of like quantity soldred thereunto or lâ—ft growing vnto the same peece and after bended in such sort that it may claspe ouer the Demicircle so doe these two peeces hold the said Demicircle straitely betwixt the same that it may mooue straitly and equally along the same in so much that the arch h b will alwaies bee carried vpon the Circumference b c in the Demi-circle C g and g h doe represent the distance of the two peeces one from the other which is the iust thickenesse of the Demicircle Neither would it be amisse to haue a small screw pinne vpon the backe or further side of this moueable Sight which would make the said Sight mooue the more steddie 10 The next thing pertinent vnto this Instrument is a Boxe to hold the Needle The Boxe The Circumference of this Boxe must agree with the circle p q in the great figure for within that ciccle must he stand the Diameter whereof must be 3 inches and in this Boxe must be placed a Needle and a glasse as the order is and the Card in the bottome which I described before 120 standing in the South 60 in the North 90 in the East and 30 in the West About this Boxe must moue the Circle that beares 4 Indexes with the Sights the which Boxe must be turned with certaine shoulderings to come halfe a quarter of an inch vpon the said circle to the end that it may keepe the same downe close to the body of the instrument and that he may mooue stedfastly about This Boxe is to be fastened through the backe side of the body of the Instrument with screw pins so may he be taken off at pleasure the two screw pins that screw on the socket vpon the backe side may also screw this Boxe by fastening a rib of Brasse vpon the bottome of the boxe with screw holes answering to the holes in the socket Vpon the Boxe aboue the glasse stands a certaine crooked wire bearing a rounâ— knob in the middest iust ouer the areltree that beares the Needle and iust vnder the plumbe when the Instrument stands vpright The Needle 11 The next thing is a Needle which must be prouided in manner following As for the Needle I would haue it made like two Needles ioyned together at right angles as you may see in the ensuing figure and you shall flâ—d it hereby more true and apt to worke then the single Needle is for it will keepe the instrument paralel and vpright without the helpe of the plumbe cut the degrée more precisely and stand more directly Now this needle must be touched with a Lord stone and it is very requisite that the said stone be good therefore make choyse of one thus The best stones be those that come from the coasts of China and Bengalia the colour whereof is like to yron or somewhat sanguine if they be right they will drawe vp their owne weight they be heauier then other there is another neere as good which commeth from Arabia they be broad like a tyle-stone and somewhat red coloured If the Magnes stone haue lost his vertue throwe it into the fire and let it lie there vntill it be neere red hot and then quench it in the oyle of Crocus Martis so shall his power bee multiplied Your stone thus ordered you shal make cleane the North end of your needle and rub the very end thereof with the stone this preconsidered that the north point of the stone touching the needle causeth that end touched to point into the South so contrariwise the end touched with the South part turneth into the North so that you must haue a care in this point After you haue touched the end of the needle if it were equiballanced before you shal find the same end to hang downwards as it were the heauier whereby the vnskilfull spoyle many needles and this is called the Declination of the needle vnder the Horizon therefore let the end that shall not be touched be the heauier before you vse the stone and after the application of the stone if it be too heauie you may amend the same The needle so touched the South end thereof will not poynt iust into the South Magneticall meridian for that the Magneticall meridian whereto the needle poynts and the common meridian wherein the iust South stands differ for the Magneticall meridian is a great circle as the other is and also passing by the Zenith diuiding the Horizon into two equall parts the intersection of which meridian with the Horizon is the point whereunto the needle turneth which is called the Variation of the needle The variation of the Needle and at London is one point of the compasse or 11. degr and 15 minuts west from our common meridian and this is the cause that in all portable sun Dials the line which the needle standeth ouer doth not point iust vnto the 12 of clocke marke nor lie vnder our common ineridian Lastly prepare a hollow socket of brasse with a screw pin the socket to be scrwed on as the order is so is your Instrument finished onely prouiding a Staffe for the same Portable Dyals the thrée footed staffe is best to place it at all heights and in all places And one speciall note you most here obserue in the delineating of this instrument that is tâ— haue a care that the body of the instrument be iust foure squarâ— and that the sides of the square lye parallel to the diameter of the circle that is diuided into 360. degrées viz. that
 Montgomery 52 ½  Shrewsbury 81  Knookin Castle 92  Morton Corbet 136 Stationary Angle is 213 ½ Whit Church 147 ½  Bridg-north 319  Beawdley C. Wigor 319  Tenbury 344  Browne-clee hill 349  Hopton 354 With these tables repaire vnto some such place whereon you would protract the worke drawing therein a circle vpon the center or point f as you see in the figure which you must diuide into 360 degrees or else by a protractor from f pul out right lines by euery grade noted in the first Table so is f p Hopton Câ—stle f e Montgomery f d Knookin Castle and so forth with the rest ending at f m Ludlow Lastly in this circle I draw the line f g by 213 ½ degrees then making g a center I describe an other such circle as before and note the larger the circle is the better it is I did vpon f and from this center g pull straight lines by the degree noted in the second Table Now note the intersection of matchy lines that is where the line of Ludlow issuing from f meeteth with the line of Ludlow running from g there make a marke thus ☉ thus prosecuting the like in the rest alwayes setting a marke vpon the concurse of correspondent right lines all other intersections not respected I haue situated all these places in due proportion noting them with these letters to auoyde here and else where often repetition of their names And now lastly to get the distance betweene euery of them diuide the line f g into 9 equall parts for so many miles by mensuration I finde betweene my two stations the Cordocke for the Wrekin then by my compasse I see how many of these 9 parts is conteined betwixt any two places whose distance is required so many miles may you conclude the distance of those two places If I haue described places both without the County of Salop as Montgomery and Bewdley and without the compasse of our Horizon as Whitchurch c. They were set downe because you should haue plenty of examples not thrust together Heere followeth for more liuelinesse the distance of euery place in this mappe from the towne of Salop the rest you may gather by your Scale in the same manner   miles Bridgnorth AH is distaÌ„t froÌ„ Shrewsbury 10¾ Bewdley AI  22 Browne-clee hil  11½ Tenbury AK  20¾ Hopton AL  15¾ Ludlow AM  28½   miles Hopton Cast AP distant froÌ„ Shrewsbury 15 Montgomery AE  12 Knookin AD  8 Whit Church AC  12½T Mort. Corbet AB  6 OrletoÌ„ Bish C.  10¾ In the order before set downe changing your stations as hauing finished all in viewe from the Cordocke and Wrekin you may goe to the Browne Clee and Stilterstone hill or any other and passing from one loftie place to another you may haue the true proportion of all Townes Castles Riuers Hilles and such like in the whole kingdome and to reduce them all into the body of one Card or Mappe you must seeke a scale proportionable to the quantitie of the paper you will drawe the map in which here for that I feare I haue beene ouer-tedious I will omit and for that it shall be taught in the Flowers of the Mathem in my 2 part of Geodetia not yet published and elsewhere is performed CHAP. XII To drawe the plat of any Region and thereby to find the distance of Townes and such like by sinicall supputation THis kind of worke To seek the proportion of countries by sinicall supputation although it be something more tedious and difficult then the former yet hath it in it selfe a most exact and certaine operation you must in performance hereof ascend the top of some high mountaine hill or such like whence you may directly behold all the adiacent townes within the circuit of that Horizon and also from that hill espie some other mountaine to whose sumunity the view of all the foresaid adiacent townes be subiect This so done make the first hill a center and the other a terme of one of the sides of euery angle and so with your Instrument by the 25 Chap. or any other Instrument take the true quantity of the angle that euery towne maketh with these two hils and note the same downe in some Table booke this so done get to the next hill and there againe obserue in like manner the quantitie of euery angle euen vpon this hill as you did vpon the fornier finally get the true distance betwixt the top of the two hils so haue you a line knowne and two angles knowne situate at the ends of a line knowne whereby get the other angle with the two lines vnknowne and then place euery towne in his due place as you shall be better taught in the Example Example Suppose I ascending to the top of Stretton hils which be certaine loftie mouÌ„tains in Salop might view al the adiacent towns set downe in the ensuing map and withall another hill called the Wrekin from whose top also I might well command the viewe of all the foresaid townes Now first I place my Topographicall Glasse at a and then viewing round about I see my eye apprehends Shrewsbury situate vpon the left hand therefore I obserue the angle g a b by the 25 Chap. and so I proceed to Oswestree taking the angle f a b and so proceed round about noting the quantitie of each seuerall angle as followeth a b being alwaies the one side    Grad Mi.  Shrewsbury GAB 46 0  Oswestree FAB 74 0 Angles obserued at Stretton hilles Welsh-pole EAB 108 0  Mont-gomery DAB 124 0  Clun CAB 168 0 These angles so obserued and noted I beare my Glasse to the Wrekin hill where planting the same making the first degree in the Periphere of the Planisphere point iust to a the Instrument so fixed viewing about I espie Clun to which I make the Alhidada point and so by the said 25 Chap. get the angle c b a 4 deg in like manner I proceede right-wards vntill I haue finished as I did at b and thereby doe I collect a table as followeth    Deg. Mi.  Clun CBA 4 0  Mont-gomery DBA 24 15 Angles obserued vpon the Wrekin Welsh-pole EBA 38 0  Shrewsbury GBA 64 0  Oswestree FBA 74 0 Now must I get the distance betwixt the hilles of Stretton and the Wrekin which you shall find to bee ten miles all these things had I get the distance of euery towne and place the same accordingly in the map thus SALOPIA To seeke the distance of townes sinically Suppose we would find how farre Shrewsbury and each o other townes is distant from the Wrekin or from Stretton hills by the former obseruations the angle g a b is 46 degrees g b a 60 degrees therfore by the 2 Booke Chap. 15 of the Geodeticall Staffe adde 46
Chapter I deduct 20. from 43. so haue I 23. remaining by which your line stationary a c 129. yards must be augmented so haue I 2967. which I part by 60. so is my quotient 49 2 6 9 6 yardes the length of b z. And if your station were at d and the one section made at n falling out in the right side o s and the other at w in the left side s p and you required to deliuer the said distance b z then must you work according vnto the 3. difference of this Chapter I did prosecute the first difference with an exaÌ„ple for that neither of the sections will be made in the right parts vnlesse you stand neerer to the base of the altitude then the length of the altitude it selfe that is vnlesse the altitude bee greater then your distance from the base Otherwise Get the three angles of the triangle b d c thus obserue the angle b d a which take from 90. for that b a d is a right angle so haue you a b d then take the angle b d c which adde to a b d taking the totall from 180 so haue you the angle b c d by the 7.60 of the Geodeticall Staffe chap. 1. p. 21. This had get the line c a or a d then protract as in the 28. chapter or work as in the 32. chapter Comp. 3. in the end thereof or as in my 7. booke of Trigonometria CHAP. XXIII To know how much one Hill or Mountaine doth exceed an other in height THis matter is not so easily performed as many ordinarily thinke it to be for to seeks how much one hill is higher then another is not to stand vpon the top of one of the hils and by your Instrument finde whether the other bee higher or lower then the leuell of your eye as in the 19. Chapter and so to iudge him higher or lower then the hill you be one But hee that will know how much one hill doth excéede another in height must finde how much both of the summities of each hil is distant from the center of the earth or at least how much either of their perpendicular altitudes excéede the Semidiameter of the earth for the lesser of this excesse taken from the greater leaueth the difference of the mountaines perpendicular altitude For you must imagine what hill soeuer you stand vpon beholding another adiacent mountaine 20. or 30. miles distant which albeit that hill you behold be equall in height with that whereon you stand yet shal he not séeme so nor fall out to be so being proued with an Instrument or line of leuell for that where you stand is alwaies the vppermost place the other hill being situate as it were vpon the side of the earth which we may proue by a Philosophical Axiome Omne graue fertur deorsum ad centrum vbi quiescit insomuch that if you trauell about the earth with a line and plumbe at the end thereof the plumbe will alwaies point towards the center so that the excesse of any mountaines altitude aboue the superficiall conuexity of the earth is altered in respect of our sight according to the position of the place and therefore when you bee asked concerning the height of two hils you must know whether they meane in respect of the position of the hils according to the apprehension of your sight or in respect of the swelling of the same aboue the true conuexity of the earth for you must vnderstand that the earth is imagined to be round as a globe and so by some it is thought that it was at the first creation and that these mountaines and hils were since made at Noahs flood by the raging of the water which forced stones trées and earth vpon diuers heapes and thereby did irregulate the globous body of the earth the which albeit being compared to the Spheares in in the heauens by the consent of most Philosophers is but as a point hauing no sensible magnitude yet to vs that inhabite vpon the superficies of the same the very hils haue an apparant and great magnitude eleuating themselues aboue the true circuit of the earth and as these hills are much aboue the true superficies of the earths circular conuexily so is it not to be doubted but that many valleyes fall within and lower then the said circular conuexity so that sometimes we may bee distant more then the earths semidiameter from the center as being vpon a mountaine and sometimes lesse as in some kinde of déepe valley all which you shall bée taught to â—â—nde and proue by the ensuing demonstration To performe what is said before you must ascend vnto the top of one of the proposed hils which let bee b let the other hill bee c and you desired to tell which of the top of the hilles is the higher that is whether b or c be furthest distant from a the center of the earth Being therefore at b by some Instrument described in this booke get the angle c b a 87 degrées then must you go to the hill c and there againe get the angle b c a 74. degrées and so adding these two angles together you haue 161. which taken from 180. leaueth 19. the angle c a b made at the center of the earth Hauing these thrée angles get their signes and finally the distance of the two hils c b so haue you a line known and thrée angles knowne and thereby as you be often taught get the side a c and a b which note for looke which is the greater and that hill may you conclude the higher which heere is c a. Now to get the height of b or c aboue the true circular conuexity of the earth subtract 34364 11. from the liue b a so haue you the height of the hill b d foure score doe so to c a so for example sake haue you the height of the hill c e seuen score aboue the true circular conuexity of the earth If you seeke how much the one hill is higher then the other then take b d 4 from c e 7 so haue you thrée score and so much is the hill c e higher then b d. And here it is apparent that if you stand at b and by a line of leuell looke towards c your sight will run to f aboue c so that the lower hill by this meanes will séeme the higher for you must note that euery line of leuell doth make right angles with the perpendicular and euery perpendicular pointeth to the center of the earth as you may perceiue in the figure for b a is a perpendicular making right angles with f b the line of â—euell On the other side standing at c you shall sée the line of leuell c g run farre aboue b and by this meanes you may seeke the altitude and difference of altitudes of hils which otherwise is difficult to be found This chapter may finely be performed by your staffe for that you haue thrée angles
conuey the Index with the sights vnto the marke or hedg vpon your right hand making the said Index point to the marke or lye paralell to the hedge note then the degrées cut by the fiduciall edge of the Index for that is the quantity of the angle This néedeth no Example CHAP. XXVI The making of a Protractor and Scale TAke a fine thinne and smooth péece of brasse of what bignes you please whereon describe a demicircle as a b c vpon the center f which diuide into 180 equall parts and so set figures thereunto as in the figure some vse to make the like circle vpon the other side the plate numbring therein the degrées from 180 to 360 but that is néedlesse now by the Diameter of this circle a c there is made a Scale according to 12 in the inch furnished with paralell lines and numbred with figures as the order is and as you may see in the figure at d e vpon the backe side the plate there is an other Scale made according to 11 parts in the inch or you may make the Scale d e according to 16 parts in the inch or more then take 12 of those parts which diuide into 11 parts make a Scale the vse whereof you may pâ—ainely see set downe in my Arte of Geodetia Chap. 53. but the best Scale is Lib. 5. of the Geodeticall Staffe Chap 2. where I treated of the Iacobs Staffe You must note further that for angles of position the one halfâ— of the circle containes the East part of the Horizon the other the West the diameter of which circle common to both the demicircles being the Meridian line or South and North points as in this figure CHAP. XXVII To protract an angle and lay downe the ends thereof HAuing learned to take the quantity of an angle by this Glasse it resteth that you also learne how to protract the same and to lay downe the sides thereof vpon paper by your â—cale and compasse First therefore to protract an angle vpon any point giuen you are to take the semidiameter of your protractor as f a This is performed with more ease and better in the 6 booke of Geodetia Cha. 2. and so placing the one fooâ—e of your compasse in g the point giuen with the other strike the portion of a circle h i k l now must you â—â—te the quantity of the angle you are to protract which let dâ— 40 degrées place therefore the one foote in a in the protractor and extend the other to 50 degrees in the demicircle a b c the which widenesse place in this circular base h i k so will the two points of your compasse fall the one at h the other at i finally draw a line from i to g and from h to g I conclude i g h is an angle of 50 degrées But say you would draw a line by some degree beyond 180 degr as by 230 degrees therefore you must take halfe 230 degr set that distance twice in the circular base viz. 115 set once there is h k which set twice there is h k l 230 degrees or deduct 230 from 360 so haue you 130 degrees and so protract an angle the contrary way of 130 degrees as h m l so shall l cut 230 degrees or be distant from h in the circular base h i k l 230 degrees and if you be regâ—â—red to lay downe the length of euery side by your Scale according as you found the same by measure first sée what length h g should be which let be 18 perches therâ—fore place the onâ— foote of your compasse in d in your Scale extâ—nding the other towards c as to 18 tâ—at widnesse of your compâ—sse place in g h fromwards g towards h as to n I conclude g n iâ— 18 pearches do so to g i g k and g l so shall you finde g o 12 pearches g k 24 and g p 6 pearches so of any other CHAP. XXVIII To obserue an angle of position and what it is as also to protract the same and to finde vpon what point of the compasse any thing seene in the Horizon lyeth AN Angle of position is such an angle that is taken in respect of the needle or in respect of the South point insomuch that the one side of euery angle of position is the Néedle and the 02 other the Alhidada the common section of the termes of which angle alwaies concurre vpon the very extrée of the Néedle differing from stationary angles because they conteine the number of degrées betwixt any two obiects proposed at euery particular station assigned and the angles of position the number of degrées distant from the Meridian To obserue therefore an angle of position you must place the Néedle ouer his true line in the bottome of the Boxe the Planisphere there resting conuey the Alhidada with the sights to the marke assigned noting the degrées cut in the Planisphere for that is the angle of position whereby you may see that the angle of position is not limited respectiuely according as hee is right acute or obtuse but onely doth extend it selfe to any degrée in the whole circle and therefore the termes of these angles might rather be called lines of position in respect of their situation and pointing into the seuerall parts of the Horizon Now to protract the said angles the difference is not any from the worke in the last Chapter onely where you beginne to protract call that point the South so forward in the other quarters of the world according as you shall be directed by the letters SEWN vpon the protractor signifying South East West North. And if you desire to know vpon what point of the compasse any thing séene in the Horizon lieth hauing planted the Glasse as before so that the Needle respect his due place turne the Index the assigned marke noting amongst the winds the wind cut by the Index for into that part of the Horizon the thing lyeth All these neede no example CHAP. XXIX To take the plat of any great Champion field or such like consisting of 1000 or 1500 acres or to take the plat of woodland and rough grounds by the Topographicall Glasse otherwise then in the eigth Chapter SVch a proposition as this cannot bee performed by one station in the midst of the field nor yet at two stations by the intersection of like lines because it is scarce possible to see all the angles at once from the said stations eithâ—r by reason of the long distance or by occasion of hils trees or such like therefore you must get the plat thereof by going round about the perimeter of the said field or wood and that you may doe either by angles stationary or angles of position but the operation by angles of position is more troublesome and lesse certaine therefore it shall bee here omitted To performe it therefore by angles stationary you must go vnto one of the corners vpon the vtter side
proportion the Radius being in the first place For as the Secant is to the Radius so is the Radius to the signe of the Complement Any other difference that may happen in any kinde of obtuse angled triangle is resolued in the manuduction in Trigonometria by making a dislocation of the oblike triangles and conuerting any one of them into two right angled triangles which for that it is briefly set downe there I will here open it with an example Example SALOPIA CHAP. XXXIII To square lands and to reduce irregular plaines into some regular Figure and that in the open Field TO square any any field is to reduce the whole body of the same into one square reiecting the corners angles and crooked hedges which must be after measured according to the figure they represent Now for the corners and fragments that doe remaine you must measure them according vnto the figure that they most resemble euen as you may best gather by the pricked lines in the demonstration without any more circumstance of words But say the irregular field lyeth most apt to be reduced into a triangle which is thus performed CHAP. XXXIIII To search out the Perpendicular in any Triangle or other Figure according as it lyeth in the open Field NOw hauing reduced any plaine irregular Poligon into a Triangle or such other Figure whose superficiall content is found by the helpe of a perpendicular and for that the length of the said perpendicular is something difficult to be attained vnto the open field because it is vncertaine vpon what part of the base your said perpendicular falleth I haue not therefore thought it much heere to deliuer you the order how to performe the same Let a b c bee a certaine plaine field which you are put to measure the question is to finde in what part of the base b c a perpendicular would happen falling from the angle a. First therefore I plant my Glasse in the line b c as néere as I can gesse vnder the angle a as at d then I moue the Indexes about vntill the Index with the shorter fights lye directly ouer the line b c then I looke through the sights in the Demicircle if then the visuall bâ—â—mes runne to a my instrument stands right and d is the place where the perpendicular a d should fall but if it had not so happened I must haue remoued the Instrument as I had seene occasion oâ— the like must you doe in any other figure in the like case whereby you may see how necessary the foure Indexes be for the false taking of perpendiculars is a chiefe occasion of those palpable errors that be daily committed and a principall cause wherefore the common practisers so often differ CHAP. XXXV To reduce many plats or all your obseruations into one and thereby to make a faire Map thereof according to the quantity assigned BRriefly to teach you to performe this chapter you must first appoint the Card of the bignesse that you intend to make your Map crossing the same with two lines at right angles that about the middest of your Map writing at the ends thereof East West North and South Now you are to seeke in your Tables gathered at your obseruations the greatest distance betwixt the most Easterne and Westerne place and also the greatest distance betwixt the most Northerne and Southernely place and so accordingly choose you a Scale that those places beeing laid down by the same may fall within your Card. ANGLIA Example Let the proposition be to describe England and therein to situate such townes as shall be required in that bignesse as is here set downe according to my Scale the first thing that I doe hauing appointed my Card I crosse the same with two right lines at right angles appointing at the end thereof the foure quarters of the world as in the Type then I find out some towne that I coniecture by conference with my tables lieth about the midst of the land which let for examples sake be Middle Wiche a towne situate in Cheshire the which towne I place vpon the very intersection of the foresaid lines and thereby write the name of the same searching in my tables for some other towne that lieth direct East West North or South I find none therefore I take some other towne as Bristow drawing a line according to the position of the same getting also from your tables the distance of that towne from Middle Wiche as 97 miles to which widenesse vpon your Scale open the feete of your compasse and then place that widenesse vpon the line of position for Bristow placing the one foote in the marke made for Middle Wiche making a note with the other in the said line of position where write Bristow These two townes so placed let vs now go situate Northampton first by my tables I finde the distance of Middle Wiche from Northampton and according to that distance the one foote resting in the marke for Middle Wiche with the other I strike the portion of any arch the like I doe with the distance of Northampton from Bristow as in the 12 Chapter is plaine Now the intersection of these two arches is the true place of Northampton where make a marke for the towne and write the name thereof by the said marke and so proceed limiting all the townes ports angles and nookes in the Island in their proper places as you may sufficiently gather by the former demonstration Hauing finished in some void place you may appoint the Mariners compasse as in the Card before and this compasse will serue you for many necessarie vses as it is not vnknowne to men seene that way CHAP. XXXVI To diuide any Empire Kingdome or Continent into Prouinces Regiments or Shires WHen you haue taken the plat of any countrey and therein situated all the townes ports and such like yet happily shall it be expedient or rather of you required to separate and distinguish the same into such Prouinces Shires or Regiments as the said Kingdome is diuided into So is England or the South part of great Britanie being a peninsula diuided into 52 parts but not equall parts which we call Shires then is euery Shire subdiuided into other certaine vnequall parts as Worcester Shire into 12 which be called Hundreds either for that there were but at first so many townes or villages therein or for that there is to be required 100 able men in euery of them Other diuisions is England yet subiect vnto as first the whole Kingdome is diuided into two Prouinces or Archbishoprickes to wit Canterbury and Yorke then these Prouinces are subdiuided into Bishoprickes and euery Bishopricke is resubdiuided into Parishes according to which diuisions I minde God willing to describe a Mappe of all England c. But now the way how to attaine vnto these subdiuisions is not knowne It is therefore to be performed after two kind of waies the first whereof is you must in your Perambulation as
55. Chapter I vtterly dislike thereof because the section of some angles fall out so acute that the conclusion cannot be without error therefore such workes that be required by the helpe of two stations are best to be wrought by the 12. or 14. Chapter Now for woodland and rough grounds or great Champion fielden grounds vse the 57. or 58. Chapter ioyntly by themselues or seuerally according as the aptnesse of the ground occasions you you may also well vse the 9. and 29. chapter to the like purpose but haue a speciall care in all your dimensions and platting that you truly obserue the angles and corners of the fields and so consequently precisely mete the hedges and limits remembring euer when you worke with the Plaine Table or Circumferentor to kéep your Instrument paralell Touching the best Instrument for this purpose I leaue that to the discretion of the Reader for my best councell is that euery good practitioner make proofe of euery instrumeÌ„t as I haue done and afterwards to apply such to vse that hee findeth to worke with the producement of fewest errours for I will nominate none least it bee censured but my peculiar affectation For euery one doth commend that instrument whereon he doth practize or in which hee is séene and therefore some affect the Plaine Table some the Theodelitus c. which are all in some respects sufficient for ordinary measurers of land so they be vsed according to Art Here shall follow another kinde of ingrossing of particulars then that in the Staffe together with the pleasant and apt placing of the foure winds and their collaterals in any plat taken which being drawne in red lines and artificially done will much beautifie the same neither néed you to extend them all ouer the body of the plat vnlesse you please but in that your owne discretion shall be your best guide THE DESCRIPTION AND vse of an Instrument called the Circumferentor vsed by some onely to measure Land whose vse was first practized by I. G. and now published and annexed hereunto in a briefe method of teaching and in some parts altered by Arthur Hopton CHAP. LXI A description of the Circumferentor and the parts thereof YOU must first prepare some péece of wood well seasoned The Circumferentor and bearing a smooth graine which let bée fire inches or there about long foure inches broad and about an inch thicke then must you stope downe the left side thereof and diuide the same into equall parts according to 12. in an inch numbring the same by 5.10.15 c. The Needle and Carde In this péece of wood must there bee a round hole cut or turned of thrée inches diameter and thrée quarters of an inch déene in the bottome whereof is placed a card as in the 4. Chapter at 3. This round hole hath a néedle therein and a glasse to couer the same as in the fourth Chapter of the Topographicall Glasse at 10. Then vpon this square péece of wood is placed a Table called Tabula Sinuum and this is such a Table that is calculated out of a quadrant whose arch is diuided into 30. degrées and the semidiameter thereof into 1000. equall parts and according vnto the totall summe are numbers placed in the said Table answering vnto euery degrée and halfe degrée And these numbers serue to expresse the length of euery right signe or of euery perpendicular let fall from the seuerall degrées and halfe degrée in the limbe vpon the semidiameter Tabula Sinuum 1 52 6 309 11 544 16 743 21 891 26 978 78 333 566 760 902 983 2 104 7 358 12 588 17 777 22 913 â—7 987 130 382 608 793 923 991 3 159 8 407 13 â—29 18 809 23 933 28 994 182 430 649 824 942 996 4 208 9 454 1â— 669 19 838 24 954 29 998 233 477 688 852 958 999 5 259 10 500 15 707 20 866 25 966 30 1000 248 522 725 878 972  Of the Sights Vpon the plaine of this square péece of wood néere vnto either end thereof is placed two sights and the one of them is but halfe the length of the other standing perpendicular Of the shorter Sight The shorter of the two sights beareth no diuisions at all The shorter sight in the top whereof is placed a pins head and vpon the side is set a péece of small wire end in the middest is hanged a plumbe The distance from the wire in this sight to the plaine is taken and diuided into 60. equall parts according vnto which diuisions is the right edge diuided beginning from the perpendicular point vnder the wire numbred by 10. as 5.10.15 c. The short sight and the wire therein represent the semidiameter of a quadrant and the wire the center thereof Now from the perpendicular point be the degrées of a quadrant perfected vpon the vpper side of the right edge numbred from 90. to 25. by 10. Of the longer Sight This sight is twice so long as the other The longer sight whereupon are placed thrée kinde of diuisions 1 First the distance from the pinnes head in the shorter sight vnto the plaine of the instrument is taken and according vnto that distance is a line stroke ouerthwarte this sight which must be called the line of leuel the distance from which vnto the pins head is taken and diuided into 100 equall parts the distance bewixt the pins head and the line of leuell is equall in length therefore diuide the longer sight into 100 equall parts according vnto which place them in the said longer fight to 50 vpwards and downewards numbring them by 10 as 5 10 15 c. Hypothenusall diuisions The second diuision is the graduation of the Hypothenusall lines according as they increase by vnits they be numbred by vnits as 1 2 3 4 c. to 12 which represent 100 102 103 104 c. and these diuisions are set from the line of leuell vpwards and downewards take the square of 100 from 102 c. from 102 103 c. Degree of a Quadrant The third sort of diuisions is the degrée of a quadrant proiected vpon the said longer sight from the line of leuell vpwards and downewards to 25. Of the slit in the sight Slit in the sight In the midst of this sight there is a slit made from the vpper and of the sight straight downe vnto the lower end Vpon this sight there is a vane of brasse made to runne equally vp and downe and in the same there is a sight hole answering to the slit and the edge of the vane Of the Index The Index Vnto this Instrument there also belongeth an Index wherein must be a center hole to put vpon the wire in the shorter sight in the other end of this Index there is a sight placed with a hole therein answering vnto the fiduciall edge of the Index the edge of this rule is diuided into such equall parts as the right edge of
of the next Manour was in doubte least the veine of coale did run towards his adioyning Manour and that they were cōmon vnder his ground wherby the coales were his To resolue this doubt descend into the pit and then by the 72 chapter get the true way that the myners haue made euen as it were a hedge still noting the degrée cut by the Néedle at euery angle where the myne runneth one way or another out of the course of a right line and also measuring the side of euery angle then ascending out of the pit by your Instrument and cheine beginning perpendicular aboue the place where you began to make obseruations in the bottome of the pit lay downe the like angles and sides so obserued which hauing so done you shall soone sée if the myne or any part thereof haue run out of the one Manour into the other for if it do you shall be forced to measure out of the one into the other And as you note these angles of deuiation being in the darke bowels of the earth you were best to haue a candle fixed vpon the end of a staffe of equall hieght with your eye and the same to be fixed in the foote of the myne at euery angle that thereby you may the better direct your sight therunto CHAP. XCI To plant barrels of powder direct vnder Castles Forts or such like and to know how farre you be vnder the same IN performing this Chapter by some Proposition formerly published you must get the Horizontall and Hypothenusall distance of the Forte from you and thereby the height thereof aboue the Horizontal line which done you are also by helpe of your Néedle placed in the glasse to find out the angle of position which is the number of degrées from any principall quarter of the world that the iourney lyeth which done you must by the same Instrument euer direct the myne direct vpon that line or part of the world and kéeping your Instrument paralel the sight vpon the diameter of the demicircle thereby alwaies cary the floore of your mine leuell with a candle fixed vpon the end of a staffe of equall height with your eye as before will helpe you to do Now when you haue gone so farre vnder the ground as you found the length of the Horizontall line to conteine you may assure your selfe that you be direct vnder the Forte and that you are so many pases vnder or below the said Forte as you found the Forte to be about the Horizontall line CHAP. XCII A Mine running vpon some certaine point yet ascending or descending to know at any time how much you are aboue or vnder the Horizotall line COncerning your iourney vnder the earth you must obserue the doctrine of the last Chapter and when the mine happeneth to fall or rise according to the doctrine of Altitudes and profundities duely note at euery seuerall station the quantity of the ascent and descent that is how much you rise aboue or fal vnder the true Horizontall line and so kéepe two seuerall tables the one of the ascents and the other of the descents Now when you desire to know how you are situate adde all the ascents together and note the product do so to the descents then must you take the lesser out of the greater so doth the remainder acquaint you how you then differ from the Horizontall line for if the ascents excéed you may be assured that you be aboue the Horizontall line if the descents excéed you be vnder the said Horizontal line according to the difference of the said ascents descents neither néed you feare any collateral declining of the way of your myne for that nothing at al altereth the ascent or descent for that is onely altered by the directing line or line that you measure insomuch that if you well obserue the premisses you may precisely know at any time or place how much you are vnder or aboue the true Horizontal line and thereby come into him againe vpon any occasion CHAP. XCIII A Myne or trench collaterally declyning how to know when you come againe into the right line of position and also how farre you be from being iust vnder any Forte proposed TO cary a myne direct forward vpon any point of the Horizon you bée sufficiently taught in the 90 Chapter and if the Mine must ascend or descend aboue or below the line of leuel you be taught in the last Chapter at all times to know how much you be aboue or vnder the said line of leuell But say you were inforced by rockes waters or other such obstacles that you méete with vnder the earth contrary to the 90 Chapter to cary your Mine side wise from the direct line of position in such a case you are first vpon a faire large shéete of paper to extend a right liue ouer the same which call the line of position béeing the direct way that the Myne should go next note the angle of deuiation from that line that is to say how many degrées the Mine doth decline from the true line of position that leadeth on directly and accordingly plat it downe vpon the paper as you be often instructed in the vse of ech seuerall Instrument procéeding so far as your Myne continues in a right line and if you be occasioned againe to direct either further of or néerer vnto the line of position alwaies protract if downe vpon your paper exactly as well in measure as angle vntill such time that you can come to make your protracted collaterall lines or lines of deuiation to intersect with the right line of position first extended ouer the paper and then by the scale with which you protracted your lines of deuiation examine how many pases or yards that point of intersection is distant frō the point where your worke began which representeth the point of your first entery into the Myne for that compared with the fundamentall distance or length of the Horizontall line informes you if you be past or not yet come vnder the proposed Forte Therefore in these cases you were best first of all to limit vpon your paper with your scale and compasse the direct length of the fundamentall or Horizontal line and so in your protracting may you call them backe if they séeme to run beyond the Forte Then in the former Chapters be you taught to know how far vnder the Forte you be whereby you may ascend néerer or descend further from the superficies of the earth as the cause shal require Certainely most exact and excellent is this kind of working for conueying of mines and of no smal importance for the due placing of Fornaces of powder to blow vp Forts Castels Townes or such like whether they bee situate high hpon an hill or low in a valey which for all purposes in these and such like cases vnder ground you shall find the Topographicall Glasse to be most requisite CHAP. XCIIII Of the building of a Citty and of the situation