Speculum Topographicum: OR THE topographical Glass. Containing The use of the topographical Glass. The use of the Theodelitus. The use of the Plain Table, and Circumferentor. With many Rules of Geometry, Astronomy, Topography perspective, and Hydrography. Newly set forth by Arthur Hopton Gentleman. woodcut, labeled circular map showing England, Wales, Scotland, and Ireland Printed at London by N. O. for Simon Waterson, dwelling at the sign of the Crown in Paul's Churchyard. 1611. TO THE RIGHT HONOURABLE, THOMAS Lord Ellesmere, Lord Chancellor of ENGLAND. THE Privilege (Right Honourable) is common, and the custom commendable, to dedicate books to Noble persons, to the end that what is effected by labour and study, may by their greatness be protected from maledictions and envy; and therefore we select one, whose eminent virtues (exempt from Rivals) is of all admired, by all observed, and with all beloved, and there the choicest wits shelter their chiefest works: the habit of which wonderful glories I find most predominating in your Honour, which are powerful inducements to animate this presumption in craving patronage for this work: The Book contains no nice or new controversies, but matter of Art, verified with demonstrations Geometrical, as still ready to confirm the truth to the ignorant, or confute the malice of the arrogant, which though it be not fashioned out with beautiful lineaments, or painted over with golden phrases: yet is there dainties sufficient to delight the eye, and recreate the mind, choice of varieties to beguile the time with gain of knowledge, and easy methods with facile documents to abjure the barbarous tyrant to understanding. Things of greatest profit, require lest praise; the white silver is wrought in the black pitch; painting better beseems rotten walls then precious stones: the Mathematics use no conference with the sence-ravishing Rhetoric, their end is to instruct, not persuade; therefore superfluous eloquence bestowed upon a matter of sufficient excellence, is rather a testimony of a trifling wit, than a token of true wisdom. A chief cause wherefore I now prove troublesome, is, my love to your Lordship, whom I ever honoured, and the advancement of the Art that I always liked. By the one I show my duty to a name emphasizing honour, and by the other my affection to an Art expressing wonders. Alexander would be painted by none but Apelles, nor have his picture cast in brass by any but Lisippus. Appelles' asked council of none but Zeuxis, and Lisippus must only censure Prisius: neither do I in this appeal to any but your Lordship, which, as yourself are honourable, so are your proceed equitable, so that all England may boast of your great justice, and all Europe rejoice of your good conscience: amongst which, myself with the best will ever bear testimony; the great son of the Macedonian king honoured Craterus, but most affected Hephestion, and I reverence all good wits, but here only appeal to your honours wisdom, that as it is exceeding the gravest, so is it more excellent than the greatest; and therefore by the inferiors fit to be admired then commended. But because your cogitations cannot but be defessed and made weary, as well in private contemplating of his majesties serious affairs, as in public negotiating for the good of the Commonwealth, I think it best to cease trouble, lest I offend your noble meditations, being leveled at more weighty entendments: not doubting but the practisers in these Arts will yield your Honour perpetual thanks, that for their good have brought this Glass to light, where they may see choice of delights to satisfy their aspiring wits. Thus committing this Book to your Honourable patronage, your Honour to the Almighty's protection, and myself to your honours command, I end, resting Your Honours in all humble duty, ARTHUR HOPTON. TO THE Mathematical Practizer. PLATO saith, there was in old time an Oracle given unto the Greeks', that they should double the Altar in the Temple of Delos (which was a piece of work for an excellent Geometritian to perform, that had the very habit, and perfection of the Art) but it was not there literally meant, (as Plutarch in his Symposiaques hath expounded) that they should do so indeed, but thereby they were injointed to study Geometry, to the end they might be able to perform any piece of work of as great consequence. Pythagoras' offered sacrifice to the Gods, to know, when two figures are given, how to find a third equal to the first, and semblable to the latter. Both which are strong arguments, invincible proofs, and persuading authorities, as well of the necessity to obtain, as of the difficulty in obtaining the Art. The remembrance whereof, (kind reder) hath oftentimes solicited me to make good my promise ●●cely granted in my Geodeticall Staff, concerning the publishing of my Topographical Glass, which having now accomplished, there remains nothing more, but to crave a kind acceptance, and favourable construction: so shall you shake off that viscosious filth of ingratitude, which so much conglomerates the heart of the envious: for the poison of malice being once fostered in men's breasts without resistance, buddeth forth daily more malignant fruits, whereby men run a malo in peius, as the fish out of the pan into the fire: it is vain then to beg applaudites of such, for Ruby stones are not found in flinty rocks nor redolent flowers amongst rough thistles. A corrupt stomach yieldeth no sweet breath, and an envious mind seldom due praise. Art may check, but can t quite change what nature cherisheth, insomuch that where ca●●●●●ation, d●samation, and vile malediction, are so consonant an● coherent in the heart, s●eking nought but contention and controversy, foolish are entreaties to win savours, or persuasions to purchase kind constructions, turning always the band of gratitude into the passion of hatred. Leaving them therefore, let us direct our speech to the benevolent reader, and kind expositor, that wrings no sense to ambiguities, that wrists no saying to amphibologies, that seeks no eversion of the word, nor aversion of the work, that urgeth no dissension in the method, or dissipation of the matter, but aims to get benefit by his reading, and to yield thanks for the writing, embracing that which yields profit, and rejecting all which is impertinent, that receiving a rose, returns a Hyacinth▪ to such do I commend this topographical Glass, as a pure transparent Crystal, wherein he may see a number of art like pleasures, and delightful conclusions, performed by the method of divers instruments, as by the Glass, by the Theodelitus by the Plain Table, and by the Circumferentor, by all which shalt thou be severally taught to describe Countries and Kingdoms to make Maps of new discoveries, to plat Manors, Lordships, or Towns, to measure parks, pastures, or enclosures, and that after divers new ways. Also here are you taught to seek the distance of Towns, the height of Turrets, to convey waters, to plant Cities and houses, to plat buildings, and to measure all k●nd of ●imber Stone, Pyramids, Columns, Cones, Spheres, Globes, and such like, in a method differing from any heretofore published: so that if you have any good Geometrical instrument already, sure also hast thou the use thereof; or if thou want, here art thou instructed to make the same according to thy affection: for I seek to tie none to my own particular humour: then might I have referred them unto my Geodeticall Staff. But happily some will say if there were sufficient instruments before, than what needeth these new inventions? would we have our own wits more excellent than our predecessors. Of such and such like, I familiarly inquire if Antiquity be only Mistress of this faculty, if modern wits may intimate or exhibit nought unto the world; if we must only believe what is set down, without contradiction, should that be so? how far had this age been from the perfect Idea of the Art, whose excellency turneth every heart after the same as the Heliotropion after the Sun? into what an intricate Labyrinth of confused errors had we run? The most ancient Philosophers were as contrary in their sects, as erroneous in their opinions, till time brought in Truth, truth Knowledge, and Knowledge perfect Understanding; what a number of sects had we? as Catonians, Peripatitians, Academians, and Epicurians. Anaximander said the earth was like a column. Anaximenes' flat, Leucippus like a Drum, Democrites like a platter; till Thales demonstrated it to be round. How grossly did they differ about the Tides? some referring the cause to rivers falling from the mountain Gaul, entering the Atlantic sea, as Tymeus: Some to a rising of certain waters, as Plato: Some to the Sun drawing the winds upon the Ocean which caused the Atlantic seas to swell, as Heraclitus, till Pytheas demonstrated the cause to be in the increasing and decreasing of the Moon; whereby Homer said: I praise not my Ancestors for their knowledge, but for that they desired knowledge. But shaking off these old differences, inquire of the commodity of the Compasses, Sea-cards, and new Maps, & of many other devices & engines, that have been lately set forth more beneficial than any heretofore, of which the old Philosophers & Mathematicians were ignorant of & I think if they were living, they would rejoice to see, as amongst many I will remember G●l●risius Astrolabe, the Mater whereof being excellent and a most Art-like proiectment resembling the true lineaments of the sphere, and is now made perpetually famous with the Edition of the Rete, by our ingenious countryman M. I. Blagra●e. But my Pen shall not be so much dismeasured to reprove ancient men▪ to the end to draw the glory to them that be present. Had we lived than we had known less than they, and were they living now, they would know more than we: every thing hath his time, before it come to maturity: neither doth nature allot like time to every like action: some things grow to perfection in a moment, when others require a month. The Bear bringeth forth in four weeks, when the Dolphin hath near 40. when one fruit faileth, another entereth into season: we must first invent, next amend, and lastly perfect: the furtherance of which perfection it behoveth every lover of the Science to cherish, which is here freely offered unto thy view, though it might have been undertaken (I confess) by some one of more experience: but he that most can, least will, and he that worst may, holds the candle: else the world must walk in darkness. There be many say, the instruments be uncertain, and this or that is better: but none seeketh to reform the same, & contrive one that shall do best: the professors themselves cry out at the erroes, but seek no reformation of the fault: so that it fareth with this, as in the civil actions of the life, every man curseth excess, yet none live temperately: every man praiseth patience, yet none will suffer every man blameth sloth, yet none will take pains: every one exclaimeth on envy, yet none leave emulating. But to remit this, and speak something of the application of Instruments. He than that will be seen in the knowledge of Geometrical instruments, must learn by contemplation, to frame his proposition, and by action to manage his instrument: for I know divers great Scholars, deeply seen in the Theorical part, though in the active, mere novices: which is a cause that such, so learned, were never able to correct and amend many defects. For as meditation causeth ability to understand, so action bringeth dexterity to perform, because the event of the work is illustrated by a precise observation, as the life of the proposition is illuminated by a plain demonstration: for as points found lines, lines surfoces, and surfaces bodies, so good instruments produce true observations, true observations symmetrical figures, and the superficial capacity of proposed platforms. And as lines bound figures, so hedges bound enclosures: and angles in the field are created by the meeting of hedges, as they be in figures by the section of lines: for you shall know that all instruments, of what kind soever, if they cannot observe the precise quantitity of the angle, their work is erroneneous: for though some instruments express the quantity of angles, as the Theodelitus. etc. and others regard not the quantity, producing angles Homogeneal, as the Plain Table: yet, if you fail in the one or the other, your conclusion is error: for it is as great an absurdity if the angle Homogeneal Mechanical delineated upon the Plain Table, Prove Heterogeneal, and accord not Symmetrially to the respondent angles in the field, as if with a graduated instrument you had falsely observed the quantity thereof: and therefore it is childish vanity, or at least self conceit, to go about to prefer the Plain Table before a large graduated instrument, and I am persuaded all good Geometricians will argue the same: though I verily believe it is possible to separate fire from heat, or the earth from the centre, at turn the obstinate will of the ignorant: for use and custom hath taken such root in such, that it will hardly be supplanted. It sufficeth them, plain men to have an Instrument, for than they presume they be Geometritians. Every horse is not Bucephalus, yet he may be are one a journey, though he be long and lame in performing it, and can hardly pass over a mountain: but let every man use his will, and so I will mine here, in saying no more at this time: only wishing myself present to open any thing that the young practizer shall doubt of. From my Lodging this 9 of April. 1611. ARTHUR HOPTON. The Author to the Reader, concerning the topographical Glass. COme you, whose eyes stand not in envious head, Whose tongue with Critic humours is not fed, And in this Glass, unto your comfort view, Such needful works that much may profit you. The grounds of Art have brought it forth for thee, Which we have sucked from famous Geometry, With Theorm's mixed and demonstrations rare, Such as in hiddan Propositions are: here's no vain show: illusions have no place, No spirit confined, no hateful painted face, No eye-deceiving glass, no Crystal brave, Which from the frozen seas we often have. But in a fair and most perspicuous light, The earthy Globe lies subject to thy sight. Whose centre's deep, whose continent is great, Here mixed with cold, and there with burning heat: As Phoebus' beams do sparkling from him glide, Or as in torrid Zones we do abide. This frame enough for all the world to view, This Glass lays in a small proportion true. That like an Eagle towering up aloft, Whole regions thou mayst view and review oft. Discoursing now of Europe in particular, And then of other Country's distant far. Now of jerusalem that was before, And then of Rome, by Tiber's silver shore. See here mount Zion and mount Moria then, That on their backs bore old jerusalem: Though now mount Caluarie hath got the grace, Which erst was but an execution place. Learn here to bond the Alps, Spain, France and all, That yields those vines whence ivycie grapes do fall. America, the new found land beside, And eke those Southern parts yet descry. See how to measure plot, and part out lands; As dressed in Flora's summer robes it stands. Or as it's rend up with the tering plough, O● hid with freezing ice or melting snow: See here the height of hills, or see their odds, That Giants meant to dart against the gods. See here the Sun, whose beams do Comets fire, That all the world with terror may admire. Behold the Moon, and fixed Stars so bright, That guide the Pilots in the darkest night. Look how the Planets with the whirling Sphere, To our Horizon fiery meteots bear. Gild the black brow of the ugly night, With burning Dragons and such fearful light. Look on them all: see in this little Glass, Their height so took as ne'er in English was. Wouldst plant a Town, or situate a house, Or force the water take a wished course. Wouldst seek the height or distance of a Town, Or undermine a Fort to blow it down▪ Or what wouldst do belonging to this Art, But in this Glass thou mayst behold apart? Then use him well, my word I pawn, he's true, Let small defects be but supplied by you. Of which few scaped my Pen, no store the Press, Where either be, in kindness use redress. So shall my Pen not be employed in vain, To please thee more when I do write again. Arthur Hopton. T: Littleton gener: ad A: Hopton, amicum suum. Illustris Lector, librum nunc cerne docentem, Eximias artes, quas tibi scire licet: Arthurus juvenis docet in iwenilibus annis, Quae priscis annis non revelata latent. Flores venturis seclis dat semper Amenos, Lectori grato suavis odorque manet. Author communem dignus velit utilitatem, Authori digno gloria dignadetur. I. Passy, in artibus Magister. COntinet hic parvus mira & secreta libellus, Quem pretio vili (lector) habere potes: Que Solis, Lunae, astrorum, coelique potestas, Inferioraregat corporaritè docet. Cultor agri, pecoris custos, pecudumque magister, Navita, mercator, caetera turba virum. Hinc petas auxilium, quo tutius exigat aewm, Authoremque operis semper amore colat. Constitit authori magno, multoque labore: Quem tibi dat gratîs gratus amicus amans. justitae virtus fama non ultima laus est. Qua semper florens, stirps generosa fuit. Arthure, liber hic lumen mortalibus adfert, Arthure Hoptonûm nomine fama viret, Hoptonides invenis generosa stirpe creatus, Arthurus, grata perlege ment●, cole. Carmina R: Stedmani, profes. Theologiae Oxoniensis, in laudem Authoris. EDidit Arthurus librum generosus amicus: Qui feret eximijs lumina clara viris. Est juvenis florens suanuis flos gloria patrum: Mirificis monstrans inelita facta modis. Lucida nubiferi decantat corpora coeli, Lucentis solis commoda magna canit. Commemorat fluvios magno cum murmur iactos, Attribuens cunctis grandia summa viris. Scriptoris memorat celeberrima facta recentis: Flammiferumque nova continet arte iubar. Carmina praedicti R. S. Praevalet antiquis geometria scripta libellis, Terminat in proprijs pascua cuncta locis. Aestivis memorat gemmantes floribus hortos: Candidulaque dabit praemia digna manu. Inque agro monstrat vestigia certa virenti: Quae proprijs dominis faedera pacis erunt. Mensurat radijs liquidos flagrantibus ignes: Solis & ardentis lumina magna capit. Pinguas describit terras cum passibus aequis: Terminat aequali iugera cuncta modo. Alios brumali mensurat tempore montes: Aestivoque canit florida prata die. Vivet opus, durabit opus, tua fama vigebit: Et tua posteritas facta sequenda canet. Errata. Pag. 3. line 7. the word) p. 13. pro. 7. place the figure in the 9 pag) 40. l. vlt. for 28 18. omit orleton) 41. omit 10. & 11. lines 68 a stands where c should) 101.8. the 61) 114.2. I can see) 117.14. one with another) 118. l. 8. your second) 150. c b should be infinite beyond b, a perpendicular falling from a, thereon at a point h) 153. the perpendiculars and bases be omitted) 161, the figure in p. 164) many times you shall find sum and sign for sine, with other small literal faults, that you may correct in the reading. VIRESCIT VULNERE VIRTUS DUM SPIRO VNICUM CHRISTUM SPERO woodcut, detailed crest THE topographical Glass. Containing the description, making, and use of the said Glass, the Theodelitus, plain Table, and Circumferentor, with many other topographical conclusions, concerning the plantation of Cities, and situation of houses, with the conveying of waters and such like: as also Propositions Astronomical, correcting thereby the altitude of the Sun and Stars, being formerly observed falsely, in respect of the omission of their Refraction and Paralax. CHAP. I. What Topography is, and how it differeth from Cosmography, and Geography. topography (with some called Corography) is an Art, Topography, whereby we be taught to describe any particular place, without relation unto the whole, delivering all things of note contained therein, as ports, villages, rivers, not omitting the smallest: also to describe the platform of houses, buildings, monuments, or any such particular thing; and therefore a topographical description ought to express every particular, which caused me the rather to call this instrument the topographical Glass, as being most apt to describe any monument, Tower, or Castle, any Manor, country, or kingdom, so do we briefly describe England thus: but we have omitted divers things, by reason of the smallness of the plat; therefore take the plat, the division, and number of shires, number of Parish Churches in every shire, etc. A Demonstration of Topography. woodcut, labeled circular map showing England, Wales, Scotland, and Ireland woodcut, stick figure castle Geography (as Vernerus in his paraphrasis saith) is an imitation of the whole earth, and his principal and most known parts differing from Topography, because it respects but places of note, and from Cosmography, because it hath no relation unto the circles in the sphere, only describing the world by hills, rivers, and therefore Geography may serve well for a general description of particular places, rivers, etc. in the world. A Demonstration of Geography. A description of the world woodcut, labeled map of the world As Apian saith, we may well gather what Cosmography is, by the bare etymology of the world. It is therefore a description of the world, which doth consi●● of the four Elements, and also of the Sun, Moon, and all the other stars both erratical and fixed, with all things else that be contained within the concavity of the heavens. First of all, it hath respect unto the circles, which are imagined in the celestial sphere: and such like circles be appointed or imagined upon the earth, also it doth demonstrate the site, symmetry, or commensuration of places, the diversity of climates, the differences of days and nights the four quarters of the world, the motion, rising, and setting of the stars, with such as are moved vertically, with all things else, that appertain to the consideration of heaven, as the elevation of the Pole, the parallels, Meridian's, circles, etc. So that like Meridian's, like parallels, etc. in heaven do corespond to their like upon the earth, hereby is every town, etc. situate under some one constellation or other: A Demonstration of Cosmography woodcut, globe and cosmological map Hereby hath every town lying into the East or West a several Meridian, and into the North or South a several parallel of latitude, etc. as you may gather by the Cosmographical demonstration before, therefore behold the figure because this is beside our intended labour. CHAP. II. Geometrical definitions of lines, angles, and figures. BEcause the ensuing terms of Geometry be often used in this book, & the understanding thereof not known (though something common) to all young practisers, as also for the they be but stenderly remembered in my Geodeticall Staff, I thought good to say something thereof. All Geometrical magnitudes take beginning from a point, which is an indivisible sign in a magnitude. 1 A magnitude is lineal or lineamentall. 2 A line is a magnitude only long, and his terms be two points bounding the terms of the line. 3 Lines are considered after two ways, first simply by themselves, than also comparatively amongst themselves. 4 Being considered simply they be right or obliqne. 5 A right line is which lieth equally betwixt his terms. 6 An obliqne line is which lieth unqually betwixt his terms. What distinctions right and obliqne lines do admit. 7 A right line being given there cannot be made a shorter betwixt his terms, and therefore they be all like unto one, but where as an obliqne line doth lie inequally betwixt his terms, they may be divided, so the the obliqne lines be simple or manifold. 8 Those be Simplex, which be terminated with a uniform and simple motion. 9 The divers or manifold obliqne lines called Helices', are such that be terminated with divers motions, and be unequally distant from the centre, such are Spirale lines, Conchales, Ouales, Lenticulares, etc. Lines compared amongst themselves be diversly affected: for either they be perpendiculars, parallels, or contrary. 10 Perpendiculars compared amongst themselves, are two right lines, whereof the one falling upon the other make two right angles, such is a Carpenter's squire, etc. 11 Parallel lines are every where one alike distant from the other be they right lines, circles or Helices'. It followeth to speak of Lineaments, and first of Angles. 12 A lineament is a magnitude more than long, consisting of lines, and is distributed into an angle or a figure. 13 An angle is a lineament, made by the common concourse of two lines, and situate within the same. 14 The two terms or lines comprehending the angles are called the sides. 15 Of angles, there be two kinds Homogencall and heterogeneal. 16 Angles Homogencall, be such that have their sides both of one kind, as both right or obliqne. 17 Angles Hetrogeneall are such that consist of mixed side as of right lines and curved. These Homogeneal angles do furthermore suffer a subdivision, and are therefore right or obliqne. 18 A right angle is where one live falleth perpendicular upon an other. 19 An obliqne angle is made when one line doth not fall perpendicular upon another, whereof there be two kinds, Acute and Obtuse. 20 An Acute angle is an obliqne angle, less than a right angle. 21 An Obtuse angle is an obliqne angle, more than a right angle. Of Triangles and Figures. 22 A triangle, is a figure, and a figure is a lineament bounded upon every side. Therefore a triangle is a figure consisting of 3. Angles, and bounded with 3. lines. 23 Every figure hath certain terms wherewith the figure is measured. 24 A Centre 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 drawn from the centre to the perimeter. 27 A Diameter is a right line inscribed in a figure, and passing by the centre, therefore Diameters in one figure may be infinite, the centre of the figure is always in the Diameter, and in the concourse of the Diameters. 28. The altitude of a figure is the length of the perpendicular let fall from the top of the figure upon 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 only broad. 31 Superficies be also plain or swelling. 32 Plain Superficies be such that lie equally betwixt their terms. 33 Plain Superficies admit a double distinction, whereof some be right lined, some curuilined. 34 Plain right lined Superficies, be such that be bounded with right lines. Of right lined plains. 35 Of right lined plains there be two kinds, as a Triangle and a Triangulate. 36 A triangle is a figure comprehended under 3. right lines. 37 Furthermore a triangle is taken two kind of ways, 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 triangle consisting of 3 equal sides, 40 An Isosceles hath only two equal sides. 41 A Scalenum, hath all his 3 sides unequal. 42 Tryangles, in respect of their angles, are distributed into two kinds, as Right Angles, and Obliqne Angles. 43 A right angled triangle is, which contains one right angle which is also called an Orthogonium. 44 An Acutangled triangle, hath all his angles acute, to wit, less than 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 mixed or comparatively, as well in respect of their sides as angles. So have we an Orthogonium Isopleuron, an Oxigonium Isosceles, or an Oxigonium Scalenum. Of Triangulates. 46 A Triangulate is a mixed figure composed of Triangles, and may be resolved into the same again, whereof there be two kinds, Quadrangles and Multangles. 47 A Quadrangle is a figure bounded with four right lines, and is twofold, as a Paralelogram, or a Trapezia. 48 A Paralelogram is a figure whose opposite sides be parallel, 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 obliqne, and they be twofold, either a Rhombus, or a rhomboids. 53 A Rhombus is an obliquangled Paralelogram equilater. 54 A rhomboids is a paralelogram obliquangled and inequilater. Of a Trapezia. 55 A Trapezia is a quadrangle being not a paralelogram. Multangles. 56 Mixed or multangled Triangulates are such figures that consist of more than four sides. Of Curuilines' superficies. 57 Curuilines' superficies be either simple or mist. 58 Simple, are circular figures & round, being every where equidistant from the centre, as a circle. 59 The terms 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 contained inwards under two right angles, making an angle either in the centre or Peripher. 62 A Section is a segment of the circle contained inwardly, with one right line, which is called the base of the Section. Lines Ascript be described Lib. 7. chap. 2. of the Geodeticall Staff. 63 Such figures be called Mist obliquilined, which be unequally distant from the midst of the figure, such figures be Ouales, Lenticulares, etc. CHAP. III. How right figures are created. I Shall not need to run into an ample discourse hereof, nor tyre you with multitudes, only some few that shall be most requisite in this intended work, I will acquaint you with, leaving apart the rest. PROPOSITION I. To create a right angle, or to rear a Perpendicular on any assigned line. A Right Angle or perpendicular is created by the extension of a right line from both the sections of two arches of equal or inequal Diameters, the centres whereof remaining in the line assigned, and this creation of right Angles or Perpendiculars is general. Example. mathematical figure corresponding to example description PROP. II. To rear a Perpendicular, or create a right angle upon the extremes of a line assigned, and not alter your compass. IN making of a Quadrant, & performing of other conclusions in this book, you shall be forced to raise a Perpendicular, or create a right angle upon the extremes of a line assigned, wherein the last Proposition would stand you in no stead: therefore work thus: mathematical figure corresponding to description A p is the line assigned, p the extremes of the line, whereon you are to erect a Perpendicular, therefore I open my compass to any reasonable scantle, placing the one foot in p, with the other I strike the arch c d e, that wideness resting, I place the one foot in c, and with the other make the point d, then with the same wideness upon d, I do describe the arch c g h f, then do I fit that wideness three times in the last described arch, as g h f: lastly, from f I draw a right line to p, so have I created a right angle f p a upon the point p. PROP. III. To draw a Perpendicular from a point assigned to a line assigned. D is the point assigned and a b the line assigned whereon a Perpendicular should fall from the point d, therefore mathematical figure corresponding to description place the one foot of your compass in d, and open the other so far, that it may reach beyond the line a b, and so strike an arch e f, which shall cut the line a b in two points, as in e and f, than your compass resting at any scantle, fix the one foot in f, and with the other strike an arch under that arch e f, the compass at the wideness, fix the one feet in e, and with the other cross the last described arch in the point g, finally draw a strait line from d to c. for that is the Perpendicular to the line a b. PROP. FOUR To make a right Angle readily. mathematical figure corresponding to description By this means may you easily open the legs of your Geodetical staff unto a right angle, or place the graduator at a right angle. To place the legs of the Geodeticall staff at a right angle. To place the legs at a right angle do thus: count 40. upon the left leg, whereto bring the centre of the Graduator, then count 30. upon the right leg amongst the equal divisions, next note 50. equal parts upon the Graduator, now the centre of the Graduator resting fixed, move the right leg, and the other end of the Graduator until 30. upon the leg, and 50 equal parts upon the Graduator, touch one the other, so shall the legs rest at a right angle. To place the Graduator at a right angle upon the left leg. Now to place the Graduator at right angles, count 40 upon the left leg, whereunto bring the centre of the Graduator, then make 50 upon the right leg, and 30 upon the Graduator intersect, so have you placed the Graduator at right angles. mathematical figure corresponding to description Otherwise: Draw a semicircle a b c d, upon the centre c, d a being the Diameter: finally from a draw a line to touch the circumference as at b or c, then from b or c draw a line to d, so have you made a right angle, as a b d, or a c d. PROP. V To make an angle like to any angle assigned. THis proposition will stand you in much stead in the use of the plain table, and is performed thus: mathematical figure corresponding to description A b c is an angle, and you be required to make another angle equal thereunto, which to do, open the feet of your compass to some reasonable scantle, placing the one foot in a, and with the other strike an arch over the sides of the angle, as d e: the feet unstirred I place the one foot in the line f c, and with the other strike the arch i h, next do I take the distance of d e, and fit it in the arch i h, from i to h: finally, I lay my Ruler upon the point i and h, drawing a line from f towards h infinitely: so have I made an angle c f g, equal to the assigned angle b a c: this proposition will stand you in singular use for translating of plaits from one paper to another, and for protraction of plaits and divers other operations performed by the Plain table. PROP. VI To draw a line parallel to any assigned line. A B is a line assigned, mathematical figure corresponding to description to which you must draw a line parallel, choose two points in the assigned line at all adventures, as c d, now open your compass to what distance you please, or rather to any distance assigned, then place the one foot in c, and with the other strike a fragment of an arch, do so at d: finally, by the extremes of these two arches produce a right line e f, which shall be parallel to a b. PROP. VII. To divide a line into two equal parts, or to find the midst of any line assigned. A B is a line assigned, which you are to divide into two equal parts, or find the midst betwixt the point a and b, which to do open your compass to any reasonable scantle, and then placing the one foot in b, with the other strike the arch d a c, that wideness remaining, place the fixed foot in the point a, and with the other strike the arch c b d: now note the intersection of these two arches, as at c and d, for a line drawn from c to d divides the line a b in two equal parts at the point e. I will not seek to deliver rules to divide a line into a number of parts, nor to give any part of any line, because it is performed else where with singular ease and dexterity. PROP. VIII. Three points being given to find the centre of a circle that shall cut all the three points. mathematical figure corresponding to description mathematical figure corresponding to description Otherwise: A b c are three points given, first from a to b I draw a right line, then from b to c: lastly by the 1. Prop. I rear a a Perpendicular in the midst of a b and another in the midst of b c, as e d, and f d, the intersection of which two Perpendiculars, as d is the true centre. Otherwise: Let the 3. points be a b c, open your compass to some reasonable scantle, let it be more than half the distance that is betwixt a b, with that wideness strike a portion of an arch upon the point a, do so upon b, and note the intersection of those two arches, as f g, do likewise to b and c, and note the intersection of the arches, as i h: lastly, produce a line from i by h infinitely, do so from f by g, noting where those two lines intersect, for that is the true centre, as at e. mathematical figure corresponding to description More of these conclusions might be set down, but here is sufficient to perform whatsoever is needful in this Book. CHAP. FOUR The making and composing of the topographical Glass. The construction of the topographical glaffe. FIrst prepare a piece of wood requisite for such a purpose, or a piece of plate, if you will have it in mettle, & let it be 9 inches square at the least, as d b or a c, then upon the centre 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 divide into 24. equal parts, as here it is, and set figures accordingly, an inch, or some reasonable distance from this circle, describe another circle, and divide the same into 32. equal parts, and there appoint the 32 winds, as you may plainly see● next unto this describe another circle, and there place the Geometrical Quadrant thus. Secondly, in the Quadrant of this last circle e g, draw the ordinary quadrant i f h, and divide the sides i f and h f into 120: or 60 equal parts, then keeping the rule upon the centre r, & each division in the sides i f, and f h, as you move him from division to division, make marks in the circle e g, so have you projected the Geometrical Quadrant i f h into the Quadrant of a circle e g. About this circle describe another circle, and divide the same into 360 degrees, as the order is, drawing parallel circles to place the figures, as in the demonstration. About this last graduated circle draw an other some half an inch distance or better, as a b c d, and betwixt the graduated circle r, and the circle a, draw five other parallel circle's equidistant one from the other, as in the figure: Lastly, upon every two degrees make an Isosceles Triangle round about the circle: you may better perceive it by the plain demonstration then understand it with multiplicity of words. These Isosceles Triangles serve aptly and precisely to express the fiftieth part of a degree from 10 to 10 etc. thus. All the sides of the Isosceles upon the right hand are divided into 60 parts with the parallel lines, reckoned by 10. as 10.20.30.40.50.60, beginning at the utter circle a, and so proceeding to k, so that if the Index cut the right side of the Isosceles in the 2. parallel it cuts so many degrees and 20. minutes, etc. But for the left side of the Isosceles the divisions do 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 parallel circled counted from k, it cuts 20. minutes. See the ensuing figures in the folded sheet. Thirdly, The Card. you must provide a Card to be placed within the circle p q, divided into an 120. equal parts, and therein draw a Dial according to the Azimuths of the Sun, for some particular latitude (but that you may omit if you will, because my new fight instrument performs it) the deliniating is performed by the Azimuth as is said: the hours be numbered, and the months wrote at every several circle, wherein you most observe the hours * See the use of the Circumferentor. according to the time of year: and because I will not trouble you over long with the making, behold the figure which I will cause to be printed in a void paper to save a labour in drawing the same. woodcut, circular table, containing degrees, roman numerals, astrological signs etc. The Index. Furthermore to this Instrument there belongeth a circle of mettle equal in Diameter to p q, all which must be cut out, only a narrow limb remaining just to contain the breadth that is intercepted betwixt the circle t u, and p q, the utter part of this limb is to be divided into 29. ½ equal parts, as in the figure, and this circle is to move about the circle p q: at every quarter of this circle there is an Index left of sufficient length to reach over the circumference a b c d in the great figure, and of sufficient strength to bear such sights as are to be placed thereon. See the ensuing figures in the folded sheet. Upon the Index i and h there be placed the two short sights marked with k and l: these two sights have two small groves or channels cut through them for one to look through, and in the top of the sights at the end of these groves there be certain things left like pings heads, as in the figure: these two sights need to be no longer then from i or h to the graovated circle, and they be made to fold down each upon his proper Index: the principal use of these sights is to square land, and to find where the Perpendicular falleth upon the base of any Triangle in the open field, as in the 31. Chap. 5 As for the Demicircle, it is a smooth piece of wood, but rather brass, whose breadth or diameter needeth not to be limited, but is best to agree with the diameter of the Planisphere. Appoint therefore a a centre, and thereupon describe a semicircle r r, something more large than the circle p q in the Planisphere: all within which circle is to be cut out, unless you please to leave sou●● certain artificial branches to hang a plumb neatly at, as in this figure, which will also beautify your Instrument; but they must be so wrought by the workman, that they let not the Semicircle to fold down. Then some two inches or better distant from this circle r r describe another, and according to that circle file all the rest of the plate that is superfluous away even to the said circle. See the precedent figures in the folded sheet. This Demicircle so made, he must be supported with Antics artificially wrought, and the said Semicircle so done must be placed upon the Indices g and f, as hereafter: but first I will teach you to describe the Astronomical circles, and horological arks, as also to project Geometrical lines, etc. upon the said demicircle. As for the Astonomicall circles, though the graduation thereof be after a new order, yet will I not stand to prove the same with Geometrical demonstrations, referring such to the fifteenth book of Ramus, where they shall find, the Circles are as the squares which are made of their diameters, and that their diameters are as the Circumferences. You shall therefore do no more but divide the circumference d c into 90. parts, and so draw parallel 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 have you no supporters at all, and this best. 6 But now for the horological arks, they be more difficult to perform: you shall therefore at each end of the former demi-circle appoint the moiety of the Zodiac, even as you be taught in the 9 place of this Chapter: let the South moiety stand at b r, and the North at o r, then upon the centre a describe the circle the r r, which we call the Equinoctial, and so upon that centre a describe the Tropic of Cancer and Capricorn, which here are both but one circle. Next upon the said centre describe other circles, as the ☉ degree of ♍ or ♐ or any other, as occasion requireth, by placing the one foot in a, and extending the other to the degree in the zodiac assigned. This so done, repair unto the Table of the Suns altitude in the first Book of the Geodeticall staff, Here serves the figure supported with Antics again. & there seek the altitude of the Sun at 12 of the clock, he being in the o degree of Aries or Libra, then finding the like altitude in the Demicircie b c, I place a ruler upon the said degree of altitude, and upon the centre a, and so I make a small prick in the circle of the Equinoctial, where the ruler cuts: this so done. I do the like to the hours of 11 10 9 8 7 and 6 of the clock, for no further do the hours extend in the Equinoctial: these pricks precisely done, and apparently noted, I do the like in the circle of the o degree of ♉ and ♋ so have you three pricks, then must you find the common centre, of each 3: match the pricks as you be taught Chap. 3. Prop. 8. and thereby describe those horological arcs. But now, for that you shall want pricks to describe the hours of 7 and 8 afternoon, and 4 and 5 before noon, you most describe another circle upon the centre a, representing the 30 degree of Taurus, making marks ●or 〈…〉 there, and so proceed as before: so have you finis●●● all the hours belonging to the North moiety of the zodiac, and they will bend towards the left hand. Then for the hour lives answering to the South moiety of the zodiac, look what you did to the Tropic of Cancer, and do the same here unto the Tropic of Capricorn: and as you used the o degree of Taurus, so deal here with the o degree of Scorpio so have you made two pricks to describe your ark by: as for the 3, they be the very same that formerly you made in the Equinoctial: and if points for the striking of the arks of 7 and 8 before noon, and 4 and 5 after noon be wanting, draw other blind parallels from some degree of Libra or Scorpio, or from both, & accordingly find the altitude in the table, and afterwards proceed as before: so have you finished the arks of the South moiety of the zodiac, and they will bend towards your right hand. So having placed a small sight fixed in the 90 degree, the forepart of this Demicircle is finished. 7 Upon the back side this Demicircle is projected the parts of the Geometrical Quadrant, and hysometrical Scale, thus. Take the Diameter of the last Demicircle, which make the Semidiameter of another circle, as a b, now making a a centre, and a b a Semidiameter, strike the Quadrant of a circle b d c, and within that Quadrant inscribe an ordinary Geometrical square, as f d e a, than divides d e and d f, each into 60 equal parts: next in the midst of the live a b make a point at g, on which, as a centre describe the Demicirle a i b equal unto the former Demicircle, now lay a Ruler upon the centre a, and every of those equal parts, both in the line f d and d e, making notes in the arch a i b where the said Ruler touched at every part. mathematical figure corresponding to description These parts so projected unto the circle a i b, you shall upon the back side your Demicircle strike an arch of the bigness of a i b, as m n o, where place all those parts, as they be in the ensuing figure, drawing parallel lines for figures accordingly. Now if you would likewise place the hysometrical Scale hereon also, because there is room sufficient and spare, draw a circle within the circle m n o, as p q, which furnish as the former with parallel lines, and so from every 15 part in the circle m n o, the one and of the ruler fixed on s, produce right lines over the parallel circles, and number them by three, as 3 6 9 ending in 12, just at 60 in the first circle: now divide every of those parts into three other parts, by pulling right lines from every 5 part in the said circled m n o, and then write umbra recta and umbra versa, as you may best perceive in the ensuing figure: this done, your Demicircle is finished. mathematical figure corresponding to description 8 This Semicircle so finished and supported (as before) with Antics, he must be placed upon two of the Indices in such sort, that the one Antic stand a● g, the other at f, bearing the foresaid Demicircle up over the Box and the Needle, provided that he may fold down at pleasure, with certain haspes or buttons likewise to fix him upright at pleasure, the Diameter thereof standing parallel with the fiducial edge of the Index g and f. 9 In the midst of this Demicircle under 12 in the hysometrical Scale may you fix a plumb, and over the Boar with the Needle a certain point just under 12, which will serve to keep the Instrument parallel and upright, which the cross Needle will as well do, but both are not amiss: the old song is, Two strings are good to one bow. figure corresponding to description Some reasonable distance, as an inch and better, draw a line b m parallel to h g, whereunto draw 11 parallels at such distance as they be in the figure, wherein must be placed the degr. figures and characters of the 12 Signs 〈◊〉 as in the demonstration they be: every parallel is divided as g k is, by placing the one foot of your compass in f, and so fetching each degree from the line g f to the other parallel, and at the ending of every third degree the line is struck quite through, so that there be two lines parallel to g m, and k l strooke quite through, and these lines do limit the beginning and ending of every sign. You must also note the South and North signs 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 piece of the movable Sight so finished, there must be another piece of like quantity soldered thereunto, or l●ft growing unto the same piece, and after bended in such sort that it may clasp over the Demicircle, so do these two pieces hold the said Demicircle straightly betwixt the same, that it may move straightly and equally along the same; in so much that the arch h b will always be carried upon the Circumference b c in the Demi-circle. C g and g h do represent the distance of the two pieces one from the other, which is the just thickness of the Demicircle. Neither would it be amiss to have a small screw pin upon the back or further side of this movable Sight, which would make the said Sight move the more steady. 10 The next thing pertinent unto this Instrument, is a Box to hold the Needle. The Box. The Circumference of this Box must agree with the circle p q in the great figure, for within that ciccle must he stand, the Diameter whereof must be 3 inches; and in this Box must be placed a Needle and a glass, as the order is and the Card in the bottom which I described before, 120 standing in the South, 60 in the North, 90 in the East, and 30 in the West. About this Box must move the Circle that bears 4 Indices with the Sights, the which Box must be turned with certain shouldering to come half a quarter of an inch upon the said circle, to the end that it may keep the same down close to the body of the instrument, and that he may move steadfastly about. This Box is to be fastened through the back 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 upon the back side, may also screw this Box by fastening a rib of Brass upon the bottom of the box, with screw holes answering to the holes in the socket. figure corresponding to description Upon the Box above the glass stands a certain crooked wire. bearing a roun● knob in the midst just over the areltree that bears the Needle, and just under the plumb when the Instrument stands upright. The Needle. 11 The next thing is a Needle, which must be provided in manner following. As for the Needle, I would have it made like two Needles joined together at right angles, as you may see in the ensuing figure, and you shall fl●d it hereby more true and apt to work then the single Needle is, for it will keep the instrument parallel and upright without the help of the plumb, cut the degree 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 choice of one thus: The best stones be those that come from the coasts of China and Bengalia, the colour whereof is like to iron, or somewhat sanguine, if they be right, they will draw up their own weight: they be heavier than other: there is another near as good, which cometh from Arabia, they be broad like a tyle-stone and somewhat red coloured. If the Magnes stone have lost his virtue, throw it into the fire, and let it lie there until it be near red hot, and then quench it in the oil of Crocus Martis, so shall his power be multiplied. Your stone thus ordered you shall make clean 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 have a care in this point. figure depicting points of a compass After you have touched the end of the needle, if it were equiballanced before you shall find the same end to hang downwards, as it were the heavier, whereby the unskilful spoil many needles: and this is called the Declination of the needle under the Horizon, therefore let the end that shall not be touched be the heavier before you use the stone, and after the application of the stone, if it be too heavy, you may amend the same. The needle so touched, the South end thereof will not point just into the South, Magnetical meridian. for that the Magnetical meridian whereto the needle points, and the common meridian wherein the just South stands, differ: for the Magnetical meridian is a great circle, as the other is, and also passing by the Zenith, dividing the Horizon into two equal 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 compass or 11. degr. and 15 minutes, west from our common meridian: and this is the cause that in all portable sun Dial's, the line which the needle standeth over, doth not point just unto the 12 of clock mark, nor lie under our common ineridian. Lastly, prepare a hollow socket of brass with a screw pin, & the socket to be scrwed on, as the order is, so is your Instrument finished only providing a Staff for the same: Portable dials. the three footed staff is best to place it at all heights, and in all places. And one special note you most here observe in the delineating of this instrument, that is, t● have a care that the body of the instrument be just four square, and that the sides of the square lie parallel to the diameter of the circle that is divided into 360. degrees, viz. that two sides apposite lie parallel to the line a c, and the other two opposite sides to the line b d: and if you work by the help of the néed●● beware that no one come about you, but such as you know kind friends, loost, otherwise of purpose they bear a Loadstone about them, which may confound you in your work. CHAP. V To set the parts of the topographical Glass together. Having now finished every part of this instrument, and being ready to set him to work, thus must the parts be joined together. First, upon the centre 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 keep down the circle, in such sort as before is said, and let the Box be furnished with his Needle, Card, & Glass, as in their proper place is taught. Next place the great Circular sight upon the Index f g, and the other sights upon the other Indices: place them artificially as you be taught before. Next put the movable sight upon the demicircle, and screw the socket to the back side: so is this instrument prepared to work as the Theodelitus, topographical instrument, Geometrical Quadrant, or as the Circumferentor, and may serve for the plain Table, as shall follow after the rest. To work as the Theodelitus, and topographical Instrument. CHAP. VI The description of the Theodelitus, and topographical Instrument, with the necessity of reformation thereof. The description of the Theodelitus. THe Theodelitus is an instrument consisting of a Planisphere and an Alhidada: upon the Planisphere there is described a circle, which is divided into 360. degrees, etc. Within this circle there is inscribed a square, which is parted into a certain number of equal parts, which do represent parts of the Geometrical Quadrant, and no more divisions or graduations be in the Planisphere of the Theodelitus. The Alhidada is a strait ruler with a fiducial edge, moving equally and truly upon the centre of the Planisphere, whose length is equal with the diameter of the circle in the Planisphere, upon the two ends of this Index or Alhidada as fixed two folding sights. The description of the Topographical instrument. But if you make this instrument like to that which Master Digges calleth the topographical Instrument, then is there a Box and a Needle placed in the centre of the Planisphere, over which there doth stand a perpendicular whereon is placed a Semicircle, to move up & down upon the perpendicular, and to move about with the Alhidada. This Demicircle is divided into twice 90 degrees, both ending in the Semidiameter, which Sediameter standeth upwards, the arch hanging towards the Planisphere: within this Semicircle is described the Geometrical Quadrant, which serveth for height whose parts, and also the degrees of the Demicircle be cut by the fiducial edge of the perpendicular: but in this Glass the Diameter of the Demi-circle lieth downwards, and always parallel to the Planisphere, whereas the other is movable: And as in the other Demicircle there be twice 90. degrees, here is but 90 degrees in all, so that they be twice so large as the other. Certainly this Demi-circle without showing further reason, is far surpassing that of M. Digges for divers good respects I might well take occasion to speak of. But, for that happily some will say the Theodelitus before was perfect, and then what needeth this alteration; it was but the Author's particlar conceit, without any necessity at all. To give such satisfaction, I answer, that there was a necessity of alteration as well in the Planisphere, as in the Demicircle. Touching the Planisphere, see how you be taught to attain to the minutes of degr. cut by the Index by help of the Isosceles Triangles, then is the Quadrant projected into a circle, and what commodity have we thereby? marry much more to me both for the needle (whose largeness is required) and other circles needful and pleasant to be added hereunto, as the Mariner's compass and other circles to tell the hour of the night by the Moon. And lastly, touching the alteration of the Planisphere, look unto the 4 Indices how requisite they be 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 Demi-circle, let every man acknowledge the necessity thereof, for that you could not take any attitude of the Sun, or any other celestial body or object situate in the heavens or upon the earth, if so the altitude thereof exceeded 60 degrees, for that you cannot look through the sight in the Diameter of the Semicircle by reason of the Planisphere, but in deed the same might be better dealt with then any other thing whatsoever: look also to she want of a plumb to keep your instrument parallel and upright for which there was no convenient place in the Theodelitus, which this Glass hath sufficient, though it be not needful, by reason of the large cross Needle which performs the same. Briefly, these impediments and defects well considered, let any ●●e of judgement spe●● i● there were not a necessity of reformation, which 〈◊〉 it is here done: so I know it by practice and continual experience to be requisite to be done (without offence be 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 we shall have no need of the needle, or such like: if you seek the proportion of a field or such like, go into that part of it, from which you may observe all the angles (and it were convenient if white papers were fixed in every angle) and there plant your instrument, beginning at what angle you please, place there for the Index that bears the circular sight, upon the o degree of the circle in the Planisphere, remaining there, erect the same by looking through the sights unto the first angle, than the instrument resting, conucy the said Index to the next angle upon your right hand, and note down what degree the fiducial edge of your Index doth cut in your table book. do so from angle to angle rightwards until you come unto the last, and write them down in your table book, in manner as followeth: the instrument resting unremooved, convey your Index towards your right hand at pleasure, observing through that sights some mark, a convenient distand from you, according to the quantiy of the field, and there must be your second place, At this place, plant your instrument by help of your back sight, in such order that the line where the degrees take beginning may point to your first station. And here likewise you must begin at the angle which at the last station was your first angle And note the degree cut by the same Index proceeding from angle to angle rightwards until you come to the last, still noting the quantity of each angle done as before. These angles thus observed at both stations resort unto some plain and smooth peace of velum or paper, and there describe a circle, and by help of a protractor (but indeed the cord divisions upon my Staff be most excellent) limit out every several angle in the circumference of the circle, and by those marks from the centre of the instrument draw lines infinitely, next protract the line directing to the second station, which properly may be called the stationary angle: upon this directing line describe another circle, as far off, or as near to the other as ye list, and upon this circle protract the angles of position, observed at the second station. Now see where the lines meet, or a like toucheth his like, so do the intersections of like lines limit the true proportion. And to get the distance, divide the stationary line, or line intercepted betwixt the centre of the two circles, into as many equal parts as you please, and with those very parts divide the lines intercepted betwixt those places whose distance is required. Now must ye multiply the parts included betwixt any two sections in the known distance, contained in the stationary line, and then divide by the number of equal parts contained betwixt the first and second station, so have you the distance required. Example. We will take Master Digges his own example, a b c are the marks in the field to be measured, d the first station, where you shall set the centre of your instrument, his Diameter or line where the divisions take beginning pointing directly to a, so do e f g the visual lines running by the angles of position of the instrument unto all the angles or marks observed, express my observations 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: Deg. o Angles observed at my first Station. 20 The stationary angle e d h 90. deg.. 40 Then going to the second station m, where ye shall now place the centre of your instrument, the line where the degrees take beginning, pointing just from m to d: so do the visual lines i k l, running to the marks before noted, cut new angles of position, which you must collect as before in a Table, thus: Deg. 55 I Angles of position collected at my second station 74 K 85 L Now if ye mark diligently where each match lines do cross 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 have the bounds and limits thereof. And if the distance betwixt any two places or marks be required, seek the space betwixt any two places or marks be required, seek the space betwixt the two stations d m, whicg as I formerly said is 300. yards, I divide d m into 18. equal parts, and demand the distance betiwxt a b, which contains 11 of those 18 parts. Then seeing I am ignorant what number of yards be contained in those 11 parts, I fly to the rule of proportion, saying mathematical figure corresponding to description thus, if 18 yield 300 yards, what shall 10 yeld, 183 and 2/4 old ⅓ therefore multiply 300 by 11, so have you 3300, which divide by 18, so have you 183 2/6 which is 1/● making a foot. Whereby I may conclude that between a and b is contained 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 encompassed with some River, or of any piece of ground subject to the sight, that lieth in such order that you cannot have access unto the same, by reason of Marshes, Fens, 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 be a piece of ground shut up within a Marsh or River in such sort that you cannot approach to the same to measure it, as the common ●●der is: you shall therefore seek out some such place as g, far without the said péete of ground, from whence you may view all the angles and corners therein, and there, as at g, observe all the angles, as you did at the first station in the last Chapter, and so seek one a second station, as h, and thence observe all these angles, as the order is: and as you may plainly perceive by the concourse 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 need not many words and therefore proceed therein, as you did in the last chapter: so shall a f, f c, e d, d c, c b, and b a, be the true bounds of the field. To do this Proposition without calculation. Appoint your first station g in a known distant from your second station h, as ten score, and when you come to protract, do not set the said two stallions g●and h down at random, as I taught you in the third chapter, but appoint their distance by your stolen according to the true measure ten score or 200. yards, even as you found it in the field then protract the angles at both stations, and note their intersection as you be wont, which done, if you desire the distance of any angles or corners, as of f e, apply the length of that time unto the scale that you set g h by: so shall you find f e 12. score. In the same order, without Arithmetic, 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 meet 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 cross line over the field for the casting up of the contents, as f d, f c, f b, or a d, a c, etc. mathematical figure corresponding to description In the same order may you perform this Chapter by the Geodeticall Staff, and by other instruments which were over tedious to repeat in the use of every instrument: and therefore I am to supply that in one, which is wanting in another, which being known you may use 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 use it in plaiting of fields, repair into some place whence you may observe all the angles, and towards the first angle upon the left hand direct the Index being placed upon the Diameter, where the degrees do take beginning, the Planisphere, or body of the Instrument resting, convey the Index from angle to angle until you have gone round, and then measure the sides containing every angle, noting the same down against the proper angle, even as you be taught in the sixth book of the Geodeticall Staff, in the third Chapter, treating of this Proposition: and then protract as there you be instructed, or as in the 27. Chapter. mathematical figure corresponding to description CHAP. X. To take a plat of Wood-ground by going round about the Circumference. mathematical figure corresponding to description Now look how you observe the angles and lines in the field, and in the same order must you protract them upon your paper. To use a Needle in the Theodelitus. And here note, that as you work by the back sight with the Theodelitus, so also may you use the Needle, by keeping the needle at every station, just over one place, and then noting the number of degrees cut, and so going round, the work is finished: so that you may hereby perceive the Circumferentor to be borrowed from this instrument, and used by a contrary application. CHAP. XI. To draw the plat of a Country, and thereby to make a true Map, and situate every Town and Village according to their true distance, that you may know the true distance without Arithmetic. To make a Map. TO perform 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 lieth adiacient round about in your Horizon, we hold the Semidiamer of the Horizon to contain 180. Stadias', and so far may one see: for we must always when we be at this work, imagine ourselves to be in the centre of the Horizon, and thence necessarily must see to the Circumference, The Semidiameter of the Horizon. which is limited from us by the Semidiameter. But to proceed, your place being appointed, there set up the topographical Glass upon his staff, ordering it in such sort, by help of the Needle, that the four Semidiameters thereof may point just East, West, North and South, every one pointing correspondantly into his like quarter of the heaven: then turn the Index with the sights to every Town, Village, or Haven, or whatsoever you desire to place in the Map, espying through the sights, the middle, or most notable mark in every of them, See Chap. 28. as commonly the Steeple, if it be a Church Town, noting at every 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 mountain or lofty place from whence again you may view all these places and appoint that to be your second station then turn thereto the Index & note also the degree cut, this done repair unto your second station formerly found, where situate your Topographical Glass in all respects as he was at the first station, turning the Index and sight about, still observing all such marks you saw before, and note again the degree cut, or angles of position, writing the name of every place and his angle by it, so have you collected a second table, which is for your second station. These things so done take a skin of velum, royal paper, or what you please, and in some place thereof appoint your first station, about which describe a circle which you must divide into 360 degrees, beginning in that quarter of the world in which the beginning of the degree in your Instrument were placed, or else protracting them by your Staff, as you be taught, Lib. 6. of the Geodeticall Staff. betaking either of the ways from the centre of this circle, to every degree noted in your first table, there must right lines be produced infinitely, noting to every of them the name of his place, now protract the line of your second station, according to the degree cut, and upon that line describe another circle, which use in all respects as you did the former, taking direction from the table, observed at your second station. To conclude, diligently note the concourse or intersection of every like lines making there on some mark, as ☉ with the name of the place correspondent, and so you have finished. Now to know how far every of these towns &c. be distant from other, do thus: measure the distance betwixt your stations by your Geodeticall Staff, or this Instrument, as you shall be after taught, or by any other Instrument, to you seeming best, and divide your stationary line, or line included betwxit the centre of the circles into so many equal parts as there be miles, furlongs, or scores, between your stations, this line so divided, measure the distance of any place thereby, as you do with an ordinary Scale in a map, taking the distance of any two places with your compass, and applying the wideness to the divided line, for so many equal parts as be then included betwixt the feet of your compass, so many miles, scores etc. is it between the two places according to the denomination of the divisions in the stationary line. Example. I am desirous to set down certain towns in the County of salop, according to their true porportion and the epact distance of every 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, removing my Index to the first town upon my right hand, and nearest to the beginning of the degree in my instrument, I find the same to be Hopton Castle, which having received through my sight, the fiducial edge of the Index cuts 18 degrees, removing the Index to the next town Montgomery it cuts 70 degrees, again the next Knookin Castle, it cuts the 134 degrees, and so I proceed rightwards from town to town, until I have finished so much as my intent was, whereof I gather a Table as followeth: Degrees Hopton Castle 18 ½ Mont-gomery 70 Knookin Castle 134 ½ Whit Church 170 Shrewsbury 171 Angles of position observed at my first station. Morton Corbet 181 ½ Browne-clee hill 289 Bewdley 290 Hopton 313 ½ Tenbury 319 Ludlow 332 Bridg-north 340 This done, I behold an other high hill as the Wrekin hill from whence I may observe 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 upon the Cordoke, I turn again my Index to the first town 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 perceive in the table ensuing. SALOPIA Milliaria AnG mathematical figure corresponding to description Degrees Angle's of position at the second Station Ludlow 1 ½ Hopton Castle 25 Montgomery 52 ½ Shrewsbury 81 Knookin Castle 92 Morton Corbet 136 Stationary Angle is 213 ½ Whit Church 147 ½ Bridg-north 319 Beawdley. C. Wigor. 319 Tenbury 344 Browne-clee hill 349 Hopton 354 With these tables repair unto some such place whereon you would protract the work, drawing therein a circle upon the centre or point f as you see in the figure, which you must divide into 360 degrees, or else by a protractor from f, pull out right lines by every grave, noted in the first Table, so is f p Hopton Castle, 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 centre I describe an other such circle as before (and note the larger the circle is, the better it is) I did upon f, and from this centre g pull strait 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 mark thus ☉, & thus prosecuting the like in the rest, always setting a mark, upon the concourse of correspondent right lines (all other intersections not respected) I have situated all these places in due proportion, noting them with these letters, to avoid (here and else where) often repetition of their names. And now lastly to get the distance between every of them divide the line f g into 9 equal parts, for so many miles by mensuration I find between my two stations, the Cordocke for the Wrekin, then by my compass, I see how many of these 9 parts is contained betwixt any two places, whose distance is required: & so many miles may you conclude the distance of those two places. If I have described places both without the County of Salop as Montgomery and Bewdley, and without the compass of our Horizon, as Whitchurch, etc. They were set down because you should have plenty of examples not thrust together. Hear followeth for more liveliness the distance of every place in this map from the town of Salop: the rest you may gather by your Scale in the same manner. mile's Bridgnorth AH is distant from Shrewsbury 10¾ Bewdley AI 22 Browne-clee hill 11½ Tenbury AK 20¾ Hopton ALL 15¾ Ludlow AM 28½ mile's Hopton Cast. AP distant from Shrewsbury 15 Montgomery A 12 Knookin AD 8 Whit Church AC 12½T Mort. Corbet AB 6 Orleton Bish. C. 10¾ In the order before set down changing your stations (as having finished all in view from the Cordocke and Wrekin) you may go to the Brown Clee and Stilterstone hill, or any other, and passing from one lofty place to another, you may have the true proportion of all Towns, Castles, Rivers, Hills, and such like in the whole kingdom, and to reduce them all into the body of one Card or Map, you must seek a scale proportionable to the quantity of the paper you will draw the map in, which here, for that I fear I have been over-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 draw the plat of any Region, and thereby to find the distance of Towns and such like by cynical supputation. THis kind of work, To seek the proportion of countries, by cynical supputation. although it be something more tedious and difficult than the former, yet hath it in itself a most exact and certain operation: you must in performance hereof ascend the top of some high mountain, hill, or such like, whence you may directly behold all the adjacent towns within the circuit of that Horizon, and also from that hill espy some other mountain, to whose sumunity the view of all the foresaid adjacent towns be subject. This so done, make the first hill a centre, and the other a term, of one of the sides of every angle, and so with your Instrument by the 25 Chap. or any other Instrument take the true quantity of the angle that every town maketh with these two hills, and note the same down in some Table book: this so done, get to the next hill, and there again observe in like manner the quantity of every angle even upon this hill as you did upon the fornier: finally, get the true distance betwixt the top of the two hills, so have you a line known and two angles known situate at the ends of a line known, whereby get the other angle with the two lines unknown, and then place every town in his due place, as you shall be better taught in the Example. Example. Suppose I ascending to the top of Stretton hills (which be certain lofty mountains in Salop) might view all the adjacent towns set down in the ensuing map, and withal another hill called the Wrekin, from whose top also I might well command the view of all the foresaid towns. Now first I place my topographical Glass at a, and then viewing round about I see my eye apprehends Shrewsbury situate upon the left hand, therefore I observe 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 quantity of each several angle, as followeth, a b being always the one side. Grad. Mi. Shrewsbury GAB 46 0 Oswestree FAB 74 0 Angles observed at Stretton hills. Welsh-pole EAB 108 0 Mont-gomery DAB 124 0 Clun CAB 168 0 These angles so observed and noted, I bear my Glass to the Wrekin hill, where planting the same, making the first degree in the Periphere of the Planisphere point just to a, the Instrument so fixed, viewing about I espy 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 proceed rightwards until I have finished, as I did at b, and thereby do I collect a table as followeth. Deg. Mi. Clun CBA 4 0 Mont-gomery DBA 24 15 Angles observed upon the Wrekin. Welsh-pole EBA 38 0 Shrewsbury GBA 64 0 Oswestree FBA 74 0 Now must I get the distance betwixt the hills of Stretton and the Wrekin, which you shall find to be ten miles, all these things had, I get the distance of every town, and place the same accordingly in the map thus. SALOPIA woodcut, labeled To seek the distance of towns sinically. Suppose we would find how far Shrewsbury and each o other towns is distant from the Wrekin, or from Stretton hills, by the former observations, the angle g a b is 46 degrees, & g b a 60 degrees: therefore by the 2 Book, Chap. 15 of the Geodeticall Staff, add 46 and 64 together, so have you 110, which taken from 180 leave 70, the quantity of the angle a g b, now having each angle, find the right sign thereof, as in the 7 Book of the Staff, so shall you see the right sign 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, do thus, multiply the sign of g b a or g a b by 10, and part the product by the sign of a g b, so have 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 sign of g a b 71933 by 10, and there is made 719330 which I part by the sign of a g b, viz. by 93969, so have I the quotient, 76/9 1/3 5/9 4/6 7/9 miles, the distance of the Wrekin hill from Shrewsbury. The like must you do to get the distance of a g, But to avoid division, work a● in the 7. book fol. 287. or chapped. 32. Compendium 3. in the end thereof. a f, a e etc. or d f, d e, b d, etc. remembering always to multiply the sign of the angle, containing the line sought, by the line known, and divide the product by the line of the angle containing the said known line. And for your better understanding, I will set down every Triangle with his respondent sign, so that you may find every side of the same. Stretton, Wrekin, Shrewsbury. Grad. Mi. Signs. GBA. 64 0 89879 The Angles. GAB. 46 0 71933 AGB. 70 0 93969 Stretton, Wrekin, Oswestree. Grad. Mi. Signs. FAB. 74 0 96126 The Angles. FBA. 74 0 96126 AFB. 32 0 52991 Stretton, Wrekin, Welsh-Pole. Grad. Mi. Signs. EBA. 108 0 95105 The Angles EBA. 38 0 78801 AEB. 34 0 55919 Stretton, Wrekin, Mont-Gomery. Grad. Mi. Signs. DBA. 24 15 41071 The Angles. DAB. 124 0 82903 ADB. 31 45 52621 Stretton, Wrekin, Clun. Grad. Mi. Signs. CAB. 170 0 98480 The Angles. CBA. 4 0 6575 ACB. 6 0 10452 To place towns in a Map truly. Having by this Cynical doctrine obtained the distance of every Town, as well from Stretton hills as from the Wrekin, according as you did Shrewsbury from the Wrekin, you shall place them proportionally in one card thus; SALOPIA woodcut, labeled We will only situate Shrewsbury in true place, proportion, & Symmetry, you shall therefore draw a line a b, which divide into so many parts as there be miles betwixt the Wrekin & Stretton hills, viz. ten miles, and according unto those parts you must make a scale as long as you please, as h i. Now place the one foot of your compass 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 compass resting at that distance, place the one foot in b, and with the other strike the portion of an arch: do so with the distance of Stretton hills from Shrewsbury upon the point a, and the conclusion will be, that the intersection of those two arches appoints the true place of Shrewsbury, as g. In like manner must you situate all the other Towns in their proper places, and then it rests at your pleasure whether you will find the distance of each one from the other, by Synical supputation, or by your new made Scale, with your compass for any three Towns not lying in one direct line, make a triangle, and so find the angles of that triangle, next, the signs, and consequently, the sides, as you may see in c d b, etc. but having placed the Towns, the application of the Scale is most speedy and ready, without more trouble to find the distance of any places. CHAP. XIII. The ground and reason of the Geometrical Quadrant, and hypsometrical Scale. BY this topographical Glass I shall teach you to deliver Altitudes, Longitudes, Latitudes, etc. 3 kinds of ways, as by the Geometrical quadrant, hypsometrical Scale, and by protraction, and because this Quadrant is used by many, and also contrived in some Instrument: I thought it not much to spend some time in acquainting you with the ground thereof: Gemma Frisius, Orontius, etc. writing of the use thereof, conceal that to themselves, but having occasion in this book (because it is projected upon my Glass) to speak of the use, I will likewise take occasion to acquaint you with the reason of the work, in a brief manner. Behold the ensuing figure, for the sites of the square s k and k l, (whereof the one is umbra recta, the other umbra versa) are no other thing then the Tangents of lesser circles in the semiquadrant. Therefore if you say. As a l the whole Seal, is to l r the equal parts of the contrary shadow: so is a c the distance, to c b the Altitude. Which is no other than 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 arks may suffice, because there is the same proportion of the Tangent to the Radius, which the Radius hath to the Tangent of the complement whereupon these consequences may be inferred. woodcut, mathematical figure corresponding to description 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. TWO 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 which p (to whom r l is equal) is to p d, so is d o, to o x, for a distance being got by the help of two stations, than 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 which 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 known, that accordingly you may make choice of the dis●drence of the Tangents at the first and second station, that is, whether those Tangents be visual lines of the angles, as which 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. TWO 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 conveniunt uni tertio, etiam inter se conveniunt, b p. CHAP. XIIII. To get the length or distance of any place from you how far soever it be. THe distance of any mark from you may be gotten by this Topographical Glass divers wa●es, the first way I will deliver is by signs, which also may as well be done by the Geodeticall Staff, or any other Instrument, truly explessing the quantity of an angle, and for your more ease in work, appoint the angle at your first station to be a right angle, which is as easy to be 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 degrees take beginning to point just to the Castle, the Planisphere so resting, move the Index to 90 degrees, and so through the said fights espy some mark 100 yards or more distant from you, or wanting a tree, cause one to place a mark by the directions of the sights, in a known distance from you, then take up your Instrument, and leaving one where he was planted, place him again at the second station, and then observe the angle betwixt your first station & the castle, which note down, and so find the signs and consequently the distance, for as the Radius is to the Tangent, so is the side given to the side sought. Example. woodcut, mathematical figure corresponding to description And if the distance of c a be required, multiply 211273 the secant of the angle b c a, by 73, so have you 15422929, which parted by the total sign, leaveth 154 22929/100000 perches your desire, and thus must you deal with any other like question. CHAP. XV. To seek the distance of any mark seen before you, by the Geometrical Quadrant. YOu must call to mind that the Geometrical Quadrant is projected upon the Planisphereof your instrument: therefore, place the Index upon that Diameter where the parts of the quadrant take beginning upon the left hand, then plant your instrument at your second station (for you must note that your observations 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 upon the beginning of those degrees, may just point to your first station, the body of your instrument resting, remove the Index to the mark whose distance is required▪ and whereas in the last Chapter you noted the angle, here only 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 upon your left or right hand. First, if the 60. parts upon the left side the Index be cut, you must increase the stationary line by the number of those parts cut, and the product divide by 60. so is the quotient, your desire, but if the Index fall upon the parts of the scale upon the right hand, you must then multiply your stationary line by 60. and divide by the parts cut. Or you may reduce the parts of the right side to the proportional parts upon the left, See chap. the 19 and so work according to the first rule in this sort: Divide the square of 60. by the parts cut in the right fide of your scale, the quotient is the parts proportional, which you must increase by the the distance of your stations, dividing by 60. so is the quotient the true distance of your mark from your first station. And if the hypothenusal, or distance of your second station from the mark be required, square your stationary line, which add to the square of the distance of your first station from the desired mark, the root quadrature whereof is your demand. Example. woodcut, mathematical figure corresponding to description A is the place whose distance is tequired, b the mark 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 half in the right side of the Geometrical parts. Now the distance of b c is found 73. yards, wherefore I increase 60. by 73. so have 5080. which divide by the parts of the square cut, as by 37. and better, so have you 135. yards, with certain odd parts more, the distance of b a. Or by dividing 3600. the square of 60. by the parts of the Quadrant cut, as 37. and better, the quotient shall produce you a a proportional 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 never cut in the parts of the Quadrant upon the left hand, unless the longitude sought be shorter than your stationary line; I mean, unless the line known, or measured, be longer than the line sought. And further note, that the more parts you divide the sides of your scale into, the easier and truer shall you work. Another way these kinds of longitudes be performed, and that is by protracting after observation of the angles, which for that it is already set down, I omit it here, only you may here protract with a circle readily divided, Lib. 6. chap. 40. Geode. as hereafter you shall be taught, if your staff be wanting. CHAP. XVI. To seek 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 certain Castle and a Fort showed you, to neither of which you could approach, and yet upon some occasion were required to deliver the distance betwixt the same. The first thing in performance hereof that you are to do, is to plant your Glass at the place where you mean to make observation, and then the Index upon the beginning of the degrees, turn the Instrument & all about until you espy the Fort upon your left hand: the Planisphere resting, convey the Index to the other Castle upon the right hand, noting the degrees cut by the fiducial edge of the Index. Now are you to view some other mark for a second station upon your right hand, whereunto turn the Index, until through the sight you espy the same, noting again the degrees cut but by the Index, (and it will be the better to let the Index upon these last degrees be about 90. for the nearer that the stationary line, running from your first station to the Fort upon the left hand be to contain a right angle, the better it is.) Now leaving some apparent mark where the centre of your Glass was, take up your Glass, & bear the same to the place appointed for your second station: and having planted him there, turn the Glass, the Index upon the beginning of the degrees, until through the sight you espy the mark or man left at your first station: the Planisphere resting, convey the Index unto the Fort upon your right hand, noting the degrees cut next to the other Castle or Turret more rightwards, noting the degrees cut: finally, measure the distance betwixt both your stations, all which note down, as well the angel's observed at your first station, as those observed at your second, as also the quantity of the line that is included betwixt both those stations, and protracting the same upon paper, by your Scale and compasses, you have finished. Example. woodcut, mathematical figure corresponding to description Admit I standing at a, am desired to deliver the distance betwixt c a certain Fort, and d a certain Turret, planting my Instrument therefore, as is said, I observe the angle c a d 49. degrees: then c a b 107. These I note down for angles at my foresaid station. Now the Index resting at a, espy through your sights some other mark for your second station, whereunto bring your Glass, as to b, where being duly situate, make the Index standing upon the o. degree in the planisphere, point just to a, the Glass resting so fixed, turn the Index with the sights to c, noting the angle a b c 54. degrees. Next remove the Index to d, noting the angle d b a 104. In conclusion; by your chain, or some other rule in this book, meet the line a b, which is 56. score, which had, protract thus: Draw a line l i, whereupon lay down by your scale and compasses, the 56 score e f, then upon 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. woodcut, mathematical figure corresponding to description If you will perform this by signs work 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 find the distance of each place from your standing, and after use reductions and extractions, so that I hold the Quadrant of himself or as he is here projected (for it is all one to work by either) most fit: so long as you may have always a right angle in your work, and therefore I will apply the same to Altitudes, and concerning this kind of Latitudes, hereby brought to find the distance of ships upon the seas, of Armies upon 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 every object, not unlike unto our year books, wherein are comprised reports of law cases, still noting all such cases of which there is no like precedent or report recorded, and as hereby they make their year books grow to a mighty volume, yet oftentimes riseth there new cases of which they have no precedent, and whether then must they fly, but to the report of the learned judges, experienced in the law: and so in this case, if I should fill a great volume with demonstrations, yet might there be found certain objects so situate that fitly would suit with none of the demonstrations: and what then is to be done, but only fly unto the grounds of the Art, therefore since I cannot suit you, according to the site of every particular plat, my drift is to acquaint you with the grounds of the work, that you may be able of yourself to pick a respondent proposition. CHAP. XVII. To take the Altitude of any accessible Tower Castle, etc. at one station. YOu may seek the Altitude of any perpendicular body, by this topographical Glass 4. kind of ways, that is, by the hypsometrical Scale, Geometrical Quadrant by Sinical working, and by protraction, three of which ways I will here deliver unto you, as for the Scale, you may work according as in the 3 book of my Staff, and first to perform the same by the Geometrical Quadrant, you are to plant the body of your Glass parallel, moving the Index until he point just to the Altitude required, then must you move the sight upon the demicircle until 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 movable sight upon the back side the semicircle, and consider if the section were made amongst those parts that stand nearest 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 nearest to the sight the desired Altitude is greater than your distance from the same so that such proportion, as 60 hath to the parts cut, the like hath the given 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 less than the Longitude assigned, and bears itself in such proportion to the said Altitude as the parts cut do to 60. 3 But if the section be made in 60, than the given Longitude is equal to the proposed Altitude. Example. woodcut, mathematical figure corresponding to description In taking of these altitudes you must note that you meddle with no part of the same, but that which is above the level of your eye, the which level you shall be taught hereafter to observe. Otherwise. Having taken the angle of Altitude, you may work this proposition by signs &c. according as you be largely taught in the 7 book of my Staff called Trigonometria, Prob, 1. Otherwise. Having taken the angle of Altitude you may perform this Chapter by protraction, as you be plainly taught in the sixth book of Geodetia, Chap. 38 whereunto for brevities sake I refer you, and the rather, for that it is performed after one and the same method, only if you please, you may protract with a circle as hereafter. CHAP. XVIII. To search out heights inaccessible, by the topographical Glass. HEights inaccessible be such to whose base we cannot approach, by reason of certain impediments, or to which we dare not go by reason of shot, so that out of this demand many demonstrations might be raised, as to seek the height of a Tower situate upon the further side of a great River or Marsh, or such like, or of a Castle fortified with shot, or such like. All which, and more, hang upon one doctrine, as followeth. Because this chapter is performed by the Quadrant in my 4. book of the Geodeticall Staff, To take all kind of altitudes by the Glass. Chap. 4. and that it differeth nothing here when you have once noted the parts cut, as in the last Chapter, I will refer you thereunto, where in deed look what is there said either in the third or fourth book of the Scale or Quadrant, the same may you perform in the same method by this Glass, when you have once observed the parts of the Scale or Quadrant cut: Therefore it would be more than needeth, here to repeat it again. But because these kind of inaccessible heights be desired of many, I will teach an excellent way. You must find out two stations in a known distance, as you please, where observe the angles of altitude, and so get the complement of the tangents of both those angles by the seventh book of my Staff, noting the difference of the said compliments: for as the difference is to the Radius, so is the difference of the stations to the altitude. Example. We have observed the angle a b d to be 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 total sign by 90. the distance betwixt. both your stations (for so many feet I found it by measure) and there woodcut, mathematical figure corresponding to description is created 9000000. which parted by 78988. the difference of Tangents, so have you 114. feet, the desired altitude. And this kind of work I hold to be most exact, and far more certain than the Quadrant, But with more ease see chap. 32. Compen. 3. for that the one side of the Quadrant here bears 100000. equal parts, which is the more certain by how much the equal parts be more in number. Note the letter d is here in the demonstration omitted. Otherwise. Having observed the angles of altitude, you may perform this Chapter by protraction, in all respects according to the 39 Chap. of my book of Geodetia. CHAP. XIX. To know what part of any Altitude is level with your eye. ADmit g c a Turret, whose height above the level of your eye, is required be therefore diligent to plant your Glass parallel, which having 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 down the woodcut, mathematical figure corresponding to description movable sight to the beginning of the degrees, as to e. Now your sight passing from a by e, will apprehend some place in the Tower, as b. I hereby conclude, that b is equal 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 upwards, as 89. feet, and to have known the height of g c, I must have 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 find the distance betwixt windows, Intteyes, etc. To perform 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 observe the angle of Altitude of the highest, and then of the lowest place, and note the degrees 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 unto the Longitude sought, therefore subtract the lesser Tangent from the greater, the remainder increase by the distance of the Tower, the product whereof, part by the total sign, and the quotient i● your desire. Example. woodcut, mathematical figure corresponding to description B c is a Tower, and you standing at a are required to deliver part of the Altitude, as c h, first therefore I observe the angle h a b, 11 degrees, then c a b 27 degrees, next I measure the distance a b, which I find 178 feet, these had, I take the Tangent of 11 degrees, viz. 19080 from 50952, the Tangent of 27 degrees, the remainder is 31872, which increased by 172 produceth 5481984, which being parted by 100000 the quotient is 54 ½ 1/● 9/● 8/● 4/●● ●●/● feet, the distance of h c, by this means you may describe stagements for buildings, and all such kind of things ingeniously. CHAP. XXI. To reduce the parts of the right side the Geometrical Quadrant, into parts proportional of the left side. IN reducing of these parts you are brought to work always as if it were by the parts of the left side the quadrant, that is, by the parts furthest from 90, which to do, you must divide the square of one of the sides by the parts cut in the right side your scale, that is by the parts nearest to 90, so is the quotient the parts proportional, which must be multiplied by the distance of the stations, and divided by the sides of your Quadrant, let the side of your Quadrant contain 60 parts, whose square is 360, let the parts of the right side cut be 30, 360 divided by 30, leaveth 12 this 12 you must agument by the stationary line, and the offcome part 60 so have you your desire. CHAP. XXII. To find lengths in heights by the Geometrical Quadrant in the Glass. IN performing hereof, you must observe the angle of Altitude made at both the marks in Altitude, whose distance is required, noting the parts of the Geometrical Quadrant cut by the movable sight, and also note if the parts cut at both your observations 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 observation 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 the which remains augment your stationary line, which parted by the whole side as by 60 leaveth 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 proportional and then work 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 divided by the whole side, yields your desire, and so must you deal, if the one section were made in 60 in the midst betwixt the right and left side. Example. woodcut, mathematical figure corresponding to description B c is a Tower, and you required to deliver a certain length in the same, as b z, a. I appoint my station 129 yards from c, now planting the Glass truly, I place my eye at a, taking the angle z a c, and note the part of the Quadrant cut by the movable sight upon the back side the demicircle, which is 20, the Glass resting parallel and equidistant to the Horizon, moving the sight until I see through the same from a to b, as at m, and so do I find the parts cut to be 43. both upon the left side the Scale k l, (which I perceive in my Glass, for that both the sections were made in the 60. parts furthest from 90. Therefore by the first difference of this Chapter, I deduct 20. from 43. so have I 23. remaining, by which your line stationary a c 129. yards, must be augmented, so have I 2967. which I part by 60. so is my quotient 49 2/6 9/6 yards 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 deliver the said distance b z: then must you work according unto the 3. difference of this Chapter. I did prosecute the first difference, with an example, for that neither of the sections will be made in the right parts, unless you stand nearer to the base of the altitude than the length of the altitude itself: that is, unless the altitude be greater than your distance from the base. Otherwise. Get the three angles of the triangle b d c thus, observe the angle b d a, which take from 90. for that b a d is a right angle, so have you a b d: then take the, angle b d c, which add to a b d taking the total from 180, so have you the angle b c d, by the 7.60. of the Geodeticall Staff, 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. book of Trigonometria. woodcut, mathematical figure corresponding to description CHAP. XXIII. To know how much one Hill or Mountain doth exceed an other in height. THis matter is not so easily performed, as many ordinarily think it to be, for to seeks how much one hill is higher than another, is not to stand upon the top of one of the hills, and by your Instrument find whether the other be higher or lower than the level of your eye, as in the 19 Chapter, and so to judge him higher or lower than the hill you be one. But he that will know how much one hill doth exceed another in height, must find how much both of the summities of each hill is distant from the centre of the earth: or at least, how much either of their perpendicular altitudes exceed the Semidiameter of the earth: for the lesser of this excess taken from the greater, leaveth the difference of the mountains perpendicular altitude. For you must imagine, what hill soever you stand upon, beholding another adjacent mountain 20. or 30. miles distant, which albeit that hill you behold be equal in height with that whereon you stand, yet shall he not seem so, nor fall out to be so, being proved with an Instrument, or line of level: for that where you stand is always the uppermost place, the other hill being situate as it were upon the side of the earth, which we may prove by a Philosophical Axiom: Omne grave fertur deorsum ad centrum, ubi quiescit, insomuch that if you travel about the earth with a line and plumb at the end thereof, the plumb will always point towards the centre, so that the excess of any mountains altitude above the superficial convexity of the earth is altered (in respect of our sight) according to the position of the place: and therefore when you be asked concerning the height of two hills, you must know whether they mean in respect of the position of the hills, according to the apprehension of your sight, or in respect of the swelling of the same, above the true convexity of the earth, for you must understand 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 mountains and hills were since made at Noah's flood, by the raging of the water, which forced stones, trees, and earth upon divers heaps, and thereby did irregulate the globous body of the earth, the which albeit being compared to the Spheres in in the heavens (by the consent of most Philosophers) is but as a point having no sensible magnitude, yet to us that inhabit upon the superficies of the same, the very hills have an apparent and great magnitude, elevating themselves above the true circuit of the earth, and as these hills are much above the true superficies of the earth's circular convexily: so is it not to be doubted, but that many valleys fall within, and lower than the said circular convexity, so that sometimes we may be distant more than the earth's semidiameter from the centre, as being upon a mountain, and sometimes less, as in some kind of deep valley, all which you shall be taught to ●●nde and prove by the ensuing demonstration. To perform what is said before, you must ascend unto the top of one of the proposed hills, which let be b, let the other hill be c, and you desired to tell which of the top of the hills is the higher: that is, whether b or c be furthest distant from a the centre of the earth. Being therefore at b, (by some Instrument described in this book) get the angle c b a 87, degrees, then must you go to the hill c, and there again get the angle b c a, 74. degrees, and so adding these two angles together, you have 161. which taken from 180. leaveth 19 the angle c a b made at the centre of the earth. Having these three angles, get their signs, and finally the distance of the two hills c b, so have you a line known, and three angles known, and thereby (as you be often taught) get the side a c, and a b, which note: for, look which is the greater, and that hill may you conclude the higher, which here is c a. Now to get the height of b or c, above the true circular convexity of the earth, subtract 34364/11. from the live b a, so have you the height of the hill b d four score: do so to c a, so (for example sake) have you the height of the hill c e, seven score above the true circular convexity of the earth. If you seek how much the one hill is higher than the other, then take b d 4, from c e 7: so have you three 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 level look towards c, your sight will run to f, above c, so that the lower hill by this means will seem the higher: for you must note that every line of level doth make right angles with the perpendicular, and every perpendicular pointeth to the centre of the earth, as you may perceive in the figure, for b a is a perpendicular, making right angles with f b the line of ●euell. woodcut, mathematical figure of Earth corresponding to description On the other side, standing at c, you shall see the line of level c g run far above b, and by this means you may seek the altitude, and difference of altitudes of hills, which otherwise is difficult to be found. This chapter may finely be performed by your staff, for that you have three angles and one side given. But if you would make experience of the height of hills only by a level, your best way is to find out a third hill alike, or near a like distance from both the other hills, & so may you more truly judge of both their heights, and also of the difference thereof from the top of that hill, even as you be directed by your line of level parallel to the Horizon. CHAP. XXIIII. To know if water will run unto any appointed place. IF you desire to know if water will be brought from any spring head unto any appointed place, you are first to consider how, and in what you mean 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 convey the same again up over the top of any hill, being not higher than the original spring: yea if it be higher, even though you should fetch it at the bottom of an hill, upon the one side, and bring it over the top of the said hill as low upon the other side: for if you once can set it running, it will never cease until 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 quills cut off at both ends, and join them close together with wax in a circular fashion, and put the one end thereof into a vessel that hath water in the bottom, let the other hang over the vessel brims, the lower the better, it sufficeth if so the one end of the quills be as low as the other. Now if with your breath you draw the water into your mouth through these quills, and so take your mouth thence, the water will run through the said quills until all be spent in the vessel: and this experience confirms 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 lie, must be lower at every miles end, by 4 ½ inches than it is at the spring head: which considered, plant your Glass at the spring head, so that the Diameter of the Demicircle lie parallel to the Horizon, and equal in height to the head of the spring, the two sights in the ends of of the Diameter, look through the same to the place whither the water should run, taking notice of what your eye apprehends through the sights, for to that place will the water run, abating 4 ½ inches for every mile the Tower is distant from you. But say there be certain hills betwixt the head of the spring and the place whereto the water should run, in such a case you must plant your Glass at the head of the spring as before, and looking through the sight, note some mark in the next hill towards the place, then go unto that mark, and if yet you cannot see to the place, observe some other mark in an other hill, and so forth, until from the last mark you may perceive the appointed place, in which make a note for abating, as before, the water will run thereunto. Example. woodcut, mathematical figure corresponding to description 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 lie parallel, and also equal in heights to a the spring head, then looking through q p towards b, I cannot see the same by reason of a certain hill, that is betwixt me and b, therefore I observe a mark in that hill through the sight, as at c, where again I plant my Glass in some part of the hill, so that the diameter of the semicircle lie parallel, and in an equal height with the conduit head a: the Instrument thus planted, look again 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 mark in the Tower, as b, which mark is just level with the spring head a, to which place the water may be brought by pipes. Note that some hold pipes of earth baked, to be better than lead, and some hold pipes of alderwood, fir, pine tree, of such wood that hath rosin in it to be better than the former. And if the ground lie reasonable level, so that you convey the water by trenches, order the said trenches so by help of a plumb, that the water may have currant 4 ½ inches at every miles end, then fill the gutter with pebble stones a foot or more deep, and upon them throw earth, so will the water run more clear to the place appointed. Note lastly, that it is best, if you bring your water by pipes, What pipes be best to bear water. to let it come by many crooked turnings, and sometimes to fall directly downwards, and then again to rise by little and little, & by this means some think one may force the waters issue to be above the head of the spring, so that in pipes you shall not need the foresaid abatement. But now whereas in conveyance of waters by this way, whether it be waters to houses or new rivers, sometimes happily you shall meet with a deep valley, out of which you cannot get the water by ditches, and to find level ground you cannot, without going a great compass: and having found it happily cannot have liberty for to cut a trench through the same: such a matter and such a difference I saw in bringing the new river from wards Beware to London, for remedy whereof if the flood be great, you must erect arches in manner of a bridge, 1609 which may extend itself over the valley even from the one bank to the other, and if the water be but for a house, posts may serve to bear the same: the like must you do, if you meet with a river, brook or such like. M. L. doth teach you how to solder your pipes of earth or wood, & that is with unquenched lime and hogs-grease, or with rosin and white of eggs, or with lime, white of eggs, and fyling of Iron. CHAP. XXV. To take the quantity of any stationary angle by the topographical Glass. YOu must put the Index with the sight upon the Diameter in the Planisphere where the degr. do take beginning (noting that a stationary angle, is such an angle that hath no respect to the needle, but to the station) The Instrument then duly planted upon his Staff, you shall move the Planisphere (the Index fixed as before) until you espy through the sight the one mark upon your left hand, and for hedges, until the Index lie parallel with the hedge, the Instrument so resting convey the Index with the sights unto the mark or hedge upon your right hand, making the said Index point to the mark, or lie parallel to the hedge, note then the degrees cut by the fiducial edge of the Index, for that is the quantity of the angle. This needeth no Example. CHAP. XXVI. The making of a Protractor and Scale. TAke a fine thin and smooth piece of brass, of what bigness you please, whereon describe a demicircle, as a b c, upon the centre f which divide into 180 equal parts, and so set figures thereunto, as in the figure: some use to make the like circle upon the other side the plate, numbering therein the degrees from 180 to 360, but that is needless: now by the Diameter of this circle a c, there is made a Scale according to 12 in the inch, furnished with parallel lines, and numbered with figures, as the order is, and as you may see in the figure at d e, upon the back side the plate: there is an other Scale made according to 11 parts in the inch, or you may make the Scale d e according to 16 parts in the inch or more, & then take 12 of those parts, which divide into 11 parts, & make a Scale, the use whereof you may plainly see set down in my Art of Geodetia Chap. 53. but the best Scale is Lib. 5. of the Geodeticall Staff Chap, 2. where I treated of the jacobs' Staff. You must note further, that for angles of position, the one halfa of the circle contains 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. woodcut, mathematical compass CHAP. XXVII. To protract an angle, and lay down the ends thereof. Having learned to take the quantity of an angle by this Glass, it resteth that you also learn how to protract the same, and to lay down the sides thereof upon paper by your ●cale and compass. First, therefore to protract an angle upon any point given you are to take the semidiameter of your protractor, as f a, This is performed with more ease and better in the 6 book of Geodetia Cham 2. and so placing the one foo●e of your compass in g, the point given, 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 degrees. place therefore the one foot in a in the protractor, and extend the other to 50 degrees in the demicircle a b c, the which wideness place in this circular base h i k, so will the two points of your compass 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 degrees. But say, you would draw a line by some degree beyond 180 degr. as by 230 degrees, therefore you must take half 230 degr. & set that distance twice in the circular base, viz. 115 set once there is h k, which set twice there is h k l 230 degrees, or deduct 230 from 360, so have 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 down the length of every side by your Scale according as you found the same by measure, first see what length h g should be, which let be 18 perches, therefore place the on● foot of your compass in d in your Scale extending the other towards c, as to 18, t●at wideness of your comp●sse place in g h, fromwards g towards h as to n, I conclude g n i● 18 perches, do so to g i, g k, and g l, so shall you find g o 12 perches, g k 24, and g p 6 perches, so of any other. woodcut, radiated circle labeled CHAP. XXVIII. To observe an angle of position and what it is, as also to protract the same, and to find upon what point of the compass any thing seen in the Horizon lieth. 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 every angle of position is the Needle, and the 02 other the Alhidada, the common section of the terms of which angle always concur upon the very extrée of the Needle, differing from stationary angles, because they contain the number of degrees, betwixt any two objects, proposed at every particular station assigned, and the angles of position the number of degrees distant from the Meridian. To observe therefore an angle of position, you must place the Needle over his true line in the bottom of the Box, the Planisphere there resting convey the Alhidada with the sights to the mark assigned, noting the degrees cut in the Planisphere, for that is the angle of position, whereby you may see that the angle of position is not limited respectively, according as he is right, acute, or obtuse, but only doth extend itself to any degree in the whole circle, and therefore the terms of these angles might rather be called lines of position, in respect of their situation, and pointing into the several parts of the Horizon. Now to protract the said angles, the difference is not any from the work in the last Chapter, only where you begin 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, upon the protractor signifying South, East, West, North. And if you desire to know upon what point of the compass any thing seen in the Horizon lieth having planted the Glass as before, so that the Needle respect his due place, turn the Index the assigned mark, noting amongst the winds, the wind cut by the Index, for into that part of the Horizon the thing lieth. All these need 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 topographical Glass, otherwise then in the eight Chapter. Such a proposition as this cannot be 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, either by reason of the long distance, or by occasion of hills, 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 do, either by angles stationary or angles of position, but the operation by angles of position is more troublesome and less certain, therefore it shall be here omitted. To perform it therefore by angles stationary, you must go unto one of the corners upon the utter side the field, where plant your Glass, and so take the quantity of that angle by the 23 Chapter, then measure from that angle unto the next angle noting it down together with the first angle in some Table book, then take the quantity of that second angle, and so measure from that angle unto the 3 angle, noting the same orderly down, & so go round about the field, still taking the quantity of every angle by the said 23 Chap. noting the same down together with the quantity of their sides, and when you have finished, upon some plain paper or velum, by the 25 Chapter, protract every several angle, laying down the containing sides thereof by your scale and compass, even as you found the same by measure, and as you be taught in the foresaid 25 Chapter: The Art of Geodetia fol. ●7. this requireth no example, for that the angles being observed the work differeth nought from the 8 Chap. part. 1. of my Art of Geodetia. CHAP. XXX. To plat meadows, plain fields, and pastures of no exceeding great quantity. THis Chapter is best to be performed by planting your Instrument in some such place in the field from whence you may command the view of all the angles in the same field, the quantity of which angel's you must observe by the 23 Chap. measuring the distance of each angle from your Instrument, and so protract the same, and lay their sides down as in the 25 Chapter, and this way I hold the best and truest in all such cases where the proposed field is not over great, and if so be that the plat be not required, but you put to measure the same, your truest work will be by this means, to get the true plat thereof first, and afterwards find 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 method as it is here, I will refer you thereunto for brevities sake. And here note, if you please, having once planted your Planisphere in such order that the Diameter where the degrees do take beginning may point into some one angle or other, you shall not need any more to move the said Planisphere, but only keep him fixed, moving the Index from angle to angle round about, noting the degrees cut by the Alhidada in the Planisphere at every several angle, and so to protract the same by the 25 chapter, These two kinds of ways remembered in this chapter and the last I do hold the best & truest, as for others many of them be uncertain, as the intersection of lines and such like, but you may find that remedied in the use of the Psaine-Table, the which if you can work it there, you need not be to seek here. Many other pretty ways may you find set down in the several use of each Instrument, which if you can perform in one, you may soon perform the same in every Instrument. CHAP. XXXI. To reduce lines hypothenusal into lines horizontal by the topographical Glass. COncerning ascending and descending grounds, I told you in the first part of my Geodetia, that it was not possible by one scale to lay down the true plat, and also by the same scale to render the true contents: for that if the plat be true, the contents will be false: And on the contrary, if the contents be true, the plat will be false, the reason whereof I acquainted you with in the ninth Chapter of the sixth book of the Geodeticall Staff, where I also taught you means to remedy the same, and that divers ways. But forasmuch as we shall not need with this instrument, to reduce these lines in the field but only when we protract, I will therefore deliver a way which you may perform by this Glass, or by your Staff. In the very place where the ascent beginneth, there plant your Glass, and in the top of the ascent, a staff of equal height with your Instrument. Now turn the Index with the sights towards the staff, the planisphere being parallel and equidistant to the Horizon, remove the movable sight in the demicircle until through that sight, and the sight in 90. you see the top of the staff, note then the degrees cut by the movable sight for the angle of ascent, then measure the distance betwixt your instrument and the staff, which increase by the sign of the complement of the angle of ascent, & part the product by the total sign, so have you your desire. For as the Secant is to the Radius, so is the Radius to the sign of the Complement of the said Secant, by the seventh book of the Geodeticall Staff, called Trigonemetria. Example. woodcut, mathematical figure corresponding to description If the grounds descend, as b d, go to b, and then work as before. But you shall not need to make this reducement in the fields, until 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 add the horizontal 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. perches, and rising 30. degrees high, if you will work truly, should be laid down not altogether 35. perches. CHAP. XXXII. Certain compendious Forms of working by my Tables in the seventh book of the Geodeticall Staff, called Trigonometria. BEcause I have taught you to perform many conclusions in this Book by my Tables in the seventh book of my Staff, and for that there were some things omitted, and some things also mistaken in the said book, I thought it not amiss here to deliver certain compendious forms of working by the said Tables: that is, how to work by any Triangle, as you be taught for the avoiding division. Compendium I. If the sign be in the first, and the Radius in the second or third, how to bring the Radius into the first place, for the avoiding of division. Regula. ●or the sign that is in the first place, put the Secant of the Complement thereof; Et voti compos eris. For as the sign is to the Radius, so is the Radius to the secant of the Complement. Compendium II. If the Tangent be in the first place, and the Radius in the second or third place, to reduce the Radius into the first place, for the avoiding of division. 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 sign of the Complement, so shall you have a true proportion, the Radius being in the first place. For as the Secant is to the Radius, so is the Radius to the sign of the Complement. Any other difference that may happen in any kind of obtuse angled triangle is resolved in the manuduction in Trigonometria▪ by making a dislocation of the oblike triangles, and converting any one of them into two right angled triangles which, for that it is briefly set down there, I will here open it with an example. Example. SALOPIA woodcut, mathematical figure corresponding to description CHAP. XXXIII. To square lands, and to reduce irregular plains 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, rejecting the corners, angles, and crooked hedges, which must be after measured, according to the figure they represent. woodcut, mathematical figure corresponding to description Now for the corners and fragments that do remain, you must measure them according unto the figure that they most resemble, even as you may best gather by the pricked lines in the demonstration without any more circumstance of words. But say the irregular field lieth most apt to be reduced into a triangle, which is thus performed. woodcut, mathematical figure corresponding to description CHAP. XXXIIII. To search out the Perpendicular in any Triangle or other Figure, according as it lieth in the open Field. NOw having reduced any plain irregular Poligon into a Triangle, or such other Figure, whose superficial content is found by the help of a perpendicular, and for that the length of the said perpendicular is something difficult to be attained unto the open field, because it is uncertain upon what part of the base your said perpendicular falleth, I have not therefore thought it much here to deliver you the order how to perform the same. Let a b c be a certain plain field, which you are put to measure: the question is to find in what part of the base b c a perpendicular would happen, falling from the angle a. First therefore I plant my Glass in the line b c, as near as I can guess under the angle a, as at d, than I move the Indices about, until woodcut, mathematical figure corresponding to description the Index with the shorter fights lie directly over the line b c, then I look through the sights in the Demicircle, if then the visual b●●mes run 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 have removed the Instrument, as I had seen occasion o● the like must you do in any other figure in the like case, whereby you may see how necessary the four Indices be, for the false taking of perpendiculars is a chief occasion of those palpable errors that be daily committed, and a principal cause wherefore the common practisers so often differ. CHAP. XXXV. To reduce many plaits, or all your observations into one, and thereby to make a fair Map thereof, according to the quantity assigned. BRriefly▪ to teach you to perform this chapter, you must first appoint the Card of the bigness that you intent to make your Map, crossing the same with two lines at right angles, & that about the midst of your Map, writing at the ends thereof, East, West, North, and South. Now you are to seek in your Tables gathered at your observations the greatest distance betwixt the most Eastern and Western place, and also the greatest distance betwixt the most Northern and Southernely place, and so accordingly choose you a Scale, that those places being laid down by the same, may fall within your Card. ANGLIA woodcut, labeled map of England Example. Letoy the proposition be to describe England, and therein to situate such towns as shall be required in that bigness, as is here set down according to my Scale: the first thing that I do, having appointed my Card, I cross the same with two right lines at right angles, appointing at the end thereof the four quarters of the world, as in the Type: then I find out some town that I conjecture (by conference with my tables) lieth about the midst of the land, which let (for examples sake) be Middle Wiche, a town situate in Cheshire, the which town I place upon the very intersection of the foresaid lines, and thereby write the name of the same, searching in my tables for some other town that lieth direct East, West, North, or South; I find none: therefore I take some other town, as Bristol, drawing a line according to the position of the same, getting also from your tables the distance of that town from Middle Wiche, as 97 miles, to which wideness upon your Scale open the feet of your compass, and then place that wideness upon the line of position for Bristol, placing the one foot in the mark made for Middle Wiche, making a note with the other in the said line of position, where writ Bristol. These two towns so placed, let us now go situate Northampton: first by my tables I find the distance of Middle Wiche from Northampton, and according to that distance, the one foot resting in the mark for Middle Wiche, with the other I strike the portion of any arch: the like I do with the distance of Northampton from Bristol, as in the 12 Chapter is plain. Now the intersection of these two arches is the true place of Northampton, where make a mark for the town, and write the name thereof by the said mark, and so proceed, limiting all the towns, ports, angles, and nooks in the Island in their proper places, as you may sufficiently gather by the former demonstration. Having finished, in some void place you may appoint, the Mariner's compass, as in the Card before, and this compass will serve you for many necessary uses, as it is not unknown to men seen that way. CHAP. XXXVI. To divide any Empire, Kingdom, or Continent into Provinces, Regiments, or Shires. WHen you have taken the plat of any country, and therein situated all the towns, ports, and such like, yet happily shall it be expedient (or rather of you required) to separate and distinguish the same into such Provinces, Shires, or Regiments, as the said Kingdom is divided into. So is England, or the South part of great Brittany, being a peninsula, divided into 52 parts (but not equal parts) which we call Shires, then is every Shire subdivided into other certain unequal parts (as Worcester Shire into 12) which be called Hundreds, either for that there were but at first so many towns or villages therein, or for that there is to be required 100 able men in every of them. Other divisions is England yet subject unto; as first, the whole Kingdom is divided into two Provinces or Archbishoprickes, to wit Canterbury and York, than these Provinces are subdivided into bishoprics, and every Bishopric is resubdivided into Parishes, according to which divisions I mind God willing to describe a Map of all England, etc. But now the way how to attain unto these subdivisions is not known. It is therefore to be performed after two kind of ways: the first whereof is, you must in your Perambulation, as in the 11 and 12 Chapter, observe the bounds and limits of each province, etc. even as there you do the towns, and to protract it accordingly, distinguishing the same with certain pricked lines. The other way is, to find by some records what towns and such like be the Meres and bounds of the said Shires, etc. The which having placed in your Map, the division is made by drawing certain lines from town to town, or if your may be small, having made a point for the place of the town, you may omit to write the name thereof, and so draw certain pricked lines from point to point. Even as you may perceive England in the ensuing Card divided into Shires. woodcut, labeled map of England CHAP. XXXVII. The reasons wherefore the Altitude of the Sun hath been hitherto falsely observed, with Tables to reform and correct the same, as well in respect of his refraction as paralax. TYcho Brahe a Dane, and diligent observer of the motions of the celestial bodies, found at last by conferring daily practice and experience with the optikes, that the Sun seemed to us elevated higher upon the vertical circle, than indeed he is, and his reason he draws from Alhazen and Vitello, saying: Quandores visibilis per diversas Diaphanitates spectatur, refracte eius formam visui occurrere, for they appoint the heavens the Elements etc. to be diaphanical, but Tycho would have the principal cause of this refraction to be in the vapours, that do continually occupy the lowest Acry Region, abounding and gathering themselves most together, whe● the Sun is ●●●test to th● Horizon, which elevating themselves by little and 〈◊〉, successively, at length merely vanish and are nothing ●ale ob●●●●●s as in the ensuing Table of the suns re●●●ction the greatest elevation of the vapours, according to the said optics of Alhazen and Vitello is 40/ 50 of which the ●●●●●●ter of the earth conten●●● 〈◊〉 which 〈◊〉 ●●●●ce 12 Ge●maino ●●les, but heaving the ample discourse hereof to the ●●pa●● of T. B. where the desirous may road at large, but us only return unto the practical use thereof, so shall it be ●●●ine by 〈◊〉 of this refraction caused by the thickness of the vapours a●●ue the Horizon, that the sight of the Sun is altered so that he seems to rest sooner and set slower thou indeed 〈◊〉 doth, insomuch that the centre 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 he seem 〈◊〉 ●●ggat at the setting th●● when he is in the 〈◊〉 whereby ●e may 〈◊〉, that by occasi●● of this refrac●●●● this who●● 〈…〉 Horizon a● vising or ●●●ting whe●●●●● there 〈◊〉 not any pa●● thereof above the sa●● and 〈…〉 pontificial, A Table of the suns Refraction. Alt. ☉ Refraction. G / // 0 34 0 1 36 0 2 20 0 3 17 0 4 15 5 14 30 6 13 30 7 12 30 8 11 45 9 10 15 10 10 30 11 9 0 12 9 30 13 8 30 14 8 0 15 7 30 16 7 0 17 6 30 18 5 45 19 5 0 20 4 30 21 4 0 22 3 30 23 3 10 24 2 50 25 2 30 26 2 15 27 2 0 28 1 45 29 1 35 30 1 25 31 1 15 32 1 5 33 0 55 34 0 45 35 0 35 36 0 30 37 0 25 38 0 20 39 0 15 40 0 10 41 0 9 42 0 8 43 0 7 44 0 6 45 0 5 continuing from Sun to Sun, be longer according to the apparent rising and setting thereof, than those our Astronomical calculations for the Poles elevation, But to set apart further discourse hereof, behold the Tablet, he use whereof is this: Seek the Altitude of the Sun in one of the rows under Alti. ☉, answering to which upon the right hand under the title of refraction is the minute and section of the suns refraction, which must be abated from the Altitude instrumentally observed, because the refraction doth always cause the sun to seem higher than indeed he is. Of the Paralaxes of the Sun. IT is evident by the paralaxes of the Sun in the circle of altitude, that the Semidiameter of the earth hath a sensible proportion, in respect of his far distance from the Sun, which (setting apart other reasons) may most plainly be seen in the time of Eclipses, especially of the Moon: by which Eclipses Coperni●us also found that the Semidiameter of the suns excentricity did contain 1142. Semidiameters of the earth, which putting aside some few minutes, agreeth with Tycho Brahe. so that the Sun, by reason of his Paralax, seems to us inhabiting the superficies of the earth lower, and more dejected in the heavens, then in deed he is; by the neglecting whereof his true altitude hath as yet been falsely observed: but here we must admit a thrée-fold distance of the Sun from the earth: to wit, most remote, as being in his Apogaeum, nearest, as being in Perigaeum, and in the mean betwixt both: that is moving about the centre of the earth. According to which thrée-fold distance from the earth, I have set down the ensuing Table by the doctrine of the foresaid Tycho, extended so far, that it may serve for every degree of the suns altitude through all great Britain, the use whereof is thus: Take the altitude of the Sun, the which altitude find in the row under Alt. G. answering to which, under the title Max. Med. or Min. is the minutes & seconds of the suns paralax, which must be added unto the suns altitude, for that the paralax doth make the Sun seem lower, and more dejected in the heavens (in respect of our fight) then indeed he is. A Table of the Paralaxe of the Sun in the vertical circle, according to his threefold distance from the earth. Alt. Max Med Min. G / // / // / // 0 2 54 3 0 3 7 1 2 54 3 0 3 7 2 2 54 3 0 3 7 3 2 54 3 0 3 7 4 2 53 2 59 3 6 5 2 53 2 59 3 6 6 2 53 2 59 3 6 7 2 52 2 58 3 5 8 2 52 2 58 3 5 9 2 51 2 57 3 4 10 2 51 2 57 3 4 11 2 50 2 56 3 3 12 2 50 2 56 3 3 13 2 49 2 55 3 2 14 2 48 2 54 3 1 15 2 48 2 54 3 0 16 2 47 2 53 2 59 17 2 46 2 52 2 58 18 2 46 2 51 2 58 19 2 45 2 50 2 57 20 2 44 2 50 2 56 21 2 43 2 49 2 55 22 2 42 2 48 2 53 23 2 41 2 46 2 52 24 2 40 2 45 2 50 25 2 38 2 44 2 49 26 2 37 2 43 2 47 27 2 35 2 41 2 45 28 2 33 2 39 2 44 29 2 31 2 37 2 43 30 2 30 2 36 2 42 31 2 28 2 34 2 40 32 2 27 2 32 2 38 33 2 25 2 30 2 37 34 2 23 2 29 2 35 35 2 22 2 27 2 33 36 2 20 2 25 2 31 37 2 18 2 23 2 29 38 2 17 2 21 2 27 39 2 15 2 19 2 25 40 2 13 2 18 2 23 41 2 11 2 16 2 21 42 2 0 2 14 2 19 43 2 7 2 12 2 17 44 2 5 2 9 2 15 45 2 3 7 2 12 46 2 1 2 5 2 10 47 1 59 2 3 2 8 48 1 57 2 0 2 5 49 1 55 1 58 2 3 50 1 52 1 56 2 0 51 1 50 1 54 1 58 52 1 47 1 51 1 55 53 1 45 1 48 1 52 54 1 43 1 46 1 50 55 1 40 1 43 1 47 56 1 38 1 41 1 45 57 1 35 1 39 1 42 58 1 33 1 36 1 39 59 1 32 1 33 1 36 60 1 30 1 30 1 33 61 1 25 1 28 1 31 62 1 22 1 25 1 28 63 1 29 1 22 1 25 64 1 29 1 22 1 25 65 1 13 1 16 1 19 66 1 10 1 14 1 17 67 1 80 1 11 1 14 68 1 5 1 8 1 11 69 1 2 1 5 1 8 70 0 59 1 2 1 5 Example. We will take Tychos own example, observed the last of june 1588., when having a large Instrument, prepared for that and other such purposes, found the Sun to be 19 degrees 17 ⅙ minutes, elevated upon the Meridian, to which in the table of refractions there did answer 4./ 5.// which taken from the apparent Altitude (because the refraction doth always make the Sun higher than indeed he is) there doth rest 19 degrees, 12./ 20.// the just Altitude by reason of refraction. But this Altitude is not as yet precise, by reason of the suns parallax, therefore in the Table of parallaxes according to the 19 degree of Altitude, and according to the suns threefold distance from the earth, find the parts there set, to wit, 2 minutes, 44 sec. these add to the former Altitude corrected by the refraction, because the parallax doth make the Sun seem lower than indeed he is, so have 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 observed instrumentally to be 19 degrees 17 ⅙ min. Hence cometh it that the Poles elevation observed by the Meridian altitude of the Sun or Stars, and by the Pole itself differ, if you confer two of these Altitudes together. CHAP. XXXVIII. Of the correcting the taking of the Altitude of the Stars, which by most hitherunto hath been falsely observed. AS the refraction of the Sun is a cause of error in the observing his true Altitude, for the causes aforesaid, so do the like causes produce the like error in the Stars, but not somuch, for to every degree of the suns Altitude the refraction of the Stars decreaseth 4 ½ minutes insomuch that at the 20 degree of Altitude the stars have no refraction at all, as for their parallax it is nothing sensible, therefore behold a Table for the refraction of the stars for every degree of their Altitude unto 20 degree, the use whereof is all one with that of the sun's refraction. A Table of the Refractions of the fixed Stars. Alt. M S Alt. M S Alt. M S Alt. M S 0 30 0 5 10 0 10 5 30 15 3 0 1 21 30 6 9 0 11 5 0 16 2 30 2 15 30 7 8 15 12 4 30 17 2 0 3 12 30 8 8 45 13 4 0 18 1 15 4 11 0 9 6 0 14 3 30 19 0 30 20 0 0 CHAP. XXXIX. To take the Altitude and Almicanther of the Sun or any star, and to find their Azimuth by the topographical Glass. PLant your Instrument parallel the Needle respecting his due place, then move the Index until the very edge of the demicircle point just unto the Sun, now for stars look through the sight in 90 moving the movable sight upon the demicircle, until through the said sight in 90, and this movable sight, you espy the Star, note then the degree cut by the movable sight, for that is the Altitude of the Star, but because you cannot view the Sun, receive his beams through the movable sight, moving the same sight upon the demicircle, until the said beams pierce 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 Sun or Star is the Azimuth or vertical circle. And what may you see further in this Glass? marry the point of the compass that the Sun or the Star lieth on. But you must remember, after you have taken this Altitude of the Sun or Stars instrumentally, to make subtractions and additions according to the refraction and paralax of the degrees of the Altitude taken, even as you be amply instructed in the last Chapter. CHAP. XL. To take the Amplitude of the Sun or any Star by the topographical Glass. BY help of the Needle, place the Planisphere of the Instrument, so that the diameters behold the four quarters of the world, the sun or star then rising, observe the same through the sights upon the demicircle, noting then the degree included betwixt the Index, which pointeth towards the star, and the diameter that respecteth the East, for that is the Amplitude: the like must you do upon the West side for a setting, also amongst the points of the compass, there may you see upon what point of the compass the rising or setting was. CHAP. XLI. To get the hour of the day, the hour of the Sun rising and setting &c. by the topographical Glass. TO get the hour of the day, you are first by the 39 Chapter to observe the Altitude of the sun, the movable sight upon the demicircle resting at the degree of Altitude, note then the degree that the sun is in, for the day proposed, which you shall find upon the movable sight, for the hour line passing by the same is your demand. But with this proviso always, that you note if the sun be in North or south signs, for the hour lines of the North signs incline to the left hand, and the hour lines of the south signs, bend towards the right hand, certainly of all kind of horological Quadrants I hold this the best, easiest and truest, not for that it is a new devise of mine own, but by reason of the exactness of working. Now for the time of the sun rising and setting is easily collected. And to get the degree of the sun, every Almanac affords you that, for towards the midst of each month you have there set down, when the sun entereth into the sign there noted, as in October the 14 day, 1610. Sol in Scorpio, and I demand the 26 day what degree of Scorpio the sun is in, beginning therefore at the 15 day I call it one, and so telling I come unto the 26 day, I end with the number of 12 whereby I conclude the sun was in the 12 degree of Scorpio. So of any other, here the loss of a day doth nothing hinder. CHAP. XLII. To find the hour of the Night by the typographical Glass, and to know the time of high water, and also the place of the Sun or Moon. SEe in any ordinary Sun dial what of the clock the shadow of the Moon yieldeth, then turn the Index that is marked with f unto the said hour in the Planispheare, which so resting, seek the age of the Moon in the circle whereto the Index is fixed, for the hour line in the innermost circle in the Planispheare passing by the said age of the Moon is the true hour of the night. So likewise doth the hour line and the foresaid Index show upon what point of the compass the Sun and Moon then be, and the number of points included betwixt the said hour line and Index acquaints you with the distance of the Sun and Moon, which the circle in the Peripher expresseth in degrees and minutes, which is more than was proposed. To know the tides or high water by the topographical Glass. Seek as hereafter what Rombe or Wind maketh a full sea at the proposed place, and then learn the age of the Moon, these two things had, put the Index where 29 ½ standeth upon the said Rombe or Wind found, which resting, seek the age oh the moon in the movable circle, for the hour in the inward circle of the Planispheare answering thereunto acquaints you with the hour of the full sea in the proposed place, and for your practice and ease behold the table of tides ensuing. The moon south or north maketh a full sea at Lands end, south and by east at the Gore end, south south west between holy Island and Tynemouth, south west and north west between Tynemouth and Flambrough head, south west and by west between Flamb. and Bridlington in the Bay, west south west between Bridligton and Laurenas, east & west between Laur. and Cromer, south east between Cromer and Yarmouth road to Laystow north road▪ south east and by south between Layst. road and Orfordenas, south south east between Orf. & Orewell woods, south & by east between Naase & the Ware head of Colnes', south south west at Gravesend, south west at Lon. bridge, south and by east at Portsmouth, east and west at Waymouth, west and by South along the coasts up to Bristol, and the coast of Ireland, from Waterford to Kynsale: if you desire more, you may have it of any skilful Mariner, or in the tables of the Regiment of the sea. One thing note, that it floweth sooner by one point of the compass in the Spring tides than it doth in any of the quarters of the moon, especially if the River have any indrafte and distance from the Sea. A note of additions to the Planispheare in the Glass. To the Planispheare in this topographical Glass you may also add the Celestial Zodiac, and another circle of the days of the month inclusively, the same or such like that are placed upon the Horizon in Sandersons' Globes, by which you may gather the sign and near the degr. that the Sun and Moon be in, and if you do but note the aspects in the Rundle of the moons age in their proper places, you may thereby find what aspect the sun and moon have one to the other at any time. Or thus you may find what sign the moon is in, place the Index marked with f upon a in the Planispheare where the degrees do take beginning, then count the age of the moon in her proper circle, under which in the Planispheare make a mark, to which mark turn the foresaid Index f, noting the degree cut in the circumference. for that is the distance of the sun and moon, which parted by 30 the quotient yields the number of signs, and the remainder the degrees; so that knowing the place of the sun by any ordinary Almanac, hereby have you also the place of the moon by adding the distance of the sun & moon unto the place of the sun in the Almanac, as in March, after the 10 day the sign is in Aries, and by the rules before I find her distant from the sun 60 degrees or 2 signs: Therefore the moons must be in Taurus, the deg. are known by the deg. that the sun is in, and by the deg. cut by the Indices as before. Finally, if in this utter circle you character the aspects, than also may you find the aspects betwixt the sun and moon. Many things Astronomical might I open in the use of this Glass, which for brevities sake I am forced to omit. At this time I will conclude the use of the topographical Glass, hoping I have said sufficient to open the whole use thereof, which contains matter sit for a great volume. THE DESCRIPTION AND use of the Plain Table, containing all such propositions as are most fit and familiar to be wrought thereon, setting aside others, as pertinent to curious demonstrations, rather than apt to produce exactness and truth. CHAP. XLIII. To use the topographical Glass as the Plain Table. To alter the topographical Glass to a plain Table. YOu must take the circular sight, box, needle, and all things of the foreside the Planisphere of the Glass, and so set the socket that is upon the backside upon the foreside the instrument, so doth the backside (being a four square plain board) stand upwards; next must you cover this smooth board with a sheet of white paper, which fasten thereunto with mouth Glue, or you may have folding Rulers, as the plain table itself, to perform the same. Lastly, have a Ruler with Sights, as in the next Chapter, to stand upon this plain Superficies, and to the one side of the board, in the thickness thereof with screw pins fix the needle and box, in such order that the South line (I mean not the line of variation) make right angles with the side of the said board, so have you finished. CHAP. XLIIII. Of the Plain Table, with a description thereof, and the parts thereunto belonging. The Plain Table. THe Plain Table or Geometrical Table is a right angled aequilater paralelogram, made of a board of half an inch in thickness, whose equal sides contain 9 or 12 inches, the superficies whereof is made smooth and plain, some use to make him represent an Oblong: all is one. Some for ease in carriage use to make this square board to consist of three pieces, which they use to join together with certain ledges, such as be at the end of Table boards, as you may gather by the figure. The edges of this table round about be abated with certain square channels to the thickness of half the board, according as you may gather by the shadowed lines about the table. a "table" with ruled edges The ribs or for rulers of the Plain Table. 2 The pieces of this table being set together, then be there certain rulers of wood joined together with small brass hinges which are made to fold, as you may perceive by this ensuing figure, which being opened and stretched square, serve just to fall into the channel made about the 4 sides of the table: & these rulers in that place have three uses principal; the first is, it ribs and binds the pieces of the table close together; the next, it holds the shéet of paper plain and strait upon the table upon every side: lastly, every side respondently is divided into a certain number of equal parts, which serve to draw cross parallel lines upon the paper when you change your sheet, as hereafter: you may gather my meaning concerning the description thereof by the ensuing figure, better than with many words. woodcut, corrisponding to description Box & Needle. 3 To this Plain Table there doth also belong a Box with a Needle, and a Card in the bottom thereof, such a one as is described in the 1 Chap. This Box must be fixed to one of the sides of the Plain Table a d or b c, in such sort, that the line in the box m n lie parallel thereto, so shall k l lie close with the side of the table a d, then is there an appendix growing to this box, through which go two holes for passage, that two screw pins may fix the said box to the body of the Plain Table: as you may gather amongst the shadowed lines in the ensuing figure. ✚ N ↓ m woodcut, circle divided into four sections 4 To the back side of this Instrument there is fixed a socket of brass, with two screw pins, as you may perceive by this figure, and in the midst of this socket there is a screw pin, which serves to wrest against the Staff that is put into this hollow socket, provided always that you have a thin piece of brass within the socket, for the screw pin, to force against the staff being put therein. 5 Also you must provide a three footed staff, that is 3 legs or feet to be fixed in a head of box like a three footed pair of compasses, & set the head of your staff that goeth into the hollow socket bewrapped in a plate of brass, so shall not the brass pin crush the same being forced here against. woodcut, cylindrical figure woodcut, mathematical figure corresponding to description woodcut, mathematical figure corresponding to description If you make the ruler and sights in brass, they were best to be made to fold up and down, as the order is in such things. Thus much of the description of the Plaine-table and his parts, which I have the rather wrote for their sakes that affect the same, for whose sake also I will something prosecute the use thereof. CHAP. XLV. Of the absurdities that many use, that affect the plain Table, and of reforming many inconveniences therein. Why the plain Table is affected by the vulgar. THis Instrument is so plain (by reason of the ocular spectation of the work still demonstrated before the eye) that it hath thereby begot itself a wonderful affectation of the vulgar, whereby they vainly think no work doth relish well unless it be served upon this plain Table. But this opinion is no less ridiculous than full of many doubts that I have heard divers plain Table men propound. Ignorance of some practisers. Saith one, I think the grounds you teach for the casting up of pieces 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 again 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 that he was a witty fellow that 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 equal sides, and double the offcome, the square root where of is the diagonal, Lib. 6. cap. 16. pro. 1. Bac. Geod. then find the perpendiculars by the 21. Chap●●● of the foresaid 6. Book, and so working according to the doctrine of triangles, you shall find the product of the two triangles to agree with the forms square. But to return to the supplements of the wants in this Instrument. Inconveniences in the plain Table reform. First, the paper of this instrument, by occasion of wet, is oftentimes blotted and blemished, insomuch that the points, lines, and other observations, be in many places taken away, and so by reason of hasty taking of the Table up, the paper is by the wet, so stretched and disfigured, that many errors grow upon a small scale. To avoid all which, and such like, I thought good to give this admonition. First let your table be covered as you be wont with a fair shéet of paper, upon which sheets let there be placed another shéet, well oiled on both sides with Lin-séede oil, so may you work notwithstanding the rain, dew, or mist, only draw the lines something more heavy upon the upper shéet, that they may also pierce the lower apparently. And this thing note further in the Table, that when you take plaits, oftentimes they will not all be contained in one or two sheeets, therefore you must draw a number of parallel lines cross wise at right angles, that thereby you may join the sheeets together truly, which the divisions upon the four rulers performs, and as you measure the circumference of any plat by this Table, oftentimes cast cross angles, & Diagonal lines over the plate, which will keep you the truer to your work: This holds not in hilly ground without reducing of hypo. lines to horizontal. for indeed a small error herein produceth a great absurdity in the closure, and you may now and then try if the chain measure of your Diagonals agree with your scale, which if it do not, is a true argument of error. You must further have a special care in reducing the lines hypothenusal to lines horizontal, as you be instructed in the sixth book. I have seen some plain table man, when he was set to measure a bank ascending round about, go unto the top thereof, and so produce lines from every corner unto one centre, measuring it thereby one station, and he thought it was a rare device, and had laid it down 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 he laid it down in a greater compass than it should, insomuch that if he had laid the plain adjacent sides precisely down by that scale likewise, the want of closure would have contained a reasonable large piece of ground: for you must needs confess that the lines he protracted upon the paper, were visual or horizontal, and the lines measured hipothenusall. Other things should be reform in this Instrument, which at this time I omit. CHAP. XLVI. Things belonging to the use of the Plain Table. Things belonging to the plain Table. TO this instrument, as to all other, appertains a chain or wire line of four perches long, according to 16. foot, and ½, or of three perches long, which is 16. yards, and ½, let the perches be noted with brass rings at the ends thereof, and then divided into halves & quarters, The Line. with lesser rings fixed at each quarter and half, that you may distinguish the same. A Scale & Compass. You must also provide a scale of brass or wood, whether you please, with a pair of brass compasses pointed with steel very neat and sharp: for it is rude to draw your lines Geometrical with Painter's kéelers, or black lead, as M. Lucar would. Also you must have such sights for this Table as be described in the use of the Circumferentor, whose use are set down in the 16. Chapter of the same book: or else you may have 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 work. CHAP. XLVII. To take any horizontal distance by the Plain Table. To take any distance by the Plain Table. IT were vain to make many demonstrations of this work, 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 unapt to perform the same, though M. Lucar have taken pains to illustrate him in that point; howbeit finding by experience the cumbersome and uncertain working thereby, I think it better omitted then remembered. You shall then understand, that you may perform any distance upon this Table in the same order as you do with my Staff, only here you must draw lines upon the paper, and measure the same by your scale, whereas the legs of the Staff represent the lines, and the divisions your scale. Therefore at the place whence the distance is required to any mark 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, whereunto bring the fiducial edge of your rule, keeping the one end of the said ruler upon the point, moving the other until through the grove or sights you espy the mark whose distance is required: the rule so resting, draw a line by the fiducial edge thereof: the Table resting, espy out a second station, & let it make, as near as you may, a right angle with the mark whose distance is required. This mark so appointed out for your second station, keep the fidutiall edge of the rule upon 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 have you finished all at your first station, with this Proviso, that you have regard to the degrees cut by the South end of the Needle in the Card in the bottom of the box before you any wise altar the table, and that you lay down your stationary line by your scale and compasses, limiting the same according to the line measured, & at the end thereof mark another prick, which call the prick of your second station. Then take up your Table, leaving a mark at your first station, under the prick made upon the table representing the same. Now must you bear your instrument to your second station, where having placed the same in such sort that the prick of your second station may directly stand over the mark representing your second station: lay then the edge of your ruler upon the stationary line keeping the prick of your second station next to your body, turning about the table, the ruler resting, as before, till through the sight you espy the mark left at your first station: which done make fast the table with your screw. A proof of the work. Now for proof of the exactness of your work, and to know if you have truly taken your back sight, have respect to the south end of the needle, for if it cut the like dgrées at this second station a● at the first, you have done well. Having so done, place again the fiducial edge of the rule upon this point of your second station, the one end being there fixed, move the other end, until through the sights you see the mark, whose distance is required: then draw a line by the fiducial edge of the rule, which will intercept with the line drawn 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 mean by the same scale you laid down your stationary line) so have you your desire. Example. woodcut, mathematical figure corresponding to description The distance a b is required, first therefore I plant my Table at b, then working as before, I find 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 find it 7●. ya●d●, which I lay down upon 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 again plant the Table, as before: So do I make the stationary line protracted, point just to b, and then noting the degrees cut a gain by the needle, I find them 40. as before, which argues I have well planted my Table. To conclunde, I place the fiducial edge of my rule upon c, moving ●he other end, until it intersect with the line representing a b; therefore by my Scale I measure the line representing b a, so have I the distance of b a 135. yards, by the same Scale might you have expressed c a. CHAP. XLVIII. The part of the distance of any thing being, given, to find the rest. Understanding the last Chapter, so we may thereby avoid many words, and may most easily be performed by the Geodeticall Staff, as may appear in the Propositions of the 18.19 or 32. Chapters of the second book of the Geodeticall Staff. But to proceed, a b is a distance required, the part of that distance given is a c, woodcut, mathematical figure corresponding to description 50. perches. 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 down my scale, upon the same line the part given. representing a c 50. perches, the instrument unremoved, I seek a second station, as in the last chapter, which is d, (but the stationary line shall not be measured.) Lastly, I note the degree cut by the South end of my needle, then leaving one at a I carry my Instrument to d, where I plant him in an respects as at a, now must I find the point upon the paper, which represented c, and thereupon lay the fiducial edge of the rule, moving the other end until through the sights you se● c, so wilt the edge of the ruler in the line upon the paper representing ● d, th●● k●●ping the ●uler ●pon that point d, I move the other 〈◊〉 until it p●●o● to 〈◊〉 shall the fiducial edge of the rule intersect● wit● the ●●●e 〈◊〉 the pap●● representing a b, from the point of which intersection to the point a, is b the terms of the line a b, which being measured by your scale and compasses is found 133 perches. CHAP. XLIX. To take the distance of any two towns 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 observing the former directions I remove my instrument to b, and so draw lines from b to point again to d and c, then do I note the concourse, or intersection of the said lines. which I measure by the scale and compasses as before, so will the stationary line a b be 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, etc. the labour is no more but to draw lines at every station, to point ●nto the distances required, and then to note the intersection of matchy lines, upon the paper, which after measure by your scale and compasses, so shall you have your stationary woodcut, mathematical figure corresponding to description line g h, 10 score, the distance f e 12 score, etc. whereby you may plat any field, and come not within the same, as in the 8 chapter. woodcut, mathematical figure corresponding to description CHAP. L. To find the horizontal distance of any place from you standing by a new way, upon the Plaine-Table. To find any horizontal distance after a new way. IN the 44 chap. at twice I told you of 4 certain rulers or ribs that were belonging unto the Plaine-Table, every one being divided into a 100 equal parts or more: by these rulers ordered in their due place upon the Plaine-Table) shall I teach you to seek the Horozintall distance of any place thus. Lay the ruler with the sights upon the very edge of one of the sides of the Plaine-table, turning the Table about until through the sights you espy the mark whose distance is required (but with this proviso that the corner of the Table, where the division take beginning, be nearest unto you:) this done take the ruler with the sights (the Table unmoned) and place the same upon the right side the Table, as before, and then looking through the sights espy your second station, in a known distance from your first station. Next shall you bear your Instrument to your second station, situating the Instrument (by help of the Needle and back sights) here in all respects as it was at the first, which being done lay the ruler over the corner or both sides of the Instrument removing the same until through the sights you espy the mark whose distance is required, lastly note the equal parts upon the ribs cut by each end of the ruler or sight having regard to those parts that do respond to the statinary line, and also to the distance required, for as the parts respondent to the stationary line, are to the line itself (being measured and known) so are the parts respondent to the distance, unto the distance required. therefore work by the golden rule, in this work 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 plain & easy. The premises being considered, and the doctrine before well understood you may produce infinite wates to perform many rare conclusions, but we cannot stand to set down a demonstration to suit to every proposition that may happen in the field, chief for that, let the demand stand howit will you may resolve the same by due regarding the prescript. Now I will briefly touch the order of taking a plat of a field, manor, etc. by the plain Table, according as we have dealt with the Geodeticall staff and other instruments before, aiming to perform some such propositions here that were omitted in the other books, for it would increase the volume over much to set down every kind, in the use of every instrument, since we understanding what is said of the one may also be performed in the other, and that much after one kind of method, as I have said before: but indeed I have here set down such propositions that will best agree with the Plain Table, and are aptest to be wrought thereon, setting aside all impertinent demonstrations. And you shall note for divers good respects that I shall omit one thing that standeth firm, and is ordinarily used in demonstrations of this nature; & that is, lines to represent the Instrument & the lines also drawn thereupon: my reason is, because I will not confound the work with multitude of lines, as also to save the cutting of many figures whereby such that served in the Glass, likewise serve in the Plain Table. CHAP. LI. To draw the plat of a piece of ground at one station where all the angles of the field may be seen from that place of standing. At one station to get a plat. FIrst, go round about the field, and in every angle set up some mark, then plant your table covered with paper, in such a place as from thence you may see all the angles of the field; that done, in a place convenient of your table make a prick or point to represent the place of standing: from the point to each mark draw a visual line by the edge of your Ruler, then from your place of standing, measure exactly with your wire line the just distance in perches to each several mark, and set those distances by the scale, each upon his own line which was drawn to those marks, noting these several points where these measures end. Lastly, from point to point by the edge of your Ruler draw lines which shall include a figure proportional to the field to be measured, and the lines so drawn shall represent the hedges of the field, as in this demonstration. woodcut, mathematical figure corresponding to description Your station is i, the lines drawn from i to point to every angle, are i a, i b, i c, i d, i k, i e, i f, i g, and i h, which are measured as is noted upon each line, as i a 27 perches, i b 9 ¾ perches, etc. then from a to b I draw a line, and so go round: so have I made a figure proportional, which was required. CHAP. LII. To draw 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 up marks in every angle of the field, as in the last Chapter of this Book, 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 upon the lines on the instrument, & from point to point draw lines to represent so much of the hedge, then appoint out some place for your second station, whereunto measure, and then draw a station line, setting the measured distance thereon: after this, remove your Table to that second station, and there fixing it, as it was at the first station, which you shall observe with your needle, as in the second proposition of the sixth book before is taught. Then from the centre of your second station draw lines to all the rest of the angles of the field, which have not lines drawn 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, (still observing that your needle stand as at the first station:) and if at this third station you cannot see all the angles yet unmeasured, you must again choose a fourth station, and the fift, if need require: and thus proceed till you have taken all the angles of the field: and at last, observing the measuring into every angle, as also the rules before taught, you shall produce a figure proportional, and equal in angles to the plat or field presented, which was the thing required to be done. woodcut, mathematical figure corresponding to description 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 again I may thence see the angles k l m n, and so proceed 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 see all the angles in the field, therefore I am forced to seek a third station, as c, and there observe the angles of oh p q r, which be all the angles, which I protract and limit upon 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 needle cut 40 deg. at a, and so must it do at b and c, and as many more stations as you should be occasioned to make: for note general, causa brevitatis. The South end of the Needle cuts like degrees in the card at every station, as at the first station. This Chapter is remembered, lib. 6. Geodetia, Chap. 7. defi. 5. CHAP. LIII. To draw the plat of a field by once placing the Instrument in an angle of the field, and measuring round about the field, because haply you cannot traverse 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 up the marks of every angle, then plant your table in such an angle, from whence you may see all the angles in the field, then make a point on your table (in a convenient place) to represent the prick of your first station, from whence draw lines into each angle, than 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 unto it, and take so many measures on the scale, and set one of the feet of your compass in the last made prick, and with the other foot strike an arch through the next line to it, and note the interception thereof, and from the prick last made thereunto draw 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 proportional to the field. woodcut, mathematical figure corresponding to description The figure is a b c d e f g, the angle where the instrument is planted is a, whence I draw lines into every 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 down, than 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 proportional to the figure proposed. CHAP. liv. To take the plat of a field by the rule of the last Chapter, where all the angles cannot be seen 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 again in an angle, where your last measures ended, in like fort situated as at the first station, which you shall do by setting the needle on the same degrees it cut at the first station, or looking back (as hath been before taught) from whence take all the rest of the field (if you can there see all the rest of the angles) if not, you may make a station or two more, as occasion shall serve: And so work till you have wholly enclosed the field; so shall you make a figure on the table proportional to the plat of the field with angles equal thereunto. This Chapter differeth little from the last, only you must perform that at many stations that there you did at one. This agreeth with the 7 Chapter, def. 4. Geodetia. CHAP. LV. To draw the plat of a piece of ground by 2 stations, and measuring but one line, where all the angles of the field may be seen from both the places. FIrst go round about the field, and set up marks in every angle thereof, A plat of two stations. then choose out your 2 stations something near the middle of the field, a good distance one from another, in such sort, that they lie not both in a strait line to any of the angles of the field, but so that they may make as great angles with every angle of the field, as may be; for the greater regard you have in the choosing out your two stations, the better will your lines intercept one from another: Wherefore having thus made choice of your stations the work is performed: As in the 31 Chapt. as you may gather by the demonstration. woodcut, mathematical figure corresponding to description 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 obtain a line like proportional 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 up marks in every angle, than point out your first station, where your instrument being placed, draw from the prick of your station lines, to as many angles as you can conveniently 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, upon that line set your distance by your scale, and then remove your Table to your second station, where plant it in his due situation, and then from the centre of that situation draw lines again to each angle whereunto you drew lines at the first, and note the interception, each with his match line, and then draw lines from point to point, which shall represent somuch of the hedges of the field as you have gotten by these two stations. Now your instrument standing thus at your second station untemoved, from the centre of your second station, again draw lines to as many new angles as you see, (that is, from whence you have not drawn lines before) then choose out a third station, from whence you may see all those angles, whereunto you drew lines last before, and then draw that station line, and then again remove your Table, and having placed it in his due form, to find the centre of this your third station do thus; lay the edge of the ruler to any point in the paper which doth represent some mark in the field, and remove your ruler to or fro, till through the sight thereof, you see that mark in the field, which the point on the paper doth represent, by which the edge of the ruler doth lie▪ and then draw a line towards you, till it cut the station line, and note the interception, for that point representeth the prick of your third station. And from the prick or centre of your second station to that point, showeth the distance betwixt the second and third station, viz. that point on the paper showeth in what part of the field your instrument is placed. Now from that centre draw lines to all the angles, which you drew to your second station, & where they intercept or cross each his match lines, make pricks 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 unto. This done, and the Table unremoved from that point or centre of your standing or third station, draw lines to as many angles as you can see, which have not lines drawn to them already. Then choose out a fourth station in such sort as you did choose 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 need shall require, till you have ended your whole work, and at last you shall produce a figure with lines proportional and equal angles to the plat of the field. woodcut, mathematical figure corresponding to description Example. My first station is a, whence I observe 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 observed at my first station as you can see as b d, d e, b f. and so noting the intersection of matchy lines, draw the lines d e and e f, which is so much of the hedges that you have observed: now the Instrument unremoved 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 find your third station c is thus, upon some point on the paper, representing some angle in the field as e, laying there the edge of your ruler, moving the other end until you observe through the sights the angle e, then note where the edge of your rule and the line b c intersect, as at c, so shall you find 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 from b, so have 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, observed 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 have included a figure proportional and like to the proposed figure. Note, I draw no figure upon 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 duly ordered, of all other is the best, because by apt choosing of your stations you may avoid acute angles. CHAP. LVII. To draw the plat of a piece of wood ground, where, for the thickness of the wood, a man cannot place his Instrument, but only in the angles of the perimeter. IN this manner of work you shall use 4 men to help you, whose labour shall he thus, two measure with the line the distance from angle, to angle, one man to go before you into every 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 every remove) look back to him. Being thus furnished you shall begin your work as followeth: First, plant your Instrument in any angle, and appoint for your place of standing some prick in your paper, then draw a line into the next angle, which line measure on the ground, and set those measures by the scale on the line drawn, then place your instrument in the angle, & say the ruler along the line drawn, & then turn the Table about, till you see the angle, or the man left in the angle from whence you came last, where screw fast the Table, and for your more assurance, you may behold your needle, The singular use of the back sight. which in this kind of plaiting will stand you in great steed. For look what degree your needle did cut your first standing, the same degree must it stand on at your other standing, wherefore it were good at the first placing of his instrument, to write down the degrees cut by the needle, for the help 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 drawn, and then plant your Table again in your third angle, and in this order work till you have compassed the whole ground, and if it fall out in the conclusion of your work, that the line and angle of your figure agree with the line & angle of the field, then is your plat perfect, if not you have some error. And herein, if I may advise you, begin your work again, to find your fault, & trust not to any help for the closing thereof, wherein you shall but deceive yourself, and haply of a small error increase a greater, for that you know not whether your fault be in the lines or in the angles, wherefore if your figures miss of closing above one perch, never trust upon your work. And be sure when you plant your Instrument in one angle & look to the next, that you so direct the sights, that the visual lines lie parallel to the hedge measured, neither observing this paralelly, is it material how far off the perimeter you place the Instrument, always provided 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 observe the mean: for in all things, observare decorum, is best. This kind of measuring is principally and commonly used for woods: for that in them a man cannot see the angles by any other means, and may serve for all kind of other grounds, and indeed commonly used of land meaters, who having but this one Chapter, and that rawly▪ presume of the full knowledge of the use of this instrument, but how they perform it, I leave to those that shall try them, which is but had, as others before me have, reported. CHAP. LVIII. To draw the Plate of a field by placing the Instrument in every 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 upon may he of reasenable length, then set up a mark in the field in such a place as from thence you may see as many angles of the field as possible may be: which mark call your principal point, and to that mark 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 down: so shall you have a point in your Table to represent that principal point in the field; This done, draw a line into the third angle, and thereto remove your instrument, and having there placed it, get the distance betwixt your two stations as you got your distance in the like case before, which you shall perform thus: Having first placed your Instrument by looking back, or by the needle, at his due situation, lay your ruler either by the prick on the paper, that representeth the principal point in the field or by any point on the paper that you know representeth some mark in the field: then turn your ruler about till you see that mark in the field which the prick by which your ruler lieth doth represent, and draw a line till it cut your stationary line, and that point of interception showeth the point on the paper, where you stand in the field. So in this order by placing your instrument in every angle you may get the length of every hedge severally with measuring but one in the whole, and the conclusion will be, that in the end you shall make a figure with equal angles and lines proportional to the plat of the field. The premises b●●ig well understood, and all things else well considered, I will ●raue pardon, and so cease further prosecution hereof, presuming that there is sufficient said to open the whose scope of this chapter: neither would I go about to fill the book with many curious demonstrations, and difficult questions, to beguile the aspiring wit of the young practitioner, but only set down some such few things that were most requisite to be known, 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 plain Table, and yet never be forced to change your paper. YEt again before I conclude, I will give you another way to seek the plat of great Champion fields, that contain 3000. or 2000 Acres, by the Plain Table, which is not much differing from your work by the Geodeticall Staff. You shall therefore place your Instrument in every angle, and so get every angle and his sides, To use the plain Table, and never change the paper. not regarding the length of the containing sides, as you be wont, then must you measure every hedge, and as you were wont to lay the same down by your scale and compass, here you shall but write the length of every hedge upon the lines drawn upon your paper, and responding thereunto, so have you finished, and you shall never be forced to shift your paper, nor have the lines to run off the same, for that you may draw them as long or as short as you please. Now when you come home, upon some sheet of paper protract all the angel's one after another, See the 3. chap. pro. 5. as you found them in the field, allowing by your Scale and Compass every line his due length, according as you find the same, note those figures upon the said respondent lines, and your conclusion will be to produce a figure like and proportional to the field proposed. This chapter is most excellent for the purpose before said, and therefore worthy of note, as they shall find it that work by the Plain Table, in countries that consist of great Champion fields. CHAP. LX. What Chapter is most fit to use in plaiting of ground, as well such whose superficies is subject to sight, as others that be rough and full of wood, as also to make choice of the best instrument to perform the same, as also to make a new kind of particular, YOu be taught before to measure and plate any piece of ground whatsoever, it resteth then for you to make choice of such Propositions that are best, as well in respect of the fashion of the field, as in respect of the aptness of the Proposition. Therefore for all grounds, whose bounds and angles may all be seen from one place, use the 51. Chapter, and if you cannot travers the same to measure it with your chain, by reason of pools, marshes, or such like, then is the 53. Chapter excellent: as for the 55. Chapter, I utterly dislike thereof, because the section of some angles fall out so acute, that the conclusion cannot be without error, therefore such works that be required by the help 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, use the 57 or 58. Chapter, jointly by themselves, or severally, according as the aptness of the ground occasions you: you may also well use the 9 and 29. chapter to the like purpose: but have a special care in all your dimensions and plaiting▪ that you truly observe the angles and corners of the fields, and so consequently precisely meet the hedges and limits, remembering ever when you work with the Plain Table or Circumferentor, to keep your Instrument parallel. Touching the best Instrument for this purpose. I leave that to the discretion of the Reader: for my best council is, that every good practitioner make proof of every instrument, as I have done, and afterwards to apply such to use that he findeth to work with the producement of fewest errors: for I will nominate none, lest it be censured but my peculiar affectation. For every one doth commend that instrument whereon he doth practise, or in which he is seen, and therefore some affect the Plain Table, some the Theodelitus, etc. which are all in some respects sufficient for ordinary measurers of land, so they be used according to Art. Here shall follow another kind of engrossing of particulars, then that in the Staff, together with the pleasant and apt placing of the four winds, and their collaterals in any plat taken, which being drawn in red lines, and artificially done will much beautify the same; neither need you to extend them all over the body of the plat, unless you please: but in that your own discretion shall be your best guide. THE DESCRIPTION AND use of an Instrument called the Circumferentor, used by some only to measure Land, whose use was first practised by I. G. and now published, and annexed hereunto in a brief method of teaching, and in some parts altered, by Arthur Hopton. CHAP. LXI. A description of the Circumferentor, and the parts thereof. YOU must first prepare some piece of wood well seasoned, The Circumferentor. and bearing a smooth grain, which let be fire inches, or there about long, four inches broad, and about an inch thick: then must you stoup down the left side thereof, and divide the same into equal parts, according to 12. in an inch, numbering the same by 5.10.15. etc. The Needle and Card. In this piece of wood must there be a round hole cut or turned, of three inches diameter, and three quarters of an inch déene, in the bottom whereof is placed a card, as in the 4. Chapter, at 3. This round hole hath a needle therein, and a glass to cover the same, as in the fourth Chapter of the topographical Glass, at 10. Then upon this square piece 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 divided into 30. degrees, and the semidiameter thereof into 1000 equal parts and according unto the total sum, are numbers placed in the said Table, answering unto every degree, and half degree. And these numbers serve to express the length of every right sign, or of every perpendicular let fall from the several degrees, and half degree in the limb upon the semidiameter. Tabula Sinuum. 1 52 6 309 11 544 16 743 21 891 26 978 / 78 / 333 / 566 / 760 / 902 / 983 2 104 7 358 12 588 17 777 22 913 ●7 987 / 130 / 382 / 608 / 793 / 923 / 991 3 159 8 407 13 ●29 18 809 23 933 28 994 / 182 / 430 / 649 / 824 / 942 / 996 4 208 9 454 1● 669 19 838 24 954 29 998 / 233 / 477 / 688 / 852 / 958 / 999 5 259 10 500 15 707 20 866 25 966 30 1000 / 248 / 522 / 725 / 878 / 972 / Of the Sights. Upon the plain of this square piece of wood, near unto either end thereof is placed two sights, and the one of them is but half the length of the other standing perpendicular. Of the shorter Sight. The shorter of the two sights beareth no divisions at all, The shorter sight. in the top whereof is placed a pin's head, and upon the side is set a piece of small wire, end in the midst is hanged a plumb. The distance from the wire in this sight to the plain, is taken and divided into 60. equal parts, according unto which divisions is the right edge divided, beginning from the perpendicular point under the wire, numbered by 10. as 5.10.15. etc. The short sight, and the wire therein represent the semidiameter of a quadrant, and the wire the centre thereof. Now from the perpendicular point be the degrees of a quadrant, perfected upon the upper side of the right edge, numbered 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 three kind of divisions 1 First, the distance from the pings head (in the shorter sight) unto the plain of the instrument, is taken, and according unto that distance is a line stroke overthwart this sight, which must be called the line of level, the distance from which unto the pin's head is taken, and divided into 100 equal parts: the distance betwixt the pin's head and the line of level is equal in length, therefore divide the longer sight into 100 equal parts, according unto which place them in the said longer fight, to 50 upwards and downwards, numbering them by 10, as 5, 10, 15, etc. Hypothenusall divisions. The second division is the graduation of the hypothenusal lines, according as they increase by units, they be numbered by units as 1, 2, 3, 4, etc. to 12, which represent 100, 102, 103, 104 etc. and these divisions are set from the line of level upwards and downwards, take the square of 100 from 102, etc. from 102 103, etc. Degree of a Quadrant. The third sort of divisions is the degree of a quadrant, projected upon the said longer sight from the line of level upwards and downwards 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 upper and of the sight strait down unto the lower end. Upon this sight there is a vane of brass, made to run equally up and down, and in the same there is a sight hole answering to the slit, and the edge of the vane. Of the Index. The Index. Unto this Instrument there also belongeth an Index, wherein must be a centre hole, to put upon the wire in the shorter sight, in the other end of this Index there is a sight placed, with a hole therein answering unto the fiducial edge of the Index, the edge of this rule is divided into such equal parts, as the right edge of the Instrument. Of the Staff. The Staff. Lastly, unto this Instrument there belongeth a Staff 4 foot long, with a good steel pike in the foot thereof: this Staff serveth to plant your Instrument upon, for which purpose in the top thereof is placed a round pin of wood or brass, and through the midst of the Instrument is bored a hole to fit the said pin: so when the Instrument is placed upon the said pin, he will move round about, but the best staff is that which is made with three staves joined together like to a 3 footed pair 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 affirm, the name was not unaptly given, for if we well consider hereof, it will be apparent that the working thereby gives or afoords the name itself, for when we work in plaiting of fields etc. we be instructed to move or bear the Instrument about, until he point unto the proposed angle, whereby you see we bear him about, upon the top of the Staff whereon he is planted, so that he is properly called a Circumferentor, of the Latin word Circum, which signifieth about or round about, and fero the verb, signifying to bear or carry, so Circumfero is to bear about, whereupon the Circumferentor taketh his name, which you may take in moving him about the Staff, or bearing him about the field, in working whereby you must always have a special care unto the paralelty thereof, so that it is not lawful for him to lean one way or another, but the plain thereof must always lie parallel to the Horizon, which the plumet in the shorter fight will help you to do one way. Then must you provide a Protractor, The Protractor, see the Chap. 26. which is a half circle divided upon the upside in the line into 60, such equal parts as the Card in the first Chapter was, the diameter whereof must agree with the diameter of the Card: the lower side of this protractor is divided into 60 such equal parts, proceeding from 60 unto 120, so that, All the divisions to 60, be upon the upside, and that is called the East side. Then all the divisions unto 120 above 60, be upon the lower side, and that is called the West side. The diameter of this Protractor representeth a Meridian. Upon the utter side of this diameter is room left for to make a scale, which is divided according unto 12 in the inch, make not your protractor, as the common order is, See the 6 book Chap. 53, of the Geod: Staff. if the scale had 12 parts in the inch upon 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 Star, by the old Circumferentor. To take the altitude and Azimuth of the Sun or Star. PLant your Instrument so that he may lie parallel unto the Horizon, then turn him about, until through the sight hole, and slit in the longer sight, and by the pins heard in the shorter sight you see the Sun or Star bringing down the vane, until through the hole therein, and by the foresaid pings head you see the said Sun or Star, than the degree cut by the si●e of the ●ane, showeth the Almicanter or altitude of the Sun, and the degree in the Card cut by the South end of the needle, showeth the Azimuth or the distance of the Sun from the Meridian. But if the Sun or Star be higher than 25 degrees, so that you cannot bring down the vane to work upon the longer sight, than put the Index upon the centre pin, looking through the sight in the Index, until through the same hole, and by the centre pin you see the Sun or Star: for the degr. then cut upon the edge of the Instrument by the edge of the Index is the altitude. And by this Proposition may you observe all the Stars in the Globe together with their motions in the heavens. Example. The 6 of October in the Morning, I made observation according unto the first difference, where having planted my Instrument parallel, and spied the Sun through the hole in the vane, and by the pin's head, I found the vane to out 12 degrees upon the sight, and the South end of the needle to cut 19 ●/60 degrees, which showed me that the Sun 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 cometh 58, which showeth me there is so many degrees included betwixt the Sun and the Meridian, and so of the rest. CHAP. LXIIII To find in what point of the Horizon any thing seen lieth, by the old Cirferentor. To know in what point of the compass any thing lieth. HEre it is requisite first to understand that 120 degrees represent the South, and that the degrees are numbered into the East, so that to find what point of the Horizon any thing lieth from you, do thus: Let the Instrument be placed parallel upon the staff, then cease not to move the same, With more ease see the 28 Chapter. until through the hole in the vane, and by the pin's head you see the thing desired, note the degree then cut by the South end of the needle, with which resort unto this Table, so have you your demand. 15 30 45 60 Sou: East East Nor: East North 75 90 105 120 Nor: West West Sou: West North Example I find the South point cut 45 degrees, I conclude the thing seen is north-east. CHAP. LXV. To find the hour of the day by sight of the Sun. To know the hour of the day Work the instrument lying parallel until the shadow of the pin's head point or fall just in the slit in the longer sight, & the intersection of the needle, I mean the South part with the parallel of the month, or sign take whether you please, is the hour, which you shall know by the hour line passing thereby. And you must understand that those circles I call parallels be such as are described about the centre of the Card, and those I call hour circles be those that pass as it were from the centre to the limb crossing the parallels. CHAP. LXVI. To find the hour of Sun 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 work in that behalf cannot be general, but it may serve without any notable error over the most part of England. You shall observe where the parallel of the month cutteth the Horizon, for the hour circle passing thereby, or the nearest thereunto showeth your demand; remembering to seek the setting upon the West side, and the rising upon the East side of the Card. So shall you find the 11 day of May the Sun to rise near 4, and set near 8: then if you would know the length of the day and night, you may repair unto the second Book, Chap. 10. of the Geodeticall Staff. CHAP. LXVII. To find the amplitude of the rising of the Sun and Stars. To find the amplitude of the Sun or stars. IT is not unknown to any man, though meanly traveled in Astronomy, that every Horizon hath four principal points, viz. East, West, North, and South; than you must understand, that there is no star, nor the sun, that riseth just East, or setteth just West, unless they be in the Equinoctial, which happeneth unto the sun but twice in the whole year: but for stars, if they rise once East, or set West, so do they always, whereof there be but a few: the star in the pinion of the left wing of the Virgin, the star in Antinous left arm, etc. come near thereunto: but as the amplitude of a star observed one day, is certain and all one in any other day for that latitude; so in the sun doth it differ every day, and is called Amplitudo ortus. This had, Observe the sun or star when they seem (as it were) to touch the earth, as being at point of rising or setting, whereunto turn the Instrument, until through the slit in the longer sight, and by the pings head you see the sun or star, 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 have you your desire, so the degr. were under 30; but if the degr. cut be above 30, multiply the degr. cut by the East end in 3, then from the total take 90, so have you your desire, and the setting shall be North from the Equinoctial. But if you seek a rising, you must consider whether the degrees cut by the East end be under 30, or above: if they be under 30, multiply them by 3, so have you your demand, and it is North: if they be above 30, see what degrees the South end cuts, which multiply by 3, subtract from 90, to have you your desire, and the rising is South from the Equinoctial. Or thus with more ease: having made your observation, see how many degrees are contained betwixt the West point of the Card, and the South end of the Needle, for arising; but for a setting, see how many degrees be included betwixt the East point of the Card, and the South end of the Needle, which triple; so have you your desire. But this Chapter is performed with far more ease & truth by my topographical Glass. CHAP. LXVIII. Of the opposite degrees, and how to find them. Opposite degrees. BY an opposite degree is meant the opposite point of a Diameter, or the point opposite unto the degr. cut by the South end of the Needle, that is the degr. which the North end should fall upon, which is always the half of a circle distant from the South end in this instrument 60 degrees; so that if the degrees be less than 60, add 60 thereunto; but if more than 60, subtract 60 from it, and the total of the ●ne, or the remainder of the other is your desire. This needeth no example. CHAP. LXIX. To find the quantity of an Angle. To find the quantity of an angle. THe quantity of an angle is the portion of a circle included betwixt the 2 sides of any angle, which is found upon this instrument, by the cutting of the Needle at two observations in one place, the lesser of which must be taken from the greater and the degrees which remain after substraction is the quantity thereof. But if the remainder after substraction exceed 60, then must the said remainder be taken from 120, so have you the quantity: if your degrees be not direct, then must you work by the opposite degrees, as in the 9 Chapter, taking the lesser of those degrees from the greater. And you must here note, that all degrees cut at divers observations in one or more places, be called direct. And such degrees as be opposite unto direct degrees, be called indirect: and here note the tediousness of taking an angle by this instrument, in respect of my Staff. CHAP. LXX. To take the distance of any mark by the old Circumferentor. To take a distance. AS I have often times said in the 2 Book of the Geodeticall staff, that there must be 3 things given, 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 always have two angles and one line given, by help of which you may seek any distance proposed thus: Plant your instrument at the place appointed, whence you desire the distance, and there looking towards the said mark, note the deg. cut by the South end of the Needle; then appoint another place for your second station to which bring the sight, as before, noting the degr. cut: that done, measure the distance betwixt the place you then stand at, and the place appointed for your second station, there again plant your instrument, looking through the fights unto the mark whose distance is required; then note the degr. cut, and so get the quantity of both the angles as in the last Chapter. When you have gotten these two angles, add them both together, which take from 60; so have you the quantity of the angle at your mark: then must you resort unto the table of signs placed in the Instrument, and there ●ind the sign of every angle, and note it down, and if the quantity of the angle exceed 30, subtract the excess or overplus from 30, and take the sign of the remainder. These signs had and noted down, work by the golden rule, wherein the sign of the angle at the mark must be the first number, the measure betwixt the two stations the second number, and the sign of the other angel's severally the third number, according to the side which is sought, and this work is grounded upon this Chapter. In all rightlined Triangles the proportion of the one side unto the other, is such as the sign representing the angles be. Or more brief. See the 7 Book Axioma 2. of the Geo. St. The sides of opposite angles be direct proportional to their signs. CHAP. LXXI. To perform the last Chap. by protracting with the old or new Circumferentor. To take a distance. Having made your observations at each station, note down the degr. cut by the South end of the needle, and then protract thus: Take a fair sheet of paper, and fasten the same upon a Plain Table, or such like, with mouth glue, then shall you make a point upon your paper to represent your first station, there keep the side of your instrument, turning him until the needle cut the degree first noted, then draw a line from that point along the edge of the instrument, then keeping the edge still at that point, move the instrument until the South end of the needle cut the degrees noted at your second observation: 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 until the south end of the needle cut the degrees you noted at the second station: then by the fiducial edge of the instrument draw a line, & note where it intersecteth with the first line, for that is the place of the mark 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 only by the old Circumferentor. To take an altitude. YOu must first get the horizontal distance unto the thing whose length is required: then plant your instrument perpendicular, and move the vane until through the hole therein, you see the top or the summitie of the altitude, note then the equal parts cut by the side of the vane, for such proportion as they bear unto 100 the like doth the altitude unto the distance: multiply therefore the distance by the parts cut, and divide by 100 the quotient showeth the height which is correspondent to the level of your eye. The ground of this work is borrowed from the jacobs' Staff, as may appear in the ninth Chapter of the fifth book of the Geodeticall Staff. An inconvenience like to that in the Theodelitus. But in taking of altitudes you shall have it oftentimes so fall out, that the altitude will be so high, that you cannot bring the vane so low as to see the top of the altitude by the hole and pins head. When it so happens, you must place the centre of the Index upon the wire in the shorter sight, looking through the sight hole in the Index, until by the wire, and through the said sight you see the summitie of the altitude: then note the equal parts cut by the fiducial edge of the Index upon the right edge of the instrument: for as those parts are in proportion to 60. the like proportion hath the distance unto the height. And so that proportion as those parts cut have to the parts cut in the Index, the very proportion hath the distance to the visual line. Therefore multiply the horizontal distance by 60. and divide by the parts cut on the right edge of the instrument, the quotient will show the height. Again, multiply the horizontal distance by the parts cut in the Index, and divide the same by the parts cut in the edge of the instrument, the quotient showeth the visual or hipothenusall line. As you seek altitudes, so must you find profundities, as I have said often in the Geodeticall Staff, but the error is great if the instrument be not exact parallel. CHAP. LXXIII. To take the plat of a piece of ground by the old or new Circumferentor. divers ways may be set down to fetch the plat of a piece of ground by this instrument: To measure woodland, or any other ground. but I hold that most easy which is to be protracted by the Instrument itself, because you shall not be troubled to seek the quantity of angles, which in this Instrument is over tedious. Having therefore a piece of ground given, you shall begin at some one corner, and there plant your Instrument, looking unto the next corner, and note what degree the south end of your needle cuts, then with a chain measure from the first corner to the second, and note down the degrees cut by the south end of the needle, and the length of the line measured. Next go to the second angle, and there convey your sight to the third angle parallel to the hedge: then measure the distance from the second corner unto the third, noting down the degrees cut by the south end of the needle, & the length of the line at your second observation. Then go unto the third angle, and note the degrees cut, and the length of them, and so proceed from angle to angle, noting the degrees cut, and the length of every line answering thereunto, until you have gone round about. And if you being at any one angle, and from thence can see two or three angles more, you shall not need to remove your instrument to any of them, but only from that angle observe all the rest, only measuring the hedges. With these notes you shall resort unto a fair sheet of paper, and there protract it down thus: In some place of the paper make a point, and there place the fiducial edge of your Instrument, turning it about until the south end of the needle cut like degrees, as he did at your first observation: then draw a line by the fiducial edge of the instrument, whereupon from the said point to wards the other end, lay down the length of the first measured line, which you must take with your compass from your scale, & where that number ends, in the said line, there make a point, where place the edge of your instrument, moving him about until the South end of your needle cut like degrees he did at your second observation: then draw a line by the fiducial 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, moving him until the South end of the needle cut like parts he did at the third observation: then draw 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 until the South end of the needle cut like parts as at the fourth observation, and so proceed, laying down the parts cut, and the length of the lines, until you have gone round about, by which means you shall lay down the plat of the piece of ground in true form, then for the casting up thereof, resort unto my book 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 kind of working is performed with as much ease as the former. You shall therefore repair into the field, and find 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 degrees cut by the South end of the needle: then direct your sight unto the second corner upon the right hand, and there again note the degrees cut by the South end of the needle, which note down, and so proceed rightwards from angle to angle noting the degrees cut by the South end, You are taught this chapter with a demonstration lib. 6. cap. 3. of the Geodeticall staff. until you have gone round about the field, of which degrees cut you shall make a little Table, to the end you may remember how many degrees were cut at the first, second, third, etc. corner. Next shall you cause one to meet with a chain the true distance of the first corner from your staff, which note down against the the first degree cut in your Table: then meet the distance of the second corner from your instrument, which note down in your Table against the number of degrees cut at the second corner, and thus proceed until you have gone round about the field, laying down the distance of every angle from your instrument against his proper degree cut, which done, fall to protracting, thus: Having prepared a fair shéet of paper, as you be taught before, about the midst thereof make a point, which call your station: then apply the edge of your instrument thereunto, moving him about until the South end of the needle cut the degrees you noted at the first corner, which done, draw a line by the edge of the instrument, from the point made in the paper out at length, then move him rightwards until the South end of the needle cut the degrees noted at the second corner, and then by the edge of the instrument, draw another line, as before, & so go forward until you have finished all the degrees cut by the south end of your needle, noted in your Table: then with your compass take from your scale the distance of the first angle from your instrument, which lay in the line first drawn 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 drawn in the paper, and so proceed from line to line according as you be taught in the third chapter of the Art of measuring ground. The length of every line laid down in such order as is said, then must you draw lines from point to point in each line; so shall you draw the limits and proportion of the ground, according as in the foresaid third chapter of the art of measuring ground by the Staff. And by this means may you measure ground at two stations measuring but one line in the whole plat, in such order as I set down in the fourth chapter of the sixth book of the Geodetical Staff. And since, what is said before may give sufficient light to perform both this way, and many other, I will omit further speech, lest I rather seem tedious to the wise, then facile to the unlearned. And you shall here note, that by taking perfect notes in the field, where one close boundeth upon another, you may take the plat of many flelds lying together, and so save a great labour. CHAP. LXXIIII. The degrees of a field being taken, to find whether the plat will close, the lines being truly taken. To know if your plat will close. NOte down the quantity of every angle at each several station, as well as you do the degrees cut, then add up all the quantities together, then multiply 60. by a number less by 2 then the number of the angles, and if your work be right, the product thereof shall be equal to the total of the quantities. Example. Let the number of angles be 8. from which take 6. which is a less, then multiply 60. by 6. and the product will be 360. which agreeing with the total of all your quantities of angles added together, is one argument that the plat will close. CHAP. LXXV. To reduce Hipothenusall lines unto horizontal, after another way, then in the 6 book 8 Chapter of the Geodeticall Staff, only by the sights in the old Circumferentor. To reduce Hypothenusall lines to Horizontal lines. PRepare a mark to be carried before you the just height of your line of level from the ground when the iustrument is planted upon his rest, this mark must be placed in the angle whereunto you look, he must stand perpendicular, and when you take the degree look your instrument stand perpendicular, and then move the vane upon the sight, until you see the top of the mark before planted through the hole, in the vane and by the pin's head, then in the hypothenusal divisions cut by the vane upon the sight, for they will show you how much that line you shall measure, will differ upon the 100: from that line you should measure, if the ground were level: therefore when you have measured that line, proportion him according to the parts cut. Example. Suppose the parts of the Hipothenusall divisions cut, to be 4 and the line measured to be 30 perches, now you are to find a number to bear 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 chain to be 30 perches, must be laid down 28 1/1 1/3 perches in your protracting. But for as much 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 perform the same by a Quadrant made of purpose. A new Quadrant to proportion lines. YOu shall prepare a Quadrant, and then divide the limb thereof in 30 equal degrees, setting number thereupon as the common order is, then shall you divide the lower side of the Quadrant the is betwixt the first degree & the centre, into 30 equal parts, raising perpendicular lines upon each division, which will be parallel unto the other side: this done prepare an Index of the length of the semidiameter of the Quadrant with a centre hole therein, this Index is to be fastened to the centre of the Quadrant with a brass pin or such like, which also must be divided into 30 such equal 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, only here you must note the degree of a circle cut by the vane in steed of the hypothenusal divisions, and then proceed thus: Put the Index to the different angles in the limb, than number the line measured upon the Index, and note the perpendicular there cut by the edge of the Index, for that shall show you the length of the horizontal 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 limb, than thereunto bring the edge of the Index, next count the line measured viz. 20. perches upon the Index fromwards the centre, so shall you there see the 19 perpendicular counted from the centre intersect, which showeth that the line measured 20 perches, must be protracted 19 And if the length of the line measured exceed 30 perches, and be less than 60, then take half the number upon the Index, and the perpendicular will answer to half the length of the horizontal line, but if the line exceed 60, take then ¼ ¾ & c. & the perpendicular will answer proportionally. CHAP. LXXVIII. To seek any altitude by this Quadrant. To seek an altitude. TAke the angle of altitude, whereunto bring the Index, the same being counted in the lymb, than number the Horizontal distance in the semidiameter, & the portion of the perpendicular to the Index showeth the height. 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, he is then declining if the wall point just East, West, North, or South, he is direct, otherwise declining. All walls decline either South or North, the quantity whereof is thus had: Set the North end of the Instrument, unto the wall, now if the needle cut 30, 60, 90, or 120, it is an East, a North, a West, or a South wall. 1 But if the needle cut betwixt 120 and 30, the wall is South East declining to the East. 2 If the Needle cut betwixt 120 and 90, that wall is South West declining. 3 If the Needle 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 degree cut from 60, and the remainder multiply by 3. 4 If North West take 60, from the degree cut and multiply by 3, so have you your desire. Com. Heref. Manerium de Sale In Superuis. manerij praed. ibid. fact. xiii. & xiv. diebus Septembris anno regni Dom. nost. jacobi Dei gran Ang. Scotiae, Fran. & Hibern. Reg. fidei defensor. etc. viz. Angl. Franc. & Hibern. sexto, & Scotiae xlij. Per B. G. gent. virtute commissionis dicti Domin. Reg. extra Scaccar. suum sibi direct. continetur inter alia, ut sequitur. viz. R. G. gent. tenet per copiam dat. xxviij. die Septembris, anno Regni Regis nunc Angl. etc. Quinto, cert. terr. & tenement. Custumar. infra manner. praed. nuper I.G. armig. ante A. Hos. gent. ante B.D. armig. patris sui. viz. Dom. mansional eight spac. unum horr. seven, unam coquinan iij. spac. unum stabulum ij. spac. unum bovile v. spac. unum columbar. unum gardinum, tria pomar, unde 2 voc. le North Orchard, & long Orchard count per estimac. iiij. acr Terr. ar. iacen. in quodam claus. inter al. voc. the West enclosure, cont. per estimat. l. acr. Parcel. unius Claus. prat. voc. le healed, per estimation. xx. acr. Parcel. unius claus. pastur. voc. le White field, count per estimat. xiii. ac. Habend. sibi & suis secund. consuetud. Manner. per Redd. per Annum xii. s. ij. d. An. val. dimit. x. l. In like manner must you deal with all the other tenements of the said Manor, noting the quantity of every particular, than the rend paid, and at the lower end a reasonable improovement. And if there be any other commodities in the said Manor accrueing to the Lord thereof, they may be noted as followeth. Manerium de Sale valet in Redd. uxx. l.xij. s. viz. Nundinum tentum annuatim ibidem die iovis proxim. post festum beatae Mariae iij. l. Nundinum tentum annuatim die Veneris proxim. post festum. etc. l. s. Markets. 3. l. x. s. Mercat. hebdomadatim ibidem tenent. dimiss. G. I. per annum 4. l. Shamellorum & scal. tam. carnium quam piscium ibid. per annum 30. s. Milles. seven. l. unius molendini aquatici iiij. l. unius molendini ventricij iij. l. Fishpools. xlij. s. una piscaria vocat. le White pool xx. s. Piscar. communis aquae ibidem vocat. le Black Moor xxij. s. Pawnage. thirty. s. Pannagio porcorum tenent. ibidem quam aliorum infra communem boscum, etc. x. s. Pannag. porcorum tenent. ibidem in parco vocat. etc. at. 3. d. the piece per annum xx. s. Swans. Cignorum in aqua Domini vocat. le Broad Poole, etc. Quarreyes'. ix. l. Quarreum lapidum vocat. le Free stone, per annum iij. l. Quarreum lapidum vocat. le Slate vj. l. Perquisites of Courts. Amerciaments, etc. iij. l. If there be any reprises wherewith the Manor is charged, as money for the yearly repairing of some bridge, high way, or any other annual pension whatsoever, let it be noted as the former: and in the conclusion say, Ei remanet clare per annum ultra repris. 306. l. 14. s. 8. d. And you must further note, that the first thing you have to deal with, is the sight of the Manor house, the buildings and demesne, than the park, parsonage, etc. if any be, and then proceed to the Tenements, as before. woodcut, mathematical topographical figure To make a plat or map, and place a Sea-Card therein. Upon the midst of your plat describe a circle, as upon a which divide into 32 parts, and then about the map describe another circle, which will likewise be divided into 30 parts, by drawing lines from the centre a by each of the 32 equal parts in the first circle: now if upon every of those intersections as a centre you describe a circle dividing every of their circumferences into 32 equal parts, extending from them right lines through the body of the map, you have finished the Sea-Card, and will beautify your map, and serve to express many pretty conclusions, which at this time I mind not to repeat: provided that you draw 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 winds in your Card otherwise, if you please by placing a circle, containing the same in some void place in your plat, as you may see in the 7 Chapter of the topographical Glass, and draw them forth only to touch the circumference of the plat, as in the 6 Book, and Chapter 49 of Geodetia. woodcut, radiated circular compass with Roman and Arabic numbering and astrological signs CHAP. LXXX. The order how to discover the true plat of any park, forest, or such like, standing upon the top of some hill, not approaching unto the same. THis Chapter is easily performed, if you do but call to mind how to seek the true proportion of any field, Island, or such like, even as you be taught in the Chapter: but indeed 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 truly than you can by the intersection of lines, for because thereby you be taught to find out the true distance of every angle from your station; in so much that if you do but observe the quantity of every angle, and again protract the same truly, you cannot err any thing at all: and for that it may be desired of many, I will not leave for their sakes to prosecute the same with an Example. Example. woodcut, mathematical figure corresponding to description Let there be a certain park c d e f g, propose the proportion and quantity whereof you are required to deliver at your standing a or b, which are certain hills from whence you may well view all the angles and corners of the said park. In performance whereof I first go unto b, making b a the one side of an angle, I observe accordingly all the angles in the said park, as a b g, a b f, a b c, a b c, and a b d, all which I note down thus: Degrees A B G 74 A B F 77 Angles observed at the first station. A B E 104 A B C 121 A B D 128 Next do I find me another station, a known distance from b, as a, 65. perches, and so repairing unto a, making a b the one fide of every angle, I again observe all the angles in the said park at a, as b a c, b a d, b a e, b ag, and b a f, and the quantity of every of which angles, I note down as before, thus: Deg. B A C 40½ B A D 44 Angles observed at the second station. B A E 65 ½ B A G 84 2/4 B A F 87 ½ woodcut, mathematical figure corresponding to description The angle a b d is 128. degrees, b a d 44. degrees, which added together, and the total taken from 180. leaveth 8. degrees, the angle b d a. Now I say by the said second Axioma, as the sign of the angle a d b 13917▪ is to the sign of the angle a b d 78801. so is the side a b to the side a d, or as the sign of a d b is to the line a b, so is the sign a b d to the line a d. Therefore multiply 78801. the sign of b, by 65. so is the product 5122065. which part by 13917. the sign of the angle d, so is the quotient 368 609/13917 perches, the line a d, which is one mile, one quarter of a mile, 8. perches and odd. In like manner must you get the lines a e, a f, and a g, as you may perceive by the ensuing example, where I have set down as well the quantity of each angle, as also the respondent sign and side that do answer or subtend the said angle. Triangle. Quantity of each angle. Signs. The subtending sides. b e a 10½ deg. 17364 a b 65 perches. A B E b a e 65½ deg. 90630 b e 65 perches. a b e 104. deg 97029 a e 363 4353/17364 perches. b g a 21 deg. 35836 a b 65 perches. A B G b a g 84¼ deg. 99496 b g 65 perches. a b g 74 deg. 96126 a g 174 12726/35836 perches. a f b 15½ deg. 29723 a b 65. perches. A B F b a f 87½ deg. 99904 b f 65. perches. a b f 78 deg. 97814 a f 239 1113/26723 perches. All these angles and sides thus gained, you shall lay down the plat, and find the contents thus: First I draw a line i k at all adventures, whereon by my scale, I appoint 65. perches, then making k i the one side of an angle, according to my observations at my second station, I protract an angle k i l of 40 ½ degrees, equal to b a c in the last figure, than I protract an angle of 44 degrees, as k i m, drawing i m infinitely, and so I proceed until I have 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. woodcut, mathematical figure corresponding to description You shall therefore note the distance of a c in the first figure obtained before, viz. 174 86047/100000 perches, the which with your scale & compass, lay down upon the line i l, from i towards l so have you the line i l, then take the distance of a d in the first figure 368 609/13917 perches, which lay down as before from i towards m, so have you the line i m, in like manner deal with i n, i p and i o, even as the precedent Table doth instruct you, so is i n 363 4753/17967 perches, i o 174 12726/35836 perches, and i o 239 1113/26823 perches, 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 again to l, so have you made a figure like and proportional to the Park proposed, as l m n o p. Certainly most exact and perfect is this kind of working, and albeit it may seem strange and difficult at the first, to young practitioners, especially because the devise is new, as for that we have not heretofore any English Treatise showing the use of right lined Tryangles, by these Signs, Secants, and Tangents, yet let none be skarred at first, for do but join the use of my 7 book of the Staff called Trigonometria herewith, and then all things will soon become easy and familiar. CHAP. LXXXI. The plat of a Park being taken, as in the last Chapter, how to cast up the contents thereof, after two manner of ways. THe first way is to resolve 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 superficial content, as in the second part of the 6 book of the Geodeticall Staff, even as you see this figure, l m n o p resolved into three tryangles l m n, l n o, and l oh 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 unto the same 2 part of the 6 book, chap. 25. woodcut, mathematical figure corresponding to description woodcut, mathematical figure corresponding to description Then having protracted your plat in plano (or reduced the same into regular figures, with your eye) divide that side whose length you found, into a certain number of small and equal divisions, then having divided the plat into such regular figures, as to you shall seem best, for the attaining of the area, you shall measure the contents superficial by those equal divisions, I mean the contents of every particular figure, adding them altogether and noting the total, then must you take the number of perches in the side of the Park, formerly measured, and also the number of equal divisions in the side of your plat, responding unto the said side of the Park measured, squaring both those two numbers. For as the square of the small divisions upon the one side, is to the square of the number of perches, responding to the same side, so is the superficial content in the former small divisions, unto the true superficial content in perches. Therefore increase the said superficial content by the number of perches squared, and divide the total by the square number of small divisions, so is the quotient the number of square perches which divided by 160 etc. leaveth the number of acres, etc. as in the Art of Geodetia, chap. 24. Example. woodcut, mathematical figure corresponding to description The figure protracted in plano, is l m n o p, Now viewing into what regular figures it may best be resolved, into which (as you may perceive by the pricked lines) it is readily converted into three Tryangles, by two right lines issuing from l to n and o, next I divide some one side as l p into 60 equal parts, and so by that divided line, as by a scale I measure how many of those divisions is in the base l o, and l m, so do I find the one to contain 88 and the other 92 of those parts, next upon 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 half 88, so have I 1100, than 59 in half 92, and there is made 2714, and lastly 53 in half 92, so is the product 2438 (or since l n was a diagonal, or a base common to both the Tryangles, you might have multiplied o r and m r in l n and all had been one) the which three aggregated sums add together, so have you 6252 produced, the whole content of l m n o p, by the which multiply 16900 (the square of 130 perches the length of the hedge in the Park c g) and the product is 105658000, the which parted by 3600 the square of 60, leaveth the number of perches viz. 29340 24/6 perches, the just 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 Axiom of Trigonometria, as if you would measure the diagonal l n, here have you two sides known i l and i n comprehending an angle known, therefore work by the said third. Axiom: the like might you do to l m, m n or l o, etc. and then the area is easily found by the 20 Chapter of Geodetia, without regard of perpendiculars. Having the 3 sides of every triangle, you may lay them down severally upon some smooth boards or such like, and so by your scale and compass, after the common order, find the perpendiculars, without regard to the quantity of the angles, for the 3 sides being known, the true intersection limits the right proportion. CHAP. LXXXII. To take the plat of any field or such like by the intersection of lines, with more truth than hath been published heretofore by any. I have 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 proves to be so acute, that you cannot precisely discover where the true point of the section was made, wherefore I have devised a way by help of three stations, (and yet measuring but one line), truly for to find the right point of intersection, which is thus performed. Look how you be taught to seek any plat at two stations, and so do here, then from your first station, beyond the second, or from the second beyond the first, observe some mark, tree, 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 the one side of the said angles, then go to the mark newly espied for a third station, and there observe some such corners that you thought would fall 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 help thereof shall you correct the acute section of the former lines, as you may best perceive by the example. Example. woodcut, mathematical figure corresponding to description The proposed field is c d e f g h, which I observed 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 perceive 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 unto i, and there again observe 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 prove acute, so shall the lines issuing from i and b make a more perfect intersection, so that when you come to protract, you shall find all the acute sections corrected, for whereas the lines a h, and b h made an acute intersection at h, the section of i h and b h do correct and reform the same, finding out the true point of intersection, and so of any other: and in your observations in the field you may well know which angel's in protracting will prove acute, for if you add the gles g b a, and g a b, or h b a and b a h together, the total taken from 180 leaveth the angle g or h, of whose acuity you may judge. Thus by means of 3 or 4 stations, or more, if occasion be, may you find the true plat of any proposed plain, by the intersection of lines, never measuring but only one line in the whole work, certainly no proposition performed by the intersection of lines is better than this, if your stations be respondently taken and the observations truly made. CHAP. LXXXIII. To seek the distance or length of any Turret, tree, town, or such like, from you, without the help of Instruments. THis conclusion, or the like unto it, is performed by Gem. Frisius, and indeed is most exact and excellent, where you may have liberty of ground at command, craving the help of no Instrument, but only some such a one that will direct you readily to set out a right angle, which the topographical Glass or Geodeticall Staff will soon perform; or for want of them, any ordinary Carpenters square, or such like, may as well serve. woodcut, mathematical figure corresponding to description Example. A is a certain Turret, whose distance is required from b, where I plant my first staff, departing thence a certain distance Orthogonally, as to c 163 yards, placing my second staff at c, d is my third staff placed backwards from b 158 yards in a right line with a b: e is the fourth staff placed Orthogonally from d, and in a right line with a c, which is measured 200 yards, so that you must departed squirewise from d towards e, until you be in a right line with c a, as is said: these staves so set, and their distances measured, deduct b c 163 the first number, from de 200, the third number, so have you 37 your Divisor, then multiply the third number 200 by 158 the second number, and there is produced 31600, which parted by your dividend 37 leaveth 854 2/37 yards, the distance betwixt d and a, whence take d b 158, so have you the distance of a b 696 2/37 yards, making ●96 yards and a little better than 2 inches. CHAP. LXXXIIII. How to find the length of any hypothenusal line, or the length of any scaling ladder. THis Chapter is so easy, and so well illustrated by Euclid in his first Book, Chap. 47 of his Elements, that it needeth no Example; for there the square of the two containing sides are proved equal unto the square of the hypothenusal, in so much that the two including sides being added together, the square root thereof produceth the hypothenusal, as is proved by the prealeaged Chap of Euclid, and else where, saying. In triangulo plano rectangulo latera includentia rectum aequà possunt Hypotenusae, penult. primi Euclid. Therefore when you are desired to deliver the distance of the top 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 the 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 near to any town of war may you tell the just length of the Scaling ladder that must reach from the brim of the counterface or ditch that encloseth the same, to the top of the curtain or wall, by adding the square of the distance of the wall unto the square of the curtains altitude above your feet, for the square root thereof (as before) yields 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 above your feet g h be 10. paces, the square whereof is 100 which added to 529. maketh 629. the square root whereof is 25 4/60 perches and better, the woodcut, mathematical figure corresponding to description length of the scaling ladder i h, or you may work as in my seventh book of the Geodetical Staff, called Altimetria, problem 5. or Trygonometria, Axioma 1. CHAP. LXXXV. To find the distance betwixt any two Towers, Castles, or such like, though you can approach to neither, and that without the help of Instrument. THis Chapter is very exact, so the observations be well made; and indeed is but wrested from the 41. chapter of my 6. book of the Geodetical Staff, howbeit it is set down much after the same method I shall deliver now, by Master Digges our countryman, and others. In performance whereof you must have any kind of triangle prepared: which had, set up a staff, whereunto apply your angle in such sort that the one of the containing sides point directly unto the mark or Castle upon the left hand: the triangle so resting, look by the containing side upon the right hand, causing one to set up a second and third staff in a right line therewith, and the further the second staff is distant from the first, the better it is: the triangle yet remaining, as before, at your first staff, make a mark in the side subtending the angle at your staff, in such sort that it may be in a right line with the said angle, and the second mark upon your right hand; so have you performed all your observations at your first staff, only as you come thence measure the distance from your first staff to the third staff. Next repair unto your second staff, where situate your triangle, in all respects as he was before at the first staff: then must you departed towards the desired distance in a right line, until such time that you come direct between the Tower or mark that was upon your left hand and your third staff, and there set up your fourth staff. Next go unto your second staff, where your triangle yet remains, and looking from the angle at your staff by the mark made before, in the side subtending the said angle, your eye will direct you in a right line, which you must continue on, until you be also in a right line with the second and third staff, & there set up your first staff. This so ordered, measure the distance betwixt the second and third staff, the quantity of which measure reserve for your divisor. Lastly increase the distance between your first and third staff in the distance betwixt the fourth & fift staff, the product whereof part by your reserved divisor, so doth the quotient yield you the desired distance, as Geometrically might be proved. For as the distance of the second and third staff is to the fourth and fifth staff, so is the distance of the first and third staff unto the distance required. Or as the distance of the second & third staff 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 staff where my triangle is placed, the one of the containing sides pointing to the castle a upon my left hand, c e the way directed by the other containing side of the triangle, d my second, and e my third staff, placed in a right line with c, by the direction of the said containing side, his a mark in the subtendiding side, found out by the visual line running from c to b the mark upon my right hand: this done, I measure the distance from c to e, which I here found 300. perches, and so take up my triangle, situating him at d my second staff in all respects as he was at c my first staff, the respondent containing side lying in a right line with the first, second, and third staff: so I standing at d, and looking thence by the other containing side, the visual line will inform you what direct course to take towards the distance until you come in a right line betwixt a and your third staff e, where set up your fourth staff f. Again, looking from d by h, the subtle mark made in the subtended side, the visull line will direct you what course to hold until you come in a right line betwixt the 3. staff e, and the other mark b: this done, measure the distance d e, which there is 100 keep that for your divisor. Next measure the distance betwixt g and f, which is 170. perches. Finally, according unto the prescript, multiply the c e 300. by f g 170. so have you 51000. which parted by d e, 100, leaveth 510. perch, the distance a b. woodcut, mathematical figure corresponding to description CHAP. LXXXVI. To seek the distance of any thing from you, howsoever situate, and that after a new way, newly devised without the help of Instrument. I Do not remember any Proposition performed without Instrument more easy, speedy, and true to seek the distance of any thing from you then this is, the 82. chapter is as true in demonstration, but not so easy and speedy in working: for here you be not tied to right angles, and such like, but only are allowed to make the observation according to the aptness of the ground, neither need you to fear whether you be situate upon hills or in valleys, more than if you were on the plain 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 look direct unto the mark whose distance is sought: the triangle remaining, look by the other containing side, causing one to set up a second and third staff in a right line with the said containing sides, where sticking a staff, carry your triangle unto the staff nearest your first station, where you set your first staff, and there situate the said triangle in all respects as at the said first station he was, so will the one containing side lie in a right line with your first staff, and also with your second and third staff: then placing your eye at the contained angle, the other containing side will direct you what course to hold until you come in a right line directly betwixt the desired distance and your third staff, and there set up your fourth staff. Now must you measure the distance betwixt your first and third staff, then betwixt your second and third staff, which reserve for a divisor, then betwixt your second and fourth staff. Finally, multiply the distance of the first and third staff, in the distance of the second and fourth staff, which parted by the distance betwixt the second and third staff, the quotient is your desire. It would be something long for me to stand to demonstrate the same Geometrically, for that I want time, but because the work 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 prove it by Ramus. lib. 7. pag. 9 so that it is needless to infer a demonstration. Therefore as b c 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. woodcut, mathematical figure corresponding to description And as in this chapter, so likewise in most other like conclusions may you so order your observations that you may avoid division, 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 staff, parting the product by the second and fourth staff, as by the distance of c d, so doth the quotient yield the distance b e. And for that your Geodeticall Staff will take or deliver any angle, or represent any triangle, he may aptly perform this Chapter according unto this kind of method. CHAP. LXXXVII. A Navy, or one Ship seen upon 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 certain space, and then observe diligently the distance thereof from you again, now if the first distance and this agree the Ship standeth still, if the first observed distance be greater, the Ship maketh towards you, but if it be the lesser, he departeth from you. CHAP. LXXXVIII. Two Ships seen one in pursuit of the other, to know whether the Ship pursued, lose way, and how long it will be before he be overtaken. BY the 84 chapter get the distance betwixt the two Ships, then stay half an hour or a quarter, observing then again the true distance, if the two distances agree the pursued Ship looseth nothing, but if the first distance exceed take the lesser distance out of the greater, multiplying the space of time betwixt your observations in the distance betwixt the two ships, the product whereof divide by the difference of the distances, so is the quotient your desire. Example. Let us 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 overtaken, first therefore I take their distance, which I will suppose 400 yards, so resting a quarter of an hour (which is 15 minutes) I observe their distance again 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 leaveth 60 minutes, the time how long it will be before the Ship pursued shall beovertaken by the Ship pursuing, which is one hour. In the like manner may you deal with Ships or Navies approaching towards any port, haven or such like. CHAP. LXXXIX. How to take the platform of any house, Castle or such like. YOu are sufficiently instructed both in this book, as also in the Geodeticall Staff to seek the true perimeter of any figure proposed, where you be also taught to seek the breadth, height and distance of any object howsoever situate, insomuch that nothing remains (those rules well understood) but to seek the true ground plat of the buildings, for the which you have divers peculiar Chapters, which had you may find the distances of Fronts, Turrets, gabel ends, returns, or such like, the breadth of windows, quadrants, and such like the heights of jutteis, Storreys, & Ascents, lengths in heigths with such like: and thus may you proceed, taking as well the ground plat as other erearements with their proportional distance, noting the same to yourself in some book, whereby you may stand in any place far off and take the plat of any house, castle, fort, or city, the situation whereof (to your great praise) you may discover, or if you please cause the like to be made. woodcut, castle with stick figure people Certainly most excellent is this book for martial discipline touching fortification, as in the delineation of royal frontiers, sconces and reinforcing old walled towns, and right necessary for battles, masters of Ordinance, etc. CHAP. XC. The order to discover how mines or trenches run under the earth being most fit for pioneers, masters of Coal, Iron, Stone mines, etc. THis Chapter is performed in best and easiest sort with any such Instrument that hath alarge Needle, wherefore the topographical Glass or Circumferentor is best, in performance whereof you shall do as followeth. Suppose there were a mine of coal in the borders of a certain Manor. which continuing, the Lord of the next Manor was in doubt lest the vein of coal did run towards his adjoining Manor, and that they were common under his ground whereby the coals were his. To resolve this doubt descend into the pit, and then by the 72 chapter, get the true way that the miners have made even as it were a hedge, still noting the degree cut by the Needle, at every angle where the mine runneth one way or another, out of the course of a right line, and also measuring the side of every angle then ascending out of the pit, by your Instrument and chain (beginning perpendicular above the place where you began to make observations in the bottom of the pit) lay down the like angles and sides so observed: which having so done, you shall soon see if the mine or any part thereof have run out of the one Manor into the other, for if it do, you shall be forced to measure out of the one into the other. And as you note these angles of deviation being in the dark bowels of the earth, you were best to have a candle fixed upon the end of a staff, of equal hieght with your eye, and the same to be fixed in the foot of the mine at every angle, that thereby you may the better direct your sight thereunto. CHAP. XCI. To plant barrels of powder, direct under Castles, Forts or such like, and to know how far you be under the same. IN performing this Chapter by some Proposition formerly published, you must get the horizontal and Hypothenusall distance of the Fort from you, and thereby the height thereof above the Horizontal line, which done, you are also by help of your Needle placed in the glass to find out the angle of position, which is the number of degrees from any principal quarter of the world that the journey lieth, which done you must by the same Instrument ever direct the mine, direct upon that line or part of the world, and keeping your Instrument parallel, the sight upon the diameter of the demicircle, thereby always carry the floor of your mine level with a candle fixed upon the end of a staff of equal height with your eye (as before) will help you to do: Now when you have gone so far under the ground as you found the length of the horizontal line to contain, you may assure yourself that you be direct under the Fort, and that you are so many paces under or below the said Fort, as you found the Fort to be about the horizontal line. CHAP. XCII A Mine running upon some certain point, yet ascending or descending, to know at any time how much you are above or under the Horizotall line. COncerning your journey under the earth, you must observe the doctrine of the last Chapter, and when the mine happeneth to fall or rise, according to the doctrine of Altitudes and profundities, duly note at every several station the quantity of the ascent and descent, that is, how much you rise above or fall under the true horizontal line, and so keep two several tables, the one of the ascents and the other of the descents. Now when you desire to know how you are situate, add 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 horizontal line, for if the ascents exceed, you may be assured that you be above the horizontal line, if the descents exceed, you be under the said Horizontal line, according to the difference of the said ascents & descents, neither need you fear any collateral declining of the way of your mine, for that nothing at all altereth the ascent or descent, for that is only altered by the directing line, or line that you measure, insomuch that if you well observe the premises, you may precisely know at any time or place how much you are under or above the true Horizontal line, and thereby come into him again upon any occasion. CHAP. XCIII. A Mine or trench collaterally declining, how to know when you come again into the right line of position, and also how far you be from being just under any Fort proposed. TO carry a mine direct forward upon any point of the Horizon, you be sufficiently taught in the 90 Chapter, and if the Mine must ascend or descend above or below the line of level, you be taught in the last Chapter at all times to know how much you be above or under the said line of level. But say you were enforced by rocks, waters, or other such obstacles that you meet with under the earth, contrary to the 90 Chapter, to carry your Mine side wise from the direct line of position, in such a case you are first upon a fair large sheet of paper to extend a right live over the same, which call the line of position, being the direct way that the Mine should go, next note the angle of deviation from that line, that is to say how many degrees the Mine doth decline from the true line of position that leadeth on directly, and accordingly plate it down upon the paper, as you be often instructed in the use of each several Instrument, proceeding so far as your Mine continues in a right line, and if you be occasioned again to direct either further of, or nearer unto the line of position, always protract if down upon your paper exactly; as well in measure as angle, until such time that you can come to make your protracted collateral lines, or lines of deviation to intersect with the right line of position, first extended over the paper, and then by the scale with which you protracted your lines of deviation, examine how many paces or yards that point of intersection is distant from the point where your work began, which representeth the point of your first entry into the Mine, for that compared with the fundamental distance or length of the horizontal line informs you if you be past, or not yet come under the proposed Fort. Therefore in these cases you were best first of all to limit upon your paper with your scale and compass the direct length of the fundamental or Horizontal line, and so in your protracting may you call them back, if they seem to run beyond the Fort. Then in the former Chapters be you taught to know how far under the Fort you be, whereby you may ascend nearer or descend further from the superficies of the earth as the cause shall require. Certainly most exact and excellent is this kind of working, for conveying of mines, and of no small importance, for the due placing of furnaces of powder, to blow up Forts, Castles, Towns or such like, whether they be situate high hpon an hill, or low in a valley, which for all purposes in these and such like cases under ground, you shall find the topographical Glass to be most requisite. CHAP. XCIIII. Of the building of a City, and of the situation thereof. IN our discourse of Topography, the building and situation of Cities, houses and such like, is right necessary to be remembered: but for Cities of defence they require a long discourse for their situation, as well in respect of their walls, etc. to defend, as of Turrets, Mounts, etc. to plant ordinauce in and upon to offend; only therefore touching health, let your city be planted by a fair and portable river, far from marshes and fenny places, for the vapours rising thence be unwholesome, and in as barren and fruitless a place (yet dry and firm) as you may, for in short time the compost and scavenger's dirt will soon make the conterminating soil batfull and fertile, as may be seen by London, which of itself, according to the nature of the soil, stands but in a dry and barren place, though it be forced rank by the abundance of compost. Sandy ground is right necessary for the plantation of a city, and for the plat ground hereof, let it not be altogether very level and plain, but have pleasant ascents, and rising banks, which will cause the city to be more pleasant to the eye, healthful to the body, and fit for warlike defence, as it may be seen by old Rome. (which now lieth ruinated) there were seven such hills, in the head of the city stood mount saturnal, towards the midst of the city were two other mounts, called Palatine and Quirinal, upon the left hand of the city was the mount Esquiline, upon the right hand Caelian, and towards the end of the city were two other mounts called viminal and Aventine, all which mounts much beautified the city, and thereon were many sports acted, where also was (and would be in other cities) fit places to erect pyramids, or other such city ornaments. Touching the streets there should be four main streets lying into the 4. Cardinals of the world, that is, one running 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 city Alexandria builded, and these streets would be spacious and broad, so shall the mind blowing from any quarter come in and pass through the whole body of the said city, and thereby purge the same of all corrupt and ill vapours, & such like, that commonly occupy cities. For the cities be ●●●y unwholesome, and apt to breed infections, where the streets be close, shutting out the open and pure air, which sure is an imperfection much to be lamented in London, & would heedfully be regarded in the new plantation in Ireland. And as you have divided the city ●nto 4. quarters, so may you appoint other collateral streets which will also receive the collateral winds, whereby any air stirring, the City shall have benefit thereof. Certainly most excellent, right pleasant and necessary would it be to see a City thus builded: for our cities 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 form or regular fashion, and now not to be reform unless it were all built anew, CHAP. XCV. Of the situation and building of a Manor house in the country. HE that will build himself a house in the country, should have a special regard that it t●e pleasant, delightful, & necessary in all respects, because he commonly spendeth a third part of his life therein: yet this Proviso would be had, that he proportion his house according to the quantity of the ground that he hath to lay thereunto, insomuch that there would be 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 judge of the quantity of his land, or viewing the land, might conjecture of the proportion of the house: for a fair house without land (such city follies that are often built out of London) are neither commendable nor necessary, and therefore they have begot themselves a nickname, or byname, as Mock-beggar. And in this point (as Pliny reporteth in his 6. Chapter, book 18. of his natural Histories, there were two men living at one time, who much halted herein, to wit, Lord Lucullus & Q. Scaevola, for Scaevola had fair lands, without a competent house and Lucullus had a competent house without lands, in which regard he was checked by the Censors (as many Londoners may) for sweeping more flowers than he ploughed lands. Touching the situation of your house, the best opinion now is, upon a hill, or hill side, having before the same a plain Champion country, for such grounds be dry and wholesome, if the air be good: for men thereby are made of a lively spirit. Pliny would not have a house situate near unto a fenny and dormant water, or over against the course or stream of a running water. Homer saith, the air and mists rising from great rivers before the the Sun rise, are unwholesome: howbeit you shall find it pleasant and necessary, to have a clear river fluant and running at a reasonable distance from your house: for besides the pleasure, you shall find it necessary for vaults, and such like, that carry filth from your house to empt themselves into. In any case situate your house distant from marshes, fens, plashy and foggy grounds, which are utter enemies to health. And in the politic situation of an house divers wise and honest men have much laboured to be far from wrangling and turbulent neighbours, which they hold as great an inconvenience as want of wholesome air, and for that part of the heaven that the face and open side of your house should behold, you must have regard unto the nature of the country, and quality of the wind issuing from that part of the heavens. Pliny would have you sittuate your house in a hot country, into the North and in a cold country to affront the South, but in temperate regions to lie open into the East. With us in England the principal coast for a house to lie open into, is Eastwards, as well for health, as in Summer for the avoiding of the extremity of heat of the midday, and afternoons Sun, which indeed is troublesome and uncomfortable, according unto Aristotle 2. Meteor, cap. 6. and also according unto Magirus, l. 6. c. 9 d. 13. The east wind with his collaterals is moderately warm and dry, and the wholesomest 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 & mind, filling the head with rheums, cathars and such like, destroying the stomach, and by the frequent blowing thereof it doth not only putrefy the bodies of living creatures, but also it corrupteth and putrefieth the fruits, whereby also quotidian fevers, pestilences, and other contagious sickness rise. Now in building a house much art is required Pliny reporteth builded a house in Cape Misenum, as he had fortified a Camp that C. Martius (who had been seven times Consul of Rome) right skilfully, that when Sylla, surnamed Foelix, saw it, said, that the rest in comparison of him were blind béetles, knowing neither how to build, or encamp. When therefore you mind to build a house, with your Scale and compass lay down the ground plate according unto your proportion, ordering your cellarage, larder, and all houses of office in as necessary form as to you shall seem most convenient, appointing places for great stairs, private stairs, houses of office, chimneys, etc. that shall be most requisite for use, and least annoying, or defacing the house, or any of the principal lights chambers, or rooms: then according to your ground plat, draw the forefront, or face side, backside, ends, and gabell ends▪ with all returns, iutteyes, soil pieces, windows, etc. even as you determine to have it made, but draw it not as commonly these Painters do proportions of houses by the eye, but lay it down by your Scale and compass, that by the application thereof at any time you may know how many foot or inches any return, any gable end, any story, or window is in length or breadth, which you shall be taught to do haply in some other place: thus upon several papers set out every several part of your house, whereby yourself, or the architector may inform the mechanical Carpenter of the length of every several piece of timber, & all things else required about the house, as the number of boards for flooring and dooring, the quantity of glass and tile, with the quantity of seeling, rough casting, paving, & other such like; whereby you may give order to the Glazier for the breadth and length of your glass, to the Tiler for tile, to the Plasterer for lime, to the Sawyer for boards, proceeding, no one thing staying for the finishing of another, thereby proportion your house according to your purse. Now for the addition of more delight unto your house, upon the South side thereof set out a fair square garden, beautified with bowers, walks and such like, as your gardener can best devise; adjoining unto which, let there be a fine orchard planted with trees: but if your climate be hot, as Spain, etc. plant your garden in the North; but for England the South is best, unless for some trees that naturally desire the shade: let there be no oxe-stall, dormant and filthy water, stable, or other thing that may breed noisome smells near unto your ga●den, make the alleys dry, for which I could teach you divers devices which here is no place for, and plant the trees in your orchard after a Chequer form, that standing at any tree, all the rest be in right line with you, which form is called a Quinqunx: within your house make your stairs large, not with these monnell posts, but with four steps and a half pace, a fair light answering to every half pace. Let the chambers be of a convenient height over head, and sufficient light, albeit the chamber you lodge in would not be over light, not yet a ground chamber, inclining rather to cold then heat; for by means of heat in sleep we may procure a swoon, because the heat of the body being become internal, and cold external, this enclosing heat and that cold will strive: let the place therefore be temperate, and free from noise, for sleep is a a cessation of the common senses, which being occupied & troubled with noise hindereth sleep: moreover keep the beams of the moon from your bed, for it is hurtful to the sight to have the moon shine upon your eyes sleeping. Touching the plaits and forms of houses, some affect the quadrant building, with a square court enclosed in the midst, like to the Colleges, or as the Royal Exchange, which indeed in respect of the columns and arches making the under walks, is more stately: again, some affect the Roman H. some other forms; but that must be partly referred unto the pleasure of him that bestoweth the cost; and for my part, I intent not at this time to lay forth the diversity of plaits, and how they should be taken or laid down by scale and compass, for that haply I shall open the same in another piece of work more proper. CHAP. XCVI. Of the sinking of a Well, and of the conveying of water in pipes. IF you desire to find a place where digging a pit you may also find water fit to maintain a well or pump, you must (ss jean Liebault writeth) early in the morning, your face into the East, look 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 found, or if such vapours rise in a dry and fair season, also if you dig trenches four or five foot deep throwing therein wool, that is clean and dry, covering the same with leaves, herbs, or such like, if then this wool having lain for a certain space still remain 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 witness. Also water is to be found under these ensuing herbs) Yarrow or Nose bleed, Vervain, wild pennyroyal, Venus' hair, Camomile, Dog's tooth, foxtaile, trifoly, Cinkefoile, Millefoile, Coliander, or as some say where abundance of green fern doth plentifully grow, or as L. saith where any other green herbs naturally flourish and abound without setting. Your springs thus found, they of longest continuance be which are in a grey or red gravelly rock, or ground, in a blackish, sandy, clayey or red stony ground, especially being mixed with stones and gravel. Now for the pipes for the conveyance of water, lead is good, earth is better, but wood of fir, Alder, or pine, or such other wood that hath rosin in it is best: such they use now in conveying of waters to houses from the new water mil in Westminster, they must be bored through with long agores, first with a less one, than with a bigger: any boughs or knotty pieces will serve, so they be large, & when the poles so bored, have not ground to lie strait upon, but lie uneven rising and falling, there be crooked pieces of wood like elbows provided of purpose, which are also bored through, being let a foot at either end into the other two poles it joineth together, & so are all the poles that be joined one to another made to go into the end of one another a foot or more, in manner as they piece bagpipes, or such like, the hole in the end of the one pole receiving the hollow end of the other pole into the same, being always for a foot deep wider than the rest of the boar, which you must join together with good cement, as you be taught before to do. CHAP. XCVII. A brief discourse how to draw the platform of any kind of building, or any other thing seen, though you cannot approach unto the same, and that according to true proportion, according as it appears or offers itself to the sight. IF you desire to project the due form of any object upon a plain superficies according as it shall offer itself to the eye at any appointed place & distance, as to describe any town, or city, any house, any flower, or any other body whatsoever, you must do thus. Take a fair piece of smooth glass, and fix the same upon a perpendicular at the end of a ruler, the which ruler let be divided into a number of equal parts, next upon this ruler must be another short perpendicular agreeing to the height of the midst of the Glass, and in the upper part of this short perpendicular must be a small and round sight hole, which done let the perpendicular be made to move equally fromwards or towards the Glass, or to stand fixed at any division upon the ruler as occasion shall be offered: this so ordered, when you desire the plat of any object as house or such like, plant the glass opposite to the proportion required, the ruler lying parallel, then move the shorter perpendicular near to or far from the Glass, even as you desire the proiectment to be great or less: this done place your eye in the small sight hole noting well through the same how every particular object doth appear upon the Glass, your eye so resting, with your pencel or diamond, draw upon the said Glass whatsoever you shall apprehend (or at the least whatsoever shall be required in your proiectment, and the work is finished. Now if you desire to make a scale for this proiectment, note the equal parts betwixt both the perpendiculars, which call your first number, then let the distance from your eye to the object be your second number, lastly draw a perpendicular upon the Glass from the summity of the object to the centre of the Glass, or rather to that part of the Glass 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 he length of that line to the equal parts upon your ruler, these 3 numbers had, multiply the second and third and divide by the first, so is the quotient the number of feet or inches, that the said perpendicular contains according as the distance of the object was expressed in feet or inches, of which make a scale and measure all the rest. And you must note in all proiectments prospectively that you can lay no more down but what you see, as in a 4 square house. you cannot possibly set down more than any of the two sides and somuch of the roof as you see and so of cities etc. and therefore you may lay down somuch of any city as your eye can apprehend, from any place where you plant your Glass. 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 form, newly devised. YOu must first prepare a ruler of brass or metal of such a length that may answer the Semidiameter of your plat (but make him long and he will be general) upon this ruler there is a socket of brass 4 square & hollow made to move equally along the ruler, and the length of the said ruler, but the lower side of the said socket is clean taken away, so that he hath but thrée-sides, and therefore he ought to be the stronger, and let the ruler be the lesser, for it is not material how small he be: in the end of a ruler there is a small centre 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 pencil, this socket you must divide into certain equal parts as 100, a 1000 or more numbering the same with figures as the order is: upon this socket there is another movable piece of brass which must hold a second pencil and that at any division required, he must have a screw pin to keep him steddyat that place he shall be appointed to stand, this done, your ruler is ready to work as thus: Fasten your plat that you intent shall be reduced, upon some plain table board, and about the midst of the said plat drive in a pin equal to fill the hole in your ruler, whereupon place the said hole, but let not the pin come above the ruler, next put your long socket upon this ruler, and let the end where the equal divisions do begin stand at the hole in the ruler, the which resting move the ruler and socket to any one of the next angles in the plat, noting how many of the equal parts cut the said angle, which let be 20, then say I would, have the plat one fourth part less, 4 in 20 is five times, therefore I place the second pencil 5 equal parts from the other in the end of the socket, where I fasten him unmovable with the screw pin, which done and the pencels being both of one length, do no more but draw the pencel in the end of the said-long socket round about the true perimeter of the plat, so will the second pencil describe you a figure lesser and proportional to the proposed figure. And if you please, you may cover 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 than the pencel next the centre must always keep the perimeter of the plat, and the other pencil shall describe a greater figure proportional to the lesser, according to the quantity assigned. And in the reducing of maps by this means may you give pricks for the situation of all towns, villages, etc. 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 never alter them until you have finished. Out of doubt this is a most exact and excellent kind of ruler, and most easy for every simple man to work withal, and if the hollowness in the socket be made like the channel in one of the legs of the Geodeticall Staff, and the ruler answerable thereunto, the socket will never 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 have a he add thereon, and a hole made in the end of the said ruler, to bury the pins said head in: when he is once knocked down to the end the socket may run over the same so will this pin's head keep the ruler from starting off the same. Let this brief description at this time suffice, which if I have said sufficient to satisfy your understanding. I doubt not but you will soon acknowledge the excellency thereof. CHAP. XCIX. To burn any thing a far of with the Sun beams. BY such a conclusion as this we read that Archimedes fired the Roman navy at Syracuse in the Island of Sicilia, which to do, you must take a number of steel glasses, made of purpose, and well polished: and then the Sun shining, place them in such sort that they may all reflect, or cast back the beams of the Sun upon the combustible matter, or subject that is to be fired: and the nearer together that the reflections fall upon, and in one point, the sooner is fire kindled. But indeed there be certain parabolical glasses placed by the aid of Geometry, more excellent for this purpose, having concaves and convexes, of which I cannot stand here to treat, neither is this conclusion so necessary in England, unless it be in Summer in the extremest of heat. CHAP. C. To make a Glass whereby to discern any small thing, as to read a written letter a quarter or half a mile off. WE have an imitation of such glasses as these about London commonly to be sold, but they be so small that they stand one in small steed, but amongst the writers of perspective, I have read that if you take a glass of the same metal that burning glasses be, and 16. or 17. inches broad, whose centre place directly against the object you look upon, and let it not incline, or hang sidewise by any means, behind this glass set a fair looking glass, the polished side beholding the said burning glass, to the intent to receive the beams that come through the same: which done, look in the looking glass, so shall you have your desire, if the burning glass were truly placed: for you must note whatsoever thing you see through the burning glass, that the further you stand from the glass, the bigger it seemeth, until you come to a certain distance, and then the object seen through the glass doth seem lesser and lesser, therefore care must be had in placing the glasses, so may you view a Town or Castle, or any window in the same, 6. or 7. miles, or see a man 4. or 5. miles, read a letter in written hand a quarter of a mile from you, etc. CHAP. CI. How you shall buy annueties, or sums of money to pay at a day yet to come. I Had thought to have concluded this Book which the divers nature of grounds, with the artificial mending of them, and the kill of Gorse that aboundeth upon cold clayey ground, and fern in a sandy and hot soil, or broom flourishing in barren ground, hot and dry, or moss spreading in a cold ground, all these, & more I thought once to have handled, but my mind altering, and withal bethinking myself of a Gentleman of worship, and my kinsman, one Master Whored, that had bestowed great cost and travel in dreyning of grounds, having now (as he told me) brought (to his great cost) the dreyning of grounds to such a perfect head, and easy method that he was able to treble the rankness of his ground only by waters, with the one quarter of the charges it was in the beginning: and indeed as the invention is new and excellent, though it hath been stumbled at by many, so no doubt but in time he will be persuaded to publish to the world (for the good of his country, and credit of himself) the form and method of the work, to which it would do well to add what I intended to have spoken of in this place which made me the rather refer it to a Gentleman, so honest, and well experienced. But to the matter. Whereas there be many annueties which some desire to buy, and some to sell, behold the ensuing Table, which telleth you what 10. pound anuety is worth for any time under 21. years, according to 10. pound in the hundred, and if you would apply the same unto any other sum more than 10. pound, use the rule of proportion, as I taught you in the 6. book of my Geodeticall Staff, chap. 51. which made me repeat it here again, because the Table is wanting there. Years. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 pounds. 9 17 24 31 37 43 48 53 57 61 64 68 71 73 76 78 80 81 83 85 86 shiling 1 7 17 14 18 11 13 7 11 8 19 2 0 13 1 4 4 0 13 2 9 pence. 10 1 4 0 2 1 0 0 ●0 11 0 9 8 4 2 9 3 3 0 9 9 Example. I have annuetie of 10. pound for 11. years, and now would know what I were worthy to have in hand for the same, by the former Table I find 10. pound for 11. years is worth 64. pound and 19 shillings, therefore by the common rule of three, if 10. give 64. pound & 19 shillings, what shall 100 pound be worth? therefore multiply 100 pound by 64. pound 19 shillings, parting the product by 10 pound, so have you 640. pound, and so much was your annuety of 100 for eleven years worth to be paid now in hand for the same. The like may you do by the ensuing Table, if a man own you 100 pound to be paid at any time under 21. years hence, and you are to buy the same of him now with present money, reckoning the sum of money that you give according to compound interest, that is, interest upon interest, as if one should owe you 100 pound, due to be paid 7. years hence, and you would know what you were worthy to give in hand for the same, according us is said, so shall you find 51. pound, 6. shillings, 13. pence. in the table under 7. years. Years. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 pounds. 90 82 75 68 62 56 51 46 42 38 35 31 28 26 23 21 19 17 16 14 13 shiling 18 12 2 6 1 8 6 13 8 11 1 17 19 6 18 15 15 19 7 17 10 pence. 2 11 8 0 10 11 13 0 2 1 0 9 4 8 10 3 8 8 0 3 3 And if the sum or number of years exceed your Table, work as in the last Table before, & so proceed, taking the sums according to the demand out of the Table, and if other sums are required, work, as I have said, by the rule of proportion, as before, making the sums in the Tables your Radix. Thus much (good Reader) have I said of the use of my topographical Glass, not doubting, joining this book and the Geodeticall Staff together, but that you have the ample use of all Geometrical, Geodeticall, and topographical Instruments, now extant, whereby you shall not be to seek further instructions, I mean such instruments that are now in most request, and by the better opinions most commended, and therefore for this time, I will conclude my Book, wishing myself present ever to explain any thing that to the young Practitioner shall seem obscure. SENECA. Nullum est tam magnum beneficium, quod non vilificare malignitas possit: nullum tam angustum, quod non bonus interpres extendat. The end of the topographical Glass. THE ORDER HOW TO MEASURE ALL KIND OF TIMBER, as well such trees as be growing, as such as be fallen and squared, as also all kind of Globes, Pyramids, Cylinders, etc. as well as excavated, with all Superficies, be they stones, pavements, boards, glasses, or such like, partly by the Geodeticall Staff, in such sort, and in such an easy method, as hath not before been published. CHAP. I. Of the falling of Timber, and the best time for the same, with the seasoning of boards, and the preserving thereof. TO tell you what wood is best for timber, in brief, the oak is principal of all trees growing, both for magnitude and duration, although in some countries, by reason of the scarcity thereof, they be forced to use other wood. Concerning the falling of timber, you are best before the retiring of the sap to cut the timber tree round about, until you come near unto the end of the sap, and so leave the tree growing until you see that the sap hath all run down, and ceased dropping, whereby you have purified the tree, and left no moisture within to corrupt or putrefy the same: and that your timber may be the more sound, void of worms, and without rifts and chaunes, let the same be fallen any time after Midsummer until the end of januarie, especially in the full moon, but not in wet and windy days, leaving such for no use but snell that were wind falls as well for that the timber is not permanent, as also for that Masters in Astrology say houses built therewith be more subject to danger in time of tempest. Now if you convert any of this timber to boards, many use to throw them into water, and there let them remain 14 or 15 days, so shall they be the sooner seasoned and better exempt from worms. And if you desire to make columns or great turned pillars of whole trees or sapplins, to the end they shall not rend or chaune, with great long agors bore out the heart thereof; but of these and such like haply I shall be occasioned to say more in another treatise. And in conclusion, of falling of great timber trees, one principal thing I would have you to observe when you fall your trees, that is to crop of all the great arms and master boughs of the tree before you fall the same, because most commonly they be the occasion of the spoiling 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 tree falling with such a weight, and pulling the rest down with such a violence, much of the tree is shattered and rend, as many may see daily to their loss. CHAP. II. Of measuring of all kind of timber. MEasuring of timber is performed after two manner of ways, the first takes respect unto all kind of timber growing or fallen of what form or fashion soever it be, and so telleth you the true square of it, and consequently how many inches or feet are required to make a square foot of timber; the other taketh no notice of the quantity of the square that the tree or timber will bear; but be it round, square, triangulate, or multiangulate, it telleth you how many feet or inches is in the same: both which ways shall you be taught to perform hereafter. And for finding of the true square of any piece of timber, M. Digges hath taken pains to calculate tables; but here you shall be taught to find the same without any kind of table or calculation, which is right fit for all masters of works, carpenters, and masons to know; for the finding of the square of a tree, is not to gird the same about, and take the fourth part thereof, as they think, therefore the 35 Chapter of my sixth Book errs in that point, being set down hastily according unto vulgar tradition. CHAP. III. To find the true square of any growing tree by the Geodeticall Staff. YOu shall in performance hereof take some packthreed gut string or such like, and with the same gird the tree about 4 or 5 feet above the ground at the least, but towards the midst of the tree were best; then shall you take the length of the string that girded the tree, and divide the same 3 into four equal parts, taking one of those equal parts, & place the same truly over in 110 and 110 equal parts, upon the legs of the Geodeticall Staff; the legs so resting, take the distance over in 70 and 70, and note that line, which you must fit over in 60 and 60 amongst the chord divisions upon the lower side the legs, and so that angle not stirred, take the distance from 90 ●o 90, and apply that distance to your rule, so have you the true square of the timber tree. Example. A b is a growing tree, whose square is required, that is, how much square that tree will bear on every side when he is squared: first therefore I gird the tree about with a gut-string as far from the ground as I can reach, as at r, then do I note the length of the line that girded the tree, as c d, which I divide into 4 equal parts, reserving one of the said 4 parts, as c e, then do I take the legs of my staff, and fit the length of c e over in 110 and 1●0 parts amongst the equal divisions: the legs resting there, I take the distance over from 70 to 70; the length of which line I keep, as l m, and turning the other side of the legs upward I place l m over in 60 and 60, and so taking the distance over from 90 to 90 (the legs being not stirred) you have the line n o the one side of the square that the tree will bear, which applying to your ruler divided into feet and inches, you shall find 2 ●/5 foot the just square, which is equal to f h i g the 4 side of the tree being squared. By this means may you find the true square and consequently the quantity of any growing tree, & so buy the same according unto your desire, by which the Carpenters may see the error they run into by taking a fourth part of the compass of the tree for the square. CHAP. FOUR Any tree or round piece of timber unsquared and match tapering, to find the square thereof as it lieth upon the ground. Or as is said, you might have girded the tree about in the midst at k, and so worked as in the last chapter. Note if you please, having a pair of Calleper compasses, you may omit to gird the tree about, whether it be standing or fallen, only by taking the true diameter or thickness of the said tree, and placing half the same over in 60 and 60, and so work as before in this Chapter. CHAP. V To find the true square of a squared piece of timber consisting of two unequal sides, and 4 right angles the one side being only known. YOu must take the length of the broader of the two sides, the which fit over in 60 and 60 amongst the chord divisions; the legs of the Staff so resting, the distance taken from 36 to 36 yields the true square of a piece of timber that being of equal longitude is also of equal quantity. But if both the sides c d and d b be known, then work by the next Chapter, for this takes no notice of the thickness. CHAP. VI To find the square of any broad or flat piece of timber that consists of 4 right angles and two equal sides. Such a piece of timber as this, the end thereof doth represent the just form of an Oblong, and is thus squared; take the longer and shorter side, and join them together in one right line, the which right line made of the length of both these lines so joined, make the diameter of a circle: lastly, upon the point where the two lines were joined, raise a perpendicular, forth length of that perpendicular to the circumference is the true side of the square. Example. A b c d is the end of a piece of timber, c d the longer side, d b the shorter, therefore I take the length of c d and d b and joint them together in v, making one right line thereof, as r s: next I part r s into two equal parts at w, then placing the one foot of my compass in w, extending the other to s or r t, describe the semicircle r t s: lastly, upon v where the two lines were joined together raise a perpendicular v t, which is equal unto p q, I conclude the square made of the line t v is equal to c d b a, and thus of any such other. Or get the square thus, multiply the breadth in the thickness, so is the square root of the product the true square, which you may easily find in the Geodeticall Staff, fol. 142. CHAP. VII. To find the true square of any piece of timber, whose ends are form like a Diamond. THe end of such a piece of timber as this, doth represent the just form of a Rombus, & therefore doth consist of equal sides and Obliqne angles, the square whereof find thus; Draw a right line betwixt any of the two opposite angles, noting the length of that line, upon which line let fall a plumb line from one of the subtended angles; so having those two lines, find the square, as in the last Chapter. Example. Or the length of the perpendicular b m is the square falling upon b c at right angles. CHAP. VIII. To find the square of any piece of timber consisting of three sides. THe true square of all kind of Triangles whatsoever are found out by the 44 Chap. Metamorphosis 7. of the 6. Book of the Geodeticall Staff, and therefore it were vain to repeat it here again. As if you join the perpendicular b d and half the base a c in one right line, according to the 6 Chap. you shall find the square of the triangle a b c to be h i. CHAP. IX. To find the square of any piece of timber containing 5, 6, 7, or 8 sides, etc. YOu must imagine in this Chapter, as also in all the other, saving for round timber, that I go not about to tell you how much square that piece of timber would bear, if it were reduced into a 4 square: but I do deliver you the side of a piece of timber being just four square, and of equal height with the piece proposed shall also be of equal quantity, which is right necessary for the attaining of the number of square feet or content of any piece of timber. Thus are you taught to find the square of any piece of timber, of what fashion soever: and if it bear none of these regular forms, or that there be wood wanting, take from one place, adding the same unto another, thereby making it perfect regular, and in such cases you must always so do. CHAP. X. The square of any piece of timber being found to tell how much of the same in length, will make a square foot of timber, and consequently how many foot is in the whole piece. A Solid or cubical foot of timber doth contain 1728 cubical inches, for so many four square inches may be taken out of one cubical foot, I mean such Inches that are square every way like unto a dye, now having the square of any piece of timber given, square the same, dividing 1728 by the product, so doth the quotient show you how much of the length of the timber must be taken to make a square foot, by the which divide the whole length or altitude of the timber, so doth the quotient acquaint you how many foot of timber is in the piece. Example. The square of the tree a b is found by the third Chapter to be 27 inches, whose square is 719 by which divide 1728 so have you 2 2/7 9/2 0/8 inches which is 2 ⅜ inches, or two inches a quarter & half a quarter, whereby I conclude, that as often as I can find 2 ⅜ inches in the length or height of the tree or timber, so many square foot of timber is in the same: the tree a b is 8 foot high, which divide by 2 ⅜ inches, or lay somuch of your rule out measuring one from a towards b, calling every 2 ⅜ inches a foot, so by either of the ways shall you find 40 foot of timber in the said tree being squared, some small quantity being over more than the same. In the like manner must you deal with all other pieces of timber of what fashion soever, first finding their square as before, & next the capacity, even as you be taught in this chapter. By this Chapter may you measure out as many foot of timber, stone, or such like, as you please, & thereby cut off any number of feet from any piece of timber as you shall be occasioned. CHAP. XI. To measure all kind of Timber, etc. after another sort, without regard of the square. THis kind of measure taketh no regard to the square feet in the timber, but unto the solid capacity thereof, but for that it is not much pertinent to the Geodeticall Staff, requiring rather numeral then instrumental operation I will be the more brief. When you have any piece of timber, stone, pillar or such like, whose solid content is required by the rules taught in the sixth book, part 2 of my Geodeticall Staff, seek the superficial contents of the end of the timber, the which augment in the altitude or length of the same, so is the product your desire. Example. The like must you do with any other piece of timber of what fashion soever, but if the timber trapeze much, use the midst or difference of the ends as in the 4 Chapter, which at this time shall suffice. And if your timber be excavate or hollow, first measure the whole as it were firm, than the excavate part by itself, lastly, the lesser taken from the greater leaveth your desire. CHAP. XII. A discourse of surfaces and solid figures, not so apt for the understanding of mechanical artificers LIke as the term of a line is a point, so is the term of a surface a line, every surface is plain or bowed, a bowed surface is either spherical or varied, a spherical surface is equally distant from the centre made by the revolution of a semicircumference upon the fixed diameter, a varied surface is either conical or cylindrical, a conical surface runneth narrower and narrower from the circular base unto a certain point in the top, a cylindrical surface is that which is equally raised from the circumference in the bottom to an equal and parallel circumference in the top made by the convolution of the side about two equal and parallel circumferences. Now from surfaces we proceed to bodies. Even as points are the terms of lines, and lines the terms of surfaces, so surfaces are the terms of bodies: a body is a lineate, broad and high, consisting of 4 dimensions, which being contained under homogeneal surfaces equal both in multitude and magnitude be also equal, and if the axe be perpendicular to the centre of the base, they be called upright solids: of solids some are plain, others bowed: the plain solids are contained under plain surfaces, and is either a pyramid or a piramidate: a pyramid is a plain solid rising equally from his right lined base and uniformedly contracting itself until it finish in a point in the top: now a piramidate is a plain solid, composed of pyramids, being either a prism or a mixed polyedron, a prism is a figure piramidate whereof two opposite plains be like equal and parallel, 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 parallel pipedon, or a trapezium, the parallel pipedon, is an hexaedron whose opposite plains are paralellograms, now every right angled parallel pipedon, is either a cube or an oblong, so that a cube is a right angle parallel pipedon consisting of 6 equal surfaces. As for your mixed ordinate polyedrons, they be but pyramidats composed of pyramids lying open in the base, and concurring with their tops in one centre, and from these mixed ordinate polyedrons are derived your regular bodies, as you may perceive by Euclid 27 D. 11, & 29, D. 11, etc. so that bodies be regular or irregular, the regular bodies contained under surfaces the one folded towards the other be only 5, as tetraedrons, hexaedrons, or cubes, Octaedrons', Dodecaedrons', & Icosaedrons', the first being a Geometrical body encompassed with 4 equal equiangled tryangles, the second with 6 equal squares, the third with 8 equal equiangled tryangles, the 4 contained under 20 equal equiangled tryangles, and the last being comprehended of 12 equal equiangled pentagonal superficies, about every of these 5 regular or platonical bodies, a comprehending or circumscribing sphere or globe may be described that shall with his concave periphery exactly touch every of their solid angles, whereby they be made bodies inscribed or contained of that sphere: also these inscribed bodies may be termed circumscribed solids of a sphere, & then the sphere is called inscribed or contained, which happening the conver superficies of the said inscribed sphere shall precisely touch all the centres of those equiangled figures wherewith the bodies are environed: touching irregular bodies they be such that be limited and described by inequal surfaces, which are twofold. to wit in respect of circular convolution and in respect of folding one towards another, the circular convolution is made two ways, as by the section of circles, or by inequal right lined figures, the section of circles are either greater or lesser than a semicircle, by the convolution of the greater lenticulare bodies are made, and by the lesser Duals are created, now by the conversion of equal right lined figures divers kinds of irregular bodies are made, & thereby divers kind of vessels: as for the irregular solids made of inequal surfaces folded one towards another, the difference that may rise therein is infinite. Like as a surface, so a solid is plain or bowed, & being bowed it is either a sphere or varied, a sphere is a round bowed solid contained under a bowed surface, made by the revolution of a semicircle upon a fixed diameter, as for the varied solids, they be contained under a varied surface and a base, and are twofold, as a cone and a cylinder, the cone is contained under a conical surface and a Glass, and the cylinder under a cylindrical surface, and opposite bases, and as for this cone and cylinder the one is made by the conversion of a right angled triangle (the one foot remaining fixed which if it be equal with that which moveth the cone is right angled, if less obtuseangled, if greater acute angled) the other by the conversion of a right angled paralellogram the one side remaining fixed: further in cones the fixed side of the triangle is called the Aris, the containing side turning about the base, and the hypothenusal the side of the cone: furthermore amongst these solids the perpendiculars, falling from the highest point of any figure upon the plain whereupon the resteth is called the altitude of the solid▪ neither is it material if the same perpendicular fall within or without the body, as the one always doth in direct solids within, and the other in declining solids without: now as plain angles are made upon superficies by the section of two lines in a point, so solid angles are made in bodies by the concourse of many supeficies in a like point, and so a right line passing from one of these solid angles unto another is called a diagonal, but passing betwixt opposite angles it is a diameter. CHAP. XIII. To measure the Contents both superficial and of Cones, Cylinders, Pyramids, 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 been said of solides: yet since you be taught to find the superficial content of any figure in my art of Geodetia, as also for that I intent to conclude, I will draw some propositions from Euclid and Ramus for the measuring of them, and so refer the further discourse unto the understanding Reader. PROP. I. To measure Cones. 1 INcrease the side in half the bases Periphere, so have you the content of the conical surface. 2 Likewise the altitude of the Cone augmented in the third part of the circular base is the content. 3 Cones of equal height are as their bases. E. l. 12. p. 11. By this Proposition you may measure Steeples and such like round or conical figures. PROP. II. To measure Cylinders. 1 THe circular base and the altitude increased one in the other yields the content of the cylindrical surface. 2 Also the plain number made of the base and the altitude is the content of the Cylinder: I mean the superficial content of the bases area augmented by the altitude yields the capacity. 3 Cylinders of equal height are as their bases. E. l. 12. p. 11. and are triple to the Cone, equal in base and altitude. R. l. vlt. pa. 7. By this Proposition may you measure all kind, of columns, pillars, or any sort of cylindrical bodies. PROP. III. To measure a Pyramid. 1 BY the art of Geodetia measure the contents of every environing triangle (of which the Pyramid consists) which add together, adding the product unto the superficial content of the base, so have you the contents of the Pyramidal surface. 2 Also increase the third part of the bases area in the Pyramid altitude, so have you the content, be the body direct or inclinate. 3 Pyramids of equal height, are as their bases. E. l. 12. p. 5. and 6. Hereby you may measure 6, 8, etc. square spire steeples, and all such Pyramidal figures. PROP. FOUR To measure a Prism. 1 GEt the area of all the paralelsgrams and bases, adding the same together, so have you the contents of the prismatical surface. 2 Also the area of the base increased by the altitude produceth the content. 3 A Prism is triple unto a Pyramid, equal in base and altitude. E. l. 12. p. 7. and Homogeneal Prismes being of equal height are as their bases. E. l. 1. p. 29, 30, & 31. PROP. V To measure a Cube. 1 GEt the superficial content of one of the equal surfaces, which multiplied by 6 produceth the content of the cubical surface. 2 Also square one of the 12 equal sides, then doth the product multiplied by the same side produce the capacity of the Cube, and after this manner any number is cubed; as 3 times 3 is 9 the square, and 3 times 9 is 27 the Cube, 3 being the square root of 9, and the cubical root of 27. PROP. VI To measure a Sphere. 1 RAmus l. 20. p. 5. hath proved that the plain number made of the greatest circumference, and the diameter, is the content of the Spherical surface: 2 or get the plain number made of the greatest circle, which increase by 4: 3 or multiply the square of the diameter by 22, parting the product by 7: by either of which ways you have the Spherical surface. 2 Furthermore the plain number made of the diameter and the 6 part of the Spherical surface is the Sphere. R. l. 26. p. 5.2. or as 21 is to 11, so is the cube of the diameter unto the sphere, ibid. therefore cube the diameter by the 5 Prop. 2, which multiply by 11, the product then parted by 21 produceth a quotient that shall contain the content of the Sphere. 3 Spheres have triple proportion unto their diameters. E. li. 12. p. 18. Hereby may you measure Globes or such other round bodies. PROP. VII. To measure part of circles. FOr the superficial or content of parts of circles, look what I have said of the whole, and proportionally understand the same of the part; as the plain number made of the circumference, and the Radius is the content of the semispherical surface; so the plain number made of the Radius and 6 part of the spherical surface is the content of the Hemisphere: so of other parts of circles, which here were tedious to recite. CHAP. XIIII. To measure the air or superficial capacity of any plain surface, as boards, glass, floors, pavements, etc. IF you would know how many foot or inches in length you must have to make a foot of board or glass, the breadth thereof being given, or how you shall cut off any number of feet proposed from any assigned board or such like, repair unto the 34 Chapter of my Art of Geodetia. But otherwise, if you have the length and breadth of any board, floor, glass, or such like, and are desirous to know the quantity of feet or yards therein, you must multiply the length in the breadth, so is the product your demand: after this sort do Painters and joiners measure the quantity of painted clothes, or wainescot for chambers: and no otherwise do Sielers meet their sieling: also in the same order may Tilers (especially those that use burnt tiles, which commonly be of one bigness) tell how many stones will cover any roof, or how many thousand be upon any covered roof, only by multiplying the number of stones that go along the eaveses of the house, by the number that go up the side from the eaveses to the top of the roof or first pole. SENECA. Plus caeteris dedit, quia sine spe recipiendi dedit. A TABLE OF THE PRINCIPAL THINGS CONTAINED in this Book. WHat Topography is: and how it differeth from Cosmography and Geography. chap. 1. pag. 1. A demonstration of topography. ibid. p. 2 A demonstration of Geography. ib. p. 3 Geometrical definitions of lines, angles, and figures. chap. 2. p. 5 How right figures are created. c. 3. p. 9 To create a right line, or rear a perpendicular. Pro. 1. p. ib. To rear a perpendicular upon extremes of a line assigned. Prop. 2. p. 10 To draw a perpendicular from one point assigned to another. prop. 3. ib. To make a right angle readily. pro. 4. p. 11 To make an angle like to an angle assigned. pro. 5. p. 12 To draw a line parallel to any assigned line. pro. 6. p. 13 To divide a line into two equal parts. pro. 7. p. ib. Three points given to find the centre of a circle that shall cut all the three points. prop. 8.14 The making of the topographical Glass. ch. 4. p. 15 To set together the parts of the topographical Glass. ch. 5. p. 26 A description of the Theodelitus. ch. 6. p. 27 To search the proportion and symmetry of countries, fields, etc. ch. 7. p. 29 To take the true plat of a small Isle compassed with a River, or of Fens not accessible to. ch. 8. p. 32 To take a plat at one station by the Theodelite. ch. 9 p. 34 To take a plat of wood-ground by going about it. c. 10. p. 35 To draw the plat or map of any country, with each towns and villages situation. ch. 11. p. 36 To draw the plat of any region, finding the distance of towns by cynical supputation. c. 12. p. 41 The ground and reason of the Geometrical Quadrant, & hypsometrical scale. c. 13.46 To get the distance of a place far off. c. 14. p. 48 To seek the distance of any mark seen, by the Geometrical Quadrant. c. 15.50 To find the distance betwixt 2 forts far off. c. 16. p. 52 To take the height of any accessible tower, castle, etc. c. 17. p. 55 To search out heights inaccessible. c. 18. p. 57 To know what part of any altitude is level with your eye. c. 19 p. 58 To search out lengths in heights. c. 20. p. 59 To reduce parts of the right side the Geometrical quadrant into parts proportional of the left side. c. 21.61 To find lengths in heights by the Geometrical quadrant in the Glass. c. 22. ibid. To know how much one hill is higher than another. 23.64 To know if water will run to the appointed place. c. 24.67 To take the quantity of any stationary angle. c. 25.70 To make a protractor & scale. c. 26. ibid. To protract an angle and lay down the ends. c. 27. p. 71 To observe an angle of position and protract it, etc. 28.73 To take the plat of a great champion, etc. c. 29. p. 74 To plat meadows, plain fields, pastures, etc. c. 30.75 To reduce lines hypothenusal into lines horizontal. 31.76 Compendious forms of working by the Geod. staff. 32.77. To square lands, and to reduce irregular forms to regular. c. 33.80 To search the perpendicular in any triangle, etc. 34.81 To reduce many plaits or observations into one map. 35.82 To divide an Empire, kingdom or continent into Prou. 36.85 Reasons why the sun's altitude hath been hitherto falsely observed. 37.87 Paralaxes of the sun. ib. 88 A table of the suns paralaxe. ib. 90 Correcting the taking of the stars altitude. 38.91 To take the altitude or Almicanther of the sun or any star, and to find their Azimuth. 39.92 To take the amplitude of sun or star. 40.93 To get the hour of the day, the hour of sun rising or setting, by the Topographical Glass. 41. ib. To find the hour of the night, & likewise high water. 42.64 Additions to the planisphere in the Glass. ib. 95 To use the topographical Glass as the Plain table. 43.97 A description of the plain table. 44.98 Absurdities used of many who affect the plain table. 45.102 Things belonging to the use of the plain table. c. 46.104 To take any horizontal distance by the plain table. c. 47. ib. Part of the distance of any thing given to find the rest. ch. 48. p. 107 To take the distances of two towns, etc. c. 49. p. 108 To find the horizontal distance from you by a new way. c. 50. p. 110 To draw the plat of a piece of ground at one station, where all the angles of the field may be seen. 51.112 To draw the plat of any field, where you can not see all the angles. c. 52. p. 113 To draw the plat of a field by once placing the instrument in an angle of the field, and measuring the field round about. c. 53. p. 115 To take the plat of a field by the rule of the foregoing Chapter, where all the angles cannot be seen from one angle. c. 54.116 To draw the plat of a piece of ground by two stations, and measuring but one line. c. 55.117 To draw the plat of a field by many stations, and yet measure but one line in all. c. 56. p. 118 To draw the plat of a piece of wood-ground, which for thickness one cannot set an instrument in. c. 57.120 To draw the plat of a field by setting his instrument in every angle, yet measuring but one line. c. 58. p. 122 To take the plat of any champion field by the plain table, yet never change paper. c. 59 p. 123 What chapter is fittest to use in plaiting of ground, & what instrument to use. c. 60.124 A description of the circumferentor and the parts thereof. c. 61. p. 126 Of the Sights longer & shorter c. ib. p. 127 The Circumferentor, his appellation, and things generally to be considered therein. c. 62. p. 129. To take the Almincanther and Azimuth of the sun. 63.130 To know in what part of the horizon any thing seen lieth. c. 64. p. 131 To find the hour of the day by sight of the sun. c. 65. p. ib. To find the hour of sun rising and setting. c. 66. p. 132 To find the amplitude of rising sun or stars. c. 67. p. ib. Of the opposite degrees, and how to find them. c. 68.133 To find the quantity of an angle. c. 69. p. 134 To take the distance of any mark by the old Circumferentor. c. 70. ib. To perform the last Chapter by protracting the Circumferentor. ch. 71. p. 135 To take an Altitude only by the Circumferentor. 72.136 To take the plat of a piece of ground by the old or new Circumferentor. c. 73. p. 137 To take a plat at one station by the circumferentor, 74.139 Degrees of a field being taken, to find the closing of the plat. ch. 75. p. Ibid. To reduce hypothenusall lines into horizontal. c. 76. p. 141 To perform the same by a Quadrant. ch. 77. p. 142 To take Altitudes by such a Quadrant, ch. 78. p. 143 To take the declination of a wall. chap. 79. p. ib. Survey of a Manor. ib. p. 144 To make a Map and sea Card. ibid. p. 146. To discover the true plat of a park, forest, etc. c. 80. p. 148 To cast the contents of a park chap. 81.153 To plat any field by intersection of lines. c. 82. p. 159 To seek the distance of a Turret. c. 83. p. 158 To find the length of any hypothenusal. c. 84. p. 160 To find the distance of two Towers. c. 85. p. 162. To find the distance of any thing from you. 86. p. 165 To know whether a ship come to you, or go from you. chap. 87. p. 167 A ship pursuing another, when it will overtake the former. chap. 88 p. Ibid. How to take the platform of a house, Castle, etc. 89. p. 168 To discover how Mines and Trenches run. ch. 90. p. 169 To place barrels of powder under Castles, etc. ch. 91. p. 170 Whether a mine be above or under the Horizon. chap. 92 p. 171 To know which way it declineth. ch. 93. ibid. To build and situate a City. ch. 94. p. 173 To build and situate a Manor. chap. 95. p. 174 To sink a well, and convey water pipes. c. 96. p. 177 To draw the plat of a building, or other thing not seen. ch. 97. p. 179 To make an excellent ruler for reducing plaits. chap. 98. p. 180 To burn any thing far off with the Sun beams, ch. 99 p. 182 To make a Glass to discern any small thing half a mile off, as to read a letter, etc. c. 100 p. ibid. How to buy annueties, or money due afterwards, ch. 101 p. 183 The measuring of Solids and Plains. THe best time to fell Timber, and to season boards. chap. 1.186. To measure solid timber. 2.187 To find the square of any tree growing. 3.188 To find the square of a tree unsquared. 4.189 To find the square of a squared piece. 5.190 To find the square of any flat piece. 6.191 To find the square of a piece like a diamond. 7.192 To find the square of pieces of 3, 5, 6, 7, or 8 sides, look the 8 and 9 chap. p. 193 To find how much timber will make a foot square. 10.195 To measure all sort of timber. 11.196 Of surfaces and figures. 12.197 Measuring contents superficial and . 13.200 To measure the air or any plain surface. 14.202 FINIS. woodcut, radiated circular compass with Roman and Arabic numbering and astrological signs You may have any of the Instruments in this book made of wood, in Hosier lane, near Smithfield in London, by john Tomson. The Glass is made in brass, in black Horse-ally, near Flecetebridge, by Elias Allin.