CHAP. I. What Topography is, and how it differeth from Cosmography, and Geography.

TOPOGRAPHIE (with some called Corography) is an Arte,Topography, whereby wee be taught to describe any particular place, without relation vnto the whole, deliue­ring all things of note contained therein, as ports, villages, riuers, not omitting the smallest: also to describe the plat­forme of houses, buildings, monuments, or any such particular thing; and there­fore a Topographicall description ought to expresse euery parti­cular, which caused me the rather to call this instrument the To­pographicall Glasse, as being most apt to describe any monu­ment, Tower, or Castle, any Mannour, country, or kingdome, so do we briefly describe England thus: but wée haue omitted di­uers things, by reason of the smalnes of the plat; therefore take the plat, the diuision, and number of shires, number of Parish Churches in euery shire, &c.

A Demonstration of Topography.

Geography (as Vernerus in his paraphrasis saith) is an imi­tation of the whole earth, and his principall and most knowne [Page 3]partes differing from Topography, because it respects but places of note, and from Cosmography, because it hath no relation vn­to the circles in the spheare, onely describing the world by hils, riuers, and therfore Geography may serue well for a general de­scription of particular places, riuers, &c. in the world.

A Demonstration of Geography.

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 [...] of the foure Elements, and al­so of the Sunne, Moone, and all the other starres both errati­call and fixed, with all things else that be contained within the concauity of the heauens.

First of all, it hath respect vnto the circles, which are ima­gined in the celestiall spheare: and such like circles be appointed or imagined vpon the earth, also it doth demonstrate the site, syme­t [...]y, or commensuration of places, the diuersity of climats, the differences of daies and nights the foure quarters of the world, the motion, rising, and setting of the starres, with such as are [Page 4]moued vertically, with all things else, that appertaine to the con­sideration of heauen, as the eleuation of the Pole, the paralels, Meridians, circles, &c. So that like Meridians, like paralels, &c. in heauen do corespond to their like vpon the earth, hereby is e­uery towne, &c. situate vnder some one constellation or other:

A Demonstration of Cosmography

Hereby hath euery towne lying into the East or West a seueral Meridian, and into the North or South a seuerall paralell of la­titude, &c. as you may gather by the Cosmographicall demon­stration before, therefore behold the figure because this is be­sides our entended labour.

CHAP. II. Geometrical definitions of lines, angles, and figures.

BEcause the insuing tearmes of Geometry bee often vsed in this booke, & the vnderstanding thereof not knowne (though something common) to all yong pra­ctizers, as also for y^e they be but stenderly remembred in my Geodeticall Staffe, I thought good to say somthing ther­of. All Geometricall 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 be­twixt his termes, and therefore they bee all like vnto one, but where as an oblique line doth lye inequally betwixt his termes, they may be deuided, so y^e 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 subdiui­sion, and are therefore right or oblique.

18 A right angle is where one liue falleth perpendicular vp­on an other.

19 An oblique angle is made when one line doth not fall per­pendicular vpon another, whereof there be two kindes, Acute 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 boun­ded vpon euery side. Therefore a Tryangle is a figure consisting of 3. 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 fi­gure.

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 infi­nite, the center of the figure is alwaies in the Diameter, and in the concurse of the Diameters.

28. The altitude of a figure is the length of the perpendicular let fall from the toppe of the figure vpon the base.

Of the kinds of figures.

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.

Of right lined plaines.

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 an­gles, as Orthogoniumes, Ambligoniumes and Oxigoniumes: so also may they be taken mixt or comparatiuely, as well in re­spect of their sides as angles. So haue we an Orthogonium Iso­pleuron, an Oxigonium Isosceles, or an Oxigonium Scalenum.

Of Triangulates.

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 equi­later.

Of Obliquangled Paralelogrames.

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 ine­quilater.

Of a Trapezia.

55 A Trapezia is a quadrangle being not a paralelogram.

Multangles.

56 Mixt or multangled Triangulates are such figures that consist of more then foure sides.

Of Curuilines superficies.

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, al­so lines Ascript, 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 vne­qually distant from the midst of y^e figure, such figures be Ouales, Lenticulares, &c.

CHAP. III. How right figures are created.

I Shall not need to run into an ample discourse hereof, nor tire you with multitudes, onely some few that shall bée most requisite in this intended worke, I will acquaint you with, leauing apart the rest.

CHAP. IIII. The making and composing of the Topo­graphicall Glasse.

The construction of the Topogra­phicall glaffe. FIrst prepare a peece of wood requisite for such a purpose, or a péece of plate, if you will haue it in mettle, & let it be 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 [Page 16]it is, and set figures accordingly, an inch, or some reasonable di­stance from this circle, describe another circle, and diuide the same into 32. equall parts, and there appoint the 32 winds, as you may plainely see [...] next vnto this describe another circle, and there place the Geometricall Quadrant thus.

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 y^e center r, & each di­uision in the sides i 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 cir­cle r, and the circle a, drawe fiue other paralel circles equidistant one from the other, as in the figure: Lastly, vpon euery two de­grées make an Isosceles Triangle round about the circle: you may better perceiue it by the plaine demonstration then vnder­stand it with multiplicitie of words.

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,The Card. you must prouide a Card to be placed within y^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 la­titude (but that you may omit if you will, because my new fight instrument performes it) the deliniating is performed by the A­zimuth as is said: the houres be numbred, and y^e moneths wrote at euery seuerall circle, wherein you most obserue the houresSee the vse of the Circumfe­re [...]tor. ac­cording to the time of yeare: and because I will not trouble you [Page 17]ouer long with the making, behold the figure which I will cause to be printed in a voyd paper to saue a labour in drawing the same.

The Index.Furthermore to this Instrument there belongeth a circle of mettle equall in Diameter to 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 o­uer the circumference a 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 [Page 18]neede to bée no longer then from 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 limi­ted, but is best to agrée with the diameter of the Planisphere. Appoint therefore a 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 sou [...] certaine artificiall branches to hang a plumbe neately at, as in this figure, which will also beautifie your Instrument; but they must be so wrought by the workeman, that they let not the Semicircle to fold downe. Then some two inches or better di­stant from this circle r 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 An­tickes artificially wrought, and the said Semicircle so done must be placed vpon the Indexes 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 de­micircle.

As for the Astonomicall circles, though the graduation there­of be after a new order, yet will I not stand to proue the same with Geometrical demonstrations, referring such to the fiftéenth booke of Ramus, where they shal find, y^e 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 di­ameter of this circular sight to agree with the diameter of the Planisphere, then haue you no supporters at all, and this best.

6 But now for the Horologicall arkes, they be more difficult to performe: you shall therefore at each end of the former demi­circle appoint the moitie of the Zodiacke, euen as you be taught in the 9. 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 cir­cle [Page 19]the r 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 cir­cles, as the ☉ degree of ♍ or ♐ or any other, as occasion requireth, by placing the one foote in a, and extending the other to the degrée in the Zodiaque assigned. This so done, repaire vnto the Table of y^e Suns altitude in the first Booke of the Geodeticall staffe, Here serues the figure supported with Antickes a­gaine. & there séeke the altitude of y^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 y^e 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 [...]or [...] there, and so procéed as before: so haue you finis [...] al the houres belonging to the North moitie of the Zodiaque, and they will bend towards the left hand.

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 after­wards proceed as before: so haue you finished the arkes of the South moitie of the zodiaque, and they will bend towards your right hand.

So hauing placed a small sight fixed in the 90 degrée, the fore­part of this Demicircle is finished.

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 De­micircle, now lay a Ruler vpon the center a, and euery of those e­quall parts, both in the line f 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 pa­ralel circles, and number them by three, as 3 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 a [...] g, the other at f, bearing the fore­said Demicircle vp ouer the Boxe and the Needle, prouided that he may fold downe at pleasure, with certaine haspes or buttons likewise to fixe him vpright at pleasure, the Diameter thereof standing paralel with the fiduciall edge of the Index g and f.

9 In the middest of this Demicircle vnder 12 in the Hyso­metricall [Page 22]Scale may you fixe a plumbe, and ouer the Bore with the Needle a certaine point iust vnder 12, which will serue to keepe the Instrument paralel and vpright, which the crosse Nee­dle will as wel doe, but both are not amisse: the old song is, Two strings are good to one bowe.

The moueable Sight.More is yet pertinent to this Demicircle, and that is a moue­able sight, which is to be moued round about y^e circumference of the same fiduciall edge therof alwaies concurring or pointing to the center a, as h g. Take therefore a péece of Brasse of the thicknesse of a shilling or better, appoint therein a center at f, then take the Semidiameter of the great circle a b in the Demicircle, and place the one foote of your compasse in f, and with the other strike the arch h b, and from h to f produce a right line; then in the Demicircle take the Semidiameter of the Tropicke of [...], placing the one foot of your compasse in f, with that widenes de­scribe the arch k l, then take the Semidiameter of the Equinocti­all a r, and with that widenesse describe the arch g m, so is g k the length of the moitie of half [...] the zodiaque. Now to place the 12

Signes and their degrees therein do thus: Vpon the point g e­rect [Page 23]a Perpendicular g i, then making g a center, and g k the Se­midiameter, with that widenesse describe the Quadrant of a cir­cle k i, which diuide into thrée equall parts, next diuide e­uery of those parts into thrée equall parts, so doth euery of those parts represent ten degrees of the zodiaque: now from euery of these degrees must you let fall a Perpendicular vpon the line or Semidiameter g k, as you may perceiue by the pricked lines q and r, so is that line diuided into 9 vnequall parts, which doth represent the one Quadrant of the zodiaque, containing ♈ ♉ and [...], and to prouide for the rest doe thus.

Some reasonable distance, as an inch and better, drawe a line b m paralel to h g, whereunto drawe 11 paralels at such distance as they be in the figure, wherein must be placed the degr. figures and characters of the 12 Signes [...] as in the demonstration they be: euery paralel is diuided as g k is, by placing the one foote of your compasse in f, and so fetching each degree from the line g f to the other paralel, and at the ending of euery third degree the line is strooke quite through, so that there be two lines paralel to g m, and k l strooke quite through, and these lines doe limit the beginning and ending of euery signe.

You must also note the South and North signes at the head, as here they be.

In the very point h there is the ordinary fight placed, such as be in Quadrants, so is the graduating of this sight finished.

This fore peece of the moueable Sight so finished, there must be another peece of like quantity soldred thereunto, or l [...]ft grow­ing vnto the same peece, and after bended in such sort that it may claspe ouer the Demicircle, so doe these two peeces hold the said Demicircle straitely betwixt the same, that it may mooue strait­ly and equally along the same; in so much that the arch h b will alwaies bee carried vpon the Circumference b c in the Demi­circle.

C g and g h doe represent the distance of the two peeces one from the other, which is the iust thickenesse of the Demicircle. Neither would it be amisse to haue a small screw pinne vpon the backe or further side of this moueable Sight, which would make the said Sight mooue the more steddie.

10 The next thing pertinent vnto this Instrument, is a Boxe to hold the Needle.The Boxe.

The Circumference of this Boxe must agree with the cir­cle p q in the great figure, for within that ciccle must he stand, the [Page]Diameter whereof must be 3 inches; and in this Boxe must be placed a Needle and a glasse, as the order is and the Card in the bottome which I described before, 120 standing in the South, 60 in the North, 90 in the East, and 30 in the West.

About this Boxe must moue the Circle that beares 4 Indexes with the Sights, the which Boxe must be turned with certaine shoulderings to come halfe a quarter of an inch vpon the said cir­cle, to the end that it may keepe the same downe close to the bo­dy of the instrument, and that he may mooue stedfastly about. This Boxe is to be fastened through the backe side of the body of the Instrument with screw pins, so may he be taken off at plea­sure: the two screw pins that screw on the socket vpon the backe side, may also screw this Boxe by fastening a rib of Brasse vpon the bottome of the boxe, with screw holes answering to the holes in the socket.

Vpon the Boxe aboue the glasse stands a certaine crooked wire. bearing a roun [...] knob in the middest iust ouer the areltree that beares the Needle, and iust vnder the plumbe when the In­strument stands vpright.

The Needle. 11 The next thing is a Needle, which must be prouided in manner following.

As for the Needle, I would haue it made like two Needles ioyned together at right angles, as you may see in the ensuing fi­gure, and you shall fl [...]d it hereby more true and apt to worke then the single Needle is, for it will keepe the instrument para­lel and vpright without the helpe of the plumbe, cut the degrée more precisely, and stand more directly.

Now this needle must be touched with a Lord stone, and it is very requisite that the said stone be good, therefore make choyse of one thus: The best stones be those that come from the coasts of China and Bengalia, the colour whereof is like to yron, or somewhat sanguine, if they be right, they will drawe vp their owne weight: they be heauier then other: there is another neere as good, which commeth from Arabia, they be broad like a tyle-stone and somewhat red coloured.

If the Magnes stone haue lost his vertue, throwe it into the fire, and let it lie there vntill it be neere red hot, and then quench it in the oyle of Crocus Martis, so shall his power bee multi­plied.

Your stone thus ordered you shal make cleane the North end of your needle, and rub the very end thereof with the stone, this preconsidered, that the north point of the stone touching the nee­dle, causeth that end touched to point into the South; so contra­riwise the end touched with the South part turneth into the North, so that you must haue a care in this point.

After you haue touched the end of the needle, if it were equi­ballanced before you shal find the same end to hang downwards, as it were the heauier, whereby the vnskilfull spoyle many nee­dles: [Page 26]and this is called the Declination of the needle vnder the Horizon, therefore let the end that shall not be touched be the heauier before you vse the stone, and after the application of the stone, if it be too heauie, you may amend the same.

The needle so touched, the South end thereof will not poynt iust into the South,Magneticall me­ridian. for that the Magneticall meridian whereto the needle poynts, and the common meridian wherein the iust South stands, differ: for the Magneticall meridian is a great circle, as the other is, and also passing by the Zenith, diuiding the Horizon into two equall parts, the intersection of which meridi­an with the Horizon is the point whereunto the needle turneth, which is called the Variation of the needle:The variation of the Needle. and at London is one point of the compasse or 11. degr. and 15 minuts, west from our common meridian: and this is the cause that in all portable sun Dials, the line which the needle standeth ouer, doth not point iust vnto the 12 of clocke marke, nor lie vnder our common ineri­dian. Lastly, prepare a hollow socket of brasse with a screw pin, & the socket to be scrwed on, as the order is, so is your Instrument finished onely prouiding a Staffe for the same:Portable Dyals. the thrée footed staffe is best to place it at all heights, and in all places.

And one speciall note you most here obserue in the delineating of this instrument, that is, t [...] haue a care that the body of the instrument be iust foure squar [...], and that the sides of the square lye parallel to the diameter of the circle that is diuided into 360. 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 [...] beware that no one come a­bout you, but such as you know kinde friends, loost, [...]herwise of purpose they beare a Loadstone about them, which may con­found you in your worke.

CHAP. V. To set the parts of the Topographicall Glasse together.

HAuing now finished euery part of this instru­ment, and being ready to set him to worke, thus must the parts be ioyned together. First, vpon the center in the body of the instrument place the circle with the 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 [Page 27]is said, and let the Box be furnished with his Néedle, Carde, & Glasse, as in their proper place is taught. Next place the great Circular sight vpon the Index f g, and the other sights vpon y^e 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 y^e plaine Table, as shall follow after the rest.

To worke as the Theodelitus, and Topographicall Instrument.
CHAP. VI. The description of the Theodelitus, and Topogra­phicall Instrument, with the necessity of refor­mation thereof.

The description of the Theodeli­tus. THe 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 cir­cle there is inscribed a square, which is parted into a certaine number of equall parts, which doe represent parts of the Geometricall 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.

The description of the Topogra­phical instrumēt.But if you make this instrument like to that which Maister 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 Se­micircle, to moue vp & downe vpon the perpendicular, and to moue about with the Alhidada. This Demicircle is diuided into [Page 28]twice 90 degrées, both ending in the Semidiameter, which Se­diameter standeth vpwards, the arch hanging towards the Pla­nisphere: 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 Demi­circle lyeth downewards, and alwaies paralell to the Planis­phere, whereas the other is moueable: And as in the other De­micircle there be twice 90. degrées, héere is but 90 degrées in all, so that they be twice so large as the other. Certainly this Demi­circle without shewing further reason, is farre surpassing that of M. Digges for diuers good respects I might well take occa­sion to speake of.

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 al­teration as well in the Planisphere, as in the Demicircle. Tou­ching the Planisphere, see how you bee taught to attaine to the minutes of degr. cut by the Index by helpe of the Isosceles Tri­angles, then is the Quadrant proiected into a circle, and what commoditie haue wee thereby? marry much more too me both for the néedle (whose largenesse is required) and other circles néedfull and pleasant to be added hereunto, as the Mariners compasse and other circles to tell the houre of the night by the Moone. And lastly, touching the alteration of the Planisphere, looke vnto the 4 Indexes how requisite they bee for the measu­ring of grounds, and to what great purpose they stand you in as in the 33 and 34 Chap. Touching the alteration of the Demi­circle, let euery man acknowledge the necessitie thereof, for that you could not take any attitude of the Sunne, or any other cele­stiall bodie or obiect situate in the heauens or vpon the earth, if so the altitude thereof exceeded 60 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 [Page 29] [...]e of iudgement spe [...] i [...] there were not a necessity of refor­mation, which [...] it is heere done: so I know it by practise and [...]ontinuall experience to be requisite to be done (without offence bee it spoken) to any friend of the composer of the Theode­llitus.

CHAP. VII. To search the proportion and symmitry of a country, fields, or such like.

To find the pro­portion of coun­tries. IN this him of worlbe wee shall haue no neede of the needle, or such like: if you séeke the pro­portion of a field or such like, go into that part of it, from which you may obserue all the an­gles (and it were conuenient if white papers were fixed in euery angle) and there plant your instrument, beginning at what angle you please, place there for the 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 fol­loweth: the instrument resting vnremooued, conuey your Index towards your right hand at pleasure, obseruing through y^t 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 an­gle which at the last station was your first angle And note the degrée cut by the same Index 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 [Page 30]in the circumference of the circle, and by those markes from the center of the instrument draw lines infinitely, next protract the line directing to the second station, which properly may be called the stationary angle: vpon this directing line describe another cir­cle, as farre off, or as néere to the other as ye list, and vpon this circle protracte the angles of position, obserued at the second station. Now see where the lines meete, or a like toucheth his like, so doe the intersections of like lines limit the true pro­portion.

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 distāce is required. Now must ye multiply the parts included betwixt any two secti­ons in the knowne distance, conteined in the stationary line, and then diuide by the number of equall parts conteined betwixt the first and second station, so haue you the distance required.

Example.

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

[Page 32]thus, if 18 yeeld 300 yards, what shall 10 yeld, 183 and 2/4 old ⅓ therefore multiply 300 by 11, so haue you 3300, which diuide by 18, so haue you 183 2/6 which is 1/ [...] making a foote. Whereby I may conclude that betweene a and b is conteined 183 paces and one soot [...]. Thus of all the other, as well d a, d b, d c, or m c, m b, m a, as c b, or c a.

CHAP. VIII. How to take the true plat of a small Island that is incompassed with some Riuer, or of any peece of ground subiect to the sight, that lyeth in such order that you cannot haue accesse vnto the same, by reason of Marshes, Fennes, or such like impediments.

THis Chapter is right necessary, as well for the act of Geodetia, and measuring grounds, as for Cosmographers and such like. Let there­fore a 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 [...]der is: you shal therfore seeke out some such place as 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 ob­serue all these angles, as the order is: and as you may plainly perceiue by the concurse of the lines at both stations, as g f, g a g e, g d, g c, g b, and h a, h f, h [...], h d, h [...], h b, 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 sta [...]ions g [...]and h 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 [Page 33]done, if you desire the distance of any angles or corners, as of 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 A­rithmeticke, may you measure the lines a 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.

CHAP. IX. To take a plat at one station by the Theodelitus.

THis Chapter is not necessary for the setting forth of great continents,To take a plat at one station. but if you would vse it in platting of fields, repaire into some place whence you may obserue all the angles, and towards the first angle vpon the left hand direct the Index being placed vpon the Diame­ter, where the degrées do take beginning, the Planisphere, 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.

CHAP. X. To take a plat of Wood-ground by going round about the Circumference.

To measure woodland. IN this kinde of worke at euery station you must looke through the backe sight as well as through the fore sight, and likewise must mea­sure the distance betwixt euery seuerall sta­tion, so that the proportion is obteyned by the direction of the backe sight. To be short, I would measure this small Tryangle a b c, I plant my instrument at c, making the Index looke to a, lying pa­ralell to a c: then I note the degrées cut, and before I remoue the same, I go to the other end of the Index, and through the backe sight I espy some marke in a right line with a c, and if there be none, I cause some to be set vp, as e, and so I take vp my instrument, and measure a c, and note that downe with the

number of the degrées cut, the Index still resting vpon those de­grées, I place the center of the instrument ouer a, mouing the body of the said instrument, the Index still resting, as at the first it was, vntill through the backe sight I espy the marke e taken at my first station through the said backe sight. The in­strument so resting, I turne the Index to point to b, he lying pa­ralell to a b, and then, as before, I note the number of degrees cut, and before I alter the Instrument or Index, I go vnto the other end of the Index, and through the backe sights I obserue some tree or bush in a right line with b a, which let bee f, then must I measure a d, and note it downe against the number of de­grees cut at a, now, as I haue shewed you in the ending of the [Page 36]first part of the sixth booke,The Geodeticall Staffe. Lib. 6. it rests at your pleasure whether you will obserue the angle b or no: for as you be there taught, if you haue wrought truly, you haue the line b c already, and as you dealt with the former two angles, so must you haue proréeded if there had beene more.

Now looke how you obserue the angles and lines in the field, and in the same order must you protract them vpon your paper.

To vse a Needle in the Theodeli­tus.And here note, that as you worke by the backe sight with the Theodelitus, so also may you vse the Néedle, by kéeping the née­dle at euery station, iust ouer one place, and then noting the number of degrées cut, and so going round, the worke is finished: so that you may hereby perceiue the Circumferentor to bee bor­rowed from this instrument, and vsed by a contrary application.

CHAP. XI. To draw the plat of a Country, and thereby to make a true Map, and situate euery Towne and Vil­lage according to their true distance, that you may know the true distance without Arithmetike.

To make a Map. 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, The Semidiame­ter of the Hori­zon. 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, See Chap. 28. as commonly the Stéeple, [Page 37]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 statiō 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, Lib. 6. of the Geodeticall Staffe. 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 [Page 38]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.

Example.

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, re­moouing my Index to the first towne vpon my right hand, and neerest to the beginning of the degree in my instrument, I find the same to be Hopton 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 finish­ed so much as my intent was, whereof I gather a Table as fol­loweth:

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 cen­ter or point f as you see in the figure, which you must diuide in­to 360 degrees, or else by a protractor from f, pul out right lines by euery grade, noted in the first Table, so is f p Hopton C [...]stle, f e Montgomery, f d Knookin Castle, and so forth with the rest, ending at. f m, Ludlow. Lastly, in this circle, I draw the line f g, by 213 ½ degrees, then making g a center I describe an other such circle as before (and note the larger the circle is, the better it is) I did vpon f, and from this center g pull straight lines by the degree noted in the second Table. Now note the intersection of matchy lines: that is, where the line of Ludlow, issuing from f, meeteth with the line of Ludlow, running from g & there make a marke thus ☉, & thus prosecuting the like in the rest, alwayes setting a marke, vpon the concurse of correspondent right lines (all other intersections not respected) I haue situated all these places in due proportion, noting them with these let­ters, to auoyde (here and else where) often repetition of their names.

And now lastly to get the distance betweene euery of them diuide the line f g into 9 equall parts, for so many miles by mensuration I finde betweene my two stations, the Cordocke for the Wrekin, then by my compasse, I see how many of these 9 parts is conteined betwixt any two places, whose distance is re­quired: & so many miles may you conclude the distance of those two places.

If I haue described places both without the County of Salop as Montgomery and Bewdley, and without the compasse of our Horizon, as Whitchurch, &c. They were set downe because you should haue plenty of examples not thrust together. Heere fol­loweth for more liuelinesse the distance of euery place in this mappe from the towne of Salop: the rest you may gather by your Scale in the same manner.

In the order before set downe changing your stations (as ha­uing finished all in viewe from the Cordocke and Wrekin) you may goe to the Browne Clee and Stilterstone hill, or any other, and passing from one loftie place to another, you may haue the true proportion of all Townes, Castles, Riuers, Hilles, and such like in the whole kingdome, and to reduce them all into the bo­dy of one Card or Mappe, you must seeke a scale proportionable to the quantitie of the paper you will drawe the map in, which here, for that I feare I haue beene ouer-tedious I will omit, and for that it shall be taught in the Flowers of the Mathem. in my 2 part of Geodetia not yet published, and elsewhere is perfor­med.

CHAP. XII. To drawe the plat of any Region, and thereby to find the distance of Townes and such like by sinicall supputation.

THis kind of worke, To seek the pro­portion of coun­tries, by sinicall supputation. although it be something more tedious and difficult then the former, yet hath it in it selfe a most exact and certaine ope­ration: you must in performance hereof ascend the top of some high mountaine, hill, or such like, whence you may directly behold all the ad­iacent townes within the circuit of that Hori­zon, and also from that hill espie some other mountaine, to whose sumunity the view of all the foresaid adiacent townes be subiect. This so done, make the first hill a center, and the other a terme, of one of the sides of euery angle, and so with your Instrument by the 25 Chap. or any other Instrument take the true quantity of the angle that euery towne maketh with these two hils, and note the same downe in some Table booke: this so done, get to the next hill, and there againe obserue in like manner the quan­titie of euery angle euen vpon this hill as you did vpon the for­nier: finally, get the true distance betwixt the top of the two hils, so haue you a line knowne and two angles knowne situate at the ends of a line knowne, whereby get the other angle with the two lines vnknowne, and then place euery towne in his due place, as you shall be better taught in the Example.

Example.

Suppose I ascending to the top of Stretton hils (which be cer­taine loftie moūtains in Salop) might view al the adiacent towns set downe in the ensuing map, and withall another hill called the Wrekin, from whose top also I might well command the viewe of all the foresaid townes.

Now first I place my Topographicall Glasse at a, and then viewing round about I see my eye apprehends Shrewsbury situ­ate vpon the left hand, therefore I obserue the angle g a b by the 25 Chap. and so I proceed to Oswestree, taking the angle f a b, and so proceed round about, noting the quantitie of each seuerall angle, as followeth, a b being alwaies the one side.

These angles so obserued and noted, I beare my Glasse to the Wrekin hill, where planting the same, making the first degree in the Periphere of the Planisphere point iust to a, the Instrument so fixed, viewing about I espie Clun, to which I make the Alhi­dada point, and so by the said 25 Chap. get the angle c b a 4 deg. in like manner I proceede right-wards vntill I haue finished, as I did at b, and thereby doe I collect a table as followeth.

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.

To seeke the di­stance of townes sinically.Suppose we would find how farre Shrewsbury and each o other townes is distant from the Wrekin, or from Stretton hills, by the former obseruations, the angle g a b is 46 degrees, & g b a 60 degrees: therfore by the 2 Booke, Chap. 15 of the Geodeti­call Staffe, adde 46 and 64 together, so haue you 110, which ta­ken from 180 leaue 70, the quantitie of the angle a g b, now ha­uing each angle, finde the right signe thereof, as in the 7 Booke of the Staffe, so shall you see the right signe of the angle g a b to be 71933, of a g b 93969, and of g b a 89879, and to get the di­stance of a 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 [Page 44]hill from Shrewsbury. The like must you doe to get the distance of a g, But to a [...]oyd di­uision, worke a [...] in the 7. booke fol. 287. or chapt. 32. Cōpendium 3. in the end there­of. 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.

To place townes in a Map truly.Hauing by this Sinicall doctrine obteined the distance of euery Towne, as well from Stretton hils as from the Wrekin, ac­cording as you did Shrewsbury from the Wrekin, you shall place them proportionally in one card thus;

Wee will only situate Shrewsbury in true place, proportion, & Symmetry, you shall therefore draw a line a b, which diuide in­to so many parts as there be miles betwixt the Wrekin & Stret­ton hils, viz. 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, [Page 46]by Synical supputation, or by your new made Scale, with your compasse for any three Towns not lying in one direct line, make a triangle, and so finde the angles of that triangle, next, the signes, and consequently, the sides, as you may see in 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 di­stance of any places.

CHAP. XIII. The ground and reason of the Geometricall Quadrant, and Hypsometricall Scale.

BY this Topographicall Glasse I shall teach you to deliuer Altitudes, Longitudes, Lati­tudes, &c. 3 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 In­strument: I thought it not much to spend some time in acquainting you with the ground thereof: Gem­ma 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 semi­quadrant.

Therefore if you say.

As a l the whole Seale, is to l r the equall parts of the contra­ry shadow: so is a 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 infer­red.

I As if you should say.

As d p is to p v so shall it come all to one purpose sayn [...]g [...]

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, so [...]s d 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 fi [...] that the Tangent p v or o x be knowns, that ac­cordingly you may make choise of the dis [...]drence of the Tangents at the first and second station, that is, whether those Tangents be visuall lines of the angles, as w 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 [...] c, 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.

CHAP. XIIII. To get the length or distance of any place from you how farre soeuer it be.

THe distance of any marke from you may be got­tē by this Topographical Glasse diuers wa [...]es, the first way I will deliuer is by signes, which also may as well bee done by the Geodeticall Staffe, or any other Instrument, truely exples­sing the quantity of an angle, and for your more ease in worke, appoint the angle at your first station to bee a right angle, which is as easy to bee made as any other angle.

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 di­stance from you, then take vp your Instrument, and leauing one where he was planted, place him againe at the second station, and then obserue y^e 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.

A is a fort, and you standing at b desire the distance of the said fort from you, wherefore I plant my Topographicall Glasse at b

[Page 49] the Index, with the sights vpon the Diameter, where the degrees doe take beginning, which so resting I turne the Planisphere of the Instrument about, vntill through the sights I espy b, the Pla­nisphere so resting, I moue the Index to 90 degrees, and then a­gaine through the sights appoint some other marke in a knowne distance from you as 673 perches, then leauing a marke at b, I beare the Instrument to c, where by the 25 Chapter I obserue the angle b c a, 61 41/60 degrees, now to get the distance b a you must multiply the Tangent of 61 degrees 45 minuts, by the distance of b c, which parted by the totall signe, yeelds the Longitude de­sired, viz. 186109, augmented by 73 produceth 13585957 which parted by 100000 leueth 135 85957/100000 perches the distance of b a.

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.

CHAP. XV. To seeke the distance of any marke seene before you, by the Geometrical Quadrant.

YOu must call to minde that the Geometricall Quadrant is proiected vpon the Planisphereof your instrument: therefore, place the Index vpon that Diameter where the parts of y^e qua­drant take beginning vpon the left hand, then plant your instrument at your second statiō (for you must note that your obseruations at your first station, & the finding out of this second station is all one in this place, as it was in the last Chapter.) So that the Index 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 required▪ and whereas in the last Chapter you noted the angle, heere onely note the parts of the Quadrant cut by the edge of the Index: then are you to con­sider, if the Index 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 propor­tionall parts vpon the left,See chap. the 19. and so worke according to the first rule in this sort:

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 de­mande.

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 a­bout a halfe in the right side of the Geometricall parts. Now the distance of b c is found 73. yards, wherefore I increase 60. by 73. so haue 5080. which diuide by the parts of the square cut, as by 37. and better, so haue you 135. yards, with certaine odde parts more, the distance of b 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 [Page 52]here protract with a circle readily diuided, Lib. 6. chap. 40. Geode. as hereafter you shall be taught, if your staffe bee wanting.

CHAP. XVI. To seeke the distance betwixt any two Forts, and yet apprach to neither of them.

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 In­dex, (and 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 be­twixt c a certaine Fort, and d a certaine Turret, planting my In­strument therfore, as is said, I obserue the angle c 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 com­passes, the 56 score e 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.

This done vpon f describe the portion of a circle l m, and so [Page 54]protract an angle e f g 54 de­grees, and an other angle of 104 degree, e f h. Finally note the intersection of the lines, as at g and h. which being applyed to your scale, yeeldes the true di­stance of c d in the first figure 131 score, which is 1 ⅝ [...]/8 miles.

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 re­ductions and extractions, so that I hold the Quadrant 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 de­mostration, according to the sight of euery obiect, not vnlike vn­to our yeare bookes, wherin are comprised reports of law cases, still noting all such cases of which there is no like president or re­port recorded, and as hereby they make their yeare books grow to a mighty volume, yet oftentimes riseth there new cases of which they haue no president, and whether then must they fly, but to the report of the learned Iudges, experienced in the law: and so in this case, if I should fill a great volume with demon­strations, yet might there bee found certaine obiects so situate that fitly would suite with none of the demonstrations: and what then is to be done, but onely flye vnto the grounds of the Arte, therefore since I cannot suite you, according to the site of euery particular plat, my drift is to acquaint you with the grounds of the worke, that you may be able of your selfe to picke a respon­dent proposition.

CHAP. XVII. To take the Altitude of any accessible Tower Castle, &c. at one station.

YOu may séeke the Altitude of any perpendi­ [...]ular body, by this 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 y^e 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 demici [...]cle vntill by that sight and the [...]ed sight in 90 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 y^e same so that such proportion, as 60 hath to the parts cut, the like hath the giuē distance to the required Altitude.

2 If the section were made in the parts of the quadrant fur­thest from the fixed sight, the Altitude required is lesse then the Longitude assigned, and beares it selfe in such proportion to the said Altitude as the parts cut doe to 60.

3 But if the section be made in 60, then the giuen Longitude is equall to the proposed Altitude.

Example.

To make proofe of what is said before, let c b be a certaine tur­ret, and you standing at f or a be required to deliuer the Altitude thereof, placing therefore my oye at a, I stir the moueable sight d, vntill from a, by d I view the simile e, noting then the parts of the Quadrant cut by the sight d, I finde them 30, then must I mea­sure f g or a b, which I also finde 178 yards, these had I conclude as 60, is to 178, so is 30 to the Altitude, therefore I multiple 178 [Page 56]

by 30, so haue I 5340 which parted by 60, there remaines 89, the Altitude b c aboue the leuel of your eye at a, if you desire a c, and the square of 178, and 89 to gather the square roote whereof is, c c.

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 me­thode, [Page 57]onely if you please, you may protract with a circle as here­after.

CHAP. XVIII. To search out heights inaccessible, by the Topographicall Glasse.

HEights inaccessible be such to whose base we cannot approch, by reason of certaine impedi­ments, or to which we dare not go by reason of shot, so that out of this demande many de­monstrations might be raised, as to seeke the height of a Tower situate vpon the further side of a great Riuer or Marsh, or such like, or of a Castle fortified with shot, or such like. All which, and more, hang vpon one doctrine, as followeth.

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 no­thing here when you haue once noted the parts cut, as in the last Chapter, 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 ma­ny, I will teach an excellent way.

You must finde out two stations in a knowne distance, as you please, where obserue the angles of altitude, and so get the com­plement of y^e 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 sta­tions to the altitude.

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 [Page 58]

is created 9000000. which parted by 78988. the difference of Tangents, so haue you 114. feet, the desired altitude. And this kinde of worke I hold to be most exact, and farre more certaine then the Quadrant,But with more ease see chap. 32. Compen. 3.for that the one side of the Quadrant heere beares 100000. equall parts, which is the more certaine by how much the equall parts be more in number. Note the letter d is heere in the demonstration omitted.

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.

CHAP. XIX. To know what part of any Altitude is leuell with your eye.

ADmit g c a Turret, whose height aboue the leuell of your eye, is required be therefore di­ligent to plant your Glasse paralell, which ha­uing done, place your eye at the end of the Se­midiameter of the Semicircle, as at a, the fixed sight in the 90, degr. then bring downe the [Page 59]

mouable sight to the beginning of the degrées, as to e. Now your sight passing from a by e, will apprehend some place in y^e 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.

CHAP. XX. How to search out Lengths in Heights.

THis Chapter is right necessary for Archite­ctors and such like, whereby they be brought to finde the distance betwixt windowes, Int­teyes, &c. To performe which you may first take the altitude of the one and the other, and so consequently subtract the lesser out of the greater, t [...]e remainder is your desire.

But if you know not the whole Altitude, first obserue the an­gle of Altitude of the highest, and then of the lowest place, and note y^e 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 y^e 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 [...] your desire.

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 de­gres, viz. 19080 from 50952, the Tangent of 27 degrees, the remainer is 31872, which increased by 172 produceth 5481984, which being parted by 100000 the quotient is 54 ½ 1/ [...] 9/ [...] 8/ [...] 4/ [...] [...]/ [...] feete, [Page 61]the distance of h c, by this meanes you may describe stagemēts for buildings, and all such kinde of things ingeniously.

CHAP. XXI. To reduce the parts of the right side the Geo­metricall Quadrant, into parts proportionall of the left side.

IN reducing of these parts you are brought to worke alwaies as if it were by the parts of y^e left side the quadrant, that is, by the parts fur­thest from 90, 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 di­uided by the sides of your Quadrant, let the side of your Qua­drant conteine 60 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.

CHAP. XXII. To finde lengths in heights by the Geometricall Quadrant in the Glasse.

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 [Page 63]I see through the same from 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 exāple, for that nei­ther of the sections will be made in the right parts, vnlesse you stand neerer to the base of the altitude then the length of the altitude it selfe: that is, vnlesse the altitude bee greater then your distance from the base.

Otherwise.

Get the three angles of the triangle b d c thus, obserue the angle b d a, which take from 90. for that b a d is a right angle, so haue you a b d: then take the, angle b d c, which adde to a b d taking the totall from 180, so haue you the angle b c d, by the 7.60. of the Geodeticall Staffe, chap. 1. p. 21. This had, get the line c a, or a d, & then protract as in the 28. chapter, or work as in the 32. chapter, Comp. 3. in the end thereof, or as in my 7. booke of Trigonometria.

CHAP. XXIII. To know how much one Hill or Mountaine doth exceed an other in height.

THis matter is not so easily performed, as ma­ny ordinarily thinke it to be, for to seeks how much one hill is higher then another, is not to stand vpon the top of one of the hils, and by your Instrument finde whether the other bee higher or lower then the leuell of your eye, as in the 19. Chapter, and so to iudge him higher or lower then the hill you be one. But hee that will know how much one hill doth excéede another in height, must finde how much both of the summities of each hil is distant from y^e center of the earth: or at least, how much either of their perpendicular al­titudes excéede the Semidiameter of the earth: for the lesser of this excesse taken from the greater, leaueth the difference of the mountaines perpendicular altitude. For you must imagine, what hill soeuer you stand vpon, beholding another adiacent mountaine 20. or 30. miles distant, which albeit that hill you behold be equall in height with that whereon you stand, yet shal he not séeme so, nor fall out to be so, being proued with an Instru­ment, or line of leuell: for that where you stand is alwaies the vppermost place, the other hill being situate as it were vpon y^e side of y^e earth, which we may proue by a Philosophical Axiome: Omne graue fertur deorsum ad centrum, vbi quiescit, insomuch that if you trauell about the earth with a line and plumbe at the end thereof, the plumbe will alwaies point towards the center, so that the excesse of any mountaines altitude aboue the super­ficiall conuexity of the earth is altered (in respect of our sight) ac­cording to the position of the place: and therefore when you bee asked concerning the height of two hils, you must know whe­ther they meane in respect of the position of the hils, according to the apprehension of your sight, or in respect of y^e swelling of y^e same, aboue the true conuexity of the earth, for you must vnder­stand that the earth is imagined to be round, as a globe, and so by some it is thought that it was at the first creation, and that these mountaines and hils were since made at Noahs flood, by the raging of the water, which forced stones, trées, and earth vp­on [Page 65]diuers heapes, and thereby did irregulate the globous body of the earth, the which albeit being compared to the Spheares in in the heauens (by the consent of most Philosophers) is but as a point hauing no sensible magnitude, yet to vs that inhabite vpon the superficies of the same, the very hils haue an apparant and great magnitude, eleuating themselues aboue the true cir­cuit of the earth, and as these hills are much aboue the true su­perficies of the earths circular conuexily: so is it not to be doub­ted, but that many valleyes fall within, and lower then the said circular conuexity, so that sometimes we may bee distant more then the earths semidiameter from the center, as being vpon a mountaine, and sometimes lesse, as in some kinde of déepe val­ley, all which you shall bée taught to [...]nde and proue by the en­suing demonstration.

To performe what is said before, you must ascend vnto the top of one of the proposed hils, which let bee b, let the other hill bee c, and you desired to tell which of the top of the hilles is the higher: that is, whether b or c be furthest distant from a the cen­ter of the earth. Being therefore at b, (by some Instrument de­scribed in this booke) get the angle c b a 87, degrées, then must you go to the hill c, and there againe get the angle b c a, 74. de­grées, and so adding these two angles together, you haue 161. which taken from 180. leaueth 19. the angle c a b made at y^e cen­ter of the earth. Hauing these thrée angles, get their signes, and finally the distance of the two hils c b, so haue you a line known, and thrée angles knowne, and thereby (as you be often taught) get the side a c, and a b, which note: for, looke which is the grea­ter, and that hill may you conclude the higher, which heere is c a.

Now to get the height of b or c, aboue the true circular con­uexity of the earth, subtract 34364/11. from the liue b a, so haue you the height of the hill b d foure score: doe so to c a, so (for ex­ample sake) haue you the height of the hill c e, seuen score aboue the true circular conuexity of the earth.

If you seeke how much the one hill is higher then the other, then take b d 4, from c e 7: so haue you thrée score, and so much is the hill c e higher then b d.

And here it is apparent, that if you stand at b, and by a line of leuell looke towards c, your sight will run to f, aboue c, so that the lower hill by this meanes will séeme the higher: for you must note that euery line of leuell doth make right angles [Page 66]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 [...]euell.

On the other side, standing at c, you shall sée the line of leuell c g run farre aboue b, and by this meanes you may seeke the al­titude, and difference of altitudes of hils, which otherwise is difficult to be found.

This chapter may finely be performed by your staffe, for that you haue thrée angles 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.

CHAP. XXIIII. To know if water will run vnto any appointed place.

IF you desire to know if water will be brought from any spring head vnto any appointed place, you are first to consider how, and in what you meane to bring it: that is, either by trenches and gutters, or in pipes of lead, or such like: for those waters that will come in pipes of lead, will not also come in gutters, because the pipes may bring the water into a valley, and so con­uey the same againe vp ouer the top of any hill, being not higher then the originall spring: yea if it bee higher, euen though you should fetch it at the bottome of an hill, vpon the one side, and bring it ouer the top of the said hill as low vpon the other side: for if you once can set it running, it will neuer ceasse vntill the pipes be burst, or all the water spent, the reason is, because 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 suf­ficeth if so the one end of the quilles bee as lowe as the other. Now if with your breath you draw the water into your mouth through these quilles, and so take your mouth thence, the water will run through the said quilles vntill all be spent in the vessel: and this experience confirmes their opinion well that say: A­qua 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 eue­ry miles end, by 4 ½ inches then it is at the spring head: which considered, plant your Glasse at the spring head, so that the Di­ameter of the Demicircle lye paralell to the Horizon, and equall in height to the head of the spring, the two sights in the ends of of the Diameter, looke through the same to the place whither the water should runne, taking notice of what your eye appre­hends through the sights, for to that place will the water runne, [Page 68]abating 4 ½ inches for euery mile the Tower is distant from you.

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 to­wards the place, then go vnto that marke, and if yet you cannot sée to the place, obserue some other marke in an other hill, and so foorth, vntill from the last marke you may perceiue the appointed place, in which make a note for abating, as before, the water will runne thereunto.

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 a­gaine [Page 69]I plant my Glasse in some part of the hill, so that the diame­ter of the semicircle lye paralell, and in an equall heigth with the cundit head a: the Instrument thus planted, look againe through the sights towards the Tower, and for that there is no other hill betwixt your sight and the same, therefore through the sight espy some marke in the Tower, as b, which marke is iust leuell with the spring head a, to which place the water may bee brought by pypes.

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, What pipes bee best to beare wa­ter. 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, 1609 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.

CHAP. XXV. To take the quantity of any stationary angle by the Topographicall Glasse.

YOu must put the Index with the sight vpon y^e Diameter in the Planisphere where y^e 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 Instru­ment then duly planted vpon his Staffe, 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 con­uey the Index with the sights vnto the marke or hedg vpon your right hand, making the said Index point to the marke, or lye pa­ralell to the hedge, note then the degrées cut by the fiduciall edge of the Index, for that is the quantity of the angle. This néedeth no Example.

CHAP. XXVI. The making of a Protractor and Scale.

TAke a fine thinne and smooth péece of brasse, of what bignes you please, whereon describe a demicircle, as a b c, vpon the center f which di­uide into 180 equall parts, and so set figures thereunto, as in the figure: some vse to make the like circle vpon the other side the plate, numbring therein the degrées from 180 to 360, but that is néed­lesse: now by the Diameter of this circle a c, there is made a Scale according to 12 in the inch, furnished with paralell lines, and numbred with figures, as the order is, and as you may see in the figure at d e, vpon the backe side the plate: there is an other Scale made according to 11 parts in the inch, or you may make the Scale d e according to 16 parts in y^e inch or more, & then take 12 of those parts, which diuide into 11 parts, & make a Scale, the vse whereof you may p [...]ainely see set downe in my Arte of [Page 71] Geodetia Chap. 53. but the best Scale is Lib. 5. of the Geode­ticall Staffe Chap, 2. where I treated of the Iacobs Staffe.

You must note further, that for angles of position, the one half [...] of the circle containes the East part of the Horizon, & the other the West, the diameter of which circle common to both the de­micircles, being the Meridian line, or South and North, points as in this figure.

CHAP. XXVII. To protract an angle, and lay downe the ends thereof.

HAuing learned to take the quantity of an angle by this Glasse, it resteth that you also learne how to protract the same, and to lay downe the sides thereof vpon paper by your [...]cale and compasse.

First, therefore to protract an angle vpon any point giuen you are to take the semidiame­ter of your protractor, as f a, This is perfor­med with more ease and better in the 6 booke of Geodetia Cha. 2. and so placing the one foo [...]e of your compasse in g, the point giuen, with the other strike the portion of a circle h i k l, now must you [...]te the quantity of the angle you are to protract, which let d [...] 40 degrées. place therefore the one foote in a in the protractor, and extend the other to 50 degrees in [Page 72]the demicircle a b c, the which widenesse place in this circular base h i k, so will the two points of your compasse fall, the one at h, the other at i, finally draw a line from i to g, and from h to g I conclude i g h is an angle of 50 degrées.

But say, you would draw a line by some degree beyond 180 degr. as by 230 degrees, therefore you must take halfe 230 degr. & set that distance twice in y^e 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 reg [...]red to lay downe the length of euery side by your Scale according as you found the same by measure, first sée what length h g should be, which let be 18 perches, ther [...]fore place the on [...] foote of your compasse in d in your Scale ext [...]nding the o­ther towards c, as to 18, t [...]at widnesse of your comp [...]sse place in g h, fromwards g towards h as to n, I conclude g n i [...] 18 pear­ches, do so to g i, g k, and g l, so shall you finde g o 12 pearches, g k 24, and g p 6 pearches, so of any other.

CHAP. XXVIII. To obserue an angle of position and what it is, as also to protract the same, and to finde vpon what point of the compasse any thing seene in the Horizon lyeth.

AN Angle of position is such an angle that is taken in respect of the needle or in respect of the South point, insomuch that the one side of euery angle of position is the Néedle, and the 02 other the Alhidada, the common section of the termes of which angle alwaies concurre vp­on the very extrée of the Néedle, differing from stationary angles, because they conteine the number of degrées, betwixt any two obiects, proposed at euery particular station as­signed, and the angles of position the number of degrées distant from the Meridian.

To obserue therefore an angle of position, you must place the Néedle ouer his true line in the bottome of the Boxe, the Pla­nisphere there resting conuey the Alhidada with the sights to the marke assigned, noting the degrées cut in the Planisphere, for that is the angle of position, whereby you may see that the angle of position is not limited respectiuely, according as hee is right, acute, or obtuse, but onely doth extend it selfe to any degrée in the whole circle, and therefore the termes of these angles might rather be called lines of position, in respect of their situation, and pointing into the seuerall parts of the Horizon.

Now to protract the said angles, the difference is not any from the worke in the last Chapter, onely where you beginne to pro­tract, call that point the South, & so forward in the other quar­ters of the world, according as you shall be directed by the letters SEWN, vpon the protractor signifying South, East, West, North.

And if you desire to know vpon what point of the compasse any thing séene in the Horizon lieth hauing planted the Glasse as before, so that the Needle respect his due place, turne the Index the assigned marke, noting amongst the winds, the wind cut by the Index, for into that part of the Horizon the thing lyeth. All these neede no example.

CHAP. XXIX. To take the plat of any great Champion field or such like, consisting of 1000 or 1500 acres, or to take the plat of woodland and rough grounds, by the Topo­graphicall Glasse, otherwise then in the eigth Chapter.

SVch a proposition as this cannot bee perfor­med by one station in the midst of the field nor yet at two stations by the intersection of like lines, because it is scarce possible to see all the angles at once from the said stations, eith [...]r by reason of the long distance, or by occasion of hils, trees or such like, therefore you must get the plat thereof by going round about the perimeter of the said field or wood, and that you may doe, either by angles stationary or angles of position, but the operation by angles of position is more troublesome and lesse certaine, therefore it shall bee here omitted.

To performe it therefore by angles stationary, you must go vnto one of the corners vpon the vtter side 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 no­ting it downe together with the first angle in some Table booke, then take the quantity of that second angle, and so measure from that angle vnto the 3 angle, noting the same orderly downe, & so go round about y^e field, stil taking the quantity of euery angle by the said 23 Chap. noting the same downe together with y^e quan­tity of their sides, and when you haue finished, vpon some plaine paper or vellam, by the 25 Chapter, protract euery seuerall an­gle, laying downe the conteining sides thereof by your scale and compasse, euen as you found the same by measure, and as you be taught in the foresaid 25 Chapter:The Arte of Ge­odetia fol. [...]7. this requireth no example, for that the angles being obserued the worke differeth naught from the 8 Chap. part. 1. of my Arte of Geodetia.

CHAP. XXX. To plat medowes, plaine fieldes, and pastures of no exceeding great quantity.

THis Chapter is best to be performed by plāting your Instrument in some such place in y^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 y^e 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 eue­ry seuerall angle, and so to protract the same by the 25 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 vn­certaine, as the intersection of lines and such like, but you may finde that remedied in the vse of the Psaine-Table, the which if you can worke it there, you néede not be to seeke here. Many o­ther pretty waies may you finde set downe in the seuerall vse of each Instrument, which if you can performe in one, you may soone performe the same in euery Instrument.

CHAP. XXXI. To reduce lines Hypothenusall into lines Horizontall by the Topographicall Glasse.

COncerning ascending and descending grounds, I told you in y^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 ren­der the true contents: for that if the plat bee true, the contents will bee false: And on the contrary, if the contents bee true, the plat will be false, the reason whereof I acquainted you with in the ninth Chapter of the sixth booke of the Geodeticall Staffe, where I al­so taught you means to remedy the same, and that diuers waies. But forasmuch as wee shall not néede with this instrument, to reduce these lines in the field but onely when we protract, I will therefore deliuer a way which you may performe by this Glasse, or by your Staffe.

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 vn­till through that sight, and the sight in 90. you sée the top of the staffe, note then the degrées cut by the moueable sight for the an­gle of ascent, then measure the distance betwixt your instrument and the staffe, which increase by the signe of the complement of the angle of ascent, & part the product by the totall signe, so haue you your desire. For 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.

In platting of grounds, I come vnto an ascending banke, as a b, I therefore plant my staffe at a, as before, and another at b, no­ting the angle of ascension c a b, 30. degrees. Now doe I take 30. from 90. so haue I 60. the complement of the said angle, the sum whereof is 86602. and the line Hypothenusal a b, 40. yards: therefore I multiply 86602. by 40. so haue I 3464080. which [Page 77]

parted by the totall sum leaueth 34 6408/10000 yards, the Horizontal line a c.

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.

CHAP. XXXII. Certaine compendious Formes of working by my Tables in the seuenth booke of the Geodeticall Staffe, called Trigonometria.

BEcause I haue taught you to performe many conclusions in this Booke by my Tables in the seuenth booke of my 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 Tri­angle, as you be taught for the auoiding diuision.

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 auoi­ding of diuisi [...]n.

Regula. [...]or the signe that is in the first place, put the Secant of the Complement thereof; Et voti compos eris.

For as the signe is to the Radius, so is the Radius to the se­cant of the Complement.

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 se­cond or third, how to bring the Radius into the first place, for more ease in working.

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 con­uerting any one of them into two right angled triangles which, for that it is briefly set downe there, I will here open it with an example.

Example.

Suppose you had obserued the angle g a b, 46. degrees, and g b a 64. degrees, according vnto the doctrine of the 12. Chap­ter, and so adding these two angles together, and taking the to­tall from 180. you haue 70. degrees, the angle a g b. Now let [Page 79]

the side a g bee required, that is, the distance of Stretton hilles from Shrewsbury, therefore I multiply a b 10. miles by the signe of the angle g b a, 64. viz. by 89879. so haue I 898790. which I part by 100000. and the quotient is 8 9879/10000 miles, the length of the perpendicular a k. Now to get the line a g, I take the angle a g b, 70. degrees from 90. so doth there rest 20. degrees, the angle g a k, whose secant is 106417. which I must multiply by a k, which was not altogether 9. miles, (yet here wee will admit it 9. miles,) so haue we in the off come 937723. which parted by the total summe 100000. the quotient is 9 37723/100000 miles, the distance of Stretton hils from Shrewsbury, which is 9. miles, 3. quarters, and better.

CHAP. XXXIII. To square lands, and to reduce irregular plaines into some regular Figure, and that in the open Field.

TO square any any field, is to reduce the whole body of the same into one square, reiecting the corners, angles, and crooked hedges, which must be after measured, according to the figure they represent.

Therefore, briefly to teach you how to square any peece of land by the Topographicall Glasse, you must ima­gine a b f e g d a péece of ground, consisting of many angles to reduce the body of which into a square, your instrument must be twise planted in such places as you (by the direction of your eye) shall thinke most conuenient for the reiecting of the angles. I

plant the Glasse first at a, then doe I moue the demicircle, vntill through the sights in the same, I may sée close by the inward an­gles, and so I note in the furthest part of the field, whither my sight runnes as to b, where I cause one to stand: the instru­ment vnremoued, I looke through the shorter sights, noting againe whither, my sight runnes, as to d, where also I let a marke. Finally, choosing out some conuenient place in the field, as c, where the visuall beames running thr ugh the sights in the Demicircle▪ may concurre with b, and also run­ning through the short sights, may in like manner point to d: [Page 81]you haue by that meanes reduced the irregular field into a per­fect oblong.

Now for the corners and fragments that doe remaine, you must measure them according vnto the figure that they most re­semble, euen as you may best gather by the pricked lines in the demonstration without any more circumstance of words.

But say the irregular field lyeth most apt to be reduced into a triangle, which is thus performed.

A b c d e f g h i is the irregulars plaine: first therfore I go into some such corner of the field, as to me seemeth most conuenient for the reducing of the field into the largest triangle, which let be

e, where I note diligently vpon both my hands wher [...] the visu­all lines runne by the very point of the hedges giuing out, exclu­ding thereby all corners and other angles, the which two lines let be e c, and e a, the like I do at a, looking to c, so haue I made a triangle e a c of the irregular poligon. As for the other figures, you must measure them as before, according to that they re­present.

CHAP. XXXIIII. To search out the Perpendicular in any Tri­angle or other Figure, according as it lyeth in the open Field.

NOw hauing reduced any plaine irregular Po­ligon into a Triangle, or such other Figure, whose superficiall content is found by the helpe of a perpendicular, and for that the length of the said perpendicular is something difficult to be attained vnto the open field, because it is vncertaine vpon what part of the base your [Page 82]said perpendicular falleth, I haue not therefore thought it much heere to deliuer you the order how to performe the same.

Let a b c bee a certaine plaine field, which you are put to mea­sure: the question is to finde in what part of the base b c a per­pendicular would happen, falling from the angle a. First there­fore I plant my Glasse in the line b c, as néere as I can gesse vn­der the angle a, as at d, then I moue the Indexes about, vntill

the Index with the shorter fights lye directly ouer the line b c, then I looke through the sights in the Demicircle, if then the visuall b [...]mes runne to a, my instrument stands right and d is the place where the perpendicular a d should fall, but if it had not so happened, I must haue remoued the Instrument, as I had seene occasion o [...] the like must you doe in any other figure in the like case, whereby you may see how necessary the foure Indexes be, for the false taking of perpendiculars is a chiefe occasion of those palpable errors that be daily committed, and a principall cause wherefore the common practisers so often differ.

CHAP. XXXV. To reduce many plats, or all your obseruations into one, and thereby to make a faire Map thereof, ac­cording to the quantity assigned.

BRriefly▪ to teach you to performe this chapter, you must first appoint the Card of the bignesse that you intend to make your Map, crossing the same with two lines at right angles, & that about the middest of your Map, writing at the ends thereof, East, West, North, and South.

Now you are to seeke in your Tables gathered at your obser­uations the greatest distance betwixt the most Easterne and Westerne place, and also the greatest distance betwixt the most Northerne and Southernely place, and so accordingly choose you a Scale, that those places beeing laid down by the same, may fall within your Card.

The next thing to doe is, to find out some such a towne or castle that you thinke lieth in the middest of the countrey you would describe (which you may easily performe by comparing your ta­bles together) and that towne place vpon the interfection of the crosse lines before drawne, then if you can find some other towne that lieth direct East. West, North or South from you▪ place the same in the line answering to that quarter, according to the scale wherewith you intend to set forth your plat; but if so there be no town obserued that lieth direct East, West, North, or South from the same, then choose some other towne, and according to the position of the place extend a right line, and so place the same

towne vpon the said line by your Scale and compasse, according to the true distance thereof from the town formerly placed vpon [Page 84]the intersection of the two lines, and so is your greatest labour fi­nished. As for the situating of the other townes: performe one and doe all; for the rule is generall: therefore search what distance the port towne or village you desire to place, is from either of the townes before situate, and so opening your compasse vpon your Scale to the like number of miles, scores, &c. placing the one foote in either of the places already situate, with the other describe se­uerall arches, noting the intersection of the same, for the true place of the said towne. And thus may you deale with all ports, villages, townes, or what else you haue obserued, and here would situate, passing from one place to another at pleasure, with this prouiso, that the firme foote of your compasse bée fixed in his re­spondent place.

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 [Page 85]of Northampton, where make a marke for the towne, and write the name thereof by the said marke, and so proceed, limiting all the townes, ports, angles, and nookes in the Island in their pro­per places, as you may sufficiently gather by the former demon­stration.

Hauing finished, in some void place you may appoint, the Ma­riners compasse, as in the Card before, and this compasse will serue you for many necessarie vses, as it is not vnknowne to men seene that way.

CHAP. XXXVI. To diuide any Empire, Kingdome, or Continent into Prouinces, Regiments, or Shires.

WHen you haue taken the plat of any countrey, and therein situated all the townes, ports, and such like, yet happily shall it be expedient (or rather of you re­quired) to separate and distinguish the same into such Prouinces, Shires, or Regiments, as the said Kingdome is diuided into. So is England, or the South part of great Britanie, being a peninsula, diuided into 52 parts (but not equall parts) which we call Shires, then is euery Shire subdiui­ded into other certaine vnequall parts (as Worcester Shire into 12) which be called Hundreds, either for that there were but at first so many townes or villages therein, or for that there is to be required 100 able men in euery of them. Other diuisions is England yet subiect vnto; as first, the whole Kingdome is diui­ded into two Prouinces or Archbishoprickes, to wit Canterbu­ry and Yorke, then these Prouinces are subdiuided into Bishop­rickes, and euery Bishopricke is resubdiuided into Parishes, according to which diuisions I minde God willing to de­scribe a Mappe of all England, &c. But now the way how to at­taine vnto these subdiuisions is not knowne. It is therefore to be performed after two kind of waies: the first whereof is, you must in your Perambulation, as in the 11 and 12 Chapter, ob­serue the bounds and limits of each prouince, &c. euen as there you doe the townes, and to protract it accordingly, distinguishing the same with certaine pricked lines.

The other way is, to find by some records what townes and [Page 86]such like be the Meres and bounds of the said Shires, &c. The which hauing placed in your Mappe, the diuision is made by drawing certaine lines from towne to towne, or if your may be small, hauing made a point for the place of the towne, you may omit to write the name thereof, and so draw certaine pricked lines from point to point. Euen as you may perceiue England in the ensuing Card diuided into Shires.

CHAP. XXXVII. The reasons wherefore the Altitude of the Sunne hath bene hitherto falsly obserued, with Tables to reforme and correct the same, as well in respect of his refraction as paralax.

TYcho Brahe 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 for­mam visui occurrere, for 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 [...] the Sunne is [...]test to th [...] Horizon, which eleuating themselues by little and [...], successiuely, at length meerely vanish and are nothing [...]ale ob [...]s as in the ensuing Table of the Sunnes re [...]ction the grea [...]est eleuation of the vapours, according to the said optiks of Alhazen and Vi­tello is 40 / 50 // of which the [...]ter of the earth conten [...] [...] which [...] [...]ce 12 Ge [...]maino [...]les, but heauing the ample discourse hereof to the [...]pa [...] of T. B. where the de­sirou [...] may roade at large, but vs onely returne vnto the pr [...]ti­call vse thereof, so shall it bee [...]ine by [...] of this refraction caused by the thickenesse of the vapours a [...]ue the Horizon, that the sight of the Sunne is altered so that he seemes to rest sooner and set slower thou indeed [...] doth, insomuch that the center of his body seeming to touch the Horizon, the same is then 30 mi­nutes below the Horizon, neither doth the magnitude thereof hinden the same, although hee seeme [...] [...]ggat at the setting th [...] when he is in the [...] wherby [...]e may [...], that by occasi [...] of this refrac [...] this who [...] [...] Horizon a [...] vising or [...]ting whe [...] there [...] not any pa [...] thereof abo [...]e the sa [...] and [...] [...] ­tificial,

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 al­titude, that the Semidiameter of the earth hath a sensible pro­portion, in respect of his farre distance from the Sunne, which (setting apart other reasons) may most plainely bee séene in the time of Eclipses, especially of the Moone: by which Eclipses Coperni [...]us also found that the Semidiameter of the Sunnes [Page 89]excentricity did containe 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 a­bout the center of the earth. According to which thrée-fold di­stance from the earth, I haue set downe the insuing Table by the doctrine of the foresaid Tycho, 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 pa­ralax doth make the Sunne séeme lower, and more diected in the heauens (in respect of our fight) then indeed he is.

Example.

We will take Tychos owne example, obserued the last of Iune 1588, when hauing a large Instrument, prepared for that and o­ther such purposes, found the Sunne to bee 19 degrees 17 ⅙ mi­nutes, eleuated vpon the Meridian, to which in the table of re­fractions there did answere 4./ 5.// which taken from the appa­rant Altitude (because the refraction doth alwaies make the Sun higher then indeed he is) there doth rest 19 degrees, 12./ 20.// the iust Altitude by reason of refraction. But this Altitude is not as yet precise, by reason of the Sunnes parallax, therefore in the Table of parallaxes according to the 19 degree of Altitude, and according to the Sunnes threefold distance from the earth, finde the parts there set, to wit, 2 minutes, 44 sec. these adde to the former Altitude corrected by the refraction, because the parallax doth make the Sunne seeme lower then indeed he is, so haue you gotten his true and perfect Altitude, as well in respect of the refraction, as the paralax, to wit, 19 degrees 15 ¼ minutes which you obserued instrumentally to be 19 degrees 17 ⅙ min.

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.

CHAP. XXXVIII. Of the correcting the taking of the Altitude of the Starres, which by most hitherunto hath bene falsly obserued.

AS the refraction of the Sunne is a cause of error in the obseruing his true Altitude, for the causes aforesaid, so doe the like causes produce the like errour in the Stars, but not somuch, for to euery degree of the Sunnes Al­titude the refraction of the Starres decreaseth 4 ½ minutes in­somuch that at the 20 degree of Altitude the starres haue no re­fraction at all, as for their parallax it is nothing sensible, therefore behold a Table for the refraction of the starres for euery degree of their Altitude vnto 20 degree, the vse whereof is all one with that of the suns refraction.

CHAP. XXXIX. To take the Altitude and Almicanther of the Sun or any starre, and to finde their Azimuth by the Topographicall Glasse.

PLant your Instrument parallel the Néedle re­specting his due place, then moue the Index vntill the very edge of the demicircle point iust vnto y^e Sun, now for starres looke through y^e 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 Al­titude of the Starre, but because you cannot view the Sunne, receiue his beames through the moueable sight, mouing the same sight vpon the demicircle, vntill the said beames pearce through that sight, and also the sight in 90, the degrees then cut is the Al­titude, & the degree cut in the Planisphere by the Index that poin­teth to the Sunne or Starre is the Azimuth or verticall circle.

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 de­grees of the Altitude taken, euen as you be amply instructed in the last Chapter.

CHAP. XL. To take the Amplitude of the Sun or any Starre by the Topographicall Glasse.

BY helpe of y^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 in­cluded betwixt the Index, which pointeth to­wards the starre, and the diameter that respecteth the East, for that is the Amplitude: the like must you doe vpon the West side for a setting, also amongst the points of the compasse, there may you sée vpon what point of the compasse the rising or setting was.

CHAP. XLI. To get the howre of the day, the howre of the Sunne rising and setting &c. by the Topogra­phicall Glasse.

TO get the houre of the day, you are first by the 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 Horologi­call Quadrants I hold this the best, easiest and truest, not for that it is a new deuise of mine owne, but by reason of the exact­nesse of working.

Now for the time of the sunne rising and setting is easily col­lected.

And to get the degrée of the sunne, euery Almanacke affoords you that, for towards the middest of each month you haue there [Page 64]set downe, when the sunne entreth into the signe there noted, as in October the 14 day, 1610. Sol in Scorpio, and I demand the 26 day what degree of Scorpio the sunne is in, beginning there­fore at the 15 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.

CHAP. XLII. To finde the houre of the Night by the Tpographicall Glasse, and to know the time of high water, and also the place of the Sunne or Moone.

SEe in any ordinarie Sunne diall what of the clocke the shadow of the Moone yeeldeth, then turne the 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.

CHAP. XLIII. To vse the Topographicall Glasse as the Plaine Table.

To alter the To­pographicall Glasse to a plain Table. YOu must take the circular sight, boxe, needle, and all things of the foreside the Planisphere of the Glasse, and so set the socket that is vp­on the backside vpon the foreside the instru­ment, so doth the backeside (beeing a foure square plaine board) stand vpwards; next must you couer this smooth board with a sheet of white paper, which fasten thereunto with mouth Glewe, or you may haue folding Rulers, as the plaine table it selfe, to performe the same. Lastly, haue a Ruler with Sights, as in the next Chapter, to stand vpon this plaine Superficies, and to the one side of the board, in the thicknesse thereof with screw pins fixe the néedle and boxe, in such order that the South line (I meane not the line of variation) make right angles with the side of the said board, so haue you fi­nished.

CHAP. XLIIII. Of the Plaine Table, with a description thereof, and the parts thereunto belonging.

The Plaine Ta­ble. THe Plaine Table or Geometricall Table is a right angled aequilater paralelogram, made of a board of halfe an inch in thickenesse, whose e­quall sides containe 9 or 12 inches, the superfi­cies whereof is made smooth and plaine, some vse to make him represēt an Oblong: al is one.

Some for ease in cariage vse to make this square board to consist of thrée péeces, which they vse to ioyne to­gether with certaine ledges, such as bee at the end of Table boards, as you may gather by the figure.

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.

The ribs or for rulers of the Plaine Table. 2 The peeces of this table being set together, then be there certaine rulers of wood ioyned together with small brasse hinges which are made to fold, as you may perceiue by this ensuing figure, which beeing opened and stretched square, serue iust to fall into the channel made about the 4 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.

Boxe & Needle. 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 de­scribed in the 1 Chap. This Boxe must be fixed to one of y^e 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 fi­gure, and in the midst of this socket there is a scrue pinne, which serues to wreste against the Staffe that is put into this hollow socket, prouided alwaies that you haue a thin péece of brasse within the socket, for the scrue pin, to force against the staffe be­ing put therein.

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 compas­ses, & set the head of your staffe that goeth into the hollow socket bewrapped in a plate of brasse, so shall not the brasse pin crush the same being forced here against.

The Rule 6 Next to this Table there doth belong a ruler of the full length of one of the sides: this ruler hath two sights, one stan­ding in either end, iust ouer the fiduciall edge of the rule, they vse such sights as bee described in the 9 Chapter, but thus I would wish them made, let a ruler bee prepared of the length a­foresaid, [Page 101]

better then an inch broad, wherein along the middest, within an inch of either end let there bee drawne a fiduciall line c d, then cut the one halfe of the rule away and kéepe the other: lastly, in either end make two square holes to beare the sights as a b, as in the figure.

The two sights. 7 Next prouide two sights for those two square holes, as e f for a, and g h for b, let the one sight be but halfe so long as the o­ther, as you he taught in the 9. chapter, and let the pins head h, and the groue or channell to looke through in f, stand iust ouer c d: lastly, for that we will auoide the number of diuisions in the longer sight, which I described in the vse of the circumferentor,

[Page 102]let the quadrant in the 11 chapter of my sixt booke of the Geo­deticall Staffe be placed with a screw pin in the longer sight, in the slit i k, which you may take thence, vntill you haue occasion for the vse thereof.

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.

CHAP. XLV. Of the absurdities that many vse, that affect the plaine Table, and of reforming many inconueniences therein.

Why the plaine Table is affected by the vulgar. THis Instrument is so plaine (by reason of the ocular spectation of the worke still demonstra­ted before the eye) that it hath thereby begot it selfe a wonderfull affectation of the vulgar, whereby they vainly thinke no worke doth re­lish well vnlesse it bee serued vpon this plaine Table. But this opinion is no lesse ridiculous then full of many doubts that I haue heard diuers plaine Ta­ble men propound.

Ignorance of some practizers.Saith one, I thinke the grounds you teach for the casting vp of péeces of ground, be false: for take a square, and measure the same by your scale and compasses, and note the contents, and then reduce the same square into triangles, and so againe mea­sure the same by the doctrine of triangles and the contents pro­duced by the square measure will differ from the triangular measure: for you must note y^t he was a witty fellow y^t wrought by the doctrine of a curious small scale, and a goodly blun [...] [...]re of compasses, as you may gather by the error he was in: for get the contents of any square, by multiplying the one side in the o­ther, as is taught, and note the product, then square one of the equall sides, and double the offcome, the square root where of is y^e diagonall,Lib. 6. cap. 16. pro. 1. Bac. Geod. then finde the perpendiculars by the 21. Chap [...] of the foresaid 6. Booke, and so working according to the doctrine of triangles, you shall finde the product of the two triangles to [Page 103]agrée with the formes square.

But to returne to the supplements of the wants in this In­strument.

Inconueniences in the plaine Ta­ble reformed.First, the paper of this instrument, by occasion of wet, is of­tentimes blotted and blemished, insomuch that the points, lines, and other obseruations, bee in many places taken away, and so by reason of hasty taking of the Table vp, the paper is by the wet, so stretched and disfigured, that many errors grow vpon a small scale. To auoyd all which, and such like, I thought good to giue this admonition. First let your table be couered as you bee wont with a faire shéet of paper, vpon which shéet let there be pla­ced another shéet, well oyled on both sides with Lin-séede oyle, so may you worke notwithstanding the raine, deaw, or miste, only draw the lines something more heauy vpon the vpper shéet, that they may also pierce the lower apparantly.

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:This holds not in hilly ground without reducing of hypo. lines to horizontall. for in­déede a small error heerein produceth a great absurdity in the closure, and you may now and then try if the chaine mea­sure of your 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 hy­pothenusall to lines horizontall, as you be instructed in the sixth booke. I haue seene some plaine table man, when he was set to measure a banke ascending round about, go vnto the top there­of, and so produce lines from euery corner vnto one center, mea­suring it thereby one station, and hee thought it was a rare de­uice, and had laid it downe in Plano truly. But surely by how much the higher the ground was, by so much the greater was his error,But small banks, and ridges (as some suppose) this respects not. for that hee laid it downe in a greater compasse then it should, insomuch that if he had laid the plaine adiacent sides pre­cisely downe by that scale likewise, the want of closure would haue contained a reasonable large péece of ground: for you must néedes confesse that the lines he protracted vpon the paper, were visuall or horizontall, and the lines measured hipothenusall. [Page 104]Other things should be reformed in this Instrument, which at this time I omit.

CHAP. XLVI. Things belonging to the vse of the Plaine Table.

Things belong­ing to the plaine Table. TO this instrument, as to all other, appertaines a chaine or wire line of foure pearches long, ac­cording to 16. foote, and ½, or of thrée pearches long, which is 16. yards, and ½, let the pearches be noted with brasse rings at the ends thereof, and then diuided into halfes & quarters,The Line. with lesser rings fixed at each quarter and halfe, that you may distinguish the same.

A Scale & Com­passe.You must also prouide a scale of brasse or wood, whether you please, with a paire of brasse compasses pointed with stéele very neate and sharpe: for it is rude to draw your lines Geometricall with Painters kéelers, or blacke lead, as M. Lucar would.

Also you must haue such sights for this Table as bee described in the vse of the Circumferentor, whose vse are set downe in the 16. Chapter of the same booke: or else you may haue such a qua­drant as is spoken of in the first part of my Art of Geodetia, as in the 26. Chapter. These things had, you may fall to worke.

CHAP. XLVII. To take any Horizontall distance by the Plaine Table.

To take any di­stance by the Plaine Table. IT were vaine to make many demonstrations of this worke, since a few may as well suffice, for this Instrument is but only fit to take lon­gitudes and latitudes, as for altitudes I hold him very troublesome and vnapt to performe the same, though M. Lucar haue taken paines to illustrate him in that point; howbeit finding by experience the cumbersome and vncertaine working thereby, I thinke it better omitted then remembred.

You shall then vnderstand, that you may performe any di­stance [Page 105]vpon this Table in the same order as you doe with my Staffe, onely heere you must drawe lines vpon the paper, and measure the same by your scale, whereas the legges of the Staffe represent the lines, and the diuisions your scale.

Therefore at the place whence the distance is required to any marke proposed▪ place your Table, which place call your first station: then your Table lying parallel with your Compasses, make a point in the paper to represent that first station, where­vnto bring the fiduciall edge of your rule, kéeping the one end of y^e 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 y^e fiduciall edge thereof: the Table resting, espye out a second station, & let it make, as néere as you may, a right angle with the marke whose distance is re­quired. This marke so appointed out for your second station, kéep the fidutiall edge of the rule vpon the foresaid point, and so draw a line to point to your second station, then let one measure the distance betwixt your first and second station (which were best to be 1/10 part of the distance required.

So haue you finished all at your first station, with this Prouiso, that you haue regard to the degrées cut by the South end of the Néedle in the Card in the bottome of the boxe before you any wise alter the table, and that you lay downe your stationary line by your scale and compasses, limiting the same according to the line measured, & at the end thereof marke another pricke, which call the pricke of your second station.

Then take vp your Table, leauing a marke at your first stati­on, vnder the pricke made vpon the table representing the same.

Now must you beare your instrument to your second station, where hauing placed the same in such sort that the pricke of your second station may directly stand ouer the marke representing your second station: lay then the edge of your ruler vpon the stationary line kéeping the pricke of your second station next to your body, turning about the table, the ruler resting, as before, til through the sight you espy the marke left at your first station: which done make fast the table with your screw.

A proofe of the worke.Now for proofe of the exactnesse of your worke, and to know if you haue truly taken your backe sight, haue respect to the south end of the néedle, for if it cut the like dgrées at this second stati­on a [...] at the first, you haue done well.

Hauing so done, place againe the fiduciall edge of the rule vpon [Page 106]this point of your second station, the one end being there fixed, moue the other end, vntill through the sights you sée the marke, whose distance is required: then draw a line by the fiduciall edge of the rule, which will intercept with the line drawne from your first station thereunto, therefore note the point of intersection, and by your scale measure the distance from any one point to the other, (I meane by the same scale you laid downe your stationa­ry line) so haue you your desire.

Example.

The distance a b is required, first thererfore I plant my Table at b, then working as before, I finde c my second station, and so draw a line to point from b to a, and another from b to c. Next, I measure the line b c, and finde it 7 [...]. ya [...]d [...], which I lay downe vpon my paper with my Scale and Compasses. Lastly, I note the degrees cut by the South end of the needle, which let be 40. This done, I go to c, and there againe plant the Table, as be­fore: So do I make the stationary line protracted, point iust to b, and then noting the degrees cut a gaine by the needle, I finde thē 40. as before, which argues I haue well planted my Table. To conclūde, I place the fiduciall edge of my rule vpon c, mouing [...]he other end, vntill it intersect with the line representing a b; [Page 107]therefore by my Scale I measure the line representing b a, so haue I the distance of b a 135. yards, by the same Scale might you haue expressed c a.

CHAP. XLVIII. The part of the distance of any thing being, giuen, to finde the rest.

VNderstanding the last Chapter, so wee may thereby auoid many words, and may most ea­sily be performed by the Geodeticall Staffe, as may appeare in the Propositions of the 18.19 or 32. Chapters of the second booke of the Geo­deticall Staffe. But to procéed, a b is a distance required, the part of that distance giuen is a c,

[Page 108] 50. pearches. Then do I plant my instrument at a, as I did in the last chapter at my first station, drawing a line to represent a b infinitely, then laying downe my scale, vpon the same line the part giuen. representing a c 50. pearches, the instrument vnre­moued, I séeke a second station, as in the last chapter, which is d, (but the stationary line shall not be measured.)

Lastly, I note the degrée cut by the South end of my néedle, then leauing one at a I cary my Instrument to d, where I plant him in an respects as at a, now must I finde the point vpon the paper, which represented c, and thereupon lay the fiduciall edge of the rule, mouing the other end vntill through the sights you se [...] c, so wilt the edge of the ruler in the line vpon the paper repre­senting [...] d, th [...] k [...]ping the [...]uler [...]pon that point d, I moue the other [...] vntill it p [...]o [...] to [...] shall the fiduciall edge of the rule intersect [...] wit [...] the [...]e [...] the pap [...] represen [...]ng a b, from the point of which intersection to the point a, is b the termes of the line a b, which being measured by your scale and compasses is found 133 pearches.

CHAP. XLIX. To take the distance of any two townes or such like.

COnsider well the premises, and this labour is already effected, therefore plant your Instru­ment at a, Latitudes. as you were directed in the 29 chapter, and let the latitude required be d c, no [...] [...]raw lines from a to point to d and c, and also to b your second station, now obseruing the former directions I [...]emoue my instru­ment to b, and so draw lines from b to point againe to d and c, then doe I note the concurse, or intersection of the said lines. which I measure by the scale and compasses as before, so will the stationary line a b bee 316 perches, and the distance required d c 131 perches.

A note for many distances,And here note, if you had sought more distances▪ as the di­stance of f e, e d, d c, &c. the labour is no more but to draw lines at euery station, to point [...]nto the distances required, and then to note the intersection of matchy lines, vpon the paper, which after measure by your scale and compasses, so shall you haue your sta­tionary [Page 109]

line g h, 10 score, the distance f e 12 score, &c. whereby you may plat any field, and come not within the same, as in the 8 chapter.

CHAP. L. To finde the Horizontall distance of any place from you standing by a new way, vpon the Plaine-Table.

To finde any Ho­rizontall distance after a new way. IN the 44 chap. at twice I told you of 4 certaine rulers or ribs that were belonging vnto the Plaine-Table, euery one being diuided into a 100 equall parts or more: by these rulers ordered in their due place vpon the Plaine-Table) shall I teach you to séeke the Horozintall distance of any place thus.

Lay the ruler with the sights vpon the very edge of one of the sides of the Plaine-table, turning the Table about vntill through the sights you espy the marke whose distance is required (but with this prouiso that the corner of the Table, where the diuisi­on take begining, be neerest vnto you:) this done take the ruler [Page 111]with the sights (the Table vnmoned) and place the same vpon y^e right side the Table, as before, and then looking through the sights espy your second station, in a knowne distance from your first station. Next shall you beare your Instrument to your second station, situating the Instrument (by helpe of the Needle and backe sights) here in all respects as it was at the first, which being done lay the ruler ouer the corner or both sides of the Instru­ment remoouing the same vntill through the sights you espy the marke whose distance is required, lastly note the equall parts vpon the ribs cut by each end of the ruler or sight hauing regarde to those parts that doe responde to the statinary line, and also to the distance required, for as the parts respondent to the statio­nary line, are to the line it selfe (being measured and knowne) so are the parts respondent to the distance, vnto the distance requi­red. therefore worke by the golden rule, in this worke the line of distance and stationary line alwiaes cut at right angles, this nee­deth no example, for as it is most exact, so it is most plaine & easy.

The premises béeing considered, and the doctrine before well vnderstood you may produce infinite wates to performe many rare conclusions, but we cannot stand to set downe a demonstra­tion to suite to euery proposition that may happen in the field, chiefly for that, let the demand stand howit will you may resolue the same by due regarding the prescript. Now I will briefly touch the order of taking a plat of a field, mannor, &c. by the plain Table, according as we haue dealt with the Geodeticall staffe and other instruments before, ayming to performe some such propositions here that were omitted in the other bookes, for it would increase the volume ouer much to set downe euery kind, in the vse of euery instrument, since wee understanding what is said of the one may also be performed in the other, and that much after one kind of method, as I haue said before: but indeed I haue here set downe such propositions that will best a­grée with the Plaine Table, and are aptest to be wrought there­on, setting aside all impertinent demonstrations.

And you shall note for diuers good respects that I shall omit one thing that standeth firme, and is ordinarily vsed in demon­strations of this nature; & that is, lines to represent the Instru­ment & the lines also drawne thereupon: my reason is, because I will not confound the worke with multitude of lines, as also to saue the cutting of many figures whereby such that serued in the Glasse, likewise serue in the Plaine Table.

CHAP. LI. To draw the plat of a peece of ground at one station where all the angles of the field may be seene from that place of standing.

At one station to get a plat. FIrst, goe round about the field, and in euery angle set vp some marke, then plant your ta­ble couered with paper, in such a place as from thence you may sée all the angles of the field; that done, in a place conuenient of your table make a pricke or point to represent the place of standing: from the point to each marke draw a visuall line by the edge of your Ruler, then from your place of standing, measure exactly with your wire line the iust distance in pearches to each seuerall mark, and set those distances by the scale, each vpon his own line which was drawne to those markes, noting these seuerall points where these measures end. Lastly, from point to point by the edge of your Ruler drawe lines which shall include a figure proportio­nall to the field to be measured, and the lines so drawne shall re­present the hedges of the field, as in this demonstration.

Your station is i, the lines drawne from i to point to euery an­gle, are i a, i b, i c, i d, i k, i e, i f, i g, and i h, which are measured as [Page 113]is noted vpon each line, as i a 27 pearches, i b 9 ¾ pearches, &c. then from a to b I drawe a line, and so go round: so haue I made a figure proportionall, which was required.

CHAP. LII. To drawe the plat of any field by the rule taught in the last Chapter, where you can­not from one place of the field see all the angles thereof.

To get a plat at many stations by the doctrine of the last Chapter, FIrst, set vp markes in euery angle of the field, as in the last Chapter of this Booke, then place your instrument in some place of the field, and from thence draw lines from as many angles as you can see, that are together, and those lines measure, and set the measured distances vpon the lines on the instrument, & from point to point drawe lines to represent so much of the hedge, then ap­point out some place for your second station, whereunto measure, and then drawe a station line, setting the measured distance ther­on: after this, remooue your Table to that second station, and there fixing it, as it was at the first station, which you shall ob­serue with your néedle, as in the second proposition of the sixt booke before is taught. Then from the center of your second sta­tion drawe lines to all the rest of the angles of the field, which haue not lines drawne to them before; or at least so many of them as you can see, and measure the lines as at the first station: this done, choose a third station (if before you could not see all the angles) as you did a second, (stil obseruing that your néedle stand as at the first station:) and if at this third station you cannot sée all the angles yet vnmeasured, you must againe choose a fourth station, and the fift, if néed require: and thus proceed till you haue taken all the angles of the field: and at last, obseruing the measuring into euery angle, as also the rules before taught, you shall produce a figure proportionall, and equall in angles to the plat or field presented, which was the thing required to bee done.

The field is f g h i k l m n o p q r s, wherein I plant my In­strument at a, 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 ob­serue the angles of o p q r, which be all the angles, which I pro­tract and limit vpon the paper, as before: and you must note, that the line a 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 oc­casioned to make: for note generall, causa 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.

CHAP. LIII. To drawe the plat of a field by once placing the In­strument in an angle of the field, and measuring round about the field, because happly you cannot trauerse the same, by reason of waters and such like impediments.

To plat by measuring about the field, and yet but once placing the Table. FIrst set vp the markes of euery angle, then plant your table in such an angle, from whence you may sée all the angles in the field, then make a point on your table (in a conuenient place) to represent the pricke of your first stati­on, from whence drawe lines into each angle, then measure first the hedges, which are on the containing sides of that angle in which your table standeth, and set those measures by the scale, on that line which representeth it, then measure the next hodge vnto it, and take so many measures on the scale, and set one of the féete of your compasse in the last made pricke, and with the other foote strike an arch through the next line to it, and note the interception thereof, and from the pricke last made thereunto drawe a line which shall represent the second hedge, and in this order measure about the field, and strike an arch from line to line, as in the first: so shall you produce a figure proportionall to the field.

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.

CHAP. LIIII. To take the plat of a field by the rule of the last Chapter, where all the angles cannot be seene from one angle.

FIrst angle out the field then plant your table in one angle, as in the last Chapter, and from that angle get as great a part of the field as you can, in order as in the last Chapter: then plant your Table a­gaine in an angle, where your last measures ended, in like fort situated as at the first station, which you shall doe by setting the néedle on the same degrées it cut at the first station, or looking backe (as hath béene before taught) from whence take all the rest of the field (if you can there sée all the rest of the angles) if not, you may make a station or two more, as occasion shall serue: And so worke till you haue wholly inclosed the field; so shall you make a figure on the table proportionall to the plat of the field with angles equall thereunto. This Chapter differeth litle from the last, onely you must performe that at many stations that there you did at one. This agréeth with the 7 Chapter, def. 4. Geodetia.

CHAP. LV. To drawe the plat of a peece of ground by 2 stations, and measuring but one line, where all the angles of the field may be seene from both the places.

FIrst goe round about the field, and set vp markes in euery angle thereof,A plat of two stations. then choose out your 2 stations something neere the middle of the field, a good distance one from another, in such sort, y^t 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 e­uery angle of the field, as may be; for the greater regard you haue in the choosing out your two stations, the better will your lines intercept one from another: Wherefore hauing thus made choyse of your stations the worke is performed: As in the 31 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.

CHAP. LVI. To draw the plat of a field by many stations, and and yet to measure but one line in the whole.

FIrst set vp marks in euery angle, then point out your first station, where your instrument being placed, draw from the pricke of your station lines, to as many angles as you can conueniently see, then appoint out your sta­tion in such a place, from whence you may see all those marks to which you draw lines at your first station, to which station draw a line, and measuring the distance betwixt those two stations, vpon that line set your distance by your scale, and then remoue your Table to your se­cond station, where plant it in his due situation, and then from the center of that situation draw lines againe to each angle whereunto you drew lines at the first, and note the interception, ech with his match line, and then draw lines from point to point, which shall represent somuch of the hedges of the field as you haue gotten by these two stations. Now your instrument stan­ding thus at your second station vntemoued, from the center of your second station, againe draw lines to as many new angles as you sée, (that is, from whence you haue not drawne lines be­fore) then chuse out a third station, from whence you may sée all those angles, whereunto you drew lines last before, and then draw that station line, and then againe remoue your Table, and hauing placed it in his due forme, to find the center of this your third station doe thus; lay the edge of the ruler to any point in the paper which doth represent some marke in the field, and re­moue your ruler to or fro, till through the sight thereof, you sée that marke in the field, which the point on the paper doth repre­sent, by which the edge of the ruler doth lye▪ and then draw a line towards you, till it cut the station line, and note the interception, for that point representeth the pricke of your third station. And from the pricke or center of your second station to that point, sheweth the distance betwixt the second and third station, 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 [Page 119]the angles, which you drew to your second station, & where they intercept or crosse each his match lines, make prickes or points there, and so from point to point draw lines, which shall repre­sent so much of the hedges of the field, as there you could see and draw lines vnto. This done, and the Table vnremoued from that point or center of your standing or third station, draw lines to as many angles as you can sée, which haue not lines drawne to them already. Then chuse out a fourth station in such sort as you did chuse out your third, and to this get your distance, as there you did, and then intercept those lines as before is taught, and in this order make so many stations as neede shall require, till you haue ended your whole worke, and at last you shall produce a figure with lines proportionall and equal angles to the plat of the field.

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 statiō as you can see as b d, d e, b f. [Page 120]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, be­cause I will omit the multitude of lines and letters; and this kind of intersection of lines being duely ordered, of all other is the best, because by apt chusing of your stations you may auoide a­cute angles.

CHAP. LVII. To draw the plat of a peece of wood ground, where, for the thickenesse of the wood, a man can­not place his Instrument, but onely in the angles of the perimeter.

IN this manner of worke you shall vse 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 Instru­ment, because you must (for the more precise planting your Instrument at euery remoue) looke backe to him. [Page 121]Being thus furnished you shal begin your worke as followeth:

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 y^e scale on y^e line drawn, then place your in­strument in y^e angle, & say y^e ruler along y^e line drawne, & thē turne y^e Table about, til you sée y^e angle, or the man left in the angle frō whence you came last, where screw fast the Table, and for your more assurance, you may behold your néedle,The sing [...]lar vse of the backe sight. which in this kind of platting will stand you in great stéed. For looke what degrée your néedle did cut your first standing, the same degrée must it stand on at your other standing, wherefore it were good at the first placing of his instrument, to write downe the degrées cut by the néedle, for the helpe of memory in the rest of the angles, I say this done and your Table made fast from the point of your standing, draw a line into the next angle, and measure the di­stance thither, which measure set on the line drawne, and then plant your Table againe in your third angle, and in this order worke till you haue compassed the whole ground, and if it fall out in the conclusion of your worke, that the line and angle of your figure agrée with y^e line & angle of the field, then is your plat per­fect, if not you haue some errour. And herein, if I may aduise you, begin your worke againe, to finde your fault, & trust not to any helpe for the closing thereof, wherein you shall but deceiue your selfe, and happly of a small errour increase a greater, for that you know not whether your saulte be in the lines or in the angles, wherefore if your figures misse of closing aboue one pearche, ne­uer trust vpon your worke.

And be sure when you plant your Instrument in one angle & looke to the next, that you so direct the sights, that the visuall lines lye paralell to the hedge measured, neither obseruing this paralelly, is it materiall how far off the perimeter you place the Instrument, alwaies prouided that you take your measure in the true & direct place, where the very hedge or bounds go, for if you measure much within the hedge, your lines fall too short, if with­out, too long, therefore obserue the meane: for in all things, ob­seruare decorum, is best.

This kind of measuring is principally and commonly vsed for woods: for that in them a man cannot see the angles by any other me anes, and may serue for all kind of other grounds, and indéed commonly vsed of land meaters, who hauing but this one [Page 122]Chapter, and that rawly▪ presume of the full knowledge of the vse of this instrument, but how they performe it, I leaue to those that shall try them, which is but had, as others before me haue, reported.

CHAP. LVIII. To draw the Plat of a field by placing the Instru­ment in euery angle thereof, as in the last Proposition, and yet measuring but one line in the whole Perimeter.

To draw the pl [...]t of any field by going round a­bout, & yet me a­suring but one line. FIrst place your instrument in that angle where you will begin, which let be in such a place that the first line you go vpon may he of reasenable length, then set vp a marke in the field in such a place as from thence you may see as many angles of the field as possible may bee: which marke call your principall point, and to that marke get the di­stance by your first and second station, which let be in the first & second angle in the field, in order as before is set downe: so shall you haue a point in your Table to represent that principall point in the field; This done, draw a line into y^e third angle, and thereto remoue your instrument, and hauing there placed it, get the di­stance betwixt your two stations as you got your distance in the like case before, which you shall performe thus: Hauing first placed your Instrument by looking backe, or by the néedle, at his due situation, lay your ruler either by the pricke on the paper, that representeth the principall point in the field or by any point on the paper that you know representeth some marke in the field: then turne your ruler about till you sée that marke in the field which the pricke by which your ruler lyeth doth represent, and draw a line till it cut your stationary line, and that point of interception sheweth the point on the paper, where you stand in the field. So in this order by placing your instrument in euery angle you may get the length of euery hedge seuerally with mea­suring but one in the whole, and the conclusion will bee, that in the end you shall make a figure with equall angles and lines pro­portionall to the plat of the field.

The premises b [...]ig well vnderstood, and all things else well considered, I will [...]raue pardon, and so cease further prosecution [Page 123]hereof, presuming that there is sufficient said to open y^e whose scope of this chapter: neither would I go about to fill the booke with many curious demonstrations, and difficult questions, to beguile the aspiring wit of the yong practitioner, but onely set downe some such fewe things that were most requisite to bee knowne, lest otherwise I should be held rather tedious then com­pendious, and therefore I will hast to an end.

CHAP. LIX. To take the plat of any Champion Field con­taining 2000. or 3000. Acres of ground, by the plaine Table, and yet neuer bee forced to change your paper.

YEt againe before I conclude, I will giue you another way to séeke the plat of great Champi­on fields, that containe 3000. or 2000. Acres, by the Plaine Table, which is not much differing from your worke by the Geodeticall Staffe.

You shall therefere place your Instrument in euery angle, and so get euery angle and his sides,To vse the plain Table, and neuer change the pa­per. not regar­ding the length of the conteining sides, as you be wont, then must you measure euery hedge, and as you were wont to lay the same downe by your scale and compasse, heere you shall but write the length of euery hedge vpon the lines drawne vpon your paper, and responding thereunto, so haue you finished, and you shall ne­uer be forced to shift your paper, nor haue the lines to runne off the same, for that you may draw them as long or as short as you please.

Now when you come home, vpon some shéete of paper pro­tract all the angles one after another,See the 3. chap. pro. 5. as you found them in the field, allowing by your Scale and Compasse euery line his due length, according as you finde the same, note those figures vpon the said respondent lines, and your conclusion will be to produce a figure like and proportionall to the field proposed. This chap­ter is most excellent for the purpose before said, and therefore worthy of note, as they shall finde it that worke by the Plaine Table, in countries that consist of great Champion fields.

CHAP. LX. What Chapter is most fit to vse in platting of ground, as well such whose superficies is subiect to sight, as others that be rough and full of wood, as also to make choyce of the best instrument to performe the same, as also to make a new kinde of particular,

YOu bee taught before to measure and plat a­ny péece of ground whatsoeuer, it resteth then for you to make choyce of such Propositions that are best, as well in respect of the fashion of the field, as in respect of the aptnesse of the Proposition.

Therefore for all grounds, whose bounds and angles may all bée séene from one place, vse the 51. Chapter, and if you cannot trauers the same to mea­sure it with your chaine, by reason of pooles, marshes, or such like, then is the 53. Chapter excellent: as for the 55. Chapter, I vtterly dislike thereof, because the section of some angles fall out so acute, that the conclusion cannot be without error, therefore such workes that be required by the helpe of two stations, are best to be wrought by the 12. or 14. Chapter.

Now for woodland, and rough grounds, or great Champion fielden grounds, vse the 57. or 58. Chapter, ioyntly by them­selues, or seuerally, according as the aptnesse of the ground occa­sions you: you may also well vse the 9. and 29. chapter to the like purpose: but haue a speciall care in all your dimensions and plat­ting▪ that you truly obserue the angles and corners of the fields, and so consequently precisely mete the hedges and limits, re­membring euer when you worke with the Plaine Table or Cir­cumferentor, to kéep your Instrument paralell.

Touching the best Instrument for this purpose. I leaue that to the discretion of the Reader: for my best councell is, that euery good practitioner make proofe of euery instrumēt, as I haue done, and afterwards to apply such to vse that hee findeth to worke with the producement of fewest errours: for I will nominate none, least it bee censured but my peculiar affectation. For euery [Page 125]one doth commend that instrument whereon he doth practize, or in which hee is séene, and therefore some affect the Plaine Table, some the Theodelitus, &c. which are all in some respects suffi­cient for ordinary measurers of land, so they be vsed according to Art.

Here shall follow another kinde of ingrossing of particulars, then that in the Staffe, together with the pleasant and apt pla­cing of the foure winds, and their collaterals in any plat taken, which being drawne in red lines, and artificially done will much beautifie the same; neither néed you to extend them all ouer the body of the plat, vnlesse you please: but in that your owne dis­cretion shall be your best guide.

CHAP. LXI. A description of the Circumferentor, and the parts thereof.

YOU must first prepare some péece of wood well seasoned,The Circumfe­rentor. and bearing a smooth graine, which let bée fire inches, or there about long, foure inches broad, and about an inch thicke: then must you stope downe the left side there­of, and diuide the same into equall parts, ac­cording to 12. in an inch, numbring the same by 5.10.15. &c.

The Needle and Carde.In this péece of wood must there bee a round hole cut or tur­ned, of thrée inches diameter, and thrée quarters of an inch déene, in the bottome whereof is placed a card, as in the 4. Chapter, at 3. This round hole hath a néedle therein, and a glasse to couer the same, as in the fourth Chapter of the Topographicall Glasse, at 10. Then vpon this square péece of wood is placed a Table, cal­led Tabula Sinuum, and this is such a Table that is calculated out of a quadrant, whose arch is diuided into 30. degrées, and the semidiameter thereof into 1000. equall parts and according vn­to the totall summe, are numbers placed in the said Table, an­swering vnto euery degrée, and halfe degrée. And these numbers serue to expresse the length of euery right signe, or of euery [Page 127]perpendicular let fall from the seuerall degrées, and halfe degrée in the limbe vpon the semidiameter.

Of the Sights.

Vpon the plaine of this square péece of wood, néere vnto either end thereof is placed two sights, and the one of them is but halfe the length of the other standing perpendicular.

Of the shorter Sight.

The shorter of the two sights beareth no diuisions at all,The shorter sight. in the top whereof is placed a pins head, and vpon the side is set a péece of small wire, end in the middest is hanged a plumbe. The distance from the wire in this sight to the plaine, is taken and di­uided into 60. equall parts, according vnto which diuisions is the right edge diuided, beginning from the perpendicular point vnder the wire, numbred by 10. as 5.10.15. &c.

The short sight, and the wire therein represent the semidia­meter of a quadrant, and the wire the center thereof.

Now from the perpendicular point be the degrées of a qua­drant, perfected vpon the vpper side of the right edge, numbred from 90. to 25. by 10.

Of the longer Sight.

This sight is twice so long as the other,The longer sight. whereupon are placed thrée kinde of diuisions

1 First, the distance from the pinnes head (in the shorter sight) vnto the plaine of the instrument, is taken, and according [Page 128]vnto that distance is a line stroke ouerthwarte this sight, which must be called the line of leuel, the distance from which vnto the pins head is taken, and diuided into 100 equall parts: the distance bewixt the pins head and the line of leuell is equall in length, therefore diuide the longer sight into 100 equall parts, according vnto which place them in the said longer fight, to 50 vpwards and downewards, numbring them by 10, as 5, 10, 15, &c.

Hypothenusall diuisions.The second diuision is the graduation of the Hypothenusall lines, according as they increase by vnits, they be numbred by vnits as 1, 2, 3, 4, &c. to 12, which represent 100, 102, 103, 104 &c. and these diuisions are set from the line of leuell vpwards and downewards, take the square of 100 from 102, &c. from 102 103, &c.

Degree of a Quadrant.The third sort of diuisions is the degrée of a quadrant, pro­iected vpon the said longer sight from the line of leuell vpwards and downewards to 25.

Of the slit in the sight.

Slit in the sight.In the midst of this sight there is a slit made from the vpper and of the sight straight downe vnto the lower end.

Vpon this sight there is a vane of brasse, made to runne e­qually vp and downe, and in the same there is a sight hole answe­ring to the slit, and the edge of the vane.

Of the Index.

The Index.Vnto this Instrument there also belongeth an Index, where­in must be a center hole, to put vpon the wire in the shorter sight, in the other end of this Index there is a sight placed, with a hole therein answering vnto the fiduciall edge of the Index, the edge of this rule is diuided into such equall parts, as the right edge of the Instrument.

Of the Staffe.

The Staffe.Lastly, vnto this Instrument there belongeth a Staffe 4 foote long, with a good stéele pike in the foote thereof: this Staffe ser­ueth to plant your Instrument vpon, for which purpose in the top thereof is placed a round pin of wood or brasse, and through the midst of the Instrument is bored a hole to fit the said pin: so when the Instrument is placed vpon the said pin, hée will moue round about, but the best staffe is that which is made with thrée staues ioyned together like to a 3 footed paire of com­passes.

CHAP. LXII. Of the Circumferentor, his appellation, and such things as are to be considered generally therein, and of the protractor.

The definition of the Circumferen­tor. WHat the intention of the first composer of this In­strument was in calling it a Circumferentor, I know not, but this I affirme, the name was not vnaptly giuen, for if we well consider hereof, it will be appa­rent that the working thereby giues or afoords the name it selfe, for when we worke in platting of fields &c. wée bée instructed to moue or beare the Instrument about, vntill hée point vnto the proposed angle, whereby you sée wée beare him about, vpon the top of the Staffe whereon he is planted, so that he is properly called a Circumferentor, of the Lattin word Circum, which sig­nifieth about or round about, and fero 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 an­other, but the plaine thereof must alwaies lye paralell to the Ho­rizon, which the plumet in the shorter fight will helpe you to do one way.

Then must you prouide a Protractor,The Protractor, see the Chap. 26. which is a halfe circle diuided vpon the vpside in the line into 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 pro­tractor is diuided into 60 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,See the 6 booke Chap. 53, of the Geod: Staffe. if the scale had 12 parts in the inch vpon the one side, and 11 on the other it would do you pleasure.

CHAP. LXIII. To take the Almicanter and Azimuth of the Sun, or Starre, by the old Circumferentor.

To take the al­titude and Azi­muth of the Sun or Starre.PLant your Instrument so that hee may lye paralell vnto the Horizon, then turne him a­bout, vntill through the sight hole, and slit in the longer sight, and by the pins hend in the shorter sight you see the Sunne or Starre bringing downe the vane, vntill through the hole therein, and by the foresaid pinnes head you see the said Sunne or Starre, then the degrée cut by the si [...]e of the [...]ane, sheweth the Almicanter or al­titude of the Sun, and the degrée in the Card cut by the South end of the néedle, sheweth the Azimuth or the distance of y^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 y^e 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 [...]/60 degrees, which shewed me that the Sunne was 1: degrees high, and that he wanreth 58 degrees of the Meridian, for the South point cut­ting 29 degrees, & 20 minutes, I multiply the same by 3 there commeth 58, which sheweth me there is so many degrees inclu­ded betwixt the Sun and the Meridian, and so of the rest.

CHAP. LXIIII. To finde in what point of the Horizon any thing seene lyeth, by the old Cir­ferentor.

To know in what point of the cō ­passe any thing lyeth. HEre it is requisite first to vnderstand that 120 degrées represent the South, and that the de­grées are numbred into the East, so that to find what point of the Horizon any thing ly­eth from you, do thus:

Let the Instrument be placed paralel vpon the staffe, then cease not to moue the same,With more ease see the 28 Chap­ter. vn­till through the hole in the vane, and by the pins head you sée the thing desired, note the degrée then cut by the South end of y^e néedle, with which resort vnto this Table, so haue you your de­mande.

Example

I find the South point cut 45 degrees, I conclude the thing seene is North-east.

CHAP. LXV. To finde the houre of the day by sight of the Sunne.

To know the houre of the day WOrke the instrument lying paralell vntill y^e shadow of the pins head point or fall iust in the slit in y^e lon­ger sight, & the intersection of the néedle, I meane y^e South part with the paralell of the month, or signe take whether you please, is y^e 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.

CHAP. LXVI. To find the houre of Sunne rising and setting at any time proposed, and the length of the day and night.

HEre you must note, that this Card is made but for one latitude, and therefore his worke in that behalfe cannot be generall, but it may serue without any notable error ouer the most part of England.

You shall obserue where the paralel of the moneth cutteth the Horizon, for the houre circle passing thereby, or the néerest there­vnto sheweth your demand; remembring to séeke the setting vp­on the West side, and the rising vpon the East side of the Card.

So shall you find the 11 day of May the Sunne to rise néere 4, and set néere 8: then if you would knowe the length of the day and night, you may repaire vnto the second Booke, Chap. 10. of the Geodeticall Staffe.

CHAP. LXVII. To find the amplitude of the rising of the Sunne and Starres.

To find the am­plitude of the Sunne or stars. IT is not vnknowne to any man, tho meanely traueld in Astronomie, that euery Horizon hath foure principall points, viz. East, West, North, and South; then you must vnderstand, that there is no starre, nor the sunne, that ri­seth iust East, or setteth iust West, vnlesse they be in the Equinoctiall, which happeneth vnto the sunne but twice in the whole yeare: but for starres, if they rise once East, or set West, so doe they alwaies, whereof there be but a few: the starre in the pinion of the left wing of the Virgin, the starre in Antinous left arme, &c. come néere thereun­to: but as the amplitude of a starre obserued one day, is certaine and all one in any other day for that latitude; so in the sunne doth it differ euery day, and is called Amplitudo ortus. This had,

Obserue the sunne or starre when they seeme (as it were) to touch the earth, as béeing at point of rising or setting, where­vnto turne the Instrument, vntill through the slit in the longer sight, and by the pinnes head you sée the sunne or starre, then note the degr. cut.

If you sought the setting, multiply the degr. cut by the West end in 3, which take from 90, so haue you your desire, so the degr. were vnder 30; but if the degr. cut be aboue 30, multiply the de­gr. cut by the East end in 3, then from the totall take 90, so haue you your desire, and the setting shall be North from the E­quinoctiall.

But if you séeke a rising, you must consider whether the de­grées cut by the East end be vnder 30, or aboue: if they be vn­der 30, multiply them by 3, so haue you your demand, and it is North: if they be aboue 30, sée what degrées the South end cuts, which multiply by 3, substract from 90, to haue you your desire, and the rising is South from the Equinoctiall. Or thus with more ease: hauing made your obseruation, sée how many de­grées are contained betwixt the West point of the Card, and the South end of the Néedle, for arising; but for a setting, sée how many degrées be included betwixt the East point of the Card, and the South end of the Néedle, which treble; so haue you your desire.

But this Chapter is performed with farre more ease & truth by my Topographicall Glasse.

CHAP. LXVIII. Of the opposite degrees, and how to find them.

Opposite de­grees. BY an opposite degrée is meant the opposite point of a Diameter, or the point opposite vnto the degr. cut by the South end of the Néedle, that is the degr. which the North end should fall vpon, which is alwaies the halfe of a circle distant from the South end in this instrument 60 degrées; so that if the degrées be lesse then 60, adde 60 thereunto; but if more then 60, substract 60 from it, and the total of the [...]ne, [Page 134]or the remainder of the other is your desire. This néedeth no ex­ample.

CHAP. LXIX. To find the quantitie of an Angle.

To find the quantitie of an angle. THe quantitie of an angle is the portion of a cir­cle included betwixt the 2 sides of any angle, which is found vpon this instrument, by the cutting of the Néedle at two obseruations in one place, the lesser of which must be taken from the greater and the degrées which remain after substraction is the quantitie thereof.

But if the remainder after substraction excéed 60, then must the said remainder be taken from 120, so haue you the quantitie: if your degrées be not direct, then must you worke by the oppo­site degrees, as in the 9 Chapter, taking the lesser of those de­grées from the greater.

And you must here note, that all degrées cut at diuers obserua­tions in one or more places, be called direct.

And such degrées as be opposite vnto direct degrées, be called indirect: and here note the tediousnesse of taking an angle by this instrument, inrespect of my Staffe.

CHAP. LXX. To take the distance of any marke by the old Circumferentor.

To take a di­stance. AS I haue often times said in the 2 Booke of the Geodeticall staffe, that there must be 3 things giuen, as 2 lines and one angle, or 2 angles and one line, by which all dimensions are per­formed; so in this kind of working you must alwaies haue two angles and one line giuen, by helpe of which you may séeke any distance proposed thus:

Plant your instrument at the place appointed, whence you desire the distance, and there looking towards the said marke, note the deg. cut by the South end of the Néedle; then appoint [Page 135]another place for your second station to which bring the sight, as before, noting the degr. cut: that done, measure the distance be­twixt the place you then stand at, and the place appointed for your second station, there againe plant your instrument, looking through the fights vnto the marke whose distance is required; then note the degr. cut, and so get the quantitie of both the an­gles as in the last Chapter.

When you haue gotten these two angles, adde them both to­gether, which take from 60; so haue you the quantity of the an­gle at your marke: then must you resort vnto the table of signes placed in the Instrument, and there [...]ind the signe of euery an­gle, and note it downe, and if the quantity of the angle excéed 30, substract the excesse or ouerplus from 30, and take the signe of the remainder.

These signes had and noted downe, worke by the golden rule, wherein the signe of the angle at the marke must be the first num­ber, the measure betwixt the two stations the second number, and the signe of the other angles seuerally the third number, ac­cording to the side which is sought, and this worke is grounded vpon this Chapter.

In all right-lined Triangles the proportion of the one side vn­to the other, is such as the signe representing the angles be. Or more briefe.

See the 7 Booke Axioma 2. of the Geo. St. The sides of opposite angles bee direct proportionall to their signes.

CHAP. LXXI. To performe the last Chap. by protracting with the old or new Circumferen­tor.

To take a di­stance. HAuing made your obseruations at each station, note downe the degr. cut by the South end of the needle, and then protract thus:

Take a faire sheet of paper, and fasten the same vpon a Plaine Table, or such like, with mouth glewe, then shall you make a point vpon your pa­per to represent your first station, there keepe the side of your in­strument, turning him vntill the needle cut the degree first no­ted, [Page 136]then draw a line from that point along the edge of the instru­ment, then kéeping the edge still at that point, moue the instru­ment vntill the South end of the néedle cut the degrées noted at your second obseruation: then draw another line by the edge of your instrument, whereupon lay the line measured betwixt both your stations counted, from the point first made towards the end of the said line, and where that number ends, there make a point, which let represent your second station, where place the edge of your instrument, turning him about vntill the south end of the néedle cut the degrées you noted at the second station: then by the fiduciall edge of the instrument draw a line, & note where it intersecteth with the first line, for that is the place of y^e marke whose distance is required, the distance of which from either of your stations may you measure by the Scale that you expressed the length of your stationary line by.

CHAP. LXXI. To take an altitude onely by the old Circumferentor.

To take an alti­tude. YOu must first get the horizontall distance vn­to the thing whose length is required: then plant your instrument perpendicular, and moue the vane vntill through the hole therein, you sée the top or the summitie of the altitude, note then the equall parts cut by the side of the vane, for such proportion as they beare vnto 100. the like doth the altitude vnto the distance: multiply therefore the distance by the parts cut, and diuide by 100. the quotient she weth y^e height which is correspondent to the leuell of your eye.

The ground of this worke is borrowed from the Iacobs Staffe, as may appeare in the ninth Chapter of the fifth booke of the Ge­odeticall Staffe.

An inconueni­ence like to that in the Theodeli­tus.But in taking of altitudes you shall haue it oftentimes so fall out, that the altitude will be so high, that you cannot bring the vane so low as to sée the top of the altitude by the hole and pins head.

When it so happens, you must place the center of the Index vpon the wire in the shorter sight, looking through the sight hole in the Index, vntill by the wire, and through the said sight you [Page 137]sée the summitie of the altitude: then note the equall parts cut by the fiduciall edge of the Index vpon the right edge of the instru­ment: for as those parts are in proportion to 60. the like propor­tion hath the distance vnto the height.

And so that proportion as those parts cut haue to the parts cut in the Index, the very proportion hath the distance to the visuall line.

Therefore multiply the horizontall distance by 60. and diuide by the parts cut on the right edge of the instrument, the quoti­ent will shew the height.

Againe, multiply the horizontall distance by the parts cut in the Index, and diuide the same by the parts cut in the edge of the instrument, y^e quotient sheweth the visual or hipothenusall line.

As you séeke altitudes, so must you sinde profundities, as I haue said often in the Geodeticall Staffe, but the errour is great if the instrument be not exact paralell.

CHAP. LXXIII. To take the plat of a peece of ground by the old or new Circumferentor.

DIuers wayes may bee set downe to fetch the plat of a péece of ground by this instrument:To measure woodland, or a­ny other ground. but I hold that most easie which is to be pro­tracted by the Instrument it selfe, because you shall not bee troubled to séeke the quantitie of angles, which in this Instrument is ouer te­dious.

Hauing therefore a péece of ground giuen, you shall begin at some one corner, and there plant your Instrument, looking vnto the next corner, and note what degrée the south end of your née­dle cuts, then with a chaine measure from the first corner to the second, and note downe the degrées cut by the south end of the néedle, and the length of the line measured.

Next go to the second angle, and there conuey your sight to the third angle paralell to the hedge: then measure the distance from the second corner vnto the third, noting downe y^e 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 [Page 138]the length of them, and so procéede from angle to angle, noting the degrées cut, and the length of euery line answering thereun­to, vntill you haue gone round about. And if you being at any one angle, and from thence can sée two or three angles more, you shall not néed to remooue your instrument to any of them, but onely from that angle obserue all the rest, onely measuring the hedges.

With these notes you shall resort vnto a faire shéete of paper, and there protract it downe thus:

In some place of the paper make a point, and there place the fiduciall edge of your Instrument, turning it about vntill the south end of the néedle cut like degrées, as he did at your first ob­seruation: then drawe a line by the fiduciall edge of the instru­ment, whereupon from the said point to wards the other end, lay downe the length of the first measured line, which you must take with your compasse from your scale, & where that number ends, in the said line, there make a point, where place the edge of your instrument, mouing him about vntill the South end of your néedle cut like degrées hee did at your second obseruation: then drawe a line by the fiduciall edge thereof, whereupon lay the length of your second line, and where that number ends, make a point, where (as before) place the edge of your instrument, mo­uing him vntill the South end of the néedle cut like parts hee did at the third obseruation: then drawe a line by the edge thereof, whereupon lay the third line, and where that number ends make a point as before, placing there the edge of your Instrument, turning him vntill the South end of the néedle cut like parts as at the fourth obseruation, and so procéed, laying downe the parts cut, and the length of the lines, vntill you haue gone round a­bout, by which meanes you shall lay downe the plat of the péece of ground in true forme, then for the casting vp thereof, resort vnto my booke of the art of measuring ground.

CHAP. LXXIIII. To take a plat at one station, from whence you you may see all the angles in the field, by the old or new Circumferentor.

To take a plat at one station. THis kinde of working is performed with as much ease as the former. You shall therefore repaire into the field, and finde some such place from whence you may behold all the corners in the said field, where plant your instrument, and then begin at some one angle, whereunto direct your sight, noting the degrées cut by the South end of the néedle: then direct your sight vnto the second corner vpon the right hand, and there againe note the degrées cut by the South end of the née­dle, which note downe, and so procéed rightwards from angle to angle noting the degrées cut by the South end,You are taught this chapter with a demonstration lib. 6. cap. 3. of the Geodeticall staffe. vntill you haue gone round about the field, of which degrées cut you shall make a little Table, to the end you may remember how many degrées were cut at the first, second, third, &c. corner.

Next shall you cause one to mete with a chaine the true di­stance of the first corner from your staffe, which note downe a­gainst y^e y^e first degrée cut in your Table: then mete y^e distance of the second corner from your instrument, which note downe in your Table against the number of degrées cut at the second cor­ner, and thus procéed vntill you haue gone round about the field, laying downe the distance of euery angle from your instrument against his proper degrée cut, which done, fall to protracting, thus:

Hauing prepared a faire shéet of paper, as you be taught be­fore, about the middest thereof make a point, which call your sta­tion: then apply the edge of your instrument thereunto, mo­uing him about vntill the South end of the néedle cut the de­grées you noted at the first corner, which done, draw a line by y^e edge of the instrument, from the point made in the paper out at length, then moue him rightwards vntill the South end of the néedle cut the degrées noted at the second corner, and then by the edge of the instrument, draw another line, as before, & so go forward vntill you haue finished all the degrées cut by the south end of your néedle, noted in your Table: then with your com­passe [Page 140]take from your scale y^e distance of the first angle from your instrument, which lay in the line first drawne from the point made in the paper towards the other end of the line: then take the distance of the second corner from your instrument, which apply to the second line drawne in the paper, and so procéed from line to line according as you be taught in the third chapter of the Art of measuring ground. The length of euery line laid downe in such order as is said, then must you draw lines from point to point in each line; so shall you drawe the limits and proportion of the ground, according as in the foresaid third chapter of the art of measuring ground by the Staffe.

And by this meanes may you measure ground at two stati­ons measuring but one line in the whole plat, in such order as I set downe in the fourth chapter of the sixth booke of the Geodeti­cal Staffe. And since, what is said before may giue sufficient light to performe both this way, and many other, I will omit further spéech, least I rather seeme tedious to the wise, then facile to the vnlearned.

And you shall heere note, that by taking perfect notes in the field, where one closse boundeth vpon another, you may take the plat of many flelds lying together, and so saue a great la­bour.

CHAP. LXXIIII. The degrees of a field being taken, to finde whether the plat will close, the lines being truly taken.

To know if your plat will close. NOte downe the quantity of euery angle at each seuerall station, as well as you doe the degrées cut, then adde vp all the quantities together, then multiply 60. by a number lesse by 2 then the number of the angles, and if your worke be right, the product thereof shall be equall to the totall of the quantities.

Example.

Let the number of angles be 8. frō which take 6. which is a lesse, then multiply 60. by 6. and the product will be 360. which agr [...] ­ing with the totall of all your quantities of angles added toge­ther, is one argument that the plat will close.

CHAP. LXXV. To reduce Hipothenusall lines vnto Horizontall, after another way, then in the 6 booke 8 Chapter of the Geodeticall Staffe, onely by the sights in the old Circumferentor.

To reduce Hypo­thenusall lines to Horizontal lines. PRepare a marke to bee carried before you the just height of your line of leuel from the ground when the iustrument is planted vpon his rest, this marke must be placed in the an­gle whereunto you looke, hee must stand per­pendicular, and when you take the degrée looke your instrument stand perpendicular, and then moue the vane vpon the sight, vntill you sée the top of the marke before planted through the hole, in the vane and by the pins head, then in the Hypothenusall diuisions cut by the vane vpon the sight, for they will shew you how much that line you shall measure, will differ vpon the 100: from that line you should measure, if the ground were leuel: therefore when you haue mea­sured that line, proportion him according to the parts cut.

Example.

Suppose the parts of the Hipothenusall diuisions cut, to be 4 and the line measured to be 30 pearches, now you are to finde a number to beare like proportion to 30, as 100 beareth to 104 which you shall find to be 28 1/5 1/1, so that the line measured by the cheine to be 30 pearches, must be laid downe 28 1/1 1/3 pearches in your protracting.

But for asmuch as these calculations be tedious in the field, your best way is to note the Hipothenusall parts cut, and then re­duce them when you come home.

CHAP. LXXVII. To performe the same by a Quadrant made of purpose.

A new Quadrant to proportion lines. YOu shal prepare a Quadrant, and then diuide the limbe thereof in 30 equall degrees, set­ting number therupon as the cōmon order is, then shall you diuide the lower side of the Quadrant y^e is betwixt the first degrée & the center, into 30 equall parts, raising perpen­dicular lines vpon each diuision, which will be paralell vnto the other side: this done prepare an Index of the length of the semidiameter of the Qua­drant with a center hole therein, this Index is to be fastened to y^e center of the Quadrant with a brasse pin or such like, which al­so must be diuided into 30 such equall parts, as the semidiameter was: the Quadrant thus prepared you shall fore shorten the lines thus:

First for the taking of your notes in the field, you must work as in the last Chapter, onely here you must note the degrée of a circle cut by the vane in stéed of the Hypothenusall diuisions, and then procéed thus:

Put the Index to the different angles in the limbe, then num­ber the line measured vpon the Index, and note the perpendicular there cut by the edge of the Index, for that shall shew you the length of the Horizontall line which must be protracted.

Example.

Let the different angles from the Horizon be taken 18 degr. and the line measured 20 perches, first count 18 degrees in the limbe, then thereunto bring the edge of the Index, next count the line measured viz. 20. pearches vpon the Index fromwards the center, so shall you there 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.

CHAP. LXXVIII. To seeke any altitude by this Quadrant.

To seeke an alti­tude. TAke the angle of altitude, whereunto bring the In­dex, the same being counted in the lymb, then num­ber the Horizontal distance in the semidiameter, & the portion of the perpendicular to the Index sheweth the heigth.

CHAP. LXXIX. To take the declination of any wall, by the old or new Circumferentor.

To get the decli­nation of any wall. BY the declination of any wall, is meant the bending or leaning of the surface from the Me­ridian.

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, y^e quantity whereof is thus had:

Set the North end of the Instrument, vnto the wall, now if y^e 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 remain­der multiply by 3.

4 If North West take 60, from the degrée cut and multiply by 3, so haue you your desire.

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 improoue­ment.

And if there be any other commodities in the said Mannor ac­crewing to the Lord thereof, they may be noted as followeth.

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 for­mer: and in the conclusion say, Ei 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 pro­céed to the Tenements, as before.

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 de­scribe another circle, which wil likewise be diuided into 30 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 [Page 147]center you describe a circle diuiding euery of their circumferences into 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 conclusi­ons, which at this time I mind not to repeat: prouided that you drawe the lines there of in some colour, as red, or such like, that they may be readily distinguished from the lines of the map or plat.

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.

CHAP. LXXX. The order how to discouer the true plat of any parke, forrest, or such like, standing vpon the top of some hill, not approaching vnto the same.

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 stan­ding a 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 e­uery of which angles, I note downe as before, thus:

Now, according to the doctrine of seeking dimensions at two stations with these two Tables of degrees collected, you should go and protract, euen as you be taught, Lib. 6. cap. 5. of the Geodecicail Staffe. But your see the angles at d and f bee so a­cute, that you shall hardly finde the true interfection, therefore hauing the quantity of euery angle at b and a, as before, your best way is to supputate the true distance of euery angle in the parke, from one of your stations, as from a, and so protract as hereafter: for hauing the two angles in any triangle, the third is easily found by adding the quantity of those 2. angles together, & taking the totall from 180. which is the quantity of the three angles by the first chapter, Deffinition 31. lib. 7. of the Geodetical Staffe. And hauing the three angles, with the one side of any [Page 150]

triangle, you may easily finde any of the other two fides remai­ning, by the said seuenth booke called Trigonometria, and that diuers wayes, but the easiest way for all oblike triangles is in the said booke, pag. 287. As for the purpose, after this easie man­ner I would finde the length of a e, first I adde the angle h a c 40½ degrees, & a b c 121. degrees together, so haue I 161 ½ degrees, which taken from 180, there remaineth 18½ degrees. These 3. angles, and the side a b knowne, I say, as a b the radius is to b h the signe of the angle a b h (which by the first chapter, Lib. 7. deffini­tion 26. of my Staffe is) 59. degrees. So is a b 65. pearches to a h, therefore multiply 85716. the signe of a b h, by 65. so haue you 5571540. which parted by 100000. the quotient is 55 71540/100000 pearches, the side a h, then say, as a h the totall signe is to a c the secant of the angle h a c, so is the side a h to a c, therefore I take the angle a c h from 90. viz. 18. ½ from 90. so is the remai­ner 71 ½, whose secant is 315154. therefore multiply 315154 by 55 ½ pearches, so haue you 17486047. which parted by the radius, the quotient is, 174 86042/100000 pearches, the l [...]ngth of the line a c. In like manner get the line a d, a e, a f, and a g, or get the length thereof, according to the second Axioma in Trigonometria, [Page 151]that is, by right signes thus:

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 13917▪ is to the signe of the angle a 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 pro­duct 5122065. which part by 13917. the signe of the angle d, 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.

And now, whereas you shall go vnto k, and there likewise protract all the angles obserued at the first station, and so note [Page 152]

the intersection of the lines, heere wee will auoyd that kinde of vncertaine working.

You shal therfore note the distance of a c in the first figure obtei­ned before, viz. 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 per­ches, i 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 fi­gure like and proportionall to the Parke proposed, as l 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.

CHAP. LXXXI. The plat of a Parke beeing taken, as in the last Chapter, how to cast vp the contents thereof, after two manner of waies.

THe first way is to resolue the plat into regular figures, such as it is most aptest for, and so find the bases, perpendiculars, sides, and diagonals, and thereby get the superficiall content, as in the second part of the 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.

As for the other kind of finding of the superficiall capacity of this or any such other kind of figure, you shall not néed the vse of the first booke, in performance whereof you must séeke the tue length of the sides of the said Parke, as g c which is [Page 154]

130 pearches, the which you be often taught before to performe, and hereafter shall be againe otherwise.

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 di­uisions, then hauing diuided the plat into such regular figures, as to you shall séeme best, for the atteining of the area, you shall measure the contents superficiall by those equall diuisions, I meane the contents of euery particular figure, adding them alto­gether and noting the totall, then must you take the number of pearches in the side of the Parke, formerly measured, and also the number of equal diuisions in the side of your plat, responding vnto the said side of the Parke measured, squaring both those two numbers. For as 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. There­fore increase the said superficiall content by the number of pear­ches squared, and diuide the totall by the square number of smal diuisions, so is the quotient the number of square pearches [Page 155]which diuided by 160 &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 [Page 156]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.

CHAP. LXXXII. To take the plat of any field or such like by the intersection of lines, with more truth then hath bene published heretofore by any.

I haue told you often, that I did not much affect working at two stations, and so to get the plat of any field by the intersection of lines, for that the angle made at the section often times prooues to be so acute, that you cannot precise­ly discouer where the true point of the section was made, wherefore I haue deuised a way by helpe of thrée stations, (and yet measuring but one line), truely for to finde the right point of intersection, which is thus per­formed.

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 y^e one side of the said angles, then go to the marke [Page 157]newly espied for a third station, and there obserue some such cor­ners that you thought would fal out acute by the section of lines issuing from the two first stations, and thereby get the angles, they make with your first or second station, for by helpe thereof shall you correct the acute section of the former lines, as you may best perceiue by the example.

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 [Page 158]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.

CHAP. LXXXIII. To seeke the distance or length of any Tur­ret, tree, towne, or such like, from you, without the helpe of Instru­ments.

THis conclusion, or the like vnto it, is performed by 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 Geodeti­call Staffe will soone performe; or for want of them, any ordina­ry Carpenters square, or such like, may as well serue.

Go therefore vnto some such place, whence you may sée the proposed Tower, where set vp some Staffe, whence depart a certaine distance orthogonally or Squire-wise, at the end of which distance set vp a second staffe: now repaire vnto the first staffe, whence go backe or forward at pleasure, and there set vp a third staffe in a right line (by the iudgement of your eye) with your first staffe and the desired distance: now againe from this third staffe depart as before, sidewise, in a right angle and right line, vntill the second staffe and your eye agrée also in aright line with the desired longitude, where set vp a fourth staffe: [Page 159]

these 4 staues so placed, get the distance betwixt the first staffe & the second, which call the first number, the like doe betwixt the [Page 160]first and third staffe, which call the second number, and the di­stance betwixt the third and fourth staffe call the third number; then must you take the first number from the third, and that which remaineth call your Diuisor: lastly, increase the third number by the second, making partition of the product by your Diuisor; so doth the quotient acquaint you with the true longi­tude required.

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 frō d, and in a right line with a c, which is measured 200 yards, so that you must depart squire-wise from d towards e, vntill you be in a right line with c a, as is said: these staues so set, and their distan­ces measured, deduct b c 163 the first number, from de 200, the third number, so haue you 37 your Diuisor, then multiply the third number 200 by 158 the second number, and there is pro­duced 31600, which parted by your diuidend 37 leaueth 854 2/37 yards, the distance betwixt d and a, whence take d b 158, so haue you the distance of a b 696 2/37 yards, making [...]96 yards and a lit­tle better then 2 inches.

CHAP. LXXXIIII. How to finde the length of any Hypothenusall line, or the length of any scaling ladder.

THis Chapter is so easie, and so well illustrated by Euclide in his first Booke, Chap. 47 of his Ele­ments, that it néedeth no Example; for there the square of the two containing sides are prooued e­quall vnto the square of the Hypothenusall, in so much that the two including sides béeing added together, the square roote thereof produceth the Hypothenusall, as is proued by the prealeaged Chap of Euclide, 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 [Page 161]toppe of any tower, castle, or such like, from your foot to the end, you may find what length a Scaling ladder to reach the same should be; you most first get the distance of the turret from you, and also y^e altitude, both which distances square, adding the pro­ducts together, the square root whereof is the length of the Sca­ling ladder.

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

[Page 162]length of the scaling ladder i h, or you may worke as in my se­uenth booke of the Geodetical Staffe, called Altimetria, probleme 5. or Trygonometria, Axioma 1.

CHAP. LXXXV. To finde the distance betwixt any two Towers, Castles, or such like, though you can approach to neither, and that without the helpe of Instrument.

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 y^e 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 [Page 163]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 loo­king thence by the other containing side, the visuall line wil in­forme you what direct course to take towards the distance vn­till you come in a right line betwixt a 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. pear­ches. Finally, according vnto the prescript, multiply the c e 300. by f g 170. so haue you 51000. which parted by d e, 100, leaueth 510. pearch, the distance a b.

Here note for your more ease, it were best (if so conueniently you may) to make the distance d e 100. (as here it is) or 1000. pa­ces, or pearches, &c. and no other number, so may you auoid di­uision, [Page 164]

as you may perceiue in the 7. booke of my Staffe, called Trigonometria, in the end of the 7. Probleme. And this Chapter may be wrought by the Geodeticall Staffe most exact without any triangle, for that he gets you the angles a c b, and b c e at your sta­tion c, and deliuers them againe at d, as f d g, equall to a c b, and g d e equall to b g e.

CHAP. LXXXVI. To seeke the distance of any thing from you, howsoeuer situate, and that after a new way, newly deuised without the helpe of Instrument.

I Doe not remember any Proposition perfor­med without Instrument more easie, spéedy, and true to séeke the distance of any thing frō you then this is, the 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 re­maining, looke by the other containing side, causing one to set vp a second and third staffe in a right line with the said containing sides, where sticking a staffe, carry your triangle vnto the staffe néerest your first station, where you set your first staffe, and there situate the said triangle in all respects as at the said first station hee was, so will the one containing side lye in a right line with your first staffe, and also with your second and third staffe: then placing your eye at the contained angle, the other containing side will direct you what course to hold vntill you come in a right line directly betwixt the desired distance and your third staffe, and there set vp your fourth staffe. Now must you measure the di­stance betwixt your first and third staffe, then betwixt your se­cond and third staffe, which reserue for a diuisor, then betwixt your second and fourth staffe. Finally, multiply the distance of the first and third staffe, in the distance of the second and fourth staffe, which parted by the distance betwixt the second and third staffe, the quotient is your desire.

It would bee something long for mee to stand to demonstrate [Page 166]the same Geometrically, for that I want time, but because the worke is new, I will acquaint you with the proposition whence it was gathered.

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 néed­lesse to inferre a demonstration.

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.

Suppose e were a ship lying at rode in the sea, and I standing vp­on a high rocke at a, were required to deliuer the distance there­of, f is my triangle, which being situate at a the one containing side directing vnto e, and looking by the other containing side in the right visual line, I cause two staues to be set vp, as a second and third staffe at c and b, then leauing my first staffe at a, I take the triangle, situating him at c my second staffe, in all respects as

at a, so will the one containing side lye in a direct line with a b my first and third staffe: then looking by the edge of the other con­taining side, the visuall line directs me what course to hold to­wards the shippe vntill such time that I come iust betwixt e (the [Page 167]wall of the hill next the sea where I formerly tooke my aime) & b my third staffe: this done, I measure the distance betwixt my first and third staffe, finding 47. yards, then betwixt c and b, 10. yards, which is my diuisor. Next I measure the distance betwixt c and d my second and third staffe, which is 36 yards. Now multi­plying 47. by 36. I haue 1592. which parted by 10. leaueth 152 2/10 yards, the distance of the said ship.

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.

CHAP. LXXXVII. A Nauy, or one Ship seene vpon the seas, to know if they make towards you or not.

BY the last chapter or some other get the true distance of the Ship from you, now rest for a certaine space, and then obserue diligently the distance thereof from you againe, now if the first distance and this agrée the Ship standeth still, if the first obserued distance be greater, the Ship maketh towards you, but if it be the lesser, he departeth from you.

CHAP. LXXXVIII. Two Ships seene one in pursuit of the other, to know whether the Ship pursued, loose way, and how long it will be before he be ouertaken.

BY the 84 chapter get the distance betwixt the two Ships, then stay halfe an houre or a quarter, obseruing then againe the true di­stance, if the two distances agree the pursued Ship looseth nothing, but if the first distance excéed take the lesser distance out of the grea­ter, multiplying the space of time betwixt [Page 168]your obseruations in the distance betwixt the two ships, the pro­duct whereof diuide by the difference of the distances, so is the quotient your desire.

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.

CHAP. LXXXIX. How to take the platforme of any house, Castle or such like.

YOu are sufficiently instructed both in this booke, as also in the Geodeticall Staffe to séeke the true perimeter of any figure propo­sed, where you be also taught to séeke the breadth, height and distance of any obiect howsoeuer situate, insomuch that nothing re­maines (those rules well vnderstood) but to séeke the true ground plat of the buildings, for the which you haue diuers peculiar Chapters, which had you may find the distances of Fronts, Turrets, gabel ends, returnes, or such like, the breadth of windowes, quadrants, and such like the heights of Iutteis, Storreys, & Ascents, lengths in heigths with such like: and thus may you procéed, taking as well the ground plat as other erearements with their proportionall di­stance, noting the same to your selfe in some booke, whereby you may stand in any place far off and take the plat of any house, ca­stle, forte, or citty, the situation whereof (to your great praise) you [Page 169]may discouer, or if you please cause the like to be made.

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.

CHAP. XC. The order to discouer how mines or trenches run vnder the earth being most fit for pyoners, maisters of Coale, Iron, Stone mynes, &c.

THis Chapter is performed in best and easiest sort with any such Instrument that hath alarge Néedle, wherefore the 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 cōmon vn­der his ground wherby the coales were his. To resolue this doubt descend into the pit, and then by the 72 chapter, get the true way that the myners haue made euen as it were a hedge, still noting [Page 170]the degrée cut by the Néedle, at euery angle where the myne runneth one way or another, out of the course of a right line, and also measuring the side of euery angle then ascending out of the pit, by your Instrument and cheine (beginning perpendicu­lar aboue the place where you began to make obseruations in the bottome of the pit) lay downe the like angles and sides so obser­ued: which hauing so done, you shall soone sée if the myne or any part thereof haue run out of the one Manour into the other, for if it do, you shall be forced to measure out of y^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.

CHAP. XCI. To plant barrels of powder, direct vnder Castles, Forts or such like, and to know how farre you be vnder the same.

IN performing this Chapter by some Proposi­tion formerly published, you must get the Ho­rizontall and Hypothenusall distance of the Forte from you, and thereby the height thereof aboue the Horizontal line, which done, you are also by helpe of your Néedle placed in y^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 Instru­ment euer direct the myne, direct vpon that line or part of the world, and kéeping your Instrument paralel, the sight vpon the diameter of the demicircle, thereby alwaies cary the floore of your mine leuell with a candle fixed vpon the end of a staffe of equall height with your eye (as before) will helpe you to do: Now when you haue gone so farre vnder the ground as you found the length of the Horizontall line to conteine, you may assure your selfe that you be direct vnder the Forte, and that you are so many pases vnder or below the said Forte, as you found the Forte to be about the Horizontall line.

CHAP. XCII A Mine running vpon some certaine point, yet ascending or descending, to know at any time how much you are aboue or vnder the Hori­zotall line.

COncerning your iourney vnder the earth, you must obserue the doctrine of the last Chapter, and when the mine happeneth to fall or rise, according to the doctrine of Altitudes and pro­fundities, duely note at euery seuerall station the quantity of the ascent and descent, that is, how much you rise aboue or fal vnder the true Horizontall line, and so kéepe two seuerall tables, the one of the ascents and the other of the descents. Now when you desire to know how you are situate, adde all the ascents together, and note the product: do so to the descents, then must you take the lesser out of the greater, so doth the remainder acquaint you how you then differ from the Horizontall line, for if the ascents excéed, you may be assured that you be aboue the Horizontall line, if the de­scents excéed, you be vnder the said Horizontal line, according to y^e difference of y^e said ascents & descents, neither néed you feare any collateral declining of y^e way of your myne, for that nothing at al altereth the ascent or descent, for that is onely altered by the di­recting line, or line that you measure, insomuch that if you well obserue the premisses, you may precisely know at any time or place how much you are vnder or aboue the true Horizontal line, and thereby come into him againe vpon any occasion.

CHAP. XCIII. A Myne or trench collaterally declyning, how to know when you come againe into the right line of position, and also how farre you be from being iust vnder any Forte proposed.

TO cary a myne direct forward vpon any point of the Horizon, you bée sufficiently taught in the 90 Chapter, and if the Mine must ascend or descend a­boue or below the line of leuel, you be taught in the [Page 172]last Chapter at all times to know how much you be aboue or vn­der the said line of leuell. But say you were inforced by rockes, waters, or other such obstacles that you méete with vnder the earth, contrary to the 90 Chapter, to cary your Mine side wise from the direct line of position, in such a case you are first vpon a faire large shéete of paper to extend a right liue ouer the same, which call the line of position, béeing the direct way y^t the Myne should go, next note the angle of deuiation from that line, that is to say how many degrées the Mine doth decline from the true line of position that leadeth on directly, and accordingly plat it downe vpon the paper, as you be often instructed in the vse of ech seuerall Instrument, procéeding so far as your Myne continues in a right line, and if you be occasioned againe to direct either fur­ther of, or néerer vnto the line of position, alwaies protract if downe vpon your paper exactly; as well in measure as angle, vn­till such time that you can come to make your protracted collate­rall lines, or lines of deuiation to intersect with the right line of position, first extended ouer the paper, and then by the scale with which you protracted your lines of deuiation, examine how many pases or yards that point of intersection is distant frō the point where your worke began, which representeth the point of your first entery into the Myne, for that compared with the fundamentall distance or length of the Horizontall line informes you if you be past, or not yet come vnder the proposed Forte.

Therefore in these cases you were best first of all to limit vp­on your paper with your scale and compasse the direct length of y^e fundamentall or Horizontal line, and so in your protracting may you call them backe, if they séeme to run beyond the Forte.

Then in the former Chapters be you taught to know how far vnder the Forte you be, whereby you may ascend néerer or descend further from the superficies of the earth as the cause shal require. Certainely most exact and excellent is this kind of working, for conueying of mines, and of no smal importance, for the due placing of Fornaces of powder, to blow vp Forts, Ca­stels, Townes or such like, whether they bee situate high hpon an hill, or low in a valey, which for all purposes in these and such like cases vnder ground, you shall find the Topographicall Glasse to be most requisite.

CHAP. XCIIII. Of the building of a Citty, and of the situation thereof.

IN our discourse of 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 con­terminating soyle batfull and fertill, as may be seene by London, which of it selfe, according to the nature of the soyle, stands but in a dry and barren place, though it be forced ranke by the abun­dance of compost. Sandy ground is right necessary for y^e plan­tation of a citty, and for the plat ground hereof, let it not bee al­together very leuell and plaine, but haue pleasant ascents, and rising bankes, which will cause the citty to bee more pleasant to the eye, healthfull to the body, and fitter for warlike defence, as it may be séene by old Rome. (which now lyeth ruinate) there were seuen such hilles, in the head of the citty stood mount Satur­nall, 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 there­on were many sports acted, where also was (and would be in o­ther citties) fit places to erect pyramides, or other such citty or­naments.

Touching the stréets there should be foure maine stréetes ly­ing into the 4. 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 [Page 174]be spacious and broad, so shall the minde blowing from any quar­ter come in and passe through the whole body of the said citty, and thereby purge the same of all corrupt and ill vapours, & such like, that commonly occupy citties. For the citties be [...]y vn­holsome, and apt to breede infections, where the stréetes bee close, shutting out the open and pure aire, which sure is an imperfectiō much to be lamented in London, & would héedfully bee regarded in y^e new plantation in Ireland. And as you haue diuided the cit­ty [...]nto 4. 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 multi­plied, whereby there be a confused number of houses ranged and thrust together without forme or regular fashion, and now not to be reformed vnlesse it were all built anew,

CHAP. XCV. Of the situation and building of a Mannor house in the country.

HE that will build himselfe a house in the coun­trey, should haue a speciall regard that it t [...]e pleasant, delightfull, & necessary in all respects, because hee commonly spendeth a third part of his life therein: yet this 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 be­twixt the house and the ground, or the ground and the house, that a wise man building, and a wise man also viewing the edifices, might iudge of the quantity of his land, or viewing the land, might coniecture of the proportion of the house: for a faire house without land (such cittie follies that are often built out of Lon­don) are neither commendable nor necessary, and therefore they haue begot themselues a nick-name, or by-name, as Mocke-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. [Page 175]Scaeuola, for Scaeuola had faire lands, without a competent house and Lucullus had a competent house without lands, in which re­gard he was checked by y^e 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 Cham­pion countrey, for such grounds be dry and holsome, if the aire be good: for men thereby are made of a liuely spirit. Pliny 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 plea­sant and necessary, to haue a cleare riuer fluant and running at a reasonable distance from your house: for besides the pleasure, you shall finde it necessary for vaults, and such like, that carry filth from your house to empt themselues into. In any case situ­ate your house distant from marshes, fennes, plashy and foggy grounds, which are vtter enemies to health. And in the politicke situation of an house diuers wise and honest men haue much la­boured to bee sarre from wrangling and turbulent neighbours, which they hold as great an inconuenience as want of holsome aire, and for that part of the heauen that the face and open side of your house should behold, you must haue regard vnto the na­ture of the country, and quality of the winde issuing from that part of the heauens. Pliny would haue you sittuate your house in a hoat country, into y^e 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 a­uoyding of the extremity of heate of the mid-day, and afternoons Sunne, which indéed is troublesome and vncomfortable, accor­ding vnto Aristotle 2. Meteor, cap. 6. and also according vnto Magirus, l. 6. c. 9 d. 13. The east wind with his collaterals is mo­derately warme and dry, and the holsomest of all, much exhilara­ting the mind, and making the body apt for any action, whereas the South wind doth diminish the strength of the body & minde, filling the head with rheumes, cathars and such like, destroying the stomacke, and by the frequent blowing thereof it doth not onely putrifie the bodies of liuing creatures, but also it corrup­teth and putrifieth the fruits, whereby also quotidian feuers, pestilences, and other contagious sicknesse rise.

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 of­fice in as necessary forme as to you shall séeme most conuenient, appointing places for great staires, priuate staires, houses of of­fice, chimnies, &c. that shall be most requisite for vse, and least an­noying, or defacing the house, or any of y^e principall lights cham­bers, or roomes: then according to your ground plat, drawe the forefront, or face side, backe-side, ends, and gabell ends▪ with all returnes, iutteyes, soyle péeces, windowes, &c. euen as you de­termine to haue it made, but draw it not as commonly these Painters do proportions of houses by the eye, but lay it downe by your Scale and compasse, that by the application thereof at a­ny time you may know how many foote or inches any returne, any gable end, any story, or window is in length or bredth, which you shall be taught to do haply in some other place: thus vpon seuerall papers set out euery seuerall part of your house, where­by your selfe, or the architector may informe the mechanical Car­penter of the length of euery seuerall péece of timber, & all things else required about the house, as the number of boords for floo­ring and dooring, the quantity of glasse and tile, with the quan­tity of séeling, rough casting, pauing, & other such like; whereby you may giue order to the Glasier for the bredth and length of your glasse, to the Tiler for tile, to the Plaisterer for lime, to the Sawyer for boords, procéeding, no one thing staying for the fini­shing of another, thereby proportion your house according to your purse.

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 de­uise; adioyning vnto which, let there be a fine orchard planted with trées: but if your clymate be hote, as Spaine, &c. plant your garden in the North; but for England the South is best, vnlesse for some trées that naturally desire the shade: let there be no oxe­stall, dormant and filthy water, stable, or other thing that may [Page 177]bréed noysome smelles néere vnto your ga [...]den, make the alleys drie, for which I could teach you diuers deuises which here is no place for, and plant the trées in your orchard after a Chequer forme, that standing at any trée, all the rest be in right line with you, which forme is called a Quinqunx: 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 suffi­cient light, albeit the chamber you lodge in would not be ouer light, not yet a ground chamber, inclining rather to cold then heat; for by meanes of heat in sléepe we may procure a swoune, because the heate of the body beeing become internall, and cold externall, this inclosing heat and that cold will striue: let the place therefore be temperate, and frée from noyse, for sleepe is a a cessation of the common senses, which béeing occupied & trou­bled with noyse hindereth sléepe: moreouer kéepe the beames of the moone from your bed, for it is hurtfull to the sight to haue the moone shine vpon your eyes sléeping.

Touching the plats and formes of houses, some affect the qua­drant building, with a square court inclosed in the middest, like to the Colledges, or as the Royall Exchange, which indéed in respect of the columnes and arches making the vnder walkes, is more stately: againe, some affect the Romane H. 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.

CHAP. XCVI. Of the sinking of a Well, and of the con­ueying of water in pipes.

IF you desire to find a place where digging a pit you may also find water fit to maintaine a well or pumpe, you must (ss 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 [Page 178]found, or if such vapo [...]rs rise in a dry and faire season, also if you dig trenches foure or fiue foote déepe throwing therein wooll, that is cleane and dry, couering the same with leaues, hearbes, or such like, if then this wool hauing lyen for a certaine space still remaine dry, there is no water thereabouts to be found, but if it be little wet, or greatly wet, there is little or great store of wa­ter to be found, according as the wool was in wetnesse.

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 plen­tifully grow, or as L. saith where any other gréene hearbs natu­rally flourish and abound without setting. Your springs thus found, they of longest continuance be which are in a gray or red grauelly rocke, or ground, in a blackish, sandy, clayie or red sto­ny ground, especially being mixed with stones and grauell.

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, thē with a bigger: any boughes or knotty péeces wil serue, so they bée large, & when the poles so bored, haue not ground to lye straight vpon, but lye vneuen rising and falling, there be crooked péeces of wood like elbowes prouided of purpose, which are also bored through, being let a foote at either end into the other two poles it ioyneth together, & so are all the poles that be ioyned one to ano­ther made to go into the end of one another a foote or more, in manner as they péece bag-pipes, or such like, the hole in the end of the one pole receiuing the hollow end of the other pole into the same, being alwaies for a foote déepe wider then the rest of the bore, which you must ioyne together with good cement, as you be taught before to do.

CHAP. XCVII. A briefe discourse how to draw the platforme of any kind of building, or any other thing seene, though you cannot approach vnto the same, and that ac­cording to true proportion, according as it appeares or offers it selfe to the sight.

IF you desire to proiect the due forme of any ob­iect vpon a plaine superficies according as it shal offer it selfe to y^e eye at any appointed place & distance, as to describe any town, or citty, any house, any floure, or any other body whatsoe­uer, you must do thus.

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 y^e perpendicular be made to moue equally fromwards or towards the Glasse, or to stand fixed at any diuision vpon the ru­ler as occasion shall be offered: this so ordered, when you desire the plat of any obiect as house or such like, plant the glasse oppo­site to the proportion required, the ruler lying paralel, then moue the shorter perpendicular néere to or far from the Glasse, euen as you desire the proiectment to be great or lesse: this done place your eye in the smal sight hole noting well through the same how euery particular obiect doth appeare vpon the Glasse, your eye so resting, with your pencel or dyamond, draw vpon y^e 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 ra­ther to that part of the Glasse that is of the same height from your ruler as the sight hole is where you place your eye, and this shall be your third number, which number is found by applying [Page 180]he length of that line to the equall parts vpon your ruler, these 3 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 ob­iect was expressed in féete or inches, of which make a scale and measure all the rest.

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 appre­hend, from any place where you plant your Glasse.

CHAP. XCVIII. The making of a most excellent Ruler whereby you may speedily reduce any plat proportionally, from a lesser to a greater, or from a grea­ter to a lesser forme, newly deuised.

YOu must first prepare a ruler of brasse or mettall of such a length that may answer the Semidiameter of your plat (but make him long and he will be generall) vpon this ruler there is a socket of brasse 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 se­cond pencell and that at any diuision required, hee must haue a screw pin to kéepe him steddyat that place he shall be appointed to stand, this done, your ruler is ready to worke as thus:

Fasten your plat that you intend shall bee reduced, vpon [Page 181]some plaine table boord, and about the midst of the said plat driue in a pin equall to fill the hole in your ruler, whereupon place the said hole, but let not the pin come aboue the ruler, next put your long socket vpon this ruler, and let the end where the equall di­uisions do begin stand at the hole in the ruler, the which resting moue the ruler and socket to any one of the next angles in the plat, noting how many of the equall parts cut the said angle, which let be 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 fa­sten him vnmoueable with the screw pin, which done and the pencels being both of one length, do no more but draw the pen­cel in y^e end of y^e said-long socket round about y^e 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 propor­tion you must neuer alter them vntill you haue finished. Out of doubt this is a most exact and excellent kind of ruler, and most easy for euery simple man to worke withall, and if the hollow­nesse in the socket be made like the channell in one of the legges of the Geodeticall 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 y^e 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.

CHAP. XCIX. To burne any thing a farre of with the Sunne beames.

BY such a conclusion as this wee read that 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 pur­pose, and well polished: and then the Sunne shining, place them in such sort that they may al reflect, or cast back the beames of the Sunne vpon the combustible matter, or subiect that is to be fiered: and the néerer together that y^e 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 Eng­land, vnlesse it be in Sommer in the extreamest of heat.

CHAP. C. To make a Glasse whereby to discerne any small thing, as to reade a written letter a quarter or halfe a mile off.

WE haue an imitation of such glasses as these about London commonly to bee sold, but they be so small that they stand one in small stéede, but amongst the writers of perspectiue, I haue read that if you take a glasse of the same mettall that burning glasses be, and 16. or 17. inches broad, whose center place directly against y^e obiect you looke vpon, and let it not incline, or hang sidewise by a­ny meanes, behinde this glasse set a faire looking glasse, the poli­shed side beholding the said burning glasse, to y^e 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 cer­taine [Page 183]distance, and then the obiect séene through the glasse doth séeme lesser and lesser, therefore care must bee had in placing the glasses, so may you view a Towne or Castle, or any window in the same, 6. 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.

CHAP. CI. How you shall buy annueties, or summes of money to pay at a day yet to come.

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, rec­koning the summe of money that you giue according to com­pound interest, that is, interest vpon interest, as if one should owe you 100. pound, due to be paid 7. yeares hence, and you would know what you were worthy to giue in hand for the same, according us is said, so shall you finde 51. pound, 6. shil­lings, 13. pence. in the table vnder 7. yeares.

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 [Page 185]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.

The end of the Topographicall Glasse.

CHAP. I. Of the falling of Timber, and the best time for the same, with the seasoning of boards, and the preseruing thereof.

TO tell you what wood is best for timber, in briefe, the oake is principall of all trées grow­ing, both for magnitude and duration, although in some countreys, by reason of the scarcitie thereof, they be forced to vse other wood. Con­cerning the falling of timber, you are best be­fore the retyring of the sappe to cut the timber trée round about, vntill you come néere vnto the end of the sappe, and so leaue the trée growing vntill you sée that the sap hath all runne downe, and ceased dropping, whereby you haue purified the trée, and left no moysture within to corrupt or putrifie the same: and that your timber may bee the more sound, voyd of wormes, and without rifts and chaunes, let the same be fallen a­ny time after Midsommer vntill the end of Ianuarie, especially in the full moone, but not in wet and windie daies, leauing such for no vse but snell that were wind falles as well for that the [Page 187]timber is not permanent, as also for that Masters in Astrologie say houses built therewith be more subiect to danger in time of tempest.

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 princi­pall thing I would haue you to obserue when you fall your trées, that is to crop of all the great armes and master boughes of the tree before you fall the same, because most commonly they be the occasion of the spoyling of much timber in the tree, and many times of most or all of the body of the tree, because the great top of the trée falling with such a waight, and pulling the rest downe with such a violence, much of the tree is shattered and rent, as many may sée daily to their losse.

CHAP. II. Of measuring of all kind of Solide timber.

MEasuring of solide timber is performed after two manner of waies, the first takes respect vnto all kind of timber growing or fallen of what forme or fashion soeuer it be, and so tel­leth you the true square of it, and consequently how many inches or feet are required to make a square foote of timber; the other taketh no notice of the quantitie of the square that the trée or timber will beare; but be it round, square, triangulate, or multiangulate, it telleth you how many solide féete or inches is in the same: both which waies shall you be taught to performe hereafter. And for finding of the true square of any péece of timber, M. 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 [Page 188]right fit for all masters of workes, carpenters, and masons to knowe; for the finding of the square of a tree, is not to gird the same about, and take the fourth part thereof, as they thinke, therefore the 35 Chapter of my sixt Booke erres in that point, béeing set downe hastely according vnto vulgar tradition.

CHAP. III. To find the true square of any growing tree by the Geodeticall Staffe.

YOu shall in performance hereof take some pack­threed gut string or such like, and with the same gird the trée about 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 [...]o 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 squa­red: first therefore I gird the tree about with a gut-string as farre from the ground as I can reach, as at r, then doe I note the length of the line that girded the tree, as c d, which I diuide into 4 equall parts, reseruing one of the said 4 parts, as c e, then doe I take the legges of my staffe, and fit the length of c e ouer in 110 and 1 [...]0 parts amongst the equall diuisions: the legges resting there, I take the distance ouer from 70 to 70; the length of which line I keepe, as l m, and turning the other side of the legges vpward I place l m ouer in 60 and 60, and so taking the distance ouer from 90 to 90 (the legges beeing not stirred) you haue the line n o the one side of the square that the tree will beare, which applying to your ruler diuided into feete and inches, you shall find 2 [...]/5 foot the iust square, which is equall to f h i g the 4 side of the tree bee­ing squared.

By this means may you find the true square and consequent­ly the quantity of any growing trée, & so buy the same according vnto your desire, by which y^e Carpēters may sée y^e error they run into by taking a fourth part of the cōpasse of the trée for y^e square.

CHAP. IIII. Any tree or round peece of timber vnsquared and match tapering, to find the square thereof as it lieth vpon the ground.

YOu are not ignorant but you may gird the same about, and so worke as in the last Chapter; but this may bee performed otherwise: let the Semidiameter of both the ends of the tree be a b and d e, both which ioyne toge­ther in one right line, as f g, so is f h equall to a b, and h g to d e, [Page 190]then diuide f g into two equall parts at

i, then the labour is no more but to fit f i or i g ouer in 60 and 60 amongst the coard diuisions, and so keeping the legs at that angle, take the distance ouer from 90 to 90, noting that line downe, as l m; so may you conclude that l m is iust the true square of the said tree.

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 stan­ding or fallen, onely by taking the true diameter or thicknesse of the said tree, and placing halfe the same ouer in 60 and 60, and so worke as before in this Chapter.

CHAP. V. To find the true square of a squared peece of timber consisting of two vnequall sides, and 4 right angles the one side being onely knowne.

YOu must take the length of the broader of the two sides, the which fit ouer in 60 and 60 amongst the coard diuisions; the legges of the Staffe so resting, the distance taken from 36 to 36 yeelds the true square of a peece of timber that béeing of equall longitude is also of equall quantitie.

But if both the sides c d and d b be knowne, then worke by the next Chapter, for this takes no notice of the thicknesse.

CHAP. VI. To find the square of any broad or flat peece of timber that consists of 4 right angles and two equall sides.

SVch a péece of timber as this, the end thereof doth represent the iust forme of an Oblong, and is thus squared; take the longer and shor­ter side, and ioyne them together in one right line, the which right line made of the length of both these lines so ioyned, make the diameter of a circle: lastly, vpon the point where the two lines were ioyned, raise a perpendicular, forthe length of that perpendicular to the circumference is the true side of the square.

Example.

A b c d is the end of a peece of timber, c d the longer side, d b the shorter, therefore I take the length of c d and d b and ioynt them together in v, making one right line thereof, as r s: next I part r s [Page 192]into two equall parts at w, then placing the one foote of my compasse in w, extending the other to s or r t, describe the semi­circle r t s: lastly, vpon v where the two lines were ioyned toge­ther raise a perpendicular v t, which is equal vnto p q, I conclude the square made of the line t v is equall to c d b a, and thus of any such other.

Or get the square thus, multiply the breadth in the thicke­nesse, so is the square roote of the product the true square, which you may easily find in the Geodeticall Staffe, fol. 142.

CHAP. VII. To find the true square of any peece of tim­ber, whose ends are formed like a Diamond.

THe end of such a peece of timber as this, doth represent the iust forme of a Rombus, & there­fore doth consist of equall sides and Oblique angles, the square whereof find thus; Drawe a right line betwixt any of the two opposite an­gles, noting the length of that line, vpon which line let fall a plumb line from one of the subten­ded angles; so hauing those two lines, find the square, as in the last Chapter.

Example.

The end of the peece of timber

is a b c d, the line drawne from one opposite angle vnto the other is a c, the line falling perpendicular vp­on a c from the angle b, subtended by a c, is b c: therefore according vnto the last Chapter, I ioyne a c, and b e together in one right line, and then describing a demicircle vpon that diameter, and noting the length of the perpendicular, I find it to be i k, which is the iust side of a true square that will be e­quall to a b c d.

Or the length of the perpendi­cular b m is the square falling vpon b c at right angles.

CHAP. VIII. To find the square of any peece of timber consisting of three sides.

THe true square of all kind of Triangles whatso­euer are found out by the 44 Chap. Metamor­phosis 7. of the 6. Booke of the Geodeticall Staffe, and therefore it were vaine to repeat it here againe.

Or else take the perpen­dicular

in any triangle and halfe the base, which two lines ioyne together, conti­nuing the same forth in one right line, as you doe the 2 sides of an Oblong in the 6 Chapter, and thereby find the square accordingly.

As if you ioyne the per­pendicular b d and halfe the base a c in one right line, ac­cording to the 6 Chap. you shal find the square of the tri­angle a b c to be h i.

CHAP. IX. To find the square of any peece of timber containing 5, 6, 7, or 8 sides, &c.

YOu must imagine in this Chapter, as also in all the other, sauing for round timber, that I goe not about to tell you how much square that peece of timber would beare, if it were reduced into a 4 square: but I doe deliuer you the side of a péece of timber béeing iust foure square, and of equall height with the [Page 194]peece proposed shall also be of equall quantitie, which is right ne­cessarie for the attaining of the number of square feet or solide content of any péece of timber.

Hauing therefore a péece of timber of 5, 6, 7, or 8 sides, &c. as b c d e f g, adde all the sides together b c, c d, d e, e f, f g, and g b, so haue you 78, halfe which is the line n o 39, to which ioyne the perpendicular h a, which you be taught to find: so haue you o p,

next vpon o reare a perpendicular o q, lastly diuide n p into two equall parts at r, placing the one foote of your compasse in r, ex­tending the other to p or n, and vpon r strike the arch s q t, which shall cut the perpendicular q o at q, I conclude q o is the true square

Thus are you taught to find the square of any peece of tim­ber, of what fashion soeuer: and if it beare none of these regular formes, or that there be wood wanting, take from one place, ad­ding the same vnto another, thereby making it perfect regular, and in such cases you must alwaies so doe.

CHAP. X. The square of any peece of tymber being found to tell how much of the same in length, will make a square foot of tymber, and consequently how many foote is in the whole peece.

A Solid or Cubicall foote of tymber doth con­taine 1728 cubicall inches, for so many foure square inches may be taken out of one cubi­call foot, I meane such Inches that are square euery way like vnto a dye, now hauing the square of any péece of tymber giuen, square the same, diuiding 1728 by the product, so doth the [Page 196]quotient shew you how much of the length of the tymber must be taken to make a square foote, by the which diuide the whole length or altitude of the tymber, so doth the quotient acquaint you how many foote of tymber is in the péece.

Example.

The square of the tree a b is found by the third Chapter to be 27 inches, whose square is 719 by which diuide 1728 so haue you 2 2/7 9/2 0/8 inches which is 2 ⅜ inches, or two inches a quarter & halfe a quarter, whereby I conclude, that as often as I can finde 2 ⅜ inches in the length or height of the tree or tymber, so many square foote of tymber is in the same: the tree a b is 8 foote high, which diuide by 2 ⅜ inches, or lay somuch of your rule out measuring one from a towards b, calling euery 2 ⅜ inches a foote, so by either of the waies shall you finde 40 foote of tymber in the said tree being squared, some small quantity being ouer more then the same.

In the like manner must you deale with all other peeces of timber of what fashion soeuer, first finding their square as before, & next the solide capacity, euen as you be taught in this chapter.

By this Chapter may you measure out as many foote of tym­ber, stone, or such like, as you please, & thereby cut off any num­ber of feet from any peece of tymber as you shall be occasioned.

CHAP. XI. To measure all kind of Tymber, &c. after ano­ther sort, without regarde of the square.

THis kind of measure taketh no regard to the square feete in the tymber, but vnto the solid capacity thereof, but for that it is not much per­tinent to the Geodeticall Staffe, requiring ra­ther numerall then instrumentall operation I will be the more briefe.

When you haue any péece of tymber, stone, pillar or such like, whose solid content is required by the rules taught in the sixt book, part 2 of my Geodeticall Staffe, seeke the superficiall contents of the end of the tymber, the which aug­ment in the altitude or length of the same, so is the product your desire.

Example.

Let the peece of tymber be pro­posed,

whose end a b c d superficial­ly by the 22 Chap. Prop. 3. of the foresaid book, is found to containe 225 inches, now the length or al­titude of the timber b g or a f is 33 inches, 33 inches mutiplyed in 225 produceth 7425, the number of solid inches in the peece, which if you part by 1728, you haue 4 5/1 1/7 3/2 [...]/8 foote, which is 4 foote 513 inches the solid content of the said peece of tymber.

The like must you do with any other peece of tymber of what fa­shion soeuer, but if the tymber trapeze much, vse the middest or difference of the ends as in the 4 Chapter, which at this time shall suffice.

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.

CHAP. XII. A discourse of surfaces and solid figures, not so apt for the vnderstanding of mechanicall artificers

LIke as the terme of a line is a point, so is the terme of a surface a line, euery surface is plaine or bowed, a bowed surface is either sphericall or varied, a sphericall surface is equally distant from the center made by the reuolution of a semicircumference vpon the fixed diameter, a varied surface is either conicall or cylindricall, a conicall surface runneth narrower and narrower [...]rom the cir­cular base vnto a certaine point in the toppe, a cylindricall sur­face is that which is equally raised from the circumference in the [Page 198]bottome to an equal and paralel circumference in the top made by the conuolution of the side about two equall and paralell cir­cumferences.

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 li­neate, broad and high, consisting of 4 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 y^e 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 pa­ralellograms, now euery right angled paralell pipedon, 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 ir­regular, the regular bodies conteined vnder surfaces the one fol­ded towards the other be onely 5, as tetraedrons, hexaedrons, or cubes, Octaedrons, Dodecaedrons, & Icosaedrons, the first being a Geometricall body encompassed with 4 equall equiangled try­angles, the second with 6 equall squares, the third with 8 equall equiangled tryangles, the 4 contained vnder 20 equall equian­gled tryangles, and the last being comprehended of 12 equall e­quiangled pentagonall superficies, about euery of these 5 regu­lar or platonicall bodies, a comprehending or circumscribing sphere or globe may be described that shall with his concaue peri­phery exactly touch euery of their solid angles, whereby they be made bodies inscribed or conteined of that sphere: also these in­scribed [Page 199]bodies may be termed circumscribed solids of a sphere, & then the sphere is called inscribed or conteined, which happening the conuer superficies of the said inscribed sphere shall precisely touch all the centers of those equiangled figures wherewith the bodies are inuironed: touching irregular bodies they be such that be limited and described by inequall surfaces, which are twofold. to wit in respect of circular conuolution and in respect of folding one towards another, y^e 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 con­teined vnder a bowed surface, made by the reuolution of a se­micircle vpon a fixed diameter, as for the varied solids, they be cō ­tained vnder a varied surface and a base, and are twofold, as a cone and a cylinder, the cone is contained vnder a conicall sur­face and a Glasse, and the cylinder vnder a cylindrical surface, and opposite bases, and as for this cone and cylinder the one is made by the conuersion of a right angled tryangle (the one foote re­maining fixed which if it be equal with y^t which moueth y^e cone is right angled, if lesse obtuse-angled, if greater acute angled) y^e other by the conuersion of a right angled paralellogram y^e one side re­maining fixed: further in cones the fixed side of the tryangle is caled the Aris, 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 fi­gure vpon the plaine whereupon the solide resteth is called the al­titude of the solid▪ neither is it materiall if the same perpendicu­lar fall within or without the body, as the one alwaies doth in di­rect solids within, and the other in declining solids without: now as plaine angles are made vpon superficies by the section of two lines in a point, so solid angles are made in bodies by y^e 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.

CHAP. XIII. To measure the Contents both superficiall and so­lide of Cones, Cylinders, Pyramides, Prismes, Cubes, Polyedrons, Spheres, Globes, and such like.

I Would say nothing of these matters in this place, were it not for that it may haply be ex­pected, since somewhat already hath béene said of solides: yet since you be taught to find the superficiall content of any figure in my art of 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.

CHAP. XIIII. To measure the aire or superficiall capacitie of any plaine surface, as boards, glasse, floores, pauements, &c.

IF you would know how many foote or inches in length you must haue to make a foot of board or glasse, the breadth thereof beeing giuen, or how you shall cut off any number of féet pro­posed from any assigned board or such like, re­paire vnto the 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 [Page 203]their sieling: also in the same order may Tilers (especially those that vse burnt tiles, which commonly be of one bignesse) tell how many stones will couer any roofe, or how many thousand be vp­on any couered roofe, onely by multiplying the number of stones that go along the eaues of the house, by the number that go vp the side from the eaues to the top of the roofe or first pole.