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Speculum Topographicum: OR THE Topographicall Glaſſe.
Containing
With many Rules of Geometry, Aſtronomy, Topograperſpectiue, and Hydrography.
Newly ſet forth by Arthur Hopton Gentleman.
Printed at London by N. O. for Simon Waterſon, dwelling at the ſigne of the Crowne in Paules Church-yard. 1611.
Geometricall, as ſtill ready to confirme the truth to the ignorant, or confute the malice of the arroMathematikes vſe no conference with the ſence-rauiſhing Rhetorique, their end is to inſtruct, not perſwade; therefore ſuperfluous eloquence beſto
A chiefe cauſe wherefore I now proue troubleſome, is, my loue to your Lordſhip, whom I euer honored, and the aduancement of the Art that I alwayes liked. By the one I ſhew my duty to a name emphaſizing honor, and by the other my affection to an Art expreſſing wonders. Alexander would be painted by none but Apelles, nor haue his piLiſippus. Appelles asked counZeuxis, and Liſippus muſt onely cenſure Priſius: neither do I in this appeale to any but your LordEngland may boaſt of your great iuſtice, and all Europe reioyce of your good conſciMacedonian king honored Craterus, but moſt affected Hepheſtion, and I reueGlaſse to light, where they may ſee choiſe of delights to ſatisfie their aſpiring wits.
Thus committing this Booke to your Honorable patro
PLATO ſaith, there was in old time an Oracle giuen vnto the Greekes, that they ſhould double the Altar in the Temple of Delos (which was a peece of worke for an excellent Geometritian to performe, that had the very habit, and perfection of the Arte) but it was not there literally meant, (as Plutarch in his Sympoſiaques hath expounded) that they ſhould do ſo indeed, but thereby they were iniointed to ſtudy Geometry, to the end they might be able to perPythagoras offered ſacrifice to the Gods, to know, when two figures are giuen, how to finde a third equall to the firſt, and ſemblable to the latter. Both which are ſtrong arguments, inuincible proofes, and perſwaGeodeticall Staffe, concerning the publiſhing of my Topographical Glaſſe, which hauing now accompliſhed, there remaines nothing more, but to craue a kind acceptance, and fauourable conſtruction: ſo ſhall you ſhake off that viſcoſious filth of ingratitude, which ſo much conglomerates the heart of the ena malo in peius, as the fiſh out of the pan into the fire: it is vaine then to begge applaudites of ſuch, for Ruby ſtones are not found in flinty rockes nor redolent flowers amongſt rough thyſtles. A corrupt ſtomacke yeeldeth no ſweet breath, and
Topographicall Glaſſe, as a pure tranſparent Chriſtall, wherin he may ſee a numGlaſſe, by the Theodelitus by the Plaine Table, and by the Circumferentor, by all which ſhalt thou bee ſeuerally taught to deſcribe Countries and Kingdomes to make Mappes of new diſcoueries, to plat Mannors, Lordſhips, or Townes, to meaſure parkes, paſtures, or incloſures, and that after diuers new wayes. Alſo here are you taught to ſeeke the diſtance of Townes, the height of Turrets, to conuey waters, to plant Cities and houſes, to plat buildings, and to meaſure all kGeodeticall Staffe. But happily ſome will ſay if there were ſufficient instrumeHeliotropion after the Sun? into what an intricate Laborinth of coCatonians, Peripatitians, Academians, and Epicurians. Anaximander ſaid the earth was like a columne. Anaximenes flat, Leucippus like a Drum, Democrites like a platter; till Thales demonſtraGaule, entring the Atlantique ſea, as Tymeus: Some to a riPlato: Some to the Sunne drawing the winds vpon the Ocean which cauſed the Atlantique ſeas to ſwell, as Heraclitus, till Pytheas demonstrated the cauſe to bee in the increaſing and decreaſing of the Moone; whereby Homer ſaid: I praiſe not my Anceſters for their knowledge, but for that they deſired knowledge. But ſhaking off theſe old differeCompaſſes, Sea-cards, and new Maps, & of many other deuiſes & engines, that haue beene lately ſet forth more beneficial then any heretofore, of which the old PhiG
the Mater wherof being excellent and a moſt Art-like proiectment reſembling the true lineaments of the Rete, by our ingenious countriman M. I. Blagra
But my Pen ſhall not bee ſo much diſmeaſured to reproue ancient men40. when one fruite faileth, another entreth into ſeaſon: we muſt firſt inuent, next a
He then that will be ſeene in the knowledge of Geometricall inTheodelitus. &c. and others regard not the quanHomogeneall, as the Plaine Table: yet, if you faile in the one or the other, your concluſion is error: for it is as great an abſurdity if the angle Homogeneall Mechanicall delineated vpon the Plaine Table, Proue Heterogeneall, and accord not Symmetrially to the reſpondent angles in the field, as if with a graduated inſtrument you had falſly obſerued the quaPlaine Table before a large graGeometritians. Euery horſe is not Bucephalus, yet he may be are one a iourney, though hee bee long and lame in performing it, and can hardly paſſe ouer a mountaine: but let euery man vſe his will, and ſo I will mine heere, in ſaying no more at this time: onely wiſhing my ſelfe preſent to open any thing that the yong practi
Pag. 3. line 7. the word) p. 13. pro. 7. place the figure in the 9. pag) 40. l. vlt. for 28 18. omit orleton) 41. omit 10. & 11. lines 68. a ſtands vvhere c ſhould) 101.8. the 61) 114.2. I can ſee) 117.14. one vvith another) 118. l. 8. your ſecond) 150. c b ſhould be in
DVM SPIRO VNICVM CHRISTUM SPERO
TOPOGRAPHIE (with ſome called Corography) is an Arte,Topographicall deſcription ought to expreſſe euery partiTopographicall Glaſſe, as being moſt apt to deſcribe any monuEngland thus: but wée haue omitted di
Geography (as Vernerus in his paraphraſis ſaith) is an imiTopography, becauſe it reſpects but places of note, and from Coſmography, becauſe it hath no relation vnGeography may ſerue well for a general de
As Apian ſaith, wée may well gather what Coſmography is, by the bare etimologie of the world. It is therefore a deſcription of the world, which doth conſi
Firſt of all, it hath reſpect vnto the circles, which are ima
Hereby hath euery towne lying into the Eaſt or Weſt a ſeueral Meridian, and into the North or South a ſeuerall paralell of laCoſmographicall demon
Geometry bee often vſed in this booke, & the vnderſtanding thereof not knowne (though ſomething common) to all yong prae they be but ſtenderly remembred in my Geodeticall Staffe, I thought good to ſay ſomthing therGeometricall magnitudes take beginning from a point, which is an indiuiſible ſigne in a magnitude.
1 A magnitude is lineall or lineamentall.
2 A line is a magnitude onely long, and his termes bee two points bounding the termes of the line.
3 Lines are conſidered after two waies, firſt ſimply by themſelues, then alſo comparatiuely amongſt themſelues.
4 Being conſidered ſimply they be right or oblique.
5 A right line is which lyeth equally betwixt his termes.
6 An oblique line is which lieth vnqually betwixt his termes.
What diſtinctions right and oblique lines do admit.
7 A right line being giuen there cannot be made a ſhorter bee the oblique lines be ſimple or manifold.
8 Thoſe be Simplex, which be terminated with a vniforme and ſimple motion.
9 The diuers or manifold oblique lines called Helices, are ſuch that be terminated with diuers motions, and be vnequally diſtant from the center, ſuch are Spirale lines, Conchales, Ouales, Lenticulares, &c.
Lines compared amongſt themſelues be diuerſly affected: for either they be perpendiculars, paralels, or contrarie.
10 Perpendiculars compared amongſt themſelues, are two right lines, wherof the one falling vpon the other make two right angles, ſuch is a Carpenters ſquire, &c.
11 Paralell lines are euery where one alike diſtant from the other be they right lines, circles or Helices.
It followeth to ſpeake of Lineaments, and firſt of Angles.
12 A lineament is a magnitude more then long, conſiſting of lines, and is diſtributed into an angle or a figure.
13 An angle is a lineament, made by the common concurſe of two lines, and ſituate within the ſame.
14 The two termes or lines comprehending the angles are called the ſides.
15 Of angles, there be two kinds Homogencall and Heterogencall.
16 Angles Homogencall, bee ſuch that haue their ſides both of one kind, as both right or oblique.
17 Angles Hetrogeneall are ſuch that conſiſt of mixt ſide as of right lines and curued.
Theſe Homogeneall angles do furthermore ſuffer a ſubdiui
18 A right angle is where one liue falleth perpendicular vp
19 An oblique angle is made when one line doth not fall perAcute and Obtuſe.
20 An Acute angle is an oblique angle, leſſe then a right angle.
21 An Obtuſe angle is an oblique angle, more then a right angle.
Of Triangles and Figures.
22 A Tryangle, is a figure, and a figure is a lineament boun3. Angles, and bounded with 3. lines.
23 Euery figure hath certaine termes wherwith the figure is meaſured.
24 A Center is a point in the midſt of any figure.
25 A Perimeter is that which bounds or comprehends the fi
26 A Radius is a right line drawne from the center to the perimeter.
27 A Diameter is a right line inſcribed in a figure, and paſſing by the center, therefore Diameters in one figure may be infi
28. The altitude of a figure is the length of the perpendicular let fall from the toppe of the figure vpon the baſe.
29 There be two kinds of figures, to wit, Superficies and Bodies.
30 A Superficies is a figure onely broad.
31 Superficies be alſo plaine or ſwelling.
32 Plaine Superficies bee ſuch that lye equally betwixt their termes.
33 Plaine Superficies admit a double diſtinction, whereof ſome be right lined, ſome curuilined.
34 Plaine right lined Superficies, bee ſuch that bee bounded with right lines.
35 Of right lined plaines there be two kinds, as a Triangle and a Triangulate.
36 A Tryangle is a figure comprehended vnder 3. right lines.
37 Furthermore a Tryangle is taken two kind of waies, as in reſpect of his ſides or angles.
38 In reſpect of ſides they be either an Iſopleuron, Iſoſceles, or Scalenum.
39 An Iſopleuron called alſo an Equicrurum, is a Tryangle conſiſting of 3 equall ſides,
40 An Iſoſceles hath onely two equall ſides.
41 A Scalenum, hath all his 3 ſides vnequall.
42 Tryangles, in reſpect of their angles, are diſtributed into two kinds, as Right Angles, and Oblique Angles.
43 A right angled Tryangle is, which containes one right angle which is alſo called an Orthogonium.
44 An Acutangled triangle, hath all his angles acute, to wit, leſſe then right angles, and is called an Oxigonium.
45 So that there be triangles in reſpect of their ſides, as Iſopleurons, Iſoſceles and Scalenumes, and in reſpect of their anOrthogoniumes, Ambligoniumes and Oxigoniumes: ſo alſo may they be taken mixt or comparatiuely, as well in reOrthogonium Iſopleuron, an Oxigonium Iſoſceles, or an Oxigonium Scalenum.
46 A Triangulate is a mixt figure compoſed of Triangles, and may be reſolued into the ſame againe, whereof there be two kinds, Quadrangles and Multangles.
47 A Quadrangle is a figure bounded with foure right lines, and is twofold, as a Paralelogram, or a Trapezia.
48 A Paralelogram is a figure whoſe oppoſite ſides be paralel, and may be Rectangled, or Obliquangled.
49 A right angled Paralelogram is a figure whoſe angles be all right, and is twofold, as a Quadrate, or an Oblonge.
50 A Quadrate or Square, is a right angled Paralelogram equilater.
51 An Oblonge is a right angled Paralelogram, not equi
52 Obliquangled Paralelogrames are ſuch whoſe angles be all oblique, and they be twofold, either a Rhombus, or a Rhomboides.
53 A Rhombus is an obliquangled Paralelogram equilater.
54 A Rhomboides is a paralelogram obliquangled and ine
55 A Trapezia is a quadrangle being not a paralelogram.
56 Mixt or multangled Triangulates are ſuch figures that conſiſt of more then foure ſides.
57 Curuilines ſuperficies be either ſimple or miſt.
58 Simple, are circular figures & round, being euery where equidiſtant from the center, as a circle.
59 The termes of circles are the parts meaſuring the ſame, which are Segments of the Peripher, Sectores and Sections, alAſcript, and Inſcript, as Tangents, Secants, Radius, and Diameter.
60 The Segment of a circle is a portion comprehended with the Peripher, and a right line which may be a ſubtenſion.
61 A Sector is a Segment conteined inwards vnder two right angles, making an angle either in the center or Peripher.
62 A Section is a ſegment of the circle conteined inwardly, with one right line, which is called the baſe of the Section. Lines Aſcript be deſcribed Lib. 7. chap. 2. of the Geodeticall Staffe.
63 Such figures bee called Miſt obliquilined, which bee vnee figure, ſuch figures be Ouales, Lenticulares, &c.
A Right Angle or perpendicular is created by ye extention of a right line from both the ſections of two arches of equall or inequall Diameters, the centers whereof remaining in the line aſſigned, and this creation of right Angles or Perpendiculars is generall.
IN making of a Quadrant, & performing of other concluſions in this booke, you ſhall be forced to raiſe a Perpendicular, or create a right angle vpon the extreames of a line aſſigned, where
A p is the line aſſigned, p the extreames of the line, whereon you are to erect a Perpendicular, therefore I open my compaſſe to any reaſonable ſcantle, placing the one foote in p, with the oc d e, that wideneſſe reſting, I place the one foote in c, and with the other make the point d, then with the ſame wideneſſe vpon d, I doe deſcribe the arch c g h f, then doe I fit that wideneſſe thrée times in the laſt deſcribed arch, as g h f: laſtly, from f I drawe a right line to p, ſo haue I created a right angle f p a vpon the point p.
D is the point aſſigned and a b the line aſſigned whereon a
d, therefore
d, and open the other ſo farre, that it may reach beyond the line a b, and ſo ſtrike an arche e f, which ſhall cut the line a b in two points, as in e and f, then your compaſſe reſting at any ſcantle, fixe the one foote in f, and with the other ſtrike an arch vnder that arch e f, the compaſſe at the wideneſſe, fixe the one foots in e, and with the other croſſe the laſt deſcribed arch in the point g, finally drawe a ſtraight line from d to c. for that is the Pera b.
By this meanes may you eaſily open the legges of your Geodetical ſtaffe vnto a right angle, or place the graduator at a right angle.40. vpon the left legge, whereto bring the center of the Graduator, then count 30. vpon the right leg amongſt the equall diuiſions, next note 50. eGraduator, now the center of the Graduator reſting fixed, mooue the right legge, and the other end of the Graduator vntill 30. vpon the legge, and 50 equall parts vpon the Graduator, touch one the other, ſo ſhall the legges reſt at a right angle.
Graduator at right angles, count 40 vpon the left legge, whereunto bring the center of the Graduator, then make 50 vpon the right leg, and 30 vpon ye
Graduator interſect, ſo haue you placed the Graduator at right angles.
Otherwiſe:
Drawe a ſemicircle a b c d, vpon the center c, d a béeing the Diaa drawe a line to touch the circumference as at b or c, then from b or c drawe a line to d, ſo haue you made a right angle, as a b d, or a c d.
THis propoſition wil ſtand you in much ſtead in the vſe of the plaine table, and is performed thus:
A b c is an angle, and you be required to make another angle
a, and with the other ſtrike an arch ouer the ſides of the angle, as d e: the feete vnſtirred I place the one foote in the line f c, and with the other ſtrike the arch i h, next do I take the diſtance of d e, and fit it in the arch i h, from i to h: finally, I lay my Ruler vpon the point i and h, drawing a line from f towards h infinitely: ſo haue I made an angle c f g, equall to the aſſigned angle b a c: this pro
A B is a line aſſigned,
c d, now open your compaſſe to what diſtance you pleaſe, or rather to any diſtance aſſigned, then place the one foot in c, and with the other ſtrike a fragment of an arch, doe ſo at d: finally, by the extreames of theſe two arches produce a right line e f, which ſhal be paralel to a b.
A B is a line aſſigned, which you are to diuide into two equal parts, or find the middeſt betwixt the point a and b, which to doe open your compaſſe to any reaſonable ſcantle, and then placing the one foot in b, with the other ſtrike the arch d a c, that wideneſſe remaining, place the fixed foot in the point a, and with the other ſtrike the arch c b d: now note the interſection of theſe two arches, as at c and d, for a line drawne from c to d diuides the line a b in two equall parts at the point e.
I will not ſeeke to deliuer rules to diuide a line into a num
A b c are three points giuen, firſt from a to b I drawe a right line, then from b to c: laſtly by the 1. Prop. I reare a a Perpena b and another in the middeſt of b c, as e d, and f d, the interſection of which two Perpendiculars, as d is the true center.
Let the 3. points be a b c, open your compaſſe to ſome reaſonaa b, with that wideneſſe ſtrike a portion of an arch vpon the point a, doe ſo vpon b, and note the interſection of thoſe two arf g, doe likewiſe to b and c, and note the interſection of the arches, as i h: laſtly, produce a line from i by h infinitely, doe ſo from f by g, noting where thoſe two lines interſect, for that is the true center, as at e.
More of theſe concluſions might be ſet downe, but here is ſuf
9. inches ſquare at the leaſt, as d b or a c, then vpon the center r deſcribe a circle of three inches diameter, as p q, a quarter of an inch, from that make two other circles, as t u, which diuide into 24. equall parts, as here
32. equall parts, and there appoint the 32 winds, as you may plainely ſee
Secondly, in the Quadrant of this laſt circle e g, drawe the ordinarie quadrant i f h, and diuide the ſides i f and h f into 120: or 60 equal parts, then kéeping the rule vpon ye center r, & each dii f, and f h, as you mooue him from diuiſion to diuiſion, make markes in the circle e g, ſo haue you proiected the Geometricall Quadrant i f h into the Quadrant of a circle e g.
About this circle deſcribe another circle, and diuide the ſame into 360 degrées, as the order is, drawing paralel circles to place the figures, as in the demonſtration.
About this laſt graduated circle drawe an other ſome halfe an inch diſtance or better, as a b c d, and betwixt the graduated cirr, and the circle a, drawe fiue other paralel circles equidiſtant one from the other, as in the figure: Laſtly, vpon euery two deIſoſceles Triangle round about the circle: you may better perceiue it by the plaine demonſtration then vnder
Theſe Iſoſceles Triangles ſerue aptly and preciſely to expreſſe the fiftieth part of a degrée from 10 to 10 &c. thus.
All the ſides of the Iſoſceles upon the right hand are diuided into 60 parts with the paralel lines, reckond by 10. as 10.20.30.40.50.60, beginning at the vtter circle a, and ſo procéeding to k, ſo that if the Index cut the right ſide of the Iſoſceles in the 2. paralel it cuts ſo many degrees and 20. minutes, &c.
But for the left ſide of the Iſoſceles the diuiſions doe begin at the circle k, and ſo proceed by 10 to the circumference, as 10.20.30.40.50.60, ſo that if the Index cut the ſecond paralel circle counted from k, it cuts 20. minutes.
See the enſuing figures in the folded ſheet.
Thirdly,e circle p q, diuided into an 120. equall parts, and therein drawe a Diall according to the Azimuths of the Sunne, for ſome particular laAzimuth as is ſaid: the houres be numbred, and ye moneths wrote at euery ſeuerall circle, wherein you moſt obſerue the houres
p q, all which muſt be cut out, only a narrowe lymbe remaining iuſt to containe the breadth that is intercepted betwixt the circle t u, and p q, the vtter part of this lymbe is to be diuided into 29. ½ equall parts, as in the figure, and this circle is to mooue about the circle p q: at euery quarter of this circle there is an Index left of ſufficient length to reach oa b c d in the great figure, and of ſufficient ſtrength to beare ſuch ſights as are to be placed thereon.
See the enſuing figures in the folded ſheet.
Vpon the Index i and h there bee placed the two ſhort ſights marked with k and l: theſe two ſights haue two ſmall groues or channels cut through them for one to looke through, and in the top of the ſights at the end of theſe groues there bee certaine things left like pinnes heads, as in the figure: theſe two ſights
i or h to the graouated circle, and they be made to fold downe each vpon his proper Index: the principall vſe of theſe ſights is to ſquare land, and to find where the Perpendicular falleth vpon the baſe of any Triangle in the open field, as in the 31. Chap.
5 As for the Demicircle, it is a ſmooth peece of wood, but rather braſſe, whoſe breadth or diameter needeth not to be limia a center, and thereupon deſcribe a ſemicircle r r, ſomething more large then the circle p q in the Planiſphere: all within which circle is to be cut out, vnleſſe you pleaſe to leaue ſour r deſcribe another, and according to that circle fyle all the reſt of the plate that is ſuperfluous away euen to the ſaid circle.
See the precedent figures in the folded ſheet.
This Demicircle ſo made, he muſt be ſupported with AnIndexes g and f, as hereafter: but firſt I will teach you to deſcribe the Aſtronomicall circles, and Horologicall arkes, as alſo to proiect Geometricall lines, &c. vpon the ſaid de
As for the Aſtonomicall circles, though the graduation thereRamus, where they ſhal find, ye
Circles are as the ſquares which are made of their diameters, and that their diameters are as the Circumferences.
You ſhall therefore doe no more but diuide the circumference d c into 90. parts, and ſo drawe paralel lines, and figure the ſame accordingly, as in the figure: and note if you make the di
6 But now for the Horologicall arkes, they be more difficult to performe: you ſhall therefore at each end of the former demi9. place of this Chapter: let the South moitie ſtand at b r, and the North at o r, then vpon the center a deſcribe the cirr r, which we call the Equinoctiall, and ſo vpon that center a deſcribe the Tropique of Cancer and Capricorne, which here are both but one circle. Next vpon the ſaid center deſcribe other cira, and extending the other to the degrée in the Zodiaque aſſigned. This ſo done, repaire vnto the Table of ye Suns altitude in the firſt Booke of the Geodeticall ſtaffe,
e Sun at 12 of the clocke, he béeing in the o degrée of Aries or Libra, then finding the like altitude in the Demicircie b c, I place a ruler vpon the ſaid degree of altitude, and vpon the center a, and ſo I make a ſmall pricke in the circle of the Equinoctiall, where the ruler cuts: this ſo done. I doe the like to the houres of 11 10 9 8 7 and 6 of the clocke, for no further doe the houres extend in the Equinoctiall: theſe prickes preciſely done, and apparantly noted, I doe the like in the circle of the o degree of ♉ and ♋ ſo haue you thrée prickes, then muſt you find the common center, of each 3: match ye pricks as you bée taught Chap. 3. Prop. 8. and thereby deſcribe thoſe horologicall arkes.
But now, for that you ſhall want pricks to deſcribe the houres of 7 and 8 afternoone, and 4 and 5 before noone, you moſt deſcribe another circle vpon the center a, repreſenting the 30 degrée of Taurus, making markes
Then for the houre liues anſwering to the South moitie of the zodiaque, looke what you did to the Tropicke of Cancer, and doe the ſame here vnto the Tropicke of Capricorne: and as you vſed the o degree of Taurus, ſo deale here with the o degree of Scorpio ſo haue you made two prickes to deſcribe your arke by: as for the 3, they bee the very ſame that formerly you made in the Equinoctiall: and if points for the ſtriking of the arkes of 7 and 8 before noone, and 4 and 5 after noone be wanting, drawe other blind paralels from ſome degrée of Libra or Scorpio, or from both, & accordingly find the altitude in the table, and after
So hauing placed a ſmall ſight fixed in the 90 degrée, the fore
7 Vpon the backe ſide this Demicircle is proiected the parts of the Geometricall Quadrant, and Hyſometricall Scale, thus.
Take the Diameter of the laſt Demicircle, which make the Semidiameter of another circle, as a b, now making a a center, and a b a Semidiameter, ſtrike the Quadrant of a circle b d c, and within that Quadrant inſcribe an ordinarie Geometricall ſquare, as f d e a, then diuids d e and d f, each into 60 equall parts: next in the middeſt of the liue a b make a point at g, on which, as a center deſcribe the Demicirle a i b equall vnto the former Dea, and euery of thoſe ef d and d e, making notes in the arch a i b where the ſaid Ruler touched at euery part.
Theſe parts ſo proiected vnto the circle a i b, you ſhall vpon the backe ſide your Demicircle ſtrike an arch of the bigneſſe of a i b, as m n o, where place all thoſe parts, as they be in the enſuing figure, drawing paralel lines for figures accordingly.
Now if you would likewiſe place the Hyſometricall Scale hereon alſo, becauſe there is roome ſufficient and ſpare, drawe a circle within the circle m n o, as p q, which furniſh as the former with paralel lines, and ſo from euery 15 part in the circle m n o, the one and of the ruler fixed on ſ, produce right lines ouer the pa3 6 9 ending in 12, iuſt at 60 in the firſt circle: now diuide euery of thoſe parts into thrée other parts, by pulling right lines from euery 5 part in the ſaid circle m n o, and then write vmbra recta and vmbra verſa, as you may beſt perceiue in the enſuing figure: this done, your Demicircle is finiſhed.
8 This Semicircle ſo finiſhed and ſupported (as before) with Antickes, he muſt be placed vpon two of the Indexes in ſuch ſort, that the one Anticke ſtand ag, the other at f, bearing the foreIndex g and f.
9 In the middeſt of this Demicircle vnder 12 in the Hyſo12, which will ſerue to keepe the Inſtrument paralel and vpright, which the croſſe NeeTwo 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 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 lh b will alwaies bee carried vpon the Circumference b c in the Demi
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 Circumference of this Boxe muſt agree with the cirp q in the great figure, for within that ciccle muſt he ſtand, the
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 cir
Vpon the Boxe aboue the glaſſe ſtands a certaine crooked wire. bearing a roun
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 fi
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 multi
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 nee
After you haue touched the end of the needle, if it were equi
The needle ſo touched, the South end thereof will not poynt iuſt into the South,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 ineri
And one ſpeciall note you moſt here obſerue in the delineating of this inſtrument, that is, t360. degrées, viz. that two ſides appoſite lye paralell to the line a c, and the other two oppoſite ſides to the line b d: and if you worke by the helpe of the néed
Index f g h i, and within this circle place the box with ſcrew pins to keepe downe the circle, in ſuch ſort as before
Circular ſight vpon the Index f g, and the other ſights vpon ye other Indexes: place them artificially as you be taught before. Next put the moueable ſight vpon the demicircle, and ſcrew the ſocket to the backe ſide: ſo is this inſtrument prepared to worke as the Theodelitus, Topographicall inſtrument, Geometricall Quadrant, or as the Circumferentor, and may ſerue for ye plaine Table, as ſhall follow after the reſt.
Theodelitus is an inſtrument conſiſting of a Planiſphere and an Alhidada: vpon the Planiſphere there is deſcribed a circle, which is diuided into 360. degrées, &c. Within this cirGeometricall Quadrant, and no more diuiſions or graduations bee in the Planiſphere of the Theodelitus.
The Alhidada is a ſtraight ruler with a fiducial edge, mouing equally and truly vpon the center of the Planiſphere, whoſe length is equall with the diameter of the circle in the Planiſphere, vpon the two ends of this Index or Alhidada as fixed two folding ſights.
Digges calleth the Topographicall Inſtrument, then is there a Boxe and a Néedle placed in the center of the Planiſphere, ouer which there doth ſtand a perpendicular whereon is placed a SeAlhidada. This Demicircle is diuided into
90 degrées, both ending in the Semidiameter, which SePlaniſphere: within this Semicircle is deſcribed the Geometricall Quadrant, which ſerueth for height whoſe parts, and alſo the degrées of the Demicircle be cut by the fiduciall edge of the perpendicular: but in this Glaſſe the Diameter of the DemiPlaniſphere, whereas the other is moueable: And as in the other De90. degrées, héere is but 90 degrées in all, ſo that they be twice ſo large as the other. Certainly this DemiDigges for diuers good reſpects I might well take occa
But, for that happily ſome will ſay the Theodelitus before was perfect, and then what needeth this alteration; it was but the Authors particlar conceit, without any neceſſitie at all. To giue ſuch ſatiſfaction, I anſwer, that there was a neceſſitie of alIndex by helpe of the Iſoſceles Tri4 Indexes how requiſite they bee for the meaſu33 and 34 Chap. Touching the alteration of the Demi60 degrees, for that you cannot looke through the ſight in the Diameter of the Semicircle by reaſon of the Planiſphere, but in déede the ſame might be better dealt with then any other thing whatſoeuer: looke alſo to ſhe want of a plumbe to kéepe your inſtrument paralell and vpright for which there was no conuenient place in the Theodelitus, which this Glaſſe hath ſufficient, though it be not néedfull, by reaſon of the large croſſe Néedle which performes the ſame. Briefly, theſe impediments and defects well conſidered, let any
Theodellitus.
Index that beares the circular ſight, vpon the o degrée of the circle in the Planiſphere, remaining there, erect the ſame by looking through the ſights vnto the firſt angle, then the inſtrument reſting, conucy the ſaid Index to the next angle vpon your right hand, and note downe what degrée the fiduciall edge of your Index doth cut in your table booke. doe ſo from angle to angle rightwards vntill you come vnto the laſt, and write them downe in your table booke, in manner as folIndex towards your right hand at pleaſure, obſeruing through yt ſights ſome marke, a conuenient diſtand from you, according to the quantiy of the field, and there muſt be your ſecond place, At this place, plant your inſtrument by helpe of your backe ſight, in ſuch order that the line where the degrées take beginning may point to your firſt ſtation. And here likewiſe you muſt beginne at the anIndex proceeding from angle to angle rightwards vntill you come to the laſt, ſtill noting the quantity of each angle done as before.
Theſe angles thus obſerued at both ſtations reſort vnto ſome plaine and ſmoth peace of vellam or paper, and there deſcribe a circle, and by helpe of a protractor (but indeed the cord diuiſions vpon my Staffe be moſt excellent) limit out euery ſeuerall angle
And to get the diſtance, diuide the ſtationary line, or line intercepted betwixt the center of the two circles, into as many equall parts as you pleaſe, and with thoſe very partes diuide the lines intercepted betwixt thoſe places whoſe diſta
We will take Maiſter Digges his own example, a b c are the markes in the field to be meaſured, d the firſt ſtation, where you ſhall ſet the center of your inſtrument, his Diameter or line where the diuiſions take beginning pointing directly to a, ſo doe e f g the viſuall lines running by the angles of poſition of the inſtrument vnto all the angles or markes obſerued, expreſſe my obſeruations at the firſt ſtation d. Laſtly doth h note 90. degrees, which directeth to the ſecond ſtation m. ſo is d m my ſtationary line, which muſt be meaſured, and is 300. yard: s whereby I gather a Table, thus:
Then going to the ſecond ſtation m, where ye ſhall now place the center of your inſtrument, the line where the degrees take beginning, pointing iuſt from m to d: ſo do the viſuall lines i k l, running to the markes before noted, cut new angles of poſition, which you muſt collect as before in a Table, thus:
Now if ye marke diligently where each match lines do croſſe one the other, there is the true proportion of ſuch places, and ſo by drawing right lines from thoſe interſections, if it were a field, or ſuch like, you haue the bounds and limits thereof. And if the diſtance betwixt any two places or markes bee required, ſeeke the ſpace betwixt any two places or markes bee required, ſeeke the ſpace betwixt the two ſtations d m, whicg as I formerly ſaid is 300. yards, I diuide d m into 18. equall parts, and demande the diſtance betiwxt a b, which conteines 11 of thoſe 18 parts. Then ſeeing I am ignorant what number of yards be conteined in thoſe 11 parts, I fly to the rule of proportion, ſaying
Geodetia, and meaſuring grounds, as for Coſmographers and ſuch like. Let therea b c d e f bee a peece of ground ſhut vp within a Marſh or Riuer in ſuch ſort that you cannot approach to the ſame to meaſure it, as the common g, farre without the ſaid péete of ground, from whence you may view all the angles and corners therein, and there, as at g, obſerue all the angles, as you did at the firſt ſtation in the laſt Chapter, and ſo ſeeke one a ſecond ſtation, as h, and thence obg f, g a g e, g d, g c, g b, and h a, h f, h
I néede not many words and therefore proceede therein, as you did in the laſt chapter: ſo ſhall a f, f c, e d, d c, c b, and b a, bee the true bounds of the field.
To doe this Propoſition without calculation.
Appoint your firſt ſtation g in a knowne diſtante from your ſecond ſtation h, as ten ſcore, and when you come to protract, do not ſet the ſaid two ſtah downe at randon, as I taught you in the third chapter, but appoint their diſtance by your ſtale according to the true meaſure ten ſcore or 200. yards, euen as you found it in the field then protract the angles at both ſtations, and note their interſection as you bee wont, which
f e, apply the length of that time vnto the ſcale that you ſet g h by: ſo ſhall you finde f e 12. ſcore. In the ſame order, without Aa f, e d, d c, c b, and b a, which is all the bounds of the field. After the ſame order may you mete g f, g a, g e, g d, g c, g b, or h a, h f, h e, h c, h d, h b, or any croſſe line ouer the field for the caſting vp of the contents, as f d, f c, f b, or a d, a c, &c.
In the ſame order may you performe this Chapter by the Geodeticall Staffe, and by other inſtruments which were ouer tedious to repeate in the vſe of euery inſtrument: and therefore I am to ſupply that in one, which is wanting in another, which being knowne you may vſe in any.
Index being placed vpon the DiamePlaniſphere, or body of the Inſtrument reſting, conuey the Index from angle to angle vntill you haue gone round, and then meaſure the ſides conteining euery angle, noting the ſame down againſt the proper angle, euen as you bee taught in the ſixth booke of the Geodeticall Staffe, in the third Chapter, treating of this Propoſition: and then protract as there you be inſtructed, or as in the 27. Chapter.
Now looke how you obſerue the angles and lines in the field, and in the ſame order muſt you protract them vpon your paper.
Theodelitus, ſo alſo may you vſe the Néedle, by kéeping the néeCircumferentor to bee bor
TO performe this, you muſt aſcend to the top of ſome high Hill, Eliffe, or Tower, from whence you may directly behold the ſituation of the Country, as it lyeth adiacient round about in your Horizon, we hold the Semidiamer of the Horizon to conteine 180. Stadias, and ſo farre may one ſee: for we muſt alwaies when wee be at this worke, imagine our ſelues to be in the center of the Horizon, and thence neceſſarily muſt ſee to the Circumference,
which is limited from vs by the Semidiameter. But to proceede, your place being appointed, there ſet vp the Topographicall Glaſſe vpon his ſtaffe, ordering it in ſuch ſort, by helpe of the Needle, that the foure Semidiameters thereof may point iuſt Eaſt, Weaſt, North and South, euery one pointing correſpondantly into his like quarter of the heauen: then turne the Index with the ſights to euery Towne, Village, or Hauen, or whatſoeuer you deſire to place in the Map, eſpying through the ſights, the middle, or moſt notable marke in euery of them,
as commonly the Stéeple,
if it be a Church Towne, noting at euery of thoſe places the degrees cut by the Index in great circle, and alſo the parts of the degrees, which are properly called angles of poſition, and collect you a Table of your firſt ſtation thereby.
Then caſting your eye round about, ſearch ſome mountaine or lofty place from whence againe you may view all theſe places and appoint that to be your ſecond ſtation then turne thereto the Index & note alſo y e degree cut, this done repaire vnto your ſecond ſtatio ̄ formerly found, where ſituate your Topographical Glaſſe in al reſpects as he was at the firſt ſtation, turning the Index and ſight about, ſtill obſeruing all ſuch markes you ſaw before, and note againe the degrée cut, or angles of poſition, writing the name of euery place and his angle by it, ſo haue you collected a ſecond table, which is for your ſecond ſtation.
Theſe things ſo done take a ſkinne of vellam, royall paper, or what you pleaſe, and in ſome place thereof appoint your firſt ſtation, about which deſcribe a circle which you muſt diuide into 360 degrées, beginning in that quarter of the world in which the beginning of the degrée in your Inſtrument were placed, or elſe protracting them by your Staffe, as you be taught,
betaking either of the waies from the center of this circle, to euery degree noted in your firſt table, there muſt right lines be produced infinitely, noting to euery of them the name of his place, now protract the line of your ſecond ſtation, according to the degrée cut, and vpon that line deſcribe another circle, which vſe in all reſpects as you did the former, taking direction from the table, obſerued at your ſecond ſtation.
To conclude, diligently note the concurſe or interſection of euery like lines making there on ſome marke, as ☉ with the name of the place correſpondent, and ſo you haue finiſhed.
Now to know how farre euery of theſe townes &c. bée diſtant from other, doe thus: meaſure the diſtance betwixt your ſtations by your Geodeticall Staffe, or this Inſtrument, as you ſhall bée after taught, or by any other Inſtrument, to you ſeeming beſt, and diuide your ſtationary line, or line included betwxit the center of the circles into ſo many equall parts as there be miles, furlongs, or ſcores, betwéene your ſtations, this line ſo diuided, meaſure the diſtance of any place thereby, as you doe with an ordinary Scale in a mappe, taking the diſtance of any two places with your compaſſe, and applying the widneſſe to the diuided line, for ſo many equall parts as bee then included betwixt the
féete of your compaſſe, ſo many miles, ſcores &c. is it betwéene the two places according to the denomination of the diuiſions in the ſtationary line.
I am deſirous to ſet downe certaine townes in the County of Sallop, according to their true porportion and the epact diſtance of euery place from other, chooſing therefore a lofty place for this purpoſe, as the Cordocke hill, from whence I may behold all my deſired places. My inſtrument there ſituated as is declared, reHopton Caſtle, which hauing receiued through my ſight, the fiduciall edge of the Index cuts 18 degrees, remoouing the Index to the next towne Montgomery it cuts 70 degrees, againe the next Knookin Castle, it cuts the 134 degrees, and ſo I proceed rightwards from towne to towne, vntill I haue finiſh
This done, I behold an other high hill as the Wrekin hill from whence I may obſerue all theſe places, and turning the Index thereunto I find the degree cut to be 213 ½
Then carrying my Inſtrument to the Wreking, and placing him in all points there as it was vpon the Cordoke, I turne againe my Index to the firſt towne before noted as Hopton Caſtle, and noting the degree cut, I find it 25. then to the next Montgomery 52 ½, and ſo to the reſt, as ye may perceiue in the table enſuing.
With theſe tables repaire vnto ſome ſuch place whereon you would protract the worke, drawing therein a circle vpon the cenf as you ſee in the figure, which you muſt diuide inf, pul out right lines by euery grade, noted in the firſt Table, ſo is f p Hopton C
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 let
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 re
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 folSalop: the reſt you may gather by your Scale in the ſame manner.
In the order before ſet downe changing your ſtations (as ha
THis kind of worke,
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.
Suppoſe I aſcending to the top of Stretton hils (which be cerWrekin, 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 ſitug 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.
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 Alhic 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.
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.
g a b is 46 degrees, & g b a 60 degrees: therfore by the 2 Booke, Chap. 15 of the Geodetia g b, now hag a b to be 71933, of a g b 93969, and of g b a 89879, and to get the dia g, or g b, doe thus, multiply the ſigne of g b a or g a b by 10, and part the product by the ſigne of a g b, ſo haue you a g or g b in the ſame meaſure as a b is expreſſed: as if I deſire the length of b g, firſt I multiply the ſigne of g a b 71933 by 10, and there is made 719330 which I part by the ſigne of a g b, viz. by 93969, ſo haue I the quotient, 76/9 1/3 5/9 4/6 7/9 miles, the diſtance of the Wrekin
a g,
a f, a e &c. or d f, d e, b d, &c. remembring alwayes to multiply the ſigne of the angle, conteining the line ſought, by the line knowne, and diuide the product by the line of the angle containing the ſaid knowne line. And for your better vnderſtanding, I will ſet downe euery Triangle with his reſpondent ſigne, ſo that you may finde euery ſide of the ſame.
Wee will only ſituate Shrewsbury in true place, proportion, & Symmetry, you ſhall therefore draw a line a b, which diuide inviz. ten miles, and according vnto thoſe parts you muſt make a ſcale as long as you pleaſe, as h i. Now place the one foot of your compaſſe in h, and extend the other to the diſtance of Shrewsbury, and from the Wrekin, according to the doctrine you found it before, viz. 76/9 1/3 5/9 4/6 7/9 miles, the compaſſe reſting at that diſtance, place the one foote in b, and with the other ſtrike the portion of an arch: do ſo with the diſtance of Stretton hils from Shrewsbury vpon the point a, and the concluſion will bee, that the interſection of thoſe two arches appoints the true place of Shrewsbury, as g. In like maner muſt you ſituate all the other Townes in their proper places, and then it reſts at your pleaſure whether you will finde the diſtance of each one from the other,
c d b, &c. but hauing placed the Townes, the application of the Scale is moſt ſpeedy and ready, without more trouble to finde the di
Topographicall Glaſſe I ſhall teach you to deliuer Altitudes, Longitudes, Lati3 kinds of waies, as by the Geometricall quadrant, Hypſometricall Scale, and by protraction, and becauſe this Quadrant is vſed by many, and alſo contriued in ſome InGemma Friſius, Orontius, &c. writing of the vſe thereof, conceale that to themſelues, but hauing occaſion in this booke (becauſe it is proiected vpon my Glaſſe) to ſpeake of the vſe, I will likewiſe take occaſion to acquaint you with the reaſon of the worke, in a briefe mannner.
Behold the inſuing figure, for the ſites of the ſquare ſ k and k l, (whereof the one is vmbra recta, the other vmbra verſa) are no other thing then the Tangents of leſſer circles in the ſemi
Therefore if you ſay.
As a l the whole Seale, is to l r the equall parts of the contraa c the diſtance, to c b the Altitude.
Which is no other then if you ſhould ſay.
As a l the Radius, is to l r the Tangent, ſo is a c the diſtance to b c, the altitude,
Therefore the Tangents in the Semiquadrant of the leſſer arkes may ſuffice, becauſe there is the ſame proportion of the Tangent to the Radius, which the Radius hath to the Tangent of the complement wherevpon theſe conſequences may be infer
I As if you ſhould ſay.
As d p is to p v ſo ſhall it come all to one purpoſe ſayn
As t o is to o d.
II IF the Tangent p v or o x be altogether required, you may eaſily find the one or the other — For
As t o is to o d, ſod p to p v, —and as w p (to whom r l is equall) is to p d, ſo is d o, to o x, for a diſtance being got by the helpe of two ſtations, then oftentimes the Tangent p v, or o x, may be defited: in ſuch a caſe when it ſhall happen ſay.
As v w is to w p, ſo is a d to d c, or
As t x is to t o, ſo is a d to d c, take which way you pleaſe.
But it is fip v or o x be knowns, that acw p (that is to ſay r l) and p v, or of the complement as o f and o x — becauſe
I The Tryangles compoſed of d x o, and b
be equiangles:
Therefore
As x l is to t o, ſo is a d to d c.
II The Tryangles made of d v p, and d b c, are equiangled,
Therefore
As v w, is to w p, ſo is b z to z c, and
And as b z, is to z c, ſo is a d to a c, becauſe d z, is parallel to the baſe a b, in the tryangle a b c
Therefore to conclude.
As v w, is to w p, ſo is a d, to a c.
Nam quae conueniunt vni tertio, etiam inter ſe conueniunt, b p.
Topographical Glaſſe diuers waGeodeticall Staffe, or any other Inſtrument, truely expleſ
You ſhall therefore plant your Inſtrument at the place from whence the diſtance of the Caſtle or ſuch like is ſought, making the diameter where the degrées take beginning to point iuſt to the Caſtle, the Planiſphere ſo reſting, moue the Index to 90 degrees, and ſo through the ſaid fights eſpy ſome marke 100 yards or more diſtant from you, or wanting a tree, cauſe one to place a marke by the directions of the ſights, in a knowne die angle betwixt your firſt ſtation & the caſtle, which note downe, and ſo find the ſignes and conſequently the diſtance, for as the Radius is to the Tangent, ſo is the ſide giuen to the ſide ſought.
Example.
And if the diſtance of c a be required, multiply 211273 the ſecant of the angle b c a, by 73, ſo haue you 15422929, which parted by the totall ſigne, leaueth 154 22929/100000 perches your deſire, and thus muſt you deale with any other like queſtion.
Geometricall Quadrant is proiected vpon the Planiſphereof your inſtrument: therefore, place the Index vpon that Diameter where the parts of ye quaIndex vpon the beginning of thoſe degrées, may iuſt point to your firſt ſtation, the body of your inſtrument reſting, remoue the Index to the marke whoſe diſtance is requiredQuadrant cut by the edge of the Index: then are you to conIndex touch amongſt the 60. parts vpon your left or right hand. Firſt, if the 60. parts vpon the left ſide the Index be cut, you muſt increaſe the ſtationary line by the number of thoſe parts cut, and the product diuide by 60. ſo is the quotient, your deſire, but if the Index fall vpon the parts of the ſcale vpon the right hand, you muſt then multiply your ſtationary line by 60. and diuide by the parts cut.
Or you may reduce the parts of the right ſide to the propor
Diuide the ſquare of 60. by the parts cut in the right fide of your ſcale, the quotient is the parts proportionall, which you muſt increaſe by the the diſtance of your ſtations, diuiding by 60. ſo is the quotient the true diſtance of your marke from your firſt ſtation.
And if the Hypothenuſall, or diſtance of your ſecond ſtation from the marke be required, ſquare your ſtationary line, which adde to the ſquare of the diſtance of your firſt ſtation from the deſired marke, the roote quadrature whereof is your de
Example.
A is the place whoſe diſtance is tequired, b the marke where my inſtrument was firſt diſpoſed, whence (as in the laſt Chapter) I depart orthogonally to c, the Index cutting 37. parts, and a
Or by diuiding 3600. the ſquare of 60. by the parts of the Quadrant cut, as 37. and better, the quotient ſhall produce you a a proportionall number: which number part by b c 73. the quotient whereof is the longitude of a b, as before.
And take this note with you, that the Index will neuer cut in the parts of the Quadrant vpon the left hand, vnleſſe the longitude ſought be ſhorter then your ſtationary line; I meane, vnleſſe the line knowne, or meaſured, bee longer then the line ſought. And further note, that the more parts you diuide the ſides of your ſcale into, the eaſier and truer ſhall you worke.
Another way theſe kindes of longitudes be performed, and that is by protracting after obſeruation of the angles, which for that it is already ſet downe, I omit it heere, onely you may
here protract with a circle readily diuided,
as hereafter you ſhall be taught, if your ſtaffe bee wanting.
LEt it be ſuppoſed that you were ſtanding in an open field, and a certaine Caſtle and a Fort ſhewed you, to neither of which you could approach, and yet vpon ſome occaſion were required to deliuer the diſtance betwixt the ſame.
The firſt thing in performance hereof that you are to doe, is to plant your Glaſſe at the place where you meane to make obſeruation, and then the Index vpon the beginning of the degrées, turne the Inſtrument & all about vntill you eſpy the Fort vpon your left hand: the Planiſphere reſting, conuey the Index to the other Caſtle vpon the right hand, noting the degrées cut by the fiduciall edge of the Index. Now are you to view ſome other marke for a ſecond ſtation vpon your right hand, whereunto turne the Index, vntil through the ſight you eſpy the ſame, noting againe the degrées cut but by the Inand it will bee the better to let the Index vpon theſe laſt degrées bee about 90. for the neerer that the ſtationary line, running from your firſt ſtation to the Fort vpon the left hand be to conteine a right angle, the better it is.) Now leauing ſome apparant marke where the center of your Glaſſe was, take vp your Glaſſe, & beare the ſame to the place appointed for your ſecond ſtation: and hauing planted him there, turne the Glaſſe, the Index vpon the beginning of the degrées, vntill through the ſight you eſpye the marke or man left at your firſt ſtation: the Planiſphere reſting, conuey the Index vnto the Fort vpon your right hand, noting the degrées cut next to the other Caſtle or Turret more rightwards, noting the degrées cut: finally, meaſure the diſtance betwixt both your ſtations, all which note downe, as well the angles obſerued at your firſt ſtation, as thoſe obſerued at your ſecond, as alſo the quantity of the line that is included betwixt both thoſe ſtations, and protracting the ſame vpon paper, by your Scale and compaſſes, you haue finiſhed.
Example.
Admit I ſtanding at a, am deſired to deliuer the diſtance bec a certaine Fort, and d a certaine Turret, planting my Inc a d 49. degrees: then c a b 107. Theſe I note downe for angles at my foreſaid ſtation. Now the Index reſting at a, eſpye through your ſights ſome other marke for your ſecond ſtation, wherunto bring your Glaſſe, as to b, where being duly ſituate, make the Index ſtanding vpon the o. degree in the planiſphere, point iuſt to a, the Glaſſe reſting ſo fixed, turne the Index with the ſights to c, noting the angle a b c 54. degrees. Next remoue the Index to d, noting the angle d b a 104. In concluſion; by your chaine, or ſome other rule in this booke, mete the line a b, which is 56. ſcore, which had, protract thus:
Draw a line l i, whereupon lay downe by your ſcale and come f, then vpon e ſtrike the portion of an arch, as k i, whereon protract an angle of 107 degrees, i e g, then one other of 49 degrees, as g c h.
If you will performe this by ſignes worke in all reſpects as in the 12 Chapter, for look how you found the diſtance of a g, ſo muſt you here of c d.
To teach you to ſeek Latitudes, as ſome do, by the Geometrical Quadrant, I hold it too tedious, for that you muſt firſt go finde the diſtance of each place from your ſtanding, and after vſe reQuadrant of himſelfe or as he is here proiected (for it is all one to work by either) moſt fit: ſo long as you may haue alwaies a right angle in your worke, and therefore I will apply the ſame to Altitudes, and concerning this kind of Latitudes, hereby brought to finde the diſtance of ſhippes vpon the ſeas, of Armies vpon the land, and ſuch like, for indeed as it is tedious, ſo is it ſcarce poſſible to ſit you with a de
Topographicall Glaſſe 4. kind of waies, that is, by the Hypſometricall Scale, Geometricall Quadrant by Sinical working, and by protraction, thrée of which waies I will here deliuer vnto you, as for ye
Scale, you may worke according as in the 3 booke of my Staffe, and firſt to performe the ſame by the Geometricall Quadrant, you are to plant the body of your Glaſſe paralell, mouing the Index vntill hee point iuſt to the Altitude required, then muſt you moue the ſight vpon the demici90 you eſpy the ſimile of the required Altitude, which done behold the parts of the quadrant cut by the moueable ſight vpon the backe ſide the ſemicircle, and conſider if the ſection were made amongſt thoſe parts that ſtand néereſt to the ſight that is fixed in 90, or in the parts furtheſt from the ſight, or in 60, the midſt betwixt both.
1 If the ſection were made in the parts neereſt to the ſight the deſired Altitude is greater then your diſtance from ye ſame ſo that ſuch proportion, as 60 hath to the parts cut, the like hath the giue
2 If the ſection were made in the parts of the quadrant fur60.
3 But if the ſection be made in 60, then the giuen Longitude is equall to the propoſed Altitude.
Example.
In taking of theſe altitudes you muſt note that you meddle with no part of the ſame, but that which is aboue the leuell of your eye, the which leuell you ſhall bee taught hereafter to obſerue.
Otherwiſe.
Hauing taken the angle of Altitude, you may worke this propoſition by ſignes &c. according as you be largely taught in the 7 booke of my Staffe called Trigonometria, Prob, 1.
Otherwiſe.
Hauing taken the angle of Altitude you may performe this Chapter by protraction, as you be plainely taught in the ſixt book of Geodetia, Chap. 38 whereunto for breuities ſake I referre you, and the rather, for that it is performed after one and the ſame me
Becauſe this chapter is performed by the Quadrant in my 4. booke of the Geodeticall Staffe, To take all kinde of altitudes by the Glaſſe. Chap. 4. and that it differeth noChapter, I will referre you thereunto, where in deed looke what is there ſaid either in the third or fourth booke of the Scale or Quadrant, the ſame may you performe in the ſame method by this Glaſſe, when you haue once obſerued the parts of the Scale or Quadrant cut: Therfore it would be more then néedeth, here to repeate it againe.
But becauſe theſe kind of inacceſſible heights be deſired of ma
You muſt finde out two ſtations in a knowne diſtance, as you pleaſe, where obſerue the angles of altitude, and ſo get the come tangents of both thoſe angles by the ſeuenth booke of my Staffe, noting the difference of the ſaid complements: for as the difference is to the Radius, ſo is the difference of the ſta
Example.
We haue obſerued the angle a b d to bee 29. degrees 40. min. and a c d to be 46. degrees, the Tangent of the Complement of 29. deg. 40. min. is 175556. and of 46. degrees, 96568. Now ſubtract 96568. from 175556. and there reſts 78988. Then multiply 100000. the totall ſigne by 90. the diſtance betwixt. both your ſtations (for ſo many feet I found it by meaſure) and there
Otherwiſe.
Hauing obſerued the angles of altitude, you may performe this Chapter by protraction, in all reſpects according to the 39. Chap. of my booke of Geodetia.
g c a Turret, whoſe height aboue the leuell of your eye, is required be therefore dia, the fixed ſight in the 90, degr. then bring downe the
e. Now your ſight paſſing from a by e, will apprehend ſome place in ye Tower, as b. I hereby conclude, that b is equall in height with my eye fixed at a, and that if I had taken the altitude of this Turret. I had told you no more but from b vpwards, as 89. feete, and to haue knowne the height of g c, I muſt haue thereto added the length of a f, or b g.
But if you know not the whole Altitude, firſt obſerue the ane degrées of both theſe angles, next meaſure the diſtance of the Tower from you, and then conclude, as the Radius is to the diſtance ſo is the difference of the Tangents of each angle vnto ye Longitude ſought, therefore ſubtract the leſſer Tangent from the greater, the remainder increaſe by the diſtance of the Tower, the product whereof, parte by the totall ſigne, and the quotient i
Example.
B c is a Tower, and you ſtanding at a are required to deliuer part of the Altitude, as c h, firſt therefore I obſerue the angle h a b, 11 degrees, then c a b 27 degrees, next I meaſure the diſtance a b, which I find 178 feete, theſe had, I take the Tangent of 11 de
e left ſide the quadrant, that is, by the parts fur90, which to do, you muſt diuide the ſquare of one of the ſides by the parts cut in the right ſide your ſcale, that is by the parts néereſt to 90, ſo is the quotient the parts proportional, which muſt be multiplyed by the diſtance of the ſtations, and di60 parts, whoſe ſquare is 360, let the parts of the right ſide cut be 30, 360 diuided by 30, leaueth 12 this 12 you muſt agument by the ſtationary line, and the offcome parte 60 ſo haue you your deſire.
IN performing hereof, you muſt obſerue the angle of Altitude made at both the markes in Altitude, whoſe diſtance is required, noting y e parts of the Geometricall Quadrant cut by the moueable ſight, and alſo note if the parts cut at both your obſeruations were in the ſide towards 90, which is the right ſide, or in the ſide fromwards 90 which is the left ſide, or if at the one obſeruation the parts cut were in one ſide, and the other in the contrary ſide.
1 If the parts cut at both the angles were in the left ſide (which is furtheſt from 90) ſubtract the leſſer from the greater, and with y e which remaines augment your ſtationary line, which parted by the whole ſide as by 60 leaueth the deſired Longitude
2 Now if the parts cut at both lines were in the right ſide towards 90, by the laſt Chapter, reduce thoſe parts into parts proportionall and then worke as in the laſt difference.
3 Or if the parts cut, be at one line in the left and at another in the right, then muſt you reduce the parts cut by the right ſide into parts proportional, as in the laſt Chapter, which done (as in the firſt difference) deduct the leſſer from the greater, & the product increaſe by your ſtationary diſtance, which diuided by the whole ſide, yeelds your deſire, and ſo muſt you deale, if the one ſection were made in 60 in the midſt betwixt the right and left ſide.
Example.
B c is a Tower, and you required to deliuer a certaine length in the ſame, as b z, a. I appoint my ſtation 129 yards from c, now planting the Glaſſe truely, I place my eye at a, taking the angle z a c, and note the part of the Quadrant cut by the moueable ſight vpon the backe ſide the demicircle, which is 20, the Glaſſe reſting paralell and equidiſtant to the Horizon, mouing the ſight vntill
a to b, as at m, and ſo doe I finde the parts cut to be 43. both vpon the left ſide the Scale k l, (which I perceiue in my Glaſſe, for that both the ſections were made in the 60. parts furtheſt from 90. Therfore by the firſt difference of this 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 exa
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.
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 ye center of the earth: or at leaſt, how much either of their perpendicular al20. 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 Inſtrue ſide of ye 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 ſupere ſwelling of ye ſame, aboue the true conuexity of the earth, for you muſt vnderNoahs flood, by the raging of the water, which forced ſtones, trées, and earth vp
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 cenb, (by ſome Inſtrument dec b a 87, degrées, then muſt you go to the hill c, and there againe get the angle b c a, 74. de161. which taken from 180. leaueth 19. the angle c a b made at ye cenc 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 greac a.
Now to get the height of b or c, aboue the true circular con34364/11. from the liue b a, ſo haue you the height of the hill b d foure ſcore: doe ſo to c a, ſo (for exc 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
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 al
This chapter may finely be performed by your ſtaffe, for that you haue thrée angles and one ſide giuen.
But if you would make experience of the height of hils onely by a leuell, your beſt way is to finde out a third hill alike, or néere a like diſtance from both the other hils, & ſo may you more truly iudge of both their heights, and alſo of the difference thereof from the top of that hill, euen as you bee directed by your line of leuell paralell to the Horizon.
non poteſt eſſe vacuum in rerum natura. And for a familiar example: Take a number of quilles cut off at both ends, and ioine them cloſe together with waxe in a circular faſhion, and put the one end thereof into a veſſel that hath water in the bottome, let the other hang ouer the veſſell brims, the lower the better, it ſufAqua aſcendit, quantum deſcendit.
But to know if water will come in pipes, after the ordinary faſhion to any appointed place, you muſt firſt know, that the ground whereupon the pipes lye, muſt be lower at eue
But ſay there be certaine hils betwixt the head of the ſpring and the place wherto the water ſhould runne, in ſuch a caſe you muſt plant your Glaſſe at the head of the ſpring as before, and looking through the ſight, note ſome marke in the next hill to
Example.
A is a place where water is found, and the queſtion is, to know if it will be brought to the Tower at b, which is diſtant from a 4 miles, I plant my Inſtrument therefore at c, ſo that the diameter p q lye paralell, and alſo equall in heights to a the ſpring head, then looking through q p towards b, I cannot ſee the ſame by reaſon of a certaine hill, that is betwixt me and b, therefore I obſerue a marke in that hill through the ſight, as at c, where a
Note that ſome hold pipes of earth baked, to be better then lead, and ſome hold pipes of alderwood, firre, pine tree, of ſuch wood that hath roſen in it to be better then the former.
And if the ground lye reaſonable leuell, ſo that you conuey the water by trenches, order the ſaid trenches ſo by helpe of a plumbe, that the water may haue currant 4 ½ inches at euery miles end, then fill the gutter with pibble ſtones a foote or more deepe, and vpon them throw earth, ſo will the water run more cleere to the place appointed.
Note laſtely, that it is beſt, if you bring your water by pipes,
to let it come by many croked turnings, and ſometimes to fall directly downewards, and then againe to riſe by little and little, & by this meanes ſome thinke one may force the waters iſſue to be aboue the head of the ſpring, ſo that in pipes you ſhall not need the foreſaide abatement.
But now whereas in conueiance of waters by this way, whether it be waters to houſes or new riuers, ſometimes happely you ſhall meete with a deepe valley, out of which you cannot get the water by ditches, and to finde leuel ground you cannot, without going a great compaſſe: and hauing found it happely cannot haue liberty for to cut a trench through the ſame: ſuch a matter and ſuch a difference I ſaw in bringing the new riuer from wards Ware to London, for remedy whereof if the floud be great, you muſt erect arches in manner of a bridg,
which may extend it ſelfe ouer the valley euen from the one banke to the other, and if the water be but for a houſe, poſtes may ſerue to beare the ſame: the like muſt you do, if you meete with a riuer, brooke or ſuch like.
M. L. doth teach you how to ſolder your pipes of earth or wood, & that is with vnquenched lime and hogs-greaſe, or with roſen and white of egges, or with lime, white of egges, and fylings of Iron.
Index with the ſight vpon ye Diameter in the Planiſphere where ye degr. doe take beginning (noting that a ſtationary angle, is ſuch an angle that hath no reſpect to the néedle, but to the ſtation) The InſtruStaffe, you ſhall mooue the Planiſphere (the Index fixed as before) vntill you eſpy through the ſight the one marke vpon your left hand, and for hedges, vntill the Index lye paralell with the hedge, the Inſtrument ſo reſting conIndex with the ſights vnto the marke or hedg vpon your right hand, making the ſaid Index point to the marke, or lye paIndex, for that is the quantity of the angle. This néedeth no Example.
a b c, vpon the center f which di180 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 néeda 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 ye inch or more, & then take 12 of thoſe parts, which diuide into 11 parts, & make a Scale, the vſe whereof you may pGeodetia 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 half
Glaſſe, it reſteth that you alſo learne how to protract the ſame, and to lay downe the ſides thereof vpon paper by your
Firſt, therefore to protract an angle vpon any point giuen you are to take the ſemidiamef a,
g, the point giuen, with the other ſtrike the portion of a circle h i k l, now muſt you 40 degrées. place therefore the one foote in a in the protractor, and extend the other to 50 degrees in
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 ye 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 regh g ſhould be, which let be 18 perches, therd in your Scale extc, as to 18, tg h, fromwards g towards h as to n, I conclude g n i18 pearg 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.
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 proSEWN, 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.
To performe it therefore by angles ſtationary, you muſt go vnto one of the corners vpon the vtter ſide the field, where plant your Glaſſe, and ſo take the quantity of that angle by the 23 Chapter, then meaſure from that angle vnto the next angle no3 angle, noting the ſame orderly downe, & ſo go round about ye field, ſtil taking the quantity of euery angle by the ſaid 23 Chap. noting the ſame downe together with ye quan25 Chapter, protract euery ſeuerall an25 Chapter:8 Chap. part. 1. of my Arte of Geodetia.
e field from whence you may command the view of al the angles in the ſame field, the quantity of which angles you muſt obſerue by ye
23 Chap. meaſuring the diſtance of each angle from your Inſtrument, and ſo protract the ſame, and lay their ſides downe as in the 25 Chapter, and this way I hold the beſt and trueſt in all ſuch caſes where the propoſed field is not ouer great, and if ſo be that the plat be not required, but you put to meaſure the ſame, your trueſt worke will be by this meanes, to get the true plat thereof firſt, and afterwards finde the contents, by the 2 part of Geodetia. and for that this chapter is alſo already performed in my firſt part of Geodetia, Chap. 3. after the ſame methode as it is heere, I will referre you therevnto for breuities ſake.
And here note, if you pleaſe, hauing once planted your Planiſphere in ſuch order that the Diameter where the degrees doe take beginning may point into ſome one angle or other, you ſhall not neede any more to moue the ſaid Planiſphere, but onely keep him fixed, moouing the Index from angle to angle round about, noting the degrees cut by the Alhidada in the Planiſphere at eue25 chapter,
Theſe two kinds of waies remembred in this chapter and the laſt I do hold the beſt & trueſt, as for others many of them be vnPſaine-Table, the which if you can worke it there, you néede not be to ſeeke here. Many o
e firſt part of my Geodetia, that it was not poſſible by one ſcale to lay downe the true plat, and alſo by the ſame ſcale to renGeodeticall Staffe, where I al
In the very place where the aſcent beginneth, there plant your Glaſſe, and in the top of the aſcent, a ſtaffe of equall height with your Inſtrument. Now turne the Index with the ſights towards the ſtaffe, the planiſphere being paralell and equidiſtant to the Horizon, remoue the moueable ſight in the demicircle vn90. you ſée the top of the ſtaffe, note then the degrées cut by the moueable ſight for the anFor as the Secant is to the Radius, ſo is the Radius to the ſigne of the Complement of the ſaid Secant, by the ſeuenth booke of the Geodeticall Staffe, called Trigonemetria.
Example.
If the grounds deſcend, as b d, go to b, and then worke as before.
But you ſhall not need to make this reducement in the fields, vntill you come to protract.
And note further, that if the ground aſcend as a b, and alſo deſcend as b d, then muſt you in your protracting adde the Horizontall line a c, and d c together, and protract that in ſtead of the lines a b and b d, ſo that by what is ſaid before, you may gather that the line which you meaſured in the field, to be 40. pearches, and riſing 30. degrees high, if you will worke truely, ſhould be laid downe not altogether 35. pearches.
Staffe, and for that there were ſome things omitted, and ſome things alſo miſtaken in the ſaid booke, I thought it not amiſſe heere to deliuer certaine compendious formes of working by the ſaid Tables: that is, how to worke by any Tri
Compendium I.
If the ſigne bee in the firſt, and the Radius in the ſecond or third, how to bring the Radius into the firſt place, for the auoi
Regula.
For as the ſigne is to the Radius, ſo is the Radius to the ſe
Compendium II.
If the Tangent bee in the firſt place, and the Radius in the ſecond or third place, to reduce the Radius into the firſt place, for the auoiding of diuiſion.
Regula. For the Tangent placed in the firſt place, take the Tangent of the Complement; Et negotium confectum erit.
For as the Tangent is to the Radius, ſo is the Radius to the Tangent of the Complement.
Compendium III.
If the Secant be in the firſt place, and the Radius in the ſe
Regula. For the ſecant put in the firſt place, take the ſigne of the Complement, ſo ſhall you haue a true 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 con
Example.
Now for the corners and fragments that doe remaine, you muſt meaſure them according vnto the figure that they moſt re
But ſay the irregular field lyeth moſt apt to be reduced into a triangle, which is thus performed.
Let a b c bee a certaine plaine field, which you are put to meab c a pera. Firſt thereb c, as néere as I can geſſe vna, as at d, then I moue the Indexes about, vntill
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 ba, 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 oIndexes 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.
Eaſt, Weſt, North, and South.
Now you are to ſeeke in your Tables gathered at your obſer
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.
peninſula, diuided into 52 parts (but not equall parts) which we call Shires, then is euery Shire ſubdiui12) 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 diui11 and 12 Chapter, ob
The other way is, to find by ſome records what townes and
a Dane, and diligent obſeruer of the motions of the celeſtiall bodies, found at laſt by conferring daily practiſe and experience with the optikes, that the Sunne ſeemed to vs eleuated higher vpon the verticall circle, then indeed he is, and his reaſon hee drawes from Alhazen and Vitello, ſaying: Quandores viſibilis per diuerſas Diaphanitates ſpectatur, refracte eius forfor they appoint the heauens the Elements &c. to be Diaphanicall, but Tycho would haue the principall cauſe of this refraction to be in the vapours, that doe continually occupy the loweſt Acry Region, abounding and gathering themſelues moſt together, whe
Alhazen and Viis 40 / 50 // of which the
12 Ge
T. B. where the deſirou
30 minutes below the Horizon, neither doth the magnitude thereof hinden the ſame, although hee ſeeme
continuing from Sunne to Sunne, bee longer according to the apparent riſing and ſetting thereof, then thoſe our Aſtronomicall calculations for the Poles eleuation,
But to ſet apart further diſcourſe hereof, behold the Tablet, he vſe whereof is this:
Séeke the Altitude of the Sunne in one of the rowes vnder Alti. ☉, anſwering to which vpon the right hand vnder the title of refraction is the minute and ſection of the Sunnes refraction, which muſt be abated from the Altitude inſtrumentally obſerued, becauſe the refraction doth alwaies cauſe the ſun to ſeeme higher then indeed he is.
Of the Paralaxes of the Sunne.
IT is euident by the paralaxes of the Sunne in the circle of alCoperni
alſo found that the Semidiameter of the Sunnes
1142. Semidiameters of the earth, which putting aſide ſome few minutes, agreeth with Tycho Brahe. ſo that the Sunne, by reaſon of his Paralax, ſéemes to vs inhabiting the ſuperficies of the earth lower, and more deiected in the heauens, then in déed hée is; by the neglecting whereof his true altitude hath as yet béene falſly obſerued: but héere wee muſt admit a thrée-fold diſtance of the Sunne from the earth: to wit, moſt remote, as being in his Apogaeum, néereſt, as being in Perigaeum, and in the meane betwixt both: that is mouing aTycho, extended ſo farre, that it may ſerue for euery degrée of the Sunnes altitude through all great Brittaine, the vſe whereof is thus:
Take the altitude of the Sunne, the which altitude finde in the row vnder Alt. G. anſwering to which, vnder the title Max. Med. or Min. is the minutes & ſeconds of the Sunnes paralax, which muſt be added vnto the Sunnes altitude, for that the pa
Example.
We will take Tychos owne example, obſerued the laſt of Iune 1588, when hauing a large Inſtrument, prepared for that and o
Hence commeth it that the Poles eleuation obſerued by the Meridian altitude of the Sunne or Starres, and by the Pole it ſelfe differ, if you conferre two of theſe Altitudes together.
4 ½ minutes in20 degree of Altitude the ſtarres haue no re20 degree, the vſe whereof is all one with that of the ſuns refraction.
Index vntill the very edge of the demicircle point iuſt vnto ye Sun, now for ſtarres looke through ye ſight in 90 moouing the moueable ſight vpon the demicircle, vntill through the ſaid ſight in 90, and this moueable ſight, you eſpy the Star, note then the degree cut by the moueable ſight, for that is the Al90, the degrees then cut is the AlPlaniſphere by the Index that poin
And what may you ſee further in this Glaſſe? mary the point of the compaſſe that the Sunne or the Starre lyeth on.
But you muſt remember, after you haue taken this Altitude of the Sunne or Starres inſtrumentally, to make ſubſtractions and additions according to the refraction and paralax of the de
e Néedle, place the Planiſphere of the Inſtrument, ſo that the diameters behold the foure quarters of the world, the ſuune or ſtarre then riſing, obſerue the ſame through the ſights vpon the demicircle, noting then the degree inIndex, which pointeth to
39 Chapter to obſerue the Altitude of the ſun, the moueable ſight vpon the demicircle reſting at the degree of Altitude, note then the degree that the ſunne is in, for the day propoſed, which you ſhall finde vpon the moueable ſight, for the howre line paſſing by the ſame is your demande.
But with this prouiſo alwaies, that you note if the ſunne be in North or ſouth ſignes, for the houre lines of the North ſignes incline to the left hand, and the houre lines of the ſouth ſignes, bend towards the right hand, certainely of all kind of Horologi
Now for the time of the ſunne riſing and ſetting is eaſily col
And to get the degrée of the ſunne, euery Almanacke affoords you that, for towards the middeſt of each month you haue there
14 day, 1610. Sol in Scorpio, and I demand the 26 day what degree of Scorpio the ſunne is in, beginning there15 day I call it one, and ſo telling I come vnto the 26 day, I end with the number of 12 whereby I conclude the ſunne was in the 12 degree of Scorpio. So of any other, here the loſſe of a day doth nothing hinder.
Index that is marked with f vnto the ſaid houre in the Planiſpheare, which ſo reſting, ſeeke the age of the Moone in the circle whereto the Index is fixed, for the houre line in the innermoſt circle in the Planiſpheare paſſing by the ſaid age of the Moone is the true houre of the night.
So likewiſe doth the houre line and the foreſaid Index ſhewe vpon what point of the compaſſe the Sunne and Moone then be, and the number of points included betwixt the ſaid houre line and Index acquaints you with the diſtance of the Sunne and Moone, which the circle in the Peripher expreſſeth in degrées and minutes, which is more then was propoſed.
Seeke as hereafter what Rombe or Wind maketh a full ſea at the propoſed place, and then learne the age of the Moone, theſe two things had, put the Index where 29 ½ ſtandeth vpon the ſaid Rombe or Wind found, which reſting, ſeeke the age o ye moone in the mooueable circle, for the houre in the inward circle of the Planiſpheare anſwering thereunto acquaints you with the houre of the full ſea in the propoſed place, and for your practiſe
The moone ſouth or north maketh a full ſea at Lands end, ſouth and by eaſt at the Gore end, ſouth ſouth weſt betwéene hoe Ware head of Colnes, ſouth ſouth weſt at Graueſend, ſouth weſt at Lon. bridge, ſouth and by eaſt at Portſmouth, eaſt and weſt at Waymouth, weſt and by South along the coaſts vp to Briſtow, and the coaſt of Ireland, from Waterford to Kynſale: if you deſire more, you may haue it of any ſkilfull Mariner, or in the tables of the Regi
One thing note, that it floweth ſooner by one point of the compaſſe in the Spring tides then it doth in any of the quarters of the moone, eſpecially if the Riuer haue any indrafte and di
To the Planiſpheare in this Topographicall Glaſſe you may alſo adde the Celeſtiall Zodiacke, and another circle of the daies of the moneth incluſiuely, the ſame or ſuch like that are placed vpon the Horizon in Sanderſons Globes, by which you may ga
Or thus you may find what ſigne the moone is in, place the Index marked with f vpon a in the Planiſpheare where the deIndex f, noting the degree cut in the circumference. for that is the diſtance of the ſunne and moon, which parted by 30 the quotient yeelds the number of ſignes, and the remainder the degrées; ſo that knowing the place of the
10 day the ſigne is in Aries, and by the rules before I find her diſtant from the ſunne 60 degrees or 2 ſignes: Therefore the moons muſt be in Taurus, the deg. are knowne by the deg. that the ſunne is in, and by the deg. cut by the Indexes as before. Fi
Many things Aſtronomicall might I open in the vſe of this Glaſſe, which for breuities ſake I am forced to omit.
At this time I will conclude the vſe of the Topographicall Glaſſe, hoping I haue ſaid ſufficient to open the whole vſe ther
aequilater paralelogram, made of a board of halfe an inch in thickeneſſe, whoſe e9 or 12 inches, the ſuperfi
Some for eaſe in cariage vſe to make this ſquare board to conſiſt of thrée péeces, which they vſe to ioyne to
The edges of this table round about be abated with certaine ſquare channels to the thickneſſe of halfe the board, according as you may gather by the ſhadowed lines about the table.
2 The peeces of this table being ſet together, then be there certaine rulers of wood ioyned together with ſmall braſſe hin4 ſides of the table: & theſe rulers in that place haue three vſes principall; the firſt is, it ribs and binds the péeces of the table cloſe together; the next, it holds the ſhéet of paper plaine and ſtraight vpon the table vpon euery ſide: laſtly, euery ſide reſpondently is diuided into a certaine number of equall parts, which ſerue to drawe croſſe paralel lines vpon the paper when you change your ſheete, as hereafter: you may gather my meaning concerning the deſcription thereof by the enſuing figure, better then with many words.
3 To this Plaine Table there doth alſo belong a Boxe with a Néedle, and a Card in the bottome thereof, ſuch a one as is de1 Chap. This Boxe muſt be fixed to one of ye ſides of the Plaine Table a d or b c, in ſuch ſort, that the line in the boxe m n lie paralel thereto, ſo ſhall k l lie cloſe with the ſide of the table a d, then is there an appendix growing to this boxe, through which go two holes for paſſage, that two ſcrew pins may fixe the ſaid boxe to the body of the Plaine Table: as you may gather amongſt the ſhadowed lines in the inſuing figure.
4 To the backe ſide of this Inſtrument there is fixed a ſocket of braſſe, with two ſcrew pins, as you may perceiue by this fi
5 Alſo you muſt prouide a thrée footed ſtaffe, that is 3 legges or feete to be fixed in a head of box like a thrée footed paire of compaſ
If you make the ruler and ſights in braſſe, they were beſt to be made to fold vp and downe, as the order is in ſuch things.
Thus much of the deſcription of the Plaine-table and his parts, which I haue the rather wrote for their ſakes that affect the ſame, for whoſe ſake alſo I will ſomthing proſecute the vſe thereof.
t he was a witty fellow yt wrought by the doctrine of a curious ſmall ſcale, and a goodly blune diagonall,21. Chap6. Booke, and ſo working according to the doctrine of triangles, you ſhall finde the product of the two triangles to
But to returne to the ſupplements of the wants in this In
And this thing note further in the Table, that when you take plats, oftentimes they will not all bee contained in one or two ſhéetes, therefore you muſt drawe a number of paralell lines croſſe wiſe at right angles, that thereby you may ioyne the ſhéetes together truly, which the diuiſions vpon the foure rulers performes, and as you meaſure the circumference of any plat by this Table, oftentimes caſt croſſe angles, & Diagonal lines ouer the plate, which will kéepe you the truer to your worke:Diagonals agrée with your ſcale, which if it doe not, is a true argument of errour.
You muſt further haue a ſpeciall care in reducing the lines hyPlano truly. But ſurely by how much the higher the ground was, by ſo much the greater was his error,
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,
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 quaGeodetia, as in the 26. Chapter. Theſe things had, you may fall to worke.
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 di
Therefore at the place whence the diſtance is required to any marke propoſede 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 ye 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 re1/10 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 ſtati
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.
Hauing ſo done, place againe the fiduciall edge of the rule vpon
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 7
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 vnred, (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 ſec, ſo wilt the edge of the ruler in the line vpon the paper repre
thd, I moue the other 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.
a,
29 chapter, and let the latitude required be d c, noa to point to d and c, and alſo to b your ſecond ſtation, now obſeruing the former directions I 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.
f e, e d, d c, &c. the labour is no more but to draw lines at euery ſtation, to point 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.
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 diuiſie 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 Inſtru
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 demonſtra
And you ſhall note for diuers good reſpects that I ſhall omit one thing that ſtandeth firme, and is ordinarily vſed in demon
Your ſtation is i, the lines drawne from i to point to euery ani a, i b, i c, i d, i k, i e, i f, i g, and i h, which are meaſured 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.
The field is f g h i k l m n o p q r ſ, wherein I plant my Ina, whence I cauſe the angles ſ f g h i, and no more: therefore I finiſh all thoſe angles, as in the laſt chapter, and then find out another ſtation, as b, where I plant my inſtrument as at a in all reſpects, whence againe I may thence ſée the angles k l m n, and ſo procéed in the ſame order with thoſe angles drawn to the ſecond ſtation b, as I did at the firſt ſtation a: but for that at this ſecond ſtation I cannot yet ſée all the angles in the field, therefore I am forced to ſéeke a third ſtation, as c, and there obo p q r, which be all the angles, which I proa b, and b c muſt be meaſured as well as a h and a i, with all the reſt: and note that the néedle cut 40 deg. at a, and ſo muſt it doe at b and c, and as many more ſtations as you ſhould be occauſa breuitatis. The South end of the Needle cuts like degrees in the card at euery ſtation, as at the firſt ſtation. This Chapter is remembred, lib. 6. Geodetia, Chap. 7. defi. 5.
The figure is a b c d e f g, the angle where the inſtrument is planted is a, whence I drawe lines into euery angle, as a b, a c, a d, a e, a f, and a g, then I meaſure the line a b, and lay it downe, then I meaſure the line b c, and note where it interſects with a c, as at c, then I meaſure c d, and note where it interſects with d, & ſo I go round about the field, and produce a figure proportionall to the figure propoſed.
7 Chapter, def. 4. Geodetia.
2 ſtations ſomething neere the middle of the field, a good diſtance one from another, in ſuch ſort, yt they lie not both in a ſtraight line to any of the angles of the field, but ſo that they may make as great angles with e31 Chapt. as you may gather by the demonſtration.
C d f g h i k is the Peripher of the field, a is my firſt ſtation, b my ſecond, and ſo working by the doctrine of the 31 Chapter. I obtaine a line like proportionall to the field, which was required.
viz. that point on the paper ſheweth in what part of the field your inſtrument is placed. Now from that center draw lines to all
Example.
My firſt ſtation is a, whence I obſerue the angles d e f k l m, my ſecond ſtation is b, whence I draw lines to point to as many of the angles I obſerued at my firſt ſtatio ̄ as you can ſee as b d, d e, b f.
and ſo noting the interſection of matchy lines, draw the lines d e and e f, which is ſo much of the hedges that you haue obſerued: now the Inſtrument vnremoued at b, I eſpy as many more angles from b as I well can, as g h and i, and ſo draw lines to repreſent b g b h and b i. Laſtly, I eſpy ſome other place whence I may ſee all theſe three former angles: but the way to finde your third ſtation c is thus, vpon ſome point on the paper, repreſenting ſome angle in the field as e, laying there the edge of your ruler, mouing the other end vntill you obſerue through the ſights the angle e, then note where the edge of your rule and the line b c interſecte, as at c, ſo ſhall you finde the true place where your inſtrument ſtands, your inſtrument reſting ſituate at c, in all reſpects as at the other ſtations, draw lines to point from c to g h and i, and ſo note the interſection of theſe lines, with their matchy lines drawn frō
b, ſo haue you another part of the perimeter, by drawing lines from one interſection to another, as g h, and h i: and for that you may ſee from c to all the reſt of the angles k l and m, obſerued at a, therefore I draw lines to point from c to k, to l and m, and ſo noting the interſections as before, and drawing lines I haue included a figure proportionall and like to the propoſed figure.
Note, I draw no figure vpon a b or c, to repreſent the Table, be
4 men to helpe you, whoſe labour ſhall he thus, two meaſure with the line the diſtance from angle, to angle, one man to go before you into euery angle: and the fourth man to be left ſtanding in the place where you planted your Inſtru
Firſt, plant your Inſtrument in any angle, and appoint for your place of ſtanding ſome pricke in your paper, then draw a line into the next angle, which line meaſure on the ground, and ſet thoſe meaſures by ye ſcale on ye line drawn, then place your ine angle, & ſay ye ruler along ye line drawne, & thee Table about, til you ſée ye angle, or the man left in the angle froe line & angle of the field, then is your plat per
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 withobſeruare 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
e third angle, and thereto remoue your inſtrument, and hauing there placed it, get the di
The premiſes be 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 com
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,
Now when you come home, vpon ſome ſhéete of paper pro
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 mea53. Chapter excellent: as for the 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 them9. and 29. chapter to the like purpoſe: but haue a ſpeciall care in all your dimenſions and plat
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 inſtrume
Here ſhall follow another kinde of ingroſſing of particulars, then that in the Staffe, together with the pleaſant and apt pla
12. in an inch, numbring the ſame by 5.10.15. &c.
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, calTabula 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 vn
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,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 ſemidia
Now from the perpendicular point be the degrées of a qua90. to 25. by 10.
Of the longer Sight.
This ſight is twice ſo long as the other,
1 Firſt, the diſtance from the pinnes head (in the ſhorter ſight) vnto the plaine of the inſtrument, is taken, and according
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.
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.
25.
Of the ſlit in the ſight.
Vpon this ſight there is a vane of braſſe, made to runne e
Of the Index.
Index, whereIndex 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 the Inſtrument.
Of the Staffe.
4 foote long, with a good ſtéele pike in the foote thereof: this Staffe ſer3 footed paire of com
Circumferentor, I know not, but this I affirme, the name was not vnaptly giuen, for if we well conſider hereof, it will be appaCircumferentor, of the Lattin word Circum, which ſigfero the verbe, ſignifying to beare or cary, ſo Circumfero is to beare about, whereupon the Circumferentor taketh his name, which you may take in mouing him about the Staffe, or bearing him about the field, in working whereby you muſt alwaies haue a ſpecial care vnto the paralelty thereof, ſo that it is not lawfull for him to leane one way or an
Then muſt you prouide a Protractor,60, ſuch equall parts as the Carde in the firſt Chapter was, the diameter whereof muſt agree with the diameter of the Carde: the lower ſide of this pro60 ſuch equall parts, procéeding from 60 vnto 120, ſo that,
All the diuiſions to 60, be vpon the vpſide, and that is called the Eaſt ſide.
Then all the diuiſions vnto 120 aboue 60, be vpon the lower ſide, and that is called the Weſt ſide.
The diameter of this Protractor repreſenteth a Meridian.
Vpon the vtter ſide of this diameter is roome left for to make a ſcale, which is diuided according vnto 12 in the inch, make not your protractor, as the common order is,12 parts in the inch vpon the one ſide, and 11 on the other it would do you pleaſure.
e Sun from the Meridian.
But if the Sunne or Star be higher then 25 degrees, ſo that you cannot bring downe the vane to worke vpon the longer ſight, then put the Index vpon the center pin, looking through the ſight in the Index, vntill through the ſame hole, and by the center pin you ſee the Sun or Star: for the degr. then cut vpon ye edge of the Inſtrument by the edge of the Index is the altitude.
And by this Propoſition may you obſerue all the Stars in the Globe together with their motions in the heauens.
Example.
The 6 of October in the Morning, I made obſeruation according vnto the firſt difference, where hauing planted my Inſtrument paralell, and ſpied the Sunne through the hole in the vane, and by the pins head, I found the vane to out 12 degrees vpon the ſight, and the South end of the needle to cut 19
120 degrées repreſent the South, and that the de
Let the Inſtrument be placed paralel vpon the ſtaffe, then ceaſe not to moue the ſame,e néedle, with which reſort vnto this Table, ſo haue you your de
Example
I find the South point cut 45 degrees, I conclude the thing ſeene is North-eaſt.
e ſhadow of the pins head point or fall iuſt in the ſlit in ye lone South part with the paralell of the month, or ſigne take whether you pleaſe, is ye houre, which you ſhall know by the houre line paſſing thereby.
And you muſt vnderſtand that thoſe circles I call paralels be ſuch as are deſcribed about the center of the Carde, and thoſe I call houre circles bée thoſe that paſſe as it were from the center to the limbe croſſing the paralels.
You ſhall obſerue where the paralel of the moneth cutteth the Horizon, for the houre circle paſſing thereby, or the néereſt there
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.
viz. Eaſt, Weſt, North, and South; then you muſt vnderſtand, that there is no ſtarre, nor the ſunne, that riAntinous left arme, &c. come néere thereunAmplitudo 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, where
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 de3, then from the totall take 90, ſo haue you your deſire, and the ſetting ſhall be North from the E
But if you ſéeke a riſing, you muſt conſider whether the de30, or aboue: if they be vn30, 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 de
But this Chapter is performed with farre more eaſe & truth by my Topographicall Glaſſe.
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
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 oppo9 Chapter, taking the leſſer of thoſe de
And you muſt here note, that all degrées cut at diuers obſerua
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.
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 per
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
When you haue gotten theſe two angles, adde them both to60; ſo haue you the quantity of the an30, 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 num
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.
The ſides of oppoſite angles bee direct proportionall to their ſignes.
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 pae 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.
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 ye 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.
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
Index vpon the right edge of the inſtru60. the like propor
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 quoti
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, ye 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.
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 née
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 ye 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
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 ob
Next ſhall you cauſe one to mete with a chaine the true die ye firſt degrée cut in your Table: then mete ye diſtance of the ſecond corner from your inſtrument, which note downe in your Table againſt the number of degrées cut at the ſecond cor
Hauing prepared a faire ſhéet of paper, as you be taught bee 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 come 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 ſtatiGeodetical 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 la
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. fro ̄ which take 6. which is a leſſe, then multiply 60. by 6. and the product will be 360. which agr
100: from that line you ſhould meaſure, if the ground were leuel: therefore when you haue mea
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.
30 equall degrees, ſete is betwixt the firſt degrée & the center, into 30 equall parts, raiſing perpenIndex of the length of the ſemidiameter of the QuaIndex is to be faſtened to ye center of the Quadrant with a braſſe pin or ſuch like, which al30 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 numIndex, 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 ſee the 19 perpendicular counted from the center interſect, which ſheweth that the line meaſured 20 perches, muſt be protracted 19.
And if the length of the line meaſured exceed 30 pearches, and be leſſe then 60, then take halfe the number vpon the Index, and the perpendicular will anſwere to halfe the length of the Horizontall line, but if the line exceed 60, take then ¼ ¾ &c.& the perpendicular will anſwer proportionally.
Index, the ſame being counted in the lymb, then numIndex ſheweth the heigth.
If a wall be not direct, hee is then declining if the wall point iuſt Eaſt, Weſt, North, or South, he is direct, otherwiſe declining.
All walles decline either South or North, ye quantity whereof is thus had:
Set the North end of the Inſtrument, vnto the wall, now if ye néedle cut 30, 60, 90, or 120, it is an Eaſt, a North, a Weſt, or a South wall.
1 But if the néedle cut betwixt 120 and 30, the wall is South Eaſt declining to the Eaſt.
2 If the Néedle cut betwixt 120 and 90, that wall is South Weſt declining.
3 If the Néedle cut betwixt 30 and 60, that wall is North, declining to the Eaſt.
4 It betwixt 60 and 90, the wall is North, declining to the Weſt.
1 If the wall decline South Eaſt, multiply the degr. cut by 3.
2 If Weſt, take the degr. cut from 120 and the remainder multiply by 3, which produceth your deſire.
3 If North Eaſt, take the degrée cut from 60, and the remain3.
4 If North Weſt take 60, from the degrée cut and multiply by 3, ſo haue you your deſire.
In Superuiſ. manerij praed. ibid. fact. xiij. & xiiij. diebus Septembris anno regni Dom. noſt. Iacobi Dei gra ̄ Ang. Scotiae, Frā. & Hibern. Reg. fidei defenſor. &c. viz. Angl. Franc. & Hibern. ſexto, & Scotiae xlij. Per B. G. gent. virtute commiſſionis dicti Domin. Reg. extra Scaccar. ſuū ſibi direct. continetur inter alia, vt ſequitur. viz.
R. G. gent. tenet per copiam dat. xxviij. die Septembris, anno Regni Regis nunc Angl. &c. Quinto, cert. terr. & teneI.G. armig. ante A. Hoſ. gent. ante B.D. armig. patris ſui. viz.
2 voc. le North Orchard, & long Oriiij. acr
l. acr.
xx. acr.
xiij. ac.
Habend. ſibi & ſuis ſecund. conſuetud. Maner. per Redd. per Annum xij. s. ij. d.
In like manner muſt you deale with all the other tenements of the ſaid Mannor, noting the quantitie of euery particular, then the rent paid, and at the lower end a reaſonable improoue
And if there be any other commodities in the ſaid Mannor ac
iij. l.
l. s.
3. l. x. s.
G. I. per annum 4. l.
30. s.
vij. l.
iiij. l.
iij. l.
xlij. s.
xx. s.
xxij. s.
xxx. s.
x. s.
3. d. the peece per annum xx. s.
ix.l.
iij. l.
vj. l.
iij. l.
If there be any repriſes wherewith the Mannor is charged, as money for the yeerely repairing of ſome bridge, high way, or any other annuall penſion whatſoeuer, let it be noted as the forEi remanet clare per annum vltra repriſ. 306.
l. 14. s. 8. d.
And you muſt further note, that the firſt thing you haue to deale with, is the ſight of the Mannor houſe, the buildings and demeſne, then the parke, parſonage, &c. if any bee, and then pro
To make a plat or map, and place a Sea-Card therein.
VPon the middeſt of your plat deſcribe a circle, as vpon a which diuide into 32 parts, and then about the map de30 parts, by drawing lines from the center a by each of the 32 equal parts in the firſt circle: now if vpon euery of thoſe interſections as a
32 equall parts, extending from them right lines through the body of the map, you haue finiſhed the Sea-Card, and will beautifie your map, and ſerue to expreſſe many prettie concluſi
You may diſtinguiſh all the windes in your Card otherwiſe, if you pleaſe by placing a circle, containing the ſame in ſome voyd place in your plat, as you may ſée in the 7 Chapter of the Topographicall Glaſſe, and drawe them forth onely to touch the circumference of the plat, as in the 6 Booke, and Chapter 49 of Geodetia.
THis Chapter is eaſily performed, if you doe but call to mind how to ſéeke the true proportion of any field, Iſland, or ſuch like, euen as you be taught in the Chapter: but indéed I hold this Chapter (for that it is to be performed only by two ſtations) beſt to be wrought ſinically, ſo ſhall you place and ſituate the angles more truely then you can by the interſection of lines, for becauſe thereby you be taught to find out the true diſtance of euery angle from your ſtation; in ſo much that if you doe but obſerue the quantitie of euery angle, and againe protract the ſame truely, you cannot erre any thing at all: and for that it may be deſired of many, I will not leaue for their ſakes to proſecute the ſame with an Example.
Example.
Let there be a certaine parke c d e f g, propoſe the proportion and quantitie whereof you are required to deliuer at your ſtana or b, which are certaine hilles from whence you may well view all the angles and corners of the ſaid parke. In performance whereof I firſt go vnto b, making b a the one ſide of an angle, I obſerue accordingly all the angles in the ſaid parke, as a b g, a b f, a b c, a b c, and a b d, all which I note downe thus:
Next do I finde me another ſtation, a knowne diſtance from b, as a, 65. pearches, and ſo repairing vnto a, making a b the one fide of euery angle, I againe obſerue all the angles in the ſaid parke at a, as b a c, b a d, b a e, b ag, and b a f, and the quantity of e
The angle a b d is 128. degrees, b a d 44. degrees, which added together, and the totall taken from 180. leaueth 8. degrees, the angle b d a. Now I ſay by the ſaid ſecond Axioma, as the ſigne of the angle a d b 13917a b d 78801. ſo is the ſide a b to the ſide a d, or as the ſigne of a d b is to the line a b, ſo is the ſigne a b d to the line a d.
Therefore multiply 78801. the ſigne of b, by 65. ſo is the prod, ſo is the quotient 368 609/13917 pearches, the line a d, which is one mile, one quarter of a mile, 8. pearches and odde.
In like manner muſt you get the lines a e, a f, and a g, as you may perceiue by the inſuing example, where I haue ſet downe as well the quantity of each angle, as alſo the reſpondent ſigne and ſide that doe anſwer or ſubtend the ſaid angle.
All theſe angles and ſides thus gained, you ſhall lay downe the plat, and finde the contents thus:
Firſt I draw a line i k at all aduentures, whereon by my ſcale, I appoint 65. pearches, then making k i the one ſide of an angle, according to my obſeruations at my ſecond ſtation, I protract an angle k i l of 40 ½ degrees, equall to b a c in the laſt figure, then I protract an angle of 44 degrees, as k i m, drawing i m infinitely, and ſo I proceed vntill I haue finiſhed all the angles taken at my ſecond ſtation, as 65 ½, 84 ¼ 87 ½ and thereby protract the angles k i n, k i p, and k i o, ſtill producing thoſe ſides infinitely.
You ſhal therfore note the diſtance of a c in the firſt figure obteiviz. 174 86047/100000 pearches, the which with your ſcale & compaſſe, lay downe vpon the line i l, from i towards l ſo haue you the line i l, then take the diſtance of a d in the firſt figure 368 609/13917 pearches, which lay downe as before from i towards m, ſo haue you the line i m, in like manner deale with i n, i p and i o, euen as the precedent Table doth inſtruct you, ſo is i n 363 4753/17967 peri o 174 12726/35836 pearches, and i o 239 1113/26823 pearches, theſe lines ſo limited you muſt draw lines from l to m, from m to n, from n to o, from o to p, and from p againe to l, ſo haue you made a fil m n o p.
Certainely moſt exact and perfect is this kind of working, and albeit it may ſeeme ſtrange and difficult at the firſt, to yong practitioners, eſpecially becauſe the deuiſe is new, as for that we haue not heretofore any Engliſh Treatiſe ſhewing the vſe of right lined Tryangles, by theſe Signes, Secants, and Tangents, yet let none be ſkarred at firſt, for do but ioyne the vſe of my 7 book of the Staffe called Trigonometria herewith, and then all things will ſoone become eaſy and familiar.
6 booke of the Geodeticall Staffe, euen as you ſée this figure, l m n o p reſolued into thrée tryangles l m n, l n o, and l o p, with their baſes l o, and l n, and their perpendiculars p q, o r and m r, made ready to be meaſured according vnto the ſame 2 part of the 6 booke, chap. 25.
Then hauing protracted your plat in plano (or reduced the ſame into regular figures, with your eye) diuide that ſide whoſe length you found, into a certaine number of ſmall and equall diarea, you ſhall meaſure the contents ſuperficiall by thoſe equall diuiſions, I meane the contents of euery particular figure, adding them altoas the ſquare of the ſmall diuiſions vpon the one ſide, is to the ſquare of the number of pearches, reſponding to the ſame ſide, ſo is the ſuperficiall content in the former ſmall diuiſions, vnto the true ſuperficiall content in pearches. There160 &c. leaueth the number of acres, &c. as in the Arte of Geodetia, chap. 24.
Example.
The figure protracted in plano, is l m n o p, Now viewing into what regular figures it may beſt be reſolued, into which (as you may perceiue by the pricked lines) it is readily conuerted into three Tryangles, by two right lines iſſuing from l to n and o, next I diuide ſome one ſide as l p into 60 equall parts, and ſo by that diuided line, as by a ſcale I meaſure how many of thoſe diuiſions is in the baſe l o, and l m, ſo do I find the one to conteine 88 and the other 92 of thoſe parts, next vpon thoſe baſes I let fall perpendiculars, which in like manner I meaſure, as p q 25, o r 59, and m r 53, now do I multiply 25 in halfe 88, ſo haue I 1100, then 59 in halfe 92, and there is made 2714, and laſtly 53 in halfe 92, ſo is the product 2438 (or ſince l n was a diagonal, or a baſe common to both the Tryangles, you might haue multiplyed o r and m r in l n and all had bene one) the which three aggregated ſums adde together, ſo haue you 6252 produced, the whole content of l m n o p, by the which multiply 16900 (the ſquare of 130 perches
the length of the hedge in the Parke c g) and the product is 105658000, the which parted by 3600 the ſquare of 60, leaueth the number of perches viz. 29340 24/6 perches, the iuſt content of the plat which you may reduce into acres by the 24 Chap. of the Art of Geodetia, or if you pleaſe, you need not to protract this figure at all, but meaſure all the diagonals & perpendiculars by the 3 Axiome of Trigonometria, as if you would meaſure the diagonall l n, here haue you two ſides knowne i l and i n comprehending an angle knowne, therefore worke by the ſaid third. Axiome: the like might you do to l m, m n or l o, &c. and then the area is eaſily found by the 20 Chapter of Geodetia, without regard of perpendiculars. Hauing the 3 ſides of euery tryangle, you may lay them downe ſeuerally vpon ſome ſmoth boords or ſuch like, and ſo by your ſcale and compaſſe, after the common order, find the perpendiculars, without regard to the quantity of the angles, for the 3 ſides being knowne, the true interſection lymits the right proportion.
Looke how you bee taught to ſéeke any plat at two ſtations, and ſo doe here, then from your firſt ſtation, beyond the ſecond, or from the ſecond beyond the firſt, obſerue ſome marke, trée, or ſuch like, inclining towards the propoſed field (which let make rather a right then obtuſe angle with your ſtationary line,) and ſo get the angles, it makes with both your ſtations, making the ſtationary line ye one ſide of the ſaid angles, then go to the marke
Example.
The propoſed field is c d e f g h, which I obſerued at two ſtatitions a and b and ſo when I come to protract, note the interſection of matchy or like lines as in ſuch a caſe I am wont, but for that I perceiue the ſection at g and h falleth out very acute, therefore (as is ſaid) I eſpy a third ſtation, I noting the angle i b a and b a i, whereby is gotten the diſtance of i my ſecond ſtation, then do I go vnto i, and there againe obſerue the angle b i f and b i g and b i h, or any other angle that by the ſection of lines iſſuing from the two firſt ſtations would proue acute, ſo ſhall the lines iſſuing from i and b make a more perfect interſection, ſo that when you come to protract, you ſhall finde all the acute ſections
corrected, for whereas the lines a h, and b h made an acute interſection at h, the ſection of i h and b h do correct and reforme the ſame, finding out the true point of interſection, and ſo of any other: and in your obſeruations in the field you may well know which angles in protracting will proue acute, for if you adde the gles g b a, and g a b, or h b a and b a h together, the totall taken from 180 leaueth the angle g or h, of whoſe acuity you may iudge.
Thus by meanes of 3 or 4 ſtations, or more, if occaſion be, may you finde the true plat of any propoſed plaine, by the interſection of lines, neuer meaſuring but onely one line in the whole worke, certainely no propoſition performed by the interſection of lines is better then this, if your ſtations be reſpondently taken and the obſeruations truely made.
Gem. Friſius, and indéed is moſt exact and excellent, where you may haue libertie of ground at command, crauing the helpe of no Inſtrument, but onely ſome ſuch a one that will direct you readily to ſet out a right angle, which the Topographicall Glaſſe or Geodeti
Example.
A is a certaine Turret, whoſe diſtance is required from b, where I plant my firſt ſtaffe, departing thence a certaine diſtance Orthogonally, as to c 163 yards, placing my ſecond ſtaffe at c, d is my third ſtaffe placed backewards from b 158 yardes in a right line with a b: e is the fourth ſtaffe placed Orthogonally fro
Euclide in his firſt Booke, Chap. 47 of his EleEuclide, and elſe where, ſaying. In triangulo plano rectangulo latera includentia rectum aequà poſſunt Hypotenuſae, penult. primi Euclid.
Therefore when you are deſired to deliuer the diſtance of the
e altitude, both which diſtances ſquare, adding the pro
After this order when you come néere to any towne of warre may you tell the iuſt length of the Scaling ladder that muſt reach from the brim of the counterface or ditch that incloſeth the ſame, to the top of the curtaine or wall, by adding the ſquare of the diſtance of the wall vnto the ſquare of the curtaines altitude aboue your feet, for the ſquare root thereof (as before) yeelds you the longitude of your Scaling ladder.
Example.
Let the diſtance of the baſe of the wall i g be 23. paces, whoſe ſquare is 529. the altitude of the wall aboue your feet g h be 10. paces, the ſquare whereof is 100. which added to 529. maketh 629. the ſquare roote whereof is 25 4/60 pearches and better, the
THis Chapter is very exact, ſo the obſeruations be well made; and indéed is but wreſted from the 41. chapter of my 6. booke of the Geodetical Staffe, howbeit it is ſet downe much after the ſame methode I ſhall deliuer now, by Maiſter Digges our countriman, and others.
In performance whereof you muſt haue any kinde of triangle prepared: which had, ſet vp a ſtaffe, whereunto apply your angle in ſuch ſort that the one of the containing ſides point directly vnto the marke or Caſtle vpon the left hand: the triangle ſo reſting, looke by the containing ſide vpon the right hand, cauſing one to ſet vp a ſecond and third ſtaffe in a right line therewith, and the further y e ſecond ſtaffe is diſtant from the firſt, the better it is: the triangle yet remaining, as before, at your firſt ſtaffe, make a marke in the ſide ſubtending the angle at your ſtaffe, in ſuch ſort that it may bee in a right line with the ſaid angle, and the ſecond marke vpon your right hand; ſo haue you performed all your obſeruations at your firſt ſtaffe, onely as you come thence meaſure ye diſtance from your firſt ſtaffe to the third ſtaffe.
Next repaire vnto your ſecond ſtaffe, where ſituate your triangle, in all reſpects as he was before at the firſt ſtaffe: then muſt you depart towards the deſired diſtance in a right line, vntil ſuch time that you come direct betwéene the Tower or marke that was vpon your left hand and your third ſtaffe, and there ſet vp your fourth ſtaffe.
Next go vnto your ſecond ſtaffe, where your triangle yet remaines, and looking from the angle at your ſtaffe by the marke made before, in the ſide ſubtending the ſaid angle, your eye will direct you in a right line, which you muſt continue on, vntill you bee alſo in a right line with the ſecond and third ſtaffe, & there
ſet vp your firſt ſtaffe. This ſo ordered, meaſure the diſtance betwixt the ſecond and third ſtaffe, the quantity of which meaſure reſerue for your diuiſor.
Laſtly increaſe the diſtance betwéene your firſt and third ſtaffe in the diſtance betwixt the fourth & fift ſtaffe, the product whereof part by your reſerued diuiſor, ſo doth the quotient yéeld you the deſired diſtance, as Geometrically might be proued.
For as the diſtance of the ſecond and third ſtaffe is to the fourth and fifth ſtaffe, ſo is the diſtance of the firſt and third ſtaffe vnto the diſtance required. Or as the diſtance of the ſecond & third ſtaffe is to the firſt and third, ſo is the diſtance of the fourth and fifth to the latitude, or propoſed diſtance.
Example.
Ab, is a diſtance required, k a triangle prepared, e my firſt ſtaffe where my triangle is placed, the one of the containing ſides pointing to the caſtle a vpon my left hand, c e the way directed by the other containing ſide of the triangle, d my ſecond, and e my third ſtaffe, placed in a right line with c, by the direction of the ſaid containing ſide, h is a marke in the ſubtendiding ſide, found out by the viſuall line running from c to b the marke vpon my right hand: this done, I meaſure the diſtance from c to e, which I heere found 300. pearches, and ſo take vp my triangle, ſituating him at d my ſecond ſtaffe in all reſpects as he was at c my firſt ſtaffe, the reſpondent containing ſide lying in a right line with the firſt, ſecond, and third ſtaffe: ſo I ſtanding at d, and looa and your third ſtaffe e, where ſet vp your fourth ſtaffe f. Againe, looking from d by h, the ſubtile marke made in the ſubtended ſide, the viſull line will direct you what courſe to hold vntill you come in a right line betwixt the 3. ſtaffe e, and the other marke b: this done, meaſure the diſtance d e, which there is 100. keepe that for your diuiſor.
Next meaſure the diſtance betwixt g and f, which is 170. pearc e 300. by f g 170. ſo haue you 51000. which parted by d e, 100, leaueth 510. pearch, the diſtance a b.
82. chapter is as true in demonſtration, but not ſo eaſie and ſpéedy in working: for heere you bee not tyed to right angles, and ſuch like, but onely are allowed to make the obſeruation according to the aptneſſe of the ground, neither néede you to feare whether you be ſituate vpon hilles or in valleyes, more then if you were on the plaine ground.
In performance whereof you ſhall make a triangle of what ſides and angles you pleaſe, the which triangle you ſhall ſituate at the place whence the diſtance is deſired, in ſuch ſort that one of the ſides containing the angle that is towards you may looke direct vnto the marke whoſe diſtance is ſought: the triangle re
It would bee ſomething long for mee to ſtand to demonſtrate
Triangular aequiangula habent latera circa aequales angulos proportionalia, & contra, Eucl. 4. p. 6.
Or you may proue it by Ramus. lib. 7. pag. 9. ſo that it is néed
Therefore as b c is to a b, ſo is c d to e a, or as c d is to a e, ſo is b d to b e.
Example.
And as in this chapter, ſo likewiſe in moſt other like concluſions may you ſo order your obſeruations that you may auoyde diuiſion, according as in the end of the 84. Chapter.
If you deſire the diſtance b e, multiply the diſtance of a e, by the diſtance of your third and fourth ſtaffe, parting the product by the ſecond and fourth ſtaffe, as by the diſtance of c d, ſo doth the quotient yeeld the diſtance b e.
And for that your Geodeticall Staffe will take or deliuer any angle, or repreſent any tryangle, hee may aptly performe this Chapter according vnto this kind of methode.
84 chapter get the diſtance betwixt the two Ships, then ſtay halfe an houre or a quarter, obſeruing then againe the true di
Example.
Let vs ſuppoſe we ſaw two Ships, the one in chaſe with the other, and we were required to know whether the Ship in purſuit did win any thing of the Ship purſued, and if he did, when the Ship chaſed ſhould be ouertaken, firſt therefore I take their diſtance, which I will ſuppoſe 400 yards, ſo reſting a quarter of an houre (which is 15 minuts) I obſerue their diſtance againe which I find 300 yards, laſtly I ſubtract 300 from 400, the remainder is 100, therefore I multiply 400 by 15, the product is 6000 which parted by 100 leaueth 60 minutes, the time how long it will bee before the Ship purſued ſhall beouertaken by the Ship purſuing, which is one houre.
In the like manner may you deale with Ships or Nauies approaching towards any porte, hauen or ſuch like.
Geodeticall Staffe to ſéeke the true perimeter of any figure propo
Certainely moſt excellent is this booke for martiall diſcipline touching fortification, as in the delineation of royall frontiers, skonces and reinforcing old walled townes, and right neceſſary for battailes, maiſters of Ordinance, &c.
Topographicall Glaſſe or Circumferentor is beſt, in performance whereof you ſhall do as followeth.
Suppoſe there were a myne of coale in the borders of a certaine Mannour. which continuing, the Lord of the next Manour was in doubte leaſt the veine of coale did run towards his adioyning Manour, and that they were co72 chapter, get the true way that the myners haue made euen as it were a hedge, ſtill noting
e 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.
e 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 Inſtru
e difference of ye ſaid aſcents & deſcents, neither néed you feare any collateral declining of ye way of your myne, for that nothing at al altereth the aſcent or deſcent, for that is onely altered by the di
90 Chapter, and if the Mine muſt aſcend or deſcend a90 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 yt 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 fur
Therefore in theſe caſes you were beſt firſt of all to limit vpe 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, CaTopographicall Glaſſe to be moſt requiſite.
Topography, the building and ſituation of Citties, houſes and ſuch like, is right neceſſary to bee remembred: but for Citties of defence they require a long diſcourſe for their ſituation, as well in reſpect of their walles, &c. to defend, as of Turrets, Mounts, &c. to plant ordinauce in and vpon to offend; onely therefore touching health, let your cittie bee planted by a faire and portable riuer, farre from marſhes and fenny places, for the vapours riſing thence be vnholſome, and in as barren and fruitleſſe a place (yet dry and firme) as you may, for in ſhort time the compoſt and ſcauengers durt will ſoone make the cone planSaturnall, towards the middeſt of the citty were two other mounts, called Palatine and Quirinal, vpon the left hand of the cittie was the mount Eſquiline, vpon the right hand Caelian, and towards the end of the citty were two other mounts called Viminall and Auentine, all which mounts much beautified the citty, and there
Touching the ſtréets there ſhould be foure maine ſtréetes ly4. Cardinals of the world, that is, one runing North and South, the other Eaſt and Weſt, croſſing each other, about which croſſing ſhould the Forum, or common market place ſtand. So was the citty Alexandria builded, and theſe ſtreetes would
e new plantation in Ireland. And as you haue diuided the cit4. quarters, ſo may you appoint other collaterall ſtréetes which will alſo receiue the collaterall windes, whereby any aire ſtirring, the Cittie ſhall haue benefite thereof. Certainly moſt excellent, right pleaſant and neceſſary would it bee to ſee a Cittie thus builded: for our citties at firſt commonly were villages, or ſuch like, and ſo increaſed and augmented, as the people multi
Prouiſo would be had, that hee proportion his houſe according to the quantity of the ground that he hath to lay thereunto, inſomuch that there would bee ſuch a proportion beMocke-begger. And in this point (as Pliny reporteth in his 6. Chapter, booke 18. of his naturall Hiſtories, there were two men liuing at one time, who much halted herein, to wit, Lord Lucullus & Q.
Scaeuola, for Scaeuola had faire lands, without a competent houſe and Lucullus had a competent houſe without lands, in which ree Cenſors (as many Londoners may) for ſwéeping more floures then he plowed lands.
Touching the ſituation of your houſe, the beſt opinion now is, vpon a hill, or hill ſide, hauing before the ſame a plaine ChamPliny would not haue a houſe ſituate néere vnto a fenny and dormant water, or ouer againſt the courſe or ſtreame of a running water. Homer ſaith, the aire and miſts riſing from great riuers before the the Sunne riſe, are vnholſome: howbeit you ſhall finde it pleaPliny would haue you ſittuate your houſe in a hoat country, into ye North and in a cold countrey to affront the South, but in temperate regions to lye open into the Eaſt. With vs in England the principall coaſt for a houſe to lye open into, is Eaſtwards, as well for health, as in Sommer for the aAriſtotle 2. Meteor, cap. 6. and alſo according vnto Magirus, l. 6. c. 9 d. 13. The eaſt wind with his collaterals is mo
Now in building a houſe much arte is required Pliny reporteth builded a houſe in Cape Miſenum, as hee had fortified a Campe that C. Martius (who had béene ſeuen times Conſull of Rome) right ſkilfully, that when Sylla, ſurnamed Foelix, ſaw it, ſaid, that the reſt in compariſon of him were blinde béetles, knowing neither how to build, or incampe.
When therefore you minde to build a houſe, with your Scale and compaſſe lay downe the ground plat according vnto your proportion, ordering your ſellerage, larder, and all houſes of ofe principall lights cham
Now for the addition of more delight vnto your houſe, vpon the South ſide thereof ſet out a faire ſquare garden, beautified with bowers, walkes and ſuch like, as your gardener can beſt deQuinqunx: within your houſe make your ſtaires large, not with theſe monnell poſts, but with foure ſteps and a halfe pace, a faire light anſwering to euery halfe pace. Let the chambers be of a conuenient height ouer head, and ſuffi
Touching the plats and formes of houſes, ſome affect the quaH. ſome other formes; but that muſt bee partly referred vnto the pleaſure of him that beſtoweth the coſt; and for my part, I intend not at this time to lay forth the diuerſitie of plats, and how they ſhould be taken or laid downe by ſcale and compaſſe, for that haply I ſhall open the ſame in another péece of worke more proper.
Iean Liebault writeth) earely in the morning, your face into the Eaſt, looke cloſe by the ground, if then you eſpy any vapour like to a little cloud riſe out of the ground, there if you dig is water to be
Alſo water is to be found vnder theſe inſuing herbes) Yarrow or Noſe bleed, Veruaine, wild peniroyall, Venus haire, Cammomill, Dogs tooth, foxtaile, trifoly, Cinkefoile, Millefoile, Coliander, or as ſome ſay where aboundance of gréene Ferne doth plen
Now for the pipes for the conueyance of water, lead is good, earth is better, but wood of fir, Alder, or pine, or ſuch other wood that hath roſen in it is beſt: ſuch they vſe now in conueying of waters to houſes from the new water mil in Weſtminſter, they muſt be bored through with long agores, firſt with a leſſe one, the
e eye at any appointed place & diſtance, as to deſcribe any town, or citty, any houſe, any floure, or any other body whatſoe
Take a faire péece of ſmooth glaſſe, and fixe the ſame vpon a perpendicular at the end of a ruler, the which ruler let be diuided into a number of equall parts, next vpon this ruler muſt be another ſhort perpendicular agréeing to the height of the midſt of the Glaſſe, and in the vpper part of this ſhorte perpendicular muſt be a ſmall and round ſight hole, which done let ye perpendicular be made to moue equally fromwards or towards the Glaſſe, or to ſtand fixed at any diuiſion vpon the rue ſaid Glaſſe whatſoeuer you ſhall apprehend (or at the leaſt whatſoeuer ſhall be required in your proiectment, and the worke is finiſhed.
Now if you deſire to make a ſcale for this proiectment, note the equall parts betwixt both the perpendiculars, which call your firſt number, then let the diſtance from your eie to the obiect bée your ſecond number, laſtly draw a perpendicular vpon the Glaſſe from the ſummity of the obiect to the center of the Glaſſe, or ra3 numbers had, multiply the ſecond and third and diuide by the firſt, ſo is the quotient the number of féete or inches, that the ſaid perpendicular containes according as the diſtance of the ob
And you muſt note in all proiectments proſpectiuely that you can lay no more downe but what you ſée, as in a 4 ſquare houſe. you cannot poſſibly ſet downe more then any of the two ſides and ſomuch of the roofe as you ſée and ſo of cities &c. and therefore you may lay downe ſomuch of any citty as your eye can appre
4 ſquare & hollow made to moue equally along the ruler, and the length of the ſaid ruler, but the lower ſide of the ſaid ſocket is cleane taken away, ſo that he hath but thrée-ſides, and therefore hée ought to bée the ſtronger, and let the ruler be the leſſer, for it is not materiall how ſmall he be: in the end of a ruler there is a ſmall center hole, and in the end of the ſocket that ſhall be towards the perimeter of your plat is a place made to hold a ſmall pencell, this ſocket you muſt diuide into certaine equall parts as 100, a 1000 or more numbring the ſame with figures as the order is: vpon this ſocket there is another moueable péece of braſſe which muſt hold a ſe
Faſten your plat that you intend ſhall bee reduced, vpon
20, then ſay I would, haue the plat one fourth part leſſe, 4 in 20 is fiue times, therefore I place the ſecond pencell 5 equall parts from the other in the end of the ſocket, where I fae end of ye ſaid-long ſocket round about ye true perimeter of the plat, ſo will the ſecond penſell deſcribe you a figure leſſer and proportionall to the propoſed figure.
And if you pleaſe, you may couer the figure to be reduced with white paper or ſuch like, and ſo draw the figure that is ſo reduced thereupon.
And if you would reduce the plat from a leſſer to a greater then the pencel next the center muſt alwaies kéepe the perimeter of the plat, and the other pencell ſhall deſcribe a greater figure proportionall to the leſſer, according to the quantity aſſigned.
And in the reducing of mappes by this meanes may you giue prickes for the ſituation of all townes, villages, &c. in the ſaid map, and thereby place them in your reduced map in their true place, for when your pencels be once placed at their true proporGeodeticall Staffe, and the ruler anſwerable thereunto, the ſocket will neuer come of the ruler: which is better, and the pin that goeth through the hole in the end of the ruler were beſt to be but ſhort and haue a he ad thereon, and a hole made in ye end of the ſaid ruler, to bury the pins ſaid head in: when he is once knocked downe to the end the ſocket may runne ouer the ſame ſo will this pins head kéep the ruler from ſtarting off the ſame. Let this briefe deſcription at this time ſuffice, which if I haue ſaid ſufficient to ſatisfie your vnderſtanding. I doubt not but you will ſoone acknowledge the excellency thereof.
Archimedes fiered the Romane nauy at Syracuſa in the Iſland of Sicilia, which to do, you muſt take a number of ſtéele glaſſes, made of pure reflexions fall vpon, and in one point, the ſooner is fire kindled. But indéed there bee certaine parabolicall glaſſes placed by the aid of Geometry, more excellent for this purpoſe, hauing concaues and conuexes, of which I cannot ſtand here to treat, neither is this concluſion ſo neceſſary in Eng
16. or 17. inches broad, whoſe center place directly againſt ye obiect you looke vpon, and let it not incline, or hang ſidewiſe by ae intent to receiue the beames that come through the ſame: which done, looke in the looking glaſſe, ſo ſhall you haue your deſire, if the burning glaſſe were truely placed: for you muſt note whatſoeuer thing you ſée through the burning glaſſe, that the further you ſtand from the glaſſe, the bigger it ſéemeth, vntill you come to a cer6. or 7. miles, or ſée a man 4. or 5. miles, reade a letter in written hand a quarter of a mile from you, &c.
I Had thought to haue concluded this Booke w c the diuers nature of grounds, with the artificiall mending of them, and the killing of Gorſe that aboundeth vpon cold clayie ground, and Ferne in a ſandy and hote ſoyle, or broome flouriſhing in barren ground, hote and drye, or moſſe ſpreding in a cold ground, all theſe, & more I thought once to haue handled, but my minde altering, and withall bethinking my ſelfe of a Gentleman of worſhip, and my kinſman, one Maiſter Whorde, that had beſtowed great coſt and trauell in dreyning of grounds, hauing now (as he told me) brought (to his great coſt) the dreyning of grounds to ſuch a perfect head, and eaſie methode that he was able to trebble the rankneſſe of his ground onely by waters, with the one quarter of the charges it was in the beginning: and indéede as the inuention is new and excellent, though it hath béene ſtumbled at by many, ſo no doubt but in time hee will bee perſwaded to publiſh to the world (for the good of his country, and credit of himſelfe) the forme and methode of the worke, to which it would doe well to adde what I intended to haue ſpoken of in this place which made me the rather referre it to a Gentleman, ſo honeſt, and well experienced. But to the matter.
Wheras there be many annueties which ſome deſire to buy, and ſome to ſell, behold y e inſuing Table, which telleth you what 10. pound anuety is worth for any time vnder 21. yeares, according to 10. pound in the hundred, and if you would apply the ſame vnto any other ſumme more then 10. pound, vſe the rule of proportion, as I taught you in the 6. booke of my Geodeticall Staffe, chap. 51. which made me repeate it here againe, becauſe the Table is wanting there.
Example.
I haue annuetie of 10. pound for 11. yeares, and now would know what I were worthy to haue in hand for the ſame, by the former Table I finde 10. pound for 11. yeares is worth 64. pound and 19. ſhillings, therefore by the common rule of three, if 10. giue 64. pound & 19. ſhillings, what ſhall 100. pound be worth? therfore multiply 100. pound by 64. pound 19. ſhillings, parting the product by 10 pound, ſo haue you 640. pound, and ſo much was your annuety of 100. for eleuen yeares worth to bee paid now in hand for the ſame.
The like may you do by the enſuing Table, if a man owe you 100. pound to bee paid at any time vnder 21. yeares hence, and you are to buy the ſame of him now with preſent money, rec
And if the ſumme or number of yeares exceede your Table, worke as in the laſt Table before, & ſo proceed, taking the ſums according to the demand out of the Table, and if other ſummes are required, worke, as I haue ſaid, by the rule of proportion, as before, making the ſummes in the Tables your Radix.
Thus much (good Reader) haue I ſaid of the vſe of my Topographicall Glaſſe, not doubting, ioyning this booke and the Geodeticall Staffe together, but that you haue the ample
vſe of all Geometricall, Geodeticall, and Topographicall Inſtruments, now extant, whereby you ſhall not be to ſéeke further inſtructions, I meane ſuch inſtruments that are now in moſt requeſt, and by the better opinions moſt commended, and therefore for this time, I will conclude my Booke, wiſhing my ſelfe preſent euer to explaine any thing that to the yong Practitioner ſhall ſéeme obſcure.
SENECA.Nullum eſt tam magnum beneficium, quod non vilificare malig
nitas poſſit: nullum tam anguſtum, quod non bonus interpres extendat.
The end of the Topographicall Glaſſe.
Now if you conuert any of this timber to boards, many vſe to throwe them into water, and there let them remaine 14 or 15 daies, ſo ſhall they be the ſooner ſeaſoned and better exempt from wormes.
And if you deſire to make columnes or great turned pillars of whole trees or ſapplins, to the end they ſhall not rent or chaune, with great long agors bore out the heart thereof; but of theſe and ſuch like happly I ſhall bee occaſioned to ſay more in another treatiſe.
And in concluſion, of falling of great timber trees, one princi
Digges hath taken paines to calculate tables; but here you ſhall be taught to find the ſame without any kind of table or calculation, which is
35 Chapter of my ſixt Booke erres in that point, béeing ſet downe haſtely according vnto vulgar tradition.
4 or 5 feet aboue the ground at the leaſt, but towards the middeſt of the trée were beſt; then ſhall you take the length of the ſtring that girded the trée, and diuide the ſame 3 into foure equall parts, taking one of thoſe equall parts, & place the ſame truely ouer in 110 and 110 equall parts, vpon the legs of the Geodeticall Staffe; the legges ſo reſting, take the diſtance ouer in 70 and 70, and note that line, which you muſt fit ouer in 60 and 60 amongſt the coard diuiſions vpon the lower ſide the legges, and ſo that angle not ſtirred, take the diſtance from 90
90, and apply that diſtance to your rule, ſo haue you the true ſquare of the timber trée.
Example.
A b is a growing tree, whoſe ſquare is required, that is, how much ſquare that tree will beare on euery ſide when he is ſqua
By this means may you find the true ſquare and conſequente Carpee error they run into by taking a fourth part of the coe ſquare.
Or as is ſaid, you might haue girded the tree about in the middeſt at k, and ſo worked as in the laſt chapter.
Note if you pleaſe, hauing a paire of Calleper compaſſes, you may omit to gird the tree about, whether it be ſtan60 and 60, and ſo worke as before in this Chapter.
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.
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 ſemi
Or get the ſquare thus, multiply the breadth in the thickeGeodeticall Staffe, fol. 142.
Rombus, & there
Example.
Or the length of the perpendicular b m is the ſquare falling vpon b c at right angles.
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 perb d and halfe the baſe a c in one right line, ac6 Chap. you ſhal find the ſquare of the tria b c to be h i.
4 ſquare: but I doe deliuer you the ſide of a péece of timber béeing iuſt foure ſquare, and of equall height with the
Thus are you taught to find the ſquare of any peece of tim
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 tym
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 fa
And if your timber be excauate or hollow, firſt meaſure the whole as it were firme, then the excauate part by it ſelfe, laſtly, the leſſer taken from the greater leaueth your deſire.
ſemicircumference vpon the fixed diameter, a varied ſurface is either conicall or cylindricall, a conicall ſurface runneth narrower and narrower
Now from ſurfaces we procéed to bodies.
Euen as points are the termes of lines, and lines the termes of ſurfaces, ſo ſurfaces are the termes of bodies: a body is a li4 dimenſions, which being contained vnder homogeneall ſurfaces equall both in multitude and magnitude be alſo equall, and if the axe be perpendicular to the center of the baſe, they be called vpright ſolids: of ſolids ſome are plaine, others bowed: the plaine ſolids are contained vnder plaine ſurfaces, and is either a piramis or a piramidate: a pyramis is a plaine ſolid riſing equally from his right lined baſe and vniformedly contracting it ſelfe vntill it finiſh in a point in ye top: now a piramidate is a plaine ſolid, compoſed of piramides, being either a priſme or a mixt polyedron, a priſme is a figure piramidate whereof two oppoſite plaines be like equall and paralell, the reſt being paralellograms, and is a pentaedron, or made of pentaedrons, and being ſo made, is either an hexaedron, or a poliedron, the hexaedron being a paralell pipedon, or a trapezium, the paralell pipedon, is an hexaedron whoſe oppoſite plaines are papipedon, is either a cube or an oblong, ſo that a cube is a right angle paralell pipedon conſiſting of 6 equall ſurfaces.
As for your mixt ordinate polyedrons, they be but pyramidats compoſed of pyramides lying open in the baſe, and concurring with their tops in one center, and from theſe mixt ordinate polyedrons are deriued your regular bodies, as you may perceiue by Euclide 27 D. 11, & 29, D. 11, &c. ſo that bodies be regular or ir5, as tetraedrons, hexaedrons, or cubes, Octaedrons, Dodecaedrons, & Icoſaedrons, the firſt being a Geometricall body encompaſſed with 4 equall equiangled try6 equall ſquares, the third with 8 equall equiangled tryangles, the 4 contained vnder 20 equall equian12 equall e5 regue circular conuolution is made two waies, as by the ſection of circles, or by inequall right lined figures, the ſection of circles are either greater or leſſer then a ſemicircle, by the conuolution of the greater lenticulare bodies are made, and by the leſſer Duals are created, now by the conuerſion of equall right lined figures diuers kinds of irregular bodies are made, & thereby diuers kind of veſſels: as for the irregular ſolids made of inequall ſurfaces folded one towards another, the difference that may riſe therein is infinite.
Like as a ſurface, ſo a ſolid is plaine or bowed, & being bowed it is either a ſphere or varied, a ſphere is a round bowed ſolid cont which moueth ye cone is right angled, if leſſe obtuſe-angled, if greater acute angled) ye other by the conuerſion of a right angled paralellogram ye one ſide reAris, the conteining ſide turning about the baſe, and the hypothenuſall the ſide of the cone: furthermore amongſt theſe ſolids the perpendiculars, falling from the higheſt point of any fie concurſe of many ſupeficies in a like point, and ſo a right line paſſing from one of theſe ſolid angles vnto another is called a diagonall, but paſſing betwixt oppoſite angles it is a diameter.
Geodetia, as alſo for that I intend to conclude, I wil draw ſome propoſitions from Euclid and Ramus for the meaſuring of them, and ſo referre the further diſcourſe vnto the vnderſtanding Reader.
1 INcreaſe the ſide in halfe the baſes Periphere, ſo haue you the content of the Conicall ſurface.
2 Likewiſe the altitude of the Cone augmented in the third part of the circular baſe is the ſolide content.
3 Cones of equall height are as their baſes. E. l. 12. p. 11.
By this Propoſition you may meaſure Steeples and ſuch like round or conicall figures.
1 THe circular baſe and the altitude increaſed one in the other yeelds the content of the Cylindricall ſurface.
2 Alſo the plaine number made of the baſe and the altitude is the ſolide content of the Cylinder: I meane the ſuperficiall conarea augmented by the altitude yeeldes the ca
3 Cylinders of equall height are as their baſes. E. l. 12. p. 11. and are treble to the Cone, equall in baſe and altitude. R. l. vlt. pa. 7.
By this Propoſition may you meaſure all kind, of columnes, pillars, or any ſort of Cylindricall bodies.
1 BY the art of Geodetia meaſure the contents of euery inui
2 Alſo increaſe the third part of the baſes area in the Pyra
3 Pyramides of equall height, are as their baſes. E. l. 12. p. 5. and 6.
Hereby you may meaſure 6, 8, &c. ſquare ſpire ſteeples, and all ſuch Pyramidall figures.
1 GEt the area of all the paralelsgrams and baſes, adding the ſame together, ſo haue you the contents of the Priſmati
2 Alſo the area of the baſe increaſed by the altitude produceth the ſolide content.
3 A Priſme is treble vnto a Pyramis, equall in baſe and altiE. l. 12. p. 7. and Homogeneal Priſmes being of equal height are as their baſes. E. l. 1. p. 29, 30, & 31.
1 GEt the ſuperficiall content of one of the equall ſurfaces, which multiplied by 6 produceth the content of the Cubi
2 Alſo ſquare one of the 12 equall ſides, then doth the product multiplied by the ſame ſide produce the ſolide capacitie of the Cube, and after this manner any number is cubed; as 3 times 3 is 9 the ſquare, and 3 times 9 is 27 the Cube, 3 being ye ſquare roote of 9, and the cubicall roote of 27.
1 RAmus l. 20. p. 5. hath proued that the plaine number made of the greateſt circumference, and the diameter, is the con2 or get the plaine number made
4: 3 or multiply the ſquare of the diameter by 22, parting the product by 7: by either of which waies you haue the Sphericall ſurface.
2 Furthermore the plaine number made of the diameter and the 6 part of the Sphericall ſurface is the Sphere. R. l. 26. p. 5.2. or as 21 is to 11, ſo is the cube of the diameter vnto the ſphere, ibid. therefore cube the diameter by the 5 Prop. 2, which multiply by 11, the product then parted by 21 produceth a quo
3 Spheres haue treble proportion vnto their diameters. E. li. 12. p. 18. Hereby may you meaſure Globes or ſuch other round bodies.
FOr the ſuperficiall or ſolide content of parts of circles, looke what I haue ſaid of the whole, and proportionally vnderſtand the ſame of the part; as the plaine number made of the circumfe6 part of the ſphe
34 Chapter of my Art of Geodetia.
But otherwiſe, if you haue the length and breadth of any board, floore, glaſſe, or ſuch like, and are deſirous to know the quantitie of feet or yards therein, you muſt multiply the length in the breadth, ſo is the product your demand: after this ſort doe Painters and Ioyners meaſure the quantity of painted clothes, or waineſcot for chambers: and no otherwiſe doe Sielers mete
SENECA.Plus caeteris dedit, quia ſine ſpe recipiendi dedit.
You may haue any of the Inſtruments in this booke made of wood, in Hoſier lane, neere Smithfield in Lon
The Glaſſe is made in braſſe, in blacke Horſe-ally, neere Flecetebridge, by Elias Allin.