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BACVLVM GEODAETICVM, sieve VIATICVM. OR The Geodeticall staff, Containing eight books: The Contents whereof follow after the Epistles.  
Newly devised, practised, and published by Arthur Hopton Gentleman.  
AT LONDON, Printed by Nicholas oaks for Simon Waterson, dwelling at the sign of the crown in S. Pauls Church-yard. 1610.   

TO THE RIGHT honourable, ROBERT earl OF SALISBVRY, KNIGHT OF THE MOST honourable ORDER OF THE GARTER, VIScount Cranborne, Lord cecil of Esingdon, Lord high Treasurer of England, &c. master of his majesties court of Wards and liveries, Chancellor of the university of Cambridge, and one of his majesties most honourable privy council.  
NO man right honourable, findeth a precious ston, bearing the splendour of any rich margarite, but strait hasteth unto the best lapidiste, whose happy allowance thereof begetteth a rare affectation, and inestimable value of the gem: so standeth it with your Honour, the curious Lapidiste of our most excellent wits, to whose touch every refined tongue offereth the quintessence of his best inventions. And therefore some, whose industries haue laboured in the treasury of divinity, some in the necessity of humanity, some in the delectation of the politics, some in the affectation of the economics, some in Philosophy, others in Poetry, haue all brought the depth of their golden studies, to bide the touch of your noble allowance: so that after-ages may rightly admire what noble maecenas it was that so inchayned the aspiring wits of this understanding age to his only censure, which will not a little magnoperate the splendour of your well known Honour, to these succeeding times. Which seeing,& having found this spark of the mathematics, never yet seen before, am bold to crave your noble approbation, that it may go( though unworthy) amongst the rest of those worthy jewels, that already haue past the proof of your singular iudgmet: So shall we haue great hope, that the radiant beams of your resplendent wisdom, favourably shining a little hereon, in short time will revive the mathematics( the Phoenix of Arts) from the ashes of oblivion, that they may sit in this kingdom amongst the sweet inventions of the age, and in our own language sing forth the eternal glory of learning, as well as in other nations it doth. every Muse( noble patron of virtue) hath long since found a double-topped Pernassus in England to solace vpon, save onely urania; every Muse hath her delicious Helicon to sport in, save only urania; every muse hath sipped of the sweet Nectar, save onely urania. Erato hath won a crown, urania wants a garland: then let me humbly implore your Honour, graciously to protect the fruits of this estranged Muse under the shadow of the sweet branches of your never decaying glory, which is all-sufficient to infatuate and annihilate each viperous malediction of the envious Pierides. I know there be books extant treating of the art of measuring ground of, which a part of this book also consisteth; but they be lame and defective, even as a number of our surveyors be, that thrust themselves into businesses without ability to perform: wherefore I haue endeavoured( my good Lord) as well to redress wants in the one, as replenish art in the other, not doubting( seeing it becomes so requisite& necessary a faculty) but to make the serious student able hereby to benefit his country and common wealth very much.  
Lastly( to cease troubling your Honour) if the dedication of books in any part may give acknowledgement to great men for peculiar favours and many singular graces shown, then doth duty bind me obsequiously to remunerate your honourable fathers respect of our name and poor family, to whom my ancestors were,( and of a necessary consequence, myself to your Honour am) ever obliged. And vpon that foundation I builded the hopes of this presumption, which if it prove firm, shall animate me hereafter, to aim to reach in some higher strain the Diatessaron of your memorable virtues. Thus in service at command, in duty bound, and in prayer continual to the Almighty for your good Honour, I end, resting  
In all humble duty at your Honours command, ARTHVR HOPTON.   

To the benevolent Reader.  
THe opinions be manifold, and the instruments multitudes, gentill Reader, that the diversity of pregnant wits haue brought forth in this age, and every of those instruments serve for some singular conclusions; but I happily finding the contriving of an instrument not bound to any particulars, but free, apt and general, so that he is most fit to work, most facile in working, and most easy to carry; out of my love to those that long after these arts, I could do no less then commit it unto their reading, hoping as my intent was to benefit them with my labours, so likewise they will graint a favourable vouchsafe to requited my pains, and that is all I seek; which finding, I will answer then that this book shall teach them to seek dimensions, both astronomical and geometrical, to perfit and measure grounds, both arithmetical and mechanical, in such a compendious method, that the unlearned never trained in the studies of these artes before, may with admiration accomplish those things that learned men haue passed over the moety of their lives to find out. And if this instrument please you not, you may turn further, and look vpon the Topographical glass, an instrument no less singular and commendable then the former, and for your practise whereon I haue set down many topographical and Geodeticall conclusions, which because I aim to suit you with varieteis, behold the use of the Theodelitus, the circumferent●r, with a reformation of the plain Table, comprehended all under the title of the topographical glass.  
And since every one that writeth a book of any material consequence, disputeth first, and shows what necessary commodity may accrue to the reader, lest other wise it be held impertinent and nothing commodious for the common wealth,( the aduancement whereof, all men are bound in Christianity to tender;) it shall therefore nothing distaste, briefly to remember the same. First then enqure of the soldier, a chief defendant of his country, if he be not taught hereby to dispose of plaits, castles, forts, cities, and such like, to appoint places convenient for the camp,& to proportion every part according to the number of men and horse: let the Politician speak of proportions, and allowances, let Architectors speak of plattes and symmetries, let the the pioners speak of conducting mines and vaults under the earth, for the blowing up of cities, forts and castles; let the Nauagators speak of true directions in the unknown path of the seas, yea let all men speak of what vocation soever, and they shall find diversity of matter, as well to exercise the head, as recreate the wit. Amongst many, I will shut up this argument with the Geodetor or measurer of land, whose antiquity is great, as we may perceive by Zachary, Chap. 2, questioning with a man that was meating jerusalem; and whose commodity is no less great then ancient, as may appear by the exundation of Nilus in egypt: for when al the bounds and limits of every several mans particular was drowned and butted in the slime and mud remaining after the recourse of the flood, by the true direction of this science, every man was allotted his proper quantity of ground in the same proportion it was before. Then let none grudge if we teach them the mystery and hide secrets of so necessary and rare an art, so beneficial for man and so requisite in a common wealth, so convenient to be known of some, and expedient to be credited of al, teaching nothing but truth and certainties, approved by invincible grounds of infallible geometrical demonstrations; and therefore we will not stagger to inserte it, as an associate to iustice, one of the four cardinal virtues, whose office, as Ambrose saith, 1 de Officiis: Quae nihil alienum vendicat, quae cuilibet dat, quod suum est, quae negligit propriam utilitatem, vt seruet communem aequitatem; Et secundum ipsum, prima iusticia in Deum, secunda in patriam, tertia in parentes, quarta in omnes. For what else doth this art of Geodetia but teach directly cuilibet dare quod suum est, to give every man his own? as well the Lord as the tenant, as well the buyer as the seller, as well one man as another: it brooks no deceit in exchange, no oppression of the poor, no wrong to the Lord, but aims in all points to give every man his own:& therfore the office of a just& skilful Ge●de●or, is a perfect glass, wherein as a true mirror we may behold a number of other virtues although some will say, they be bad members grant( in this as all other faculties) some be, shall a particular subvert the general? A particulari ad general, non est tenenda ratio: for we mean none of those that in base bills like players, paisted vpon posts, impudently affirm, they will survey any manor, and find it more then formerly it was sur●eyed to be, such paltry bills I saw myself in London some three or four yeares since, much disgracing the arte: for either they must affirm they may lie by arte, or else can enlarge the true bounds of the manor by art. But the first perfit being truly taken, this assertion must needs be false● but I can no more blame them to avouch this in their bills, then they are to be condemned for the paisting of them vpon every post: shall we now turn the M●thematicks a begging? where is Alphonsus, Aristarchu●, Archimides, or some of the noble pr●fessors? if this bold, it is time to make the painters correct their Tables, and take the Globe out of the king Ptolomies band and there place a poor ●iquis, such as forlorn foreci●ers use to haue in Pauls Church. We call the mathematic the liberal sciences, because they should be freely taught, or for that they be most fit to be studied by great men, that haue large purses& riot wits for indeed these kind of artes he both chargeable, and difficult. But now for soath, he that can but get an old plain Table, and measure a meadow( how vn●rue soever) is a Geometritian, though by profession a homely Carpenter or rude Mason, and he that can but find the ordinary place of a planet in an Ephemerides, is an Astronomer, though otherwise a poor tradesman, of wit scarce enough to get his own living; thus is art trodden down with presumption, and the world possessed with errors. I haue heard diuers assertions and childish assumptions of such artless fellowes, though myself being unknown to them; some will produce you a pretty small brazen instrument forth of their pocket, and will affirm with that to go into the midst of any field, and presently tell how many acres be in the same before he stir, neither will he use the help of any cheine: which is nothing but mere foolish surmises and rude presumption, as far out of their ability to perform, as they are from the understanding of the art, possessing the world with impossibilities and boasting affirmations. I confess by Art such a thing may be performed, so the meadow be plain, and of a small quantity, but is not to be spoken in a generality, or as a necessary or speedy conclusion.  
Now some there are that will abruptly say, this surveying maketh the poor Tenants haue hard bargains, whereby all things grow scarce and dear. I answer, no: In what part of iustice is it that the Lord should be ignorant of his own? doth his knowledge abridge his conscience? how then is ignorance the mother of transgression? is it not seen that many men of good conscience by over estimating their lands, let their Tenants more then hard peniworths? the randon is no scantell● can the tradesman set his friend a reasonable price of his merchandise, when he is both ignorant of the quantity and quality of the commodity he selleth, whenas! knoweth both? the using of his friend relleth in the goodness of his conscience, so is it with the Lord and Tenant, and that the measuring of grounds should bring dearth, is as false as the former. For if the tenant will covenant with the Lord, that he shall haue all commodities at the price they were in yeares past, the Lord will likewise promise the Tenant the like rate of his lands; but if all things grow to a far higher rate then formerly they were, shall onely land be bound to rest at the old stint? shall every tradesman enhance the prizes of their wears, and onely the reuenews of gentlemen be ranted? must every one haue free liberty to make the best of his own, and onely Landlords apportioned? It is a peevish observation and foolish lenity so to do,& is a principal cause wherefore so many tradesman creep into the ancient possessions of Gentlemen; yet notwithstanding it is most requisite that the Lord use a favourable improvement, for so to do is acceptable before God and man: and therefore the Geodetor ought to yield a conscionable& true information to the Lord. But gentill Reader to conclude, let sparing reproof assure me of kind acceptance, and then haply I shall be induced the rather to publish the flowers of the mathematics, not yet finished, with another instrument called clavis Mathematica or the key of the mathematics; for that it vnlocketh as it were all mathematical questions, laying them in a plain method open unto thy view; the mean time I rest ready to do thee any favour I may,  
Arthur Hopton.   

joannes Passy in Artibus Magister, in lauds huius operis.  
STemmata quid prosunt, seriem referentia longam?  
Quid genus antiquum nobilitate potens?  
Si non quae scriptis ornet stirpem generosam,  
Verturesque virum, dextera doctaforet?  
Est domus Hoptoni( fateor) per saecula durans,  
Virtute exultans,& sine tabe virens:  
Quae nunc Arthuri vigili decorata labour  
Emicat,& florens tollit ad astra caput.  
Agricolae, Mercatores, Nautae, Medicique,  
Et genes omne hominum sedulitate tua  
Debebunt Arthure tibi, dum voluitur axis  
Aethereus dum Sol splendour,& astra micant.  
Desine livor edax benefacta reprendere dictis,  
Et been promeritis gloria digna sonnet.  
Ingenium Phoebus, Pallasmi contulit artes,  
juno decus, lauds vita polita dedit.  
Arthure o foelix viuas ter Nestoris annos,  
Accedat laudi viuida fama tuae.   

I. P. madge. artium, in laudem huius operis Arthuri Hoptoni Generosi  
OVi legis Hoptoni perscriptum nomine librum,  
Perlege sincere, consulito queen boni,  
Multa fauente Deo, permulta volumina posthac,  
Confecturus erit, dummodo vita sinat.   

In Laudem M. Arthuri Hoptoni Mathematici.  
WAke gentle Muse and give me leave a space,  
Inspire my wit with thy all-quickning grace:  
Direct my pen to paint forth Hoptons name,  
whose merits do deserve immortal famed.  
immortal famed and honor of our time,  
Whose kind endeavours do our art refine:  
Teaching proportion Mistris of each art,  
Vpon the stage to act an unknown part.  
An unknown part, for never did I hear,  
That Stagerite in arts high Theatere,  
Breathing dark grounds from aphorisms strange,  
Could act a scene, so frought with pleasing change.  
Thou rarest man, and more then Englands pride,  
Hast brought to pass what art before denied:  
Hold on thy course, for ever shalt thou live:  
Thy bodies death, life to thy virtues give.  G. Giff. Gen.  

Rogerus Stedinanus Capellanus reuerendi in Christo patris, Roberti permissione diuina Hereford.  
Episcopi,& publicus praedicator, in laudem authoris& singularis industriae suae.  
QVod domus Hoptoni, multis durauerit annis;  
longaeui recolunt tempora long a senes.  
Cognita praeclaram referunt insignia laudem:  
antiquam resonant inclyta facta Domum.  
Hoptoni de Hopton vigeat celeberrima fama,  
Arthuri ac feriat sydera clara poli.  
Laus sua iustitia celebratur digna fideli,  
digna quidem meritis, dignaque fama fide.  
Est inuenis vultu: divino Pallas ab ore  
profluit,& meritis inclyta fama venit.  
Sydera describit notis distincta figuris:  
dumliquidis lymphis aurea luna nitet,  
Arthurus numeris numerabit tempora certis:  
ordine cantabit sydera celsa suo.  
Splendida perlustrat supremae sydera sphaerae,  
somnigenis tribuens lumina magna viris.  
Denique saint Pater, specialem mitte fauorem.  
vt faueat coeptis candida musa suis.  
Coelestique libro scribatur vita beata.  
inter& electos numina sancta choros  
Prospera cuncta precor, sibi sint foelicibus annis,  
postque suam mortem non peritura salus.   

To master Arthur Hopton.  
PYthagoras in teaching did maintain,  
When bodies di●de, their souls returned again,  
And kept the minds that they were vs'de to do,  
Within those bodies they did run into:  
And now I see his maxim true and just,  
Hath ben too long abused by the most:  
For who shall with judicial eye ore-looke  
The curious art composed in this book,  
Cannot deny in Hopton to arise,  
The soul of great Archimedes the wise.  
sweet Arthur, well deserust that princely name,  
Which from the regal constellation came,  
Arcturus bright, which in himself contains,  
The very nature that in thee remaines.  
And doubtless at thy blessed birth so shone,  
That all the stars were dimd by him alone,  
Whilst he about the northern pole did glide,  
As Lord comander of the heauens wide,  
Distilling down vpon thee without spare,  
Ioues divine nature and the God of war:  
As thou by wisdom mighst thy famed maintain,  
So well as with the honour of thy name,  
Which hath ben long and ancient without doubt:  
witness the house from whence thou camest out,  
And from whence each best Hopton of old time  
By law of arms ought to derive their line.  
Then what apt ornament is't I might choose  
To feed the height of thy aspiring muse?  
A crown of bays for Poets, not for thee,  
Thy Temples scorn to bear Apollos three:  
A weath of stars, robes of immortal famed,  
shall girte thy brows, shall cloth thy loved name.  I. M.Gen.  

The contents of each several book.  
1 The first declares the composition, construction& definition of the staff, with certain necessary Tables and propositions to be foreknown.  
2 The second contains the quantity of all measures both English and other, with Tables of the Iame, to seek all kind of angles and their sides,& thereby to find all dimensions, as longitudes, latitudes, altitudes, and profundities with the ground thereof, also to work as M. D. Hoods sector &c.  
3 In the third book is set down the use of the hypsometrical scale, according to the doctrine of the best writers in that behalf, to which is added many other propositions performed by glasses, &c.  
4 In the fourth, is the large use of the geometrical Quadrant.  
5 In the fifth, the use of the Iacobs staff with the making of a new kind of scale, and thereby to work the rule of proportion, called the golden rule or rule of three, and to work the same rule inversed.  
6 The sixth contains the art of Geodetia, or measuring of grounds, which is divided into two parts, in the first you be taught to seek the true proportion and symmetry of grounds, according as it lieth without the doors, with rules to know if you haue wrought true or false; also to reduce the hypothenusal lines to lines horizontal, and to protract after a new order, reforming many errors formerly practised in plaiting grounds.  
The second part doth teach you how to know the contents of any piece of ground how irregular soever, with Tables diligently calculated to avoid arithmetic, as well to know the exact contents as to divide or separate land, with rules to perform the same instrumentally; also to measure solids as ston, timber, &c. or superficies, as boards, glass, &c. with M. digs his Tables for that purpose; to seek all kind of dimensions by protracting, the rare metamorphosing of all kind of figures, and finding of their proportions one to the other, with the theorical ground of all these; also to make a Lord of a Mannour to express each several tenor, &c. newly& compendiously, with certain notes requisite for all Geodetors to know, &c.  
7 The seventh contains Trigonometria or the doctrine of right lined triangles, by sins, Tangents, and Secants, with new Tables.  
8 The eight, contains the use of the sphere in plano, with the making of many kind of Dials onely by the staff.  
There is an other book called the topographical glafle yet to come forth, containing the large use of the Theodelitus, topographical instrument, circumferentor, and plain Table; teaching you also to describe countries, and to find the hour of the day and night, to know the altitude of the sun or any star after a new order also to make a new kind of perfit, and to take a survey after another order, all which you may better see in the general Table.   

The Printer to the Reader.  
For errors past, or faults that scaped be,  
Let this collection give content to thee:  
A work of art, the grounds to us unknown,  
May cause us err, though all our skill be shown.  
When points and letters, do contain the sense.  
The wise may halt, yet do no great offence:  
Then pardon here, such faults that do befall;  
The next edition makes amends for all.   

Errata.  
PAg 3 line 26. for graduator red graduation, p. 7 l. 17, for please the, r. please of the, p. 8. l. 6.8 for Dr. R: p. 9 l. 18 for 4 red D, p. 9 l. 26 for Q Nr. Q W, p. 10 l. 5 for Br. O, p. 11 l. 23 for derivation, r. graduation, p. 11 l. 4 for the description r. the end of the description, p. 14 l. 12 for other of r. to the kind of, p. 17 l. 19 for adumbiatr. abumbrat, P. 18. li. 12. for equal red equat, p. 18 l. 29 for other where r. otherwise p. 18 l. 40 for lacob and r. lacobs staff and, p. 23 l. 3 for spoked r. spoken, P. 24 l. 3 for food. r foot, p. 37 I. 11 for until the point r. till he point, p. 3● l. 7 for let NAB, r. let A B, p 4● l 7 for all r. al vaies, p. 50 l. 21 foreightr. right, p. 56 l. 17 for I take, r must be out. p. 64 l. 4 for 29 r. 28 p. 72 l. 1 for ACK r. AIK, p. 87 l. 7 for AH r. H I, p. 90 l. 11 for SKE r, ●R P, p 91 l. 15 for A Cr. A one, p. 97 l. 22 for 4 in 5 r. 4 times 5. p. 99 l. 30 for G Dr. GE, p 99 l. 34 for of the I Rr. of 1 K, p 100 l. 24 for proportional 801. Proportionte arms. p. 102 l. 19 for the fill r. the light, p, 104 l. 5 for tranluersing r. transueriary, p 107 l 32 for right as the, 1. as the segment of the index shall be to the, p. 112 l. 18 for as B D r, as B E, p. 115 l. 〈◇〉 for of degreer, of 39 degrees, p. 121 l 18 for then BG C r. then BIC, p. 122 l. 20 for AM r. AN, p. 123 l. 1 for compass E r. compass in E, p. 124 l. 30 for you are 〈◇〉. that are, p. 128 l. 26 for CB CB r. AB AB. r. 131 l. 4 for seem curious r. see me so curious. p. 133 l. 25 for answered r. assuted, p. 143 l. 12 for the chapter r. the last chapter. p. 14● l. 14 for compose halfer. composed halt, p. 131 l. 12 for Isophtron r. Isopluron, p 171 l. 3 for opposite r. proposed, p 171 l. 4 for a trapezia r. Isoscles crapezia, p. 171 l. 20 for leiser r greater, p. 172 l. 26 for half three degrees r. half third parte, p. 216 l. 17 for Crowards Dr. H to wards G, p. 216 l. 18 for C D and vpon Dr. H G vpon G, p 216 l. 19 r. then vpon H of 6 degrees as F H E, then note &c. p. 217 l. 8 for 31 chap 1.39 chap, p. 218 l. 18 for as L F I r. as I F R, p. 222 l. 39 for E F K H r. E I K H, p. 224 l. 3 for E C F B r. E C F A, p 239 l. 1 for D I r. GI, p. 239 l. 6 for to taker take, p. 252 l, 27 for omit ear. committed, p. 257 l. 22 for taking r. laking, p. 258 l. 16 for chargeth r. chensheth, p. 260 l. 2 for the side r. the site, p. 263 l. 25 for but took r. but take, p. 276 l. 21 for lines r. sins, p. 281 l 31 for F N r. F M, p. 285 l. 6 for B E be equal, r. D E be pa●alled, Ibid. l. 7 for P 29 r. P 30, p. 297 l. I for Regromiontanus, p. 298 l. 11 for G F r. G F, p. 299 l. 39 for finding r. flood, p. 335 l. 1 for F K r. L K, p. 336 l. 1 for by the 20 degree of 8 ar, for the 29 degree of 81, p. 336 l. 15. for M r. my, p. 337 l. 1 for go I r. to S, Ibid: Ibid for B I r. B S, p. 339 l: 8 for parts r. poles, p. 239 l. 14 for I L r. K L, Ibid F. 30 for ⊙ degree ♎ r. 0 degree r. p. 3 44 l. 21 for 8 degrees r. 18. degrees p. 361 l. 19. for on L r. on B, p. 363 l. 2 for and in any r, in any, p. 363 l, 5 for F E G r. G E I, p. 363 l. 7 for E F r. E I. p. 363 l. 7 for declination r. declinano.  
Scheames or figures misplaced or omitted.  
Fel. 55 l. 17 scheame in the 22 Chap. fol. 93 chap 17 scheame in the chap. 7 l. 4 fol. 151 pro 2 place the figure in the 3 pro. fol. 151 pro. 3 place the scheame in the 2, pro. fol. 165 scheame in the 24 chap. fol. 347 chap, 21 the figure in the chap 17 fol 360 scheame in the 357 fol.  
Faults by the cetter.  
These fault is be something general and not well to be reformed by the Printer, as in the second broken their is many pictures of Castles, Treas, &c. printed in diuers scheames and there and else where, some letters omitted and others mistaken, which the discreet reader will best correct.   

THE FIRST book OF THE GEODETICAL staff, showing THE COMposition and making, with the definition and explication thereof, and every thing therein contained, with their vocables: also certain Tables and propositions, necessary to be first known.  

CHAP. I.  
Of the framing, composing, and quantity of mettall or wood required to make the Geodeticall staff.  
TO set a part all circumstances in speech whatsoever, you shall therefore first prepare two rulers or neat pieces of silver, 
What stuff the staffs is of.  bone, brass or wood, which you please: Being wood, let it be such that is firm, solid, and of a close grain, well plained and truly tried to 3. quarters of an inch broad,& a quarter and half quarter thick, in each of these rulers must be two hollow channels, Rabboth, or Transumes, as Carpenters call thē, they must be under hollowed douetaile wise, so that the two hollowed sides being turned together, there may be a concavity or hollowness of a quarter of an inch square, representing this figure.  
2 Then must you provide a Ruler of like wood or stuff, 
The graduator.  made feite and equal to run into the foresaid hollow channel, so that the two hollow sides being turned together, and the Ruler put thereinto, the legs cannot bee opened by reason of the dove tail channels.  
3 Next must you join these two rulers( which bear the hellow channels) together, as you do the Sector or Circular Scale or such kind of Instruments, 
The center pin▪  and the Center pin that joins them must be hollow, 
The Center pin,  sufficient to contain a strong screw pin, neither is it material whether these two rulers so joined together be left square, or made round.  
4. This so contrived, there must be a Courser prepared, as B, which is a hollow socket of brass made fit and pliant to move along one of the rulers lately toyned together, 
The Courser.  vpon the top of which Courser answering to the fidutiall edge of the Ruler must strongly be fixed a screw pin better then an inch long, as B A, so that this pin is always moved just over the fidutiall edge of the Ruler.  
If this Courser bee made half round for the round legs, or square, for square legs: yet must it be cut full of holes, to the end to see the figures vpon the legs through the same, as you may perceive by the figure. In the lower side, this courser must be placed a screw pin, 
The Comfer for square legs.  as C to fasten the courser at any place assigned, with a thin plate of brass( as the order is) within the said socket, for the said screw pin to force against the ruler.  
5. Next must you haue a little square piece of brass, the thickness of which must be equal to the thickness of the third Rule or graduator: through the prece must be bored a hole, 
The square pieces of brass.  as E equal unto the bigness of the screw pin A, so that the said piece may slip down unto the foot of the pin B, then through the fore side of this piece of brass must bee bored another great hole, on the bigness of a large wheaten straw, as E, the centers of which holes F and E must be so far sidewise one off the other, as half the thickness of the Graduator E, standing equal be twixt the vpper& lower side of the piece of brass, as in this figure: When these holes be bored, you may file much of the superfluous brass away, provided that it retain the foresaid thickness.  
6. Next unto this must be fastened at the end of the third rule, a pin of brass, 
The pin at the graduators end.  equally to move in the hole E, in the last figure. This pin at the one end must haue a screw for the piece of brass A, in the 7. figure to screw on the back side, being put through the hole E in the 5. figure, to the end he may not slip out of the foresaid hole, but be fastened by help of the screw, the pin is L O, and to the and C must be firmly fastened or filled two lippets or cheeks of brass, K& M, to rivet the said pin fast unto the graduator or third ruler, which being artificially made, it will represent this figure.  
7. Next must you provide two neat pieces of brass, with screw holes therein, justly to fit the screw pin A B in the 2. figure, and L O before described, 
The screw piece of brass.  so that the hole F in the 5. figure being put on the pin A, and down to the foot thereof at B, this said screw piece B being turned down hard, and wrested thereunto, may stay the piece of brass, that he cannot move any way: they would be made in maner of the screw piece in the dividing compass, representing these figures.  
8 The sixth figure must be so contrived, that whether you turn the side K or M, they may either answer to the midst of the pin A B, so that which side the graduator foeuer be turned thereunto, the fidutiall edge thereof may alwaiss agree with the center of the said pin, the hole F being put on the pin A B, and the pin L O being put through E in the 5. figure.  
9. Next must you prepars a staff 4. foot in length, he must be bored hollow 2. foot and a half deep, that the foresaid 3. rules being put together, 
The hollow staff.  they may be all three likewise put into the said hollow staff, and then you must haue a sine top to go on with a screw in maner of an Aqua vitae bottle, so the staff being made round raper wise, and small to carry with ease, the ioyners work thereof is finished. Then proceed unto the graduator thereof: but because you shall be acquainted with the names and parts of the staff thus set together, I will first show you the destinition and vocation thereof.   

CHAP. II.  
Of the Definition, vocation or nomination of the Geodeticall staff, and his principal parts.  
BEfore wee proceed unto the divisions& projecting of circles and such like vpon the Geodeticall staff, we will first lay open the several parts thereof: if any man therefore deinaund what this Instrument is it may be answered, that it is an Isosceles triangle of two equal containing sides, coupled together in maner of a Sector, which do open& close as an ordinary pair of compa●ses, every degree or part therein contained or imagined, is so proportionally, congruently and artificially projected thereon, that it performeth and containeth in dimention, the whole use of any Geometrical or Topographical Instrument whatsoever, not onely by sundry new and rare ways of himself: but also after the self same method and form, as those instruments are wrote by their several Authors, for the Appellations of this instrument, having shewed you that it is a triangle, the rest follows.  

1 
The legs.  The two sides of this triangle are distinguished by the right and left leg, and each leg hath his vpper and lower side, and each side hath his inward and outward side.  
2 
The vpper side of the legs.  The vpper sides of these two legs, inward& outward, bear equal divisions,& are numbered by 5. 10. 15. 20. &c. from the Center towards each end, on the outer side onely.  
3 
The cites.  In each leg is placed an equicurrall sight, and to the sight in the left leg is fixed the end of the graduator, these sights serve to direct the legs unto any mark assigned.  
4. In each leg is a line drawn round about the same,& where this line and the fidutialedge of the legs intersect, there is a point called the point respective: 
Point respective.  because without respect unto this point, the quantity of no angle can be expressed: therefore you must haue a special regard thereunto.  
5 
Lower side of the legs.  Upon the lower side inward of this instrument are set unequal divisions, onely vpon the right leg( as in the 4. pro. they be numbered) from the point respective towards the center by 5. as 95.100.105.110. &c. and serve to express the quantity of an Angle obous 90. degrees, see more Pro. 7.  
6 To this Instrument also there belougeth a rule 4. square hollow on two of his sides, 
The Graduator.  as in the 1. propostrion. This rule is called the graduator, because of the manifold divisions projected thereupon, he is so fastened unto the sight in the left leg, that by help of a screw,& screw pin he may be fixed at any angle given, and yet will move, so that you may turn any side downwards you please, 
Pro. 4.  so that he admitteth a triple motion: upon this graduator are projected 4. maner of divisions.  
7 
Degrees of a circled.  First are placed the degrees of a circled to 90 numbered from the end where the brass pin is, towards the other end by 5, as 5. 10. 15. &c. and serve to express the quantity of any angle under 90. degrees.  
8 
equal parts of the Graduator.  Secondly, are placed equal divisions numbered as the degrees of a circled, and serve to express the side of any triangle given, and are placed in the hollow side of the graduator, the equal parts are such as be placed vpon the vpper side of either legs.  
9 Thirdly, are placed the part of the geometrical quadrant, 
Parts of the geometrical Quadrant.  projected on the graduator, and are numbered from either end thereof towards the midst by 5, as 5. 10. &c. ending in the midst at 60. they stand opposite unto the degrees of a circled.  
10 
Part of the Hypsometrical Scale.  Fourthly, are placed the parts of the hypsometrical Scale, in the hollow side opposite unto the equal divisions, numbered vpon the Graduator from either end towards the midst by 3, as 3.6.9. ending at 〈◇〉 in the midst. 
The center of the Instrumēr.  Through the center of the 2. legs, there is a hole called the Center of the Instrument, through which hole is a brass pin to be thrust, called the Center pin, three inches long, as in the 5. pro. at this latter W.   

The center of the graduator  The screw pin that fastens the graduator unto the left leg, is called the Center of the graduator.  
In the right leg of this instrument, must be a little movable sight to run in the hollow channel which I call the right Equicurrall sight, so haue you the vocables.  

The name of the staff,  As for the name of this staff, bee is called the Geodeticall staff, of the word Geodetia, which signifies a science concerning sensible greatness and figure, as hereafter I might rather haue called him the mathematical staff, in respect of this generality of work but I gave him this name in respect of his aptness to measure and plate ground, he hath the addition of Viaticum in respect of his portability or aptness to travell with.   

CHAP. III.  
Of the divisions of the vpper and lower sides of both the legs, together with the Graduating of the two sides of the Graduator, and of the Triple motion he admitteth.  
having finished the instrument, as before, you shall divide the opper side inward of both the legs in two equal parts. 
division of the legs.  thus: The legs being 2. foot and a half long, divide each foot into 12. inches, and then subdeuide each inch into 8. parts( or more if you will) then draw parallel lines, and there place figures, as you be wont in such works, numbering the said parts by 5. fromwards the Center, as 5. 10. 15. &c.( as before) the equal parts must be placed vpon the inward side of the legs, and vpon the outer side, numbered onely on the outer, as in the end of the 7. pro.  

The equal division vpon the Graduator.  Such divisions as these are placed vpon the Graduator, as in the 2. Pro. Du●tie. You must haue a care that the parts be truly divided, so is this kind of Graduating finished, considering well the end of the 7. Prop.   

CHAP. IIII.  
To project the degrees of a circled unto 90. upon the Graduator, and of the triple motion he admitteth.  

The division of the Graduator for degrees of a circled to 90.  First take the length of one of the legs, which here let B C represent, then making B C the side subtending a right angie, thereupon frame an 
See the 6. Pro.  Orthogonium Isosceles, that is, a right angled triangle of two equal containing sides, A B, A C, here represents then making A. the Center, with the length of A B or A C deseribe a Semycircle, then draw C A infinitely beyond A, until he intersect with the said semicircle at D so is C D the Diameter thereof, then divide the said semycircle into 180. equal degrees, so shall each Quadrant contain 90. degrees: Now to project these degrees, vpon the graduator, take with your compass the length of every cord in the quadrant C B, and place them in the line C B, representing the Graduator, still keeping the one foot of your compass in the point C.  
Example.  
I take the length of O N, C O, C P, C Q, C R, C S, C T, C V, which is 80. 70. 60. 50. 40. 30. 20.& 10. degrees,& then I put the figures vpon the graduator C B, as they be here placed, so is the degree of a circled to 90. projected vpon the Graduator: here note that you must take every degree in the circled, though I haue put them down in the demonstration by 10.20. &c.  
Now to project the degrees from 90. unto 180. do thus. Make C the Center, and with the length of C B describe the arch B F: then keeping the one end of a Ruler vpon the Center A, moon the other end from degree to degree, in the quadrant B D, noting at every degree where the ruler cuts the arch B. F: having ●o done, you must new understand that these degrees are to be projected vpon the right leg of the instrument, and vpon the lower side inward thereof, represented by A Z: therefore first take the length of A B, 
Of projecting the degrees to 180. vpon the leg.  and place that from the Center of your Instrument towards the end of the right leg, and where it ends; make an apparent score or race round about the said right and left leg, noting the same with some mark 〈◇〉 vpon the outer side of both the legs, and where this line so drawn doth intersect with the Ad●tiall edge of each leg, there is a point called the point respective, as in the second Proposition, Definition 4. having thus done, take the length, of A A, 
Point respective.  and place the same in the right leg from the Center towards the point respective, as A S in the line A B: do so to A R, A Q, A L L, A P P,& place the same in the right leg A Z from A toward B, so shall A R be equal to A T, A Q equal to A Y, A L L to A D, and so of all the rest until you haue finished each degree unto 180, you may put the figures thereunto where you see best occasion. I hold the fittest place to be the hollow channel in the right leg, because they may there bee seen best, so haue you finished all the graduations belonging unto the legs, saving those in the 7. Proposition: the graduator divided as before, and placed vpon the leftleg, 
The triple motion of the Graduator.  he admitteth a triple motion: the first is along the left leg from one division to the other, made by moving his center: secondly, to make any angle with the least leg proposed, as in the second book, Chap. 26.  
3. To turn any division on any of the 4. sides, you please, the Graduator towards the legs: all which are performed simply of themselves without the let or help the one of the other, as you may ●●due the Graduator along the left leg, and not alter the Angle he maketh with the said leg, as in the 27. Chapter, book 2. or turn any side of the Graduator downward, and not move him vpon the left leg, or alter the Angle, &c.   

CHAP. V.  
The order of projecting the geometrical quadrant, and hypsometrical Scall vpon the Graduator, with the figuring of the hollow staff.  

The projecting of the geometrical Quadrant.  DEscribe a Semycircle of the just bigness of the former in the 4 Proposition, then place the quadrant therein thus, divide the arch B into two equal parts at D, and from D let fall 2. perpendiculars, the one vpon the line A B at E, the other on the line A C at F, then divide the line D E and D F into 60. equal parts a piece.  
Now it rests at your pleasure, whether you will place these parts of the quadrant vpon the legs, or vpon the Graduator, but in my conceit the Graduator is far the better, by reason the divisions be there more large.  
having therefore divided the quadrants, as before, into 60. equal parts, lay a ruler vpon each part in the line E D and D F, and where it cuts the limb make notes, and put figures thereunto, as in the figure, thē as in the 4. Proposition making C the center, where you still place the one foot of your compass, take with the other the length of each cord or degree noted in the limb C B, which place vpon the Graduator C B, so shall you find C F equal to C H, C I to C K, C L to C M, C N to C O, C P to C Q, C R, to C S, you may better perce●ue the placing of the figures by the demonstration then with many words, place these divisions according unto the second Pro. and 9. Deff.  
 
Then for the projecting of the hypsometrical Scale vpon the graduator, 
The projecting of the Hypsomericall Scall.  the order observed therein is nothing different from this of the geometrical quadrant, onely where the sides of the quadrant were divided into 60. parts, here the sides of the scale must be divided but into 12. equal parts, and then project unto the limb, and from thence unto the graduator, as before, as is plain by the seal.  
In the Quadrant B D, so is B D the Graduator, divide into twice 12. parts; the figure being, placed thereunto; as in the 2. Proposition, 10. Definition, the Graduator is finished together with the whole graduating of the Iustrument.   

CHAP. VI.  
Now followeth the staff and his parts, with their application.  
A B is the left leg, A C the right leg, G is the courser, and must run, and A B the hole: R in the figute F must go on the pin B in the figure G.  
D E is the Graduator, the pin at the end thereof marked 〈◇〉 this figure 4. must be put through the hole P. in the figure F,& then the screw piece marked with 4. must be put on the end of the said pin or courser, and then you may blunt the end of the pin so that the screw shall not come off, nor the pin come forth of the hole P, letting the piece F always hang at the end of the Graduator with the screw 4. H. is to screw on the pin B in the figure G. under which screw K the bellow pin is to be put vpon the said pin B, so that they may the better force down F. w is the site for the center of the legs; A R is the right equicurall sight.  
Q M and Q N are the points respective, which you shall find to be ●●st of like distance from A, as the length of A B is in the figures of the 4. and 5. Propo.  
Z is a pin to place the new quadrant, or in the 6. book on the saffe with, else not material. The pieces marked with these figures, 3. are two screw pieces of brass to be butted in the top of the staff, I L O is the hollow haffe, I is the top to go on and of l like an Aqua vitae stopple.  
P P is a right angle made with a foot meet for the channel in ●●e of the legs and is to place the right or left leg to lye parrall●ll vpon any occasion, by hely of the plum or perpendicular in the courser G. M is a screw pin.  
In the figure W. the lower end N N should be hollowed some small quantity as you may perceive by the shadowed lines, to the end when the pin is wrested hard, it may force the joint of the legs together, and so stay them at any angle: for if it were not hollow, it would stay vpon the riuetting put of the leg, which the lower end marked with B goes through.  
F Is the legs closed with the Graduator therein, which being thrust home should be put all into the hollow staff.  

Simulacrum Baculi Geodaetitii siue viatici: Astronomis, Astrologis, Geometris Medicis, Politicis, Oeconomicis, Cosmographis, Nauigantibus,& Mensoribus commodissimum, Tyronibusque facillimum: Per Arthurum Hopton de hospity Sancts Clementis societate conditum& editum. 1610.   
A A Is the Graduation of the equal divisions, which must be put 5. times vpon this instrument, one vpon the one side of the Graduator, and twice on each leg, as in the 3. and 8. Proposition.  
B B Are chord divisions, and are placed 3. times vpon this Instrument, viz. once on the Graduator, as in the 3. Proposition, and once on each leg vpon the lower side outwards, as in the 8. Proposition, and as they be placed in the figure F.  
C C Are degrees of a circled from 90. to 180. projected vpon the right leg A C from M towards A.  
D D Are parts of the geometrical quadrant placed one by one, on one of the sides of the Graduator, as in the 5. Proposition.  
E E Are parts of the hypsometrical Scale, placed also on the one of the sides of the graduator, as in the 5. Pro.  
These things so ordered, you must place of three Tables ensuing vpon the outer side of the hollow staff, which will serve you for many pleasant and necessary uses; and if your staff be too small to hold the Table of the suins altitude, as he may if you make the legs very small( as the use is being in brass or silver) then may you omit to stamp the minutes on the right hand, each row putting one degree more unto the degrees, if the minutes answering thereunto much exceed 30. and so may you place the Table on the smallest staff that is.  

The derivation of all the lines in the staff.  
The description of the Geodeticall staff and the parts thereof, which you may haue ready made by M. read and John Tomson in Hosier lane, near Smithfield in London.   

The Table for board and glass measure &c. called the Table of Planimetria.  P. Y.   VI XII XVIII XXIIII XXX foot yn: foot yn: yn: par: yn: par: yn: par: ync: par:     2 0 12 0 8 0 8 0 4 ⅘ ¼ 48   1 11 1/ 25 11 3/ 3 7 ⅞ 5 15/ 16 4 ¾ 7/ 2 24   1 10 11/ 7 11 ½ 7 ⅘ 5 ⅞ 4 5/ 7 ¾ 16   1 9 l⅓ 11 2/ 7 7 ⅔ 5 ⅘ 4 ⅔   I VII XIII XIX XXV XXXI 1 12   1 8 4/ 7 11 1/ 16 7 4/ 7 5 ¾ 4 ⅝ ¼ 9 7 ⅕ 1 7 ⅔ 10 ⅞ 7 ½ 5 ⅔ 4 4/ 7 ½ 8 0 1 6 ⅕ 10 ⅔ 7 ⅜ 5 ⅝ 4 ½ ¾ 6 10 2/ 7 1 6 4/ 7 10 ½ 7 2/ 7 5 ⅝ 4 ½   II VIII XIIII XX XXVI XXXII 2 6 0 1 6 10 2/ 7 7 ⅕ 5 ½ 4 ½ ¼ 5 4 1 5 3/ 7 10 3/ 2 7 ½ 5 ½ 4 ½ ½ 4 9⅗ 1 4 15/ 16 9 ⅞ 7 l⅓ 5 3/ 7 4 3/ 7 ¾ 4 4⅜ 1 4 3/ 7 9 ¾ 6 15/ 16 5 ⅜ 4 ⅜   III IX   XXI XXVII XXXIII 1 4 0 1 4 9 ⅝ 6 1/ 7 5 l⅓ 4 l⅓ ¼ 3 8 l⅓ 1 3 4/ 7 9 3/ 7 6 ⅘ 5 2/ 7 4 2/ 7 ½ 3 5⅛ 1 3 1/ 7 9 2/ 7 6 5/ 7 5 2/ 9 4 ⅙ ¾ 3 2 ⅖ 1 2 ¾ 9 ⅛ 6 ⅝ 5 ⅕ 4 ¼   IIII X XVI XXII XXVIII XXXIIII 1 3 0 1 2 ⅖ 9 0 6 ½ 5 13/ 8 4 ¼ ¼ 2 9 ⅞ 1 2 ½ 8 6/ 7 6 ½ 5 3/ 32 4 13/ 16 ½ 2 8 1 1 ¾ 8 ¾ 6 ⅜ 5 2/ 16 4 ⅙ ¾ 2 6 ⅔ 1 1 ⅜ 8 ⅝ 6 l⅓ 5 0 4 1/ ●   V   XI XVII XXIII XXIX XXXV 1 2 4 ⅘ X 1 1/ 11 8 ½ 6 ¼ 5 0 4 ⅛ ¼ 2 3 3/ 7 1 1 ⅘ 8 7/ 3 6 ⅕ 4 ⅞ 4 5/ 32 ½ 2 2 ⅕ 1 ½ 8 ⅕ 6 ⅛ 4 ⅞ 4 1/ 16 ¾ 2 1 2/ 23 1 2/ 7 8 3/ 12 6 1/ 16 4 ⅚ 4 1/ 32  
A Table to be placed vpon the hollow staff, serving for the number of Timber measure, and therefore I aptly call him  

The Table of Solidmerria. 
This Table was calculated by M. DIGS, and also the former, which are best to be placed vpon the backeside the legs, as ordinarily they be set vpon the Carpenters Rule, as is taught in M. DIGS his Tectonicon.   S Fo Inc: S Inc: Pat S In: P S In: Pa 1 144 0 10 17 2/ 7 19 4 25/ 32 2● 2 ⅕ 2 36 0 11 14 2/ 7 20 4 5/ 16 29 1 1/ 16 3 16 0 12 12   21 3 11/ 12 30 1 11/ 12 4 9 0 13 10 11/ 5 22 3 4/ 7 31 1 ⅘ 5 5 9 3/ 25 14 8 113/ 16 23 3 ¼ 32 1 11/ 16 6 4 0 15 7 2/ 7 24 3 0 33 1 ⅝ 7 2 11 2/ 7 16 6 ¾ 25 2 ¾ 34 1 ½ 8 2 3 17 6 0 26 2 9/ 16 35 1 3/ 7 9 0 21 l⅓ 18 5 l⅓ 27 2 ⅜ 36 1 l⅓  
In the foot of this staff, there must be a steel pick, and in the top, two pieces of brass, with a screw hole therein for the center pin, w to fasten A the center of the instrument thereunto, that on of these screw pieces, must stand in the top of the staff, the other in the side, as near to the top as you can, the first serveth to take angles of latitude, that which is in the side serveth to place the legs to take Angles of altitude, as will appear in the second book. Propo.  

PRO. VII.  
A needful geometrical Proposition to bee understood for the performing of the 4. Pro.  
To make an Isosceles triangle where the line subtending the right angle is given.  
LEt B C bee the line given, representing the base of an Isosceles, then open your compass unto any reasonable distant( as somewhat more then half the length of B C, making then B the center, strike the arch O P, do so to C, and make the arch M N, now these two arches shall intersect one the other twice, as at I. and K. laying a rule on those two points I K draw the line I A infitite beyond K, so is C B divided into two equal parts at R, then place the length of R C or R B in the perpendicular R A, placing the one foot of the compass in R, and when the other falls, make a point as at A. Lastly draw the line A C and A B, so is your Isosceles triangle made, A being a right angle in the triangle B A C. This Proposition I placed here for those that be but meanly red in Geometry, because that should not be scaled in any thing necessary for the performing of this Instrument. Here I should proceed in opening further the terms of Geometry, which I will omit because it is grown something common, and also for that I haue collected them into one volume in my Epitome of Geometry, namely in the first part called Euthymetria.    

CHAP. VIII.  
These former Tables orderly placed vpon the outer side of the staff, yet there remaineth other of divisions, which are to be placed vpon the lower and outer sides of the legs, and vpon the vpper and outer side of the said legs, that is the chords of a circled to 60. and the equal parts, which shall serve for many pleasant uses▪ as hereafter.  
The order of placing the Chords of a circled and equal divisions, vpon the utter side of the legs of the Geodeticall staff.  

The chord divisions.  WE will take the figure in the 4. Prop. but first prepare the staff so that he may be apt for those divisions: you shall first therefore at the point respective, make a mark in each leg on the lower side, half a quarter of an inch, or thereabouts, from the fidutiall edge of the legs. Which done, draw a straight line from the Centre, direct to the two former points, made at the points respective, then draw parrallelled lines thereunto, to place the figures at, as the common order is.  
Now for graduating the chords of a circled, on each leg, the work is nothing different from the projecting of the degrees of a circled unto 90 vpon the Graduator, as in the 4. Prop. therefore the center of the Graduator, being placed at the Center of the leges,& the degree of a circled being turned towards the legs, you might truly thereby mark out the chords vpon each leg, unto 90. so will the 60. degrees vpon the Graduator, just touch the point respective. Otherwise you may fetch the said degree from the circled, with your compass, placing them vpon the one of the legs from the center. The one leg thus divided, you shall then divide the other leg in the same order, putting figures thereunto, and numbering the degrees by 10. as 5. 10. 15. 20. &c. as you did vpon the Graduator unto 90. You shall find that these degrees will help you to perform many singular Propositions, as will plainly appear in the 6. book.  
Of the equal divisions.  

Of the equal divisions.  To place the chords of a circled vpon the lower and outer side of the legs, I bade you draw a line in each leg that should be produced from the Center, so that the said right lines might run sloping wise a certain distance from the fidutiall edge of each leg: I said half the quarter of an inch at the points respective, would suffice, and so it will; and here I let you to wit that the like sloping lines as these must be placed vpon the vpper and outer side of each leg, with parralel lines drawn for the divisions and figures, as you did for the chords; these sloping lines must be divided according unto 8. in the inch, and numbered by 5. as 5. 10. 15. 20. &c. they be the same divisions that are placed vpon the vpper inward side of each leg, and are called equal divisions, as in the 3. Proposition: and note if you please, you need not set any figures unto the equal divisions on the inward side of the leg, because those on the outer will serve for both: onely draw parralel lines producing the division at every 5. and 10. notwithstanding if the room will admit figures, stamp them there.  

A Table containing the degrees and minutes of the suins altitude for every hour in the year, exactly calculated for the latitude of London.  Capri.   june. 69 12 11 10 9 8 7 6 5 4 G M G M G M G M G M G M G M G M G M   0 12 0 62 0 59 43 53 45 45 42 36 42 27 23 18 11 9 28 1 31 20 22 10 61 37 39 21 53 26 45 24 36 25 27 6 17 54 9 9 13   10 33 20 60 30 58 17 52 28 44 32 35 35 26 16 17 3 8 16 0 16 Gemi.   july. lo. 12 1 2 3 4 5 6 7 8 12 11 10 9 8 7 6     0 13 0 58 42 56 34 50 55 43 6 34 13 24 56 15 41 6 50 0 0 20 24 10 56 17 54 15 48 48 41 10 32 22 23 6 13 50 4 55 0 0 10 34 20 53 21 51 26 46 12 38 46 30 6 20 52 11 34 2 34 0 0 Taur.   Aug. Vir. 12 1 2 3 4 5 6 7   12 11 10 9 8 7 6     0 13 10 50 0 48 11 43 11 35 33 27 2 18 1 8 9   20 24 10 46 20 44 37 39 51 32 53 24 32 15 27 6 8   10 34 20 42 23 40 51 36 18 29 34 21 24 12 25 3 6   Arie.   Sept. Libr. 12 1 2 3 4 5 6   12 11 10 9 8 7     0 13 0 38 30 36 58 32 37 26 7 18 8 9 16   20 23 10 34 32 33 4 28 55 22 33 14 51 6 7   10 33 20 30 40 29 16 25 18 19 14 31 38 3 2   Pisces.   Octo. Scor. 12 1 2 3 4 5   12 11 10 9 8 7     0 14 0 27 0 25 40 21 51 15 50 8 3 6 0   20 24 10 23 39 22 22 18 42 13 1 5 5 0 0   10 33 20 20 43 19 29 15 55 10 23 3 17   Aqu.   now. Sagta. 12 1 2 3 4 5   12 11 10 9 8     0 12 0 18 18 17 6 13 38 3 3 1 15   Capri. 20 22 10 15 30 14 48 11 55 6 36 0 0   10 32 20 15 23 14 13 10 52 5 36 0 0      

THE SECOND book OF the Geodeticall staff, showing the searching and speedy finding out of all Longitudes, Latitudes, Altitudes and Profundities, together with the Angles and sides of triangles( without the help of arithmetic, Protractor, Scale& compass, or any other cumbersome appendent whatsoever) as well astronomical, as geometrical, and that after two maner of ways; both pleasant, speedy and necessary: also you are taught with facility to work the Golden rule on the said staff.  

CHAP. I.  
Of the Art of Geodaesia, together with the maner of using of the Geodeticall staff, with the things therein to be considered.  
AS the Elements of Geometry serve to express the magnitudes in determined species or figures, and their diverse affections by definitions& proposed theormes, whereby we haue the theorical or contemplative part; so this art of Geodaesia doth explicate or adumbrate the use of those propositions and theormes, certainly& by infallible grounds, in any material subject, whereby wee haue the active and practical part. 
Geodaesia quid.  Therefore may we define Geodaesia to be all art, which doth search out the quantity of any material magnitude proposed to be measured, expressing the same to us in a known measure; so that it is written, Geodaesia est ars, 
D. Ryff.  quae propositae magnitudinis quantitatem nota quadam mensura invenit. But the very emphasis of the word itself doth imply a proper application, in respect of our intended measuring of ground; for the word signifieth Terrae distributio, or Terrarum partitio: 
In Geodaesia du● consideranda ●eniunt.  and therefore we may aptly call the art of measuring ground, the art of Geodaesia. So now in this art of measuring two things happen to be considered, that is, the material subject, proposed to be measured, and the instrumental cause, by which it is measured. This material subject is considered as well according unto his kind and form, 
De subiecto Geo. daesiae.  as according unto his adjuncts.  

Genera magnitudinis quid,  By the kindes of magnitude is understood, whether or not the thing proposed to be measured, be lineal or limamentall, and thereby whether an Angle or a figure? and if a figure, whether plain or solid? if plain, whether a right line or a curued line? if right lined, whether a triangle or Tryangulated? and so forth, for it is a Geodeticall ground: 
Axioma Geodaeti.  Omnis maguitudo cognomine mensurae genere mensuratur. So lines are measured by lines: for the purpose, it is required to know the length of London bridge, that is laid open by a right line, which doth equal the longitude of the bridge: but superfices are measured by superficies, as an acre is so many feet square &c. so likewise solids are measured by solids, as hereafter.  
Now in the knowledge of these kinds of magnitude, the affection whereof doth offer itself to be considered, that is to say, whether or no the lines to be measured be right or obliqne, 
Affectio magnitudinis.  and in those liniaments whether the proposed figure be ordinary or inordinary &c. as whether the given triangle do answer unto an Orthogonium, Amblygonium, or an Oxigonium &c. for you must note that you may measure right lines by this geodetical Instrument after 2. 
Mensuratio. lineae duplex.  ways, that is, either naturally or artificially.  
The measuring of right lines naturally by this Instrument, is perfected by the congruency of the thing measured with the measure itself; 
Naturalis mensislin.  so a longitude 4. foot long( as the staff) is absolved by the appellation in measure of four foot; so a yard being measured, is called three foot, or 36. inches, or 180. barley cornes, &c.  

Artificiaelis Mens●lin. In vsu Baeculi Geodaetici dvo consideraenda.  But the artificial measuring of things being most commodious, is performed by my Geodeticall staff otherwhere; wherein two things be occurrant or seem fit to be preconsidered, that is, the aptness of the measurer, and the diversity or anomality of triangles, as in the 12. Chapter: but for the aptness of the measurer, 
Aptitudo menso ris.  first he must haue a care that the sight be not infinite, that is, if the thing whose distance from you is sought, be not so far off, 
Visus infinitus.  that the eye cannot bear the visual beams thereunto forcibly.  
2 That you always wink on the one eye; in so doing, the sight is made more strong and forcible for a long distance. vis nanque optica e duobus oculis in vnum conducta, firmius collimat. Here we shall not need manus quietae, 
Obductus oculus.  as the jacob, and such other staues require.  
3 That your stationary line, or line that you measure, 
Linea Data.  be not too short; for the longer your line is, the truer you work; one 8. part of the distance is a good length: and note if the distance be a mile or 2, or 3. &c. never express your stationary line by feet or yards, but by perches or scores, or such great measures, because there be not so many divisions vpon the leg haply as there be feet or yards in the distance &c. though there may be as many score or perches, 
M. Blagraue.  yet you may make shift to help that by imagining every division in each leg, to be subdeuided into 2. 3. 4. or 5. parts, and so proceed.  
4 Let the angle betwixt the 2. stat. and the mark whose distance is required, be as near unto a right angle as you may, rather a little less then more: and as in the 3. note I shewed you how to increase the divisions for long distance, so for short distances you may reckon ever 10. for one, so in levying one cipher from each number, your staff is readily divided.   

CHAP. II.  
Of English measures, both old and such as be allowed by our statutes, and of the comparing of them together, with the difference that riseth betwixt any of them, in an acre of ground.  
I go about here to speak of English measures, because through the want of knowledge thereof many absurdities by the vnskillfull practitioners be daily committed: for some count the Italian miles and ours to agree; some seeming to be more wise give a little difference betwixt the same, which they say is not apparent or sensible, all which is a most palpable absurdity. For the true Italian mile contains 5000 feet, 
Milli. Italicum.  and the true English mile contains 5280. feet: 
Milli. Anglicum.  so that our English miles be longer then the Italiā miles by 280 feet: which I think is a difference sensible enough for any man of iudgement to perceive. If some of our more then curious heads were no better persuaded of themselves, they would never go about to affirm they can fetch you a true distance to the inch( nay I haue heard some vouch the barley corue) vpon their scale with the plain table Theodilitus &c, when I haue smiled at their simplicities, seeing them maintain such impossibilities: for how can they express the inch upon their scale, when the 16. or 17. perhaps the 30. part of an Inch stands for apearch or mile? I will maintain against such, 
unless they use the scale in the 5. book, or lib. 6. Chap. 42.  that vpon a scale of above ●0. parts in the inch, they can hardly express a distance unto the yard, but for fear of digression, let v●ouerpasse such sulphureous conceited Artists.  
As many use a false mile for our english mile, so diverse use false perches, 
False perches.  when we haue one onely perch allowed by Statute: for in some places in this kingdom, notwithstanding the* Statute provided for the contrary, 
Anno. 33. Ed. 1  they use twelve foot in a perch, unto the great loss of the buyer, wherewith they bee accustomend to meate meadows, 
Tenant-right measure.  calling it Tenant-right measure; of no word of art, but onely implying( as I take it) to be a right and proper measure belonging unto Tenants: for so the word itself imports. Others more proper and agreeing unto the nature of the said measure, 
Curt measure.  call it Curt measure; likewise before the said Statute( which many unto this day use) a perch of 18. 20. and 24. feet, 
Wood-land measure.  called Woodland measure: all which differ from the true and allowed measure, in such sort as ensueth.  
One acre of 12. feet in the perch, contains of Statute measure Ro. Da. 76   2. 1. 121. of a perch. One acre of 18. foot in the perch, contains of Statute measure Ac. Da. Perch. used in most places for wood-land measure yet. 1. 7. 2. 50   1099 One acre of 20. foot in the perch, contains in Statute measure Ac. Ro. Per. 603 1. 1. 3. 1089 One acre of 24. foot in the perch, contains of Statute measure Ac. Da. Per. 60 2. 4. 2. 121  
By this that is said, you may gather the diversities of measures used by many, together with the great loss or gain to the buyer or seller, according as the less or greater measure is used: but for the true measure which should bee measured through England, take it thus.  
thee barley cornes dried, 
Compositio vlnarum perdicarum &c. perch. Day-worke.  and taken out of the midst of the ear, make one inch of assizet 12. such inches make one foot; one foot and a half make a cubi●e: two cubits or three feet make one yard: two yards a fathom: 5. yards and a half, a perch: one perch in breadth, and four in length make a day-worke: ten day-workes make a rood( of some called Farthendell:) but more plain the quarter of an acre, 
Rood,  containing 4. perches in breadth,& ten in length, or 5. in breadth, and 8. in length, all is one: one perch in breadth, an 160. in length, make anaker: or four perches in breadth, and forty in length, make a true acre; 
acre.  so that an acre by Statute ought to contain 6272640. square inches, which is 43560. square feet, 4840. square yards, or 160. square perches.  
again for measures in longitude, 4. feet is an ell, 16. elles or 20. yards a score, 40. perches a furlong, 8. furlongs a mile, as in the ensuing Table.  
Fabian a Chronogapher writing of the conqueror, sets down in the history thereof another kind of measure, very necessary for all men to understand; four acres( saith he) make a yard of land, five yards of land contain a hid, 
Yard of land. hid of land. A Knights see.  and 8. hides make a Knights fee, which by his conjecture is so much as one plough can well till in a year; in yorkshire and other countries they call a hid an ox skin.  

A new and true Table of English measures, according unto the Statute.  barley corn. 3 36 54 108 135 216 594 1460 23760 190080   Inch. 12 18 36 45 72 198 720 7920 63360   foot 1½ 3 3¾ 6 16½ 60 660 5280   Cubit 2 2½ 4 11 40 440 3520   Yard. 1¼ 2 5½ 20 220 1760   ell. 1 3 foot 4 In 18 16 176 1408   Fadō 2¾ 10 110 880   Perch 3 Cu. 7 40 320   Score 11 88   Furlong 8   Mile.  
NOw I haue given you a perfect Table of our English measures, I will proceed unto such measures as be used by the Geometritians, and men of other countries, to the end you may both see the difference thereof, and also apply your Statte thereunto, being occasioned by any means whatsoever.   

CHAP. III.  
Of the parts of geometrical mensurations.  
A Pian in his Cosmography saith, that measure is a finite or determinated longitude, 
Mensura quid.  whereby is measured the unknown distance of places: but a late german Doctor, calleth all those measures that do not exceed the height of a perfect man, 
P. Ryff. Francefur.  mensurae minores, wherewith saith he, wee measure things naturally, setting aside arte: but those as exceed the ordinary stature of a man, he calleth mensurae maiores, which are composed of the less measures. And here you shall note that the parts wherewith Geometricians use to measure with, are called:  
1 Granum hordeaceum, 
Paries mensura.  a grain or corn of barley.  
2 Digitus, 
Digitus.  a finger containing 4. grains.  
3 Vncia seu pollex, 
Vincia.  an iuch containing 1. ½ fingers, but after Apian 3. fingers.  
4 Palmus minor, 
Palmus.  of Apian called palmus onely, containing 4. fingers.  
5 Spithama, 
P. R. Spithama.  alii dicunt, palmus maior, aspanne, the distance betwixt the little finger and the thumb stretched out, which contains 3. hand-breadths, or 12. fingers. Apian puts a measure called Dichas, after palmus, which contains 8. fingers.  
6 Pes, 
Dichas. Pes. Sesquipes.  a foot containing four palms, or 12. inches.  
7 Cubitus seu sesquipes, which is a measure of one foot and a half, or 6 palms.  
8 Gradus siue gressus, 
Gradus.  a Gréese, a step, containing two feet, but with Apian 2 feet: otherwise it were all one with his passus simplex.  
9 Passus, a place, 
Passus Geodaeticus.  two grades containing 5. geodeticall feet: but 〈◇〉 Apian this measure is called Passus Geometricus, 
Passus Geometricus.  which he saith is used in Cosmometra.  
10 Orgya, 
Orgya.  a fathom, containing 4. cubits or 6. feet: here some make a division, 
Mensurae maiores.  calling the measures henceforth, mensurae maiores.  
11 Pertica siue virga, 
Pertica.  a rod, a perch, a pole, which among the romans was 10. foot, and so doth Apian take it: but with the germans now it is commonly taken for 16. feet, agreeing near with our perch.  
12 Stadium, a measure of ground, containing 125. paces, 
Stadium.  or 625. feet. There be three sorts of this measure, 
Ital●●ura.  the first is that of italy or Germany, as we haue spoked of alteadie: the secand Olympicum, of 600. feet, 
Olympicum.  which is 120. paces: and the third is Pythicum, of 100. feet, that is 200. paces: of these Stadia, 
Pythicum.  8. make an Italian mile, containing 1000. paces, 
Milliare Italicum.  every place being 5. feet.  
I haue red of a difference betwixt Plinie and Diodorus, 
Plinie,& Diderus.  Siculus, in the discerning of Sicile, which haply may grow by occasion of the diversity of these stadia.  
Apian henceforth proceeding calls first,  
13 Leuca, 
Leuca.  a league, with some containing 1500. paces.  
14 Miliare Italicum: the Italian mile hath 1000. paces, 
Miliare Italicum.  or 8. stadia.  
15 Miliare Germanicum, 
Germanleum.  the german mile hath 4000. paces.  
16 Miliare Germanicummagnum, 
Germanicum magnum.  hath 5000 paces, which they hold a sufficient journey for a reasonable hodeper or traveler to go in two houres.  
17 Miliare Germanicum commune, 
Miliare commune.  the common german mile hath a 32. stadia.  
And you must understand that the Italians( which Apian called the latins) and we of great britain, do measure our great distances vpon the earth by miles, the Grecians by stadia, or furlongs, the French and Spanish by leucas or leagues; and yet they all differ in the length of their leagues: 
Leuca gull.  for the French league contains two Italian miles, 
Leuca Hispa  and the Spanish league 3. Italian: 
Mag. Gorma. lieu. Leuca Germa.  yea in some places of high Germany their league is so long as a man can scarce ride three of them in a whole day, as in Sueuia, for the great league of Germany contains five Italian miles, which is nothing in comparison of the leagues near Sueuia. Now the Egyptians measure the great space of the earth, by signs, the Persians by Parasangus, 
Parasange quid.  which as I conjecture is a measure of ground containing 30. furlongs, but as some writ, 50. furlongs.  
Thus you fee the dinersitie of measures according unto the custom of the country, but I marvel how our yards &c. and the French& Spanish, so differ, seeing the original of them both is all one: for as well they, as we, haue 3. barley cornes allowed unto one inch, and so forth, as in the second Chapter; and yet it is found that a French foot at Paris, is more then ours by one inch, and the Italian foot to be longer then ours by 2. inches and near a quarter; and yet their miles be shorter then ours, as in the second Chapter. 
Stophilerus.  Some writ that the german food is less then ours by two inches and a half, so that the Italian foot by this account, should be longer then the german by 4. inches and near three quarters: but I would with thee to apply thy staff onely unto our English measures before, unless thou wouldest make any map or such like for other countries, or confer former works with thy own, to the which end I haue wrote this Chapter, and set down the ensuing Table.  

A Table of the lesser sort of Geodeticall measures.  Granum Digit● Vncia Palm' Spitha ma Pes Cubitus gressus Passus Orgia   Granum 4 5 11/ 5; 16 48 64 96 160 320 384 640   Digiti 1⅓ 4 12 16 24 40 8 96 160   Vncia 3 9 12 18 30 60 72 120   Palmi 3 4 6 10 20 24 40   Spithamae 1⅓ 2 3½ 6⅓ 8 13⅓   Pedes 1½ 2½ 5 6 10. alias 12. vel 16   Cubiti 1⅔ 3⅔ 4 6⅔   Gressus 2 2⅖ 4   Passus 1⅕ 2   Orgiae 1⅔   

CHAP. IIII.  
Of Geodeticall Angles, and how to use the staff in taking them.  
THese angles which I call Geodeficall, be such as haue relation unto things situate vpon the earth, containing in them sensible greatness, and figure, in the using whereof, 
Angul. Geodael.  wee attain unto the practical part of Geometry: they be 4. in number, that is, Angles of Longitude, Latitude, Altitude, and Prosunditie, which indeed is but a reversed Altitude.  

PRO. I. Angles of Longitude. 
Angul. longitudinis.   
ANgles of Longitude be such as be taken vpon plain or hilly ground, lying before you in a right line, and be thus performed: place the legs vpon the side of the staff,& then bring the left leg even with the hollow staff, placing the said staff perpendicular, by the 5. Chap. then move the right leg up and down, until by the center pin, and the sight in the fidutiall edge thereof, you spy the mark whose distance from you is required: so is the Angle taken, which if you will express in degrees, repair unto the 5. Chapter.   

PRO. 2. Geodeticall Angles of Latitude. 
Anguli geodaetisiae latitudinis.   
ANgles of Latitude, be such as bee taken in seeking the distance betwixt two towers, trees, Stéeples, or such like, wherein there is no regard to be taken of the perpendicularity, or the parallitie of the legs, or the staff: but always are placed at the pleasure of the Geodetor, or Mensor; you may choose whether you will place the legs vpon the top of the staff, or hold thē in your hand only: the staff is Manus quietae, the maner of their taking is thus: turn the left leg towards the mark vpon your left hand, until by the top of the center pin, and the sight in the left leg, you see the said mark vpon your left hand; that leg resting, remove the right, until( likewise) by the ceter pin& right sight, you see the mark vpon the right hand: the legs resting, you haue the quantity of the Angle, which if you will express in degrees or parts of the quadrant, resort unto the 5. Chapter. No otherwise are 
Augul. Astrono. latitudinis.  Angles astronomical taken.   

PRO. 3. Geodeticall Angles of Altitude.  
ANgles of Altitude, be such as bee made in the taking the height of Turrets, 
Angul. Geodae. altitudinis.  houses, trees, &c. Wherein it is convenient that the left leg lye always parallel unto the Horizon, as in the 5. Chapter: then turning the right leg unto the summite or top of the Altitude, looking by the sights, as before the Angle is taken.  

Altitudinem stellarum invenire.  And no otherwise are astronomical Angles of Altitude taken: for there it is most requisite that the left leg lye always parallel unto the Horizon, and the right leg point unto the sun or star: but because you cannot look vpon the sun it his beams be forcible, take either of these ensuing ways.  
having place the left leg parallel to the Horizon, the end thereof pointing towards the sun, 
Solis altitudinem extrahere.  heave up or put down the right leg, until the shadow that the sight in the right leg yeeldeth, agree with the fidutiall edge of the said right leg: so is the Angle taken; which to express in degrees, resort as before unto the 5. Chapter.  
Or if you had a square piece of read glass placed at the end of the right leg, as Sea-men do vpon the cross staff, then might you take the Altitude of the sun, as you would of a star, or such other thing: because his beams could not annoy you, be they never so forcible.   

PRO. 4. Geodeticall Angles of profundity.  
ANgles of depth, are such as be taken in measuring of valleys, wells, 
Profunditas.  or any such other depth; they be but plain reversed Altitudes, as will better appear when we come to treat thereof. In the taking of them, the left leg should ly parallel, and let him stand perpendicular, the end pointing downwards. What should I say further? 
Vide lib. 4. cap. 9.& 10.  you see they be but Visaversa, to Altitudes, as I haue said.    

CHAP. V.  
The leg of the Geodeticall staff, being opened unto any angle to find the quantity thereof in degrees.  
A degree in latin called Gradus, is the 360. part of a circled: into which portions, all circles astronomical, or geometrical, 
Gradus. Circulus.  are imagined to be divided; and they make choice of that number, because of any so little a number, it will be divided by most numbers.  
Now for to find the quantity of an angle, you shall first understand what the quantity of any Angle is.  
The quantity of an Angle is the portion of a circled included betwixt the two marks, 
quamtitas A●● gulor.  whose distance is required: the center of which circled, is imagined to be at your eye, the proportion& semitry of which Angle is represented by the legs of the staff: so that to find the quantity of any Angle, is no more but to find the portion of the circled that is included betwixt the two points respective in each leg, which is thus had.  
Place the center of the graduator vpon the Equicurrall sight in the left leg: then bring the same center unto the point respective, 
Angulorum quantitatem, calculare.  by the 4. Definition, in the 2. Prop. of the first book: and by the 7. Definition, turn the degrees of a circled towards the legs, the center of the Graduator so resting, bring the other end of the Graduator unto the right point respective: then the degrees cut vpon the Graduator by the said point respective, is the true quantity of the Angle.  
But it may so fall out, the Angle being greater then 90. that the Graduator will not reach unto the point respective in the right leg: then must you note among the degrees, or unequal divisions, in the lower side the said right leg, where the very end of the Graduator cuts( the other end resting at the left point respective,) which by the 5. Definition, in the 2, Prop. 1. book, is the quantity of the Angle.  
Example.  
 
LET C A B bee an Angle, whose quantity is required, here placing the Graduator in the left leg A C, at the point respective D, then moving the other end of the graduator F, until the fidutiall edge cut the point respective E, in the right leg A B: so shall you see the quantity of the Angle C A B, numbered vpon the graduator, to be included betwixt the two points respective D E, videl. 57. degrees, which the right point respective E. cuts.  
But if the feet be opened to the Angle C A M, then the end of the graduator F will not reach to L the point respective in the right leg A M, therefore move that end lower towards A, until the very aper, or end of the graduator touch among the unequal divisions in the lower side of the right leg, as at P, so shal you sóe the said end of the graduator, among those unequal divisions, point out the just quantity of the Angle C A M, 104. degrees.   

CHAP. VI.  
To open the legs unto any Angle proposed.  
BRing the graduator unto the point respective, in the left leg: then if your Angle be under 90. degrees, bring the point respective in the right leg, unto the quantity of the said Angle in the graduator, so is the Angle made: but if the Angle proposed he above 90, degrees by the 3. Defi. in the 2. Prop. book 1. seek the said Angle, and thereunto bring the movable end of the graduator, so is the angle made. I will make short of this Chap. because he is but visa versa unto the former.  
Before we proceed any further in dimensions, wee will lay down some few astronomical propositions, because they be desired by most, and make the staff more detectable to many.   

CHAP. VII.  
To take the height of the Sun, stars, Comer, or any other thing above the Horizon in the heauens.  
BY the Altitude of the sun or stars, is meant( not as the vulgar imagine, that is to say, how far they are distant from the center of the earth, 
De altitudine so● lis,& stellarum. ) but how many degrees the said sun or star is elevated above the Horizon towards our Zenith, which is easily had thus.  
Place the legs vpon the side of the staff, so that the left leg lye parallel to the Horizon: then turn the right leg towards the sun or star, as in the 3. Prop. of the 4. Chapter is plainly set down, so haue you the angle.   

CHAP. VIII.  
The Altitude of the sun being given, to find the hour of the day easily.  
having taken the Altitude of the sun, repair unto the Table vpon the outer side of the staff, 
Horae inuentio interdius.  which Table is also set down in the 5. Chap. book 1. and there in one of the ten rows, vpon the left hand amongst the moneths, find the month, and day of the month( or the nearest thereunto,) and you shall understand that all the moneths from the beginning of June to the end of november, haue their daies set under them from 10. to 10. and all the other moneths from the beginning of December, to the end of May, haue their daies put above the month, counted from 10. to 10. or there abouts. having found the day of the month, note the row wherein it stands, in which row proceed rightwards, going about the staff, until in the Table answering thereunto, you find the degrees& minutes, if there be any of the sins true altitude before taken: and you must note, that the letter D in the top of the staff, and the letter M placed in each column, stands for degrees and minutes, serving for all the rows in each Table, as they stand under the said letters. Now the suins altitude being found in the Table agreeing unto the month, and then in the said Table answering to the day of the month, to find the hour of the day, look in the head of the said Table, inst over the degrees of the suins altitude, if your demand were before noon, or in the foot of the Table, just under the degrees of the suins altitude, if your observation were after noon, so will the hour of the day appear: and héere note, if you cannot find the just altitude of the sun in the Table, take the nearest by proportion, which by the 28. Chapter, you may work as near as you will.  
Also note in some moneths, there be placed the 33.35.34. day of the month, as in June, July, August &c. which show the 3.4. &c. day of the ensuing month: as in August there is placed, the 35. day, August hath but 31. daies, therefore it stands for the 4. day of the ensuing month, which is September.  
Example.  
The 12. of June I find the sun to be 53. degrees high, and 45. minutes: then resorting unto my Table vpon the high outerside of the staff, I find the said 12. day of June; and then proceeding rightwards in the same row, I find 53. degrees, 45. minutes, answering to which, in the top of the Table is 10. of the clock, and in the foot, two of the clock: but forasmuch as my observations were in the forenoon, I conclude it therefore to be 10. of the clock before noon.  
Another Example.  
The 16. day of August, 1606. I find the sun to be 40. degrees high, the nearest day to the 16. that I can find, in the 13. day, answering to which, in the rowerightwards, I seek 40. degrees, the height of the sun: but forasmuch as I cannot find 40. degrees in that row, I take therefore the nearest thereunto, which is 43. degrees, and 11. minutes: then for as much as my observation was after noon, I seek the hour in the foot of the Table, right under the degrees of altitude, as before, answering whereunto is two of the clock after noon.  
Now if you desire to haue the just and precise hour of the day, when the just day of the month, nor the true degree of the suins altitude cannot be found in the Table, work by the 28. Chapter, making proportion according to the difference.  
Note, you may make a Cylinder by the foresaid Table, and place the same vpon the back of your staff in stead of the said Table, and so save the stamping of so many figures, 
Gruntȳ Finei delft, horologio rum, lib. 3. pro. 3  as you be instructed.   

CHAP. IX.  
To take the distance of any two stars unknown.  
PLace the legs vpon the top of the staff, 
To take the distance of stars.  and then resort unto the 2. Prop. 4. Chap. of the Geodeticall Angles of Latitude, where this Chap. is performed in all respects.  
Example.  
I desire the distance betwixt Oculus Tauri, and Canis Maior; working according unto the 4. Chapter, and 2. Prop. you shall find they be distant 46. ¼ degrees: so and in like maner may you take the length of the tail of any Comet, or such other like impression seen in the airs; you shall always find the Comet and the sun to be in one great circled, 
Cometa.  he being opposite unto the sun, towards whom the tail of the said Comet is extended: from which reason, Cornelius Gemma draws the foundation of his argument, when he affirms that there is no fiery Region, and that Comets, &c. are set on fire by the onely heat of the sun: but the ancient Philosophers, affirm that a Comet is engendered in the highest Region of the air, being of a viscosious, or gross, warm, and slimy substance, 
Cometa quid.  apt to nourish fire, there kindled by the fiery Region, &c. But I run here from my own intention into a new discourse, therefore I refer those inquisitive hereof, unto Aristotle. I haue drawn a book of Meteors myself, 
Liber. 1. cap. 4. Meteorologicorum;.  not yet Printed.   

CHAP. X.  
To find the hour of sun rising and setting, the length of the day and night by the Geodeticall staff.  
having found the day of the month as before, proceeding in that row, go round about the staff rightwards, until you come unto the last degrees and minutes of the suins altitude, 
Ortum& occasum solis, nec non arcum eius diurnum& nocturnum, siue quantitatem diel& noctis observare.  that is placed there: then see what hour is just over the head thereof in the hours of forenoon, and that observe for the hour of sun rising. The hours in the foot of the Table, just under the said degrees, is the hour of sun setting, or a little after, according to the quantity of the degrees.  
Then if you double the hours of sunset, thereof cometh the length of the day, which if you take from 24. you haue the length of the night, as if the sun set at 8 of clock, that doubled is 16. the length of the day, which taken from 24 leaveth 8. the length of the night.   

CHAP. XI.  
To find the Altitude of the north Pole by the stars, as neuerset, a necessary Proposition for Sea-men.  

Altitudinem poli ngnoscere diversis modis sub nocte.  TAke the Meridian Altitude of the said star when he is just above the Pole, and again when he is just under the pole, right in the Meridian, then add these two Altitudes together, the half whereof is the Elevation of the Pole. In this working there must be always just 12. hours betwixt the taking of the two altitudes: for so long will the star be descending from the Meridian above the Pole, unto the part of the Meridian under the Pole: Now if you take the elevation of the Pole from 90. 
Altitudinem Equinoct. supputare.  there remaines the equinoctial height.   

CAP. XII.  
To do the last Chapter another way, by the stars set, or by the sun.  
FIrst take the Meridian altitude of the star, then learn the stars declination: if it be North, you must subduct the said declination out of his Meridian altitude, so haue you the Equinoctials height: but if the declination were South, add the same unto the Meridian altitude, so likewise haue you the Equinoctials height, which if you take from 90, you haue the Poles elevation: for the latitude and the equinoctial height are always one, the compliment unto the other, both making 90.  

Altitudinem poli in die perscrutari.  As in the night you take the elevation by the stars, in the self same manner may you find the elevation in the day by the sun: for there is no difference at all, therefore I will make short hereof.   

CHAP. XIII.  
The Latitude of any place being found, how to get the declination of the sun, or any Planet, star, or Comet out of the heauens, by observation.  
TAke the Altitude of the star or Planet, by the 7. Chap. then note whether the said altitude be more or less then the Equinoctials height: if it be more then the Equinoctials height, 
Declinationem solis stellarum,& planetarum elicer●.  then take the height of the equinoctial from the said Meridian altitude, the remainder is the declination of the said star, which you may conclude to be north, because the altitude was greater then the equinoctial height.  
If the altitude had been less then the equinoctial height, you must haue taken it out of the greater: so should you haue had the declination which you must haue called south, because the Meridian altitude was less then the height of the equinoctial: but you shall find some stars as never come into the south, for that they always keep vpon the north side of the Zenith. The Meridian altitude of such stars may be twice taken, as in the 11. Chapter: for the declination of these, take the latitude out of the greater Meridian altitude, and the remainder from 90. so haue you the declination: but if you will work by the less Meridian altitude, take then the said altitude out of the elevation, and that which remaines out of 90, so haue you the declination.  
Or if you take the lesser Meridian altitude from the greater, and half the remainder thereof from 90. the declination also appears: and take this general to make up the end of astronomical matters, that all those stars, whose declination is equal to the latitude, being both north or south, once in 24. hours, touch our Zenith, and those stars declining fromwards the Pole elevated, 
Quae stellae orlantur,& o cidant, aut per ver ticem aut infrae horizontem semper m●uentur, manif●stare.  whose declination is greater( never so little) thē the Equinoctials height( or compliment of the elevation) are never seen in that country; and in like maner those stars whose declination towards the Pole elevated, exceedeth the height of the equinoctial, do never set, as Casiopaeia, Vrsa maior, and minor, with us do.  
Such as never appear unto us, be Ara, pars equina Centauri. the bottom of the ship, and Canopus, with the cross and Triangle to Constellations newly found out, the being whereof is doubted of many.  
Let this suffice for matters astronomical at this time, hereafter it may be, I shall be occasioned to speak further: now we will onely hast to the Geodeticall use of the staff, for to find the pralax of Comets, Planets, &c. and the distance from the center of the earth, is easy.   

CAAP. XIIII.  
Of the diversity of geometrical Triangles.  
BY the diversity of Triangles, 
diversity of triangles.  is not meant in respect of their quantity, according as they be greater or lesser: but in respect of the sides or Angles that be given or required: for in all Geodeticall mensurations, you shall haue but two lines, and one Angle, or two Angles,& one line given; the other unknown lines and Angles be sought for, and may happen diverse ways.  
1 You may haue two known angles to stand at the end of a line known, 
Situs angulorum& linearum rectarum.  as A B C, and B C A, at the ends of the line C B, known, as in the 15. Chap. or 30.  
2 Or one of the Angles known, may be situate at the end of a line known, as C A B, at the end of A B the line known, and the other angle may stand at the end of a line unknown, as A C B at the end of A C an unknown line, as in the 18. and 19. Chapter, or 32.  
3 For the second difference, where you haue two lines and one angle given, thus may they be situate, either both the lines known may contain the angle known, as C B and C A, do the angle A C B, as in the 32. Chapter.  
Or else the one line known, 
Vide chap. 34.  may be the side of the angle known, as A C is to the angle A C B,& the other line known, may subtend the angle known, as A B doth the angle A C B,& this is the greatest diversity that may happen in any dimension vpon the staff, as in the 33. Chapter.   

CHAP. XV.  
If a Castle, Fort, or army of men, &c. be seen far before you, how to fetch the distance of them from you with great speed, and facility, without arithmetic,  
Propositio.  
Two Angles known, 
Vide chap. 30.  being situate at the ends of a line known, to find the quantity of the 3. Angle, and the other two sides unknown.  
IT is a common received ground amongst all Geometricians, 
En. 32. p. 1. Ra. 9. cl. 6.  that in all Triangular figures, the third angle is taken together, are equal unto two right angles, and two right angles make 180. degrees of a circled; therefore subtract the two known angles from 180. the remainder is the quantity of the 3. angle. 
Longitude.  Now for the distance of sides of Triangles, work thus: having a place assigned, whose distance is required, appoint two stations, in a known distance from you, then go unto the first station, and turning the left leg towards the second station, take the quantity of the angle betwixt the said 2. stations,& mark whose distance is required, by the 4. Chap. Prop. 2. Wrest the center pin there hard, so that the angle made by the legs be not altered: next take up your staff; but first remove the center of the graduator vpon the left leg, so many parts from the center of the Instrument, as the distance betwixt the first and second station is found to be: now go unto the second station, and there place the center of the Graduator, inst over the said station, in such sort, that the sidutiall edge of the left leg may lye in a right line with the stationary line( the angle before made looking fromwards the first station,) which you may eastly do, by moving the staff, you standing at the end of the left leg, until by the center of the graduator, and the center pin of the Instrument, you see the first station: but you must haue a special care not to alter the angle made vpon the legs at the first station: this done, turn the movable end of the graduator which lieth vpon the right leg( but not stirring the center from his place) until the fidutiall edge thereof, point just unto the mark, whose distance is required, this being done, you haue your desire: for the equal parts betwixt the center of the graduator, and the instrument, is the distance betwixt your two stations, and the parts cut by the fidutiall edge of the graduator vpon the right leg, is the distance of the mark from the second station, and the equal parts of the graduator cut by the right leg, or so many parts of the graduator, as be included betwixt the right& left leg( for all is one) is the just distance of the 2. station from the mark desired, in such parts of measure as you expressed your stationary line by, so haue you the two unknown sides of the triangle.  
Example.  
Let A B be a distance, A a three, B your station from whence you desire the true distance of the three, appoint C for the second station on the left hand, and there set up a visible mark: now take the quantity of the angle A B C by the 4. Chapter, Prop. 2. which let be 75. degrees, then count the length of B C vpon the left leg from the center, which let be 40. perches, whereunto bring the center of the graduator, then take up the staff, and go unto the second station C, over which place the center of the graduator, then turn the left leg K M, until the center of the graduator C, and the center of the instrument K point just to B the instru: so resting, remove the end of the graduator O, until the fidutiall edge thereof C& O point just to A; so is there 55. perches betwixt C and A: for that there be so many equallparts included betwixt the left foot K M, and the right foot K R vpon the graduator, then the graduator cuts the right foot at R, which is 49. parts, and so many perches is B from A. So you see we haue made a triangle vpon the legs, as K C R equal and proportional unto B C A, whereunto we might infer proportional terms, with the geometrical demonstration, which we will reserve for another place.   

CHAP. XVI.  
To take the height of any Castle, Tower, three, &c.  
FOrasmuch as all altitudes stand perpendicular, 
Altitude.  and therefore make right angles with the line of level, by occasion whereof, you shall need to seek but one angle and one line, for that the second angle is before hand known to be right.  
Take therefore the distance from your standing, unto the base of the thing whose altitude is required, this done, open the legs unto a right angle,& then place them vpon the side of the staff, so that the left leg lye parallel to the Horizon, next remove the center of the graduator, to so many parts of the left leg, as the base of the foresaid altitude is distant from you, then heave up, or put down the other end of the graduator, until the fidutiall edge thereof agree with the summite or top of the altitude; the parts then betwixt the graduator and the center, counted vpon the right leg, is the altitude of the thing required, and the parts of the graduator included betwixt the two legs, is the distance of the top of the altitude to your foot, whereby you may find the length of any scaling ladder.  
 
Let N A B be a height, B C the distance, 27. ½ yards, therefore I remove the center of the graduator C, 26. ⅕ equal parts fromwards N, videl. to C, then I open the right leg N M to a right angle on M, then I heave up the other end of the graduator F, until the point just to A, so will he cut 29. parts at Z vpon the right leg M N. I conclude therefore briefly, that A B is 29-yards high: and forasmuch as there be 39. equal parts counted on the graduator C F, betwixt the two legs M N, and N O: I therefore affiune the length of the Hipothemsall line C A to be 39. yards, and so long must the scaling ladder haus been.   

CHAP. XVII.  
You standing vpon a Cleffe, Tower, or rock,& to tell how far any thing seen before you, is distant from you.  
OPen the legs to a right angle, 
Longitudoc  and then place the right leg to lye parallel, so shall the left leg point just to the Zenith, now measure how high the Tower whereon you stand is, according to which height, place the center of the graduator in distance from the center of the instrument; the legs standing as before, and the right angle not altered, move the other end of the graduator, until by the fidutiall edge thereof, you see the mark whose distance is required. The parts then vpon the right leg, intercepted betwixt the graduator, and the center of the Instrument, is the true distance of the mark whose longitude was sought.  
 
Here A B is the distance sought, C B the Tower, 32. foot high; then working as in the last Chapter you shall find the distance A B 57. foot: for the working is all one, if the terms of arte do but interchange, as distance for altitude, &c. as you may easily perceive by the letters in each demonstration, or figure. Note that in the former, and also in this Chapter, you might haue taken away the graduator, and onely worked by the left sight, unless you had required the hypothenusal line, as in the next Chapter will appear.   

CHAP. XVIII.  
The height of a Fort, Castle, Tower, or such like given, to fetch his distance from you by onely taking his angle of altitude.  
Propositio.  
Two known angles of a triangle, being situate, the one at the end of a line known, and the other opposite to the said line, to find the other angle and two lines unknown.  
YOu be taught in the 15. 
longitude.  Chapter, to find the quantity of the third angle, and for the lines work thus.  
Open the legs unto a right angle, and then turning the center of the instrument towards the height, place the left leg to lye parallel, and the right to point just unto the Zenith: now remove the sight in the right leg, so far from the center as the height of the thing instructs you, the right sight resting there, draw back the sight in the left leg, until he be in a right line with the summite of the altitude, and sight in the right leg: then so far as he is from the center, that is, so many equal parts as be included betwixt him and the center, so many perches, scores &c. is the thing from you.  
Example.  
Suppose you lay encamped in a field by a river side, on the further side whereof there were a Castle, against which you would lay battery, and would be resolved whether your Drdinance that throws a bullet point blank 50. yards, would reach thereunto, or whether it were requisite to mount the said piece to any degrees of random: now suppose likewise you durst not( by reason of shot) stir abroad to perform it by the 15. Chap. but onely by some friend in the adverse Camp, you had notice that the Castle were 30. yards high above the level of vs. Well then proceed thus.  
Place the left leg C D to lie parallel, and then remove P C the right leg to a right angle: now for that B R is 30. yards high, I place the sight A in the right leg C P, 30. equal parts from C, thē I draw back F the sight in the left leg C D, until by F and the site A, I see the top of the Tower R: so that F A R lie all three in a right line, then I count the equal parts betwixt F and C, and find them to be 50: so then I may conclude, that it is just 50. yards to the Castle, and that the piece of Ordinance will bear a bullet thereunto point plank.   

CHAP. XIX.  
The last Chapter performed by the Author another way.  
OK thus, you may apply the last Chapter, Suppose you had a piece of Ordinance that would carry point blank; 50, yards, and that you were vpon a present occasion to place him to endanger the adversary, here remove the sight in the right leg to 30. the height of the Tower( as before) and then draw back the sight in the left legs to 50. the length that the piece will carry: now cause the piece to be drawn after you, you going direct towards the Tower, and bearing the left leg( as before parallel, until by the cites in both the legs, you see the very top, and there may you surely plant your piece, so that he shall throw the bullet full point blank vpon the Castle.   

CHAP. XX.  
You seeing two ships vpon the seas, or two Castles, Towers, Trees, or such like vpon the land, to seek the distance thereof without arithmetical calculation.  
Propositio.  
One angle, and two sides of a triangle being known, which contain the angle known, to find the other two angles, and third side unknown.  
AS for the finding of the angles, they bee taught in the 33. Chapter hereafter, onely thus you shall find the third side.  
By the 15. Chapter, take the distance of the two towers from you then by the 4. Chapter, 2. Prop. take the quantity of the angle betwixt the 2. marks, whose distance is required: then count the distance of the mark vpon the left hand, on the left leg, and thereunto bring the center of the graduator; then count the distance of the mark vpon your right hand from you, vpon the right leg, and thereunto bring the fidutiall edge of the Graduator: the equal part then included betwixt both the legs vpon the graduator, is the true distance sought.  
Example.  
Let C A in the figure of the 15. Chapter, be to mark whose distance is required, let B be your sta. first I take the distance of C B, and find it to be 40. perches, then of A B, and find it to be 49. perches, which noted, I take the angle A B C, then I count 40. vpon the left leg, and thereunto bring the center of the graduator, then I count 49. on the right leg, where I lay the fidutiall edge of the graduator: so then there are 55. equal parts vpon the graduator cut by the right leg, which I pronounce for the true distance of C A.   

CHAP. XXII.  
A Castle, Fort, tower, or such like seen before you, so that by reason of diverse impediments, you cannot take his distance from you by the 15. Chapter, how to perform it here by the help of two stations, though you cannot discern the base or any part thereof, but some pinnacle or such like; nor go sidewise, but strait forward or backward.  
Propositio.  
Si Triangula sunt aequiangula seu similia, cruribus homologis propertionalia, 
En. 45.6.& 7. Prop.  & contra.  
OPen the legs unto a right angle, and then take away the graduator, placing both the cites in the left leg, this done, go as near unto the tower as you will, or place the lower sight in the left leg at soms even number vpon the leg, going to and fro, or stirring the said site until the extremes of the end of the right leg, and the foresaid sight agree with the very summite of the altituds, keeping the right leg parallel, the site resting so, remove the other site some even number from the first placed site, going fromwards or towards the mark, as occasion serveth, until you see the summite of the altitude: again by the end of the right leg and the last placed site, the cites so resting, measure the distance betwixt your two stations, and note how many even parts be contained betwixt your two cites upon the left leg: for so many times as those even parts be contained betwixt the uppermost site, and the center of the Instrument, so many times may you safely conclude the distance betwixt your two stations, to be contained betwixt the tower,& the station furthest off the said tower.  
 
Suppose the distance of A B is required, now by reason of woods and hills you can see nothing but the top thereof C; well then, I open the legs of my instrument unto a right angle, and appoint my first station at D: the right leg lying parallel, I place my eye at the end there of E, removing the site in the left leg D F, until he be in a right line with C E, which falleth out to be in the top of the leg at F: so if you could measure the distance of B E, you had the altitude B E( as it falleth out here) but for that I cannot, I go back to G, and there I appoint my 2. stations, keeping the legs at a right angle( as before) and the right leg G A, parallel pointing just to B: then I place again my eye at A, removing the other site until he be in a right line with C A, as at I, then I note how far I the second site is off H my first sire, and I find it 96. equal parts( the séete being 2. foot long) then I note how often that is to be found betwixt H G, which hath 192. equal parts, I find it is twice there, therefore measure the stationary line E A, and it shall be twice so far from A to B A as 35. perches, therefore A B is 70. perches, which is twice so much as 35.   

CHAP. XXII.  
To take the altitude of a Castle, Steeple, tower, three, or such like, where you dare not by reason of shot, or cannot by occasion of waters, &c. come near unto the base thereof, according to the 16. Chapter, without arithmetical calculation.  
OPen the legs unto a right angle, then go towards the base of the altitude( so it be not over near) there plant your Instrument, 
Altitudo.  so that one of the legs lye parallel, and the other stand perpendicular, as they did in the former Chapter: Now place the site in the perpendicular leg, about the midst of the leg, or as occasion leads you, so that your eye being placed at the extremes of the parallel leg, you may thence, and by the site in the perpendicular leg, see the summite or top of the Altitude required; this done, bring the site in the le●ell leg, from the end towards the center, a certain e distance, as 3. quarters, half a quarter, or just so much as the site in the perpendicular leg is from the center: let the cites and right angles all stand fixed, then choose you a second station( going towards the wall) until you see again the top of the tower by both these last fixed cites( still keeping the foresaid legs, the one parallel, the other perpendicular.) Lastly measure the distance betwixt your two stations, that is betwixt the places where your seete did stand at time of observation, and the same distance shal be either the third part, the half the quarter, or just the height you sought, according as you did place the site in the level leg, the third part, half the quarter, or whole length of the site in the perpendicular leg from the center. Example.  
 
I place the left leg to lye parallel, and the right perpendicular at E, making E my first station, E D the left leg, E F the right: then I draw G the site down from F towards E, until he be in a right line with D A. viz. at G, which falleth out in the midst of the right leg E F. then I make a mark under the end of the left leg D, viz. at O, next I remove the staff, bringing the site in the left leg so far from the end M, as G was from E, that is to K, so is K the left site so far from M, as I is from L: then I go toward the altitude, until K I be in a right line with A, as D G was before, then I make a mark just under K at S, then I measure the distance betwixt S O,& find it 6. perches. I conclude, because K M is just as much as H I, or F G, that S O is just so much as the height A C, S O is 6. pearthes, therefore A C is 6. perches high, 
Vide Chap. 36.  so that such proportion as K M hath to H I, the same hath S O to the altitude C A, so may you work by the 29. Chapter.  
Note, in all these dimensions you fetch so much of any height as is above the level of your eye: for if you would know B A, you must add K S or D O thereunto.   

CHAP. XXIII.  
To measure any valley of such like prosunditie.  
we will make short of this Chapter, because of the facility thereof.  
Suppose you were to measure the valley A B C, first get the length of A B, then of A C by some proposition in this book, which had, the line C B cannot be hide, if you work by the 20. Chapter, for it is all one therewith.  
Example.  
Take the quantity of the angle C A B, then count the length of A B 46. perches vpon the left leg, and thereunto bring the center of the graduator: then count C A 40. perches vpon the right leg, to which bring the fidutiall edge of the graduator; the parts then intercepted betwixt both the legs, is the length of C B 43. perches.   

CHAP. XXIIII.  
A Church or such like, situate vpon the top of a high hill, and you standing vpon a lower hill, and a great valley betwixt, so that you cannot place the left leg to make right angles with any part of the Altitude, yet to find the height of the said tower, Church, &c.  
FIrst get the length of the line A C in the last figure, then by the. Chapter, get the quantity of the angle A C B, then take the said quantity from 180. so haue you the quantity of the angle E C A; 
En. 13.14. p. 1. Ra. 1, cons. 8. E. 5.  then open the legs unto the quantity of the said angle C by the 6. Chapter, next count the line C A vpon the left leg from the center, and thereunto bring the site or center of the graduator, then make the fidutiall edge of the said left leg so lye in a right line with A C: then placing your eye at the site in the left leg, heave up or put down the other end of the graduator, or the fite in the right leg, if the graduator be away, until the two cites be in aright line with the summite; the part then betwixt the site in the right leg and the center, is the altitude required: so may you do by a Castle standing vpon a high rock by the sea fide, and you in a ship far below.  
Example.  
A C is 40. perches, and so working as before, you shall find C E 24. perches, as is plain by the figure, without the multiplicity of words.  
  

CHAP. XXV.  
To take the profundity of any Well, or such like.  
IN this work, 
Profundities. See the 4. Book Chap. 10.11.& 12.  you may take away the graduator, then open the legs to a right angle, making B F the left leg lye parallel to the top of the Well B C, thē shall the right leg B G stand perpendicular; then move the two sights in each leg, as A& D, until your sight passing in a right line by both those sights concur just with the point E the bottom of the Well, then shall you note the equal parts on each leg, where both the cites stand, let D stand at 11. and A at 30. lastly measure B C, which let bee 60. now to seek the profundity or length of C E, work thus.  
Multiply B A in B C, and divide by D B, the quotient is your desire; I increase B A 30. by B C 60. so haue I 1800. which partend by D B 11. yields 135. the profundity.  
A note for another working by the staff then before.  
In all the Chapters before, I advertised you to open the leg unto a right angle, and here I let you to wit, that you may do it as well if you place the graduator all at right angles with the left leg, therefore you may work by either, according as you shall be occasioned.  
And note further, that as you wrought any of the former propositions instrumentally, so likewise may they be performed on the staff Arithmetically, which will nothing differ from the nature of the 5. book, and therefore I haue here let it pass: for consider what part of your staff doth correspond with the thing measured, which increase by those parts of the legs answering to the thing required, and the product part by the line measured; so haue you your desire.  
Example.  
The question is to know the distance of B F, the altitude B R being given 30. feet, open the legs to a right angle, as P C D, then put the sight A in the left leg at any number of equal parts, as at 15. which represents the line given: then draw back the other sight, until he bee in a right line with A and R, as R A F: note the parts, cut them by F on the right leg C D, which let be 25; so doth C D correspond to B F the distance sought, therefore as A C/ 15 is to C F/ 25, so is R B/ 30 to B F/ 50, or as A C/ 15 is to R B/ 30, so is C F/ 25 to B F/ 50 feet. work therefore by the common golden rule, or by the 28. Chapter.  
To work in maner of the Sector by the Geodeticall staff.  
Before wee can come to work by the Geodeticall staff after the maner of a Sector, it is most requisite to understand the Chapter ensuing, without which, things hereafter cannot so well be attained unto.   

CHAP. XXVI.  
Of the Equation of angles in the Geodeticall staff.  
BY the Equation of angles, is meant to make an angle at the center of the graduator equal unto the angle proposed at the center of the Instrument, or contrariwise to make an angle at the center of the Instrument, equal to the angle at the center of the Graduator, both which are performed after one and the self same maner; therefore when henceforth I bid you make Equation, remember that thereby is meant the making or coequating of an angle at the center of the graduator, equal to the angle at the center of the Instrument, or contrary, as before.  
Example.  
Let the legs of the Instrument be opened unto an angle of what quantity you will, as C A B. Now to make one at the center of the graduator equal thereunto, bring the center of the graduator unto what point you will in the left leg, viz. to P, then move the other end of the graduator K to and fro, until A N and P N be equicurrall, or make an Isosceles, that is to say, until the parts of the graduator cut by the right leg A B, be equal unto the parts of the right leg cut by the graduator, and so will there bee as much distance betwixt the center of the graduator P,& the place where he falls on the right leg, as there is betwixt the center of the legs and the graduator, counted on the right leg, 
Ra. lib. 6.10. Pr●.  viz. to N; and so haue you made the angle N P A equal to P A N, as may be proved.  
And so must you work if the angle made at the center of the Instrument be required to be made equal to the angle at the center of the graduator, onely you must keep the graduator at one stay, as you did the right leg, and move the right leg as you did the graduator.   

CHAP. XXVII.  
Two numbers being given, or two lines assigned, to find the third in proportion.  
OPen the legs to any angle what you will, then on the left leg count the lesser of the numbers or shorter of the lines, 
Duobus numeris datis, innenire tertium proportionalem.  whereto bring the center of the graduator; then count the greater number on the right leg, and thereunto bring the fidutiall edge of the graduator, he resting at that angle, wrest the screwe at the center of the graduator straight, so that the said angle be not altered: then count the greater number vpon the left leg, and thereto draw the socket and center of the graduator, not altering the angle; so shall the parts cut by the graduator vpon the right leg, be the third number or line in proportion.  
Example.  
 
Let the proposition be to find a number to bear such proportion to 30. as 30. beareth to 20. count 20. from A towards B, as to C, there set the center of the graduator, remove the other end E, until the fidutiall edge cut 30. in the right leg A G, as at F, then wrest the screw at C hard, so that you do not alter the angle A C F, next count 30. vpon the left leg A B, and thereto draw the center of the graduator, as to H: so shall the fidutiall edge of the graduator cut the right leg at O,& the parts there cut shall be in such proportion to 30. as 30. is to 20. which you shall find to be 45, as may be proved 
Eu. 2. pro. 6▪& 17. Prop. 11.  in civil affairs.  
And thus may you apply this Chapter: if 20. horses eat 30. bushels of oats in a day, what shall 30. eat in a day? and you shall find 45.   

CHAP. XXVIII.  
To perform the Golden number, called the Rule of 3. or rule of Proportion, where you haue three numbers given, and be required to find the fourth in proportion.  
THis rule of 3. is so called, for that as much as by 3. proportional numbers known, it always findeth out the fourth, 
De potestate 4. numerorum proportionalium.  and is of such force, that well-nie, all as well civil as mathematical negotiations depend thereupon: but we will onely open it for the brief mathematical use thereof. The work.  
Count the first number given vpon the left leg, and thereunto bring the center of the graduator: then( the legs being opened to any angle, as in the last Chap.) count the third number given vpon the right leg, whereto bring the edge of the graduator, and so wrest the screw at the center of the graduator hard, that the angle made be not altered: next count the second line vpon the left leg, and thereunto draw the center of the graduator, not altering the angle: 
Eu. lib. 6. Pro. 10.  note the parts then cut by the fidutiall edge thereof in the right leg, for that shall be the 4. number in proportion.  
Example.  
 
Let the proposition be, if 48. give 96. what shall 40. give?  
First count 48. in the left leg A B, and where that number ends, bring thereto the center of the graduator, as to C: then count the third number given( which must always be of the same denomination as the first) vpon the right leg A G, which is 40. whereunto bring the edge of the graduator C D, as to D: next count the second number given, that is 96. vpon the leg A B, and where it ends make the point E, whereunto bring the center of the graduator, not altering his angle: so shall the fidutiall edge thereof, cut the right leg A G at the fourth number in proportion, viz. at F, which is 80, bearing such proportion to D, as E doth to C.  
To apply it Geodetically.  
If 48. inches, give 96, what shall 40. inches give? and you shall find 80. as before, the height sought for.  
In civil affairs.  
If 48. yards of velvet cost me 96. pounds, what then shall I pay for 40. yards? 
E. 19. Pro. lib. 7.  you( as before) shall find 80. pound, as is demonstrated.  
As I haue noted before, so must you take heed in the placing of the numbers vpon the legs of the staff: for the 1. and 3. must be of one denomination, 
Conditio regulae animaduertenda.  so must the second and fourth, and the first and second must always be counted vpon one leg, and the third and fourth on the other, as vpon the eight, so as the placing of these numbers differeth not from the working Arithmetically, as it is written:  Debent nihilominus ipsi numeri, in vsum practicum eo modo reuocari, vt ignotus& optatus numerus quartum possideat ordinem: 
Orontius lib. 3. cap. 4. De Arith. Parct.  ac ipse primus, ré& nomine conveniat cum ipso tertio, secundus autem cum acquisito quarto. 
You may use the third note in the first Chapter, if need be.   

CHAP. XXIX.  
If a Tower, Castle, or army of men be seen far before you, how to fetch the distance thereof from you.  
The Proposition.  
Two Angles known, and situate at the ends of a line known, to find the quantity of the third angle,& the other two sides unknown, in any triangle.  
AS in the former working by the staff, 
Longitude.  of necessity you performed every Proposition before you could remove the staff, so in this kind of working may you do; notwithstanding it rests at your pleasure so to do, or onely take the angles, and perform the rest at your best leisure. I will make short hereof, let A be a tree whose distance is required, B your first station, and C your second; first, by the 4. Chapter, Prop. 2. take the angle C B A, which is 79. degrees, note that done, then take the angle B C A, which is 60. degrees, note that done: lastly measure the line C B( which I call your stationary line, because it is included betwixt your two stations) and you shall find it 40. perches, and note that done; so haue you two angles and one line: now to find the third angle, and the other two lines, work thus.  
add the two angles together, that is 79.& 60. which makes 136, which take from 180, so haue you the quantity of the third angle 44. degrees, as is proved: then for the two sides, B A, 
Chap. 15.  and C A, work thus.  
By the 6. Chapter, open the legs to the quantity of the angle C B A, which is 79. degrees, then make equation by the 26. Chapter, which done, wrest the screw at the center of the graduator hard, so that the angle made at the center of the graduator be not altered then open the right leg to the 
Vide lib. 6. Cap. 13.  angle of 60. degrees, and there let him rest, lastly draw the center of the graduator, so far from the center of the Instrument as the line C B instructs you, that is 40. perches; then the parts cut vpon the right leg by the graduator, is the distance of B A, which will be 49. perches, and the equal parts of the graduator included betwixt both the legs, is the length of C A, 55. perches, and so of any other.  
So haue you the third angle, and the two sides of the unknown triangle, known, which was required.   

CHAP. XXX.  
To take the height of any accessible thing, standing perpendicular.  
TAke by the 4. Chapter, 
Altitude.  3. Proposition, the angle of altitude, viz. A C B, then measure the line C B, which is 27. ½ yards. Now you need not to seek the angle C B A, because he is known to be right, therefore you need to do no more but remone the graduator 27. ½ equal parts from the center, and there place him at a right angle, then the legs resting at the angle A C B, the parts of the graduator cut by the right leg shows the altitude, which will be 29. yards.  
 
Or if you cannot place the graduator at a right angle, then make equation by the 26. Chapter: which done, wrest the screw at the center of the graduator hard, so removing the right leg from the former angle unto a right angle, the parts then cut by the graduator vpon the right leg is the foresaid altitude, which will fall out to be 29. yards. And here you must note that you may not remove the center of the graduator from the place he was fixed first, as at 27. ½.  
If you understand this Chapter, the working of the 17. Chapter after this maner cannot be hide, onely the angle A C B is taken as an angle of longitude, see therefore the 4. Chapter, Prop.   

CHAP. XXXI.  
The height of a Tower, &c. given to fetch his distance from you, and the length of a scaling ladder, without measuring to the base.  
The Proposition.  
Two angles of a triangle being known, whereof the one is situate at the end of a line known, and the other opposite to the said line known, to find the third angle, and other two lines unknown.  
SUppose I were standing at F, and that I were desirous to know the distance of F B and F R: 
Longitude.  first I take the angle of altitude by the 4. Chapter, Prop. 3. viz. R F B, and now the other angle R B F is already known: wherefore the third angle F R B is found by the 30. Chapter: but forasmuch as R B F is known to be right, I need therefore but to take R F B out of 90. and the remainder is the angle F R B: R F B is found to bee 30. or there abouts, take 30. from 90. there remaineth 60. the angle F R B, then for the sides, work thus.  
Open the legs to the angle F R B, viz. 60. then remove the refer of the graduator so far from the center of the legs as the line B R instructs you, that is 30. yards, he resting there, make equation by the 26. Chapter, not stirring the right leg: which done, fasten the graduator at the angle: lastly remove the right leg unto a right angle, so is the parts of the right leg cut by the Graduator the length of B F, viz. 50. yards: and the equal parts of the graduator which are included betwixt the two legs, the length of F R, the hipothenusall or scaling ladder.  
And so might you haue fetched the length of R B standing at B, by knowing the line B F.   

CHAP. XXXII.  
Two ships seen vpon the seas, or two Towrets vpon the land, to fetch the distance betwixt them both.  
The Proposition.  
An angle known, and two lines known, containing the said known angle, to find the other two angles unknown, and the third line.  
LEt C A be two marks, 
Latitude.  whose distance is required, first by the 30. Chapter take the distance of B C, and you shall find if 40. perches, then by the same Chapter, take the distance of B A, which you shall find 49. perches, then take the angle C B A y the 2. Prop. Chap. 4. this done, stay the legs at that angle, and count the length of C B vpon the left leg, thereto bring the center of the graduator, then count the line B A vpon the right leg, and thereto bring the edge of the graduator: the equal parts then vpon the graduator included betwixt both the legs, is the distance of C A, viz. 55. perches. Now to find the quantity of the other two angles, wrest the screw at the center of the graduator hard, so that the angle made before be not altered; the graduator so resting, make equation by the end of the 26. Chap.  
This so done, the right leg resting at that angle, remove the graduator to the point respective, and so find the quantity of the angle made by the said legs, which is 60. degrees: so now haue you 2. angles, viz. C B A first taken, which is 79. degrees, and B C A now taken, which is 60. degrees; so is the third angle easily found to bee 44. degrees, by the 15. or 30. Chapter.   

CHAP. XXXIII.  
To find the distance of any 2. places, though you cannot come to take the angle of latitude, as in the last chapter.  
The Proposition.  
Two lines and one angle being given, so that the one line is the side of the angle known, and the other doth subtend the said angle, yet to find the third line, and the other two angle unknown.  
SUppose you knew how far B were off C, and also how far C were off A: but you desired to know how far A were off B, now suppose you durst not go to C to take the angle of latitude but were compelled to stand at B.  
First therefore take the angle C B A, the feet resting at the angle 79. degrees, remove the center of the graduator vpon the left leg to the length of B C, viz. 40. perches; this done, count the line C A vpon the graduator, viz. 55. stirring the movable end of the graduator, until the number of 55. agree just with the fidutiall edge of the right leg: the parts then cut by the graduator vpon the said right leg, is the true length of A B, 49. perches.  
Now to find the other two angles, the center of the graduator fastened at his former angle, by the end of the 26. Chapter, make Equation, not stirring the graduator; which done, bring the graduator to the point respective, and so by the 5. Chapter, find the quantity of the angle made at the center of the legs, which you shall find to be 60. so haue you two angles, A B C, 79. degrees, and B C A 60: then by the 15. or 30. Chapter, the other is found easy to be 44. degrees,* or otherwise, 
Lib. 6. chap. 13.  as is said.  
I haue worked in this Chapter something from the nature of the Prop. though nothing from the truth: for the words of the proposition be, that the one of the sides should subtend the angle known, which you shall ice me briefly work, because you may see the diversities.  
having, taken the angle C B A to bee 79. degrees, as before, count the fide C A vpon the left leg of the instrument, whereto bring the center of the graduator,( hither unto the work is all one.) Now must you make Equation by the 26. Chapter, not stirring the right leg: so that now by transposition, the center of the graduator contains the angle known, which angle must be kept truly; then count the line C B 40. perches vpon the right leg, this done, open the right leg or close him, as occasion serveth, until the fidutiall edge and the point where the 40. perches before counted, did end just cut the edge of the graduator: the equal parts then vpon the graduator, included betwixt the two legs, is the length of the line B A, 49. perches, as before.  
Now to find the two unknown angles, you must measure the quantity of the angle which is made at the center of the legs by the 5. Chapter, 
Vide. lib. 6. chap. 13.  which you shall find to be 60. degrees, so haue you the first angle 79. degrees, this 60. degrees; the third therefore cannot be unknown, as before.  
  

CHAP. XXXIIII.  
Any Tower of such like seen before you, to fetch the altitude& hipothenusall distance thereof at two stations.  
IN this kind of dimension, I am to forewarn you of one thing, 
Conditio reguloe ●nimaduertenda.  which ought to be observed in all propositions before, where the angle of altitude is twice taken; that is, that the leg of the instrument which lieth parallel, be placed direct in the same line of level at the second station, as he was at the first, so that it is not lawful at the first station to plant the instrument high, and at the second low, although at both stations you haue respect to the parallitie of the leg; therefore look what part of the altitude is level with your eye at the first station,& that make again level with your eye at the second station: ordering your staff thus, you may proceed; ede without the producement of any error at all.  
Let the Proposition be to fetch the altitude of A C: 
Altitudo.  first I take the angle of altitude A K C, which I find 45. degrees; next I take( going backward some reasonable distance) I take the angle of altitude A D C, 23. degrees: then I measure the distance of K D, which is 6. perches: these three had, I note them down, and proceed thus.  
Open the leg by 
Lib. 6. Cap. 13.  the 6. Chapter to 45. degrees, the angle taken at my first station K: then make equation by the 26. chap. but first draw the center of the graduator to some point in the left leg, as to Z in the ensuing figure, so by transposition is the angle at the center of the graduator D Z P, equal unto the angle at the center of the instrument F P Z: the graduator so resting, remove the right leg unto a right angle, then diligently observe where the fidutiall edge of the graduator cutteth the right leg, which will fall out at the point D; this done, make a note where the center of the graduator was as at Z, then by the 6. Chapter 
Lib. 6. chap. 13.  open the legs unto the angle made at the second station D, 23. degrees; which done, make equation, thē wrest the screw hard, that the angle at the center of the graduator be not altered: now remove the right leg unto a right angle, and draw back the center of the graduator, until the fidutiall edge thereof fall again vpon the point D before noted in the right leg Y P: then note the parts included betwixt D P on the right leg, 
The legs with the geometrical ground of the work.  and N Z vpon the left leg.  
Now thus may you infer the proportional terms, as D P is to N P, so is C D in the 22. Chapter to A C: but you see part of C D is unknown, therefore argue it thus from the parts known, saying, as N Z, the parts included vpon the left leg N P, is to D P the sigment of the right leg, so is D K in the 22. Chapter, the distance betwixt the two stations, to C A the altitude: N Z contains 144. parts, and D P 144. parts also; the distance betwixt the 2. stations is 6. perches: therefore works by the 29. Chap. and you shall find the altitude A C to be 6. perches, equal with the length of the two stations, and so of any other, length or depth, thus far the sector.  
The geometrical ground.  
Si Triangula sunt aequiangula seu similia, 
Eu 4.5.6.& 7. Prop. 6.  cruribus homologis proportionalia,& contra.  
The triangle A C D and D P N are alike, and the angle A C D and D P N are equal, therefore D P is also parallel to A C, as may be proved 
Eu. 31. P. 1. Ra. 5. confi. 3. pro. 12. : therefore by the same reason 
Eu. 2. Pro. 6.& 17, p. 11. , the thighs of both the triangles bee proportional among themselves, so that as A C is to C D, 
Fundamentum est. 4.5.6.7.8.9.10.11.& 12. p. 6 R. lib. 5. P. 13. lib. 6. p. 8.  so is D P to P N: and as N P is to N D, so is C D to A C: in like maner as N P is to P D, so is D C to C A, and as N P is to D C, so is P D to C A.  
And again for our purpose, as N Z is to P D, so is D K to C A, as before: or as N Z is to D K, so is P D to C A.  
As the first of these two last demonstrations serve for this last Chapter, so also doth it hold in the 22. Chapter, and therefore may the said 22. Chapter be wrought by the 29. Chapter,  
The end of the second book of the Geodeticall staff.  
Note generally, that the equal parts inserted vpon the vpper side of the legs outwards, are always to work by the help of a pair of compasses, and the equal divisions inwards to work by the graduator: therefore if I bid you see what number of equal parts the graduator cuts, you must seek it amongst the equal parts vpon the inward side of the legs.    

THE THIRD book OF the Geodeticall staff, concerning the use of the hypsometrical Scale.  

CHAP. I.  
To place your staff vpon plain ground, so that he may stand just perpendicular on the line of level, or vpon slopy or hilly ground to make right angles with the ground line.  
I Place this Chapter first, 
To place the staff at right angles with the ground line.  because of the great use thereof in every Proposition ensuing; for in the taking of many kindes of dimensions, I was often driven to carry a plumb live with me for the placing of the staff perpendicular, and in other dimensions, a squire for the placing of the said staff at right angles with the ground line, until at last the staff yielded a better and more easy way unto me himself, which is thus.  
From the place where you will plant your staff towards the mark, whose distance is desired, measure the just length of your staff, and there make an apparent mark: your staff then being placed in the extremes of the last measured line, and in the end furthest distant from the mark whose distance is required, open the legs unto an angle of 45. degrees, by the 6. Chapter, lib. 2. then bring the fidutiall edge of the left leg to agree with the edge of the hollow staff being there fixed: and the angle before made not altered, move the top of the staff towards or fromwards the mark, until by the fidutiall edge of the right leg you see the mark before noted vpon the ground to agree therewith, so is the staff ready planted.   

CHAP II.  
To measure the distance of any mark seen before you in a right line.  
Orontius. Chap. 4.  
TAke the legs out of the staff, and place them vpon the side of the staff, placing the hollow staff so; that he may not lean towards or fromwards the mark whose distance is required, 
Qualiter linea recta in plano metiatur.  as in the last Chapter: then by the 4. Chapter, Prop. 1. book 2. take the angle of longitude, and wrest the screw pin hard, so that the angle be not altered: next bring the center of the graduator unto the left point respective, and then turn the parts of the scale towards the leg*; 
Lib. 1. Pro. 2. Desi. 10. ; and note what parts are there cut by the right point respective: for such proportion as the parts cut haue to 12. the same hath the hollow staff to the length required.  
Example.  
 
Let G the point respective cut D B the graduator at 3. parts, and for because 3. is to 12. in a Quadruplate proportion, I conclude E F the line required, to contain the length of the staff 4. times, whereupon the staff being four foot, I may inter the line to be 16. foot.  
Hopt. Or work thus by the 
Lib, 2,  29. Chapter, saying, it 3. give 12. what shall 4. give? and you shall find 16. and here it were fit to use the fourth note of the first Chapter, book 2.   

CHAP. III.  
To perform the former Proposi. where a distance is required, by the onely help of a vulgar gnome, or Carpenters squire, represented on the staff.  
Orontius Chap. 5.  

●●a operandi vulgati Gnomo●i●.  IT hath pleased some to adjoin a maner of measuring without the hypsometrical scale, whereby the length of the foresaid lines lying vpon plain ground, are obtained as it were at an instant by a gnomon or right angled figure, such as mechanics vulgarly use, called a squire. I would not let pass thus kind of measuring( saith my Author) as well for that it is facile and easy, as also because it seldom happens that such measures haue always a scale at hand; let therefore the right line be given whose longitude thou dost desire, and let it be A B: then by the first Chapter erect your staff A C, then take the graduator of the legs, and open them unto a right angle, as D C E, then place the center C vpon the side of the staff, not altering the angle, lift up or depress down both the legs until by the fidutiall edge of C D, the visual beam concur just with the further terms or end of the line whose length is sought, as to B: the legs so resting and the right angle not altered, wrest the center pin hard, and there let the instrument remain, then look by the fidutiall edge of the other leg C E, and note where the visual beams touth the ground, as at F, which noted, say thus:  
Such proportion as the erected staff A C hath to the parts A F, the same shall the line A B haue to A C, the length of the staff.  
As if A C be 6. foot, and A F 2. foot, because 6. to 2. hath a triple proportion, therefore the proposed longitude A B shall three times contain six feet the length of the staff, which is 18. feet.  
Hopt. But my staff is but 4. feet, and A F falleth out to be 1. foot and four inches, which is also in a triple proportion to 48. inches, which is four foot: work by the twenty eight Chappter, book 2. 
Deductio praedioctorum geometricae, Eu. lib, 1, pro, 32   
The Geometicall demonstration hereupon.  
In the triangle B C F, the 3. angles bee equal unto a right angle, as may be proved*: but B C F is a right angle, therefore the other two, C B F, and B F C together are equal unto a right angle: also by the same reason the two angles A C F, and C F A in the triangle A C F, are together equal to one right angle: for the third C A F is right, therefore the two angles C B F and B F C, are in like maner equal to the two right angles A C F and C F A, because they be coequated by a like angle, that is to say, with a right angle: but if you do take the angle B F C, common to both, from those equal angles, the remainder C B A by common sentence will be equal to the other A C F, and the angle B C A is equal to the angle C A F: for they be both right, and therefore the other angle A C B shall in like maner be equal to C F A: therefore the two triangles A B C and A C F, are equangle( saith my Author, 
Orontius. 4. Eu. lib. 6. ) quare& quae circum aequales angulos laterae proportionalia: therefore as A C the staff is to A F the measured line, so is the proposed longitude A B to the erected staff A C, which was required to be done.  
Hopt. As you haue fetched lengths, so by the same reason may you fetch heights, as in the demonstration: for C D will hold always the same proportion to D E, that B F doth to A B; so by measuring of F B, and working by the 29. Chopter the altitude cannot be hide.   

CHAP. IIII.  
To take the height of Altitudes perpendicular, elevated by the means of shadows, with the Geodeticall staff.  
Orontius Chap. 8.  
ALthough( saith Orontius) we haue determined to set down the difference of shadows to the thing ensuing shadow, in their place, that is in our fourth book of Cosmography, howbeit we shall not bring it out of season, hereto deliver a pregustation in brief, how things directed Orthogonaliter vpon the plain of the earth, do seem to exceed their altitudes. Therefore of shadows we must understand which be called right, 
Vmbrarum recta. rum Definitio.  that is to say, which are extended in length and direct on the plain of the earth, and do make right angles with the thing causing shadow; such are the shadows of Turrets or any other thing perpendicular, elevated vpon plain ground: but all right shadows are protended infinite when the sun is in the east or west, 
De vmbrarum rectarum cremento atque decremento. & by how much the more the sun ascends upwards, by so much the more the longitude of the said shadow doth decrease, and so continue proportionally until such time as the Sun shall come unto the Meridian, and then all right shadows be at their shortest. Furthermore, the sun declining from the Meridian into the west, the foresaid shadows be augmented in right order, and are made so much the greater by how much the more the sun shall be nearer unto setting: but on this condition, that the sun being in points distant alike from the Meridian, the longitude of the shadow shall fall out alike.  
1 Now to fetch the altitude of any thing by the shadow thereof, 
Operandi modus quand● vmbra maior ●st vmbr●si longitudine per B●c. Geod.  first by the 4. Chapter, Prop. 3. book 2. take the height of the sun, and fasten the legs at that angle; then bring the graduator unto the point respective in the left leg, and note the parts of the scale cut by the right point respective 
Lib. 1. Pro. 2. Defi. 10. : if it fall out in the contraris shadow, you may safely conclude the length of the shadow to be longer then the altitude sought; and the same proportion as 12. hath to the parts cut, the like hath the length of the shadow to the height required.  
Example.  
we will take Orontius his example, let 6. bee the parts cut on the graduator by the right point respective,& let the altitude causing shadow be F G, and the shadow it yeeldeth G I, the beams of the Sun H I: now zbecause 12. is to 6. in a duplatiue proportion, I conclude, the altitude F G to be twice contained in G I the shadow; therefore because I G is 20. paces, G F of consequence must needs be 10. paces.  
Or multiply the length of G I by the parts cut, and make division by 12. so haue you the altitude, as may bee proved by the demonstration of Orontius in this Chapter.  
Example.  
Multiply 20. by 6. there riseth 120. which divide by 12.& there comes 10. so many perches or paces high is F G, according ●a you did measure G I by perches or perches.  
Hopt. But with more ease then any of these, work by the 29. Chapter, book 2, thus: if 12. give 6. what shall 20. give? so shall the graduator cut 10, as before.  
2 If you take the altitude of the Sun, 
Dum vmbra suo aequatur vmbroso.  as before, and the right point respective, cut 12. parts just, then the altitude is equal unto the length of the shadow, therefore measure the shadow,& the desired altitude cannot be hide.  
Example.  
Let F G be the altitude required, the Sun being in K, whose beams A K L, and the shadow G L, which shall bee equal to G F.  

De umbrae altitudine velvmbros● minore.  If you take the altitude of the sun by the 2. book, Chap. 4. Prop. 3. as before, bringing the graduator unto the points respective, and see the parts cut to be in the right shadow or parts next to the movable end of the graduator, then is the shadow of the body causing shadow shorter then the body itself.  
Therefore such proportion as the parts cut haue to 12. the same hath the length of the shadow to the height.  
Example.  
Let the right point respective cut the graduator in 6. parts of the right shadow, let the shadow be G N, the sun beams M N, let the length of the shadow G N be 5. paces: then for as much as 6. is to 12. in a subduplatiue proportion, therefore G N is half the height of G F: which doubled, maketh the just altitude, viz. 10. paces.  
Generally multiply the length of the shadow by 12. and denied by the parts cut: the quotient is the altitude.  
Example.  
Multiply 5. by 12. there riseth 60. which divide by 6. the parts cut, there remaines 10. and so many paces high I conclude the altitude S G as before, and so for all the parts of the right shadow.  
Hopt. But without arithmetic, work 
Lib, 2, cap, 39.  as before, saying, if 6. give 12. what shall 5.? and in so doing by the foresaid doctrine, the graduator will cut 10. the height, as before.  
And take this general in all your observations, if 〈◇〉 altitude of the sun be under 45. degrees, then is the length of the shadow greater then the body causing shadow, and the point respective shall cut the graduator in the contrary shadow: which in our latitude falleth out from the 24. of August to the beginning of april, because in all that time the sun cometh not at the nearest within 45. degrees of our Zenith.  
But if the altitude fall out to be above 45. degrees, then is the shadow shorter thē the perpendicular boy causing shadow:& the parts cut by the point respective on the graduator, shall be in the right shadow, which falleth out in the summer time, from the beginning of april until the 24. of August, because the sun at that time towards the midst of the day moves nearer unto the Zenith then to the Horizon, because his meridian altitude is greater then the equinoctials height: which happen to regions on the north side the equinoctial, when the sun is included betwixt the equinoctial and the vernal Solstitiall parallel or tropic of Cancer, and with us in England being 7. degrees north from the equator: and further, at half an hour after three of the clock after noon, or so much after 8. in the forenoon, you shall never find the point respective to cut in the right shadow,( especially towards the midst of England) tho you make observation even on the solstitiall day: so that after 3. a clock in the after noon, and after 8. before noon, you shall never use the parts of the right shadow for the cause abovesaid: but these differences I will refer unto a larger volume, where it shall bee handled more artificially, onely here I give a taste thereof for the better understanding of the staff.  
And you shall note that the first side of this scale whereof the one is called Vmbra recta, 
Note.  the other Vmbra versa, be no other thē the Tangents of a lesser mark in a semiquadrant, as you may perceive in the appendix of the 7. book.   

CHAP. V.  
To perform the last chapter with more ease.  
having taken the angle of altitude, note the part cut in the right or contrary shadow, as before you did in the last Chapter: if the parts cut be in the right shadow, thence sort unto the two towes of figures placed vpon the back side of the right leg near the movable end, 
Tabula vmbrarum.  and in the aguies vpon the left hand, find the parts cut, answering to which is the figures set to express how often the length of the shadow must bee taken to find the length of the thing causing shadow, as in the Table in the margin appears, which Table must bee placed vpon the right leg, as he is here.  
1 12 2 6 3 4 4 3 5 2 ⅖ 6 2 7 1 ●/ 7 8 1 ½ 9 1 ⅔ 10 1 ⅗ 11 1 〈◇〉 12 1  
Example.  
I find, having taken the suins angle of altitude, that the point respective cuts 4. parts of the right shadow, which 4. I find in the towe vpon the left hand, answering to which is 3. placed, signifying unto you, that you must take the length of the shadow 3. times to make the altitude causing the shadow.  
2 If the point respective cut the graduator in the contrary shadow, resort also unto the said Table as before,& the parts answering to the parts cut, sheweth thee how often the height is contained in the shadow, that is, how often the length of the shadow doth contain the height of the thing causing shadow.  
Example.  
having taken the angle of altitude as before, I find the point respective to cut 8. parts of the contrary shadow, answering to which in the Table of shadows is set 1 ½, which shows the shadow contains the altitude causing shadow, once and a half; therefore if the shadow were 12. foot, the height was but 9. foot.   

CHAP. VI.  
To find the altitude of any body perpendicularly elevated by the visual beams, without the help of the sun.  
Orontius. Chap. 9.  
IT often falleth out when we would measure altitudes after the foresaid maner, 
Generalis operands via.  that by reason of the interposition of some dark cloud, the beams of the sun are made so weak, that there can no shadow happen: therefore wee must use the visual beams of the eye thus.  
Take the angle of altitude 
Cap. 4. Pro. 3. lib. 2. , as before, then as you be accustomend bring the center of the graduator unto the left point respective,& note the partacut 
Pro. 2, Desi. 10. lib. 1.  by the right point respective, which done, proceed therein according to the 8. Chapter: for there it is performed in all respects, in which for the better opening of the use of the staff, I will proceed more at large, taking the demonstration in the said 8. Chapter.  
1 Let the tower proposed to be measured be F G, let the angle of altitude be F N G, 
Vide cap. 4. apud. 3.  and let the right leg cut the graduator at 6. parts of the right shadow: therefore such proportion as the parts cut haue to 〈◇〉, 
Quando altitud● propositae mato● est intercaped●ne plani.  the same hath the distance of N G to the altitude F G: N G is 5. paces, the parts cut be 6. therefore F G must needs be 10. paces.  
 
2 
De plani longitudine, proposita altitudine aquali,  If the parts cut( when the angle of altitude is taken) be 12, then is the altitude equal with the length betwixt your staff& the base, adding( as you must before also) the distance of your eye from the ground thereunto: see the 4. Chap, at the figure 2.  
If the parts cut by the point respective fall in the contrary shadow, then must you work contrary unto the operations in the right shadow, 
Dum altitudo ab intercapedine plani supercttur.  after this maner: such proportion as 12 hath to the parts cut, the same hath the length from your staff unto the base to the desired height, always adding the height of your eye above the base thereunto. Example.  
Let the parts cut be 6. in the contrary shadow, which is to 12 in a duplatiue proportion: therefore the distance of your eye from the base as G I; is twice so much as the altitude F G: G I is, 20. paces, therefore G F is but 10. paces: work therefore by arithmetic or without, as in the 8. Chapter at the figure 1. You shall also find this Chapter to agree with the 9. Chapter: therefore the Table of shadows set there may also be used here in the same maner.   

CHAP. VII.  
To search out inaccessible height, with no less case then before.  
Orontius. Chap. 12.  

Canon generalis ●perationis.  IT often falleth out that you cannot haue access to the base of the perpendicular, whose altitude is required, and yet you are desirous of the height therof, which you may easily perform thus.  
Appoint thy first station, and there place thy staff, and take the angle of altitude by the second book, Chap. 4. Prop. 3. and note the parts of the scale cut as before: then go towards or fromwards the altitude as occasion serveth, and there make thy 2. station in a known distance from thy first: which done, take again the angle of altitude( as before at the first station)& likewise note the parts cut at this station: then see senerally what proportion the parts cut at either station haue to 12. and subtract consequently ●he lesser out of the greater( for they will bee always unequal) and the residue reserve for the number of the altitude then measure the distance betwixt the falling of the first visual became and the second,( or betwixt the two places where your staff was, if after you add the distance of your eye from the base to the altitude) the occurrant or former number is to be divided by this number of measures betwixt your two stations: the quotient engendered by the division of this number, sheweth the altitude, in those parts or measures, as before you measured the distance of the two stations by.  
It sometime happeneth that the distance intercepted between both the stations is to be taken for the altitude, 
Corolarium.  which comes to pass so often as after subtraction of the former numbers there remaineth but one, because it is frustrate to divide a number by one.  
 
Let the altitude difficult to haue access unto, be F G, and let your first station be where you placed your staff, at the point H, and let the visual beams fall at I, and let the right leg B A cut twelve parts on the graduator at C; here the parts cut are equal to 12. therefore you shall keep but one for the denominator in the first number.  
Afterwards go back to the second station, as at K, and there again take the angle of altitude, and note the parts cut as before, viz. 4. by D O, the right leg at F: then because 12, to 4. is in a triple proportion, keep therefore 3. and in like maner th● visual beams vpon that plain do concur to the point L: the● subtract 1. from 3. the remainder is 2. which keep for the height then measure the interval or distance of I L, which let be 20 cu●bits, which divide by 2. there is made 10. for the quotient, therefore let us conclude F G so many cubits high.  

Alius ad idem operandi modus.  Hopt. Here note, if you measure the distance betwixt H K, the places where your staff was, and omit the reversed visual sights D L and B I; then working as before, the altitude taken is so much of the Tower as was above the level of your eye at either station, viz. Z: to which if you add the length of your staff B H, or B K, the whole altitude from the base to the top appears as before: and note further that the left leg D Y and B R must ly parallel, and at both stations point to one mark, as to Z; as is in the 36. Chapter, book, 2.   

CHAP. VIII.  
To perform the last Proposition with more ease, after Gemma Frisius.  
Gemma Frisius. Chap. 8.  
Gemma Frisius in his book called Vsus Annuli Astronomici, teacheth this way for such that be ignorant in arithmetic.  
seek two stations in going towards or departing from the mark whose altitude is required, so that in the one station the point respective cut 12. the other 6. of the right shadow; then double the distance between both the stations, and the altitude appears: or if in the one station 12. be cut at the other 8 of the right shadow, then triple the distance or in the one at 12, the other at 9. umbrae rectae quadruplate the distance.  
In like maner if at the one station the point respective cut 12. in the other 8. of the contrary shadow: then double the distance of both the stations, or one in 12. the other in 6. of the same shadow; then the space betwixt both the stations is equal to the height.  
Also if the point respective cut 6. parts, umbrae rectae, and at the other station 8. umbrae versae, or 6. umbrae versae, and four of the same shadow at the other station, or 4. and 3. umbrae versae; in all these the distance betwixt both the stations is equal to the height: remembering to add the height of your eye above the base thereunto, as in the end of the last Chapter.   

CHAP. IX.  
To take any perpendicular height by the onely help of the hollow staff, without the aid of the legs.  
Orontius Chap. 10.  
LEt your staff be divided into 12. equal parts, then erect him orthogonaliter vpon the given plain, 
Modus metiendi resum altitude●s per Baculum G●adaetum ab sque crurum be nesicio.  so that he lye parallel to the perpendicular altitude: then placing your eye at the ground, go backward or forward, as occasion serveth, until such time as thou dost behold the summite or highest part of the thing to be measured by the head of the staff: which done measure the distance betwixt thy eye and the foot of the staff by the same parts of measure as the staff is divided into: for such proportion as the staff hath unto the said distance, the same hath the proposed altitude unto the longitude intercepted betwixt thy eye and the base of the said altitude.  
1 Whereupon if the staff and the foresaid distance be equal, so mayst thou conclude the proposed altitude to be so much as it is betwixt thy eye and the base thereof, as in the figure with this example is apparent.  
Example.  
C D is the staff equal to the distance A C, 
Prima exempli differentia.  included betwixt your eye A, and the foot of your staff C; whereupon it may be inferred, that the proposed altitude B F is equal to the plain A H, included betwixt the points A and B, whereof both contain just 6. staues length.  
2 But if it so happen that the foresaid interval be less then the height of the staff, 
Secunda exempli differentia.  then the proposed altitude is greater then the intercapedicall plain, which is comprehended betwixt thy eye and the base of the altitude required; and the altitude hath the same proportion unto the longitude of the plain, as the staff hath unto the interval, betwixt thy eye and the foot of the staff as is not difficult to perceive by the staff E G, and the interval A E: for even as the staff E G doth contain the interval A E once and a half, after the same maner the altitude B H doth comprehend A B the longitude once and a half; A B the longitude doth contain 6 such parts, therefore B H shall be 9. so that the half of A B is to be added the whole longitude, that the foresaid altitude may appear.  
3 Furthermore if the foresaid interval be greater then the length of the said staff, 
Tertia exempli differentia.  then the foresaid longitude shall also be greater then the altitude proposed, and in such proportion the length shall exceed the height, as the interval doth the staff.  
Example.  
As for the purpose, such proportion as I K hath to the internal A I, the like hath the height L B to A B: I K is to A I in a sesquialiter proportion, whereupon it cometh to pass that A B the measured longitude, containeth B L the altitude once and a half: therefore it A B be 6. cubits, B L the altitude is 4: so that a third part of the said A B is to be taken away, that the proposed altitude B L may appear.  
The reason of this and all other such examples whatsoever, 
Praedictorum confirmatio Geometrica.  do hang on the equal proportion of angles& sides of triangles: and that we may summarily, 
Eu. l. 4.5& 7. p. 6.  or compendiously comprehend all things before recited, behold A C D, and A B H together with A C K, 
Ra. 7. p. 9.10, 12  and A B L, are alike equianguled: therefore as the side A C is to the side C D in the triangle A C D, so is the right line A B to B E: and also as A E is to E G, so is A B to B H: and as A I is to I K, so is A B to B L,  
Relatiuarelatiuis singulorum triangulorum, comparando latera. 
Orontius.   
  

CHAP. X.  
To setch altitudes by the visual beams reflected into a glass pleasantly and with great facility.  
Orontius Chap. 10.  
TAke a plain glass, and throw the sane vpon the ground, 
Altitude by a glass.  then depart from or draw near thereunto, until such time as you see the top of the Tower in the glass to answer with the top of your hollow staff held perpendicular: then measure the distance betwixt the foot of your staff, and the mark in the glass where you fixed your eye, for such proportion as the said internal or distance hath to the staff, the like hath the length from the glass unto the base of the perpendicular, unto the height desired.  
Example.  
Let A B be a Turret whose altitude is required, let the glass be C, the center of the eye E, the staff erected perpendicular D E: therefore it cometh to pass, as C D is to D E, so is C B to B A: the altitude C D is 6. foot, and so is D E: therefore C B is equal to B A, C B is 6. yards, therefore C A is 6. yards; work by the 29. Chapter of the second book.   

CHAP. XI.  
To fetch the altitude of a Tower or such like, after a new way not spoken of heretofore, and that with great ease.  
SUppose that you were vpon the top of the tower F, 
Altitude by a glass.  where you might behold another tower, as A: now suppose there were a man standing at C in the open field betwixt the two towers A F, who desired the altitude of the tower A B, from the place where he stands, he could tell you no more but how far the tower was from him, how far from you he or the tower was he could not tell, and yet he required the altitude: now imagine also you might not stir from F to fetch the altitude of F G, or the longitude of G C, nor G B to perform it by any proposition heretofore, so then must you work it in this maner.  
Cause the man in the field to throw the glass on the ground, just where his feet were: then cause him to move it up or down( keeping it parallel) until you see the top of the tower A B reflected therein, so will the visual beams be A C F: then with the legs of your instrument by the 4. Chapter, Prop. 4. book 2. take the quantity of the angle C F G, which note down, viz. 4●. deg●ées: so this is all the observations that you shall make, and yet we will fetch the quantity of every angle in the 2. triangles, which is more then was proposed.  
First take the angle C F G from 90.( for C G F is known to be a right angle) so haue you the quantity of the third angle F C G 
Ra. lib. 6. Pro. 9. , which is equal to the angles A C B, as may be proved Per sextam secundae parts perspectiuae communis, at queen decimam 12& 13. 
Eu. 32. p. 1.  perspectiuae Vitellionis.  
having taken the angle C F G, as before, viz. 45. take that from 90. so haue you the angle A C B, viz. 45. degrees: this had, count the line B C vpon the left leg of the instrument, viz. 6. cubits, where make a mark: then open the legs unto an angle of 45. degrees by the second book, Chapter 6, and bring the center of the graduator unto the point before noted, viz. 6. cubits the length of C B from the center of the instrument: the graduator resting there, 
Lib. 2. Cap. 26.  and the angle before made vpon the feet not altered, make equation,* and then remove the right leg unto a right angle: the parts then included betwixt the graduator and the center of the instrument is the just altitude, viz. 6. cubits.  
Now if you require the quantity of the other two angles, take 45. from 90, so haue you the angle B A C, because the angle A B is known to be right.  
In like maner, if the man had given you the altitude of the Tower, and required the distance thereof from him, it must haue been performed as before, as you may perceive, Lib. 2. Chap. 14. deft. 2.   

CHAP. XII.  
To fetch heights after another way( by their shadow) then was spoken of before.  
ERect your hollow staff perpendicular, and note the length of the shadow, 
Altitudes by their shadow.  then see what proportion it beareth unto the said staff: next measure the length of the shadow of the thing, whose altitude is required: for such proportion as the length of the shadow had to the hollow staff, the same shall the shadow of the altitude haue to the altitude itself.  
Example.  
The staff is D E, his shadow D C, the shadow of the altitude is B C: therefore such proportion as C D hath to D E, the like hath B C to B A: D E is equal to D C, therfore A B is equal to B C: work as before. 
Lib. 2. Chap. 29.   
Many things more might bee said of the hypsometrical Scale: but it were tedious to writ all. Here is all set down that Orontius speaks of, whose limits I intend not here to exceed in that behalf, because I haue chosen him for my Author: onely profundities and lengths in heights are wanting, which he hath wrote of, the which I here omit, because I will show their use in the geometrical quadrant ensuing now next: and as I haue applied the staff unto Orontius, so likewise haue I as well made choice of diverse other Authors, as you may perceive by the Title of the Chapter, as also added many new conclusions:& for that you shall the better understand the divisions, proportions and subdiuisions herein contained, I haue inserted a compendious Table, which may much enlighten a weak understanding.  
The end of the third book.   

A general Table of Number, declaring not onely all their kindes of number, and in what division or subdivision, any number whatsoever is comprehended: but also sundry varieties of proportion arising by the comparison of numbers together, as well in quantity as in quality.  
The branch most necessary for this third and fourth book.  

〈◇〉 

Absolute. which is 

Solte, by themselves. 

even 

Primes as two onely.  
compounds 

Evenly euven.  
Unevenly even.  
Perfect.  
Imperfect. 

abounding.  
defective.        
odd. 

Primes.  
Compounds     

among themselves. one to another. 

Prismes. usually.  
Compounds usually.      
relative. 

Inquantitie. 

Of difference.  
Of Proportion 

Of equality.  
Of inequality.      
In quality. 

Of proportion. 

arithmetical. 

continual.  
discontinuall. 

Proportion of equality is 

The greater which compareth the greater to the lesser being 

Sunplex. 

Multiplex. 

Dupla, as 4 to 2.6 to 3.10 to 5.  
Tripla, as 3 to 1.6 to 2, 9 to 3.  
Quadrupla, as 4 to 1.8 to 2.16 to 4.  
Quintupla, as 5 to ●. 10 to 2.15 to 3. &c.    
Superparticular. 

Sesquialtera, as 3 to 2.6. to 4.9. to 6.  
Sesquitertia, as 4. to 3.8 to 6 12 to 9.  
Sesquiquarta, as 5 to 4.10 to 8.15 to 12.  
Sesquiquinta, as 6 to 5.12 to 10.18 to 15 &c.    
Superpartiens. 

Superbitertias, as 5 to 3 10 to 6.15 to 9.  
Supertriquartas, as 7 to 4.14 to 8.21 to 12  
Superqui●●●qu. as 9 to 5 18 to 10 27 to 15  
Superquintusextas, as 11 to 6 22 to 12, &c.      
Multiplex. 

Multiplex superparticuler 

Dupla 

sesquialtera, as 5 to 2 10 to 4.  
sesquitertia, as 7 to 3.14 to 6.  
sesquiq. as 9. to 4 18 to 8 11 to 5. &c.    
Tripla 

sesquialt. as 7 to 2 14 to 4 19 to 6  
sesquiter. as 10 to 3 20 to 6 22 to 7  
sesquiq. as 13 to 4.26 to 8.25 to 8 &c.    
Quadrupla 

sesquialtera, as 37 to 9 18 to 4.  
sesquiter. as 41 to 10 26 to 6.  
sesquiquarta, as 17 to 4 34 to 8. &c.      
Or multiplex superpartiens. 

Dupla 

superbitertias, as 8 to 3 16 to 6.  
supertriquar, as 11 to 4 22 to 8.  
superquadri, as 14 to 5, 28 to 10. &c.    
Tripla 

superbitertias, as 11 to 3 22 to 6  
supertriquar, as 15 to 4.30 to 8.  
superquadriq. as 19 to 5.38 to 10 &c.    
Quadrupla 

superbitertias, as 14 to 3.28 to 9.  
supertriquar, as 19 to 4.38 to 8.  
superquadriquartas, as 24 to 5. &c.          
The lesser which compareth the lesser to the greater 

simplex. 

submultiplex. 

Subqua 

dupla, as 2 to 4.3 to 6.5 to 10.  
tripla, as 1 to 3.2 to 6.3 to 9.  
drup, as 1 to 4.2 to 8.3 to 12.  
quintu. as 1 to 5.2 to 10, 3 to 15 &c.      
subsuperparticuler. 

Sub 

sesquial. as 2 to 3.4 to 6 8 to 12.  
sesquiter. as 3 to 4 6 to 8.9 to 12  
sesquiq. as 4 to 5.8 to 10 12 to 15  
sesqu●●quinta, as 5 to 6.10 to 12. &c.      
subsuperpartiens. 

Sub 

superbit. as 3 to 5 6 to 10, 9 to 15  
supert. as 4 to 7 8 to 14.12 to 21  
supper. as 5 to 9.10 to 18, 15 to 27  
superquintu, as 6 to 11.12 to 22 &c.        
Multiplex. 

submultiplex superparticular. 

Subdupla 

sesquialtera, as 2 to 5.4 to 10.  
sesquitertia, as 3 to 7.6 to 14.  
sesquiquarta, as 4 to 9.8 to 18. &c.   

sesquialtera, as 2 to 7.4 to 14.  
sesquitertia, as 3 to 10.6 to 20.  
sesquiquarta, as 4 to 13.8 to 26. &c.    
Subquadrupla 

sesquialtera as 2 to 9.4 to 18.  
sesquiter. as 3 to 13.6. to 26  
sesquiquar. as 4 to 17.8 to 34. &c.      
submultiplex superpartiens. 

Subdupla 

superbitertias, as 3 to 8.6 to 16.  
supertriquar. as 4 to 11.8 to 22.  
superquadr. as 5 to 14.10 to 28 &c.    
Subtripla 

superbitertias, as 3 to 11, 6 to 2●.  
supertriquart, as 4 to 15, 8 to 30.  
superquadriqu. as 5 to 19.10 to 38 &c.    
Subquadrupla 

superbitertias, as 3 to 14,  
supertriquartas, as 4 to 19.  
superquadriquintas, as 5 to 24. &c.                    
geometrical. 

continual.  
discontinuall 

Direct.  
conversed.  
Alternate.  
inversed.  
Compounded  
partend.        
Of disproportion.      
figural. 

Linearie numbers:  
superficial mibers. 

Trianguler. The art of which numbers is called arithmetic figural not mentioned in this .3 book.  
Circuler. The art of which numbers is called arithmetic figural not mentioned in this .3 book.  
Square. The art of which numbers is called arithmetic figural not mentioned in this .3 book.  
Long square. The art of which numbers is called arithmetic figural not mentioned in this 3. book.  
Like flats. The art of which numbers is called arithmetic figural not mentioned in this 3. book.  
Diamerrals. The art of which numbers is called arithmetic figural not mentioned in this .3 book.  
Multianguler, &c. The art of which numbers is called arithmetic figural not mentioned in this .3 book.    
Solid numbers. 

Cubes.  
Squared squares.  
Sursolids.  
Zenzicubes, &c.       

Contract. 

To vulgar quantities, as 

Weight. not pertinent to this treatise.  
Measure. not pertinent to this treatise.  
Tune. not pertinent to this treatise.  
coin. not pertinent to this treatise.  
Men or beasts, &c. not pertinent to this treatise.    
To costick quantities 

figural whose proper figures are 

{potestas} signifying an unit or an ace. all which belong also to arithmetic figural.  
{powerof1} signifying a Roote. all which belong also to arithmetic figural.  
{powerof2} signifying a Square. all which belong also to arithmetic figural.  
{powerof3} signifying a Cube. all which belong also to arithmetic figural.  
{powerof4} signifying a squared square all which belong also to arithmetic figural.  
{powerof5} signifying a sursolide, &c. all which belong also to arithmetic figural.    
or radical whose proper figures are 

√ signifying a roote square. all which appertain to arithmetic radical.  
● signifying a roote Cubicke. all which appertain to arithmetic radical.  
● signifying a roote Zenzizenzike. all which appertain to arithmetic radical.  
● signifying the roote square of a roote square. all which appertain to arithmetic radical.  
● signifying the roote square of a cubic roote. all which appertain to arithmetic radical.  
● signifying a roote universal, &c. all which appertain to arithmetic radical.         
which is so much as ½ 3/ 1 4/ 1 5/ 1 1 ½ 1 l⅓ 1 ¼ 1 ⅕ 1 ⅔ 1 ¾ 1 ⅘ 1 ⅚ 2 ½ 2 l⅓ 2 ⅕ 3 ⅙ 3 1/ 7 3 ⅛ 4 1/ 9 4 1/ 10 4 ¼ 2 ⅔ 2 ¾ 2 ⅘ 3 ⅔ 3 ¾ 3 ⅘ 4 ⅔ 4 ¾ 4 ⅘ This proportion of the lesser inequality is when the antccedent exceeds the consequent, as 2 to 3, as 5 to 7, if you divide 2 by 3, the quot ent is 2 thirds, if 5 by 7, the ●u●●● 5 7 parts, so that this d●●●ereth not from the former, onely to every name this word sub is joined.    

THE fourth book of the Geodeticall staff, concerning the use of the geometrical quadrant:  
According as it is set down by Orontius: so that in practising by this staff, you may work by the quadrant itself at first inspection; set down both arithmetical, according to the said Orontius doctrine, and mechanical, agreeing to the vulgar capacity.  

CHAP. I.  
To take the horizontal distance of any mark from you, with or without the help of arithmetical calculation, and that at one station.  
Orontius, Chap. 3.  

De metienda linea supper idenplanum.  BEcause we will save a labour in the cutting of new figures, we will take the demonstration in the third book, Chap. 2.  
Let therefore the mark whose distance is required be F, the place where you stand E: where erect your staff at right angles with the ground line, by the third book, Chap. 1. then take the angle of longitude F A E by the 2. book, Chap. 4. Prop. 1. next see what parts of the 
vide lib. 1. Pro. 2. Defi. 9.  geometrical quadr●nt are cut by the right point respective: for such proportion as 60. hath to the parts cut, the like hath the length of the staff unto the altitude.  
Let the right point respective cut 15. parts at G, then such proportion as 60. hath to 15, the like hath A E the staff to E F the longitude: 60 is to 15 in a quadruplate proportion: therefore A E must needs contain E F 4. times, A E is sour foot, therefore E F is 16 foot: bear this Prop. in mind, for he cometh often in use hereafter.  
Example.  
 
Hopt. And here we may work better then Orontius, because the staff in these operations represents the one of the sides of the quadrant, which by how much it is larger, by so much it works truer then the quadrant.  
And to put by arithmetic, work 
Lib. 2. Chap. 29.  thus, saying: if 60. give 15. what shall 4. give?   

CHAP. II.  
You standing vpon the top of a Turret, to measure the distance of any mark seen before you on the ground.  
Orontius. Chap. 3.  

Qualiter ex alto linea recta in plano metiatur.  IF thou standing on the top of a Turret or any such like,& dost see any mark vpon the ground, whose distance thou wouldest measure, the ground being plain, and making right angles with the said Turret, do thus.  
Let the Turret be B E, the line proposed E F, or E H, or E K, whose longitude from the foot of the tower B is required by the geometrical quadrant.  
Take therefore the angle of longitude by the first book, placing the left leg of your staff by the side of the Turret, and heaving up the right leg until by the fidutiall edge thereof, you see the mark, whose distance is required: then bring the graduator unto the points respective, and note the parts cut in the geometrical quadrant: for either the section will be at C, which is the midst of the graduator, or betwixt B and C, or betwixt C and D, of necessity.  
1 
Quando linea data aequalis est alti ab oculo.  First let the parts cut be in C, and let the proposed line to be measured be E F; I say the line E F to be equal with A E the the perpendicular, because the graduator is cut in 60. parts at C therefore measure A E, so haue you E F; which happens so often as the point respective cuts the graduator at 60. parts.  
 
2 But if the segment by the right point respective be made betwixt B and C, as in G, 
De linea cadem altitu, ab oculo minore.  then the proposed line to bee measured E H, is shorter then the perpendicular A E, and hath such proportion to the longitude E H, as 60 hath to the parts cut: therefore if B G be 40. parts, because 60 is to 40 in a sesquialiter proportion, in the same maner A E shall contain E H once and a half: therefore measure A E by a thread, and take thence a third part of the length thereof, that is of A E; so haue you E H.  
Example.  
As if A E were 24. cubits, then E H the desired line should be 16 cubits  
Hopt. Or say, if 60 give 40, what shall 24. give? 
Vide lib. 2, cap. 29.   
3 But if the segment bee made betwixt C and D, as in I, and the line to be measured be E K, 
Cum cadem linea altitu. ab oculo superat.  then the said E K shall be longer then A E the perpendicular; and such proportion as the parts cut haue to 60, the like hath the-altitude A E to E K, the proposed longitude.  
Example.  
Let D I be 40 parts, 40 is to 60 in a sesquialiter proportion, whereupon the foresaid E K contains A E once and a half: so as if A L be 24 cubits, E K is 36 such cubits.  
Hopt. work by the legs as before, saying: if 40 give 60, what shall 24 give? you find 36 cut.   

CHAP. III.  
You standing vpon the ground, how to measure the perpendicular altitude of any thing standing orthogonaliter vpon plain ground.  
Orontius, Chap. 8.  
Because the working of this Chapter is but 'vice versa to the former, we will therefore turn the figure thereof upside down, and so make him serve our purpose.  
FOr the avoiding of proffers, let the altitude be A K, A F, or A H, 
Generalis operands modus.  erected perpendiculary vpon the plain A E: then placing your staff at A, take the angle of altitude 
Vide lib. 2. cap. ●. Pro. 3. : which done, consider the section made in the parts of the geometrical quadrant by the right point respective, that is, whether it be made in the point C, which is at 60 in the midst of the graduator, or betwixt B C, or else betwixt C D: for because it cannot bee made in any other manner.  
1 
Quando linea data maior est intercapedine plani.  First therefore let the section be made betwixt C D, as in I, and let the altitude to be measured be A K: then the altitude shall be greater then the space intercepted betwixt thy staff and the base: and such proportion as the parts cut haue to 60, the like hath the line A E to E K, the altitude.  
Example.  
As if D I were 40 parts, because 60 to 40 is in a sesquialiter proportion, so likewise A E shall contain E K once and a half: therefore if A E were 18 cubits, E G must contain 27 such cubits.  
2 
De linea aequalidistantiae ab ocuio ad basim.  But let the parts cut be 60 in the point C, and let the altitude be E F: then such proportion as the said parts cut haue to 60, the like hath A E to E F.  
Example.  
A E is 18 cubits, therefore it may be inferred that the said line E F proposed to be measured, is also 18 such cubits: therfore measure A E, so haue you E F.  
3 
Quando plani longitude datam superat altitud.  When the foresaid section is made betwixt C and B, as in G: then the proposed altitude shall be less then the line A E, intercepted betwixt thy eye and the base of the said altitude E H: so that such proportion as 60 hath to the parts cut, the like hath A E to the altitude E H.  
Example.  
Let B G be 40 parts, then because 60 is to 40 in a sesquialiter proportion, therefore E A shall contain E F once& a half, so that if you measure A E, and take away a third part thereof, you haue E F: as if A E were 18 cubits, you may conclude E F to be 12 such cubits, and so of any other of those parts. Therefore if 60 give 40, what shall 18 give? you shall find 12, and so work as before.   

CHAP. IIII.  
To measure inaccessible heights by the geometrical quadrant.  
Orontius, Chap. II.  
THere be many altitudes whose base we cannot come unto, by reason of waters or ditches in the earth, or such other prohibiting impediments: when such altitudes happen whose height be required, in this maner shalt thou work.  
Choose some convenient place, and there plant thy staff, 
Operandi regula.  and take the angle of altitude by the 2. book, 4 Chapter, as heretofore: then note the parts of the geometrical quadrant cut by the right point respective, observing what proportion they haue to 60, and those keep in mind: then go direct backward, or draw near unto the altitude, even as the aptnes●e of the place requires, and there again plant thy staff, taking the angle of altitude: and then note the parts of the quadrant cut, and what proportion they haue to 60. Now consider whether the parts kept at this station of the first be greater, and then subtract the lesser from the greater,& that which remaineth keep in mind: then measure the interval or stationary line, which divide by the former or occurrent number, which after subtraction was the denominator, and then the quotient thereof engendered declareth the altitude.  
Whereupon if there were a unite left, the interval comprehended betwixt the two stations, 
Corolarium.  is to be taken for the altitude: for because a unite neither in dividing or multiplying, doth alter the number, as is said before.  
Example.  
 
By example what we haue said, it may be, it will be the better understood; let therefore the proposed Turret be E F compassed about with water, whose altitude is required: let the first angle of altitude be at G, and the parts cut vpon the graduator be D H, 20 at the point H: 20 to 60, is in a triple proportion, therefore keep 3 for the denominator. This so done, go to the second station, viz. to I, and there again take the angle of altitude, noting the parts cut, and then see what proportion they haus to 60: the parts cut here are 12, betwixt D K, at the point K: 60 to 12, is in a quintuplate proportion: therefore keep 5 for the second denominator: next take 3 from 5, there remaineth 2, which keep for the upright, next measure the distance G I, and let it be 24 cubits: which divide by 2, so is the quotient 12: so many cubits high is E F the altitude.   

CHAP. V.  
You being vpon the top of a high Turret or such like, how to measure the altitude of any other Turret, though he be far lower.  
Orontius. Chap. 13.  

Ex altiori breuicrem altitudinem met●ri.  LEt the greater altitude be A E, from the top whereof you are put to measure the lower altitude G F: therefore place the left leg to point just to F, and open the right leg to the angle F A G: the right and left leg so resting, and pointing to the former marks, place a thread with a plumb at the end, in some part of the right leg, and to cut the left leg A D at what parts you will, as H I: then consider what proportion A I hath to the part of the thread, intercepted betwixt the two legs of the instrument: for the same proportion shall the visual beam A F haue to the proposed and less altitude F G.  
For there be two triangles, A H I, and A F G, alike equal:& because the angle at A is common to both the triangles, 
Probatio Geoemtrica.  and the angle A H I intrinsical, and of the same parts is equal to the angle A G F, and also the angle A I H is equal to the intrinsical angle A F G, as may be proved 
Eu. 1. Pro. 29. Eu. lib. 4. p. 5.6. Ra lib. 7 pro. 9. : therefore such proportion as A I hath to A H, that keep: for A F shall haue the like, to F G the altitude, as may be proved.  
Therefore it shall be necessary to teach you how to find the quantity of the line F A, which thou shall do after this maner: take the longitude of A E by some plumb line, then measure E F by the second Chapter, or any other in this book: 
Inuentio. Diagonalis.  afterwards multiply A E and E F in themselves quadrately, and reduce the product thereof into one number, and then extract the quadrant roote from the foresaid aggregated number: for that shall be the side A F of the right angled triangle A E F. For example, let A E be 8 perches, and E F 6: therefore 8 multiplied in 8, makes 64, afterwards 6 times 6 is 36: put 64 and 36 together, so will there result 100, the roote quadrature whereof is 10: therefore 〈◇〉 many perches A F: the thread H I in the midst of A D, the left leg, and A I, is double to I H: therefore A F is double to F G: so of consequence F G is 5 perches, such as A F was 10.   

CHAP. VI.  
To work the last Chapter versavice.  
Orontius, Chap. 13.  
IF you would work the former Chapter contrary, as to stand at O the less altitude, and fetch R E the greater, do thus: place the left leg O Z to ly parallel, then hsaue up the right leg O P, until you haue taken the angle R O L: so O L E F is a parallelogram, whose opposite sides are equal, as may bee proved*: 
Eu. lib. 3.4. p. 1. Ra. 10. p. 6. cons, 2  therefore measure O F by a thread, so haue you L E: then take the length of E F by the 2 Chapter, so haue you O L by the former alleged reason:* then take the altitude L R by the third Chapter, which added to L E, the altitude appears: for such proportion as 60 hath to the parts cut, the like hath O L to L R, by the third Chapter, Defi. 3.   

CHAP. VII.  
How to measure the longitude of any mountain, whose top stands not perpendicular over the base, but hangs sidewise.  
Orontius, Chap. 13.  

To seek the height or length of mo●taines.  THe longitude of any mountain whose top bends from, or hangs not over the base perpendicular, is had no otherwise then you found the length of lines lying vpon plain ground by the first Chapter: therefore you must observe that maner of working: and here you shall see the necessary use of the first Chapter, book 3.  

Summarius operandi modus.  Let E F be the mountain proposed to be measured, F the top thereof, hanging like the reoffe of a house from E: place therefore your staff by the preallegated Chapter, so that he may make right angles with the ground line E F: then take the angle of longitude 
Vide lib. 2. Chap 4. pro. 1,  A F E, and note the parts of the geometrical quadrant cut by the right point respective.  
For such proportion as the parts cut haue to 60, the like hath the length of the staff to the longitude E F.  
Example.  
Let the right leg A B cut the graduator at 10 parts, viz. at G: then because 60 to 10 is in a sextuplate proportion, in like maner E F the longitude contains A E 6 times; whereupon if the staff were 4 foots, the longitude is 24 foots.   

CHAP. VIII.  
To take the altitude of a Tower &c. placed vpon the top of a hill, or such like.  
Orontius, Chap. 15.  

Quomodo altit●do linearum disquiratur.  LEt the Turret be P R, placed vpon the top of the mountain S R: then by the last Chapter, take the longitude of the mountain S R betwixt the foot thereof, and the base of the tower, which let be 18: then turn the left leg S H to point just to the base of the tower R, heaving up the right leg S G, until he point just to P, the top of the tower: the legs resting in this maner, let fall a thread with a plumb from the right leg to cut the left leg S H in what part you will, as G H the thread, deu●ding the leg S G into two equal parts at the point H: then measure the part of the thread G H intercepted betwixt both the legs, and also the part of the leg S H intercepted betwixt S H: for such proportion as the part S H hath to the part of the thread G H, the like hath S R, the length of the mountain, to R P.  

Geometrica probatio.  For the two triangles S G H, and S P R be alike equal, as may be proved 
Eu. 29.1. Ra. 7. P. 9. Eu. 4& 5.6. P. : and for because the angle S H G intrinsical, is equal to the like parts of the angle S R F, it is proved, that as S H is to H G, so is S R to R P, the proposed altitude.  
Example.  
Let for example S H cut half the left leg, and H G bee a quarter, that is 15 parts, because the leg is imagined to be 60, so as S H will bee 30: then because 30 is to 15, in a duplatine proportion, therefore S R the longitude shall contain R P the altitude of the Turret twice: and we haue supposed S R the longitude to contain 18 cubits, therefore R P the altitude must bee but 9 cubits; which you may work by the golden rule, saying:  
First multiply 18 by 15, there is 270, which divide by 30, so haue you 9 for your quotientingendred, the altitude; or work as I haue often advertised you by the 2 book, 29. Chapter without arithmetic.   

CHAP. IX.  
How lengths in heights are found out.  
Orontius, Chap. 15.  

Length: in height are found.  WE will make short of this Chapter, because of his facility in working. Suppose that you were standing at F, and desired the length of A L in the altitude A E, first take the altitude of E L, then of E A; lastly take the lesser out of the greater, the remainder is your desire.  
To perform the same by the staff other ways.  
First take the height of L E as before, then open the legs unto a right angle, turning the right leg towards the height,& holding the left parallel, then remove the left sight so far from the center as you be from the base of the Tower; next remove the right site to the height of L E vpon the right leg: so you placing your eis at the left site, you may see the right site, and the said left to be in a right line with the said latitude L: the legs so resting, heave up the movable end of the graduator not stirring the center thereof, until the fidutiall edge thereof point just in a right line, to A: the parts then intercepted betwixt the right site and the edge of the graduator vpon the right leg, is the height of L A: so of any other.   

CHAP. X.  
To take the depth or profundity of any Well or such like.  
Orontius. Chap. 16.  
I Haue often  told you that profundities bee but reversed altitudes, therefore we will make short of Orontius, and onely proceed to the matter.  
Let therfore the oblated pit be quadrangular B E F G, 
profundity.  whose depth B G, or E F is proposed to be measured: place therefore the left leg to lye in a right line with B G: he resting so, remous the right leg until by the center A C, the site therein, you see the further side of the pit F, which is opposite to you at B: then see what parts of the geometrical quadrant be cut, as you be wont: for such proportion as the parts cut haue to 60, the like hath the diameter E B to the profundity B G, or E F; so measuring E B, you haue your desire.  
Example.  
For example, B H is 20 parts cut, 
Eu. lib. 34. P. 1. Ra. li. 10. P. 6. c. 2  B E measured is 6 cubits, and so many cubits is F G, as is proved: therefore multiply 6 by sixty, there is made 360; which divide by 20, the quotient is 18, and so many cubits is A G; from which if you take A B, the length of the left foot, you haue B G.  
  

CHAP. XI.  
To perform the last Chapter by the equal parts of the legs.  
FIrst take the angle F B E 
Lib. 2, cap. 4. P. 4.  as you are often instructed, next measure the line B E; this done, count the line E B vpon the left leg, whereunto bring the center of the graduator, and then make equation, 
Lib, 2, cap, 26. * not stirring the legs: which done, wrest the screw at the center of the graduator hard, so that the angle be not altered: lastly remaue the right leg unto a right angle; the parts then intercepted betwixt the graduator and the center of the initrument, is the length of B G.   

CHAP. XII.  
To perform the foresaid Chapter otherwise.  
Open the leg unto a right angle, and there fasten them, remove then the center of the graduator so far from the center of the instrument as the line B E shall instruct you; then place the said left leg to lye parallel to the line B E, so will the right leg be in a right line with G B: the legs so resting, move the movable end of the graduator up and down, until you see by the fidutiall edge thereof, the bottom of the pit opposite to you, as to E: the equal parts then that the fidutiall edge of the graduator cut vpon the right leg, is the depth of the pit, as may be proved* 
Eu. 4& 5. P. 6  
R. 7. P. 9. .   

CHAP. XIII.  
To measure prosundities by the hypsometrical Scale.  
Orontius, Chap. 16.  
LEt the pit or Well whose depth is sought, be E B G F: 
Profundities by the hypsometrical seal.  place the left leg of your instrument A B to stand perpendicular vpon the line B G: then remove the right leg until he point just to the opposite and profundest part of the pit, as to F: which done, bring the graduator unto the points respective, and note the parts of the hypsometrical scale cut: 
Lab. 1. pro. 2.  next measure the line or diameter E B: 
Desi, 10.  for such proportion as the parts cut haue to 12, the like hath E B the diameter to B G, the profundity.  
Example.  
E B is 6 foot, the parts cut be 4: multiply 6 by 12, there cometh 72, which divide by 4; there is 18 foot the depth of B G or E F: or work as I haue often instructed you, by the legs, saying, if 4 give 12, what shall 6?   

CHAP. XIIII.  
To fetchaswell the length of the sides, as also the perpendicular depth of any value or such like.  
Orontius, Chap. 17.  
IT is available oftentimes to know the depth and latitude of valleys, or such like, which you shall do in this maner. Let the valley be F, EE, FF, such as they be wont to dig about the walls of cities, whose greatest latitude F, FF is required, and also his greatest profundity GG 
De Vallis aut fossae latitudine.  EE: take therefore the longitude of D F by the second Chapter, which let be 18 cubits: then by the doctrine of the said second Chapter, take the dorsine longitude, or the length of F, EE, which let be 15 cubits.  
Multiply then 15 in itself, and there is made 225, afterward multiply the half of F, FF in itself, that is F, GG, 9 cubits; and there arifeth 81: take then 81 from 225, there remaineth 144, the quadrant roote whereof is 12: so many cubits may you conclude the profundity GG, EE to be; so of any such like.  
And here we will conclude this fourth book, together with the first book of Orontius, presuming that there can no right line happen howsoever situate, but he may be measured by the doctrine beforesaid.   
The end of the fourth book.   

THE FIFTH book OF the Geodeticall staff, containing the use of the Iacobs staff, together with the working of the Golden rule by scale and compasse● without arithmetic.  
Also you are taught to work the rule of proportion after another way then in the second book, that is, with scale and compass, with such speed as hath not been seen or heretofore published by any; also it teacheth you the golden rule inversed, and the double golden rule without Arith-meticke , with the construction of a new scale.  

CHAP. I.  
Of the Iacobs staff, of the comparing him with the Geodetical staff, together with the things requisite for to be observed in the use thereof.  
THe Iacobs staff is an instrument consisting of two rules made 4 square, 
Descriptie Baculi jac.  and truly tried: the longer of the two rules is called the Index or yard: the shorter is called the Transum, or Transuersarium, that is the transverse or overthwart rule: for because he must always lye cross on the yard, making right angles. The yard with some is divided into 2100 equal parts,& the transverse into 1000 such like parts, with other less, according as the use thereof occasions them: the transverse& the yard are coupled together, so that the one may move along the other at right angles.  
This instrument by some is said to take his name of jacob, 
Nominatio Eaculi jac.  grounding their reasons vpon these words in the Bible, For with my staff came I over this jordan: 
Gen, Cap. 32.  and the name might be well taken, the nee vpon diverse reasons, though the staff he used there, be interpnted to no mathematical use: others say he taketh his name of jacob, for because he was invented of that Patriarch himself, or his use was most in request about the time of jacob: and there is no one of these prealiegated reasons, but there may be sufficient causes urged to induce some credit to the foundation of their opinions that so called the said staff: but howsoever, it is written of the better sort, this instrument Esse omnium reliquorum geodaeticorum instrumentorum commodissimum, which also is well demonstruted by P. Ramus 
Lib. 9. Pro. 1.2 3.4.5.6.7. &c. . Now for the affinity that this staff hath to the Geodeticall staff, I say, the equal divisions are alike vpon both, but those of the legs of my staff be not so many: 
Lib. 2. cap. 1 no. 3  notwithstanding you be* instructed to make them as many as you will, they also agree because they be both right angles: now the diffecence of these two staues is this, the Iacobs staff hath a transverse to move along towards or fromwards the end of the yard: which my staff hath not, but is as well supplied by the site in the left leg, which doth represent the yard: for you may mo●ue him towards or fromwards the right leg at pleasure, as well as you could the transverse vpon the yard. The Iacobs staff hath also a yard to move up and down vpon the transverse, so that he may be near unto the one end,& further off the other end, or contrary, as occasion requires. My staff cannot do so, howheit you may move the equicurrall site in the right leg in such fort, so that the right leg may supply the use of the transverse in as good and east sort as the transverse itself.  
The things requisite to bee observed herein, and in thè use hereof, are no other then such as be remembered in the 1 Chapter and second book, in the end thereof: whereunto for brevities sake I refer you, onely here the legs must still be kept at a right angle by help of a pin put through the hole near the center, for so much I intend in every proposition.   

CHAP. II.  
Of the making of a new Scale, and to work the golden rule thereby.  
In my second book I spake of the delivering the 400 part of an inch, which I held no instrument could do: howbeit I will affirm, that by this scale you may give the smallest part as can be required, and possibly laid down: and indeed this kind of scale above all others for exactness and truth is best, as you shall find by practising thereon.  
Prepare therefore a fine piece of wood, or rather brass well smoothed& polished of 6 inches long, as M O, or thereabouts; and an inch or inch and quarter broad, as M N: now you must consider according unto how many parts in the inch you would make your scale, I make this according unto 40, divide your ruler first into inches, as R. S.T.V.W.O: then divide every inch into 4 equal parts, as well on the side N B, as on M O; thē draw the Isosceles triangles, according as in the figure, still skipping one division, so shall you make N 10 and 20: the first Isosceles; then 10 20 and 30, the next Isosceles, and so proceed. Lastly divide the side M N into 10 equal parts( for because 4 times 10 maketh 40) and from each division draw parallel lines to M O, according as you see: then put figures at every angle of the Isosceles, according as in the demonstration: and if you would haue made your scale according unto 20, or 16, &c. in the inch, you must then haue divided M N but into 5 or 4 equal parts, because 4 in 5 is 20, or 4 times 4 is 16, &c.  
To work by the scale.  
The numbersincreasing by 10, are set down vpon the side M O and N B, which you may take thence with your compass: but if any of those numbers haue an unite annexed, as 7 to 10 making it 17, or 5 to 20 making 25 &c. then must you place the foot of your compass in the line M N, extending the other along one of the parallel lines according unto the number proposed: and you must note that the figures increasing to 9 at the top of the rule M N, serve for the side N B, and the figures below for the side M O. I would express a line 45 parts long: according unto the prescript, place one foot of your compass in the line M N, at the end of the parralel marker with 5, extending the other foot along that parallel, until it cut the side of the I sosceles running from 40 to 50; which will fall at Z your desire.  
I would haue alone 98 parts long, according unto the prescript, I place the one foot of my compass in the line M N, at the end of the parallel answering to 8 in the lower end O B, extending the other foot of my compass to the intersection of that eight parallel, with the side of the Isosceles extended betwixt 90 and 100, which will fall out at the point marked with 98; which is your desire, and so of all the rest.  
 
Now if you would apply this scale unto 20 in the inch, you must omit every other parallel, &c.  
And note, the broader you make your scale, the truer shall your work be.  
The ruler made, you shall work the golden rule thereby thus.  
Draw two lines, which let intersect, making what angle you will at all adventures, that is, it forceth not how big or little a quantity so ever the angle be, as for the purpose B A C; thess two lines so drawn vpon a faire sheet of paper, of some smooth board or state &c. proceed thus,  
To work the golden rule.  
Let the proposition be, if 36 give 72, what shall 40 give?  
First with your compass take the length of 36 degrees, and place that in the line A C( being one of the sides of the foresaid angle) by putting the one foot of your compass in A, and extending the other towards C, and where he falls, make the point D: next take the third number given, that is 40, which taken with your compass from your scale as before, place it in the line A B, the one foot of your compass resting at A, the other extended towards B, as to F: next draw a straite line from D to F, and so beyond F at your pleasure, then with your compass take from your scale the length of your second number given, 72, which place in the line A C, from A towards C, as at E: this done, from the point E draw a line E G parallel to the line D F,( which you be after taught how to do,) but to find the fourth number in proportion, I say it is already found: for note where the line E G intersects with the line A B, as at G: then take the length of A G, and apply it unto the scale among the divisions, as you worked before: and so many equal parts as be contained betwixt the two feet of your con, pass, so much is the fourth number in proportion, which you shall find to be 80.  
You shall draw the line G D parallel to D F thus: take the length of D E with your compass, the same wideness remaining, place the one foot in the line F D at all adventures, as at H: with the other describe the portion of a circled, as K I: then lay a rule vpon E, and the very edge of the I K, as at L: now by the fidutiall edge of your scale or rule, 
Ra. lib. 2. P. 11. & Ra. 5. Pr. 11. Eu. 2. P. 6.& 17. l.  line as long as you will, viz, E G, which shall be parallel to F. D. as may be proved 
Eu. lib. 1. defi. 35.   
The ground of this working may be gathered from in fallible geometrical demonstrations*. 
Ra. 5. P. 13. con. 1 2& 3.  
Eu. 2 P, lib 6. R. 8. P. lib. 6.    

CHAP. III.  
To work the golden rule reversed, called in latin, Regula euersa.  

Regula euersa.  THis rule is termed reversed or inversed, when the question is so propounded, that according unto the course of numbers, the fourth should surmount the third, and yet in reason it must needs be less; or contrary according unto course of numbers diminishing, when it should increase: as for the purpose, if 12 Masons can be able to make a certain wall in 20 daies, how many shall make the same in fine daies?  
Here it is apparent that by course of numbers, as 12 is less then 20, so the fourth sought should be less then the third, and therefore consequently much more then 12 daies: but reason teacheth that if 12 men be 20 daies about a piece of work, if you would haue the same dispatched in less time, it is requisite to haue more then 12 men; the course of numbers therefore,& the truth in reason being repugnant, you must work by the rule of proportion inversed.  
This reciprocal or conversed proportionalitie is such a proportion, which beginneth with the first number, comparing it forward directly, but not to the next term, but unto the third and then with the fourth or last term, comparing it unto the second conuersely, as it were backward: as for the purpose, comparing these 4 terms before remembered, 12, 20, 5, 48, I say thus, as 12 exceedeth 5, so doth 48, 20, whereby you see there is a true proportion; otherwise if the proportional 80 terms had been inferred as they stand, there had been an utter disproportionality: therefore to work this rule, we must reduce the numbers given into a direct proportion thus, not euersed, put the third term first, the first second, and the second last, so will they stand thus: if 5 give 12, what shall 20: and so working by the doctrine of the second Chapter, you shall find 48 your demand: for as 5 is to 12, so is 20 to 48, or as 5 is to 20, so is 12 to 48; but conversed in direct proportion thus, they should stand 12, 5, 48, 20, so as 12 is to 5, so is 48 to 20: so that by the prescript you see, there is 48 men required to finish that wall in 5 daies, which 12 men were about 20 daies.  
And this rule may be easily wrought vpon the legs of the staff, even as it is here.  
Another example.  
If 20 men can undermine a town in 30 daies, how many will do it in 8 daies.  
The numbers 

proposed 20-30-8  
reduced to reversed proportion, 20-8-30  
reduced to direct proportion, 8.20, 30. Ans. 75.    
And here I could easily teach you to work the double ruls, 
Re●ul●● diplex.  but for because we shall haue small use thereof in these intended works, and also for that the former operations well understood, the working thereof is very facile, we will here omit the same until I we be further occasioned that way by some fit incitation.  
And here you must note that this kind of working meddleth not with fractions or irrationalities, for because it is the office of geometry to proceed no further then to prove that there are irrationals meddling onely with rational and whole numbers. Infinite ways could I teach you to apply the use of this scale, as to find out the proportion of lines one unto another, to draw figures, mays, sea-cards &c. proportional in a less or greater figure, &c. all which wee will put over until I haue accomplished my present intendements.   

CAAP. IIII.  
How many byways the Iacobs staff will perform Dimensions; whereupon he dependeth, and what he requireth.  
The Iacobs staff finds out the height and length of any thing three several ways, and breadths two ways; the two first ways require you to make observations but once, the third way must be performed at two observations, that is at two stations: and you must understand that all the measures performed vpon the Iacobs staff depend vpon three numbers given to find out the fourth, whereby you may perceius how apt the second Chapter is to bee coupled to the use of this said staff. Two of these given numbers are always found vpon the staff of the one, vpon the Inder or yard, the other vpon the transverse: but in our working the one is vpon the right leg, the other vpon the left leg; the third number assigned in proportion is some thing which is measured: for in the measuring of lengths, heights, &c. you must haue some thing or other measured, sured, which must be the third number in proportion; which 3 numbers had, and all things considered, fall to work thus.   

CHAP. V.  
By some known altitude to find out the unknown longitude at one station.  
Proposition Bac. jac.  
Si visus sit ab indicis recti in metam longitudinis: 
Rae. lib, 9 P. 7. ex 21, P. optic. Eucl.  vt tunc erit segmentum indicis ad segmentum Transuersais, sic mensoris altitudo ad quaesitam longitudinem.  
IF the sight pass from the beginning of the inder, being right to the thing measured, unto the end of the length; as the segment of the inder shall be unto the segment of the Transuersarie, so is the height of the measure unto the length.  
And here you must understand that by the height of the measurer, is not meant the height of the man that measureth simply, but the height or distance of his eye from the ground, be he vpon the ground or on a tower, &c.  
The Proposition for my staff.  
As the equal parts included on the left leg, betwixt the site and the center, are to the parts on the right leg included betwixt the file and the center, so is the length to the height, and contrary.  

E statione unica, longitudinem ignotam invenire, ex alto.  OPen therefore the legs of your staff unto a right angle, and there stay them by putting a pin through the hole near to the center, as you may perceive in the figure in the first book; this done, take away the graduator, for here wee haue no use thereof, and then proceed thus: place your staffs with the legs, so that the left leg may lye parallel, and the right perpendicular: as you be wont 
vide lib. 2. cap. 17.  remove the cites in the right and left leg until they both agree in one right line from your eye to the mark whose longitude is required; which done, note the equal parts included betwixt the two cites, and the center vpon each leg: for such proportion as the parts cut haue one to the other, the like hath the height to the length or length to the height.  
Example.  
Suppose the parts included betwixt the site and the center vpon the right leg were 60, and the parts cut vpon the left leg by the site were 180, let the height of your eye from the ground be 4 pards: then work with ease by the second Chapter, saying, if 60 the parts included vpon the right leg, give 180, the parts included vpon the left leg, what shall 4 yards give? so shall you find 12,& so many yards may you conclude the height.  
In the same maner may you measure heights standing below.   

CHAP. VI.  
To measure lengths standing vpon the ground at one station.  
PLace your staff to make right angles with the ground line as 
Lib. 3. cap. 1  you be taught; the legs resting at right angles( as they always must) place the left leg to lye parallel, 
Qualiter ex alto line 〈◇〉 in plane metiatur.  then remone the cites in both the legs, until they agree in a right line with the thing whose distance is required: then note the parts included betwixt the cites and the center vpon the right and left leg; for such proportion as the part cut vpon the right leg hath to the parts cut vpon the left leg, the like hath the height of your eye from the ground to the longitude.  
Example.  
Let the parts cut vpon the right leg be 20, and the parts of the le●t leg be 60, let the altitude of your eye be 4 foot: then say by the second Chap. if 20 give 60, what shall 4 give? so shall you find 12 foot, the longitude desired.  
Neither is it any matter whether the longitude be on plain ground or vpon hilly, so as you place your staff at right angles with the ground line, as you haue* done before, 
Lib. 4. cap. 6.  as it is written.  
Nec quidquaminterest, siue longitude sit in subiecto plano, siue in ascensu descenuve montis, sursum deorsumve collimando: dummo do index in lineam longitudinis sit rectus.   

CHAP. VII.  
By the known altitude to find the unknown longitude at one station, the left leg lying parallel, and the right standing perpendicular.  
Proposition Bac. jac.  
Sivisus sit ab initio Indicis paralleli in metam longitudinis: vt erit segmentum transuresarii ad segmentum Indicis, 
Ra. lib. 9. P. 8.  fie data sen cognita altitudo ad longitudinem.  
If the fight pass from the end of the Inder, lying p●●allell unto the thing measured, as the segment of the transnerss●● shall be unto the segment of the Inder, so is the given height to the length.  
The proposition for my staff.  
As the included parts vpon the right leg are to the parts included vpon the left leg, so is the given altitude to the longitude desired.  
PLace, the left leg to lye parallel, and the right perpend cular, as before, and let the right leg he towards the perpendicular altitude, as in the second book, Chap. 16; then move the fites up and down, until they he both in a right time with the altitude required: then note what equal parts be included betwixt the cites in each leg, and the center for such proportion as the parts included vpon the right leg haue to the parts included vpon the left leg, the like hath the altitude to the longitude desired.  
Example.  
Suppose you had the height of a Castle given you, and you being in the field far off, were required to tell the diffance of the said Castle, let the parts included vpon the right leg be 120, or 60, let the altitude of the Towore given be 400, or 20 feet, let the parts included vpon the left leg us 210, or 10●: then if you work by the second Chapter, you shall find the longitude or distance of the said tower from you to be 700, or 350 feet, and so of any other such like.   

CHAP. VIII.  
By the known altitude to seek any longitude at two stations, when you can work by neither of the former ways.  
Propositio Bacu. jac.  
Si visus sit ab initio transuer sarii par alleli, in met am longitudinis, per terminum Indicis recti ad metam in alto propositam: vt erit in Indice differentia maioris segmenti ad minus, 
〈◇〉. lib. 9. Pro 9.  sic differentia inter priman& secundam stationem ad longitudinem quaesitam  
If the sight pass from the end of the Transuersarie being parallel, &c. as in the inder the difference of the greater segment is unto the lesser, so is the difference of the second distance unto the length.  
The Proposition for my staff.  
AS the difference of the greater parts included vpon the right leg, is unto the lesser, the sight passing from the end of the left leg, or any part thereof, so shall the length between both the stations be to the distance required.  
In the Iacobs staff, here the yard standeth perpendicular,& the Trans. parallel, notwithstanding though we haue appointed the left leg of our instrument to represent the yard, &c. yet let him lye parallel: as he was wont, for all is one.  
Choose thee out therefore one station as near unto the altitude as thou wilt, and there plant thy staff, so that the left leg lye parallel, and the right stand perpendicular, looking towards the altitude: then place thy eye at the end of the left leg, or in any place thereof, as you will, and look by the very end of the left leg, or by the site therein placed any where, removing the site in the right leg, until by the end of the left leg or site therein placed and the said right site, you see the altitude or summite thereof: make a note in the right leg what parts the site cut, then the legs resting at a right angle, and the cites not stirred, go backward in a right line from your first station the further the better, and there again plant your instrument as before 
vide lib. 2. cap. 36. ; then place your eye again at the end of the left leg or the site therein, as you did before, drawing down the site in the right leg until he be in a right line with your eye and the summite of the foresaid altitude: note again the parts of the said right site then stands at: for such proportion as the former parts where the site was at the first station, haue thereunto, the like hath the distance betwixt both the stations to the longitude required.  
Example.  
Suppose you make your first station, placing your eye at thee and of the left leg, and remove the site in the right leg, until he be in a right line with the altitude; let the parts the cites then stands at, be 72: then go backward in a right line, which let be 40. feet, and there again make collimation, drawing down the site in the right leg, until he bee in a right line with your eye( placed at the end of the left leg) and the summite of the altitude, as before, let the parts the site then stands at, be 36: then must I take the lesser out of the greater, vi●. 36 from 72, there remaineth 36: therefore I conclude thus, if 36 the difference of the segments on the right leg, give 40 feet, then 72 must needs give 8 feet, because the proposition saith, as the difference of the greater parts included, be to the lesser, so is the distance betwixt my two stations to the length propounded, therefore work by the second Chapter.  
You may perform the last Proposition, and never remove the site in the right leg, as you be there required.   

CHAP. IX.  
By the known longitude to find out the unknown altitude at one station, the right leg being perpendicular, and the left parallel.  
Propositio Bac. jac.  
Si visus sit ab initio Transuersarii rectiterminum altitudinis, 
Ra, lib, 9, P. 10,  per terminum indicis paralleli in metam altitudinis: vt erit segmentum Transuersarii ad segmentum indicis, sic longitude data ad altitudinem.  
If the sight pass from the end of the Transuersarie being right, &c. as the segment of the Transuersarie shall be unto the segment of the Inder, so is the length given unto the heights,  
For the Geodeticall staff, Prop.  
IF the sight pass from the extremes or end of the left leg being parallel, or from any part thereof, as the parts included vpon the left leg are to the parts included vpon the right leg, so is the longitude given unto the proposed altitude.  
Place the left leg to lye parallel, then place your eye at the end thereof, or in any part of the left leg, as at 60: then move the site in the right leg, till he be in a right line with the left site: and the summite of the altitude: then note the parts cut, which let be 36: let the distance of the tower from you be 20 feet, work then by the golden rule, or the rule 2 Chap. so shal you find the altitude 12 feet.  
But if your feet were level with the base, you must add hereunto the length of your hollow staff, which is 4 foot, so the altitude would be 16 foot: for by this kind of working you always tell how much higher the summite is then the level of your eye, I mean that part of the altitude which is level with your eye.   

CHAP. X.  
By knowing of the breadth of any Pit, Well, Turret, &c. to fetch the unknown profundity thereof.  
Propositio, Bac. jac.  
Sivisus sit ab initio indicis paralleli in metam profunditatis: 
Ra. lib, 9. con. 10.  vt erit segmentum Transuersarii ad segmentum indicis, sic data longitude ad profunditatem.  
If the sight pass from the end of the Inder being parallel unto the thing measured, as the segment of the Transuersarie shall be unto the segment of the Inder, so is the length given unto the profundity.  
For the Geodeticall staff.  
Proposition.  
If the sight pass from the end of the right leg, being perpendicular, or from any part thereof, as the parts included vpon the left leg, are to those parts included vpon the right leg, so is the breadth to the depth.  
Place therefore the left leg to lye parallel, and the right to stand perpendicular in a right line with the profundity: let the parts included vpon the right leg be 13, and let the parts included vpon the left be 5: let the breadth of the Well be 10 feet: then if; the parts of the left leg give 13, what shall 10 give? work therefore by the golden rule, or according to the doctrine of the second Chap. so haue you the depth from your eye to the bottom of the Well, which will be 26 feet: from which you must take the height of your eye above the mouth of the Well, which let be one foot, 6 inches: which taken from 26, there remaineth 24, 2/ 4 feet.   

CHAP. XI.  
By the given longitude to find the unknown Altitude of any thing.  
Propositio Bac. jac.  
Si visus sit ab initio indicis recti in terminum altitudinis: 
Ra, lib, 9, Pr, 11  vt erit segmentum indicis ad segmentum Transuersarri, sic data longitude ad altitudinem.  
If the sight pass from the beginning of the Inder being right as the Transuersarie, so is the assigned longitude unto the height.  
The Geodeticall staff.  
Proposition.  
IF the sight pass from the extremes of the legs, or by any parts equally included on each leg, as they be equal one to the other, so is the longitude to the height, and contrary; therfore the legs being opened to a right angle, as you be accustomend, if you draw near or depart from the altitude( keeping the left leg parallel) until you see the top of the altitude by the two ends of the legs, then may you safely conclude the distance from you to the base of the altitude, to be equal to the altitude itself; always adding the length of your eye above the base thereunto, and so would it come to pass, the sight passing from any parts equally on each leg included.  
Example.  
Let the parts on the right and left leg included betwixt the cites and the center of the Instrument, be 60, and let the distance of the tower be 250 feet; and so working by the golden rule, or the second Chapter, the altitude will also appear to be 250 feet; to which add 4 feet the length of my staff, so there is 254 feet the true altitude.   

CAAP. XII.  
By the known part of some altitude, to find our the whole altitude itself.  
Propositio.  
Sivisus sit ab initio Indicis, 
Ra. lib, 9, con. 11  per pinnas seudioptras Transuersarii& cursoris, in terminos notae parts: vt erit interuallum pinnaerum ad reliquum supereminentis Transuersarii, sic notae desuper altitudinis pars adreliquam.  
If the sight pass from the beginning of the Index, being right by the vanes of the Transuersarie to the terms, as of some known parts; as the distance of the vanes is unto the rest of the Transuersarie above the Index, so is the part known unto the remainder.  
The Geodeticall staff.  
Proposition.  
IF the sight pass from the beginning of the left leg, by both the cites in the right, to the terms of some known parts, as the parts included betwixt the two cites are to the parts included betwixt the site nearest to the center, and the center, so is the known part of the altitude to the part unknown.  
Place therefore both the cites in the right leg standing perpendicular: then set your eye at the end of the left leg, moning the cites in the right, until they agree with the extremes of the known parts of the altitude: note then the equal parts contained betwixt both the cites, for as they are to the parts betwixt the lower site and the center, so is the altitude known to the rest unknown.  
Example.  
Let the parts included betwixt the cites be 20, and the parts betwixt the site and the center 30: let the known part of the altitude be 15 foot: I conclude by the former doctrine, that the part unknown is 22, ½ feet.   

CHAP. XIII.  
To find the altitude of any thing by help of two stations.  
Propositio, Bac. ●ac.  
Sivisus sit ab initio Indicis recti in terminum altitudinis, vt erit in In dice differentia segmenti ad differentiam distantiae( primae videl. ac secundae stationis) sic segmentum Transuersarit ad altitudinem. 
Ra. lib, 9. P. 12,   
If the sight pass from the beginning of the Index being right, as in the Index the difference of the segment shall be to the difference of the distance,( that is, of the first and second station) so is the segment of the Transuersarie unto the height of the thing measured.  
For the Geodeticall staff.  
Proposition.  
If the sight at two several stations pass by two points in the left leg, and by some one point in the right; as the difference of the parts vpon the left leg be to the distance of the two stations, so are the parts included vpon the right to the altitude desired.  
having placed your staff as you be often instructed before, 
vide lib, 2, ca, 36  then put the site in the right leg at what parts you will, as at 44 parts; then make observation at the first station, and note the parts cut by site in the left leg; then go* back a certain space, as 30 feet, drawing back the site in the left leg, until again the right site and he agree in one line with the altitude: note there also the parts included betwixt the place where your site was at the first station, and where he now is; which let be 23 parts: then if 23 give 30, 44. shall give 57 feet, and 9/ 23.   

CHAP. XIIII.  
To find out latitudes by 2 stations, that is, the distance betwixt two towns, Turrets, &c.  
Proposition Bac. jac.  
Si visus sit ab initio Indicis recti, per pinnas Transuersarii, in terminos latitudinis: 
Ra. lib, 9. P. 13.  vt erit Indice differentia segmentorum( per duplicē distantiam facta) ad differentiam ipsam stationum seu distantiarum, sic interuallum primarum Trausuersarii ad latitudinem quaesitam.  
If the sight pass from the beginning of the Inder, being right by the deigns of the Transuersarie to the terms of the breadth: as in the Inder the difference of the segment( made by the double station) is to the difference of the distance, so is the distance of the veins to the breadth.  
IT would be something troublesone to teach, you to seek out breadths after this Proposition, therefore without any more trouble, perform them as you would an altitude: for which purpose the 9 Chapter shall at this time serve, because I will hast to that which is more fingular, and performed with great ease, that is, the art of measuring ground, and fetching of longitudes, latitudes, &c. by protracting: and you shall note that you may perform the proposition joined with perspective in the former books after great ease, by using this staff as a Iacobs staff: all which and far more the ingenious practiser will find out with ease, whose studies that way, I could wish myself present to further, to the uttermost of my ability: because aiming at facility, I doubt I haue written so rude, as it will seem difficult to a judicial eye, not for that the matter is hard, but because the method is hearth, in wanting of convenient phrases to explicate the same: but my excuse is, that my pen was applied to the understanding of the ignorant, and not for the ripe wits of the learned.   
The end of the fift Boooke.   

To the equal Reader.  
FRiendly Reader, after I had finished my former books, I was drawn on, as well by diverse my friends, as also with a certain affectation which I bear thereunto, for to set forth a book of the art of measuring grounds by this my new devised instrument: for that I see daily errors continually practised, even by those which be in most practise: whereby it gives the ignorant occasion, and not without cause, to bring in question the truth of that infallible and noble science of geometry, the mistress of all Arts. Let no one be offended though I affirm that all ascending, or descending grounds heretofore laid down in plano by the plain Table, Theodelitus, &c. are falsely laid down. If they follow the doctrine of any one writer now extant, I know the rules they publish to cast up the contents, be true, but the rules they teach instrumentally to fetch the contents is false, as may appear in the ninth Chapter of this book. Some it may be will say, we haue instruments sufficient already, and then what needeth this? which I affirm also conditionally, that they adjoin the use of the II Chapter of this book thereunto: but this let such understand, that they haue not extant a better for portabilitie, and true rendering of the quantity of an angle; for the whole instrument is to be carried in a small walking staff, of one inch diameter in the biggest place. And as thereby thou art taught to perfit and measure grounds, both arithmetical,& mechanical, so also hast thou rules to know whether thou haue wrought true of false; which heretofore was never published, but thou left in that point, as it were to work at random, so that thou mightest hope to haue well done, though thou hadst no certainty thereof. Then if this Book give thee full satisfaction, and bring the art of measuring grounds unto a perfect head in one entire volume, which never heretofore hath been; accept him kindly,& give the Author good words for his pains:& so I end, wishing myself present when any doubt happens to arise by means of hasty writing, or ill phrased terms,  
Arthur Hopton.   

THE sixth book OF THE Geodeticall staff, containing the Arte of Geodesia, or mesuring ground, divided into two parts; the first most rarely and after sundry new ways, teaching you to take the true perfit of any piece of ground, according to the true proportion& symmetry, without the help of needle, protractor or any other cumbersome appendent, with the correcting of diverse errors heretofore practised.  


CHAP. I.  
Of measuring of ground, and what instrument is most fit for the same.  
MEasuring or plotting of ground, is an Art of well measuring, 
Ars metiendi.  and therefore it might be called Geometry, which properly signifies, ars been metiendi. But we vndersiand in the learning of geometry, we obtain a theorical knowledge, by the contemplating with demonstations and figures; and in this arte of measuring ground, we haue as it were the active part by practise and use thereof: for which practise& use, there haue diverse men studied to find fit instruments for the true, easy and speedy working thereof; amongst whom I seem with the best to compare this Geodeticall staff, not doubting, having reason my judge, but to show sufficient cause why this instrument should be preferred. Let us therefore enter into consideration what is required of that instrument that should take place, and that I hold to be, which can truest express the quantity of an angle, and render it again: for so to do is the drift of all instruments: then it refteth to prove what instrument is best able to express the quantity of an angle. First for instruments guided with the needle, I say they be uncertain, because the sides which represents the one side of the angle vpon the instrument, be so short, that the loss of a degree is never perceived; so that when they come to protract with a large protractor, they produce most absurd errors: for in a Theodelitus of 18 inches diameter, I say, the work is with as much error as if he were but half a foot diameter: my reason is, because that which representeth the one side of the angle is so short, that he hath no congruencie with the other, which is the Inder. To what purpose falls it, to seek the degrees of the one side in a large circled, and of the other in a small circled? haply some will say, the needle is but to fix some part of the instrument just under the Meridian, and then the Inder serves in a large circled to express the angle of position, which I grant: but yet I say, the uncertainty of a short needle produceth an error as before: as for the purpose, take an angle of position, and note it down, then let another, nay, come you yourself and take the said angle; against I say, you shall hardly find him to agree with the former: and what is the reason? first the uncertainty of the needle, whereby you are not able to place that part of the instrument under the Meridian as was before, then the disparallelitie of the planting of your instrument &c.  

But being made large, are as good.  This granted, to descend then unto other instruments without néedles: let the smallness of their divisions be a sufficient argument of their uncertainty without urging further. Lastly, to proceed unto the Geodeticall staff, our now intended labour, whose sufficiency it rests to prove, which I had rather refer unto the censure of the wise, then seem to heap many words together in the praise thereof: but this I will say, that he shall with more speed and truth fetch the quantity of an angle, then instruments with néedles can. First for speed, while you be waiting for the motion of your needle to cease, I will haue finished, nay, before you haue planted your instrument parallel, for my instrument respect no parallelity at all: then for the yielding up of any true angle, I hope this instrument is best able to do the same, because his sides be longest, and consequently his degrees largest: but let these things pass, onely return we to the brief use thereof in measuring ground.   

CHAP. II.  
Of protracting of Angles, and finding out of the quantity of angles protracted.  

Protracting without a circled divided.  FIrst therefore to protract an angle, is to down vpon the paper an angle, just to contain the number of degrees given: all angles in this instrument are protracted by the chords of a circled graduated vpon the back side of the legs, in this maner.  
Describe a circled or a portion thereof, place then the semidiameter of the said circled betwixt the two points respective, in the points there made, as in the first book, Pro. 7: then take over in each leg the quantity of the angle given, that is, place the one foot in the number of the angle given, vpon the one leg, and extend the other foot to the like number in the other leg, which place in the portion of the circled: so haue you the quantity of the angle, by drawing lines from the center to these 2 points made in the circled.  
Example.  
 
I would protract an angle of 30 degrees, first describe a portion of a circled, as BD, vpon A the center: then I place the length of A B the semidiameter over in the points respective; next among the chord divisions, I take the distance over from 30 to 30, in each leg, which I place in the foresaid circled, the one foofe in B, the other in C: and then draw the line A B, and the right of B, and there take the quantity of the said angle A I B, as you be taught*, which I find to be 39 degrees: next I direct the left leg towards B, 
Lib. 2. Cap 9, Prop. ●.  and the right to C,& then take the quantity of that angle as before, which I find to be 67 degrees, B I C: then I direct the left foot to C, and the right to D, and take the quantity of the angle C I D, which I find 56 dedegrées; so I proceed to all the rest, and take their several quantities, as of D I K, which I find 16 degrees, of K I E, 30 degrees; of E I F, 20 degrees, of F I G, 9 degrees, of G I H 72 degrees: and last, of H I A, 51 degrees. Of these I make a Table, and set the same down thus, with figures to show which was the first, second, third angle, &c.  
The quantity of the angles of position or station.   Degrees. 1 39 2 67 3 56 4 16 5 30 6 20 7 9 8 72 9 51  
Then I fall to measuring the several distance of each angle from my staff, and note the same orderly down by my angles of position: first I measure with a line from I to A, and find it 27 perches, then I measure from I to B, and find it 9 perches and ¾ parts; then I measure I C, and find it 7 ¾ perches, then I D, 19 perches, I K, 12 ½ perches, I E 28 perches, I F 25 perches, I G 14 perches: and lastly I H, which I find to be 21 perches, and these I join to the former Table, thus.  
The quantity of each angle of station. Sides of each angle.  
  Ang.   1 39 27 2 67 9 ¾ 3 56 7 ●/ 4 4 16 19 5 30 12 ½ 6 20 28 7 9 25 8 72 14 9 51 21   total. proof.   360 Cap. 4.  
This table taken, and in this order noted down, you shall lay down the plot vpon paper when you come home, or in the field. if you please, so the weather be faire, thus.  
Draw a line at all adventures, as A B: then taking from the legs of your staff, as is said, or from your scale, the length of 27 perches, which lay vpon the said line, A B; then protract by the second Chapter an angle of degrees, as M A N: then protract an angle of 67 degrees, as N A O; then of 56 degrees, as o A P, then of 16 degrees, P A Q, and so round to all the rest: then draw lines from each mark to the center A, M A, N A, O A, P A, Q A, R A, S A, T A, V A: next from your scale, take 9 ¼& place that in the line A N, from A to D; then count 7 ¾ perches in the line A O, placing the one foofe at A, always, as from A to E: then from A to P, count 19 perches, as to F, there make a mark: then from A to Q, count 12 ½ perches, as to G: do so to all the rest, so will A H be 28 perches, A I, 25, A K 14, A L 21 perches: then from B to D draw a line, then from D to E draw another line; do so from E to F, from F to G, from G to H, from H to I, from I to K, from K to L, and lattly from L to B, where you began: so haue you a perfit agreeing in true symmetry to the former, the contents whereof, you may easily cast up in acres.   

CHAP. IIII.  
To know if your perfit will close, and whether you haue done true or not.  
WHen you haue set down all your angles of station, add the said quantity together, according to the common order of addition, and if the total surmount, or be under 360, you haue falsely done, and your perfit shall never close: as for the purpose, I add the former angles together, and after addition, I find the total to be 360: which gives me warrantise, I haue wrought truly, for if it had been more or less, my work had béeene false.   

CHAP. V.  
To take the perfit of a field at two stations, and to lay down the true proportion thereof, according unto the situation of the angles in the said field, so that you may make a faire card, as before, in maner of a map.  
YOu shall repair into the field, or to some corner thereof, from whence you may see all the angles in the said field, 
A perfit taken at two stations.  as before; there plant your instrument, and take the quantity of every angle, according as you did in the third Chapter, whereof make you a Table, and call that the Table of your first station: but in this sort must you begin to take the first angle: when you haue planted the staff where you intend to make the first station, spy out the place where you will make the second station, then go unto the end of the left leg, moving the same, until by the site and center pin you see the place where your second station shall be; the left leg so resting, go then unto the center, and remove the right leg to point to the next corner of the right hand, and take the angle, and then proceed as you be wont.  
When you haue thus taken all the angles, and made a Table thereof, as I haue said, take up your staff, leaving some apparent mark, or some body to stand in the place where your staff was: then go you unto the place that you appointed before, for your second station, and there again plant your staff, directing the end of the left leg to the first station, or place where you left the foresaid mark, and turn the right leg unto the angle or corner vpon the right hand, taking the quantity thereof: then proceed round about, taking the quantity of every angle as you be accustomend, and note them all down, making a Table thereof, which call the Table of your second station this done, measure, or by some proposition take the distance betwixt the two places where you made your observations, and note that line down, which you shall call your stationarte line: this done, you shall take a faire shéere of paper, on the midst whereof, draw a straste line cross over the paper. then towards the one end, make a point, which call your first station, whereupon by the second Chapter, protract all the angles noted in your first Table of station, beginning at the line that you drew cross the paper, and proceeding rightwards round about: then take from your scale with your compass the length of your stationarie line, and apply the one foot of your compass to the former point in the line, whereupon you protracted the angles before, and extend the other towards the other end of the line, and where he falleth, make a mark, which call the second station, whereupon as before, describe all the angles noted in the Table of your second station: draw out the sides of all these angles at both stations infinitely. finally, you shall note diligently the concourse or crossing of every line like line with his match, according as you noted them down in their proper Tables, with figures of purpose, from which crossing, if you draw lines, that is to ●ay, from one crossing unto to the other, you haue the true perfit of the field: the length of every hedge, or part therein, may you measure by the scale, which before you laid out the stationarie line by.  
Example.  
Suppose there were a field to bee measured, whose corners were DEFGHIK, the perfit and length of every hedge is here required: I cannot hour-glass from my staff to every angle in the field, by reason of some quabdes or martsh, or some such like.  
Therefore go I into the field, and choose me a place, from whence I may see all the corners in the said field: which let bee at A: there I plant my instrument, and take notice of some other mark, whence also I may see all the corners in the said field, which let be at B: now I open the legs and go unto the end of the left leg, moning the same about until by the said end and the center I see to B: then I 〈◇〉 unto the center: so will the end of the left leg point just to C: then I direct the other leg to the first angle on my right hand, and so take the quantity of the angle C A D: which I note down in my Table of the first station, viz. 57 degrees: then I take the angle D A E, directing the left foot to D. and the right to E, which I find 62 degrees: this I note down in my table for the second angle, as in the table, and so I do to all the other angles, and note them down, viz. to the third, 4.5, 6, 7, and 8: as E A F: F A G, G A H, H A I, I A K, and K A G: so shall you collect as Table in such maner as I haue here set down.  
A Table of my first station.   Degrees. 1 57 2 62 3 33 4 ●4 5 37 6 32 7 100 8 36 total 360 proof. prove this Table by the 4. Chapter.  
 
I having thus done, I go unto my second station B, where turning the left egg to A my first station, I take the angle A B D, 33 degrees, and note it down in my Table of the second station; and so I procce●e right wards to the second, third, fourth &c. angles, taking their several quantities, and noting it down as before as of D B E E B F, F B G, G B H, H B I, I B K, K B A, so shall I haue made such: a Table as the former.  
A Table of my second station.   Degrees. 1 33 2 35 3 30 4 39 5 69 6 77 7 58 8 19 total proof. 360 Chap. 4  
Then with these Tables I resort to some plain& smooth superficies, as paper or such like, whereupon I protract all the former angles in maner as is said before.  
Example.  
 
First I draw the line A B at all adventures, then towards the end A, make a point for your first station C, whereon describe a circled A F G I K: then protract the angles of the first table therein beginning at A, as A C E, E C F, F C G, G C H, H C I, I C K, and K C A, and draw lines from every of those letters to C, as from E to C, from F to C, &c. and put the figures thereunto for to note the first second third, &c. angle as you see: then from C rightwards in the line A B, set out the length of your stationary line 15 perches, as C D, and vpon D describe another circled, L O Q R, and beginning at L, protract the angles of the second table, as L D M, M D N, N D O, O D P, P D Q, Q D R, R D S, and S D L, and then draw lines from each letter to D, as M D, N D, O D, P D, &c. then put figures thereunto as before, to give thee notice which was the first, 2, 3, 4, &c. Now note the concourse of the semblable lines, that is to say, where the second line E issuing from C, doth meet with the second line M, issuing from D; and thus I proceed to all the rest, still noting the concourse of the correspondent lines, which I note with these letters, T, V, w, X, Y, Z, L, as in the figure you may see.  
Then draw lines from T to V, from V to w, from w to X, from X to Y, from Y to Z, from Z to L, and from L to F, so haue you made the true perfit of the former figure in symmetry& proportion, which you may measure by the scale you laid out the line C D by; so shall you find T V 21 perches, V w nine perches, and so go round about.   

CHAP. VI.  
To take the perfit of a piece of ground, by measuring round about the same.  

A perfit taken nt one station otherwise then before.  THere may a piece of ground so happen, that you can neither measure the distance from every corner to your staff, as is required in the third Chapter, by reason of marshes, fens and such like, nor yet find two places any thing distant one from the other: from whence you may behold perfectly all the corners in the said piece of ground: for your help therein, work after this maner, you shall go unto some such place in the field, whence you may behold all the angles, and there plant your stasfe, and take the quantity of every angle severally, and note them down as you be accustomend: then let one measure the length of every line which do subtend those angles, that is, let every hedge particularly be measured, that is included betwixt each corner in the field, which note down against his proper angle; this done, measure the distance of the first corner from your staff, which also set down: these things had, protract the foresaid angles vpon a faire board or clean shéets of paper; then on the line which points unto the first angle, which you measured and noted down before, lay down with your scale and compass, the length thereof: then take from your scale the length of the first line which you measured on the outer side the file, and put the one foot in the point made in the former line, and turn the other about, until he intersect with the second line; then take the length of the second line measured, and do so to the third line protracted, and so go round about, as will better appear in the ensample.  
Example.  
Suppose A B C D E F G H, a piece of ground to be measured, I place my staff at I, and there begin and take the angle A I B, 39 degrees, then B G C, 67 degrees; and so I go round, taking every angle, and noting the same down as you see.  
 
The quantity of each angle of station. 1 39 2 67 3 56 4 16 5 30 6 20 7 9 8 27 9 51 total proof. 360 Chap. 4  
Then I measure the distance of A, the angle, from my staff I, viz. 27 perches: then I measure the length of A B, B C, C D, D E, E F, F G, G H, and H A; which is found, as in the Table.  
The quantity of each angle of station, with sides subtending the said angles. A, I, 27 perches.  
Ang. Sides. 1 39 20¼ A B 2 67 9¾ B C 3 56 16 C D 4 16 7½ D K 5 36 18½ K E 6 20 10 1/ ● E F 7 9 11 F G 8 27 22 G H 9 51 21 H A  
Then in the midst of some sheet of paper, I describe a large circled, and there I begin to draw the line M A, at all adventures from A the center, infinite beyond the circumference: then I take the length of A I, and place it from A in the line A M in this figure, and there make the point B: then I protract an angle of 39 degrees, as M A N, then of 67 degrees, as N A O, and so round as before 
vide 3. Cap. : and then I draw lines from A to each point in the circumference, as from A to M, from A to N, &c. next I take from my scale the length of A B, 20¼ perches,& place the one foot of my compass in the point B in this figure, and move the other about, until he cut the next line A M, and there make the point D: then take the length, 9¾ perches vpon your scale, placing the one foot of your compass in the point D, with the other strike an arch or portion of a circled,& where it intersects with the line A O, make the point E: next take the length of 16 perches, placing the one foot of your compass E, and with the other strike some small portion of a circled,& where it cuts the line A P, make the point F: do so with all the rest, as with 7½, from F to G in the line A G, with 18½ perches, from G to H in the line R A, with 10¾ perches, from H to I in the line A S, with 11 perches, from I to K, in the line A T, with 22 perches, from K to L, in the line A V, and with 27 perches from L to B in the first line M A: then draw lines from B to D, from D to E, from E to F, from F to G, from G to H, from H to I, from I to K, from K to L, and from L to B: so haue you made the true perfit of your ground, which you may measure or know the contents, according as you bee hereafteer instructed.  
Infinite be the ways that I could teach you here to fetch the platies of ground: but I will here cease, lest that I confounded the memory of some with multiplicities, and only give them acatalogue of the diversities of plaiting by this staff.   

CHAP. VII.  
diverse ways of taking plattes of ground, by this Geodeticall staff.  


diversities of plaiting. vide cap, 3,  1 AT one station, where all the angles in the field may be seen, as before.  
2 At two stations, 
Cap. 5.  where the angles may be all seen, as before.  
3 At one station by measuring round about the field. 
Cap, 6,   
4 At many stations by measuring round about the field.  
5 At many stations by measuring from every angle to your staff, where all the angles cannot be seen from any one station.  
6 At diuers stations, yet measuring but one line in the whole field.  
7 At diverse stations by going round about the field, as with the geometrical Table, or circumferencer, after a new maner never yet practised vpon any instrument.  
8 At diverse stations by taking the angles in the midst of the field, and also going round about the field as before, yet measuring but one line in the whole perfit.   
Any of all which ways are easily performed, if you understand but what is said before: therefore it would be to too tedious and superfluous to demonstrate them all.   

CAAP. VIII.  
To measure wood-land ground, or such other rough grounds, where you cannot run up and down with your chain within, by reason of bushes, or cannot see the corners because of Trees.  

To measure wood-land. confer this Chapter with the end of the 48.  I Had well near forgotten to set down this Chapter, which is more requisite then the former, for because it often cometh to pass, that you be put to measure woods and such like, whose perfit must be fetched by going round about the same, because through the impediment of trees and such like you cannot make observations as before, and here I must show you to work after such a way as hath not been yet published, in the Theodelitus circumferēror, and all other instruments, you are now onely guided by the needle and back site, but here we will hold the needle as needless, and to no purpose: let there be a piece of ground full of wood, assigned to be measured, as A B C D E, &c. here you see by reason of woods and such like, I cannot work according unto any of the former Chapters.  
In the taking of this perfit for your more ease in working, you be to consider 4 things, that is, whether the angle proposed do incline into the field, as D, or F, or decline from the field, as A, B, or C, &c. thē whether he be more then 90 degrees, or less: this considered, you shall take their quantity thus.  
 
1 It the angle be more then 90 degrees, as B, and decline, thē place the left leg B H, to lye parallel to the side of the hedge A B, and move the right leg I, until the fidutiall edge thereof direct your sight to C: then take the quantity of the angle as the legs be opened unto, which let be 35, I B H. But you will ask me how I will come by the angle A B C, which is all our drift in this place? you shall double the angle I B H, made vpon the legs, which was 35, which being doubled there is 70: take that from 360, there remaineth 290: half that is the quantity of the angle required, viz. 145: the reason of this work is grounded vpon this proposition.  
Si duae rectae inter secantur, aequant angulos verticales inter se, 
Eu. lib. 1. Pro. 15 Ra, l, 5, P. 8, co, 2  & omnes quatuor rectis.  
But forasmuch as I B H and his vertical angle are known to be equal, and A B C and his vertical angle also known to be equal: therefore you shall not need to double them, and so make subtraction, but onely take the angle found vpon your staff, viz. 35 from 180: so haue you 145, as before, which is the true quantity of A B C, and this work is grounded vpon this proposition.  
Si recta obliqne insistit rectae, 
Eu lib. 1 Pro. 13& 14, Ra, l, 5, P 8, con, l,  anguli deinceps positi duobus rectis aequentur,& contra.  
2 If the angle decline and be less then 90, as E, then shall you make the left leg H E ly parallel to the hedge E F, and the right I C to the hedge E D, so haue you the true quantity of the angle.  
3 If the angle incline and be more then 90 as F, you must go into the field, and there make the left leg E H lye parallel to F G: then move the right leg E l, you so standing at l, until the fidutiall edge thereof con●●y your sight parallel by the hedge side to E; then see what quantity the angle ● H made vpon the leg contains, which let be H, which take from 180, so haue you 143, the true quantity of the angle E F G.  
4 If the angle incline and bee less then ●0, then work as in the second difference of declining angles.  
If you ask me why I put you thus to so many differences, and to use subtraction, since I might take the quantity of each several angle with the leg, for that they will express the quantity of any angle to 180 degrees, and no angle can be bigger. I say that is true: yet my reason is two sold, first in respect of the smallness of the degrees beyond 140 degrees; secondly,& most especially, because of the ●n aptu●sse of the p●●ting o● the legs at so great an angle, as well in ●espect of your standing, as the due ordering of the legs: is for the purpose, you would take the angle G F E, you must then thrust your ●elfe betwixt the instrument and the hedge, otherwise you could not come unto the center, and when you were there; the further discomoditie you may easily perceive in searching the degrees cut, and such like: yet he that liketh this may better then the former, may use it at his pleasure, for it will well serve for any angle. You having thus learned to take the quantity of the several angles, you shall fetch the platt thus.  
Plant your staff at any angle, as at A, and take that angle, then let one with a chain measure so far as your fight by the fidu●iall edge of the right leg passed in a right line, as to B, then note the angle G A B, and the line A B down, then plant your staff at B, and take that angle, causing one to measure from that angle you be at, unto the next, viz. C; then note the angle A B C, and the line B C down, next remove your staff to C, and there take that angle, causing one to measure from your staff, unto the next angle, as to D, then go unto D, but you must go over into the field, and there take that inclining angle, causing one to measure from D to the next corner, as E, then note down the angle C D E, and the length of the line D E, then go to the declining angle E, and take his quantity, causing ove to measure to F, both which note down as before then go unto F, the inclining angle and take his quantity, causing one to measure to G which line and angle note down: now if you work truly, you need not to take the angle G, or measure the line G A, because in protracting, they will be expressed of necessity, therefore I would counsel you for the saving o● labour so to begin that you might haue the largest side or hedge to do last, because you may ●uo●d the labour of measuring him.  
Now all the angles and side had, and noted down in a Table, as you bee ●ont, beg●●ne then and protract the quantiti● of the angle G, B, drawing the two sides A G and A B at length then take the length of A B from your scale: and place it in the said line, as at B from A, and on that point making A B one side, protract the angle A B C, and so go round as your table directs you: and when you come to protract the angle F, you shall see that you need not to measure the line G A or take the angle G; because the intersection of A G and F G, ●quates the line A G, and the angle G nay that which is more you need not measure the line F G but onely take the angle G F E, for that the intersection of the side F G and A G, limits asw●ll the length of these two lines, as it doth the quantity of the angle G; let this ensuing rule therefore be inserted to this kind of mesuring general.  
If the figure or perfit proposed to be measured, 
Regula obseruanda.  be a triangle of what sides soever, as an I sopleur●n, I so●●eles, or Salenum, or of what angles soever, as an orthogon●um, ambligoni●●n, or origonium you shall measure but one side and two angles in quadrangles of what kind soever, as right angled parallel● grams and oblonges, or obliqne angled paralle lograms as Rōbes and ●omboide: measure but 2 lines and 3 angles, and so of a Trapezia, which is quadrangulate, but more brief, take it thus.  
In all figures, be they multanguled or otherwise, you may omit the measuring of any one angle and his two sides.   

CHAP. IX.  
Of errors daily practised in plaiting ground, and of the reformation thereof.  

Of errors before practised, reformed.  IN taking and plaiting of vneuen grounds, ourlan● measurers continually practise and commit great errors, because when they are put to measure a hill, or an ascending ground, when they go round about the said field, as is said before, and so measure from corner to corner, I say, the ground lines they measure with their chain, be hypothenusal, and the right lines they protract be horizontal: so that it is not possible that their perfit should close. I haue seens diverse using the Geometrical Table, called of some the plain Table, impute the not closing of their perfit to the ill ordering of the needle& the back sight when it was nothing else but the want of skill to reduce the hypothenusal lines unto lines horizontal, though indeed there may be great errors unawares committed by means of the needle and back sight: but to return unto our purpose, the onely means to help these forepractised errors, is to reduce the lines hypothenusal unto lines horizontal.  
It is certain that in this kind of working the hypothenusal line A B, 
Eu, lib, 1, P, 18& 19.  is a line subtending a right angle; and therefore is, the longest line that may be made in any orthogonium, as may be proved*: 
Ra, lib, 6, p, 11 Lu, lib, 1, p, 47& 48, R, lib, 12, P, 5  for that a right angle is the greatest angle, that can be made in any or thogonium try angle: this granted, then of consequence must the line A C called here the horizontal line being the base of the said triangle, needs be less then the line C B: for that the hypothenusal C B, is in power equal to the horizontal A C, and the perpendicular B C*, and so of necessity more then the line A C, be the perpendicular or Cathetum B C never so short: then let them confess how far they haue erred heretofore, which haue laid down the hypothenusal line in stead of the horizontal, since here it is apparent that if they measure ascending or descending grounds, it is not possible for them to make their perfit close, unless the lines hypothenusal be reduced into lines horizontal: so then it resteth for us, having the length of the line A B given, by measure or otherwise, to find the length of the line A C, and him protract vpon our paper, which is thus performed.  
Let there be a mark placed before you in the angle, whereunto you would direct your sight, just so high from the ground as your eye is placed, viz. 4 foot: now are you to consider if the ground ascend, as A B, or descend, as B E.  
If the ground ascend, as A B, open the legs of your instrument unto a right angle, turning the right leg towards the mark, and causing the left leg to lye parallel: then bring the center of the graduator unto any number of divisions where you will, as to 30, heaving up the other end, until the fidutiall edge thereof convey your sight direct unto the top of the staff before planted in the angle at B: then shall you note the equal parts cut by the right leg vpon the graduator, 
Here note if the ground both ascend& descend, as A B and B D, you need but take the angle D B A, then measuring the two sides containing the angle, viz. A B,& D B then work as in the 2. Bocke, Chap. 20. imagining D A a distance, or book 7, cap 3, Axioma 4, de. 2 Vide lib, 7, axi●m 3, sub, 3,  which let bee 34 and better: lastly measure the hypothenusal line A B, which let be 40¼ perches: then multiply the equal parts cut by the center of the graduator vpon the left leg, viz. 30 in 40¼, the length of A B: and the product divide by the parts cut vpon the graduator by the right leg, viz. 34, and better: so haue you 35 the length of the line A C, whereby you may infer that the line A B, which was measured 40 perches, must bee laid down in protraction: but 35 perches& this proposition may you work by the 5 book, Cap. 2.  
This Proposition may also be wrought Synnically by the table of sins: for if you take the angle A, which here is 30 from 90, so haue you 60 the angle C B A, as may be proved, 
Eu, lib, 1, P, 32 Ra, lib, 6, to 9  because one of the angles is known to be right: having the angles, then find the Synes, and proportion them accordingly.  
And as you worked for grounds that did ascend, so must you work for grounds that do descend: onely it were best that the left leg should stand perpendicular, and the right parallel, which before was placed perpendicular.  
But forasmuch as some will hold this way tedious to stand calculating in the field, and for that arithmetic is not common unto every one, nor the brief use thereof known unto al men, you shall haue a way ensuing to avoid all arithmetical calculations.   

CHAP. X.  
To find the difference betwixt the hypothenusal and the horizontal lines without calculation.  
IT is our onely intent in this staff to avoid all kind of arithmetical calculations, 
The difference of the horizontal and hypothenusal line,  as hitherunto we haue done, and therfore I should do the staff wrong, if in this I should leave him destitute of such an necessary conclusion, being offered vpon his own accord.  
You shall therefore plant the legs vpon the side of the staff, so that the left leg ly parallel unto the Horizon: then heave up the right leg, until by the center pin and the site in the said left leg, you spy the top of the staff placed in the angle, as in the last Chapter: Next, place the graduator at a right angle, and there fasten him by help of the screw: then measure the length of the hypothenusal line A B, which let be 40 perches, and count that vpon the right leg making a mark there: next shall you draw the center of the graduator along the left leg, not altering him from his right angle, until the fidutiall edge thereof just cut the point before noted, at 40 in the right leg: the equal parts then vpon the left leg, included betwixt the center of the graduator, and the center of the instrument, is the length of the said horizontal line, which you shall find to be 35 perches as in the last Chapter.  
If the ground descend, as B D, then must you make the left leg hang perpendicular, pointing towards the ground, and the graduator to lye parallel at right angles, with the said left leg.  
And here let if be noted, that your best way is onely in the field to take the angles of altitude, and measure the hypothenusal line, noting whether the line did ascend, or descend: and then may you reduce the same at home when you protract, as is fought.   

CHAP. XI.  
Of the making of a new quadrant to be used with the geodeticall staff, for the finding out of the horizontal lines in the plaiting of grounds.  
SOme it may be will seem curious, 
A new Quadrant to find the horizontal and hypothenusal lines.  that they will hold it much labour to take off the legs from the top of the staff, and place them vpon the side when they measure vneuen grounds, which I am persuaded can be no labour at all: howbeit, because I would set my staff so near as I might unto the disposition of each several mind, they shall here haue a quadrant most ap● for to perform the same.  
You shall therefore prepare a quadrant of brass, or some fine grained woods, whose two sides let be 6 inches, or there abouts: them must you divide the limb thereof into 90 degrees( if you please) as the common order is then shall you divide the two sides thereof in to 90 equal parts, drawing parallel lines by every of those 90 divisions in each side, so will there be in the quadrant a number of just 4 square holes, according as you may plainly perceive by the ensuing figures.  
Next must you make an handle or such like device of brass or wood, with a s●lit through the same for the quadrant to move in: the center of which quadrant must be placed in the top of the handle, according as you see at A and then must there be a fidutiall line for to answer unto the center of the quadrant, as A B, which must be divided into 90 such equal parts as the one of the sides of the quadrant was: vpon this handle there must be struck a line C D parallel to the edge A B, whereon must hang a line with a plumb: lastly in the foot of this handle there must be a hole, as at R, through which must go a scrue pin of braffe into the hole in the side of the staff, to the end that by the help thereof, you may fasten the handle firm vpon the side of the staff, as occasion requireth, all which you may plainly perceive in viewing well the ensuing figure.  
The quadrant thus ordered, you shall fall to work therewith in this maner.  
To work by the new quadrant.  
Place the hole R vpon the side of your staff, by help of a screw pin, causing the handle D C to stand perpendicular by help of the plumb: then having prepared your staff to be carried before you, as it is said before, and duly placed in the angle, heave up or put down the side of the quadrant, until through the two sights F and G, you see the top of the staff before planted in the angle: then count the hypothenusal line vpon the handle A B, towards B: the line which then passeth by that number unto the side A K, is he that limits the length of the horizontal line, which you must protract, being numbered in the side A K.  
As for the purpose, let the quadrant stand as he doth: I count the line measured being 55 perches vpon the handle from A towards B, and there do I see a line pass by at Z, which following to the side A K, I find him numbered, 30: whereby I may conclude that the line measured 55 perches, must in protraction be laid down but 30 perches and so of any other. Or you may take the angles of ascension in the field, and measure the hypothenusal line, and then reduce them by the quadrant when you come home, onely noting in the field the angle of ascension, and the length of the hypothenusal line: for which purpose I haue appointed the limb of the quadrant to be divided into 90 degrees, as the common order is, and so work in angles of descension.   

CHAP. XII.  
To take an altitude onely by the Quadrant.  
having placed the handle is stand perpendicular as before, 
Altitudes by the new quadrant.  heave up or put down the sides of the said quadrant, until through the sights F G, you see the summite of the altitude: the quadrant so resting, count the distance of the tower from you in the side A K, and where that number ends, note there what line passeth thence unto the handle: for the line crossing there, and passing to the side F G, limits the desired altitude.  
Example.  
Let the Quadrant rest at the angle he doth, and let the distance of the tower from you be 40 feet: I count 40 vpon the side A K, and there do I see a line pass thence to the handle intersecting at P, whence do I see a cross line pass to the side F G, which is numbered 60: and so many feet high may I conclude the tower to be: then shall I see 70 feet counted vpon the handle from A to the intersection of the foresaid lines at P, which is the length of a scaling ladder, and so of any other.  
So may you find out depths: many rare conclusions might be performed by this quadrant, which at this time for want of opportunity I am forced to omit.  
So then by that which is said before, you may be answered, that it is not possible for any man vpon one scale to lay down the true perfit, and by the same scale express the true contents of ascending or descending grounds: for because if he make the perfit true, and then by the same scale cast up the contents thereof, it will be less then the ground if self is: and again if you lay down the true contents of such vneuen grounds, then the may you make will never close, nor you be able to set forth the true proportion and Symmetry of the said grounds.  
The best way therefore is, for him that will measure vneuen grounds, to provide two scales, one to express the true hypothenusal line, and the other to lay down the true horizontal line: and so by taking notice thereof, you may work with more truth then I haue seen any hitherto practise.   

CHAP. XIII.  
The legs being opened unto any angle, to find the quantity thereof without the help of the graduator in a most excellent maner, so that you may take expressly the quantity of any angle, and put aside the graduator or protractor.  
YOu must turn the back side of the legs upwards, 
To find the quantity of any angle vpon the legs without the graduator.  and then with a pair of compasses take the distance over from one leg unto the other, in the two points respective: the compass resting at that extent, apply the one foot to the center of the instrument, and turn the other towards the point respective in one of the legs, and where that foot of the compass falleth amongst the divisions, the degree there set, is the quantity of the angle: but if the angle exceed 90, then having taken the distance over betwixt the two points respective, half that distance, and apply it unto the legs, as before, so shall you find half the quantity of the angle, which double, so haue you the whole: and by this means may you take the height and distance of stars, measure grounds, &c. without the help of the graduator.  
As for to make proof of this Chapter by help of the graduator, open the legs unto an angle of 30 degrees, then with your compass take the distance over in each leg betwixt the points respective: which applying to one of the legs, as before, you shall find the foot of your compass fall just in 30 degrees, which is the angle the legs then make; allowing some small difference for the stopping of the chard divisions from the ●idutiall edge of each leg.  
To open the legs unto any angle assigned.  
Place the one foot of your compass in the center of the legs, 
To open the legs unto any angle assigned.  and extend the other unto the angle required, in one of the legs amongst the divisions vpon the lower side: which wideness of your compass still remaining, set the same over in the two points respective, opening the legs until 60 degrees, or each point respective agree with the said wideness of the foot of your compass, and so is the angle made: and so must you work in any other such demand: this proposition is most excellent for to find angles in the 2 book, Cap. 29, &c. whereby you may perform it with great ease and speed.   

CHAP. XIIII.  
having taken the quantity of each angle, and measured the lines truly, in going round about wood grounds, as Lordships or such like, to know if you haue wrought truly or not.  
having gone round about the wood or Lordship proposed to be measured, 
To know if you haue wrought truly or not.  add all the quantities of each several angle together, and note the total: then multiply 180 by a number, which shall be less by 2, then the number of angles or corners in the said field, and note the product; for if it agree with the former total, you haue wrought truly, otherwise not, and your perfit shall not close.  
Example.  
The total of all the angles being added together, is 540, the number of angles is 5, from which I take 2, the remainder is 3: 180 multiplied by 3, produceth 540, agreeing with the former total: which argues my perfit will close,& that I haue wrought truly, and so of any other.  
The geometrical ground hereof is grounded vpon these Propositions ensuing.  
1 In omni triangulo trees anguli simul sumpti, sunt duobus rectis aquales. Euc. lib. 1. P. 32. Ram. lib. 6. P. 9.  
The 3 angles in any triangle are equal unto two right angles, therefore 180 is multiplied.  
2 Cuiuscunque triangulaeti latera sunt binario pluratriangulis, è quibus constat. Ram. lib. 10. Pr●. 1. cons. 1.  
The sides of a tryangulate are more by 2, then the triangles whereof he is made.  
Therefore multiply by two less then the number of sides or angles, in the multangled figure.  
The end of the first part of the art of Geodetia.    

THE SECOND PART OF Geodetia, containing the measuring of the superficial content or Area of any perfit, piece of ground, map or such like, howsoever formed: and that two kind of ways, as arithmetical, or vulgarly without calculation, with diverse new tables, as well for the casting up or dividing of grounds, as also to extract the square roote, or find the true square of any number, with the measuring of all kind of superficies, as boards, glass, &c. or solids, as timber, ston, &c. after diverse ways, both old& new, together with the ground of the work: also to seek dimensions of heights, bredths, &c. by protracting after my new kind of working, to separate or enclose grounds, or make any figure according unto any proportion assigned onely by the staff, to metamorphize or transform any figure from one shape unto any other, for the more speed in measuring to find what proportion pieces of ground, glass, boards, or any superficies one hath to the other: the making of a compendious Table to contain the legal part of measuring ground, with many Apophthegmes, requisite for all Geodetors to understand.  

CHAP. XV.  
Of such things that are to be considered in the casting up of grounds, as also to extract the square roote, or find the true square of any number by a Table.  
I Haue before given you rules how to fetch the true perfit of grounds out of the field as they lie: 
Of things which are to be preconsidered.  it resteth now to declare by what means wee shall find the superficial content thereof: but before we speak thereof, having a perfit of a piece of ground given, you must consider whether it be a triangle, a quadrangle or a multanguled figure: and if it be a triangle, how he is distributed, as well in respect of his ●●des, as in respect of his angles: in respect of his ●ides, whether it be an Isopleu●o●. Isosceles, or a Scalenum: in respect of his angles, whether an Orth●gonium, Ambligonium or Origonium.  
If the perfit be a quadrangle or figure of 4 angles, then must you consider whether it be a parallelogram, or a Trapezia: if a parallelogram, whether a quadrant, an oblong, or an obliquangle: if obliquangled, whether a Rombus, or a Romboides.  
If your perfit be a figure multangled, then must he be reduced into some regular figure, and so measured as hereafter, and you shall note that it is best to reduce any multangled figure into triangles, because they be soonest done, and easiest measured: but first behold briefly what is said before, for Ignorant in the terms, and ignorant also in the Art.  

Geometriae Radix.  
Geometry is distributed into 2 parts, whereof the first delivereth: 

Things given, that is, the definitions of magnitudes& their species, as well 

lineal, which is either 

Right, vide A  
or obliqne B 

simplo, vide C  
or mixed D      
as E Lineamentall being 

F An angle, which is 

G homogeneal, being 

Right. vide H  
or I obliqne which are 

R Acute  
or L Obtuse      
or Hetrogeneall.    
or afigure, vide M      
Things require●, that is, the adjunct or habitudes of given maguitudes.     

A Table to find any of the prescript in euclid or Ramus their Geometry, according as you be Alphabetically directed.  Let. A B C D E F G H I K L Eu. l. 1. d. 4.                 l. 1. d. 12 l. I. d. II Ra. l. 2. p. 5. l. 2. p. 6. l. 2 p. 8 l. 2. p. 9. l. 3. p. 1. l. 3. p. 3. l. 3. p. 4. l. 3. p. 8 l. 3. p. 9 l. 3. p. 11 l. 3. p. 10 Let. m.         Ra. l. 4. p. 1.          

The forms of figures, are 

Superficies, which be 

plain being 

Right lim●: which are 

triangles consisting of 

Angles 

Right Chap. 16.  
o● obliqne, being 

Acute, cap. 17.18,& 19.  
or Obtuse, cap. 20, Pro. ●,& ●      
or sides being 

Equall● as an Isopleuron, 

cap, 17    
or inequal, as an 

Isosceles, cap, 1●  
or Scalem●n, cap. 1●        
or Triangulated being 

a quadrangle, which is 

a parale●●gram being 

Right angled which be 

quadrant●, ca. 22, P. 1.  
or Obloug●, cap. 2●. P. 2.    
or obliqne angled as a 

Rombus cap. 22 P. 3.  
or Rombeides 

ca. 22. P. 4.          
or a multangle: ca, 23 

or a Trapezia. 

Cap. 22: Pro. 5.          
or tut●ilined. 

simplo, as a circled: cap, 23:& 44  
or mixed, consisting 

of one term.  
of many terms        
or swelling.    
or boats 

Cap. 35.       
To extract the Quadrant roote, or to find the square of any number with great speed, by help of a new Table.  
IF you seek the quadrate roote of any number, enter the table following, seeking the said number whose radix is desired, in one of the colums descending, under the Title of square numbers: the which if you find precisely written there then the number answering in order on the left hand, under the title of Radix, is the just square roote of the number proposed: as if you would seek the square roote of 1600, enter the Table following, seeking therein orderly under the title of square number, the said number 1600: so shall you find in the row on the left hand under the title of Radix, 40 answering thereunto, which is the square roote of the said number 1600: and so contrariwise having the radix given, to find the square number thereof.  
But if the number whose square roote you seek be not precisely to be found in the table, then must you work with a double entering in this maner: take two square numbers expressed in the Table, whereof let the one be the next, that is less then the number proposed,& the other the immediate greater then take the lesser of those two out of the greater& that which remaineth call the first number, which indeed is the portion or fraction of the integer number, and is called the difference, and shall be the deuisor; afterwards subtract the foresaid lesser number, out of the number proposed, whose radix you seek, and the residue keep for a second number, then the third number shall always be 60: therefore augment the second number reserved, in the third, and the product divide by the first: and that which shall be gathered by this division, shall be added to the square roote of the lesser of the numbers found in the table: so shall you haue the true square roote of the number proposed.  
Example.  
Let the number proposed be 829, I cannot find this number precisely in the table, wherefore I take in the table that number which is next less then it, which is 784, and the next greater, which is 841: the difference of these numbers is 57, as you may perceive in the row on the right hand under the title Difference, and this difference is the first number, and the deuisor: to conclude, the lesser number being subtracted from the number proposed, viz. 784, from 829, there remaineth 45, that is to say, the second number: the third number( as is said) is always 60, therefore augment the second in the third, and there is produced 2700 parts of fractions, which divide by the first number that is 57 parts or fractions, so is the quotient 47 ferè, which are fractions to bee added unto the square roote, that is 28: so will there rise 28 integers; and 47 fractions, the radical square of the number proposed, viz. 829.  

A perfect Table for the extraction of square roots, or finding the true square of any number, from 2 to 14290, faithfully supputated.  Radix. Square number Differ. Radix. Square number Differ. Radix. Square number. Differ. Radix. Square numbers. Differ. 2 4 0 32 1024 63 62 3844 123 92 8464 183 3 9 5 33 1086 65 63 3969 125 93 8649 185 4 16 7 34 1156 67 64 4096 127 94 8833 187 5 25 9 35 1225 69 65 4225 129 95 9025 189 6 36 11 36 1296 71 66 4356 131 96 9219 191 7 49 13 37 1369 73 67 4489 133 97 9409 193 8 64 15 38 1444 75 68 462● 135 98 9604 195 9 81 17 39 1521 77 69 4761 137 99 9801 197 10 100 19 40 1600 79 70 4900 139 100 10000 199 11 121 21 41 1681 81 71 5041 141 101 10201 201 12 144 23 42 1764 83 72 5184 143 102 10404 203 13 109 25 43 1849 85 73 5329 145 103 10609 205 14 196 27 44 1936 87 74 5476 147 104 10816 207 15 225 29 45 2025 89 75 5625 149 105 11025 209 16 2●6 31 46 2116 91 76 5776 151 106 11236 211 17 289 33 47 2209 93 77 5929 153 107 11449 213 18 324 35 48 2304 ●5 78 6084 155 108 11664 215 19 ●61 37 49 2401 97 79 6241 157 109 11881 217 20 400 39 50 2500 99 80 6400 159 110 12100 219 21 441 41 51 2601 101 81 6561 161 111 12321 221 22 48● 4● 52 2704 103 82 6724 163 112 12544 223 23 529 45 53 2809 105 8● 6889 165 113 12769 225 24 570 47 54 2916 107 84 7056 167 114 12996 227 25 625 49 55 3025 109 85 7225 169 115 13115 229 26 676 51 56 3136 111 86 7396 171 116 13346 231 27 719 53 57 3249 113 87 7569 173 117 13579 233 28 784 55 58 3364 115 88 7744 175 118 13814 235 29 ●41 57 59 3481 117 89 7921 177 119 14051 237 30 900 59 60 3600 119 90 8100 179 120 14290 239 31 961 61 61 3721 121 91 8281 181         

CHAP. XVI.  
Of an Orthogonium, or right angled triangles,& to measure the area or superficial content thereof, with the ground and reason of the work.  
Of an Orthogonium, Isosceles. Proposition. 1.  
AN Orthogonium Isosceles is such a figure, 
De Triangule recsangulo Isosceles. ●uc. 1. Defi. 25.  that hath one right angle and two equall* sides, as B A C, and may be measured after two manner of ways, thus.  
1 Multiply the one equal side in himself, that is, square the one side half, which is the superficial content, which is eastly found by the 27 Chapter.  
Example.  
A C is 20 perches, the square whereof by the Chapter is 400, the area of A B D C: half which is 200, the content of B A C, for A B C is equal to B C D.  
2 Or thus, you must multiply A C one of the equal sides in A F the half of A C, so shall the product yield the area.  
Example.  
A F 10 perches, multiplied in A B 20, produceth 200, the contents of A B E F, which is equal unto A B C, for B E G, the triangle without the figure is equal to F C G, the triangle within the figure: therefore the one doth decrease what the other hath in excess.  
Of an orthogonium Scalenum. Proposition. 2.  
THis figure is such an one that hath but one right angle, 
De triangulo rectangulo, Scaleno Eu●. l. 1. Defi. 26. E. 1. defi. 27. Ra. 8. Pro▪ 2  as A:* and all his sides be unequal: and as the measuring of the right angled Isosceles depended on the measuring of a right angled equilater parallelogram, or square, so doth this orthogonium Scalenum rely vpon an oblong: for the two sides containing the right angle, as A B, and A C are but the 2 sides of an oblong, as you may perceive by the pricked lines,& may be measured after two maner of ways, thus.  
1 Take with your compass the length of two of the sides containing the right angle, and thē multiply one of those sides in the other half the product, whereof is the superficial content or area of the right angled Scalenum.  
Example.  
A B is 20 perches, and A C 40: 20 multiplied in 40, produceth 800, half which is 400, the true area of the said piece of ground.  
Whereby you see that A B augmented in A C, yields the content of A B B C, the oblong: but for that we desire but the area of A B C, which is but the half thereof, therefore I take but half of the product, and it sufficeth.  
2 Or you may multiply the one side in half the other, and the product is the true area.  
 
Example.  
As H I, 10, multiplied in A C 400, produceth 400, as before.  
Or E H 20, half A C multiplied in A B 20, produceth as before.  
Wherein you see you work always but as it were an oblong, or a square, and may easily perceive the ground by applying the pricked lines unto the préescript of this proposition.  
And thus far of right angled triangles, called Oxigonium.   

CHAP. XVII.  
Of Acute angled triangles called Oxigonium.  
Of acute angled triangles, there be three sorts, and every one hath his proper measuring: 
Oxygonia Trianguia tripheis,  the first hath all his sides equal, and is called an orthogonium Isopleuron: the second an oxigonium Isosceles, and the third a Scalenum oxigonium, the difference always rising in the sides.  
To measure an oxigonium Isopleuron. Proposition. 1.  
An Oxigonium Isopleuron is such a figure whose 3 sides be all equal, 
De Triangulo onigonion aquilatro Euc. 1 Defi. 24. Ra. 8. Pro 8 Euc. 1. Defi. 29.  and whose angles be all acute, and may be measured after 4 several ways.  
First, square the square of one of the sides by the Chapter: then multiply the off-come in 3, and the product divide by 16: the roote quadrate is the area: this way depends rather vpon proportion, then otherwise.  
Example.  
Admit A B one of the equal sides by your scale& compass, found to be 6 perches, the squared square whereof, is 1296: which augmented by 3, produceth 3888: that divided by 16, yieldeth in the quotient 243, the roote quadrate being 15 and better as betwixt ⅗, and l⅓ 8/ 1, yielding the true area.  
2 Or you may multiply the cube of half one of the equal sides in the semiperimeter of the triangle: the roote quadrature is the superficial content.  
Example.  
The cube of AE, 3, being half A C, is 27, which if you multiply by the Semiperimetry B A E, 9, yeeldeth 243, whose roote quadrature is the true contents, as before, viz. 15, &c.  
3 Square one of the known sides, which multiply in 13, then the number resulting, divide by 30, so haue you the contents.  
Example.  
The square of A B is 36, which multiplied in 13, produceth 468, which divided by 30, leneth 15, &c. as before.  
 
4 But arithmetical with most ease, take it thus. Take the length of the perpendicular let fall from any angle unto his opposite side subtending the said angle, as you bee taught: 
Chap. 21.& 25.  which multiply in half the length of the base, the product whereof is the area: for the perpendicular and half one of the sides represent the two sides of an oblong, equal to the triangle.  
Example.  
The perpendicular B E is 5 perches, and something better, and E C half A C, 3 perches: 3 multiplied in 5, &c. produceth 15, &c. as before, the area of the oblong B D C E, which is equal unto the oxygonium Isopleuron A B C.   

CHAP. XVIII.  
Of an oxygonium Isosceles, and to measure the area thereof.  

De Trianguio Oxygenio Isosceles.  THis triangle is such a figure that hath all his angles acute, as before, only having but two sides equal, 
Euc. l. 1. Defi. 25  wherein the difference consisteth: and hath two measurings proper to himself.  
1 First from the square of one of the equal sides, subtract the square of half the base or side which is unequal: the quadrate roote of the remainder multiplied in half the base, produceth the superficial content.  
Example.  
Let A B one of the equal sides be 6 perches, let B C the base be 4, the half whereof is 2: then take 4 the square of 2, from 36 the square of 6, there resteth 32: the roote quadrate whereof multiplied in 2, produceth 11 7/ 21 perches, the area very near.  
2 Or with more ease as before, you may reduce this figure into ablongs thus: take the length of the perpendicular as you be* taught, 
Chap ●●. Pre. 2.& Chap. 25.  and multiply that length in baise the base, and note the product for the true area.  
Example.  
Multiply D C in D A, so haue you the contents of A E C D the oblong, which is equal unto A B C the triangle.   

CHAP. XIX.  
Of an oxigonium scalenum, and to measure the superficial content thereof.  

De Onigenie See, l●n●.  THis triangle is such a figure that hath all his angles acute as before, the difference from the former rising in the sides, for that the other triangles had fides equal, 
Eu. l. 1. Desi. 16  in this there is no side* equal one unto another, but all unequal,& is measured after one way.  
 
Multiply the perpendicular in half the length of the base, the product there of is the area.  
L O, the perpendicular, augmented in M P, or P N, produceth the oblong, L Q, N O, which is equal to the triangle L, M, 
Corolarium.  N: whereby you see the measuring of triangles for the most part, differ not from oblongs, as hereafter.   

CHAP. XX.  
To measure all kind of superficial triangles generally, and of obtused Angles in particular.  
I think it not amiss in the conclusion of triangles to adds a rule, not onely to measure the Area of amblygoniums, but generally of any other kind of triangle, without the seeking of the perpendicular: and the rule is thus.  
Adds together the length of every side of the triangle, 
Regula generalis ad omnetriaugulorum. Oron.  whose superficial capacity is sought, and of the total take half: out of this half by deduction, noting the difference, that is, how every side differeth from the reserved half: then multiply that half by any one of the differences which you will,( but most fitly by the greater)& the product thereof augment by any of the other differences: which off come increase by the last and third difference, and of the total take the square roote, which note for the area.  
Neither is it material in this kind of work, which of the three differences you make the first, second, or third.  
Example.  
Let the triangle be A B C, whose left side A B is 6 perches, the right side A C 8, and the base B C 10: join therefore 10 and 8, and 6 together, and there will rise 24, whose half is 12, from which detract 6; so the remainder is 6: and 8, there resteth 4: and 10, so 2 remaines: therefore augment 12 in 6, so haue you 72, and 72 in 4, the off-come is 288: this number again multiply by 2, so shall there result 576, the square roote whereof is 24 the area.  
The same number 576 should be engendered if you had augmented 12 in 4, and the product in 6, and that product in 2; or if you had multiplied the said 12 by 2, and the off-come by 4, and the product thereof by 6: or the same 12 in 2, the product in 6: and lastly, the number engendered in 4: so is there always made 576, as you may perceive by the demonstration follow●ng.  
Demonstration. 
The form of, finding the square roote:  First. Second. Third. Fourth.   12 12 12 12   6 4 2 2   72 48 24 24   4 6 4 6 X X   288 288 96 144 X 7 6 2 2 6 4 2   4 576 576 576 576   4    
To perform the same Proposition otherwise.  
Take the former example; if you multiply 6 in 4, 
Alia via qua omnis pariter Trianguli ared vestagatur.  and that product in 2, and again that product in 12, or if you augment 6 in 2, and the product in 4, and that last product in 12, or 4 in 2, and the product in 6, and the resulting number thereof in 12; there will always arise one and the self same number, as 48: which multiply by 12, so is there produced 576, according unto the subscript following.  
Subscript. First. Second. Third. Fourth. 6 6 4 48 4 2 2 12 24 12 8 96 2 4 6 48 48 48 48 576  
The sum of this rule is but adding the three sides of any triangle together, 
Summaria huius Regula declaratio.  and of the resulting number to take half, and then to take the difference that every side severally doth differ from that; compose half( according to the premonition,) afterwards to multiply one of the differences in the other, and the product in the third difference: and again, the product in the half of the number aggregated of the composition of the sides, the roote quadrature whereof is your desire.  
 
Of an Ambly gonium, and to measure the Area thereof. Proposition 2.  
OF Amblygonium or Obtus-anguled triangles, they be found onely to consist of two differences, that is, an Amblygonia Isosceles, and an Amblygonium Scalenum.  
 
An Amblygonium Isosceles is a figure consisting of two equal sides, 
Eu. 1. def. 28. Ra. l. 8. p. 7.  having an obtuse angle. Measure him according unto the 18 Chap. Def. 2.  
Of an Amblygonium Scalenum. Proposition 3.  
 
This triangle is such a figure which hath an obtuse angle as before, differing from the former because all* his sides be unequal.  
This figure agrees with the measuring of the figure in the 19 Chapter: and therefore I will let it pass in this place. 
Eu. 1. def. 16.    

CHAP. XXI.  
To find the length of the perpendicular in all kind of Triangles.  
AS for all right angled triangles, the one side containing the right angle is the perpendicular, and the other the base: and therefore there needeth no other rule for that purpose then what is said before; for the rest both Acute and Obtuse, take these ensuing rules.  
Proposition 1.  
To find the perpendicular of an Isophiron oxigonium.  
If you will work Arithmetically without the help of the 25 Chapter, 
Oxigonium aequilatrum.  you must multiply one of the equal sides by 13, and part the offcome by 15: the quotient is your desire.  
 
Multiply 6 in 13, there is 78, which divide by 15; so haue you B E 5 perches and 3/ 15 your desire.  
Proposition 2.  
To find the perpendicular in an Isosceles Oxigonium.  
Square B D half the base B C, and then square one of the equal sides, A B or A C, 
Oxigonium Isosceles.  from which square subtract the former square of B D; the roote quadrat of the remainder is your desire.  
Example.  
4. The square of B D taken from 36 the square of A B, leaveth 32, whose square roote is 5 perches, and 94 fractions and better, the length of A D your desire.  
 
Proposition 3.  
To find the perpendicular of any Oxigonium Scalenium.  
First square every side, 
Oxigonium Scalenum,  noting the products; this done, add the square of the base unto the square of the shorter side of the triangle, and from the offcome detract the third square, noted before: half the remainder divide by the base, the quotient where of square again, which deduct from the square that you added to the square of the base, and note the square roote for the perpendicular.  
Example.  
Let the Oxigonium Scalenum be L M N, whose side L M let be 6 ½ perches, L N 7 ½ perches, and M N the base 7 perches, the square of L M: 6 ½ is 42, of L N 7 ½ 56: and of 7 is 49: then the square of 7 added to the square of L M 6 ½, produceth 91: from which if you abate the square of L N 7 ½ there remaineth 35: the half whereof is 17 ½, which divide by 7 the base, the quotient is 2 and ½: so many perches is the left section M O, whose square is 6, which take from 42( the square added to the square of the base) so there remaineth 36: the roote quadrate whereof is the length of the perpendicular, viz. 6.  
Otherwise.  
add 56 the square of L M to 49 the square of the base, so haue you 105, from which take 42, the square of L M the other side, and there remaineth 63, the half of which is 31 and ½, which part by 7 the base, the quotient is 4 ½; so many perches is the right section N O: therefore square 4 ½, so haue you 20, which if you take from 56 the square of L N, the remainder is 36, the roote quadrate whereof is 6 perches, as before.  
Prodosition 4.  
To find the perpendicular in any Ambligonium Isosceles.  
Let the figure be A B C, let the two equal sides A B and A C contain 10 perches, 
Amblygonium Isosceles.  let the base be 16; therefore square 10, so haue you 100: then multiply half the base in itself, that is B D 8: so haue you 64, which if you deduct from 100, the remainder will be 36, whose square roote is the perpendicular.  
Proposition 5.  
To find the perpendicular in any Amblygonium Scalenum.  
You must square every side, then add the square of the base or sides subtending the obtuse angle unto one of the other squares, deducting the third square from the product, 
Ambl●gonium. Scalenum.  and then of that which remaineth: half which divide by the base, whose quotient you shall square, and deduct the same from the square which before you added unto the square of the base, the square roote of the remainder is your desire.  
Example.  
 
In the triangle L M N, L M is 20, L N 34, M N 42, the square of L M 400, of L N 1156, of M N 1764:— 1764, put to 400, raiseth 2164, from whence detracting 1156, the remainder is 1008, whose half is 504, which divided by 42 the base, yeeldeth 12, the length of the left section M O, which square maketh 144, and subtracted from 400 leaveth 256, the quadrate roote thereof yeeldeth the perpendicular L O, 16: this differeth not from the 3 Propop.   

CHAP. XXII.  
Of quadrangles or pieces of ground, having 4 sides and 4 Angles.  
A Quadrangle is such a figure that is comprehended under* 4 right lines, 
Quadrangulum quid.  and that may happen two manner of mayes, 
Eu. l. 1. def. 22. Ra. 10. pro. 4.  as a Parallelogram, and a Trapezia; and the parallelogram may be right angled, and then a quadrat or an oblong, or oblique-anguled, and then a Rombus or Romboydes, as followeth.  
Proposition 1. Of a quadrate or square.  
A square is a right angled aequilater parallelogram, 
Quadratum quid. Eu l. 1. def. 30. * whose sides be all equal, as you may perceine by the figure, and is measured thus. 
Ra. 12. p. 2.   
Multiply the known side of the square in itself, 
Ra. 2. cons. 1. De quadrati area.  the product whereof is the Area.  
Example.  
 
Multiply A C 20 perches, one of the known sides, so haue you 400, the Area of A B C D, the square.  

Inuentio Diagonalis quadratorum.  And if you desire to find the diagonal B C, multiply A C in itself, and the offcome double; the Radix quadrature is your desire.  
Proposition 2. Of an oblong.  
An oblong is a rect-angled parallelogram, 
Oblongum quid. Eu. l. 1. def. 31. Ra. 13. p. 1.  not aequilater,* with some, called a long square, whose measuring followeth.  

De mensione quadranguli altera parte longioris.  Take with your scale and compass the length of one of the shorter sides, and also of one of the longer, which augment one in the other; the off-come is your demand.  
Example.  
 
Multiply A B 20 in A C 40, the product is 800 your demand, viz. the area of the oblong A B C D.  
If you desire to find the diagonal B C, square A C 40, 
Vt Diagonalis oblong invenitur.  and B A 20, which add together; the square roote whereof is your desire.  
Proposition 3. To measure a Rombus.  
A Rombus is an oblique-angled Parallelogram, aequilater, 
Rombus quid. Eu. l. 1. def. 32. Ra. 14. p. 8. * formed like a diamond, and hath four kind of measurings proper.  
1. Let there fall a perpendicular from one of the obtusest angles, 
Rombus qualiter metiatur.  to one of the sides, then multiply the same side in the length of the perpendicular, the product is the Area.  
Example.  
A B C D is the Rombus, augmented by A F, the perpendicular falling on B C, in B C; so haue you 96 perches.  
 
2. Or you may measure a Rombus by multiplying the whole distance betwixt two opposite angles in half the distance betwixt the other two opposite angles, whereby it may appear that this kind of work differeth not from an Origonium Isosceles.  
Example.  
Multiply A C 12, in D E 8, half of D E; so haue you 96 the Area.  
So that you see A C D is an Isosceles Origonium, as in the 18 Chapter: now for because A B C is equal to A D C, therefore take all the line A C, and the whole perpendicular E D, and so get the capacity of the oblong A C H G, which is equal to the Rombus A B C D; as is plain, if you consider well the pricked lines.  
3. Or you may measure him as an Ambligonium Isosceles, as in the 20 Chapter Propo. 2. for if you multiply B D the base of the Ambligonium Isosceles in A E the perpendicular, you haue the oblong B D G I, which by the prescript is equal to A B C D the Rombus.  
4. Or you may measure him as an Orthogonium Scalenum, as in the 16 Chapter Prop. 2: for A E and E D the two sides containing the right angle, augmented one in the other, produce 48, the contents of the oblong A E D G, which is equal to half the Rombus: therefore double 48, so haue you 96, as before.  
Proposition 4. To measure a Romboides.  
A Romboides is an oblique-angled parallelogram, not* aequilater. 
Romboides. Eu. l. 1. def. 33. Ra. 14. p. 9.   
 
Let fall a perpendicular from one of the longer sides unto the other, which perpendicular increase by one of those longer sides, so haue you your desire: 
De Romboides Area.  this differeth not from the 4 Prop. Def. 1.  
Example.  
A B C D is the Romboydes, B F the perpendicular, which multiplied in B C, produceth the oblong B C G F, equal to A B C D.  
2. Or you may reduce the Romboydes into two triangles by the line A C, and so measure them as an Ambligonium Scalenum: for D E the perpendicular in the Ambligonium Scalenum A C D, multiplied in A C the diagonal, produceth the contents of the Romboides.  
Proposition 5. To measure a Trapezium Isosceles.  
A Trapezium is defined* to be a Quadrilater Triangular, 
Trapezium. Eu. 1. def. 23. Ra. 14. p. 10. Trapezium diuersitat.  not Parallelogram: therefore we do call that figure a Trapezium, which is neither a parallelogram, neither aequilater, nor aequi-angled. And of Trapezias there be diverse sorts, as well in respect of the diversity of the sides, as angles among themselves: 
Isosceles Traperium.  Some do seem to bear the form of a perfect Isosceles, having two equal sides with two other unequal, yet parallel with two angles acute, and so many likewise obtuse, whereupon he is not unaptly called an Isosceles; and many although they haue two sides alike equal, or parallel, yet they do admit ●●nd right angles and therefore I will presume( and not unworthily) to call such a figure an Orthogonium Trapezium, the rest of the Trapezias, 
Orthogonium Trapezium.  which admit neither like angles or sides, I will call an Ambligonium Trapezium. And first I will speak of the Isosceles Trapezium, and then of the rest.  
You shall measure the length of the other perpendicular included betwixt the two parallel sides, which you may find by the 7 Prop. then multiply the perpendicular in half of the two parallel sides added together; the product is your desire.  
Example:  
The perpendicular by the 7 Prop. Def. 1. is found to be 8 in the Trapezia A B C D, the equal sides are A B and C D, the parallel sides B C and A D: B C and A D added together, make 20, half which is 10, 10 multiplied in 8 produce 80, the contents of the oblong 4 F E 16, which is equal to A B C D the Trapezia: or you may resolve the Trapezia into two triangles, and so measure him by the Doctrine of triangles before.  
Proposition 6. An Orthogonium Trapezium.  
And if it shall any thing avail thee to find the Area of a right angled Trapezia, 
Vt rectangulum metiatur. Traperium.  do thus. add together the two parallel sides( whereof the one contains the right angle,) half which increase by the perpendicular or other containing side, the product is your demand.  
 
Example.  
Let the Orthogonium Trapezium be F G H E, add half F G 6 perches to half E H 18, so haue you E K 12, which augment by F E 5; so haue you 60 perches the Area of the oblong F I K E, which is equal to F G H E, for that the triangle G I L exclusive, is equal to the triangle inclusive L K H.  
Or you may resolve this figure into a triangle as, before.  
Proposition 7. An Ambligonium Trapezium.  
This figure may happen after diverse manner of ways: but for all I will set down onely three differences, whose measuring are all one kind of way.  
You must resolve each figure into two triangles by the shortest diagonal you can, then measure those two triangles by the former instruction of triangles.  
 
Example.  
These three Trapeziums A D C D are resolved into two triangles by the diagonal A D, and then measured by the prescript of triangles; and A D is the shortest diagonal that can be found.  
 
Proposition 8. To find the perpendicular in an Isosceles Trapezium:  
You shall multiply one of the equal or dis-parallel sides in itself, and reserve the product; then square half the difference of the two parallel sides, and subtract the product from the first reserved number: the square side or roote whereof is the perpendicular.  
Example.  
 
Let the Isosceles Trapezia be A B C D, whose unequal side is A B 10 perches, A C 4, and the base A D 16; the square of which is 100, the difference of A C and A D is 12, whose half is 6, whose square is 36, which take from 100, the remainder is 64, whose radix is 8 your demand. And thus far of Quadrangles.   

CHAP. XXIII.  
Of a Polygonium, and to find the center and superficial content thereof, as also to measure irregular figures.  
A Multangle figure is that which is contained under more then four right lines* and is called a Poligonim, 
Multangula. Eu. l. 1. def. 23. Ra. 14. p. 11. Oron.  and of some a Multilater, for because it doth contain more then four sides and four angles: and of Multilaters, some be regular, others irregular: 
Regulares multilatrae figurae.  those be called regular Multilaters, when one may inscribe or circumscribe a circled, whose center shall be the common center both of the equal sides and angles. Those be called irregular which haue sides, 
Irregulares.  and consequently angles vnéquall.  
Proposition 1. To measure a Regular Poligonium.  
When you would find the Area of any regular Poligonium or Multilater, observe then this general rule.  Si fiat planus è perpendiculari à centro in latus,& dimidio p●ri●etri, factus est area multanguli or dinati: Oron cap. 8. lib. 2. De R●& prac. Geomet. Ra. l. 19. P. 1. 
Which is Englished thus: The plain number made of the perpendicular, drawn from the center unto the side, and of half the perimeter, is the content of a multangled ordinate figure.  
Therefore by-the-second Proposition find the center of the Poligonium, and produce a line from their perpendicular unto one of the equal sides, afterwards augment the s●yd perpendicular in half the peripher, and the product is the Area of the Poligonium.  
 
Example.  
Let the poligonium A B C D, be a Pentagon, 
E●empium de Pentagono.  whose equal sides let be 12 perches, let the center be A, from whence draw ●●ine perpendicul●r to B C, which let be 8 perches: then for because 5 times 1● make 60, the half thereof is 30 perches, ●s D E: therefore if you multiply 30 by 8, you haue the content of the oblong D E F A, equal by the prescript unto the polygonium pentagon, or figure of 5 sides.  
Proposition 2. To find the center of a polygonium having an odd side.  
If the figure haue an odd side, as the pentagon in the last Prop. find his center thus.  
Let fall a perpendicular from one of the angles unto his opposite side, as from D on the line B C; do so to some other angle, as you may perceive by the figure. I then conclude that the intersection of those two perpendiculars is the center of the figure as A; vpon which you may inscribe or circumscribe a circled, and the inscription shall answer unto the midst of each side, and the circumscription to the angles.  
Proposition 3.  
 
To find the center of any polygonium containing equal sides.  

Exemplum De Hexagon.  You must draw two lines between any opposite angles, the intersection is the center, so shall you find D to be the center of the hexagon D E F H G M K; then the perpendicular is 5 ⅖, which augment in 30 half the peripher, so haue you 93 ⅗ perches, the content of the Hexagon Poligonium.  
And thus far of regular poligoniums, presuming I haue said sufficient for the perfect understanding thereof.  
Proposition 4. Of irregular Polygoniums, called irregular figures.  
There can be no other rule be prescribed then to resolve such irregular figures into some regular form: for which purpose the 44 Chapter and 8 Metamorphis shall stand you in a singular use: therefore I will cease here to speak further thereof: and indeed by that chapter you may measure all kind of ordinate or inordinate figures without any of the precedent Propositions.  
And here I might take occasion for to amplify this work with the dimensions of circles and their parts: but it were vain so to do, since they are performed in more easy manner by my staff in the prealle gated 44 Chapter: But howsoever I hold a circled nothing necessary to treat of in the measuring of ground: for that of all figures is the perfects, so that I think it scarce possible to find a piece of ground simply round as a circled: yet for those which be persuaded to the contrary, let them take the ground of finding the Area of circles thus: Planus siue quadratus è radio& peripheriae dimidio, dabit aream circuli. Ramus lib. 19. Prop. 2. Cons. 1. which is: The plain number made of the radius and half the circumference, is the content of a circled.   

CHAP. XXIIII.  
To reduce perches unto acres, and such like Statute measure.  
WHen you haue multiplied your figures, as before, and found the true quantity of perches therein, you shall reduce them unto acres, 
To reduce perches unto acres and Statute measure.  roods, &c. according unto the second Chapter of the second book, thus. If the number of perches exceed 160, divide the said number by 160, the quotient she weth the number of acres, then if there be any thing remaining, if it exceed 40, make partition thereof by 40, so shall the quotient show the number of roodes; then if there be any thing remaining divide it by 4, the quotient is the number of day works, and the remainder the number of perches; and if after partition by 160, 40, ●24, the number remaining be less then 4, it is perches.  
Example.  
 
By the doctrine of the 20 Chapter, Prop, 3. I find the perpendicular L O 16 perches, 
By this doctrine you may augment the Tables of land measure as far as you please.  and the base M N 42, then by the foresaid doctrine of triangles I augment half M N 21 in L O 16, so haue I 336 perches, which by the prescript of this Chapter I first divide by 160, so haue I 2 for my quotient, and 16 remaining: then because 16 will not suffer partition by 40, I part it by 4, so haue I 4 for my quotient, and nought remaining: whereby I may conclude that the triangle L M N, doth contain 2 acres, 0 roodes, 4 day-workes and 0 perches.   

CHAP. XXV.  
To measure all manner of land without arithmetic, or to render the contents of the plate taken.  

To measure ground without arithmetic.  I Am not ignorant that diverse never yet studied in the Art of arithmetic or Geometry will notwithstanding be very desirous to haue a sight in this Art of Geodetia or measuring grounds; which to perform let such work thus.  
If the figure be irregular, resolve him into triangles( because experience teacheth me, and Geometry confirms the same, that there is no figure how irregular soever, but he may be resolved into triangles:) but if the figure be a triangle if self, you shall seek out the greatest angle: 
To find the perpendicular in triangles without arithmetic.  wherein place the firm foot of your compass, extending the other so that you may take the shortest extension betwixt the said angle, and the base line that doth subtend the angle: this wideness of your compass you shall apply unto your Scale, and see how many perches it contains: for so many equal parts as be included betwixt the two feet of your Commpasse, so many perches long is the perpendicular: then take the length of the base, and note the half thereof, and so by the 27 Chapter enter the table with those two numbers, and you haue your desire.  
Example.  
Let L O M be a tryangular piece of ground, whose base M N is 7 perches, and whose perpendicular L O is 6 perches, which is the shortest extension betwixt the angle L, and the line M N: then I enter my table according to the 27 Chapter, with 6 and 3 ½, which is half 7. But first because there be fractions annexed, I enter with 6 and 3, so do I find 2 day-work and 1 perch: then for the ½ or half the perch annexed to 3, I enter the table according unto the 28 Chapter: so do I find 3 perches more, which added unto the former, viz. to 2 day-worke, and one perch make 2 day-worke, and 4 perch: which by the last Chapter is just 5 day-worke, or half a rood, the content of L M N.  
Which is as if you had multiplied 6 by 3 ½, and proceeded as in the 19 Chapter.  
And hear let it be noted generally, when you reduce your plate into triangles, make them as big as you may, 
Notae.  and so consequently as few as you can: for look how many angles the plate contains, if you work truly, there will be two less triangles therein; as if the plate consist of 6 angles, the triangles angles therein will be but 4; if of 4, the triangles will be 2, as you see in the ensuing figures by the line A D.  
  

CHAP. XXVI.  
Of the measuring of a Trapezia, Rombus, Romboides, &c. without arithmetic, or an oblong, &c. also to measure all lands without plaining.  
YOu may resolve any of all these figures into two triangles by the 7 Proposition of the 22 Chapter, and so measure them as in the last Chapter; or otherwise work thus,  
Proposition 1. For a square.  
Let A B C D be a square: 
A Square.  by my scale and compass I and A B to contain 20 perches, with which entering the table by the 27, with 20 and 20 I haue my desire, viz. 2 acres and 2 roodes.  
Proposition 2. For an oblong. 
An oblong.   
Let A B D C be a long square: as before, I find A B 20 perches, and A C 40, with which entering the ensuing table according to the 27 Chapter, I find 45 acres the content of A B D C; and you might haue resolved the figure into 2 triangles by the line B C, and measured him, as in the last chapter.  
Proposition 3. For a Rombus.  
Let a figure be given A B C D, like unto a Diamond: 
A Rombus.  by my Scale and compass I find B D 16 perches, and the line A C 12: enter the table with 8, half B D, and 12, the whole of A C: so haue I 2 acres, and 4 day-workes, the contents of A B C D: or you might haue resolved it into triangles, according as you see the pricked lines, and so wrought by the 25 chapter.  
Proposition 4. For a Romboides.  
For a Romboides, 
A Romboides, Vide prop. 4. cap. 22.  look what numbers I bid you multiply one in the other, the same find by the last Chapter, and enter the table therewith; so haue you your desire. Or you may resolve the figure into 2 triangles by the line A C, and so work as in the last Chapter.  
Proposition 5. A general Rule.  
In brief look what numbers I bid you multiply one in the other, the same find by your Scale and compass, and so enter the ensuing table, and you shall haue your desire.  
Example.  
We will take this Poligonium Pentagon, or figure of five sides A D G B C H: first by the 23 Chapter Prop. 2, I find his center A, then I find the perpendicular by taking the shortest extension betwixt A and one of the equal sides, as B C: let the perpendicular be 8 perches, then I search the length of the 5 equal sides with my Scale and compass, which let be 12 apiece, the whole peripher D B C H making 60, the half whereof is 30: so I enter the table with 8 the perpendicular, and 30 half the peripher, and find one acre and two roodes, which is the contents as truly as if I had increased 30 by 8, and found 240 perches, which by the 24 chapter agrees with the former.  
Proposition 6. To measure all kind of land without taking the plot.  
But many times you shall not be put to plot ground in such order as the first part of this book directeth you, 
To measure land without plaiting.  but onely to yield the contents of the piece of land in the field according to the dimension there taken by your chain, and so to engross the particulars, as you be taught in the Appendix; which when it so fortunes, you must diligently observe the forms and diverse fashions of the said grounds, and so reduce it into some one regular figure or other, as into a square, a triangle, or triangles, a Trapezia or such like, and then measure it as before. But in so much the many, or rather most pieces of land bear no regular figure, there be diverse ways that I haue used for the measurement of the same.  
First, you may take your instrument and square the field, 
To square a piece of land.  and then measure the odd pieces by itself, according to the figure they represent: and this squaring of a piece of ground is either performed by the staff or some other instrument without an needle or with an instrument with a needle.  
To perform it by the staff, open the legs unto a, right angle, and then planting the same in four several places the one leg at the second station, respecting the first station, and so the third, the second; the irregular superficies is reduced into a true square, the fragments being measured according unto such figures as they represent: as in the figure A B F E D, opening the legs unto a right angle, I place them first at A, so doth the one leg point to B, the other to D: then placing him at D, the one leg points to A, the other to C. Lastly placing at C, the one leg points to D, the other to B, so haue you made a perfect long square parallelogram. As for the fragments and corners omitted, you must measure them as I said, according to the figure that they represent, as you may plainly perceive by the figure before, and by the pricked lines inscribed therein.  
If you conceive this form, you may also perform the same by the Theodelitus or topographical glass, &c.  
Proposition 7. To find where the perpendicular in any triangle fals vpon the base, without arithmetic, in the open field.  
Now in plaiting of grounds, 
To find where the perpendicular in any triangle fals vpon the base.  when there happen triangles, you must of necessity find the perpendicular; and for that it is troublesone to perform it Arithmetically, you shall therefore perform the same by any of these ensuing ways.  
1. Open the legs to a right angle, and then place the one leg just over the line subtending the greatest angle, moving the Instrument from place to place( keeping the said one leg still over the said line) until the other point just in to the angle opposite to that line: so then doth the instrument stand in the place where the perpendicular should fall: therefore measure from thence in to the said angle, so haue you your desire.  
2. Or observe the angle at one of the ends of the subtending line, and then( the legs refting at that angle) measure the other side containing the said angle, which count vpon the right leg: now placing the graduator at a right angle, draw him along the left leg, until the fiducial edge thereof cut the parts before numbered vpon the right leg; so doth the center of the graduator vpon the left leg cut the number of perches; that the perpendicular falleth vpon the base line, being counted from that end you observed the angle at.  
3. You may perform the same by any of the other instruments, according to the first direction. Further more the figure opposite may happen to be most easily measured, by reducing the same unto a Trapezia, and certain figures, the which perform thus.  
A B C D is a figure or field proposed to be measured. Here you see that this field may be well and easily reduced into a perfect Trapezia, which it doth of itself much represent: but is not perfect: for that C D and A B be not parallel: therefore to make a true Trapezia thereof, or to prove if the two opposite ends lie parallel, do thus.  
Plant your topographical glass at A, making the needle stand just in his true place, then move the Index or Alhiada, until through the sight you see B: note the degree cut, which let be 200, then plant him in the lesser of the angles at the end of the line C D, as at C; making again the needle to stand over his right place, bringing also the Index unto the degree cut before, viz. 200: so do the sights point to E, whereby you haue made a true Trapezia A B C E, which measure accordingly; then for the triangle D E C, measure the same according to the doctrine of triangles.  
By this means may you draws parallels in any kind of figure conceived in the fields, and so reduce the same into what figure seeming most fit and requisite, as a Quadrate, an oblong, a Rombus or Romboides, which are all figures consisting of parallel sides.  
Likewise may you draw any lines stopping out one as much as another, as from the point B, to draw the line B E to slope out as much as the line A C doth at the point A: hereby also haue you made the angle C A B. equal to A B D.  
In this order must you work when you be put to measure lands without plaiting, still by your instrument reducing the proposed field or pasture into certain exact and regular figures: 
Many measure grounds falsely.  for it is most absurd and direct contrary to the grounds of Art, to give credit unto the measuring that many country fellowes use. For you shall see a Mason, Carpenter, or some such mechanical fellow undertake to measure and truly denied his neighbors grounds, when as his rules be most false and uncertain. For some use to measure round about the ground without regard of any regular or irregular figure it represents: others use to measure all kind of lands as we do a Trapezia: others take half the sides and ends, all which to be used in a generality, is false and absurd.   

CHAP. XXVII.  
The use of the Table for land measuring.  
WHen you haue two numbers assigned, thē resort unto the Tables, and in the vpper end or head thereof seek the greater number assigned, going forward from Table to Table, as you be there directed, until you haue found the said number: then descend that column until you come so low that you are just against the lesser assigned number in the descending row vpon the left hand: so in the common angle or square where the concourse of both lines meet, you shall find placed in the said square the number of acres, roodes, day-workes or perches, as the plate contains.  
Example.  
I find by the doctrine before, that the length of some perpendicular is 15 perches, and the half of some base 20 perches: I find 15 in the side of the table, and so proceeding until I come right under 20, in the uppermost column: thers then in the common angle do I find 1 acre, 3 roodes, and 5 day-worke, and so of any other number.  
And if the numbers exceed your table, 
To work when the numbers exceed the rables.  make partition thers of taking the half 3 degree, part or quarter; and so enter the Table, and then proportien what you find according unto the parts you took.  
Example.  
My one number is 100, the other 8; I can find 8, but not 100 in my table: therefore I take the fourth part of 100, which is 25, 
The one number exceeding the Table.  entering my table with 8 and 25, so do I find one acre, and one rood; which take 4 times, because I took the fourth part, so haue I just 5 acres: or you might haue taken half and so entred with 50 and 8, so should you haue found two acres and two roodes; which doubled amounts, as before.  
Now it may happen that both the numbers shall exceed the Table, let the two numbers be 100 and 100; suppose I cannot find them in the Table, take therefore what parts you will thereof, I take half of the one, and the fourth part of the other, half 100 is 50, the fourth part is 25: I enter my table with 50 and 25, and find 7 acres, 3 roodes, 2 day-workes, and 2 perches: then because I took 2 parts of the one number, and the fourth of the other, I increase 4 by 2, so haue I 8, which argueth that I haue found but the eighth part: therefore must I take the foresaid number eight times, so shall I haue 62 acres, and 2 roodes; so of any other. And note that each square hath four numbers therein contained, fignifying as followeth:  
Of the four numbers, the 

vpper 

leftwards  
rightwards    
lower 

leftwards  
rightwards     contains 

acres.  
roodes.  
day works  
perches.    

Table 1.  
This row proceeds in the 2 Table.    1 2 3 4 5 6 7 8 9 10 1 1 2 3 1 1 1 1 2 1 3 2 2 1 2 2   2 1 1 2 2 2 2 3 3 2 4 4 2 3   3 2 1 3 3 3 4 2 5 1 6 6 3 7 2   4 4 5 6 7 8 9 1   5       1 1 1 6 1 7 2 8 3   1 1 2 2   6   1 1 1 1 9 2 2 3 2 5 1   7 1 1 1 1 2 1 4 3 3 7 2   8 1 1 2 6 8     9 2 2 1 2 2   10 2 5  
This Table doth extend from 1 perch to 10, in breadth and length; the left row proceeding in the 2 Table, at 11.  
You most note that these words below every Table,( In Table) do note in what Table both the sides of that Table are continued together.  

Table 2.  
This row proceeds in the 3. Table.    11 12 13 14 15 16 17 18 19 20 1 2 3 3 3 1 3 2 3 3 4 4 1 4 2 4 3 5 2   1 5 2 6 6 2 7 7 2 8 8 2 9 1 9 2   3   1 1 1 1 1 1 1 8 1 9 9 3 2 1 1 2 2 3 3 2 4 1 5 4 1 1 1 1 1 1 1 1 1 2 1 2 3 4 5 6 7 8 9   5 1 1 1 1 1 2 2 2 2 2 3 3 5 6 1 7 2 8 3 1 1 2 2 3 3 5   6 1 1 1 2 2 2 2 2 2 3 6 2 8 9 2 1 2 2 4 5 2 7 8 2   7 1 2 2 2 2 2 2 3 3 3 9 1 1 2 3 4 2 6 1 8 9 3 1 2 3 1 5 8 2 2 2 2 3 3 3 3 3 1 2 4 6 8 2 4 6 8   9 2 2 2 3 3 3 3 1 1 1 4 3 7 9 1 1 2 3 3 6 8 2 2 2 3 5 10 2 3 3 3 3 1 1 1 1 1 1 7 2   2 2 5 7 2   2 2 5 7 2   11 3 9 3 3 1 1 1 1 1 1 1 1 1 3 5 3 8 2 1 1 4 6 3 9 2 2 1 5   12 3 3 1 1 1 1 1 1 1 1 1 1 2 6 9 2 5 8 1 4 7     13 1 1 1 1 1 1 1 1 1 1 2 1 2 2 1 5 2 8 3 2 5 1 8 2 1 3 5   14 1 1 1 1 1 1 1 1 2 1 2 1 2 9 2 2 6 9 2 5 6 2     15 1 1 1 2 1 2 1 2 1 3 1 3 6 1   2 3 7 2 1 1 5   16 1 2 1 2 1 3 1 3 2 4 8 2 6     17 1 3 1 3 2 2 2 1 6 2 3 5   18 2 2 2 1 1 5 2     19 2 1 2 1 3 5   20 2 2  
This side proceeds from 20 in the 5 Table to 30. In Table 4.  
In Table 4.  

Table 3.  
This row proceeds in the 4. Table.    21 22 23 24 25 26 27 28 29 30 1 5 1 5 2 5 3 6 6 1 6 2 6 3 7 7 1 7 2 2 1 1 1 1   1 1 1 1 1 2 1 1 2 2 2 2 3 3 2 4 4 2 5 3 1 1 1 1 1 1 2 2 2 2 5 3 6 2 7 1 8 3 3 9 2 1 1 1 3 2 2 4 2 2 2 2 2 2 2 2 2 3 1 2 3 4 5 6 7 8 9   5 2 2 2 3 3 3 3 3 3 3 6 1 7 2 8 8   1 1 2 2 3 3 5 6 1 7 2 6 3 3 3 3 3 3 1 1 1 1 1 2 3 4 2 6 2 9 2 2 3 2 5 7 3 3 1 1 1 1 1 1 1 1 1 1 6 3 8 2 1 2 3 3 5 2 7 1 9 1 3 2 2 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 6 8   2 4 6 8   9 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 2 7 1 9 2 1 3 4 6 1 8 2 3 3 5 1 7 2 10 1 1 1 1 1 1 2 1 2 1 2 1 2 1 3 1 3 1 3 2 2 5 7 2   2 2 5 7 2   2 2 5 11 1 1 1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 3 2 ● 7 3 2 3 6 8 3 1 2 4 1 7 3 3 2 2 12 1 2 1 2 1 2 1 3 1 3 1 3 2 2 2 2 1 3 6 9 1 5 8 1 4 7   13 1 2 1 3 1 3 1 3 2 ● 2 2 2 1 2 1 2 1 8 1 1 2 4 3 3 1 1 4 2 7 3 1 4 1 7 2 14 1 3 1 3 2 2 2 2 1 2 1 2 1 2 2 2 2 3 2 7 2 1 7 2 1 4 2 8 1 2 5 15 1 3 2 2 2 1 2 1 3 1 2 2 2 2 2 2 2 3 8 3 2 ● 6 1   1 3 7 2 1 1 5 8 3 2 2 16 2 2 2 ● 2 1 2 2 2 2 2 2 2 3 2 3 3 4 8 2 6   4 8 2 6   17 2 2 1 2 1 2 2 2 1 2 3 2 3 2 3 3 3 9 1 3 2 7 3 2 6 1 2 4 3 9 3 1 7 2 18 2 1 2 1 2 2 2 2 2 3 2 3 3 3 3 1 3 1 4 ● 9 3 2 8 2 2 7 1 2 6 2 5 19 2 1 ● 2 2 2 2 3 2 3 3 3 3 1 3 1 3 2 9 3 4 2 4 1 4 8 3 3 2 8 1 3 7 1 2 2 20 2 2 2 3 2 3 3 3 3 1 3 1 3 1 3 2 3 3 5   5   5   5   5    
This left side proceeds from 20. in the 5. Table. In Table 6.  

Table 4.  
This row proceeds in the 7 Table.    31 32 33 34 35 36 37 38 39 40 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 3 ● 0 8 1 8 2 8 3 0 0 0 1 0 2 0 3 0 0 2 0 1 1 1 1 1 1 1 1 1 2 3 2 6 6 2 7 7 2 8 8 2 9 9 2   3 1 2 2 2 2 0 2 0 2 2 2 3 3 1 4 4 3 5 2 6 1 7 7 3 8 2 9 1   4 3 3 3 3 3 0 3 3 3 3 1 1 2 3 4 5 6 0 7 3 9   5 3 1 1 1 1 1 1 1 1 1 1 8 3   1 1 2 2 3 3 5 6 1 7 2 8 3   6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 6 2 8 9 2 1 2 2 4 5 2 7 8 2   7 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 3 4 1 6 7 3 9 2 1 1 3 4 3 6 2 8 1   8 1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 3 1 3 2 2 4 6 8   2 4 6 8   9 1 2 1 3 1 3 1 3 1 3 2 2 2 2 2 1 9 3 2 4 1 6 2 8 3 1 3 1 5 2 7 3   10 1 3 2 2 2 2 2 1 2 1 2 1 2 1 2 2 7 2   2 2 5 7 2   2 2 5 7 2   11 2 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 3 5 1 8 3 3 2 6 1 9 1 3 4 2 7 1   12 2 1 2 1 2 1 2 2 2 2 2 2 2 3 1 3 2 3 3 3 6 9 2 5 8 1 4 7   13 2 2 2 2 2 2 2 3 2 3 2 3 3 3 3 3 1 3 4 7 1 2 3 3 7 1 3 2 6 3   14 2 2 2 3 2 3 2 3 3 3 3 3 1 3 1 3 2 8 2 2 5 2 9 2 2 6 9 2 3 6 2   15 2 3 3 3 3 3 1 3 1 3 1 3 2 3 2 3 3 6 1   3 3 7 2 1 1 5 8 3 2 2 6 1   16 3 3 3 1 3 1 3 2 3 2 3 2 3 3 4 3 4 4 8 2 6   4 8 2 6   17 3 1 3 1 3 2 3 2 3 2 3 3 3 3 4 4 4 1 1 3 6 1 4 2 8 3 3 7 1 1 2 5 3   18 3 1 3 2 3 2 3 3 3 3 4 4 4 1 4 1 4 2 9 2 4 8 2 3 7 2 2 6 2 1 5 2   19 3 2 3 3 3 3 4 4 4 1 4 1 4 2 4 2 4 3 7 1 2 6 3 1 1 6 1 1 5 3 2 5 1   20 3 3 4 4 4 1 4 1 4 2 4 2 4 3 4 3 5 1   5   5   5   5    
This left side proceeds in the 6. Table. In Table 9.  

Table 5.  
This row proceeds in the 6. Table.    21 22 23 24 25 26 27 28 29 30 21 2 3 2 3 3 3 3 1 3 1 3 2 3 2 3 3 3 3 1 5 2 3 6 1 1 6 2 1 3 7 2 1 7 2   22 3 3 3 1 3 1 3 2 3 2 3 3 3 3 4 1 6 2 2 7 2 3 8 2 4 9 2 5   23 3 1 3 1 3 2 3 2 3 3 4 4 1 4 1 2 2 8 3 3 9 2 5 1 1 6 3 2 2   24 3 2 3 3 3 3 4 4 4 1 4 1 4   6 2 8 4     25 3 3 4 4 4 1 4 2 4 2 4 1 2 2 8 3 5 1 1 7 2   26 4 4 1 4 2 4 2 4 3 9 5 2 2 2 2 5   27 4 2 4 2 4 3 5 2 1 9 5 3 2 2   28 4 3 5 5 1 6 3     29 5 1 5 1 1 7 1   30 5 2 5  
This proceeds in the 6 Table at 31, on the left side. In Table 6.  

Table 6.  
This row proceeds in the 9 Table.    31 32 33 34 35 36 37 38 39 40 21 4 4 4 1 4 1 4 2 4 2 4 3 4 3 5 5 1 2 3 8 3 1 8 2 3 3 9 4 1 9 2 4 3   22 4 1 4 1 4 2 4 2 4 3 4 3 5 5 5 1 5 2 2 6 1 2 7 2 2 8 3 2 9 4 2   23 4 1 4 2 4 2 4 3 5 5 5 1 5 1 5 2 5 3 8 1 4 9 3 5 2 1 1 7 2 2 8 2 4 1   24 4 2 4 3 4 3 5 5 1 5 1 5 2 5 2 5 3 6 6 2 8 4   6 2 8 4   25 4 3 5 5 5 1 5 1 5 2 5 3 5 3 6 6 1 3 3   6 1 2 2 8 3 5 1 1 7 2 3 3   26 5 5 5 1 5 2 5 2 5 3 6 6 6 1 6 2 1 2 8 4 2 1 7 2 4 2 7 3 2   27 5 5 1 5 2 5 2 5 3 6 6 6 1 6 2 6 3 9 1 6 2 3 9 2 6 1 3 9 3 6 2 3 1   28 5 1 5 2 5 3 5 3 6 6 1 6 1 6 2 6 3 7 7 4 1 8 5 2 9 6 3   29 5 2 5 3 5 3 6 6 1 6 2 6 2 6 3 7 7 1 4 3 2 9 1 6 2 3 3 1 8 1 3 2 2 3   30 5 3 6 6 6 1 6 2 6 3 6 3 7 7 1 7 2 2 2   7 2 5 2 2   7 2 5 2 2   31 6 6 6 1 6 2 6 3 6 3 7 7 1 7 2 7 3 1 8 5 3 3 2 1 1 9 6 3 4 2 2 1     32 6 1 6 2 6 3 7 7 7 1 7 2 7 3 8 6 4 2   8 6 4 2     33 6 3 7 7 7 1 7 2 7 3 8 8 1 2 1 2 8 3 7 5 1 3 2 1 3     34 7 7 1 7 2 7 3 8 8 1 8 2 9 7 2 6 4 2 3 1 2     35 7 2 7 3 8 8 1 8 2 8 3 6 1 5 3 3 2 2 1 3     36 8 8 1 8 2 8 3 9 4 3 2 1     37 8 2 8 3 9 9 1 ● 1 1 2 3     38 9 9 1 9 2 1 2     39 9 2 9 3 1     40 10  
This left side proceeds in the 11. Tables In Table 11.  

Table 7.  
This row proceeds in the 8 Table.    41 42 43 44 45 46 47 48 49 50 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 1 2 1 3 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 3 3 2 4 4 2 5 3 3 3 3 3 3 3 3 3 3 3 3 1 2 2 1 3 3 3 4 2 5 1 6 6 3 7 2 4 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9   5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 2 2 2 3 5 6 1 7 2 8 3   1 1 2 2 6 1 2 1 2 1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 3 1 2 3 4 2 6 7 2 9 2 2 3 2 5 7 1 3 1 3 1 3 1 3 1 3 2 2 2 2 2 1 3 3 2 5 1 7 8 3 2 1 1 4 5 3 7 2 8 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 2 1 4 6 8   2 4 6 8   9 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 3 2 3 2 1 4 2 6 3 9 1 1 3 2 5 3 8 1 2 2 10 2 2 2 2 2 2 2 3 2 3 2 3 2 3 3 3 3 2 2 5 7 2   2 2 5 7 2   2 2 5 11 2 3 3 3 2 3 3 3 3 3 3 1 3 1 3 2 2 2 5 2 8 1 1 3 3 6 2 9 1 4 3 7 2 1 12 3 3 3 3 1 3 1 3 1 3 2 3 2 3 2 3 3 3 6 9 2 5 8 1 4 7   13 3 1 3 1 1 3 2 3 2 3 2 3 3 3 3 3 3 4 3 1 6 2 9 3 3 6 1 9 2 2 3 6 9 1 2 2 14 3 2 3 2 3 3 3 3 3 3 4 4 4 4 1 4 1 3 2 7 2 4 7 2 1 4 2 8 1 2 5 15 3 2 3 3 4 4 4 4 1 4 1 4 2 4 2 4 2 3 3 7 2 1 1 5 8 3 2 2 6 1   3 3 7 2 16 4 4 4 1 4 1 4 2 4 2 4 2 4 3 4 3 5 4 8 2 6   4 8 2 6   17 4 1 4 1 4 2 4 2 4 3 4 3 4 3 5 5 3 1 4 1 8 2 2 3 7 1 1 5 2 9 3 4 3 1 2 1 18 4 2 4 2 4 3 4 3 5 5 5 1 5 1 5 2 5 2 4 2 9 3 2 8 2 2 7 1 2 6 2 5 19 4 3 4 3 5 5 5 1 5 1 5 2 5 2 5 3 5 3 4 3 9 2 4 1 9 3 3 8 2 3 1 8 2 3 7 2 20 5 5 1 5 1 5 2 5 2 5 3 5 3 6 6 6 1 5   5   5   5   5    
This left side proceeds in the 9. Table: In Table 10.  

Table 8.  
This row is continued in the 13. Table.    51 52 53 54 55 56 57 58 59 60 1 1 1 1 1 1 1 1 1 1 1 2 3 3 3 1 3 2 3 3 4 4 1 4 2 4 3 5 2 2 2 2 2 2 2 2 2 2 3 5 2 6 6 2 7 7 2 8 8 2 9 9 2   3 3 3 3 1 1 1 1 1 1 1 8 1 9 9 3 2 1 1 2 2 3 3 2 4 1 5 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 3 4 5 6 7 8 9   5 1 2 1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 3 1 3 3 3 5 6 1 7 2 8 5   1 1 2 2 3 3 5 6 1 3 1 3 1 3 2 2 2 2 2 2 2 1 6 2 8 9 2 1 2 2 4 5 2 7 8 2   7 2 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 2 9 1 1 2 3 4 2 6 1 8 9 3 1 2 2 1 5 8 2 2 2 2 2 2 2 2 2 3 2 3 2 3 1 3 2 3 3 2 4 6 8   2 4 6 8   9 2 5 2 3 2 3 3 3 3 3 3 1 3 1 3 1 4 3 7 9 1 1 2 3 3 6 8 1 2 2 3 5 10 3 8 1 3 1 3 1 3 2 3 2 3 2 3 2 3 2 3 3 7 2   2 2 5 7 2   2 2 5 7 2   11 3 1 3 2 3 2 3 2 3 3 3 3 3 3 3 3 4 4 9 1 3 3 3 8 2 1 1 4 6 3 9 2 2 1 5 12 3 3 3 3 3 3 4 4 4 4 1 4 1 4 1 4 2 3 6 9 2 5 8 1 4 7   13 4 4 4 1 4 1 4 1 4 2 4 2 4 2 4 3 4 3 5 3 9 2 1 5 2 8 3 2 5 1 8 2 1 3 5 14 4 1 4 2 4 2 4 2 4 3 4 3 4 3 5 5 5 1 ● 2 2 5 2 9 2 2 6 9 2 3 6 1   15 4 3 4 3 4 3 5 5 5 1 5 1 5 1 5 2 5 2 1 1 5 8 3 2 2 6 1   3 3 7 2 1 1 5 16 5 5 9 1 5 1 5 2 5 2 5 2 5 3 5 3 6 4 8 2 6   4 8 2 6   17 5 1 5 2 5 2 5 2 5 2 5 3 5 3 6 6 1 6 1 6 3 1 4 1 3 3 9 2 3 3 8 6 2 3 5 18 5 2 5 3 5 3 6 6 6 1 6 1 6 2 6 2 6 3 9 2 4 8 2   7 2 2 6 2 1 5 2   19 6 6 6 1 6 1 6 2 6 2 6 3 6 3 7 7 2 1 7 1 2 6 2 1 1 6 3 3 2 1 5 20 6 1 6 2 6 2 6 3 6 3 7 7 7 1 7 1 7 2 5   5   5       5    
This left side proceeds in the 20 Table. In Table 15.  

Table 9.  
This row is continued in the 10 Table.    41 42 43 44 45 46 47 48 49 50 21 5 1 5 2 5 2 5 3 5 3 6 6 6 1 6 1 6 2 5 1 2 5 3 1 6 1 1 2 6 3 2 7 1 2 2 22 5 2 5 3 5 3 6 6 6 1 6 1 6 2 6 2 6 3 3 2 1 6 2 2 7 2 3 8 2 1 9 2 5 23 5 3 6 6 6 1 6 1 6 2 6 3 6 3 7 7 5 3 1 2 7 1 3 8 3 4 2 1 6 1 3 7 2 24 6 6 1 6 1 6 2 6 3 6 3 7 7 7 1 7 2 6 2 8 4   6 2 8 4   25 6 1 6 2 6 2 6 3 7 7 7 1 7 2 7 2 7 3 6 1 2 2 8 3 5 1 1 7 2 3 3   6 1 2 2 26 6 2 6 3 6 3 7 7 1 7 1 7 2 7 3 7 3 8 6 2 3 9 2 6 2 2 9 5 2 2 8 2 5 27 6 3 7 7 1 7 1 7 2 7 3 7 3 8 8 1 8 1 6 3 3 2 1 7 3 3 2 7 1 4 3 7 2 28 7 7 1 7 2 7 2 7 3 8 8 8 1 8 2 8 3 7 4 1 8 5 2 9 6 3   29 7 1 7 2 7 3 7 3 8 8 1 8 2 8 2 8 3 9 7 1 4 2 1 3 9 6 1 3 2 3 8 5 1 2 2 30 7 2 7 3 8 8 1 8 1 8 2 8 3 9 9 9 1 7 2 5 2 2   1 2 5 2 2   7 2 5 31 7 3 8 8 1 8 2 8 2 8 3 9 9 1 9 1 9 2 7 3 5 2 3 1 1 8 2 6 2 4 1 2 9 3 7 2 32 8 8 1 8 2 8 3 9 9 9 1 9 2 9 3 10 8 6 4 2   8 6 4 2   33 8 1 8 2 8 3 9 9 1 9 1 9 2 9 3 10 10 1 8 1 6 2 4 3 3 1 1 9 2 7 3 6 4 1 2 2 34 8 2 8 3 9 9 1 9 2 9 3 9 3 10 10 1 10 2 3 2 7 5 2 4 2 2 1 9 2 8 6 2 5 35 8 3 9 9 1 9 2 9 3 10 10 1 10 2 10 2 10 3 8 3 7 2 6 1 5 3 3 2 2 1 1   8 3 7 3 36 9 9 1 9 2 9 3 10 10 1 10 2 10 3 11 11 1 9 8 7 6 5 4 3 2 1   37 9 1 9 2 9 3 10 10 1 10 2 10 3 11 11 1 11 2 9 1 8 2 7 3 7 6 1 5 2 4 3 4 3 1 2 2 38 9 2 9 3 10 10 1 10 2 10 3 11 11 1 11 2 11 3 9 2 9 8 2 8 7 2 7 6 2 6 5 2 5 39 9 3 10 10 1 10 2 10 3 11 11 1 11 2 11 3 12 9 3 2 9 1 9 8 3 8 2 8 1 8 7 2 7 1 40 10 1 10 2 10 11 11 1 11 2 11 3 12 12 1 12 2                      
This left side proceeds in the 11 Table. In Table 12.  

Table 10.  
This row is continued in the 15. Table.    51 52 53 54 55 56 57 58 59 60 21 6 2 6 3 6 3 7 7 7 1 7 1 7 2 7 2 7 3 7 3 3 8 1 3 2 8 3 4 9 1 4 2 9 3 5 22 7 7 7 1 7 1 7 2 7 2 7 3 7 3 8 8 1 2 6 1 2 7 2 8 3 2 9 4 2   23 7 1 7 1 7 2 7 3 7 3 8 8 8 1 8 1 8 2 3 1 9 4 3 2 6 1 2 7 3 3 2 9 1 5 24 7 2 7 3 7 3 8 8 1 8 1 8 2 8 2 8 3 9 6 2 8 4   6 2 8 4   25 7 3 8 8 1 8 1 8 2 8 3 8 3 9 9 9 1 8 3 5 1 1 7 2 3 3   6 1 1 2 8 3 5 26 8 1 8 1 8 2 8 3 8 3 9 9 1 9 1 9 2 9 3 1 2 8 4 2 1 7 2 4 2 7 3 2   27 8 2 8 3 8 3 9 9 1 9 1 9 2 9 3 9 3 10 4 1 1 3 3 4 2 1 1 8 4 3 1 2 8 1 5 28 8 3 9 9 1 9 1 9 2 9 3 9 3 10 10 1 10 2 7 1 1 8 5 2 9 9 3   29 9 9 1 9 2 9 3 9 3 10 10 1 10 2 10 2 10 3 9 3 7 4 1 1 2 8 3 6 3 1 2 7 3 5 30 9 2 9 3 9 3 10 10 1 10 2 10 2 10 3 11 11 1 2 2   7 2 5 2 2   7 2 5 2 ● 31 9 3 10 10 1 10 1 10 2 10 3 11 11 11 1 11 2 5 1 3 3 8 2 6 1 4 1 9 2 7 1 5 32 10 10 1 10 2 10 3 11 11 11 1 11 2 11 3 12 8 6 4 2   8 6 4 2   33 10 2 10 2 10 3 11 11 1 11 2 11 3 11 3 12 12 1 3 9 7 1 5 2 3 3 2 1 8 2 6 3 5 34 10 3 11 11 1 11 1 11 2 11 3 12 12 1 12 2 12 3 3 2 2 2 9 8 2 6 4 2 3 1 2 35 11 11 1 11 2 11 3 12 12 1 12 1 12 2 12 3 13 6 1 5 3 3 2 2 1 1   8 ● 7 2 6 1 5 36 11 1 11 2 11 3 12 12 ● 12 2 12 ● 13 13 1 13 2 9 8 7 6 5 4 3 2 1   37 11 3 12 12 1 12 1 12 2 12 3 13 13 1 13 2 13 8 1 3 ● 1 9 2 8 3 8 7 1 6 2 5 ● 5 38 12 12 1 12 2 12 3 13 13 1 13 2 13 3 14 14 1 4 2 4 3 2 3 2 2 2 1 2 1 2   39 12 1 12 2 12 3 13 13 1 13 2 13 3 14 14 1 14 2 7 1 7 6 3 6 2 6 1 6 3 3 5 2 5 1 5 40 12 3 13 13 1 13 2 13 3 14 14 1 14 2 14 3 15                      
This left side proceeds in the 12 Table. In Table 17.  

Table 11.  
This row is continued in the 12 Table.    41 42 43 44 45 46 47 48 49 50 41 10 2 10 3 11 11 1 11 2 11 3 12 12 1 12 2 12 3 1 2 3 1 1 1 1 2 1 3 2 2 1 2 2   42 11 11 1 11 2 11 3 12 12 1 12 2 12 3 13 1 1 2 2 2 3 3 3 2 4 4 2 5   43 11 2 11 3 12 12 1 12 2 12 2 12 3 13 1 2 1 3 2 3 4 2 4 1 5 1 6 7 2   44 12 12 1 12 2 12 3 13 13 1 13 2 4 5 6 7 8 9     45 12 3 12 3 13 13 2 13 5 14 6 1 7 2 8 3   1 1 2 2   46 13 13 2 13 3 14 14 1 9 2 2 3 2 5   47 13 3 14 14 1 14 2 2 1 4 5 2 7 2   48 14 1 14 2 15 6 8     49 15 15 1 1 2 2   50 15 2 5  
This left side proceeds in the 12 Table. in Table 12.  

Table 12.  
This row is continued in the 17 Table.    51 52 53 54 55 56 57 58 59 60 41 13 13 1 13 2 13 3 14 14 1 14 2 14 3 15 15 1 2 3 3 3 1 3 2 3 3 4 4 1 4 2 4 3 5 42 13 1 13 2 13 3 14 14 1 14 2 14 3 15 15 1 15 3 5 2 6 6 2 7 7 2 8 8 2 9 9 2   43 13 2 13 3 14 14 2 14 3 15 15 1 15 2 15 3 16 ● 1 9 9 3 2 1 1 2 2 3 3 2 4 1 5 44 14 14 1 14 2 14 3 15 15 1 15 2 15 3 16 ● 16 2 1 2 3 4 5 6 7 8 9   45 14 1 14 2 14 3 15 15 1 15 3 16 16 1 16 2 16 3 3 3 5 6 1 7 2 8 3   1 1 2 2 1 3 5 46 14 2 14 3 15 15 2 15 3 16 16 1 16 2 16 3 17 1 6 2 8 9 2 1 2 2 14 5 2 7 8 2   47 14 3 15 1 15 2 15 3 16 16 1 16 2 17 17 1 17 2 4 1 1 2 3 4 2 6 1 8 9 3 1 2 3 1 5 48 15 1 15 2 15 3 16 16 2 16 3 17 17 1 17 2 18 2 4 6 8   2 4 6 8   49 15 2 15 3 16 1 16 2 16 3 17 17 1 17 3 18 18 1 4 3 7 9 1 1 2 3 3 6 8 1 2 2 3 5 50 15 3 16 1 16 2 16 3 17 17 2 17 3 18 18 1 18 3 7 2   2 2 5 7 2   2 2 5 7 2   51 16 1 16 2 16 3 17 17 2 17 3 18 18 1 18 3 19 1 3 5 3 8 2 1 1 4 6 3 9 2 2 1 5   52 16 3 17 17 2 17 3 18 18 2 18 3 19 19 2 6 9 2 5 8 1 4 7     53 17 2 17 3 18 18 2 18 3 19 19 2 19 3 2 1 5 2 8 3 2 5 1 8 2 1 3 5   54 18 18 2 18 3 19 19 2 19 3 20 1 9 2 2 6 4 2 9 6 2     55 18 3 19 1 19 2 19 2 20 1 20 1 6 1   3 3 7 2 1 1     56 15 2 19 3 20 1 20 2 2 1 4 8 2 6     57 20 1 20 2 21 21 1 2 1 6 1 5   58 21 21 21 3 1 5 2     59 21 3 22 5 1 5   60 22 2  
This left side proceeds in the 19 Table. In Table 19.  

Table 13.  
This row is continued in the 14. Table.    61 62 63 64 65 66 67 68 69 70 1 1 1 1 1 1 1 1 1 1 1 5 1 5 2 5 3 6 6 1 6 2 6 3 7 7 1 7 2 2 3 3 3 3 3 3 3 3 3 3 2 1 1 2 2 2 2 3 3 2 4 4 2 5 3 1 1 1 1 1 1 1 1 1 1 5 3 6 2 7 1 8 8 3 9 2 1 1 1 3 2 2 4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 3 1 2 3 4 5 6 7 8 9   5 1 3 1 3 1 3 2 2 2 2 2 2 2 6 1 7 2 8 3   1 1 2 2 3 3 5 6 1 7 2 6 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 1 2 3 4 2 6 7 2 2 2 2 3 2 5 7 2 2 2 2 2 3 2 3 2 3 2 3 2 3 2 3 3 3 6 3 8 2 1 2 3 3 5 2 7 1 9 3 2 2 8 3 3 3 3 3 1 3 1 3 1 3 1 3 1 3 2 2 4 6 8   2 4 6 8   9 3 1 3 1 3 2 3 2 3 2 3 2 3 3 3 3 3 3 3 3 7 1 9 2 1 3 4 6 1 8 2 3 3 5 1 7 2 10 3 3 3 3 3 3 4 4 4 4 4 1 4 1 4 1 2 2 5 7 2   2 2 5 7 2   2 2 5 11 4 4 1 4 1 4 1 4 1 4 2 4 2 4 2 4 2 4 3 7 3 2 3 1 6 8 3 1 2 4 1 7 9 3 2 2 12 4 2 4 2 4 2 4 3 4 3 4 3 5 5 5 5 1 3 6 9 2 5 8 1 4 7   13 4 3 5 5 5 5 1 5 1 5 1 5 2 5 2 5 2 8 1 1 2 4 2 8 1 1 4 2 7 3 1 4 1 7 2 14 5 1 5 1 5 2 5 2 5 2 5 3 5 3 5 3 6 6 3 2 7 2 4 7 2 1 4 2 8 1 2 5 15 5 2 5 3 5 3 6 6 6 6 1 6 1 6 1 6 2 8 3 2 2 6 1   3 3 7 2 1 1 5 3 3 2 2 16 6 6 6 1 6 1 6 2 6 2 6 2 6 3 6 3 7 4 8 2 6   4 8 2 6   17 6 1 6 2 6 2 6 3 6 3 7 7 7 7 1 7 1 9 1 3 2 7 3 2 6 1 2 4 3 9 3 1 7 2 18 6 3 6 3 7 3 7 7 1 7 1 7 2 2 2 7 3 7 3 4 2 9 2 8 2 2 7 1 2 6 2 5 19 7 7 1 7 1 7 2 7 2 7 3 7 3 8 8 8 1 ● 3 4 2 9 1 4 8 3 3 2 8 1 3 7 3 2 2 20 7 2 7 3 7 3 8 8 8 1 8 1 8 2 8 2 8 3 5   5   5   5   5    
This left side proceeds in the 15 Table. In Table 6.  

Table 14.  
This row is continued no longer.    71 72 73 74 75 76 77 78 79 80 1 1 1 1 1 1 1 1 1 1 2 7 3 8 8 1 8 2 8 3 9 9 1 9 2 9 3   2 3 3 3 3 3 3 3 3 3 1 5 2 6 6 2 7 7 2 8 8 2 9 9 2   3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 4 4 3 5 2 6 1 7 7 3 8 2 9 1   4 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 2 1 2 3 4 5 6 7 8 9   5 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 8 3   1 1 2 2 3 3 5 6 1 7 2 8 3   6 2 2 2 2 2 2 2 3 2 3 2 3 2 3 2 3 2 3 3 6 2 8 9 2 1 2 2 4 5 2 ● 8 3   7 3 3 3 3 3 1 3 1 3 1 3 1 3 1 3 2 4 1 6 7 3 9 2 1 1 3 4 3 6 2 8 1   8 3 2 3 2 3 2 3 2 3 3 3 3 3 3 3 3 3 3 4 2 4 6 8   2 4 6 8   9 3 3 4 4 4 4 4 1 4 1 4 1 4 1 4 2 9 3 2 4 1 6 2 8 3   3 1 5 2 7 3   10 4 1 4 2 4 4 2 4 2 4 2 4 3 4 3 4 3 5 7 2   2 2 2 5 7 2   2 2 5 2   11 4 3 4 3 5 5 5 5 5 1 5 1 5 1 5 2 5 3 8 2 3 3 2 6 1 9 1 3 4 2 7 1   12 5 1 5 1 5 1 5 2 5 2 5 2 5 3 5 3 5 3 6 3 6 9 1 2 5 8 1 4 7   13 5 3 5 3 5 3 6 6 6 6 1 6 1 6 1 6 2 3 4 7 1 2 3 3 7 1 3 2 6 3   14 6 6 1 6 1 6 1 6 2 6 2 6 2 6 3 6 3 7 8 2 2 5 2 9 2 2 6 9 2 3 6 2   15 6 2 6 3 6 3 6 3 7 7 7 7 1 7 1 7 2 6 1   3 3 7 2 1 1 5 8 3 2 2 6 1   16 7 7 7 1 7 1 7 2 7 2 7 2 7 3 7 3 8 4 8 2 6   4 8 2 6   17 7 2 7 2 7 3 7 3 7 3 8 8 8 1 8 1 8 2 1 3 6 1 4 2 8 3 3 7 1 1 2 5 3   18 7 3   8 8 1 8 1 8 2 8 2 8 3 8 3 9 9 2 4 8 2 3 7 2 2 6 2 1 5 2   19 8 1 8 2 8 2 8 3 8 3 9 9 9 1 9 1 9 2 7 1 2 6 3 1 2 6 1 1 5 3 2 5 1   20 8 3 9 9 9 1 9 1 9 2 9 2 9 3 9 3 10  
This left side proceeds in the 16 Table.  

Table 15.  
This row is continued in the 6. Table.    61 62 63 64 65 66 67 68 69 70 21 8 8 8 1 8 1 8 2 8 2 8 3 8 3 9 9 1 5 2 3 6 1 1 6 2 1 3 7 2 1 7 2 22 8 1 8 2 8 2 8 3 8 3 9 9 9 1 9 1 9 2 5 2 1 6 2 2 7 2 3 8 2 4 9 2 5 23 8 3 8 3 9 9 9 1 9 1 9 2 9 3 9 3 10 3 6 2 2 1 8 3 3 9 2 5 1 1 6 3 2 2 24 9 9 1 9 1 9 2 9 3 9 3 10 10 10 1 10 2 6 2 8 4   6 2 8 4   25 9 2 9 2 9 3 10 10 10 1 10 1 10 2 10 3 10 3 1 2 7 2 3 3   6 1 2 2 8 3 5 1 1 7 2 26 9 3 10 10 10 1 10 2 10 2 10 3 11 11 11 1 6 2 3 9 2 6 2 2 9 5 2 2 8 2 5 27 10 1 10 1 10 2 10 3 10 3 11 11 1 11 1 11 2 11 3 1 3 8 2 5 1 2 8 3 5 2 2 1 9 5 3 2 2 28 10 2 10 3 11 11 11 1 11 2 11 2 11 3 12 12 1 7 4 1 8 5 2 9 6 3   29 11 11 11 1 11 2 11 3 11 3 12 12 1 12 2 12 2 2 1 9 2 6 3 4 1 1 8 2 5 3 3 1 7 2 30 11 1 11 2 11 3 12 12 12 1 12 2 12 3 12 3 13 7 2 5 2 2   7 2 5 2 2   7 2 5 31 11 3 12 12 12 1 12 2 12 3 12 3 13 13 1 13 2 2 3 2 8 1 6 3 3 1 2 9 1 7 4 3 2 2 32 12 12 1 12 2 12 3 13 13 13 1 13 2 13 3 14 8 6 4 2   8 6 4 2   33 12 2 12 3 12 3 13 13 1 13 2 13 3 14 14 14 1 3 1 1 2 9 3 8 6 1 4 2 2 3 1 9 1 7 8 34 12 3 13 13 1 13 2 13 3 14 14 14 1 14 2 14 3 3 2 7 5 2 4 2 2 1 9 2 8 6 2 5 35 13 1 13 2 13 3 14 14 14 1 14 2 14 3 15 15 1 7 3 2 2 1 1   8 3 7 2 6 1 5 3 3 2 2 36 13 2 13 3 14 14 1 14 2 14 3 15 15 1 15 2 15 3 9 8 7 6 5 4 3 2 1   37 14 14 1 14 2 14 3 15 15 1 15 1 15 2 15 3 16 4 1 3 2 2 3 2 1 1 2 9 3 9 8 1 7 1 38 14 14 2 14 3 15 15 1 15 2 15 3 16 16 1 16 2 9 2 9 8 2 8 7 2 7 6 2 6 5 2 5 39 14 3 15 15 1 15 2 15 3 16 16 1 16 2 16 3 17 4 3 4 2 4 1 4 3 3 3 2 3 1 3 2 3 2 2 40 15 1 15 2 15 3 16 16 1 16 2 16 3 17 17 1 17 2  
This Table proceeds in the 17 Table. In Table 18.  
This row is continued no further.  

Table 16.    71 72 73 74 75 76 77 78 79 80 21 9 1 9 1 9 2 9 2 9 3 9 3 10 10 10 1 10 2 2 3 3 3 3 8 2 3 3 9 4 1 9 2 4 3   22 9 3 9 3 10 10 10 1 10 2 10 2 10 2 10 3 11 2 6 1 2 7 2 2 8 3 2 9 4 2   23 10 10 1 10 1 10 2 10 3 10 3 11 11 11 1 11 8 1 4 9 3 8 2 ● 2 7 2 3 8 2 4 1   24 10 2 10 3 10 3 11 11 1 11 1 11 2 11 2 11 3 12 6 2 8 4   6 2 3 4   25 11 11 1 11 1 11 2 11 2 11 3 12 12 12 1 12 2 3 3   6 1 2 2 8 3 5 1 1 2 2 3 8   26 11 2 11 2 11 3 12 12 12 1 12 2 12 2 12 3 13 1 2 8 4 2 1 7 2 4 2 7 3 2   27 11 3 12 12 1 12 1 12 2 12 3 12 3 13 13 1 13 2 9 1 6 2 3 9 2 6 1 3 9 3 6 2 3 1   28 12 1 12 2 12 3 12 3 13 13 1 13 1 13 2 13 3 14   4 1 8 5 2 9 6 3   29 12 3 13 13 13 1 13 2 13 3 13 3 14 14 1 14 2 4 3 2 9 1 6 2 3 3 1 8 1 5 2 2 3   30 13 1 13 2 13 2 13 3 14 14 1 14 1 14 2 14 3 15 2 2   7 2 5 2 2   7 2 5 2 2   31 13 3 13 3 14 14 1 14 2 14 2 14 3 15 15 1 15 2 1 8 5 3 3 2 1 9 ● 3 4 2 2 1   32 14 14 1 14 2 14 3 15 15 15 1 15 2 15 3 16 8 6 4 2   8 6 4 2   33 14 2 14 3 5 15 1 15 1 15 2 15 3 16 16 1 16 2 ● 3 4 2 1 2 8 3 7 5 2 2 2 1 3   34 15 15 1 15 2 15 2 15 3 16 16 1 16 2 16 3 17 1 2 2 2 9 7 2 5 4 2 2 3 1 2   35 15 2 15 3 15 3 16 16 1 16 2 16 3 17 17 1 17 1 1 1   8 2 7 2 6 1 5 3 3 2 2 1 1   36 15 3 1● 16 1 16 2 16 3 17 17 1 17 2 17 3 18 9 8 7 6 5 4 3 2 1   37 16 1 16 2 16 3 17 17 1 17 2 17 3 18 18 1 18 2 6 3 6 5 1 4 2 3 3 3 2 1 1 2 ● 3   38 16 3 17 17 1 17 2 17 3 18 18 1 18 2 18 3 19 ● 2 1 3 2 3 2 2 2 1 2 1 ● 2   39 17 1 17 2 17 3 18 18 1 18 2 18 3 19 19 1 19 2 2 1 2 1 3 1 2 1 1 1 2 ● 2 2 ● 1   40 17 3 18 18 1 18 2 18 3 19 19 1 19 2 19 3 20  
This Table proceeds in the 18 Table.  
No contin●nance of both sides and further.  

Table 17.  
This row is continued in the 18. Table.    61 62 63 64 65 66 67 68 69 70 41 15 2 15 3 16 16 1 16 2 16 3 17 17 1 17 2 17 3 5 1 5 2 5 3 6 6 1 6 2 6 3 7 7 1 7 2 42 16 16 1 16 2 16 3 17 17 1 17 2 17 3 18 18 1 2 1 1 2 2 2 2 3 3 2 4 4 2 5 43 16 1 16 2 16 3 17 17 1 17 2 18 18 1 18 2 18 3 5 3 6 2 7 1 8 3 3 9 2 1 1 1 3 2 2 44 16 3 17 17 2 17 2 17 3 18 18 1 18 2 18 3 19 1 1 2 3 4 5 6 7 8 9   45 17 17 1 17 2 18 18 1 18 2 18 3 19 19 1 19 2 6 1 7 2 8 3   1 1 2 2 3 3 5 6 1 7 2 46 17 2 17 3 18 18 1 18 2 18 3 19 1 19 2 19 3 20 1 2 3 4 2 6 7 2 9 2 2 3 2 5 47 17 3 18 18 2 18 3 19 19 1 19 2 19 3 20 1 20 2 6 3 8 2 1 2 3 3 5 2 7 1 9 3 2 2 48 18 1 18 2 18 3 19 19 2 19 3 20 20 1 20 2 21 2 4 6 8   2 4 6 8   49 18 2 19 19 1 19 2 19 3 20 20 2 20 3 21 21 1 7 1 9 3 1 3 4 6 1 8 2 3 3 5 7 2 50 19 19 1 19 2 20 20 1 20 2 20 3 21 1 21 2 21 3 2 2 5 7 2   2 2 5 7 2   2 2 5 51 19 1 19 3 20 20 1 20 2 21 21 1 21 2 21 3 22 1 7 3 2 3 1 6 8 3 1 2 4 1 7 ● 3 2 2 52 19 3 20 20 1 20 3 21 21 1 21 3 22 22 1 22 3 3 6 9 2 5 8 1 4 7   53 20 20 2 20 3 21 21 2 21 3 22 22 2 22 3 23 3 1 1 2 4 3 8 1 1 4 2 7 3 1 4 1 7 2 54 20 2 20 3 21 1 21 2 21 3 22 1 22 2 22 3 23 1 23 2 3 2 7 2 4 7 2 1 4 2 8 4 2 5 55 20 3 21 1 21 2 22 22 1 22 2 23 23 1 23 2 24 8 3 2 2 6 1   3 3 7 3 1 1 5 8 3 2 2 56 21 1 21 2 22 22 1 22 3 23 23 1 23 2 24 24 2 4 8 2 6   4 8 2 6   57 21 2 22 22 1 22 3 23 23 2 23 3 24 8 24 2 24 3 9 1 3 2 7 3 2 6 1 2 4 3 9 3 1 7 2 58 22 22 1 22 3 23 23 2 23 3 24 1 24 2 25 25 1 4 2 9 3 2 8 2 2 7 1 2 6 2 5 59 22 1 22 3 23 23 1 23 3 24 1 24 2 25 25 1 25 3 9 3 4 2 9 1 4 8 3 3 2 8 1 3 7 q 2 2 60 22 3 23 1 23 2 24 24 1 24 3 25 25 2 25 3 26 1 5   5   5   5   5    
This Table proceeds in the 20 Table. In Table ult.  

Table 18.  
No further Continuance.    71 72 73 74 75 76 77 78 79 80 41 18 18 1 18 2 18 3 19 19 1 19 2 19 3 20 20 2 7 3 8 8 1 8 2 8 3 9 9 1 9 2 9 3   42 18 2 18 3 19 19 1 19 2 19 3 20 20 1 20 2 21 5 2 6 6 2 7 7 2 8 8 2 9 9 2   43 19 19 1 19 2 19 3 20 20 1 20 2 20 3 21 21 2 3 1 4 4 3 5 2 6 1 7 7 3 8 2 9 9 1 44 19 2 19 3 20 20 1 20 2 20 3 21 21 1 21 2 22 1 2 3 4 5 6 7 8 9   45 19 3 20 1 20 2 20 3 21 21 1 21 2 21 3 22 22 2 8 3   1 1 2 2 3 3 5 6 1 7 2 8 3   46 20 1 20 2 20 3 21 1 21 2 22 3 22 22 1 22 2 23 6 2 8 9 2 1 2 2 4 5 2 7 8 2   47 20 3 21 21 1 21 2 22 22 1 22 2 22 3 23 23 2 4 2 6 7 3 9 2 1 1 3 4 3 6 2 8 1   48 21 1 21 2 21 3 22 22 2 22 3 23 23 1 23 2 24 2 4 6 8   2 4 6 8   49 21 2 22 22 1 22 2 22 3 23 1 23 2 23 3 24 24 2 9 3 2 4 1 6 2 8 3 1 3 1 5 2 7 3   50 22 22 2 22 3 23 23 1 23 3 24 24 1 24 2 25 7 2   2 2 5 7 2   2 2 5 7 2   51 22 2 22 3 23 1 23 2 23 3 24 24 2 24 3 25 25 2 5 1 8 3 3 2 6 1 9 1 3 4 2 7 1   52 23 23 1 23 2 24 24 1 24 1 25 25 1 25 2 26 3 6 9 2 5 8 1 4 7   53 23 2 23 3 24 24 2 24 3 25 25 2 25 3 26 26 2 3 4 7 1 2 3 3 7 1 3 2 6 3   54 23 3 24 1 24 2 24 3 25 1 25 2 25 3 26 1 26 2 27 8 2 2 5 2 9 2 2 6 9 2 3 6 2   55 24 1 24 3 25 25 1 25 3 26 26 1 26 3 27 27 2 6 1   3 3 7 2 1 1 5 8 2 3 2 6 1   56 24 3 25 25 2 25 3 26 1 26 2 26 3 27 1 27 2 28 4 8 2 6   4 8 2 6   57 15 1 25 2 26 26 1 26 2 27 27 1 27 3 28 28 2 1 3 6 1 4 2 9 3 3 7 1 1 2 5 2   58 25 2 26 26 1 26 3 27 27 2 27 3 28 1 28 2 29 9 2 4 8 2 3 7 2 2 6 2 1 9 2   59 26 26 2 26 3 27 1 27 2 28 28 1 28 3 29 29 2 7 1 2 6 3 1 2 6 1 1 5 2 2 5 1   60 26 2 27 27 1 27 3 28 28 2 28 3 29 1 29 2 30 5   5   5   5   5    
This left side proceeds in the 20 Table. No further continuance of both sides.  

Table 19.  
This row is continued in the 20 Table.    61 62 63 64 65 66 67 68 69 70 61 23 1 23 2 24 24 1 24 3 25 25 2 25 3 26 1 26 2 1 5 2 3 6 1 1 6 2 1 3 7 2 1 7 2   62 24 24 1 24 3 25 25 2 25 3 26 1 26 2 27 1 6 2 2 7 2 3 8 2 4 9 2 5   63 24 3 25 25 2 25 3 26 1 26 3 27 27 2 2 1 8 3 3 9 2 5 1 1 6 3 2 2   64 25 2 26 26 1 26 3 27 27 2 28 4   6 2 8 4     65 26 1 26 3 27 27 2 28 28 1 6 1 2 2 8 3 5 1 1 7 2   66 27 27 2 28 28 1 28 3 9 5 2 2 8 2 5   67 28 28 1 28 3 29 1 2 1 9 5 3 2 2   68 28 3 9 1 29 3 6 3     69 29 3 30 1 7 2   70 30 2 5  
This left side proceeds in the 20 Table. in Table vle.  

Table 20.    71 72 73 74 75 76 77 78 79 80 61 17 27 1 27 3 28 28 2 28 3 29 1 29 2 30 30 2 2 3 8 3 1 8 2 3 3 9 4 1 9 2 4 3   62 27 1 27 3 28 1 28 2 29 29 1 29 3 30 32 2 32 2 6 1 7 2 2 8 8 2 9 4 2   63 27 3 28 1 28 2 29 29 2 29 3 30 1 30 2 31 31 2 8 1 4 4 3 5 2 1 1 7 1 3 8 2 4 1   64 28 1 28 3 19 29 2 30 30 1 30 3 31 31 2 32 6 2 8 4   6 2 3 4   65 28 3 29 1 29 2 30 30 1 30 3 31 1 31 2 32 32 2 3 3   6 1 2 2 8 3 5 1 1 7 2 3 2   66 29 1 29 2 30 30 2 30 3 31 31 3 32 32 2 33 1 2 8 4 2 1 7 2 4 2 7 3 2   67 29 2 30 30 2 30 3 31 1 31 3 32 32 2 33 33 2 9 1 6 2 3 4 2 7 1 9 3 6 2 3 3   68 30 30 ● 31 31 1 31 3 32 1 32 2 33 33 2 34 7 4 1 8 5 2 9 6 2   69 30 2 31 30 1 31 2 32 1 32 3 33 33 2 34 34 2 4 4 2 9 1 6 2 3 3 1 8 1 5 2 2 3   70 31 31 2 31 3 32 1 32 3 33 1 33 3 34 34 2 35 2 2   7 2 5 2 2   7 2 5 2 2   71 31 2 31 3 32 1 32 3 33 1 33 2 34 34 2 35 35 2 1 8 5 3 3 2 1 1 9 6 3 4 2 2 1     72 32 1 32 3 33 1 33 3 34 34 2 35 35 2 36 6 4 2   8 6 4 2     73 33 1 33 3 34 34 2 35 35 2 36 36 2 2 1 2 8 3 7 2 5 1 3 2 ● 3     74 34 34 2 35 35 2 36 36 37 8 7 2 6 4 2 3 1 2     75 35 35 2 36 36 2 37 37 2 6 1 5 3 3 2 1 1 1     76 36 36 2 37 37 2 38 4 3 2 1     77 37 37 2 38 38 2 2 1 1 2 3     78 38 38 2 39 1 1     79 39 39 2 1     80 40    
The Table ends here.   

CHAP. XXVIII.  
How to work when there be parts of perches adjoined to  

A perfect Table of the Fractions, or parts of Integer perches: newly calculated, with easy denominations for the exact measuring of all Land, according to the Statute made by Ed. 1. Anno. 33. entitled Compositio vlnarum& perdicarum.  Quarters of perches. half perches Three quarters of the perch.   Quarters of perches. half perches Three quarters of the perch. perches. Day-worke. perch. yard. foot. Inch. Day-worke. perch. yard. foot. Inch. Day-worke. perch. yard. foot Inch.   perches. Day-worke. perch. yard. foot. Inch. Day-worke. perch. yard. foot. Inch. Day-worke. perch. yard. foot Inch. Here the denominations change to those in the foot of the table. 1 0 0 1 1 1. 0 0 2 2 3 0 0 4 0 4:   41 2 2 1 1 1: 5   2 2 3 7 2 4 0 4: 2 0 0 2 2 3 0 1 0 0 0 0 1 2 2 3   42 2 2 2 2 3 5 1       7 3 2 2 3 3 0 0 4 0 4:   1 2 2 3   2 1 1 1:   43 2 2 4   4: 5 1 2 2 3 8   1 1 1: 4 0 1 0 0 0 0 2 0 0 0   3 0 0 0   44 2 3       5 2       8 1       5 0 1 1 1 1.   2 2 2 3 3 3 4   4.   45 2 3 1 1 1: 5 2 2 2 3 8 1 4   4: 6 0 1 2 2 3   3 0 0 0   0 2 2 3   46 2 3 2 2 3 5 3       8 2 2 2 3 7 0 1 4 0 4:   3 2 2 3 3 1 ● 1 1:   47 2 3 4   4: 5 3 2 2 3 8 3 1 1 1: 8 0 2 0 0 0 1 0 0 0 0 0 2 0 0 0   48 3         6         9         9 0 2 1 1 1. 1 0 2 2 3 1 2 4 0 4   49 3   1 1 1: 6 0 2 2 3 9   4   4: 10 0 2 2 2 3 1 1       1 3 2 2 4   50 3   2 2 3 6 1       9 1 2 2 3 11   2 4 0 4: 1 1 2 2 3 2 0 1 1 1   51 3   4   4: 6 1 2 2 3 9 2 1 1 1: 12   3 0 0 0 1 2 0 0 0 2 1 0 0 0   52 3 1       6 2       9 3       13   3 1 1 1: 1 2 2 2 3 2 1 4 0 4:   53 3 1 1 1 1: 6 2 2 2 3 9 3 4 0 4: 14   3 2 2 3 1 3 0 0 0 2 2 2 2 3   54 3 1 2 2 3 6 3       1   0 2 2 3 15   3 4 0 4: 1 3 2 2 3 2 3 1 1 1:   55 3 1 4 0 4: 6 3 2 2 3 1   1 1 1 1: 16 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0   56 3 2       7         1   2       17 1 0 1 1 1. 2 0 2 2 3 3 0 4 0 4:   57 3 2 1 1 1: 7   2 2 3 1   2 4   4: 18 1   2 2 3 2 1 0 0 0 3 1 2 2 3   58 3 2 2 2 3 7 1       1   3 2 2 3 19 1   4   4: 2 1 2 2 3 3 2 1 1 1.   59 3 2 4   4: 7 1 2 2 3 1 1 0 1 1 1: 20 1 1 0 0 0 2 2 0 0 0 3 3 0 0 0   60 3 3       7 2       1 1 1       21 1 1 1 1 1: 2 2 2 2 3 3 3 4   4:   61 3 3 1 1 1: 7 2 2 2 3 1 1 1 4   4: 22 1 1 2 2 3 2 3 0     4   2 2 3   62 3 3 2 2 3 7 3       1 1 2 2 2 3 23 1 1 4   4 2 3 2 2 3 4 1 1 1 1:   63 3 3 4   4: 7 3 2 2 3 1 1 3       24 1 2       3         4 2 0 0     64 4         8         1 2         25 1 2 1 1 1 3 1 2 2 3 4 2 4 0 4:   55 4   1 1 1: 8   2 2 3 1 2   4   4 26 1 2 2 2 3 3 1       4 3 2 2 3   66 4   2 2 3 8 1       1 2 1 2 2 3 27 1 2 4   4: 3 2 2 2 3 5   1 1 1:   67 4 0 4   4: 8 1 2 2 3 1 2 2 1 1 1: 28 1 3       3 2       5 1         68 4 1       8 2       1 2 3       29 1 3 1 1 1: 3 2 12 2 3 5 1 4   4:   69 4 1 1 1 1: 8 2 2 2 3 1 2 3 4   4: 30 1 3 2 2 3 3 3       5 2 2 2 3   70 4 1 2 2 3 8 3       1 3   2 2 3 31 1 3 4   4: 3 3 2 2 3 5 3 1 1 1:   71 4 1 4   4: 8 3 2 2 3 1 3 1 1 1 1: 32 2         4         6           72 4 2       9 0       1 3 2       33 2   1 1 1: 4   12 2 3 6   4   4:   73 4 2 1 1 1: 9 0 2 2 3 1 3 2 4 0 4: 34 2   2 2 3 4 1       6 1 2 2 3   74 4 2 2 1 3 9 1       1 3 3 2 2 3 35 2   4   4: 4 1 2 2 3 6 2 1 1 1:   75 4 2 4 4: 9 1 2 2 3 1 4 0 1 1 1: 36 2 1       4 2       6 3         76 4 3     9 2       1 4 1       37 2 1 1 1 1: 4 2 2 2 3 6 3 4   4:   77 4 3 1 1 1: 9 2 2 2 3 1 4 1 4   4: 38 2 1 2 2 3 4 3       6 4 2 2 3   78 4 3 2 2 3 9 3       1 4 2 2 2 3 39 2 1 4   4: 4 3 2 2 3 7 1 1 1 1:   79 4 3 4 4: 9 3 2 2 3 rood. Day-worke perch. Yard. foot. Inch. 40 2 2 0 0 0 5 0 0 0 0 7 2         80 5 0 0 0 0 A just rood.  
under the title of Inch, one prick stands for a quarter,& two pricks for half an Inch.  
For partes wanting in 

length  
breadth   enter this Table with the 

breadth  
length   

of your piece  
of ground    
Parts wanting in breadth and length work first as it were the length. then as it were the leagth, then as it were the breadrn of the ground.& that as refulteth of both add together.  
3. Barley cornes i● an Inch. 12. Inches a foot. 3. foot a Yard. 5. Yards and a half a perch. 4. perch a day-worke. to day work a rood. 4 rood an Acker.  
Place this Table between folio 194. and 195.   

CHAP. XXIX.  
A triangle piece of ground given, in the one side whereof is a Well placed, this piece of ground is to be equally divided betwixt two men, in such sort that both parties may haue benefit thereof: my question is, how it shall be performed.  
LEt the piece of ground like a triangle be A B C: 
To divide a piece of ground from any point assigned.  let the Well be at D, then I divide that side C B into 2 equal partes at E, next I place my staff at the Well D, and take the angle C D A, keeping that angle vnatred I place again my staff at E, the midst of the side the Well standeth in, making the left leg as it was before ly parallel unto the side C B, then I note unto what place of the side A B the fidutiall edge of the right leg pointeth as to F. and there I make a mark: for a Hedge made from D to F, divides the triangle as it was proposed.   

CHAP. XXX.  
A Triangular piece of ground assigned, to cut off any number of Acres, roots, &c. as shall be required, and that by a Hedge drawn from any angle assigned.  
YOu shall first consider from what Angle the Hedge shall be drawn, 
To cut off any number of acres in any square piece of ground.  and then what side subtendeth the said angle, which side you shal measure, and then measure the whole Area of the triangle: then shall you consider how many Acres, Roodes. &c. you would cut off, which multiply by the length of the Base or Line subtending the former Angle, the Product whereof divide by the whole Area of the Triangle: the Quotient engendered thereof declares unto you how many perches you shall measure in the Base from any one of the Angles to cut off the number of Roodes desired.  
Example.  
LEt the 3 sides of the triangle be 30, 40, 50, Perches my desire is with a right line from the angle subtended by ●0 the greatest side to cut of one acre off ground: to perform this first I seek the Area of the triangle, as is taught before, which is 600 Roodes, now because I would cut off from the Figure one Acre, which is 160 Roodes, I multiply 160 by 50, the whole side subtending the said Angle; the product being. 8000, I divide by the Area of the triangle. 600, the Quotient is 13⅓, so many Perches I reckon in the Base from some of the corners, and then from the assigned Angle to the end of your measure, in the Base draw a line which cuts off one exact Acre.   

CHAP. XXXI.  
In a rectangled or oblongd Paralellogram to cut off any quan-assigned by a parallel line.  
ADmit you haue a parallelogram which contains. 65 acres, 16 perches,& that with a parallel to one of the ends, I would cut off 5● acres first therfore by some Rules before published, find the length of one of the longer parallel sides which let be 308 rood; thē I multiply 308 by 800, for so many roods are there in ● Acres the whole Area, which yieldeth. 246400, which divide by. 10416. the number of perches contained in the whole area, which yeeldeth in the Quotient 23. perches 11. foot, and so much must you measure from the end in both the longer parallel sides,& where that number ends, in each side make two marks direct from one of which unto the other draw a line which shal cut of just ● acres:& this rule is general for al quadrangle pieces of ground whose sides& ends be parallel be the figure a Rombus, an oblong &c. other ways you may find( which I omit) to separat other regulare figures,& the rather because there shal a Table of purpose ensue, and may be performed Instrumentally also.  
And here note when you haue any piece of ground how irregular soever or of what regular fashion soever, as circled, &c. you may reduce the same in to a square or such like figure as is* taught,& so speedily measure the same, or divide it into what kind or how many partes you will, as you be taught in the Cap. 30, 31, 32, 33, of this book.  

The Table whereby to separate or enclose one Statute acre of land by itself.  First side. The second side.   First side. The second side. perches. parts. feet inch.   perches. parts. feet inch. 1 160 0 0 0   25 6 ¼ 2 5: 2 80 0 0 0   26 6 0 2 6: 3 53 ¼ 1 4.   27 5 ¾ 2 10: 4 40 0 0 0   28 5 ½ 3 6: 5 32 0 0 0   29 5 ½ 0 3: 6 26 ½ 2 9   30 5 ¼ 1 4. 7 22 ¾ 1 9:   31 5 0 2 7: 8 20 0 0 0   32 5 0 0 0 9 17 ¾ 0 6.   33 4 ¾ 1 7. 10 16 0 0 0   34 4 ½ 3 4: 11 14 ½ 0 9   35 4 ½ 1 2: 12 13 ¼ 1 5.   36 4 ¼ 3 2: 13 12 ¼ 0 11:   37 4 ¼ 1 2: 14 11 ¼ 2 11:   38 4 0 3 5: 15 10 ½ 2 9   39 4 0 1 8: 16 10 0 0 0   40 4 0 0 0 17 9 ¼ 2 8:   41 3 ¼ 2 6: 18 8 ¼ 2 3.   42 3 ¾ 0 11: 19 8 ¼ 2 9:   43 3 ½ 3 7: 20 8 0 0 0   44 3 ½ 2 3. 21 7 ½ 1 11:   45 3 ½ 0 11 22 7 ¼ 0 4.   46 3 ¼ 3 9: 23 6 ¾ 3 4:   47 3 ¼ 2 6: 24 6 ½ 2 9   48 3 ¼ 1 4:  

The Table for one Acre.  First side. The second side.   First side. The second side. perches. parts. feet. inch.   perches. parts. feet. inch. 49 3 ¼ 0 3:   73 2 0 3 1 50 3 0 3 3:   74 2 0 2 5 51 3 0 2 3:   75 2 0 2 2 52 3 0 1 3:   76 2 0 1 5 53 3 0 0 3:   77 2 0 1 3 54 2 ¾ 3 6:   78 2 0 0 10 55 2 ¾ 2 7:   79 2 0 0 5 56 2 ¾ 1 9:   80 2 0 0 0 57 2 ¾ 0 9:   81 1 ¾ 3 8 58 2 ¾ 0 1:   82 1 ¾ 3 3 59 2 ½ 3 5:   83 1 ¾ 2 11 60 2 ½ 2 9   84 1 ¾ 2 6 61 2 ½ 2 0:   85 1 ¾ 2 2 62 2 ½ 1 3:   86 1 ¾ 1 9 63 2 ½ 0 7:   87 1 ¾ 1 5 64 2 ½ 0 0:   88 1 ¾ 1 1 65 2 ¼ 3 5:   89 1 ¾ 0 9 66 2 ¼ 2 10   90 1 ¾ 0 5 67 2 ¼ 2 3:   91 1 ¾ 0 1 68 2 ¼ 1 8:   92 1 ½ 3 11 69 2 ¼ 1 1:   93 1 ½ 3 7 70 2 ¼ 0 7:   94 1 ½ 3 4 71 2 ¼ 0 0   95 1 ½ 3 0 72 2 ¼ 3 8   96 1 ½ 2 9  

The Table for one Acre of ground.  First side. The second side.   First side. The second side. perches. parts. feet inch.   perches. parts. feet inch. 97 1 ½ 2 5   121 1 ¼ 1 2 98 1 ½ 2 2   122 1 ¼ 1 0 99 1 ½ 1 11   123 1 ¼ 0 10 100 1 ½ 1 7   124 1 ¼ 0 7 101 1 ½ 1 4   125 1 ¼ 0 5 102 1 ½ 1 1   126 1 ¼ 0 3 103 1 ½ 0 10   127 1 ¼ 0 1 104 1 ½ 0 4   128 1 ¼ 0 0 105 1 ½ 0 4   129 1 0 3 11 106 1 ½ 0 1   130 1 0 3 9 107 1 ¼ 4 1   131 1 0 3 7 108 1 ¼ 3 9   132 1 0 3 6 109 1 ¼ 3 7   133 1 0 3 4 110 1 ¼ 3 4   134 1 0 3 2 111 1 ¼ 3 2   135 1 0 3 0 112 1 ¼ 2 11   136 1 0 2 10 113 1 ¼ 2 8   137 1 0 2 9 114 1 ¼ 2 6   138 1 0 2 7 115 1 ¼ 2 3   139 1 0 2 5 116 1 ¼ 2 1   140 1 0 2 5 117 1 ¼ 1 11   141 1 0 2 2 118 1 ¼ 1 8   142 1 0 2 1 119 1 ¼ 1 6   143 1 0 1 11 120 1 ¼ 1 4   144 1 0 1 10  

One Acre.  first side. Second side. perches. parts. feet inch 145 1 0 1 8 146 1 0 1 6 147 1 0 1 5 148 1 0 1 4 149 1 0 1 2 150 1 0 1 1 151 1 0 0 11 152 1 0 0 10 153 1 0 0 9 154 1 0 0 7 155 1 0 0 6 156 1 0 0 5 157 1 0 0 3 158 1 0 0 2 159 1 0 0 1 160 1 0 0 0  
The end of the first Table calculated for one acre of ground.  

Two Acres.  first side. Second side. perches parts. feet inch. 1 320 0 0 0 2 160 0 0 0 3 106 ½ 2 9 4 80 0 0 0 5 64 0 0 0 6 53 ¼ 1 4 7 45 ½ 3 6 8 40 0 0 0 9 35 ½ 0 11 10 32 0 0 0 11 29 0 1 6 12 26 ½ 2 9 13 24 ½ 1 10 14 22 ¾ 1 9 15 21 ¼ 1 4 16 20 0 0 0 17 18 ¾ 1 2 18 17 ¾ 0 5 19 16 ¾ 1 6 20 16 0 0 0 21 15 0 3 11 22 14 ½ 0 9 23 13 ¾ 2 8 24 13 ¼ 1 4  

The Table for two Acres.  First side. The second side.   First side. The second side. perches. parts. feet. inch.   perches. parts. feet. inch. 25 12 ¾ 0 9   49 6 ½ 0 6 26 12 ¼ 0 11   50 6 ¼ 2 5 27 11 ¾ 1 8   51 6 ¾ 0 4 28 11 ¼ 2 11   52 6 0 2 6 29 11 0 0 6   53 6 0 0 7 30 10 ½ 2 9   54 5 ¾ 2 10 31 10 ¼ 1 2   55 5 ¾ 1 1 32 10 0 0 0   56 5 ½ 3 6 33 9 ½ 3 3   57 5 ½ 1 10 34 9 ¼ 2 8   58 5 ½ 0 3 35 9 0 2 4   59 5 ¼ 2 10 36 8 ¾ 2 3   60 5 ¼ 1 4 37 8 ½ 2 5   61 5 0 4 0 38 8 ¼ 2 0   62 5 0 2 7 39 8 0 3 4   63 5 0 1 3 40 8 0 0 0   64 5 0 0 0 41 7 ¾ 0 1   65 4 ¾ 2 10 42 7 ½ 1 11   66 4 ¾ 1 7 43 7 ¼ 3 1   67 4 ¾ 0 5 44 7 ¼ 0 4   68 4 ½ 3 4 45 7 0 1 10   69 4 ½ 2 3 46 6 ¾ 3 4   70 4 ½ 1 2 77 6 ¾ 0 11   71 4 ½ 0 1 78 6 ½ 2 9   72 4 ¼ 3 2  

The table for three acres of ground.  first side. Second side.   first side. Second side. perches parts feet Inch   perches parts feet Inch 73 4 ¾ 2 2   97 3 ¼ 0 9 74 4 ¼ 1 2   98 3 ¼ 0 3 75 4 ¼ 0 3   99 3 0 3 10 76 4 0 3 5   100 3 0 3 3 77 4 0 2 6   101 3 0 2 9 78 4 0 1 8   102 3 0 2 3 79 4 0 0 10   103 3 0 1 9 80 4 0 0 0   104 3 0 1 3 81 3 ¾ 3 3   105 3 0 0 0 82 3 ¾ 2 6   106 3 0 0 0 83 3 ¾ 1 8   107 2 ¾ 3 0 84 3 ¾ 0 11   108 2 ¾ 3 0 85 3 ¾ 0 2   109 2 ¾ 3 0 86 3 ½ 3 7   110 2 ¾ 2 0 87 3 ½ 2 11   111 2 ¾ 2 0 88 3 ½ 2 3   112 2 ¾ 1 0 89 3 ½ 2 6   113 2 ¾ 1 0 90 3 ½ 0 11   114 2 ¾ 0 0 91 3 ½ 0 3   115 2 ¾ 0 0 92 3 ¼ 3 9   116 2 ¾ 0 0 93 3 ¼ 3 1   117 2 ½ 3 0 94 3 ¼ 2 6   118 2 ½ 3 0 95 3 ¼ 1 11   119 2 ½ 3 0 96 3 ¼ 1 4   120 2 ½ 2 0  

A Table to lay out two Acres of ground.  first side. Second side.   first side. Second side. perches. parts. feet inch   perches. parts. feet inch 121 2 ½ 2 4   141 2 ¼ 0 3 122 2 ½ 2 0   142 2 ¼ 0 0 123 2 ½ 1 8   143 2 0 3 11 124 2 ½ 1 3   144 2 0 3 8 125 2 ½ 0 11   145 2 0 3 4 126 2 ½ 0 7   146 2 0 3 1 127 2 ½ 0 3   147 2 0 2 11 128 2 ½ 0 0   148 2 0 2 8 129 2 ¼ 3 9   149 2 0 2 5 130 2 ¼ 3 5   150 2 0 2 2 131 2 ¼ 3 2   151 2 0 1 11 132 2 ¼ 2 10   152 2 0 1 8 133 2 ¼ 2 6   153 2 0 1 6 184 2 ¼ 2 3   154 2 0 1 3 135 2 ¼ 1 11   155 2 0 1 0 136 2 ¼ 1 8   156 2 0 0 10 137 2 ¼ 1 4   157 2 0 0 7 138 2 ¼ 1 1   158 2 0 0 5 139 2 ¼ 0 10   159 2 0 0 0 140 2 ¼ 0 7   160 2 0 0 0  
The end of the second Table calculated for two acres of ground.  

The table for three acres of ground.  first side. Second side.   first side. Second side. perches parts feet Inch   perches parts feet Inch 1 280 0 0 0   25 19 0 3 3 2 240 0 0 0   26 18 ¼ 3 5 3 160 0 0 0   27 17 ¾ 0 5 4 120 0 0 0   28 17 0 2 4 5 96 0 0 0   29 16 ½ 0 10 6 80 0 0 0   30 16 0 0 0 7 68 ½ 1 0   31 15 ¼ 3 10 8 60 0 0 0   32 15 0 0 0 9 53 ¼ 1 1   33 14 ½ 0 9 10 48 0 0 0   34 14 0 1 11 11 43 ½ 2 0   35 13 ½ 3 6 12 40 0 0 0   36 13 ¼ 1 4 13 36 ¾ 2 1   37 12 ¾ 3 8 14 34 ¼ 0 1   38 12 ½ 2 2 15 32 0 0 0   39 12 ¼ 0 11 16 30 0 0 0   40 12 0 0 0 17 28 0 3 1   41 11 ½ 3 5 18 26 ½ 2 0   42 11 ¼ 2 11 19 25 ¼ 0 2   43 11 0 2 8 20 24 0 0 0   44 10 ¾ 4 7 21 22 ¾ 1 0   45 10 ½ 2 9 22 21 ¾ 1 0   46 10 ¼ 3 0 23 20 ¾ 1 2   47 10 0 3 6 24 20 0 0 0   48 10 0 0 0  

The Table to lay out three acres of ground.  first side. Second side.   first side. Second side. perches parts feet Inch   perches parts feet Inch 49 9 ¾ 0 9   72 6 ½ 2 9 50 9 ½ 1 7   73 6 ½ 1 2 51 9 ¼ 2 8   74 6 ¼ 3 10 52 9 0 3 9   75 6 ¼ 2 5 53 9 0 0 11   76 6 ¼ 1 1 54 8 ¾ 2 3   77 6 0 3 10 55 8 ½ 3 9   78 6 0 2 6 56 8 ½ 1 2   79 6 0 1 3 57 8 ¼ 2 9   80 6 0 0 0 58 8 ¼ 0 5   81 5 ¾ 2 10 59 8 0 2 2   82 5 ¾ 1 8 60 8 0 0 0   83 5 ¾ 0 6 61 7 ¾ 1 11   84 5 ½ 3 6 62 7 ½ 3 11   85 5 ½ 2 5 63 7 ½ 1 11   86 5 ½ 1 4 64 7 ½ 0 0   87 5 ½ 0 3 65 7 ¼ 2 2   88 5 ¼ 3 3 66 7 ¼ 0 4   89 5 ¼ 2 3 67 7 0 2 8   90 5 ¼ 1 3 68 7 0 0 11   91 5 ¼ 0 3 69 7 ¾ 3 4   92 5 0 3 0 70 6 ¾ 1 9   93 5 0 2 0 71 6 ¾ 0 2   94 5 0 1 0  

The table for three acres of ground.  first side. Second side.   first side. Second side. perches parts yard Inch   perches. parts yard Inch. 95 5 0 0 10   118 4 0 1 1 96 5 0 0 0   119 4 0 0 6 97 4 ¾ 3 3   120 4 0 0 0 98 4 ¾ 2 5   121 3 ¾ 3 6 99 4 ¾ 1 7   122 3 ¾ 3 0 100 4 ¾ 0 9   123 3 ¾ 2 6 101 4 ¾ 0 0   124 3 ¾ 1 11 102 4 ½ 3 4   125 3 ¾ 1 5 103 4 ½ 2 7   126 3 ¼ 0 11 104 4 ½ 1 10   127 3 ¾ 0 5 105 4 ½ 1 2   128 3 ¾ 0 0 106 4 ½ 0 5   129 3 ½ 3 7 107 4 ¼ 3 10   130 3 ½ 3 2 108 4 ¼ 3 2   131 3 ½ 2 8 109 4 ¼ 2 6   132 3 ½ 2 3 110 4 ¼ 1 10   133 3 ½ 1 9 111 4 ¼ 1 2   134 3 ½ 1 4 112 4 ¼ 0 7   135 3 ½ 0 11 113 4 0 4 1   136 3 ½ 0 5 114 4 0 3 5   137 3 ½ 0 0 115 4 0 3 10   138 3 ¼ 3 9 116 4 0 2 3   139 3 ¼ 3 4 117 4 0 2 8   140 3 ¼ 2 11  

A Table to lay out three Acres of ground.  first side Second side.   first side Second side. perches parts. feet inch   perches parts. feet inch. 141 3 ¼ 2 6   151 3 0 2 1 142 3 ¼ 2 1   152 3 0 2 7 143 8 ¼ 1 9   153 3 0 2 3 144 3 ¼ 1 4   154 3 0 1 11 145 3 ¼ 0 11   155 3 0 1 6 146 3 ¼ 0 7   156 3 0 1 3 147 3 ½ 0 3   157 3 0 0 11 148 3 ¼ 4 0   158 3 0 0 6 149 3 0 0 7   159 3 0 0 3 150 3 0 4 3   160 3 0 0 0  
The end of the third table for three acres of Statute measure.   

Chap. XXXII.  
The use of the former Tables.  
THE use of the three former Tables is, 
The use of the former Tables.  to lay out one acre of ground, two acres of ground, or three acres of ground, in any kind of parallelogram, which you shall thus perform.  
Consider how many perches you will haue the one side of the Acre to contain, and that measure out with your chain, and there make a mark: note then how many perches you measured in that side, which call the first side: Now, to lay out one, two or three acres of land, find the number of those perches you haue measured in the former Tables in the row of the left hand under the title of the first side: then look in the table right-mards under this title, the second side, where you shall see the l●gth of the second side of the acre, in perches, parts of perches, &c. This found, measure from one of the ends of the first side Orthogonaliter or Squyre-wise, the number of perches, &c. last found, and let that be the length of the second side, so haue you gotten two sides of the acre; then measure so many perches from the end of this last measured line(& that Squyrewise) as the length of the first side was: then from the end of that line measure so many perches as the length of the second side was, which you shal meat with the first side, so haue you finished.  
Example.  
I Haue allotted unto me 13 perches one way, for to make the one side of an acre which I am allowed: I would know how long the second side should be? repair unto the first table, and there in the first column vpon the left hand find 13 perthes, under the title of first side rightwards, answering to which under the title of second side, shall you find 12 perches& ¼ of a perch, 11 inches and better, and so long must the second side be; having these two sides, the two other are like thereunto, therefore proceed as you be taught before: The like order is to be observed in laying out of 2 or 3 acres: And if you would lay out. 4. acres, look in. 2. acres, and that you find in the second side double; or add that you find for one and 3 together: and if you would lay out 6 Acres, double that you find for the second side in three Acres: and if you would lay out 9 Acres, triple it. &c.  
Note in every Table under the title of inch, one prick standeth for half an inch,& ● 2. pricks for less then half an inch.  
Now( for a time) we will digress to show you other novelties, so shal you be more apt when we return to our former intendments of measuring ground,   

CHAP. XXXIII.  
To know if Water will be brought unto any place.  
TAke the distance betwixt the Spring-head and the place whereto you would bring the water, 
If water will be brought unto any place.  which multiply in itself; add the product therof to the Square of the earths Semidiameter; viz. to the square of 3436 and 4/ 11 talian miles, then out of the product therof extract the square roote, and then from that square root, take 3436& 4/ 21 miles, the remainder is the difference betwixt the line of unveil& the water or circular unveil.  
But this you may do with more ease and that without arithmetic by your staff, if you note but the Chapter of the difference of the Horizontal and hipothenusall lines.   

CHAP. XXXIIII.  
The Description and use of the Table set down in the end of the first book, to be placed vpon the hollow staff called the Table of Planimetria.  
THis Table is denided into 6 columns, 
To measure boards.  and one little row vpon the left hand, then every of those columns is subdeuided into two rows: in the first column is placed the parts of an inch, which happen in the taking of the breadth of your boards, glass, &c. set down thus ¼ ½ ¾, signifying one quarter, one half, and three quarters of an inch: then the Table is divided cross into 6 great columns, wherein is placed capital letters from? to 36, which stand for inches: wherein the breadth of any glass, &c. is to be sought in whole inches, as the parts of inches were before: all other pendant columns express the length of your rule the must be laid out to make a foot of boards, &c. as will better appear in the example.  

A Table for board and glass measure, &c. called the Table of Planimetria.  P. Y.   VIX II XVIII XXIIII XXX foot yn: foot ync: yn. par: yn: par: yn: par: ync: par:     2 0 12 0 8 0 6 0 4 ⅘ ¼ 48   1 11 1/ 25 11 3/ 3 7 ⅞ 5 15/ 16 4 ¾ 7/ 2 24   1 10 11/ 7 11 ½ 7 ⅘ 5 ⅞ 4 5/ 7 ¾ 16   1 9 l⅓ 11 2/ 7 7 ⅔ 5 ⅘ 4 ⅖   I VII XIII XIX XXV XXXI 1 12   1 8 4/ 7 11 1/ 16 7 4/ 5 5 ¾ 4 ⅝ ¼ 9 7 ⅕ 1 7 2/ 8 10 ⅞ 7 ½ 5 ⅔ 4 ⅘ ½ 8 0 1 6 ⅕ 10 ⅖ 7 ⅜ 5 ⅝ 4 ●/ 2; 2/ 4 6 10 2/ 7 1 6 4/ 7 10 ½ 7 2/ 7 5 ⅝ 4 ½   II VIII XIIII XX XXVI XXXII 2 6 0 1 6 10 1 2/ 17 7 ⅕ 5 ½ 4 ½ ¼ 5 4 1 5 3/ 7 10 3/ 22 7 ½ 5 ½ 4 3/ 7 ½ 4 9 ⅗ 1 4 15/ 16 9 ⅞ 6 1/ 31 5 3/ 7 4 ⅜ ¾ 4 4 ⅜ 1 4 3/ 7 9 ¾ 6 15/ 16 5 ⅜ 4 l⅓   III IX XV XXI XXVII XXXIII 1 4 0 1 4 9 1 ⅝ 6 1/ 7 5 l⅓ 4 1/ ● ¼ 3 8 l⅓ 1 3 4/ 7 9 3/ 7 6 ⅘ 5 2/ 7 4 2/ 7 ½ 3 5 ⅛ 1 3 1/ 7 9 2/ 7 6 ●/ 7 5 2/ 9 4 ⅙ ¾ 3 2 ⅕ 1 2 ¾ 9 ⅛ 6 ⅝ 5 ⅕ 4 ¼   IIII X● XVI XXII XXVIII XXXIIII 1 3 0 1 2 ⅖ 9 0 6 ½ 5 13/ 8 4 ¼ ¼ 2 9 ⅞ 1 2 1/ ●● 8 6/ 7 6 ½ 5 3/ 32 4 13/ 16 ½ 2 8 1 1 ¾ 8 ¾ 6 ⅜ 5 1/ 16 4 ⅙ ¾ 2 6 l⅓ 1 1 ⅜ 8 ⅝ 6 l⅓ 5 0 4 2/ 8   V XI XVI XXII XXIX XXXV 1 2 4 ⅘ 1 1 11/ 11 8 ½ 6 ⅔ 5 0 4 ⅛ ¼ 2 3 3/ 7 1 1 ⅘ 8 7/ 3 6 ⅕ 4 ⅞ 4 5/ 32 ½ 2 2 ⅕ 1 ½ 8 ⅖ 6 ⅛ 4 ⅞ 4 1/ 16 24 2 1 2/ 23 1 2/ 7 8 3/ 12 6 1/ 16 4 ⅚ 4 1/ 32  
Example.  
I haue a piece of glass 9 inches and ¾ broad, I would know how much in length would make a foot square. First find 9 inches in some of the cross rows amongst the great letters, which you shall find to be in the second great pendant row, then in the first row vpon the left hand I seek ¾ parts, and drawing rightwards from thence, until I come under IX, there do I find 1 foot 2 inches, and ¾ of an inch, and so much measure in length must the glass be to make one foot square: So that I may conclude a piece of glass 9 inches and ¾ of an inch broad, and one foot two inches, and ¾ of an inch long maketh one square foot, which is 144 square inches.  
Then suppose my glass were 3 foot 8 inches and ¼ of an inch long, I then take 1 foot 9 inches and ¾ parts,& measure the length of the glass, noting howe often it contains the length of 1 foot 9 inches and ¾, which I find 3 times: therefore I conclude that there is 3 square feet of glass contained in the whole, which was 9 inches, &c. long, and 1 foot, &c. broad.  
Another Example.  
I haue a piece of board 15 inches one way, I would know how much of my rule must be laid out to make a square foot. seek 15 inches, as before, and then in the margin vpon the left hand find the figure 1, from which directing your eye rightwards, until you come under 15, there shall you find 9 inches and ⅝ parts of an inch, and so much must the other side be, so that I may conclude, a board or pavement, &c. 15 inches one way, and 9 inches ⅝ parts, the other is a just square foot, and then if the board be many feet long, see how often you can find 9 inches and ⅝ parts therein, and so many feet of board may you safely conclude to be contained in the said piece of board.  
And if the breadth exceed the table, then divide the breadth into parts, and so proceed.  
The ground of this working is thus. 
The ground of board measure.  You must understand that the divisions of the board measure, is the breadth of the board given, to find how much in length will make a foot of board.  
To do which, you must also understand, that a square foot is 144 square inches. Therefore if you will know how long the second side must be, divide 144 by the side given, and the product sheweth thy demand.  
Example.  
I would know how many inches the second side should be to make a foot square, the first side which I measure being 36 inches, divide 144 by 36, and the quotient will be 4 inches, which sheweth that the second side must be 4 inches long to make a square foot.  
And so shall you find 32 to yield his second side to be four inches, and ½, &c. so of all the rest, and thus take the ground hereof briefly.   

CHAP. XXXV.  
The use of the Table, called the Table of Solidmetria.  
THis kind of measure is to meate all kind of pieces of timber, 
To measure timber.  ston, pillars, or such like solid things, which sheweth how much in length will make a foot of timber, the side of the square thereof being given; the working whereof is thus.  
First get the square thereof in equal parts of measure: for which purpose M. digs hath well calculated a Table, notwithstanding you shall go near enough to work, if you do but gird the said piece of timber about, and then take the fourth part thereof for the square of such a piece of timber; the side of the square found in inches resort to your table of Solidmetria, and there in one of the rows under this letter S, signifying square, find the square of the said piece of timber, answering to which rightwards shall you find the feet, inches, or parts of inches, which you must measure in length, to make a foot of timber.  
Example.  
The square of a piece of timber is eight inches, answering to which is two feet, three inches, and so much of my rule must I lay out to make a foot square of timber: therefore if the piece of timber had been nine foot long, there must be 5 square feet of timber therein, because 2 foot, 3 inches is fine times contained in 9 foot.  
And so if the piece had been longer, you must haue seen how often you could, haue found by measure 2 foot and 3 inches therein, and so many foot of timber might you conclude to be in the said piece, so of all the rest.  
The ground.  
Hereof is divided the cubicall inches in foot, 
The ground of timber measure.  viz. 1728 by the square of the timber given, the quotient she weth how many inches of your rule must be laid out to make a square foot.  
Example.  
The square is given to be 36 inches, I would know how much of my rule must be laid out to make a square foot: multiply 36 by 36, which maketh 1296, by which deuids 1728, and the quotient will be 1 ⅔, and so much of your rule must be laid out to make a cubicall foot, that is, a foot to contain 12 times so much as a plain square foot of 144 inches: by these tables may you place board and timber measure vpon the Carpenters rule truly and precisely.   

CHAP. XXXVI.  
Of another kind of new working in timber measure.  
THis kind of working shall show you how many foot of Timber there is contained in every foot of length, 
A new kind of timber measure.  the side of the square thereof being given, and is known by extracting the square root of so many square feet as the length contains; as if you would know how many inches square that piece of timber must be as should contain 2 foot of timber in one foot of length, work thus.  
The square inches in one foot us 144, so 2 foot 288, the square roote whereof is 16 64/ 65, which sheweth 16 64/ 65 inches square doth make 2 foot of timber in each foot of length: the like of any other.   

CHAP. XXXVII.  
Of Shide measure.  
SHide measure sheweth how many shides of timber is contained in each foot of length: 
Shide measure.  for you must note that a shide of timber is half a foot of timber, and such a piece whose end is 18 inches over, and whose length is 4 foot: and is thus wrought. Multiply the number of shides that you would know by 18, and from that product extract the square roote which sheweth how many inches square that piece of timber must be that shal contain the number of shides given therein.  
Example.  
I would know what square the piece must bear that shall contain 15 shides in 4 foot of length. First multiply 15 by 18, there cometh 270, the square roote whereof is 16 ⅘ your demand, so of all the rest.  
These divisions of shides serve for the measuring of a timber tree that standeth, thus performed. Gird the tree about 4 foot from the roote, and take ¼ of the g●●ding, then see how many shides be contained in that square, and for the next 4 foot abate 2 shides for the Trapezing of the tree, and so for all the 4 footes in the length thereof; then add all the shides together, so haue you the contents of the tree.   

CHAP. XXXVIII.  
To fetch all manner of dimensions by help of the Cord divisions placed vpon the back of the legs, with more ease then before hath been fet down, and first of altitudes.  
AL altitudes are perpendicular● e●leuated vpon the sure face of the earth: 
Altitudes by protracting.  and therefore make right angles with the line of level, the working whereof is thus.  
Go us near, or stand as ●ure off the altitude as you please, and there take the angle of altituds as you be instructed in the second: book, Chapter four, which note down, thou measure the true distance of the say● 〈◇〉 from your standing, which also note down, and then fall to work thus: Draw a straite line: and on the one end thereof erect● perpendicular, call this line the perpendicular; and the first line, the line of level: then open your compass unto so many number of perches, yards, &c. as you noted the distance of the tower from your standing, which lay vpon the ground line from the intersection of the perpendicular therewith towards the other end, Lastly, where the foot of your compass falleth there make a point, thereon by the 12 Chapter protract the ●n●●e of altitude before taken, and then ●●●●ing that side of th● angle forth in●●nitly, note at last the intersection of the said line with the perpendicular, the distance betwixt the said intersection and the base, is the altitude which is found by applying the said distance unto your former scale.  
Example.  
Let A B be an altitude, C your station, where you take the quantity of the angle A C B 60 degrees, 
Lib. 2. cap. 4. pro. 5.  then I measure my distance from C to B 16 yards, these two I note down, then I draw the line E upon faire paper in●●nitly, then upon E the extremes of E G I, erect the perpendicular F E; then I lay down 16 yards in the line E G, viz. E F, and upon G H protract an angle of 60 degrees, by the second Chapter, as I H E. Then 〈◇〉 H to T, and to infinitely I dr●●● 〈◇〉 and note, where it cuts the ●h●● E F, 〈…〉 F, thou I take the ●●●●th of E F, and apply the same unto my scale, which I find to contain 29 equal parts betwixt the two feet of my compass; therefore, I conclude the altitude 29 feet high, in like manner must I haue found the hypothenusal line C A for the length of any scaling ladder.   

CHAP. XXXIX.  
To search out inaccessible heights.  
SUppose the height A were required, 
Inaccessible heights by protracting.  and that you could not approth unto B to measure the distance from me to the base: therefore I appoint the first at C, where I take the angle of altitude, viz. 60 degrees: then I go back some certain space as to D, and there I take the angle of altitude again 30 degrees, as A D B. Lastly I measure the interval or distance betwixt C and D, which let be 33 yards: these three things had, viz. the two angles of altitude, and the distance betwixt the first and second station, seek the altitude thus.  
 
You shall draw a ground line forth so long as you may, as E G, then about the middeth thereof, as at C towards D, layout 33 yards, as C D, and vpon D ●rotract an angle of 30 degrees, as F G M: then note the intersection of F G, and H F, and there make the point F, from which point let fall a perpendicular vpon the ground line E G, as at E. Lastly, apply the length of F E unto your scale, so haue you 29 yards the altitude, to which if you add the height that your eye was above the base, the whole altitude appears: and if you require the hypothenusal or length of a scaling ladder from either of your two stations, apply the length of F H, or G F to your scale, as before; so haue you your desire.   

CHAP. XL.  
To seek the distance of any place from you.  
IN the 31 Chapter let, the distance of A from D be required: First therefore I appoint D for my first station, 
Longitudes by protracting.  and then from thence some known distance as 33 yards: I appoint my second station C, then I take the angle A D C 30 degrees, next I go to C,& there again take the angle A C D, which I find 120 degrees: these three things I note down, viz. two angles, and one line, then I protract thus.  
First as I am wont I draw a base line forth at length E G, 
The figures in the 40 Chapter serveth here.  then vpon the extreme, as vpon G protract an angle of 30 degrees, as K G L; then from G by K I draw a line forth infinitely, next I lay down the length of 33 yards from G towards E, and where the foot of my compass falleth I make the point H, whereupon I protract an angle of 120 degrees in H I( or if you will, you may take 120 from 180: so will there rest 60, the quantity of the angle F H E, 
Eu. lib. 1. p. 13& 14. * which angle you may protract in stead of the other) then note the intersection of the line G F, and G H, 
Ra. l. 5. p. 8. cons. 1.  which let be at F. Lastly by applying the length of F G to your scale, you haue the distance of D A 58 yards, and so may you measure the other line C A.  
Note in taking the angle A C B, if you had made the ●●dutiall edge of the lost l●gge C O point just to D, as O C doth, then might you have taken the angle A C B, as you did in the 31 Chapter.   

CHAP. XLI.  
To take a distance betwixt any two marks, though you cannot approach to either of them.  
THis Chapter I will commend unto you for ease, facility, and speed, before any other heretofore wrote whatsoever. Imagine that you would take the distance of A B, standing at F or E.  
 
First I appoint me two stations F E in a known distance from me, as 35 yards: then I take the quantity of the angle A F E, and note it down, viz. 30 degrees: Next I take the angle A F B, and note that down, viz. ●8 degrees: these two angles noted for my first station, then I repair to the second station as to E, and there observe the angle F E B, and A E B, which I note down for the angles of my two stations, viz. 45 and 60 degrees, then I protract thus.  
First I draw a line at all adventures, as E F, whereon I appoint 35 yards taken with my compass from my scale, then vpon the point I protract an angle of 30 degrees, as I F H, then from F by I. I draw a line infinitely as to A, next I protract and angle of 98 degrees as L F I, drawing a line from F by K out atlength, as to B, so haue I finished the angles observed at my first station F, then I go to E, and there protract an angle of 45 degrees, as L E M, drawing a li●● from E by M out at length, until it intersect with the line F B, as at E, next I protract an angle of 60 degrees, as N E M, laying a rule vpon E and N, and drawing forth a line until it intersect with the line F A, as at A, then from the intersection A, to the intersection B, I draw a straite line A B, which applied unto the scale sheweth the distance, viz. 40 yards.  
In like manner may you measure the line F B, or F A, or the line E B, or E A, by applying the length thereof unto your scale: and thus briefly of dimensions by protracting.  
And here note that you may perform any kind of Dimention after this way,( I mean by protracting) mentioned in any of my former books. And this kind of work I placed here( as it were out of order) for that you may hereby seek the length of hedges, or any distance in any field, for the speedy dividing or separating of the same without chain measure.   

CHAP. XLII.  
To use the legs of the Geodeticall staff as any kind of Scale, according to 10, 16, 20, 30, &c. in the inch.  
IN the second Chapter of the fifth book I taught you to make a Scale, 
Of the proportion of lines.  and here I let you to understand that you may use the equal parts vpon the legs in as good sort as it were a Scale, by taking the equal parts set vpon the legs thence, as you would from a Si●●le: But some will say, and truly, that the divisions he so large, that apaper will not contain any number about 100, or thereabout: for remedy whereof I will teach you a most excellent way.  
Suppose you haue a line now perches long, which is a pointer to be abase line, vpon which you must 〈◇〉 are a perpendicular 160 perches long. First open your compass to what small distance you please, and that wideness fit over in a hundred in each leg; the legs at that angle resting, take with your compass the distance over in 160 in each leg, so haue you the length of that line as truly as he had been taken from a small Scale. And so might you haue taken the length of any other number whatsoever, 10, 20, 40,& c● and by this means may you find the length of any part of a line: 
Cap. 45. pro. 6. To use the legs of any kind of Scale.  and thereby also may you use the legs as any kind of Scale, according to 8, 10, 16, 20, 30, &c. in the such, as for the purpose.  
I haue observed the angles, and measured the lines in a field, according to the doctrine of the first part of this book, and am required to protract the plate according unto a Scale of 16 in the inch, which I easily perforine, as thus.  
Fit the length of an inch( which you may take from the equal divisions on the legs, for 8 make an inch) over in 16 equal parts on each leg; the leg so resting, take the distance over in each leg from number to number, so may you express any line assigned: as if you would express a line of 20 perches by a scale according unto 16 in the inch, the leg resting as before, take the distance over from 20 to 20, which is your desire; and so of any other number.  
Now if you would apply them unto a scale of 20 or 30 in the inch, the labour is no more but to fit the inch over in the assigned number, into which it is divided, as in 20 or 30, and then work as before: whereby you may note the singular use of this Chapter, by help whereof you may cast up any assigned plate according unto any Scale, reduce maps and plaits from greater forms to less, or from less to greater, perform the Art of Little foot in proportioning images, pictures, houses, &c. and still avoid the cost and multitude of seals.   

CHAP. XLIII.  
To divide a piece of ground as shall be assigned.  
five men haue all alike title unto one tryangular piece of ground, 
To divide ground instrumentally.  the fourth man compoandeth with three of them for their titles: so then there is but two haue title to the said piece, and they agree to enclose it, the purchasor must haue four of the five parts, and the other man but one of the stue parts unto his share: I demand how I shall perform this?  
 
To performs this you must note this general, that if the base of a triangle be cut according unto a proportion assigned, a right line drawn from the angle subtended by the base to any point therein, shall divide the triangle according unto the proportion assigned; let the triangle A B C be a piece of ground to be divided, as is said.  
I divide C B in such proportion as 4 is to 3, by the 3 Prop. 45 Chapter, so that C B the base is so cut in the point E, that C E is to E B, as 4 to 5: therefore a line drawns from A to E divideth the triangle, as was proposed. In like manner could I teach you to divide a parallelogram: but for because there be tables for the purpose set down before, I will onely show you the proposition and no more.  
If the base of a parallelogram be divided according unto a proportion assigned, a perpendicular reared vpon that point in the base where the division was made, shall divide the parallelogram given, according to the proportion assigned.   

CHAP. XLIIII.  
The rare Metamorphosing of all geometrical figures, both regular or irregular, into any regular figure, for the speedy measuring thereof by the Geodeticall staff.  
Metamorphosis 1.  
OF all figures a circled is the most perfect, and hitherunto hath been most troublesone to be measured: for which causes we will begin therewith.  
A circled is a round plain, as is proved, 
Circulus quid. Eu. l. e. 15. Ra. l. 15. p. 1. * the measuring whereof by this instrument is grounded vpon Eucl. l. 12. p. 2. Ram. l. 15. p. 2. &l. 19. p. 2. Cons. 1.2.& 3.  
To reduce a circled into an oblong.  
The easies●way possible I can find to measure a circled, 
Archimedes demonstrauit aream cir●●li aqualem  is to find two lines, which being multiplied in themselves may produce the Area of the circled: 
esse triangulo rectangulo, cuius vn●m latus ex his qu●e rectum comprehendunt, angulum ipsius circulo sem●diametro, re iquum circumferentiae fuerit aequale.  and for that in all other abseruations you be required as well to know the Circumference as the Diameter, before you can find two such lines, I will teach you here a way by help of this instrument: that having but any part of the Peripher or Diameter given, how to find out the whole peripher and Diameter both.  
Therefore in this kind of working you must either haue the Circumference or the Diameter given, or else some part of the one of them.  
First, let the whole diameter be given A B, which set over in 7 and 7 equal parts on the outer side both legs; the legs so resting, take the distance with your compass betwixt 5 ½ ● and 5 ½: then the diameter A B, and the length of that line last taken B C, are two lines, of which you may make an oblong, by erecting B C perpendicular vpon one of the extremes of A B; or for the speedy casting up of the contents, measure with your compass and Scale the length of those two lines A B, and B C, and then work as in the 22 Chap. Prop. 2, or Chapter 25.  
Now if you desire the right extension of the Circumference A R B N, the legs resting at the distance they were, with your compass take the distance over in each leg betwixt 22 and 22, so shall you haue the line E I your demand.  
If you please, or if the numbers of 7 and 7, and 5 ½, and 5 ½ fall too near the center, you may take 14 and 14, and 11 and 11, and all shall be one.  
Secondly, if you haue but the Semidiameter of a circled given, as B E, set the length of that line over in 3 ½ equal parts on both legs: but because those parts fall too near the center, set the line E B over in 14 equal parts in both legs, and then take the distance betwixt 44 and 44, so shall you haue the line E G: now it resteth a● your pleasure whether you will make an oblong thereof, as E G F B, or cast it up, as in the last Metamor.  
Thirdly, you may happen to haue but the half Semidiameter H E given, which set over in 7 and 7 equal parts, or in 14 and 14, and then take the distance over in 88 and 88, or in 176 and 176, and note that line down I E, for the other fide of the oblong; then you may either make an oblong E F K H, or speedily cast it up, as in the first Metamorp.  
Fourthly, if you haue onely the Circumference E I given, set that distance over in 88, and then take with your compass the distance betwixt 7 and 7, and note that line down E H; then with these two lines work as before.  
Fiftly, if you haue onely the Semicircumference given, and be required to find the true Area of the circled, set that line as E G over in 22 and 22 equal parts in each leg, taking the distance over betwixt 7 and 7, so shall you haue the line E B; then work as before: for hereby haue you the end and side of an oblong.  
sixthly, if you haue but the quarter of the Peripher, as E P given, set it over in 11 and 11 equal parts, and then take the distance over betwixt 14 and 14, so shall you haue the line A B, then work as in the first Metamor.  
Metamorphosis 2. To reduce the Sector of any circled into an oblong.  
The Sector of a circled is a segment contained inwardly under two right lines, making an angle in the center, 
Sector. Eu. l. 3. p. 9. Ra. l. 16. p. 3.  the concourse being called the base, as is proved.  
 
Then to make a right angled Parallelogram, the semidiameter, and half the segment are the two sides thereof; and contrary, as the segment N E B is equal unto the oblong L E B M, made of the semidiameter E B, and half the segment B O.  
Or you may use half the Semidiameter, and the whole segment; so is E P the whole base of the sector, and E H the half of the semidiametient E B, the two sides of an oblong E P Q H, equal to the sector R E B.  
Metamorphosis 3. To reduce a Lunular into asquare.  
Let the Lunular be A B C D, then draw the line A B, half the square whereof is the Area of the Lunular, viz. E C F B, which measure by the 22 Chapter, Prop. 1.  
Metamorphosis 4. To reduce an oblong into a square.  
To perform this you must turn the sides of the legs upwards, which bear cord divisions, and then proceed thus.  
In all Oblonges you shall either haue one side or two sides given, and if you haue one side, it is either the longer or the shorter; and for either work as followeth.  
First, let the greater side A B us given, which set over in 60, and 60 in each leg, then take the distance betwixt 36 and 36, and note that line down A G; I say the square made thereon is equal unto the oblong, viz. A E F G is equal to A B C D, 
Eu. l, 6. p. 17.& l. 7. p. 20. Ra l. 12. p. 4.  as may be proved,* and that G B is the other side of the oblong B D unknown.  
 
Secondly, if the shorter side B D be given, set that over in 36 and 36, and then take the distance betwixt 60 and 60, and note the length of that line down A G, which is the side of the square equal to the oblong, which you may measure as the third Metamorp. Then if you desire the longer side of the oblong unknown, continue B D and A G forth in a right line, so haue you A B your desire: for B D is equal to G B, and A G is the mean proportion, 
Ra. l. 18. p. 8.  as may be* proved.  
Thirdly, if both the sides of the oblong be given, as K I and K G, join them both together, continuing them forth in one right line, so shall you haue the line A I; make that line the diameter of a circled, and on the point K, where they were both joined together, raise a perpendicular to the circumference, as K L; which is the side of the square K N M L, that is equal unto the oblong K I H G, as may be proved. 
Eu. l. 6. p. 23. Ra l. 16. p. 19.   
And in the same manner that you found K L, so must you find out the mean proportional of any two lines proposed: therefore note this difference well.  
 
Metamorphosis 5. The one side of an oblong, and the side of a square being given, to find the other side of the oblong, that is, to reduce a square into an oblong, to contain the just height of the oblong proposed.  
LEt A B be the shorter side of the oblong proposed, 
To reduce a square into an oblong,  which continue forth at pleasure, as to E; then on the point B raise the perpendicular B D equal in height to the side of the given square; then from D to A draw the line D A, in the midst whereof as at G, raise a perpendicular, and note where it intersects with the line AE, as at F, then making F a center, with the length of A F describe a circled, which shall cut the point A and D, and also the line A E; as at E; so is D E the other side of the oblong: note this proposition, for it will be in singular use hereafter.  
 
Metamorphosis 6. To reduce a square into an oblong, not regarding the height of the oblong.  
SEt the side of the square given A G over in 60, and 60 Chords of each leg: then take the distance betwixt 36 and 36, so shall you haue the line G B, which join in a right with A G: so haue you the line A B, the longer side of the oblong: now to find the other, I say it is already found: for B G is your demand, to which B D is made equal: so haue you A B D C an oblong equal to A E F G, the square which was required.  
Metamorphosis 7. To reduce a triangle into a square, and to find the mean proportion betwixt the perpendicular and the base of any triangle.  
I Would make a square equal unto the triangle A B C: 
To reduce a triangle into a square.  First I continue forth the side B C infinite, as to D, and from A the top of the triangle, I let fall the perpendicular A D on the line D C: then thus I first find the mean proportion, to A D I add half B C, as B E, which I continue forth in a right line, so that the continuation D F be equal to B E, then making A F the diameter of a circled vpon K the center thereof, I describe a half peripher, F H I A: 
Eu. l. 6 p. 13. Ra. l. 16. p. 19.  so is D H the mean proportion, as may be proved:* which if you make the side of a square, it shall be equal unto the tryaugle proposed, as the square D H I G, is equal unto the triangle, A B C?  
Metamorphosis 8. To reduce any irregular or multanguled figure into one square.  
THis is also performed by finding the mean proportion or square of each triangle, 
To square any irregular Polygonon.  and of all other is most necessary: for because you shall hardly find any figure but he may be resolved into triangles, and so consequently all those for their speedy measuring, may bee reduced into one entire square.  
Let the figure be an Hexagon, as A B C D E F, which must be resolved into four triangles, as in the end of the 8 Chapter, and as you see; then by the 7 Metamorphosis find the square or mean proportion of each triangle, so shall you find the line G to be the square of A E F H, the square of A D E, I of A C D, and K of A B C: then rear G perpendicular upon the extremes of H, as L M on L N; the Hipothenusall M N, is the side of a square equal to the two first triangles F and E: so as if your figure had been a Trapeza, it had been finished; for the square made vpon M N, is equal to the square of M L, and L N, as the pricked lines show, and as may be proved, Euc. lib. 1. Prop. 47.48.& 6. Prop. 31. Ra. l. 12. Prop. 5.  
 
But to proceed, lest haply I seem to obscure that which is easy, next you shall rear the line I perpendicular on the extreme of M N, as M N O: then the hypothenusal O M is the side of a square equal unto the three triangles F E and D, as you may perceive by help of the pricked lines. Lastly, on the extreme of O M, raise K perpendicular, as M P, I conclude, the square made vpon the hypothenusal P O, is equal unto the whole Hexagon, as P O R G, is equal unto A B C D E F: and thus might you haue proceeded unto an Heptagon, an Octagon, Eneagon, Decagon, or any kind of multanguled figure: and thereby find one line, which multiplied in himself, should produce the superficial content of any field, Lordship, or country, as for the purpose.  
 
If you work according unto the doctrine before said, 
The square or mean proportion of the County of Salop.  you shall find the mean proportional of the County of Salop, to contain 28 of our English miles, whose square is 784 miles; which augment by the square of 320, viz. by 102400, and there is produced 80, 281, 600, 
The number of perches in Salop.  the number of perches in the County of Salop; which by reduction unto statute measure, according unto the 24 Chapter, you shall find there amounts in the County of Salop 501760 acres, 
Number ofacres in SaloP  if it were plain: but it is not procise, by reason it was taken from a small Scale.  
Metamorphosis 9. To perform the last Metamorphosis in strumentally.  
having found out the square of each triangle, you may perform this Metamorphosis easily by your staff thus, drawing no figure at all.  
Open the legs unto a right angle, for which purpose there is a hole near unto the center, to put through a screw pin, and so stay the legs at a right angle; then shall you count the line G 10 perches vpon the left leg, whereunto bring the center of the Graduator, and fasten him there; then vpon the right leg count the line H 14 perches, whereunto bring the fidutiall edge of the Graduator; so shall you find 17 13/ 60 equal parts and better cut vpon the Graduator by the right leg, which is the line S, being equal to M N; and so proceed forward until you come unto the last line: as for the purpose, having found 17 13/ 60 equal parts cut vpon the Graduator, count that vpon the left leg, whereunto bring the center of the Graduator: then count the line I 12 perches vpon the right leg, whereunto bring the fidutiall edge of the Graduator, noting the parts cut thereon, which you shall find 20 and something better, which is equal to M O. Lastly, count 20 and better on the left leg, and 11 on the right, so shall yond see well néete 23 equal parts cut on the Graduator( by working as before) which is the line V equal to P O, whose square is the superficial content, as is said.  
In like manner may you work with your compass on the legs, 
Chap. 25. Pr. 1.  setting aside the. Graduator, and all diuistons, by noting the length of each line.  
Metamorphosis 10. To perform the last Metamorphosis Arithmetically.  
having found the mean proportionals, as before, viz. GH I K, work thus. add the square of G. no H together, and note the Radix by the 1● Chapter; then add the square of that Radix to the square ofl, and note again the Radix quadrature, the square whereof added to the square of K, produceth a number, whose square roote is the true square of the Hexagon, whose square is the contents.  
Example.  
The square of G is 100, and of H 196, which added together maketh 296, whose Radix is 17 13/ 60, the square whereof must be added to the square of I, as 289 169/ 600 to 144, there eiseth 433 169/ 3000, whose square must be added to the square of K; so haue you the contents, the radix quadrature being the sibe of the said square, as P O.   

CHAP. XLV.  
Of the proportion of lines and numbers one to another, with the dividing of lines, without the knowledge whereof it is not possible for any to be perfect survey or or Geodetor.  
Proposition 1. To cut a line by extreme and mean proportion.  
A Right line is said to be divided by extreme and mean proportion, when, 
Es●. l. 6. p 3. Ra. l. 14. p. 1.  as the whole shall be unto the greater segment, so the greater segment is unto the lesser; and is performed thus.  
You must first consider whether you haue given the extreme or the mean: and if the extreme, whether the greater or the less.  
1. If the greater extreme be given, as C D, fit that line over in 60 and 60 amongst the Chord divisions on each leg; then take the distance betwixt 36 and 36; count that distance in the line C D, as C H. I conclude, the line C D is cut proportionally, C D and H D being the extremes,& C H the mean, 
Metamor. 4. Def. 1.  as is proved.  
 
2. If the lesser extreme be given, as H D, fit that line in 36 and 36 among the Chord divisions in each leg, and then take the distance over in 60 and 60, and note that line down C H, which continue forth in a right line: so shall you haue C D the greater extreme, and C H the mean, as is proved Metamorphosis 4, Deft. 2.  
3. If the mean be given, and both the extremes required, work as in the 4 Metamorphosis, Def. 3.  
Proposition 2. Two numbers being given, to find two lines which shall haue such proportion one to the other, as the numbers given haue.  
THis kind of working is grounded vpon this geometrical Proposition:  
Sirectam triangulo est parallela basi, secat crura proportionaliter, 
Ra. l. 6. p. 9. & contra, Eucl. l. 1. Prop. 32.  
In this kind of working we shall not altogether digress from D. Hood in his Sector, 
Proportión of numbers to lines.  whose many labours deserve to be eternised.  
Consider if the two numbers given be under or above 19, if they be both under 19, join a cipher on the right hand, and then open the legs unto what angle you please, seeking those two numbers in the legs; and then the distance with your compass taken over from number to number, yieldeth the proportional lines.  
Example.  
Let the numbers be 13 and 6, I join a cipher to 13, and make it 130, and to 6, making it 60; then I open the legs, as I see cause, and take the distance over betwixt 130, and 130 equal parts in each leg, and note that line C D: the legs so resting, take the distance betwixt 60 and 60, and note that line down C E; so haue you two lines, Viz. C D and C E, in such proportion one to the other, as 6 is to 13, or as 60 is to 130.  
2. If either of the two proportional numbers be above 19, join no cipher, but seek the numbers out in the legs, as before.  
Proposition 3. A line being given, and a proportion assigned, to lay down a line, which shall be to the line given in suchproportion as was assigned.  
I haue a line given C E, 
Proportion of lines.  and am required to lay down a line that shall bear such proportion unto C E, as 13 doth to 6; by the doctrine of the last Proposition I join a cipher unto the two numbers given, so be they 130 and 60: fit the line given C E over in 60 equal parts of each leg, and then take the distance over from 130 to 130; so shall you haue the line C D, which is the line that is unto C E, as 13 to 6.  
In this kind of working you must call the first number the antecedent, 
Note.  and the latter the consequence, and you must see whether the antecedent or consequent be greater: If the antecedent be the greater, you may conclude that the line sought for is greater then the line given, and that the line given must be applied to the lesser number: because I seek for a great ex line, as 13 is to 6: here 13 the antecedent is greater then 6 the consequence: therefore the line sought for is greater then the line given, and must be fitted in 6 and 6, and the line sought taken from 13 to 13.  
If the consequence exceed the antecedent, then the line sought for is lesser then the line given, and the line given must be applied to the greater number, because I seek for a lesser line. Or thus briefly according to the prescript.  
The 

Antecedent  
Consequent   being Maior, apply the given line to the 

lesser number.  
greater number.    
Proposition 4. Two lines being given, to find what proportion the one hath to the other.  
In this Art of Goedetia you shall find this proposition to be no less requisite then the rest, 
What proportion lines haue among themselves.  as hereafter appears, which made me the rather devise to perform it by my staff: for there shall no supersicies that is plain, as boards, glass, pavements, pastures, &c. happen, but by this Proposition you shall speedily find what proportion the one hath unto the other: but setting aside amplifications, let us return unto the purpose.  
Fit the longer of the two lines given over in any number of equal divisions what you will: the legs so resting, with your compass take the length of the shorter line, and bring that along the graduation of the equal divisions, until you haue fit it over in like parts on both legs; which done, you may conclude that the parts wherein the feet of your compass then stand, and the parts wherein you placed the longer line, be in proportion one to the other, as the lines proposed be.  
Example.  
I haue two lines C D and C E given, and am required to thew what proportion the one beareth unto the other.  
 
First I fit C D over in 130 equal parts on both legs, then I bring the length of C E along the graduation of equal divisions, until the feet of my Compasses cut equal parts in both legs, which will fall out at 60: I conclude C D is in such proportion to C E, as 130 is to 60.  
Proposition 5. To denied a line into any number of equal parts proposed.  
Fit the line given over in the number of parts, 
To divide a line into any number of equal parts.  into which you would divide the same, and then take the distance over from division unto division, so haue you your desire.  
Example.  
I would divide an inch into 20 equal parts. Let B E be the inch, which fit over in 20 and 20 of each leg; then take the distance from 19 to 19, which apply to the line B E, as to C from E: so haue you B C, which set 20 times, denideth B E into 20 equal parts: or else proceed downwards along the graduation, taking the distance from number to number, and applying it to the line, as from 18 to 18, so haue you the line F E, &c.  
Proposition 6. A line being given to deliver any part thereof.  
I haue a line A E given, and am required to deliver a third part thereof: 
To deliver any part of a line.  to avoid arithmetic, choose some number which you may divide into three equal parts, as 60: therefore fit A E over in 60 and 60 equal parts on both legs. Then because 20 is the third part of 60, I take the distance betwixt 20 and 20, so applying that distance to A E I haue A B, a third part of A E: if you would haue found ⅔, you must haue taken the distance over from 40 to 40, so should you haue found the line A D, your desire.  
Or thus Arithmetically.  
Fit the length of the line given, as A E over in any number of equal parts, as in 60 and 60, which divide by the parts of the line required, which let be the sixth part: 60 divided by 6, yeeldeth 10: therefore I take the distance betwixt 10 and 10 in both legs: so haue I the line A G, which is contained 6 times in A E, and so often is 10 in 60.   

CHAP. XLVI.  
Of finding of the proportion that one figure or piece of ground, &c. hath to another, to find out the Diameter of a circled, or any chord or side of any regular figure, the circumference being unknown, to divide a circled, and find the proportion of any chord unto the circumference.  
Proposition 1. Two figures or two pieces of ground being given, to find what proportion the one hath to the other.  
EIther the figures proposed be all squares or oblonges, 
What proportion any two figures one hath to the other.  &c. or one a square, the other an oblong or a multangled, &c. and for either work thus. Reduce the two figures into two parallellograms of equal height, then by the 45 Chapter, Pro. 4, find what proportion their bases haue one to the other: for such haue the figures, as is proved*. 
Eu. l, 6. p. 1. Ra. l. 10. p. 13.   
Example.  
Let the one figure be a circled, and the other a multangle, by the first Metamorp. I reduce the circled into an oblong, then by the 7 Metamorp. I reduce the multangle into a square. Lastly, by the 5 Metamorp. I reduce the square into an oblong, equal in height to the oblong made of the circled. I conclude by the 45 Chapter Prop. 4, that such proportion as their bases haue the one to the other, the like hath the figures, viz. the circled and the multangle one to the other.  
In the same manex may you deal with pavements, floors, boards, glass, &c.  
Proposition 2. Any chord of a circled being given, to find the diameter of a circled, to which the inscription of the chord shall answer.  
To perform this proposition, 
To find the diameter of any circled.  you must turn the chord divisions vpon the legs upwards, and then fit the length of the chord in the parts of each leg answering thereunto; the distance then taken betwixt the two points respective, or from 60 to 60, yieldeth the just length of the semidiameter, which doubled is the diameter.  
Or fit the chord given over in the number answering to the half thereof, the distance then betwixt 60 and 60 is the diameter.  
Example.  
 
I haue B C the side of a pentagon given, and am required to deliver the diameter of the circled wherein the pentagon or figure of five sides may be inscribed: fit therefore the side of the pentagon B C over in 72 and 72 parts of each leg, which are marked with this letter P to signify a pentagon, and then take the distance from 60 to 60, which double; so haue you your desire G H.  
And by this means the chord of a circled being given, you may find out the center.  
Proposition 3.  
Also the diameter being given, you may inscribe an equilater triangle, a square, a pentagon, an heragon, &c. by fitting the semidiameter over in 60 in each leg amongst the chord divisions, and then take the distance over from number to number in each leg marked with the letter K answering thereunto.  
Example.  
I would find the side of an heragon, that might be inscribed within the circled E G F, the semidiameter E D being given, fit E D over in 60 and 60, then because 60 and 60 on the legs is marked with an H, signifying an hexagon, you haue your demand, which fit 6 times in the circumference, so will E F one of the equilaters be equal to D E the semidiameter.  
2 But if you would inscribe a square, having fitted D E in 60 and 60, take the distance from 90 to 90, marked with S Q, to signify a square, so shall you haue D I, which may be set four times in the circumference, as E I G L.  
Or having fitted the semidiameter in 60 and 60, work thus in numbers.  
For the side of 

a square  
a pentagon  
an hexagon  
an heptagon  
an octagon  
an eneagon  
a decagon   to take the distance over betwixt 

90  
72  
60  
51.20 ferè  
45  
40  
36   degrees in each leg.  
To find the side of an equilater triangle, you must put the side of an hexagon twice in the circumference, so shall you inscribe the triangle E H M. Whereby vpon a necessary consequence it followeth.  
that having the side of 

an Eq. triangle  
a Square  
a Pentagon  
an Hexagon  
an Heptagon  
an Octagon  
an Eneagon  
a Decagon,   by dividing and subdeuiding, we may find the side of figures having 6.12.48.96. 8.16.32.64.128 10.20.40.80.160 as atryangle 14.28.56.112. as a square 18.36.72 as a square sides.  
Proposition 4.  
Also having the semidiameter of any circled, you may easily divide the same into 360 parts, 
To divide any circled into 360 parts.  by fitting the semidiameter in 60 and 60, and then taking the distance from degree unto degree, and applying the same unto the circled, until you haue finished the one quadrant, which done, proceed to the next.  
Proposition 5. The chord of any circled being given, to find what proportion it beareth to the circumference.  
First, by the 45 Chapter, 
To find the proportion of a chord unto the circumference.  find a line which shall be to B A the diameter, as 7 is to 22( which yond be taught to do also in the 44 chapter, Def. 1.) vide E I: let the chord be R S, which by the 45 Chapter, Prop. 4, you may easily find what proportion R S hath to E I.  
Proposition 6. To find a right line equal to any assigned ark of a circled.  
As 360 is to the ark given, 
Of the right extension of any ark.  so is the line of the right extension of the circumference to the line required: understanding the premises, this is easy.   

CHAP. XLVII.  
Of the making of any figure like and proportional to any figure assigned, with a further discourse of dividing all maner of grounds.  
Proposition 1. A circled being given, and a proportion assigned, to deliver a circled answering to the Symmetry thereof, and the proportion assigned.  
AL circles be like one unto another, and as for the proportion, work thus.  
Because there can no figure he made proportional unto any assigned figure, 
To make a circled according to any proportion.  unless some line be given to make the same vpon: therefore in all circles take their diameter, which let be A B; then by the 45 Chapter, Prop. 3. séeks a line which shall be to the Diameter A B, according as the proportion was assigned, viz. as 4 to 3, which let be C D: join those two lines together in the point D, and let them be continued forth in a right line F F, which make the diameter of a circled E G F: then vpon the point D, where the two lines were joined, raise a perpendicular D G, to touch the circumference, as at G, at all adventures: so is D G the diameter of a circled, whereon the circled must be made, as you may perceive by the figure, where two circles be circumscribed according unto the proportion assigned, as may be proved.* 
Ra. l. 16. p. 19. Eu. l. 6. p. 13.   
Problema. 2. A greater triangle being given, and a proportion assigned, to make a triangle lesser and like the greater, according to the proportion assigned.  
Let the triangle in the last problem be D L F, 
dividing of triangles.  take either of the sides thereof, namely D F, which is equal to the line D C; let the proportion be as 3 is to 4, a subsesquitertia: find therefore by the 3 prop. of the 45 Chapter, a line which shall bear such proportion unto C D, as 3 doth to 4, as B A; continue them in one right line E F, which make the diameter of a circled, as in the last problem: and where they were joined together, raise a perpendicular, as on D, which shall cut the circumference in G; then count the line D G in the line F D, from F towards D, as F K: then from the point K draw a line parallel to D L, which let be K M. I conclude the triangle K M F is like the triangle D L F, 
Ra. l. 4. p. 14. Eu. l. 6 p. 2.& l. ●. pro. 4.& 5. Ra. l. 6. p. 8.& l. 7. p. 9.  as may be* proved, and hath such proportion to the other, as 3 to 4, as may be proved.  
Problema 3. A lesser triangle being given, and a proportion assigned, to make a greater like and proportional.  
This problem is but as it were visu verso unto the last, 
To make a triangle according to any proportion.  saving where you counted the mean proportion D G from F towards D, vpon the greater extreme, here you must count it from E towards F vpon the lesser extreme, as to K.  
Example.  
Let E I D be the triangle proposed, and let the proportion be as 4 to 3, work as before, and then count D G from E towards F, as to K: then draw the line K H without the triangle, and parallel to the said D I. I conclude the triangle E H D is to the triangle E I D, 
Problema. 2.  as 4 to 3, and is like thereunto also, as may be proved.  
Problema 4. To perform the second prob. by drawing parallel lines about the triangle.  
Let the triangle be B C D, whose center you must find by Euc. l. 4. prop. 5. or Ra. l. 17. pro. 5,* or as you be taught, 
Cap. 23. p. 1.  which let be A; whence draw right lines to each corner, 
To inscribe a triangle according to a proportion.  as from A to B, to C and D: now let the proposition be as 30 to 40, here you be required to inscribe a lesser triangle, like and proportional unto the greater B C D: take therefore the line A B, and by the 45 Chapter, Prop. 3, find a line which shall be to A B as 30 to 40, which let be P Q and R S: join P Q and R S together at F, making one right line as G E, and let the point of connection be F; make E G the diameter of the circled G H E, then on the point F raise a perpendicular, which shall cut the circumference M H, then count F H M on the line A B, from A towards B, as to I, which line also count in the line A C and A D, as A K, and A L; then draw the line I K, K L, L I, which shall be parallel to B C, C D, 
Eu. l. 6. p. 2.  and D B, 
Ra. l. 6. p. 8.  as may be 
Ra. l. 4. p. 14.& lib. 5. p. 12. Cons. 4.& l. 7. p. 9.  proved. Then I conclude that the triangle I K L is like the triangle B C D, as may be proved, and that the triangle I K L inscribed is to B C D, as 30 to 40, as is also proved, 
Eu. l. 6. p. 15.  Ra. l. 4. prop. 15. Cons. 1.  
Problema 5. To perform the third proposition, by drawing parallel lines about the figure.  
THis is not much differing from the last problem, when you haue found out the proportional lines.  
Let therefore the triangle be I K L, 
city umscribing triangles according to a proportion.  find the center thereof as in the last proposition, whence draw the lines A I A, K,& A L infinitely, now let the proportion be as 40 to 30: first by the preallegated 45 Chapter, proposition 3, find a line which shall be to A I, as 40 is to 30, which let be M E; to which anner A I, in the point M, so shall you haue the line B E,( so now the work differeth not from the last proposition:) now I make B E the dyameter of the circled B N E, raising on the point M A perpendicular, which cutteth the circumference in N, at all adventures; then shall you count the line M N, in the line A I, A K, and A L, as from A to B, from A to C, from A to D, beyond the triangle: lastly draw the line B C, C D, and D E, which shall be parallel to I K, K L, and L I, as is proved in the last problem. I conclude that the triangle B C D circumscribed, is to I K L, as 40 is to 30, or in lesser numbers, as 4 to 3( which is a sesquitertia proportion) as is proved in the 4 problem.  
Problema. 6. A greater tryangulate figure, or piece of ground being given, and a proportion assigned, to divide the lesser, so that he shal be like unto the greater, and keep the proportion assigned.  
IN this as in the former problem, you be required to perform two things, 
To divide a cryangulate.  that is to make a lesser figure like to a figure assigned, and to retain the proportion assigned,  
Let the greater Tryangulate be A w X Y Z, &c. and let the prob. be to find a lesser tryangulat that shall be to the greater as 3 to 4: first therefore from any one corner, as from A to each corner in the tryangulat draw strait lines, as A X, A Y, A Z, then take any one side of the tryangulat containing the angle A; let it be A w, and by the 45 Chapter, proposition 3, find a line which shall be to A w, as 3 is to 4: join those two lines together, as in the former problem, making them the dyameter of a circled and so find out the mean proportional, which count in the line A w, as A F: then draw alone from the point F parallel to w X as F E, and from the point E draw a line parallel to X Y as E D; from D draw a line D C parallel to Y Z, and from C draw C B parallel to Z&. I conclude the tryangulate A B C D E E, is like the tryangulate, A w X Y Z&, as 
Problema 4.  is proved, and that they bear such proportion one to the other as was assigned, as also may be 
Ra. l. ●0. p. ●2.  be proved in the end of the second and sourth problem.  
Problema 7. A lesser tryangulate being given and a proportion assigned, to deliver a greater, like and proportional to the lesser.  
If the lesser be given, and the greater required, having found the lines by the preallegated prob: 
To circumscribe a tryangulate proportional. 45. cap. p. 3.  that haue such proportion one unto the other as the numbers given haue, and then made them the dyameter of a circled, and so found out the mean proportional as before, the work differeth not from the last problem, saving that the lines must be drawn vpon the utter side the figure, as before they were on the inward: so is A F, the line given, and the proportion as 4 to 3; following the doctrines before, A w is found the mean proportional, and so is there drawn the lines, w X, X Y, Y Z, and Z, &c. parallel to F E, E D, D C, and C B, so is the greater tryangulate A w X Y Z, &c. made according to the proportion assigned.  
 
Problema 8. A field of tryangulates being given, to inscribe a tryangulate with parallel sides according unto the proportion assigned.  
The forms generally that may rise in kinds of tryangnlates be infinite, 
To inscribe a triangle.  for that the several kind of triangles that may be packed together be infinite: for a tryangulat is nothing alss but the aggregating of a certaiue number of triangles: for he is said tryangulated, because he is composed of triangles,& may be resolved thereinto again: but notwithstanding their different forms, yet the order of working in one is common to all, whether they be quadrangles nuiltaugles or any such kind: and so I would haue you to vndorstand that whatsoener I haue said of any one figure before in particular, the like is to be understood of all of like denomination in general, as of triangles whether they be Isosceles or scalenums, &c. And to be short, this work, the prenrisses understood, differeth not from what is said, but for the vnpractized Geometritian, whose cause I tender; I will not cease to prosecute the same.  
Let the maltangle be F F G G, and Z Y X w, then make a point somewhere about the midst, as at A; from whence unto each corner produce right lines, as A FF, A GG, A&, A Z, and so round as you see in the figure( of the seventh probleme●) then take any line, let it be A w, and let the proposition be as 3 to 4: find a line which shall be to A w as 3 to 4,& so as before find the mean proportion, which count in the line A w, as A F. then from the point F draw a line parallel to w X, as F E, and from the point E draw a line parallel X Y, as E D, and so go round about, so haue you inscribed a lesser tryangulate HH II, B C D E F, like and proportional to FF GG, and Z Y X w, the greater tryangulate, that is as 3 to 4.  
And if you please you may draw any of these forenamed triangles or tryangulates without circumscription or inscription, according unto any proportion assigned, onely by themselves if you do but note the angles, and make your proportion, the ground whereof you may gather by these propositions. If parts like to the parts of a figure given, and in like manner situate, be placed vpon a term given, there shall be placed vpon the famed term a figure, like to the figure given,& in like manner situate. Ra. L. 4. P. 14. Cons. 4. Like figures be figures equianguled proportional in the feet of the equal angles. Ra. l.& pro. as before.  
Problema 9. Any kind of figure being given, and a proportion assigned, to make a figure which shall haue his sides to the sides of the figure given, in such, proportion as was required.  
Let the figure be A B C D E, and let the proportion be subsesqui●erti●, as 30 to 40, 
To make a figure proportional in sides to any figure.  by which proportion you may gather that the figures sought is less then the figure given.  
Take therefore some point in the side or Area of the figure given, which let be A, from either of which points produce right lines unto each corner in the peripher of the figure, as from A to B, from A to C, from A to D, &c. then take the line A C, and by the 45 Chapter, Prop. 3, seek a line which shall be to A C, as 30 to 40, which let be A F; do so to A D, so shall you find the line A G; do so to A E, and A B, so shall you find the lines A H, and A I. Lastly, if you took your point in the Area, as at A, do as before with A A, so shall you haue A K; then draw the lines K I, or A I, I F, F G, G H and H K. If you took your point in the Area, I conclude either of the inscribed figures, as A F G H K I made on the point A, taken in the Area, or A I F G H taken in the corner A, in like unto the figure A B C D E, and that their sides be in such proportion as 30 to 40.  
 
Or thus otherwise.  
Let the numbers be 30 and 40, multiply 30 in 180, so haue you 5200, which divide by 40, the quotient yieldeth 130; then work with 180 and 130, as you did before with 40 and 30, and all will be one: and if you can no arithmetic, you be taught to perform it in the second book, or in the fist book by my scale. Let this conferred with the prescript of the Chapter be sufficient to teach you for to haue circumsc●●●ed, that is, to haue made a greater, or to haue made the lesser or greater by themselves, according unto the proportion assigned. Now I proceed according to promise, unto a further discourse of de● uiding grounds.  
Problema 10. To divide or cut off any number of acres from any parallelogram, be it square, Rombus, &c. by aright line drawn from a corner to the opposite side.  
In this work you be to consider two things, that is, 
To divide any parallelogram as shall be assigned.  whether the quantity you would enclose exceed the moiety of the figure, or whether the medietie of the Area of the figure exceed the quantity to be enclosed. If the portion inclusive be lesser then the mediety of the Area, multiply the number of perches that you would cut off in the opposite side to that line, whence your dividing line is produced, and the product part by half the area; so shall the quotient show you how many perches you shall meate on the foresaid opposite line, from whence a hedge ensuing to the angle, shall divide the figure as was assigned.  
And this number is counted vpon the opposite side, beginning at the end next unto the angle whence the hedge cometh.  
But if the portion of separation be greater then the mediety of the figure, you shall deduct it from the whole area of the parallelogram, and with the residue work as before: onely this difference is, that the part separated is at that end of the parallelogram opposite to that which was separated before.  
Problema 11. The King granted a piece of wast ground unto eight men, vpon condition that five of them shall occupy their portions together: yet to be severed from the parts of the other three, which also occupy their three parts together. I demand how it shall be performed.  
The piece of ground granted is found to contain 16 acres, 
dividing of grounds.  3 day-workes, and 2 perches, which let be A B C D: the grant is to eight men, the proportions be 3 and 5, which make 8: by the 45 Chapter. Prop. 3, find a line which shall be to A D, as 3 to 5, which let be E D: draw E F parallel to D C or A B. I conclude, A E F B is the just portion that the 5 men should haue, and E F C D the part that the 3 men should haue.  
But I said before, that the peec● of ground contained 16 acres, 3 dayes work, 2 perches, which is 2574 perches, I would know how much each man hath unto his part, and so consequently howe much each enclosure contains. divide 2574 perches by 8, so haue you 32● perches, 8 cubits, and 4 inches and better: then if you multiply this number by 3. you haue the content of E C F, or by 5, of A B F E, or ha●ing multiplied by 3 or 5, subtract the quotient from 2574 the area, so is the remainder your desire. I conclude by the prescript that each man hath; 32● pearthus, 8 cubits 4 inches and a little better, &c. and in the same order might you haue enclosed any of their parts by themselves, and this rule is general for any parallels gram: and if you do but consider what is said of proportions, there shall nothing seem so hard but your staff will help you therein with ease, be the figure ordinate or inordinate.   

CHAP. XLVIII.  
To measure mountaines and valleys, with a further discourse of measuring grounds.  
THe order of measuring mountaines, 
Mountaines and valleys.  and not reducing them into plains as before, is, first you must measure the cirenit of the base of mountaines or compass of the brim of valleys, 
But this work of the hills be-much irrregularis not precise.  which two add together: then in mountaines measure the longest and shortest ascent from the base unto the top of the said mountain, which also add together: then multiply or enter your table with half of the two ascents, and half of the circuit of the base and top, so haue you your desire so of valleys onely reckoning, or measuring the descent for the ascents.  
This is the received opinion of measuring hills and dales, but if it so fortune that the hill or dale be much irregular and not inclining to the form of a loaf of bread or sugar loaf( for so master digs in his Tectononicon intends) but confists of along side of a hill or such like, do thus.  
resolve the sides of the said bank or hill into such figures that it will most aptly receive, and then measure the same according to the ●●gure it doth represent, 
To measure hills and valleys after a new way.  and in so doing you shall ●uoide erro●s; for if I measure the side or root● of an house as it were plain I haue the true contents, and therefore I will will take a plain ensample: imagine that the roof of an house, from the wall plates upwards did stand in the midst of a plain field, now if you did measure the contents of the field not regarding the bank like the roof of an house, it were too little. Therefore not regarding the ground the roof or bank standeth on, but omitting the same I resolve the said field into apt figures to measure, and then afterwards measure the two sides of the roof or bank like unto two long squares( for such a figure the side of the roof of an house doth represent) which I add to the former, and the contents appears.  
But if you had taken the perfit of the field, and so cast up the contents without regard of the hill, then must you haue measured the hill as before, and added the contents thereof to the contents of the field; next must you also haue taken the plate of the ground that the b●nke did stand on by measuring about the fo●te of the said bank, and also haue given the contents of the said plate of ground that the hill did cover or stand vpon, which you must haue taken from the former total, and then the remainder is the true contents of the field& hill, viz. of so much ground that is to be seen, but we take no notice of small banks or valleys.  
Example.  
The contents of the field is 40 acres, of the bank 10 acres, of the ground that the hill stands vpon, 4; acres; 40 and 10 added together make 50, from which I take 4; so there remaines 46 acres the true contents, whereby the contents of the field by reason of the hill contains 46 acres, which being plain had ben but 40. acres,  
And here I would haue you note that whereas in the 8 Chapter of the first part of this book, I shewed you how to go round about the field, stil taking the quantity of every several augle, here I let you to wit that if the ground be full of angles you may measure such hedges as if it were one strait line, remembering at every small angle, to measure how much he doth give out, which note down,& against what part of the strait like the giuing out was: and this strait line I would haue you to call the visual line, and the line of giuing out must be measured into the very utter angle perpendicular from the visual line, because you may the easier protract: and as you noted the angle of giuing out, so likewise must you note the point where it cometh into a right line again, and by this means may you reduce any irregular piece of ground into a regular& easy form of measuring.  
 
Let the irregular field be, A B C D E F G H I, which by the visual lines A C, C E, and A E you see I haue reduced into a triangle, so that whereas there were fifteen angles to be observed, you see we need but three; then for the giuing out at D in the line C E, I work thus. I take the angle A C E, and meate the visual line C E,& in my meating I note where the giuing out begins, as at K 13 perches: then I meate forward until I come against D, as at M 33 perches: there the angle giveth out furthest: then I meate in a right line from M to D 11 perches which I note down( and let M D concur with C M at right angles in M) then I proceed forward from M in measuring, as to L, 56 perches, where the giuing out cometh in again; then I proceed in meating unto E, so C E the visual line is 91 perches: so that you may see that the angle of deflection began at 13 perches, as at L, and that the very tropic of deflection was at M 33 perches, containing 11 perches as M D; and that the helical deflection came into the right line A E at; 56 perches, as at L;& so by taking such notes as these be, all the rest performed with greater ease and speed.  
Many other things must be omitted to the discretion of the Geodetor or measurer, which would seem vain and prove tedious to be aggregated in one volume: such things that be not ordinary, I haue here set down, and some such as I am certain were never mentioned heretofore by any: We haue diners books extant for the Mathematical part of measuring prounds, as first master Benefe, whose labours might as well haue served as any one of his successors, if there had been no more sit to speak thereof; but such is the acuity of our refined wits, th●t they obtain a more daily affect action unto the arte by every of their new additions: First master Benese, he contrived tables and taught to measure ground well& plainly; then comes master digs and wittily addeth the measuring of superficies& solides, diligently calculating tables; master L●●, he addeth the legal part of survey. 
solace good as the best.  And lastly comes I. N. and propounds a dialogue explaining thereby, what he thought convenient, and for myself, what I haue added or devise, I refer to your consideration: Amongst many things I hau● added the order of making a perfit, with brief tables to expre●●e every several commodity rising or acrewing therein, and if this like you not, you may see another in the ending of the book of the Geographical instrument.   

CHAP. XLVIIII.  
Of the making of a Table to declare every several tenor and commodity, rising in any Manor, &c.  
IT is not sufficient for one to take the true plate of a Manor, and so render the true contents thereof in acres &c. but you must also acquaint the Lord of the manor what estate every tenant holdeth by, that is, 
The making of a neccessary table for a survey.  whether the lands they hold be free hold, or whether they haue a rearme therein for lives, yeares, at will, or such like: then must you see what meadow ground, what errable, what pasture, and what woodland is belonging unto every several tenement, and what quantity of such pastures belonging to each feueral tenement, that is, how many acces of woodland, how many of meadow ground, &c. then is it requisite to see what rent each tenement payeth, and what their tenement were worth, if their estate for life, or yeares, were expired: then should you take notice, what time of their estate is complete, next should you see what services each tenant oweth unto his Lord by force of the lease, or otherwise; then what number of sheep and other great cattle they be aloud to keep vpon the common, if there be any: then should you see what commodities rise in the Lordship, and what profit may accrue unto the Lord thereby, and in whose ground, and what ground they stand in, to the end the Lord in his chari● may the better consider the poor tenant, for the loss of his ground: for it may so fall out that a moyne may happen in a meadow of some one of his tenants, which happily may be all the meadow ground the tenant hath: and the moyne still continuing, God forbid the tenant should be so much damnified, as to lose the most commodity thereof, since he still pays his rent: these and all such other things happening it is requisite the furueior should take notice of, and acquaint the Lord, as well of the one as the other: and for the more lively understanding, I haue here drawn the epitome of a Lordship, &c, in such a compendious form as I presume the like hath not been published.  
Upon the one side, thou seest the true proportion of the Lordship or manor, and every parcel of ground therein contained, then on the other thou seest a table to expound all that we haue said before, directed therein by letters alphabettically as will be plain by an example or two.  
Example.  
William Ashbe is my tenant, I would see what tenewre &c. he holdeth by.  
On the left hand I see he is tenant at will, then proceeding rightwards until I come under the title of meadow, there do I see that he hath but one parcel of meadow ground, and that doth contain 1 acre, 2 roods, and 4 day●works of land: the letters A B telling me, that where I find those two letters in the perfit, there is the piece of meadow, and the name of the said piece, which you shall find to be a close, and so proceed until you come to the end of the table.  
Another Example.  
William Cocke● is my tenant, I desire his tenewre estate, &c. on the left hand. I see his tenewre is for yeares, and he hath in possession 2 parcels, of meadow ground, containing 4 acres, 1 rood, 3 day-works, 1 perch, and AE directs me to find how they stand in the manor, where I find their names and several contents as small acre containing 1 acre, 2 day-works, 1 pearche. And new meadow containing 3 acres, 1 rood, 1 day-worke, 1 perch: in like manner must you find wood ground, pastures and errable, as is plain in the table, and you must note in each row under the title of acres, that the uppermost figure vpon your left hand represents acres, and that on the right hand roods, the lower on the left hand day works, and  

A Table to declare all the several Commodities issuing out of the Mannor of Sale, &c.  Tenewers in special. Tenants names. Mcdo. wood pasture Errabl Rents Esta tes   Rate. parcels. Acres. parcels. Acres. parcels. Acres. parcels. Acres. now pai improu. time lef time exs. Seruisers. sheep. Beasts. At will. Willi. Ashbe. 1 B 1 2     AD 2 3 AC 3 1 S L     ▵ 40 3 1 4     1 0 0 1 5 O 9 3     At will. Robert Poine                               Ter. of ye. Will. cock. A E 4 1 of 4 1 AG 10 0 AH 7 0 s l 21 8 * 50 2 2 3 1 1 0 0 3 8 0 2 8 6 10 6 Ter. of ye. david welsh                               For yeares Charles Mae.                               For life. Ric. Rumley. AR 10 3 AP 4 1 FF II 0 PP 5 1 s 1 3 1 R 30 3 3 8 0 1 5 1 2 4 0 1 8 3 15 11 For life. roll. Dod. LL 14 1 H 1 3 HH 3 0 II 3 3 s 1 3 2 P 30 2 2 6 3 1 6 2 I 6 0 1 5 15 8 For life. Ra. Adames.                               Socage. Rich. Preene. zz 6 0 D 2 0 B 3 2 E 4 2 s 1     L 50 1 2 1 1 1 2 2 1 I 2 1 7 2 18 9 K. service george Hock                               Durgage. rob. Royde                               For life. joh. driver. BB 2 3             1 1 3 2 ▵   2 2 3 0             2 20 For years. Rich. Houte S 6 1             3 1 21 14 *   3 2 3 0             14  
services proper unto each tenant in the Manor, and where Heriots be due, &c.  
To grind at the Mill. ▵ R.P. Two dayes with his Teeme in seednesse— ▵* To reap in the harvest. ▵ R.P. P. Two daies with his Teeme in harvest to carry corn. ▵* P. To appear at his Courts one day with his Teeme in Lent seednesse. ▵* R. P.l.   * R.P. His best beast at his death. ▵* Money at his death for A Heriot. R xx. shillings.   P. x. shillings.  

Commodities rising in the Manor.  19. Acres of common pasture. 16. Acres of common woods, wherein be 600: great Limer Trees. 100. Ashes, &c.   Worth by year. The renewer. Parcel. One quarey of Sclate. Twenty marks. Wil. Ashbee. AB. Long meadow. One quarey of Free-stone. Ten pound. Wil. cock. of. Birches. A Moyne of led. An hundred pound. Ri: Rumley. FF. black field. A Moyne of Iron ston. Fifty pound. Ro. Dod. HH: The sleade. A Moyne of coal. 4. hundred pound. Ri. Preene. D. The Manor is charged. With one ANnuity of twenty pound for 18 yeares, Rent charge of one hundred pound.  
Note the Demesnes with the site of the Manor house, &c. being faire and in good reparations contains 500. acres, 220 of errable, of pasture 150, of meadow 130, left vnplatted.  
Also the Tenants claim paughnage for four pence a swine in the common wood.  
Place this Table and May between the folioes 254. and 255.  
the other perches.  
But to proceed to the estate, I find that William cock was tenant for yeares, then if you proceed in a right line rightwards until you come under the title of estates, there shall you find that his estate was for twenty one yeares, standing under time leased: next in the row under time expired, there shall you find eight yeares, and so much of his twenty one yeares were expired, at the time of taking the survey. In the next row, under suruises there shall you find this mark* which refers you to the little table of suruises, below the other great table: and so often as you find that star there, so many suruices is the said Willam cock bound to do to his Lord, which you shall find to be four times, and those four suruises be.  
First, 2 daies in the harvest to carry corn, 2. to be two daies with his téeme in séednesse. 3. one day with his téeme in lent séednesse. 4. his best beast at his death.  
Lastly under the title of Pated for, you shall find how many sheep, and great beasts he is allowed to put into the common, where you shall find that the foresaid William cock is ranted at 50 sheep, and 2 beasts, as appears under the several titles.  
Then what other commodities rise in the manor,& what the said manor is charged with, you may plainly perceive in the tables under placed of purpose.  
As there is a moyne of iron ston worth 50 pound by year, it is in the ground of Robert Dodde, and then the letter ( HH) refers me to the vpper great table, which sheweth me that the said moyne lieth in his pasture ground, which is called the Slead, containing 3 acres, &c. and if you would see where the said piece of ground standeth in the Lordship, the letters ( HH) will direct you.  
And here note that if you haue occasion to hedge or ditch& parcel in the manor, you may meat the same with your pair of compass and scale in the foot of the perfit as truly as if you did go into the field, and so may you know what the doing thereof is worth &c. and so set hedgers &c. tax work.   

CHAP. L.  
certain Apopthegmes requisite for all Geodetors or measurers of ground to understand.  

Geodeticall Apopthegemes.  AL grounds be either naturally good of themselves, and such be known by the herbs, &c. as they yield; or artificially good, and those are made by endeavours and painful travels in manuel labours.  
Grounds naturally good, bring forth 

Wall words  
Soft rushes.  
mallows.  
three leafed grass.  
strait and faire trees.  
Broad elms.  
Great oaks and Ash.  
slow, or Bullace trees.  
Wilding, called crab-trées.  
black thorns.  
Black herrie trees  
corn, thistles,& great thistles.  
White or read honisucles.  
Hony suckle leans.  
grass thick at the ground.  
Good wild fruits.  
Good wild herbs.  
Or any one of all the foresaid.    
artificial good grounds, are  
Made by continual heaping of dung thereupon, such are about London.  
Where old Castles or buildings haue been situate.  
Where pools haue heretofore been, by carrying the mud forth, but then it is best to let itly vpon heaps till it be dry, and after carry the same vpon the land and spread it.  
Or where land floods& waters be brought over the ground.  
Made by letting the grass grow 6 or 7 yeares vneaten.  
Where many sheep are folded, which they use in some countries, and so save dunging their errable grounds, by burning the ground:  
But grounds would be considered according to the nature thereof, viz. hot or could, apply to 

hot: mud.  
could, slime, &c.  
cold: lime ston.  
marle, &c.    
signs of fertile grounds, and such as be rich, are  
Where crows and Pies follow the plough.  
Mellow and fat grounds that will soon be dssolued best for corn.  
Such which be fat and stiff best for corn.  
Where fern doth grow. best for corn.  
The colour of the earth being like new war.  
The swelling of the earth, and waxing black and dirty after rain.  
A black or yellowish earth, which keepeth not rain water but drinketh it up.  
Where the face of the earth is not hard in winter.  
Where the earth doth yield a pleasant savour after sun set.  
Where after a shewer of rain following a great drought, you find pleasing smells.  
If a piece thereof haue a good taste after it hath line four houres in pure water.  
Which being wet waxeth clammy, and cleaveth unto your fingers like doughty.  
If a digged pit lye open three or four dayes, and the mould earth taken thence being cast in again, afterwards swell or rise above the brim of the pit being trodden down.  
signs of barren, lean and unfruitful ground.  
If it rise but to the brim of the pit, indifferent ground.  
If it shrink in under the brimmes of the pit, or gape or channe, barren ground.  
Where the earth wet with rain doth harden& show white.  
Which be white and hare.  
Which abound with gravel, sand or chalk, taking mixture of earth.  
Which in a temperate summer doth gape and open.  
Which aboundeth with great gravel, stony or glittering dust.  
Which abound with gravel or pebble ston.  
Which is so dry that it burneth the roots of plants put therein.  
Which is heated and gapeth with small heat of the sun beams.  
signs of bad ground, are.  
A rough, stiffer or tough, a fennish, marish or slime, a read dry and thick, a lean and could, a dry lean and stiff ground.  
Where the face of the earth is hard in winter.  
Bad grounds naturally yield  
Short trees or scrubs.  
Alder trees, hard and pricking plants, ill relishing herbs, bitter, Juniper, thorns, briars, fyrses, heath, wormwood, loathsome and ill weeds, withered fern and plants, arguing a could seyle, ill coloured herbs, weeds and plants, arguing a wet ground over abounding with moisture.  
The yearly growth of trees is known  
In some by the knots and ●oints, in some by their branches; but the general way for all is▪ give a slope streake in the rind of the tree, so that it pierce unto the body, then see in the rind so cut, how many seams or scores be therein contained betwixt the body of the tree and the utter side of the bark, and so many yeares growth is the said tree.  
Likewise may you perceive the same if a tree be fallen, by the seams that be within the body of the tree running about the heart, like a number of parallel circles about one center, for so many seams or circles, so many yeares growth is the tree.  
The air according as it is situate breedeth pure and wholesome blood, or corrupt and bad humors in the body of man, and so chargeth or infecteth the heart, the which is the most noble part.  
A sound and pure air is  

Made hote soon after sun rising.  
Made cold soon after sun setting.  
Which yieldeth a pleasant smell after rain.  
clear and not euaporated, open and lightsome.  
Where the North wind bloweth, for it cleanseth the air,  
but causeth dry colds, and hurteth plants,  
Where the East wind bloweth in the morning.   
A corrupt and infectious air, is  

Made by influence of diuers stars.  
Made by the usual blowing of the South-east and by  
South-wind, which gendereth clouds and sicknesses, or by the  
South-wind or west in the evening.  
Where the sun is long a making warm after sun rising.  
Where the air is long hote after sun set.  
Where the air is close, cloudy or thick.  
Where abundance of carrion lieth long above ground.  
Where waters filthy and defiled with watering hemp,  
throwing of carrion, emptying of vaults &c, stand long.   
Where multitudes of uncleanly people lye in small room, many ill savouring trades in narrow lanes wanting air, with vnclensed cellars and such ground rooms, abounding too much in London.  
Waters are often lead by pipes and gutters unto houses, and the finest leading of them in descending grounds, is in trenches, filled full of pebbell stones, and covered over with earth: and you must note that as there be corrupt and unwholesome airs, so be there bad and naughty waters unfit for the use of man  
That water is best and purest for mans use which,  

Runneth Northwards.  
Runneth out of a hill into the East, from the North, and  
is the best that runneth farthest in course.  
Runneth swistly through stones, rocks, &c. and from a hill.  
Hath no savour, smell or colour, but is thin and pure.  
Will take the colour of any thing cast therein.  
Will be soon hot and could.  
Is warm in winter and could in summer.   
Yeeldeth no stain after a drop or two, is let dry vpon a polished steel glass, plate of silver, brass, which boiling yieldeth least filth to be scummed off, or being poured out leaveth the vessels bottom cleanest.  
Is lightest in wait.  
Is taken in summer after a clap of thunder, or after a mildred shewer, but not such as falleth off houses and out of gutters.  
The bad and unwholesome waters be such, which are opposite unto the former, as such which runs into the South, which be heavy, ill savoured, and stain what they fall vpon, which run flow, which is long in heating.   

CHAP. LI.  
Of surveying of ground, of making a new kind of particular of apportionating lands, of ancient measure, and buying annuities.  
surveying of grounds consists of two parts, Mathematical and legal the mathematical is performed instrumentally, as in the first parte of this book, the instrumental operation is either to deliver the true contents of every particular, by plot or ingrosement; to ingrose the contents of any manor is to measure every particular truly and precisely, and to writ the same in latin orderly, in a faire book with broad margin notes, easily and quickly offering the contents to the eye, and this is performed after diuers wales, some begin with the side of the manor place, then proceed to the common fields, &c. butting and bounding every several particular, which is not material, unless there be free holders in the manor, or one man holding free hold, and copy, or leass land both. Others begin with the circuit, then go to the fight, next to the Auousonage, and then to the tenants.  
But the best way, in my fantasy, is first having given the title to begin with the Maner house, noting the buildings, &c. then to proceed to the demesnes, then to the Rectory, next to the freeholders, next to the Tenants for yeares, lives, will, &c. noting down every several particular by name, with the true quantity of acres, and what an acre is worth: For you must note that some acres be better then some, according to the place it lieth in. For meadow grounds in some vpland grounds may be worth but ii●j s. the acre, in some places worth v. s, in some x. s. in some xx. s. or xxx. s. or xl. s. yea in some I know meadows let for i●j. l. the acre, and yet the Tenants haue a good bargain. Now all this being noted, draw a right line to contain every particular belonging to that tenement, as in the end of the topographical glass.  
Writing in the right margin in the midst of the said right line the yearly rent, and at the lower end the improvement: And last by collect, how many acres of meadow, how many of pasture, &c. each tenement hath belonging thereunto, and then say, Inpratis 40 acr. in pasturis 80 ac. &c. according as you may gather by this one tenement under written, in manner and form aforesaid.  

Superuis. maner. de Sale ibidem capt. per ambulationem examinationem& mensurationem tam, A. H. general, superuisor. quod T.N. Domini maner. predict.& per mandat. eiusdem, quam per sacrament. R. B, &c. tuncibid existent. 12. die Octobris, anno Domini 1606, annoque regni lacobi, Dei gratia, Angliae Franciae,& Scotiae.   

johannes at Noke tenet per Indenturam gerent. dat. 28 Septembris, Anno regni domini Regis, &c. vnummesuag. siue tenementum cum certis terris, viz.  DOmum mansual. iacent. inter regiam viamibidem ex parte occidental.& oriental. inter tenementum A. D.& communem campum ex parte boreal.& le broad street ex parte Austral.& continet 28 perticum in latitudine& 62 perticum in longitudine,& mesuagium siue tenementum praedictum sufficienter constructum ac etiam tegulatum cum vna aula, vnum caenaculum vocat. a parlour, octo cameris& coquina cum uno stabul.& uno horreo, acle oxhouse, ibid cum stramine coopertum continen. 8 bays, ac caeteris domib. necessari is adjacent.   P s d Vnum clauss. prati voat. le broad meadow, cont. 20 ac. quaelibet acra valens 30. s P 30. 31. 7. 6. Clauss. pasture. vocat. le healed, cont. 10 acr. qualibet acr. 15 s, & redditper ann. 7. s. 6. d. 7 P 10 s Clauss. terr. ar abil. vocat. le mill field, cont. 20 ac. quaelibet acr. valens 8 s & reddit peranum 1 s 6 d 8 P 0 s Clauss. prati vocat. long meadow, cont. 2 acr. quaelibet acr. 18 s, & reddit per ann. 13 s. 6 d. 1 P 16 s And so of all the rest, as you find it: then say, Valet dimittendum. 47 P 6 s.  
Lastly say according to the true improvement.  
Acres. P. s. d. pratis— 22. 56 0 0 johannes at Noke tenet in pastuis— 10. 7 10 0 terr. arabil.— 20. 8 0 0  
But seldom times you shal come to a Manor, and find all the particulars valued by themselves, as before. unless in some certain places by an ancient reservation: 
To apportion linds.  for commonly the tenement is taken altogether, giuing a fine and some small rent, so that it is hard to come to know what they pay for every acre according to the present rent: therefore where the rent of every several particular is not known, the surneyor must apportion the value and worth of the acres; according to his discretion. For meadows lying barren and high, or low and fenny. yielding green hard rushes, flags and knotty grass, are not worth above 4 s, 6 d per annum: but whether the rent be known of every particular or not, the receined order of apportionating is thus. Get the value of an acre of arable land( but aim not at the best nor the worst, as hereafter) then double the value of the arable land, and to the product add half the value of the said acre of arable, so haue you the worth of an acre of pasture: then multiply the value of the arable acre by 4, and to the product add half the worth of the said arable acre, so haue you the value of an acre of meadow. And note, if the meadow& pasture consist most of a fertile soil, take the value of your arable acre in the best place, &c. But this rule is not certain, and therefore must be amended according as the goodness of the ground shall appear: neither be woods, orchards or gardens subject unto this apportion.  
In the second book and second Chapter I recited a measure set down by Iabian, but that is not certain in all places: For there I  told you, 
Ancient measures of grounds hid of land. Plough land.  that 8 hides make a knights fee, but in the Duchy of Lancaster they count but half so much, to wit, 4 hides, every hid 4 plough land, which is so much as one téeme may till in a year.  
And this plough land is called in the North an Oxgange, every plough land is 4 yard land, every yard land 30 acres, so that every plough land contains 120 acres, every hid 480 acres, and every knights fee 1920 acres: but some account it more, as every kuights fee 5 hides, every hid four yard land, every yard land 24 acres, but in the second book every knights fee is 8 hides, &c.  
Now two knights fee make a Cantred, which according to the first of the former accounts is 3840 acres; 
A Camred.  6 Cantreds& ½⅙ make a Barony, which is 25600 acres, whose relief is 100 marks; 
A Barony.  one Barony and a half makes an earldom, which is 38400 acres, whose relief is 100 pounds: but I mean not that every earldom or Barony of neressity must haue so much and no more: 
A earldom.  onely these were the proportions at the first institution.  
Annuities.  

To purchase Annuities.  And if any annuities be issuing forth of the Manor which you be desirous to purchase, by my tables published in my almanac 1608, seek what 10 P. per. annum is worth for the time proposed,& then making that your radix by the rule of proportion taught often before, obtain your destre, saying, if 10 P per annum be worth so much for so many yeares, what is 60 P. &c. worth? Concerning my almanac for this year 1610, 
My Alman●cke  there was in my absence a most rare kind of Tables for the computation of time omitted for the private gain, and at the especial request( as I hear) of some private man: whereby I was much injuried, my book quiter defaced, and the common weal restrained from the general benefit thereof, as though they or we were tied to a particular method: but if it might haue been a hinderer to any, more then the same might haue benefited many, I then rest well contented.   

CHAP. LII.  
To reduce Statute measure into customary measure, or any kind of measure used in any particular place: or contrary.  
IN the second book of our Geodeticall staff, I gave, certain notes concerning the diuerfitie of measures used in diverse places, and in my Art of Geodetia I omitted to deliver any proposition that might yield the proportion of one to the other: For indeed as the Chapters are there set down disorderly without any commendable consequence, so was my leisure but short for the accomplishment of the same; neither could I intend to alter the first manuscript, but did onely proceed, adding( though not in their proper places) such things I saw wanting, but took them by whole sale, one with another: for my intent indeed was not to publish the same.  

To reduce stasture measure to woodland measure.  But to proceed, when you know the contents of any piece of ground according to Statute measure, and are to deliver the contents thereof according to 18 foot in the perch or pole, commonly called woodland measure, do thus.  
You are first to consider the lowest proportional terms of their lines: for the like proportion that the square of the one term beareth to the square of the other, the like shall the acre of the one bear to the acre of the other, or the square perch of the one to the square perch of the other.  
The lowest proportional terines of 16½ is 11, and of 18, 12,( which you must find by the abbreuiating of thē by 1½:) now the square of 11 is 121. and the square of 12 is 144; by the thirteenth Chapter of the sixth book.  
So then the proportions stand thus, as 121 the square of 11, is to 144 the square of 12, so is an acre of 16½ in the perch to an acre of 18 foot in the perch.  
Therefore for the reducing of Statute measure into woodland measure, as before, multiply the given number by 121, and the product divide by 144, the quotient shall yield you the contents according to woodland measure. 
To reduce woodland measure to statute measure.   
But it may be that the woodland measure of 18 feet in the pole, will be given, and you required to reduce it into Statute measure, which perform thus. Multiply the given number by 144, and make partition of the product by 121, so haue you your desire.  
And here note, that if the given number be expressed in acres, reduce the same into perches for your more ease in working.   

CHAP. LIII.  
Of the plaiting or casting up of one manor, &c. by diverse scales, as Statute measure, and woodland measure, &c.  
I Haue seen it so fall out in plaiting of Manors, Lordships, &c. that there hath been wood grounds therein to be measured, not according to statute measure as the former was, but after 18 foot in the perch, which you may do thus. Measure it by one scale, 
To measure grounds by two scales.  as you did the rest, then afterwards reduce it by the last Chapter. But say that some stubborn people not seen in the art of our reducement, will onely haue it measured in the field, according unto 18 foot in the perch, and not suffer us about the woods to use the Statute measure: wherefore I shall be driven to work by another Scale, otherwise I should thrust the plate out of proportion.  

To measure and plate wood-grounds by 18 foot pole, so that it shall agree with the rest of the plate taken by statute measure.  Therefore I must work thus: take 12 of those parts of the scale I formerly plaited the rest with, and that length divide into 11 equal parts, so haue you made a new scale which you may make as long as you will, according to those 11 equal parts.  
Now may you safely measure according unto 18 foot in the perch, still plaiting or laying the measured lines down by this new scale, in so much the though you measured the manor by a scale of 16½ foot in the perch, and the wode by 18 foot in the perch; yet protracting by this new scale, all the grounds will about and bound in due and true proportion, even as they had been measured by one scale of 16½ foot in the perch.  
For these causes it were not amiss to haue two scales made according to the prescript,  
And take this note, that all grounds measured by a statute perch, and plaited by a scale of 12 in an inch, that you may readily find the contents according to 18 foot in a perch, if you do but cast up the perfit by a scale according to 11 parts in an inch: the reason is because a 11 perches in length of this woodland measure make 12 in length of statue measure; hereby may you render the contents of any wood grounds according to either.   

Chap. liv.  
In the surveying of a manor, how to join all the parcels in one perfit as they lie in the whole manor.  
quibbling by the plain table Theodelitus Geodetical staff, or any other instrument, the best rule I can prescribe you is, 
To join al the parcels in a manor in one plate or carded.  first to get the perfect& direct perimeter of the Lordship, and that to lay down by a large scale vpon faire shéets of paper glued together; then go into every several field in the manor taking the true perfit as you be instructed in the first pare of Geodetia: and then at every night carefully to place those parcels, within the former perimeter in their proper places; which to do you must take notice how one lieth by another, that you misplace not any: having finished, if the particulars just fill up the perimeter, you haue truly wrought.  
But this note, if you perfit by the plain table, take as many particulars vpon one shéet as you may, and when you change your sheet, work as you be taught in my Topographical glass, 
To change the paper in the plain table.  or by help of your needle at every changing of your sheet, strike a Meridian over the new sheet, for your better directing yourself in joining your shéetes, because the Meridians must lye parallel when you join the said shéets.  
Also haue a care in making a perfit to place the houses, buildings, trees, and every notable thing in your perfit.   

CHAP. LV.  
To alter the whole perfit of a manor, and every particular therein, easily and speedily.  
WHereas in the last chapter I instructed you to lay the perimeter of a manor down by a large scale for your more direct working, yet happily you shall be required to deliver the same in a far more less proportion; and tho I haue already set down diuers rules in my arte of Geodetia for the performance thereof instrumentally, yet shall it be nothing amiss to show you an other easy way, 
To alter the perfit of a manor to any proportion.  for that your instrument is not always at hand. By the ensuing Chapter, get the true plate in a faire sheet of paper in such lines that may be rubbed forth again, then to alter the plate to any other proportion do thus. Prepare a rule with a renter hole, divide this rule into as many small equal parts as you can,& number the same by 10: let the rule be of some thin plate.  
Now to work herewith, about the midst of the perfit with a small needle through the center hole of the rule, there fasten the said rule to the paper, and the board that the paper lieth on: then remove the edge of the rule to any corner in any field, and note the equal parts cut, and of these take what part you please according to the proportion assigned as the ½ l⅓ or any other: your rule stil resting at that angle, by the edge of the said rule make a prick according to the proportion taken, as if the parts cut be 60,& you mean to make the plate but half so big that is, then must you make a prick at 30, which is half 60: then turn the rule to the next angle in the said field, and do the like to the proportional part of that number cut, and so laying the rule by every angle make pricks at the proportional part, and then draw a line from prick to prick; so is a figure made less and proportional to the greater; this done rub out with crumbs of bread the black leaded lines of the field, and then proceed to the next, doing so to every field severally until you haue finished, still rubbing forth the black leaded lines as you finish the said field.  

To alter plaits from lesser to greater.  And there you shal note, that as you may by this doctrine alter a great perfit to a less, so may you one the contrary, alter a lesser to a greater, onely noting  
In altering of plaits from the 

Lesser to the greater,  
Greater to the lesser,   You must begin at the 

Perimeter of the perfit.  
midst of the perfit.    
But in this you must not alter the center of your inder from the midst of the perfit, howsoever you alter the perfit either from greater to less or less to greater: certainly this kind of work is most precise and excellent for the altering plattes of manors, sea-cards or such like, tho they use to perform it by cross lines, or squares, which may serve for that purpose, but not so well for this: others use to alter them by circles other ways; but take this tho something mechanical, yet most requisite, for many conclusions stand firm and true in demonstrations geometrical, though otherwise in practise tedious and unnecessary.   

Chap. LVI.  
To take a perfit out of one piece of paper, and place the same truly in an other piece according to the same proportion and fashion it had before, and that diuers ways.  
YOu must take the paper whereupon the perfit is drawn, and rub the back side with black led all over, 
To take any perfit out of one papar and place it in en other.  then lay a faire sheet of paper( or two or three glued together if the plate and large) vpon a plain board, and vpon that lay the leaded side of the perfit or other sheet: fasten them both vpon the board at the 4 corners, then take a sharp pointed bodkin, and with somewhat a heavy hand trace all the lines within the said perfit,& all such other things that you would express: lastly take away the uppermost paper, so will all the lines that you haue traced, remain vpon the lower sheet, which you may rub out with crumbs of bread.  
Otherwise:  
Lay the plate itself vpon the Table, then prepare a paper well oiled with Lin-séede oil, and fasten the same vpon the plate that lieth vpon the table: so shall you see all the lines and other notes in the plate through this oiled paper, as through a clear glass; then with a pen& ink, or a bodkin draw or trace all the lines required: ●o haue you taken the plate out of the lower paper into the oiled paper, which you may prick and pounce, and so place vpon any other paper at your pleasure: by such oiled paper may any simplo body take any work, flower, or such like, forth of any place as he pleaseth.  
Otherwise.  
You may prick all the lines with a small needle, as the order is, and so accordingly with pund coals in a clout pounce the same, and then draw the lines with a pen, or pensell made of black led, or a Sally coal, according as you would haue the lines to remain, or vpon any occasion blotted out. Many forms more art-like I could deliver, but for the causes before remembered these shall suffice.   

CHAP. LVII.  
To garnish your plate.  
YOur plate being finished, and all things orderly contrived, it resteth then for you to garnish and beautify the same with such water colours as is convenient, 
To garnish a perfit.  adding thereunto the due proportion of all houses, buildings, &c. and for your better instruction lay all arable lands in read, all pasture in a russet, or some such kind of colour, and all meadows in green, decking all woodlands with trees, and do not omit the port ways or any other way worthy of note nor any brook, well, cross, or tree of note or other special mark of note: for all such are true ornaments& most requisite to be placed in your carded: for it is absurd to follow many rude painted observations, making trees where none be, laying green where no meadow is, and such like;& take care that you make not your port ways, books, &c. broader then the proportion alloweth. I haue séens diverse draw a brook, being not passing one perch broad indeed, more then 5 perches broad, being applied to their scale; which was absurd: and many more such gross errors concerning the rules of proportion, which I omit. And so for this time I end this Art of Geodetia, desiring my ability were such to answer to the fullness of thy expectation in this volume: 
In my Topographical glass.  hereafter it may be I shall be occasioned to speak more.    
The end of the sixth book.  
Deus hac otiae fecit.   

THE SEVENTH book OF THE GEODETICAL staff, called Trigonometria:  
Containing Longimetria and Altimetria, performed by Synnicall supputation, with a Canon for the dimension of triangles: compendiously calculated for the performance thereof.  


CHAP. I.  
geometrical definitions, propositions and consequences of lines, angles, and triangles, &c.  

Trygonometria  TRigonometria is a doctrine treating of the dimension of triangles, of which there be two kinds, right lined and spherical; we onely at this time speaking of right lined angles.  

A point.  2 A point is a sign in a magnitude vndiuisible. Ra. l. 1. p. 6.  

A right line.  3 A line is a magnitude onely long. Ra. l. 2. p. 2. Or a line is the shortest extension between two points; or a line is the motion of a point.  
4 A touch is made in a point, Eucl. l. 13. p. 3.  

Crooked lines.  5 A crooked line is a Circumference, or Helix. Ra. l. 2. p. 7.  
6 A Circumference is that which is equally distinct from the midst of the space contained in it. ibid. p. 8.  
7 A Circumference is therefore made by the running of a line equi-distant about the renter, the one end resting fixed; or a Circumference is made by the motion of a point equally about the Center.  
8 An Helix is that which is unequally distant from the midst of the space that is contained within: but we need not this Prop.  
9 Parallel lines are such that be equally distant one from the other every where. Eu. 35. p. 11. 
Parallels.  Therefore parallel lines being exiended infinitely never intersect.  
10 If one line be parallel to many, all those lie parallel one to another, or amongst themselves.  
11 An angle is a liniate consisting in the common section of the terms. Ra. l. 3. p. 3. or an angle is made by the concourse of two lines.  
12 The feet of the angle are the terms comprehending the angles. Ra. ibid. p. 4.  
13 Angles homogeneal, are angles alike in all respects.  
14 Ifangles haue equal feet, they be also equal. R. l. 4. p. 5.  
15 An angle is either right or obliqne.  
16 A right angle is made by a perdendicular falling vpon the base, or which doth contain the fourth part of a circled.  
17 In right angled triangles the line subtending the right angle is called the hypothenusal, either of the other terms containing the right angle, may be called the perpendicular, or the base.  
18 An obliqne angle is either obtuse or acute.  
19 An obtuse angle is an obliqne angle greater then a right angle. Eu. l. 1. def. 11.  
20 An acute angle is an obliqne angle, less then a right angle. Eu. ibid. Def. 12.  

Figures.  Of a figure.  
21 A figure is a lineate magnitude, limited on every side. Ra. l. 4. p. 1.  
22 A center is the middle point in a figure.  
23 A perimeter is that which doth enclose a figure.  
24 A radius is a right line extended from the Center to the perimeter.  
25 A diameter is a right line inscribed in a figure, and drawn through the center. therefore 

1 In one figure there may be infinite diameters.  
2 The center is always in the diameter.  
3 If there be many diameters the center is in the concourse thereof.  
4 If two diameters intersect, the center is in the intersection.    
26 If a right line fall vpon a right line, the angles there made are equal unto two right angles, and contrary. Euc. l. 1. p. 14. 13.  
27 If two lines cross one the other, the vertical angles be equal, and all the angles equal to fours right angles.  

triangles.  28 A triangle is a right lined plain contained under three angles, and three right lines.  
29 Any two sides of a triangle are greater then the third side remaining. Ra. l. 6. p. 7.  
30 If a right line in a triangle be parallel to the base, it cutteth the feet proportionally, and contrariwise. Eu. l. 6. p. 2. and being parallel the angles be equal.  
31 The three angles in a triangle are equal unto two right angles. Eu. l. 1. p. 32. Therefore 

1 Any two angles in a triangle are lesser then two right angles.  
2 The side of a triangle being continued forth infinitely, the outward angle is equal to the two inward angles opposite, and greater by the same consequence then any inward angle opposite.    
32 If a triangle haue equal feet, the angles at the base are equal and contrary. Eu. l. 1. p. 5.& 6. Consequently 

1 An equiangle triangle is also equilater, and contrary.  
2 An angle of an equilater triangle is as great as ⅔ of a right angle. Ra. l. 6. p. 10. Cons. 3.    
33 The greatest side of a triangle subtendeth the greatest angle, and contrary. Eu. l. 1. p. 19.& 18.  
34 triangles of equal height are in proportion as their bases. Eu. l. 6. p. 1.  

Circles.  Circles.  
35 All Circles in Trygonometria are divided into 360 equal parts, and so into minutes, &c. accordingly. So that the quadrant of a circled contains 90 degrees.  
36 The given ark in the lesser quadrant is the compliment to the distance thereof from 90 as 40, the given ark is the compliment to 50, the distance thereof from 90.  
37 The ark in the greater quadrant is an excess above 90, as 140 is an ark in the greater quadrant exceeding 90, 50 degrees.  
38 All lines drawn from the center to the circumference be of equal length.   

CHAP. II.  
having given you some necessary geometrical instructions, for the better opening of your understanding, now shall follow the axioms of Trigonometria itself, which chiefly is effected by the rule of proportion: the first axiom shall teach what proportions shall be in the triangles or parts thereof; afterward shall be declared, how those axioms may be applied to use.  
Axioma 1. In plain rectangled triangles, any one of the three sides may be put for the radius,  
For  
FIrst, if you put the side subtending the right angle for the radius, the sides including the right angle are the sins of the acute opposite angles, As in the plain triangle A B C, if you put the side A B, subtending the right angle for the Radius, then B C, the less side including the right angle is the sine of the lesser acute angle opposite thereto, as A B C; and the greater side A C including the right angle, is the sine of the greater angle opposite as A B C.  
2 If you put the greater side including the right angle for the Radius, then the lesser side including the right angle is the Tangent, and the side subtending the right angle is the secant of the lesser acute angle.  
As if you put the greater side including the right angle A C for the Radius, then is the lesser including side B C the Tangent of B A C, the lesser angle opposite thereto, and A B, the line subtending the right angle, the secant of the said acute angle.  
3 If to conclude, you put the less side including the right angle for the Radius, the greater side including the right angle is the Tangent, and the side subtending the right angle is the secant of the greater acute angle.  
As if you put the less side B C for the Radius, then the greater side A C including the right angle, is the Tangent of the greater acute angle opposite there to A B C, and the subtending side A B is the secant of the said acute angle.  
Consectary 1. Therefore in plain right angled triangles  
The angles being given, the proportions of the side are had three manner of ways.  
One side together with the angles being given, the rest of the sides are had after a threefold proportion, according to the side that you put for the Radius.  
As in the plain right angled triangle proposed A B C, the given angle A, is 30 degrees 20 min. and thereby B 59 degrees, 40 min.( for the one acute angle is the compliment to the other, as you may gather Chapter 1, def. 30, therefore in plain right angled triangles, one of the acute angles being given, the rest are likewise had) so as I say the angle at A is 30 degrees, 20 min. and at B 59, 40 whereby the proportion of the sides stand thus.  
Either. 1 A B is the Radius. 100000 B C the sine of the acute angle B A C 50502 A C the sine of the acute angle A B C 86310 Or 2 A C, is the Radius. 100000 B C, the Tangent of the acute angle B A C 58513 A B, the secant of the said acute angle 115861 Or to conclude, 3 B C is the Radius 100000 A C, the Tangent of the acute angle A B C 17901 A B, the secant of the said acute angle 198008  
Which is to say.  

1 As A B/ 100000 the radius is to B C/ 50502 the fine of the acute angle B A C.  
2 Or, as A B/ 115861 the secant is to B C/ 58513 the tangent of the acute angle B A C.  
3 Or to conclude, as A B/ 198008 the secant of the acute angle A B C is to B C/ 100000 the radius, and so of the rest.   
1 Besides the angles given there is also given the side A B 4 feet, and the side B C is required how many feet it conaines: Therefore say  
Either.  
1 As A B/ 100000 the radius, is to B C/ 505029 the right sine, so is A B/ 24 feet the side, to A C 12 12671/ 100000 feet.  
Or  
2 As A B/ 115861 the secant, is to B C/ 58513 the tangent, so is A B/ 24 feet to B C 12 17986/ 115861 feet.  
Or to conclude:  
3 As A B/ 198008 the compliment of the secant, is to B C/ 100000 the radius, so is A B/ 24 feet the side to B C 12 23902/ 198008 feet.  
2 Likewise if the same side A B 24 feet were given, and the side A C required, Say  
Either.  
1 As A B/ 100000 the radius, is to A C/ 86310 the right sine, so is A B/ 24 feet the side, to A C 20 71444/ 100000 feet.  
Or  
2 As A B/ 115861 the secant, is to A C/ 100000 the radius, so is A B/ 24 feet the side, to A C 20 82776/ 115861 feet.  
Or to conclude.  
3 As A B/ 198008 the compliment of the secant, is to A C/ 170901 the tangent, so is A B 24 feet to A C 141465/ 198008 feet.  
3 In like manner, if the side A C 20 71444/ 100000 feet be given, and the side B C required: Say  
Either  
1 As A C/ 86310 the right sine is to B C/ 505027 so is A C 20 71444/ 100000 feet, to B C 12 16418/ 86●10 feet.  
Or,  
2 As A C/ 100000 the radius, is to B C/ 58523 the tangent, so is A C 20 71444/ 100000 feet, to B C 12 12071/ 100000 feet.  
Or to conclude.  
3 As A C/ 170901 the tangent of the compliment, is to B C/ 100000 the radius, so is A C 20 71444/ 100000 feet, to B C 12 26636/ 170901 feet.  
But the witty Artist skilful in the use of this treatise of Trigonometria, will proportion his numbers in such order that the Radius shall always haue the first place, thereby to avoid the great labour in dividing such great numbers.  
consectary 2. By the giuing of two sides which soever they be, both the acure angles are obtained by a double proportion, according to the side that you appoint for the radius.  
1 AS in the plain right angled triangle A B C, if two sides including the right angle be given, as A B and B C, the one being 5 feet, the other 3 feet, and the acute angles at A and B are required; Say  
1 As A B/ 5 Feet is to B C/ 3 feet, so is A B/ 100000 the radius, to B C/ 59995 the sine of the angle B A C, answering to which sine in the Table vpon the left side the page. is 36 degrees, 52 minutes, and the last row in the Table vpon the right hand is 53 degrees, 8 minutes, the compliment of A B C.  
Or  
2 As B C/ 3 Feet is to A B/ 5 Feet, so is B C/ 100000 the radius to A B/ 166679 the secant of the angle A B C, to which secant vpon the right hand in the right Table answers 53 degrees, 8 minutes, and in the left margin in the left Table answering thereunto is the compliment of the Angle 36 degrees, 52 minutes.  
Likewise in the triangle A B C, if there be given two sides containing the right angle, as A C 4 feet, and B C 3 feet, and the acute angles B and A, be required, you may say  
Either,  
1 As A C/ 4 feet is to B C/ 3 feet so is B C/ 100000 the radius to B C/ 74991 the tangent of the angle B A C, the ark answering in the Table is 36 degrees, 
The second figure in the first axiom serveth here.  52 minutes, and the compliment of that angle is 53 degrees, 8 minutes.  
Or,  
2 As B C/ 3 feet is to A C/ 4 feet so is B C/ 100000 the radius to A C/ 133549 the tangent of the angle A B C, the ark answering to which tangent in the right table is 53 degrees, 
The third figure in the first axiom serveth here also.  8 minutes, and the compliment of the said ark in the last row leftwards in the left Table is 36 degrees 52 min.  
Axioma 2. In all kind of plain triangles  
The sides haue like proportion unto the other, as the signs of the opposite angles haue, respecting the said sides.  
The signs be equal with the half of the subtensions, and the sides of triangles haue like proportion unto the other as the lines haue subtending the said angles: the like therefore hath the half subtension; for what proportion the whole hath to the whole, the same hath the half to the half; for the same proportion as 16 hath to 6, the like hath 8 to 3.  
 
Demonstration: if about the plain triangle A B C. there be circumscribed a circled, the side A B is made a line subtending the angle A B C, that is the ark A B, which meeteth with the angle A C B; the side B C is made the subtension of the angle B A C, because it meeteth with the angle A C B: to conclude, the side A C is made to subtend the angle A B C, that is the ark A C, because it meeteth with the angle A B C.  
Therefore the side A B bears itself in such proportion to the side B C, as the subtention of the angle A C B, doth to the subtension of the angle B A C, &c. which was required to be demonstrated.  
consectary 1.  
Therefore the angles being given, the proportion of the side● be also found.  
And by the consequence  
Besides, the angles having also one side, both the other sides are easily obtained.  
 
As in the plain obliqueangled triangle A B C, the angle at A 20 degrees, 10 minutes, at C 60 degrees, 23 min. is given, and therefore B 99 deg. 27 mini is also given, then the proportion of the sides be found thus.  
A B 86935 the sine of the angle A C B, 60 degrees, 23 min.  
B C 34475 the sine of the angle B A C, 20 degr. 10 min.  
A C 98642 the sine of the angle A B C, 99 deg 27 min.  
Therefore if further there be given one side, as A B 34 feet, the other sides B C and A C, are also to be had: For  
1 As A B/ 869●5 is to B C/ 34475● so is A B/ 34 feet to B C 13 42660/ ●6935 feet.  
In like manner.  
2 As A B/ 869●5 is to A C/ 98642● so is A B/ 34 feet to A C 38 ●9324/ 89985 feet.  
Veltranspositis terminis intermediis.  
2 As A C B/ 8663● is to A B/ 34 feet, so is B A C/ ●4475 to B C 13 42/ 8● feet.  
2 As A C B/ 8693● is to A B/ 34 et so is A B C/ 98642 to A C 38 50/ ●● feet.  
consectary 2. Two sides of a triangle being given, with an angle opposite to one of the said sides, to find the other side, and the other two angles opposite to the two sides.  
AS in the obliqne angled triangle A B C, two sides being given A B 34 feet, and B C 13 42660/ 86935 feet, with the angle A C B opposite to A B containing 60 degrees 23 min. You shall also be brought to the knowledge of the angle A B C opposite to any other of the given sides, to wit A C, for by the giuing of the angle A C B 60 degrees 23 min, the fine of that angle 86935, is also had: Therefore I say, as A B the side 34 feet is to B C the side 13 41669/ 86935 feet, so is A B the sine of the angle A C B 86935 to B C the sine of the angle B A C 34475.  
Veltransposi●is terminis intermedii●●  
As AB/ 34 feet is to A C B/ 86●35 so is B C, 13 42000/ 86935 feet to B A C/ 34475 To which sine 34475 in the Table vpon the let hand in the left margin 20 gr. and 10 min. doth answer, therefore the angle B A C is 20 gr. 10 min.  
Note, in the use of the former consectary certain doubts may arise, to wit: if there be given two sides, whereof the one is the greater side, together with the angle opposite to the less of the given sides, and there is required the angle opposite to the greater of the given sides, and for because this angle may be acute or obtuse, and so the same sine serve to which you please, having obtained the sine, it is a doubt to which angle it answea●eth.  
 
As in the obliqne triangle A B, if two sides be given A C ●0 feet, and B C 10 feet together with the angle B A C 29 degrees and 59 min. and the angle A B C is demanded opposite to the greatest side, if I should say:  
As the side B C 10 feet is to the side A C 20 feet; so is B C the sine of the angle B A C 49974 to A C to the sine of the angle A B C, for the sine of A C is easily found 99947, but because this sine is as well the sine of the acute angle A D C 88 degrees 8 min. as of the obtuse angle A B C 91 degrees 52 min. it is doubtful whether the obtuse angle of 91 degrees 52 min. or the acute angle of 88 deg. and 8 min, be found by that sine.  
Neither can this doubt otherwise be resolved, then by the doctrine before to find out the quantify of the said unknown angle.  
Axioma III. In all kind of plain triangles.  
AS the whole of two sides is to the difference of the same, so is the Tangent of the whole half of the two opposite angles, to the Tangent of the difference, that is, more or less then the half.  
Declaration: in the plain obliquangled triangle A B C, I say, that the tangent of the whole half of the two angles at A and B, is to the Tangent of the difference of the angle B more, and of the angle A less then the half, as the whole of the two sides B C, and A C opposite to those angles, is to the difference of the same.  
 
Demonstration: describe a quadrant A B C, and therein appoint the angles D A E and E A C equal to the half angles in the former scheme A B C, and B A C, and therein let be the whole half of those two angles, which is héere contained in the angle D A C; let the half of that whole be D A F, or F A C, the difference of the angle D A E more then the half of D A F, or of the angle E A C, less then the half of F A C, is the angle F A E. The subtension of the whole of those two angles is the right line D C, the sine of the angle D A E is D G, the sine of the lesser angle E A C is the right lins C H; the Tangent of the whole half of the two triangles is F N or F K, the Tangent of the difference less, or more then the half is F E: now the triangles G D P, and C P H, are equal by the Chapter 1, def. 26, so that we may infer proportional terms thus; as P D is to G D, so is P C to C H, as is proved 4. P. 6. Eucl. And so in the first figure.  
As A C is to A B C, iso is B C to B A C, by the third Axioma, therefore the sides D P and P C, haue in like manner the same proportion in this figure, as the sides A C and B, haue in the first scheme.  
Wherefore we may boldly take the right line D C for the whole of the two given ●ids A C and B C, and the partes D P and P C, for those two sides A C and B C: the which so being, N P is the difference of the two sides, but the sides A C and B C in the first, and D P and P C in the second scheme are given, therefore the difference of the same side N P is also given together with the half of the difference thereof, as O P.  
Furthermore, because the triangles composed of A K L F E M and A D N O P C, are together equiangled by reason of the parallels D C and K M, therefore both the sides and the segment of the sides be proportional:— so that,  
As D C the whole of the two sides, is to N P the difference of the same:  
So is K M the double Tangent of half of the two angles to L E the double Tangent of the difference more or less then the half.  
Or  
As O C the half of the whole of the two sides, is to O P the half of the difference thereof:  
So is F M, the Tangent of the whole half of the two opposite angles, to F E the tangent of the difference more, or less then the half,  
Or with more ease.  
As the whole of the two sides is to N P, the difference of the same, so is F M, the Tangent of the half of the two opposite angles, to F E the Tangent of the difference, more or less then the half. For as the whole is to the whole, so is the parte to the parte: therefore as the whole K M is to the part L E, so is the half F M, to the half F E.  
Consectarium. Therefore in plain obliqne angled triangles, two sides being given with the angle comprehended by the same, the other two angles are also found.  
As in the plain obliquangled try angle A B C, by the giuing of the two sides A C 5, and B C 3 feet together with the angle A C B 40 degrees, the other two angles A B C and B A C are found, thus.  
 
The whole of the two given sides is eight, the difference two.  
The whole of the angles at A and B is 140 degrees by the Chap. 1. def. 30  
 
The half of that whole is 70 grad. whose Tangent is 274747  
Therefore I say,  
As the two given sides 8, is to 2 the difference of the same, so is the Tangent of the half of the opposite angles 274747 to 68685 the Tangent of the ark, 34 degrees, 29 min, the difference of the angle A less, and of B more then the half.  
So that it is thus.  
  deg. mi.   deg. mi.   70 0   70 0 Subtr. 34: 29. add. 34: 29. The angle at A 35: 31. The angle A B C 104 29.  
Axioma V. In all kind of plain triangles,  
As the greatest side is to the other two sides, so is the difference of the rest of the sides, to the segment of the greatest side, which being taken away, the perpendicular falleth in the midst of that which remaineth.  
 
Declaration, let the obliquangle be A B C, whose lest side is A C, the greatest B C, the radius of the lief side A C, the center A, whereon describe a circled C D E F, cut in the other two sides in points at F and E, making A C the radius or semidiameter: furthermore the side A B is produced to D, and B D is the whole of the two sides, A B and A C, for A C and A D be equal. now E B is the difference of the sides A B and A C, for A E and A C again be equal by the Chapter 1. P. 37— so that I say: As C B, is to B D, so is E B to B F.  
Next the right line F G, is cut into two parte by the perpendicular A G falling in the midst thereof.  
consectary. Therefore in plain obliquangled triangles, the three sides being given, to find either of the segments in the greatest side whereon the perpendicular falleth from the greatest angle.  
As in the plain obliquangled triangle A B C, there be the 3 sides given, to wit.  

A B, 20 feet.  
B C, 22 feet.  
A C, 13 feet.   
And there is required the segment of the greatest side B C, in the concourse whereof, to wit, of B G and B C, the perpendicular falleth.  
 
The greatest side is 21 feet the whole of the other two sides viz. of A B 20 feet, and A C 13 feet; is 33 feet, the difference 7 feet.  
Therefore I conclude,  
As BC/ 21 Feet the greatest side is to BD/ 3● Feet the whole of the other two sides, so is BE/ 7 Feet the difference of the other two fids to BF/ 11 Feet, the which segment being taken from B C 21 feet, there remaines F C, 10 feet, whose one half F G, or G C is 5 feet; therefore G C is 5 feet, and G B 16 feet.  
A Manuduction or the use of the former axioms.  
IN all triangles there be 3 sides and three angles, of which any three being given, the other three are found,& that after diuers ways as you may gather in the precedent: axioms, onely excepting that you haue not onely three angles, and no side, for no side being given, no side is found; for because the three angles of one triangle may be equal unto the three angles of another triangle, although'all the sides be vnquall', as three angles in the triangles A B C, and D A E, are equal, because their bases B C and B E be equal, as in Chap. 1 P. 29,& yet the sides of the triangle A B C are far greater then the sides of the triangle D A E: and this axception is onely in Trigonometria; in all other take any of the three, and you may haue the other three, howsoever the question be propounded, the differences whereof follow.  
Aplaine triangle, is either a right angle or an obliquangle.  

1 In plain right angled triangles, either all the angles are given( to wit by the giuing of one of the acute angles) with one side, and the other two sides be sought.  
2 Or there is given two sides with one angle, to wit, the right angle, and the other two angles, and the third side is required.   
Both the which are resolved in the first axiom.  
In plain obliquangled triangles.  
1 Either all the angles be given( which is as often as two be given, for the third is always the compliment unto two right angles by the 30 P. Chap. 1) with one side, and there is sought the other two sides.  
2 Or two sides is given with the angle opposite unto one of the given sides, and there is required the angle opposite to the other of the given sides, together with the third sides.  
3 Or there is given two sides with the angle comprehended by the same, and there is required the other two angles with the third side.  
4 Or to conclude all the three sides be given, and th angles he all required.  
The two first of these differences is resolved by the second Axioma.  
In the third difference the two unknown angles are found by third Axioma, and afterwards the third side by the second Axioma.  
In the fourth difference there is made a dislocation of the plain obliquangle triangle, into two right angles, letting a perpendicular fall from the greatest angle vpon the greatest side, by the fourth Axioma, and so by these right angled triangles every angle is found by the first Axioma.  
¶ But the three former differences may be resolved by the first Axioma, if so that from any unknown angle vpon the fide opposite thereto, a perpendicular be let fall, happening within or without the triangle, according as the case shall require,& thereby the obliquangled triangle is converted into two triangles; whether the perpendicular fall within or without the triangles  
1 As if there be given this proportion; as A B the sine of the angle A C B, is to B C, the sine of the angle B A C, so is A B the side to B C the side by the second Axioma.  
 
To the same effect you may say by the first Axioma:  
1 As A B the radius is to B D, the sine of the angle B A D, so is A B the side to B D E the side.  
Secondly as B D the radius, is to B C the secant of the angle D B C( which is the compliment of the angle B C D) so is B D the side to B C the side.  
 
2 If this proportion be given, as A B the side is to C B the side, so is A B the sine of the angle A C B, to B C the sine of the angle B A C by the second Axioma.  
After the same manner may you say.  
First, as B C the radius is to B D the sine of the angle B C D, so is B C the side to B D the side.  
Secondly, as B D the side, is to A B the side: so is B D the radius to A B the secant of the angle A B D, whose compliment is the angle B A D, which angle B A D added to the angle D B C, maketh the angle A B C:  
3 Lastly, if there be such a proportion as this given: As the whole of the sides A B and A C, is to the difference of the same, so is the Tangent of the half of the angle A B C, and A C B to the Tangent of the difference, being more or less then the half, as by the third Axioma.  
To the same effect may you say by the first Axioma:  
First, as A B the radius, is to B D the sine of the angle B A D, so A B the side to D B the side.  
Secondly, as A B the radius is to A D the sine of the angle, A B D, so is A B, the side to A D the side; which being taken from the side A C, the side D C remaineth.  
 
Thirdly, as D C the side, is to the side B D: so is D C, the radius to B D the Tangent of the angle D C B, which added to the angle B A C, and the whole taken from two right angles, there remaines the angle A B C.  
And if furthermore you would know the side B C, by the 1. Axiom. say, as the radius D C is to the secant of the angle D C B, so is the side D C, to the side B C.  
¶ Thus far of the dimension of triangles by sins, tangents, and secants, which I haue for the most part drawn from the doctrine of Bartholo. Pitiscus Grunbergens. a foreign modern writer, hoping you will take pleasure in the application thereof to the singular use of the Geodeticall staff.  
By this Sinicall doctrine may you find the sides, diagonals and perpendiculars in any figure, and so thereby measure fields, woods, countries, &c. as in the sixth book.   
The end of the first part of Trigonometria.   

TRIGONOMETRIA THE SECOND PART.  
Containing Altimetria and Longimetria, by sins, Tangents and Secants.  
Problema 1. The distance of any Turret, &c. being given, to find the altitude.  
LEt the Turret be B G, and let the given distance be A C, or E G 200 feet, and you desire to know the altitude of the battlement of the said Turret B, above the eye of the observer A, or C: You shall therefore by your staff, as in the second book, Chapter 4, or by your topographical glass, or some such Instrument, observe the angle of altitude C A B 29 degrees, 40 min. and thereby get the angle A B C 60 deg. 20 min. by the second book Chap. 15. Then say.  
As AC/ 86892 the sine of the angle A B C 60/ 200 deg 20 m. feet is to B C/ 4949● the sine of the angle B A C 29 deg. 40 min, so is A C/ 200 feet to B C 113 80204/ 86892, which is well near 114 feet.  
Or As A C/ 199000 the radius, is to B C/ 56●62 the tangent of the angle B A C 29 degrees, 40 min. so is A C/ 200 feet to B C 113 922/ 2000 feet: that is near 114 feet.  
Or to conclude. As A C/ 175556 the tangent of A B C 60 degrees, 20 min. is to B C/ 109000 the radius, so is A C/ 209 feet to B C 113 162172/ 175556 feet.  
All these be referred unto the first Axioma:  
But vulgarly.  
Multiply the tangent B C 56961 by 200 feet, the distance of A C, and the product is 113( 92200, which part by the total sine 100000( as in the end of this treatise you be easily taught to do) so haue you 113 9●2/ 1000 feet, the height of B C.  
problem 2; The distance of any tower being given, to find the hypotenusall.  
A Castle is appointed, the distance of the same assigned, and you be required to deliver the length of a scaling ladder answering to the same distance.  
The given distance is A C 200 feet, and the hipotenusall or scaling ladder A B is required.  
observe therefore as before the angle C A B, &c. then say,  
As A B C/ 86892 60 deg. 20 m. is to A C/ 200 feet the distance, so is A C B/ 1●●000 90 degrees to A B the hipotenusall 230 14340/ 86892 feet.  

The figure of the first problem serveth here.  As A C/ 100000 the radius, is to A B/ 11508● the secant of the angle B A C, 29 deg. 40 min. so is A C/ 200 feet the distance to A B the hipotenusall 230 17/ 100 feet.  
Therefore compendiously,  
Multiply A B, 115085 the secant, by A C, 200 feet the distance, and the product is 23017000, which part by 200 feet, so haue you 230 17000/ 100000 feet, or 230 17/ 100 feet; the hypotenusall A B.  
problem 3. Any part of the distance of a tower being given, to find the rest.  
Let a part of the distance A C be given, as A D, or E F 90 feet, and let the rest of the distance be required, to which you cannot come to meate by reason of waters, ditches, &c. therefore at the first station A, observe the angle B A C 29 degrees, 40 min. and at the second station F, the angle B D C, and so by substraction from 90 by the 30 Prop. Chap. 1. you haue the angle A B C, 60 deg. 20 min. and the angle D B C 44 degrees.  
 
First then subtract the tangent of the angle D B C 96568, from the tangent of the angle A B C 175556, and there rests 78988 the line A D, then say  
As A D/ 789●● the difference of the Tangents is to D C/ 96568 the lesser tangent, so is A D/ 90 feet the difference of the stations to D C/ ●2●● feet the distance of the near station, viz. of D from C.  
Therefore  
Multiply D C 96568 the lesser tangent, by A D 90 feet, the difference of the stations, so haue you 8691120, which denied by A D, 78988 the difference of the stations, and the offcome is 100 feet your desire.  
problem IIII. Any parte of a distance being given, to find out the altitude.  
As AD/ 78988 the difference of the Tangents is to BC/ 100000 the radius, so is AD/ 90 feet the difference of the stations to BC/ 114 feet ferè the altitude.  
Therfore.  
Multiply the total sine 100000 by the difference of the stations 90 feet, so haue you 9000000, which parte by difference of the Tangents, and you haue 114 feet the altitude.  
problem V.  
Any parte of a distance being given, to find the hypothenufall.  
keeping ourselves unto the former angles and terms, first subtract the angle B D C, from 180, so haue you the obtuse angle A D B 134 degrees, by the 30 Pro. Chap. 2● or all is one to add the angle D B C 44 degrees, to the angle B C D 90 degrees, and there shall result the same angle A D B 834 deg. afterward add the angle A D B 134, to B A C 29 degrees, 40 min. the total of which take from 180, so haue you the angle A D B 16 deg. 20 min. then say;  

The figure of the first problem serveth here.  As AD/ 28122 the sine of the angle A B D 16 deg. 20 min. is to A B the sine of the angle A D B 134 deg./ 71933, so is AD/ 90 feet. to A B, 230 491/ 28122 feet.  
Or  
As A B D/ 28122 the sine of 16 degrees, 20 min. is to A D/ 90 feet, part of the distance, so is ● D B/ 71933 the sine of 134 degrees, to 234 401/ 28122 feet the hinothenufall.  
Or if you seek the Hypothenufall D B; say  
As A D/ 28122 is to B D 29 deg. 40 m./ 49495 so is A D/ 90 feet to D B 154 11274/ 28122 feet.  
Or  
As A B D/ 28122 is to A D/ 90 feet, so is ● A D 29 deg 40 m./ 49495 to D B 158 11274/ 28122 feet.  
problem 6. To seek any kind of distance at one station.  
LEt the distance be proposed and your station assigned, from which you may not depart, where plant your hollow staff, as you see cause, which done, you are to consider three things.  

1 If the mark whose distance is required, lie level with your foot or no.  
2 If not level, whether above the level.  
3 Or below the level of your foot.   
1 If the distance desired lys level with your foot, as F E, you shall first observe the angle F A E 75 deg. 58 min. Now for the more ease, it is best that your staff A E stand perpendicular, so shall A E F be a right angle; the angle A being had, you may conclude.  
As A E/ 100000 the total sine, is to F A E 75 deg. 58 m./ 400086, the tangent, so is A E/ 4 feet the length of my staff to E F/ 16 feet the distance.  
Therefore vulgarly,  
Multiply the secant 400086 by 4, so haue you 16( 00344, which divide by the total sine 100000, the quotient yieldeth your desire: 16 344/ 109000 feet, the distance of F E.  
2 If the distance required lie above the level of the lower end of your staff, work thus.  
Q H is your staff,& B a distance required, which lieth higher then H my feet.  
Therefore I seek the angle B Q H 101 deg. then I go to H the foot of my staff, and there observe the angle B H Q 67 degr. by adding these two angles together, and taking the total from 180, I haue the angle Q B H 12 deg. then I conclude by the 2 Axioma, Confect. 1.  
As Q B H 12 deg./ 20791 the right sine is to Q H ● 67 deg./ 92939 the fine of the said angle, so is Q H/ 4 feet the length of my staff to Q B 17 1579/ 20791 feet, the distance.  
By the same axiom may you find the line H B: for As H B Q 12 deg./ 20791 is to the sine of the angle H Q B 102 deg./ 98162( or the sine of 79 degrees, the compliment of 101 deg) so is Q H/ 4 feet the length of the staff to H B 18 18410/ 20791 feet.  
Or  
As 20791 is to, 4, so is 98162 to the distance H B, your desire.  
Therefore vulgarly  
Multiply 98162 by 4, the product whereof is 392648, which parte by 20791, and the quotient is 18 18410/ 20791 feet.  
Thus of any such like question, the second Axioma will direct you plainly.  
Thirdly, before you were placed vpon the top of some cliff or rock, where you could not stir to meet an ordinary stationary line, and yet were required to express the distance of the top of a Turret lying far above your feet.  
Now here also are you placed vpon the like rock where you haue no liberty to stir any way from the place you be in, neither to seek the déepenesse of the said rock with a plumbs line to make a stationary line of,( as in such cases the order is) nor to go any way at all, and yet are required to deliver the true distance of the base of a Turret that lieth far below the unveil of your feet, which to do.  
Erect your staff and observe the angle H Q C, and there the angle Q H C, which adding together as before, get the angle H C Q, having the angles seek their sins, and so work as before, according to the foresaid second axiom, cons. 1.  
And here observe the singularity of this problem, whereby you are taught to seek the distance of any thing, within sight at one station howsoever situate, as from the top of a castle or rock or such like to descry the distance of ships vpon the seas, of armies of men or such like, and contrary being in a low valley, to discover the distance of forts, camps, bulwarks or any other object situate vpon the top of any high hill, and that directly, speedily and readily at one station, with most approved truth after the same order may you stand in the midst of any meadow, and speedily plate the same &c.  
problem VII. To seek the distance betwixt any two towns, any two ships vpon the seas, or any other inaccessible objects whatsoever.  
THis problem is performed after this manner, first get the distance of each town from you by the 6 problem, and then observe the angle at your station, so haue you an angle,& his two containing sides, then may you find the third side,& the other two angles diuers ways by my doctrine of triangles.  
And so referring all other kind of Altitude, Longitude, Latitude or profundities to the discréete carriage of the reader, since they be but relatives to what is said before. indeed we might invent and draw diverse kinds, according to the situation of the object,& so increase the book,& confounded the memory, but the whole ground of all that can be devised or invented is already set down, and therefore I presume that no right lined triangle howsoever situate, shall happen, whose sides you shall be ignorant of, if you do but well observe my doctrine in Trygonometria.  
Note  
¶ And here let it be noted that when the total sine is the first number in proportion, you may audide division, and if it be not, you may so order the proposition, that it shall be in all right angled triangles, as in the first axiom; for if the radius be not in the first place, then either the right sine, the Tangent, or the secant is, all which the first Axioma resolveth, so that you may make which side you please the radius; therefore when I say, but compendiously, as in the end of the first problem, having multiplied the second number by the last, as 56961 in 200, the product is 113,( 92200: and where you should divide this number by the total sine, you need but cut off so many figures from the product vpon your right hand, as there be ciphers in the total sine as you see before, so is the three figures vpon your left hand the quotient, and the five figures vpon your right hand the remainder; and so must you deal in al such kind of works where the total sine is deuisor.  
Example 

The numbers to be multiplied. 

56961  
200    
The product after multiplication 113( 92200  
The total sine 100000  
The quotient 113  
The remainder. 92200    
And here note further, when I say, as such a number is to such a one, &c. so is such a one to such a one &c. you must work by the rule of three, to find out the numbers required.  
As for the purpose:  
As A D/ 2 is to A C/ 3 so is ● D/ 4 to D F/ 6, therefore  
Multiply A C, the second number in proportion by C D, the third number, to wit three in four, so haue you twelve, which parte by A B 2, the first number in proportion; so is the quotient D F; or the fourth in proportion viz, 6 the remainder: and thus of any other, and this in brief is the golden rule or rule of three.  
The description and use of the Canon or Table for the dimension of triangles, by sins, Tangents, and secants, according as I haue newly calculated the same.  
TO audid tediousness in fractions, as also for that few fractions are precisely true, Authors haue devised to divide the semidiameter of a circled into a certain number of equal parts, whereby they express the true quantity of any arch;& the more partes the semediameter is divided into, the truer is your work. Regiomtonanus used 6,000,000, equal parts, and Radium particularum 60,000,000,000. Rheticus made the semidiameter to contain 10,000,000,000, and Radium particularum, 1,000,000,000,000,000, but with us 100,000, equal partes shall suffice, according unto which there be tables of sins calculated by Wittikineus for dials, but not after this method.  
2 In this doctrine crooked or circular lines be reduced to right lines.  
3 Curued lines are reduced unto right lines by the definition of quantity, which they haue, being applied to a circled in respect of the radius.  
4 Any right lines being applied to a circled is called a subtention, which may be sins, Tangents, or Secants.  
5 A subtention is a right line inscribed in a circled dividing the whole circled into two segments, and in like manner subtending both those segments.  
9 A subtention is great or not great.  
7 The great subtention is a line, which divides the whole circled into two equal parts, and therefore doth subtend both the semicircles, as the right line G C, which is also the diameter.  
8 A subtention not great, is, which divides the circled into two unequal segments, subtending them both, as the right lines I B doth, both subtend the lesser ark in the semicircle I F B, and also the greater ark in the semicircle, as I H B.  
9 A sine is right or versed.  
10 A right sine is the half subtention of a double ark, as the right sine of the ark B C, or B G is the line D E, the half subtention of the double ark B C D or B G D, which the line B E D doth subtend.  
Consectary.  
1 Therefore the right sine of the ark in the lesser or greater quadrant even to the semicircle be all on.  
As the right sine of the ark B C and B G, be all one to the right line B E, because it is the half of the right line B E D, which doth as well subtend the ark B G D, as the ark B C D  
2 So that when we say the compliment of the right sine, there is onely meant the compliment of the right sine in the lesser quadrant.  
As the compliment of the right sine B C, to wit B F is the right line B K.  
 
Thirdly, every right sine falleth perpendicular vpon the diameter from the term of the given ark, as B E.  
Fourthly, the right sine of the compliment is equal the to radius or segment of the diameter intercepted betwixt the right sine of the ark and the center, as the right sine of the compliment B F viz. B K is equal to E A.  
11 Sinus versus a versed sine is the segment of the diameter intercepted betwixt the right sine and the circumference.  
As the versed sine of the ark B C is the segment of the diameter E C, and sinus versus of the ark B G, is the segment of the diameter G F,  
12 Therefore of versed fines some be great, others less.  
13 Sinus versus maior is the versed sine of the ark in the greater quadrant as G E, is the versed sine of the ark G F B in the greater quadrant.  
14 Sinus versus minor, is the versed sine of the ark in the lesser quadrant, as E C of the ark B C.  
15 A Tangent is a right line meeting with the secant of the one term of an ark falling perpendicular vpon the diameter, by the other term of the ark.  
Or a Tangent is a right line standing perpendicular vpon the extreme of a diameter at the one term of a given ark, and produced infinitely, until it intersect with the secant of the said ark, as L C is the Tangent of the ark B C.  
16 A secant is a right line produced from the center by the term of a given ark until it touch the Tangent, as A L is the secant of the ark B C:  
17 A Tangent is so called, for that it toucheth the circumference. A secant is so called for that it cuts the same circumference.  
18 Cordus a chord is a circled cutting the total sine at right angles extended from the one term of an arch unto the other, as B D is the chord of the arch B C D.  
19 every cord is a double sine.  
20 the greatest chord that may be in any circles is the diameter, and the greatest sine the semidiameter.  
21 The Tables of sins, Tangents, and secants extend not beyond 90 degrees, for the right sins of the greater and lesser quadrant be all one by the 10 def. as for Tangents and secants they never exceed the quadrant of a circled by the 15 and 16 def.  

1 To come unto the tables themselves, first know, that vpon the sides of every page. there be placed four tables: it would be but vain to make ample relation, since that the superscriptions be plain, onely the difficultnesse is in the degrees and minutes.  
2 Therfore you must note, that we in this calculation, haue two kind of quadrants, and thereby two kind of arkes, as the ark of the lesser quadrant, which cannot exceed 90 degrees, and the ark in the greater quadrant which may be 180 degrees.  
3 The degrees of any ark in the lesser quadrant to 45, are placed vpon the right hand in the top of every fable in great figures, so that every table hath one of these deg. the min. answering to these deg. as in the first colum or row, under these deg. proceeding by 5, as 0. 5. 10. 15. &c.  
4 The minutes are set by 5, because vpon our instruments( notwithstanding my exact dividing of the circled by help of I sosceles triangles) we cannot put a sensible difference in less divisions, and therefore we were content to draw our tables according unto a deg. divided into 12 equal parts, every parte containing 5 min. for it is vain to deliver quantities more precise in figures then we can, finding it by observation, for what would the sine or secant of three min. avail, us when instrumentally we can hardly observe 5 minutes: therefore be our tables be drawn to answer to our instrumental obsernations.  
5 If the ark of the lesser quadrant exceed 45 degrees, you shall find the same in the table vpon the right side the page., at the lower end of the said table, the minutes answering whereunto, do ascend in the last column vpon the right hand.  
6 So that if the given ark be under 45, find the deg. and min. in the table vpon the left hand, if above 45, in the table vpon the right hand.  
7 The compliment of any ark in the lesser quadrant, stands directly against the ark given, but not in the same table; as the ark given is 3 deg. 45 min. those deg. and min. are found in the table vpon the left hand, answering to which in the table vpon the right hand, in the last row, is 86 degrees, 15 minutes, the compliment of 3 deg. 45 min. or if 86& 85 min. had been the given ark, 3 deg. and 45 min. in the other table was the compliment to the same.  
8 Briefly, this table extends but to the half ark of the lesser quadrant, yet doth serve to the whole ark in the greater or lesser quadrant, as in the 10 def. consect. ●. so that you may find any ark or angle given, together with the sins, Tangents and secants answering thereunto; also the compliments of the arkes or of the sins &c. or the sine, Tangent or secant being given, you may find the ark answering thereunto.   
The given ark is 3 degrees 35 minutes, 

Sine 6250  
Tangent. 6262  
Secant 100195    
The compliment of the ark 86 deg. 25 min. whose. 

Sine 99804  
Tangent( is) 1596867  
Secant 1599995    

9 If the sine &c. be given, and the ark required, seek in the body of the tables for the sine, and then remembering the precedent rules, the ark answereth thereunto.  
10 Touching arkes in the greater quadrant, they and the compliments thereof stand in the tables vpon the right hand with the minutes answering: they begin at 90 deg. in the first row at the top of the right tables, and so proceed to 135, and then go forward in the foot of the table, being the lowermost of the two numbers placed there; but if you seek the compliment of those numbers in the other table on the left hand, in the foot of the tables, they be in the table on the left hand, as if you seek the compliment of 176 deg. 25 min. the degrees of compliment is in the table vpon the left hand, and in the row under it is the minute also, as 3 deg. 35 min. which makes 176 deg. 25 min. just 180 deg.  
11 There is no Tangent or Secant in the greater quadrant by the 15 and 16 Def.  
12 So that if you seek the compliment of any ark in the greater quadrant, if the degrees and minutes stand in the first row in the right table vpon the left hand, then the compliment thereof is the uppermost of the numbers in the foot of the said table vpon the right margin, and the minutes stand in the uttermost columbe rightwards ascending.  
13 But if you seek the compliment of any of the numbers that stand lowermost in the foot of the right table, then must the degrees of compliment be sought in the left table.  
14 Subtract the arch given from 90, the remainder is Sinus Complementi of the lesser quadrant.  
15 subtract Sinus Complementi of the arch given from the total sine, the remainder is Sinus versus, as Sinus Complementi is 35027, which take from 100000, and there remaines 64973. Sinus versus is in stead of the right sine of any ark of the greater quadrant, the compliment thereof is used: therefore the ark itself doth serve in these tables, because the true ark stands against the compliment, as well in the great quadrant as in the lesser, as is said.  
16 Deduct the ark given from 280, the remainder is Sinus Complementi of the greater quadrant.   

A Canon or Table for the dimension of Triangles compendiously calculated. 1609.  
Belonging to the 7 book of the Geodedical staff called Trigonometria.  0 sins Tang Secants   90 sins Tang. Secants. M   1     100000 1 10000 Infinite Infinite 60 5 145 145 100000 5 99999 68754887 98754060 55 10 290 290 100000 10 99999 34377371 34377516 50 15 436 436 100000 15 99999 22918166 22918384 45 20 581 581 100001 20 99998 17188540 17188831 40 25 727 727 100002 25 99997 13750735 13751108 35 30 872 872 100003 30 99996 11458865 11459301 30 35 1018 1018 100005 35 99994 9821794 9822303 25 40 1163 1163 100006 40 99993 8593979 8594561 20 45 1308 1309 100008 45 99991 7639001 7639655 15 50 1454 1454 100010 50 99989 6875009 6875736 10 55 1599 1599 100012 55 99987 6249915 6250715 5 60 1745 1745 100013 60 99984 5729869 5728996 0 89   179  

Canon Triangulorum.  1 sins Tang Secants   91 sins Tang. Secants M   5 1890 1891 100017 5 99928 5288211 5289156 55 10 2036 2036 100020 10 99978 4841208 4911406 50 15 2181 2182 100023 15 99976 4582935 4584026 45 20 2326 2327 100027 20 99972 4296408 4297571 40 25 2472 2473 100030 25 99969 4043584 4044820 35 30 2617 2618 100034 30 99965 3818846 3820155 30 35 2763 2764 100039 35 99961 3617760 3619141 25 40 2908 2909 100042 40 99957 3436777 3438232 20 45 3053 3055 100046 45 99953 3273026 3274554 15 50 3199 3200 100051 50 99948 3124158 3125758 10 55 3344 3346 100055 55 99944 2988230 2989903 5 60 3489 3492 100060 60 99939 2863625 2865371 0 88   178  

Canon Triangulorum.  2 sins Tang Secants   92 sins Tang. Secants M   5 3635 3637 100066 5 99933 2748985 2750804 55 10 3780 3783 100071 10 99928 2643160 2645051 50 15 3925 3929 100077 15 99922 2545170 2547130 45 20 4074 4074 100082 20 99917 2454176 2456212 40 25 4216 4220 100089 25 99911 2369454 2371563 35 30 4361 4366 100095 30 99904 2290377 2292559 30 35 4507 4511 100101 35 99898 2216398 2218653 25 40 4652 4657 100108 40 99891 2147040 2149368 20 45 4797 4893 100115 45 99884 2081883 2084283 15 50 4943 4949 100122 50 99877 2020555 2023028 10 55 5088 5094 100129 55 99870 1962730 1965275 5 60 5233 5240 100137 60 99862 1908114 1910732 0 87   177  

Canon Triangulorum.  3 sins Tang. Secants   93 sins Tang. Secants M   5 5378 5386 100144 5 99855 1856447 1859139 55 10 5524 5532 100152 10 99847 1807498 1810262 50 15 5669 5678 100161 15 99839 1761056 1763893 45 20 5814 5824 100169 20 99830 1716934 1719843 40 25 5959 5970 100178 25 99822 1674961 1677944 35 30 6104 6116 100186 30 99813 1634986 1638041 30 35 6250 6262 100195 35 99804 1596867 1599995 25 40 6395 6408 100205 40 99795 1560478 1563679 20 45 6540 6554 100214 45 99785 1525705 1528979 15 50 6685 6700 100224 50 99776 1492442 1495788 10 50 6830 6846 100234 55 99766 1460592 1464011 5 60 6975 6992 200244 60 99556 1430067 1433559 0 86   176  

Canon Triangulorum.  4 sins Tang Secants   94 sins Tang. Secants M   5 7120 7138 100254 5 99746 1400786 1494350 55 10 7265 7285 100265 10 99735 1372674 1376311 50 15 7410 7431 100275 15 99725 1345663 1349373 45 20 7555 7577 100286 20 99714 1319688 1323472 40 25 7700 7723 100297 25 99703 1294692 1298549 35 30 7845 7870 100309 30 99691 1270620 1274549 30 35 7990 8016 100321 35 99680 1247422 1251424 25 40 8135 8162 100332 40 99688 1225051 1229125 20 45 8280 8309 100344 45 99656 1203462 1207610 15 50 8425 8455 100356 50 99644 1182617 1186837 10 55 8570 8602 100369 55 99632 1162476 1166769 5 60 8715 8748 100381 60 99619 1143005 1147371 0 85   175  

Canon Triangulorum.  5 sins Tang. Secants   95 sins Tang. Secants M   5 8860 8895 100394 5 99606 1124171 1128610 55 10 9005 9042 100407 10 99593 1105943 1110454 50 15 9150 9188 100421 15 99580 1088292 1092876 45 20 9294 9335 100434 20 99567 1071191 1075848 40 25 9439 9482 100448 25 99553 1054615 1059345 35 30 9584 9628 100462 30 99539 1038539 1043343 30 35 9729 9775 100476 35 99525 1022942 1027818 25 40 9874 9922 100491 40 99511 1007803 1012752 20 45 10018 10069 100505 45 99496 993100 998122 15 50 10163 10216 100523 50 99482 978817 983912 10 55 10308 10363 100535 55 99467 964934 970102 5 60 10452 10510 200550 60 99452 951436 956677 0 84   174  

Canon Triangulorum.  6 sins Tang. Secants   96 sins Tang. Secants. M   5 10597 10657 100566 5 99433 938306 943620 55 10 10742 10804 100582 10 99421 925530 930916 50 15 10886 10951 100597 15 99405 913093 918553 45 20 11031 11098 100614 20 99389 900982 906515 40 25 11175 11246 100630 25 99373 889185 894790 35 30 11320 11393 100646 30 99357 877688 883367 30 35 11464 11540 100663 35 99340 866482 872233 25 40 11609 11688 100680 40 99323 855554 861379 20 45 11753 11835 100697 45 99306 844895 850793 15 50 11898 11983 100715 50 99289 834495 840465 10 55 12042 12130 100733 55 99272 824344 830388 5 60 12168 12278 100750 60 99254 814434 820550 0 83   173  

Canon Triangulorum.  7 sins Tang. Secants   97 sins Tang. Secants M   5 12331 12426 100769 5 99236 804756 810945 55 10 12475 12573 100787 10 99218 795302 801564 50 15 12619 12721 100805 15 99200 786064 792399 45 20 12764 12869 100824 20 99182 777035 783443 40 25 12908 13017 100843 25 99163 768207 774689 35 30 13052 13165 100862 30 99144 759575 766129 30 35 13196 13313 100882 35 99125 751131 757759 25 40 13340 13461 100901 40 99106 742870 749571 20 45 13485 13009 100921 45 99086 734786 741559 15 50 13629 13757 100941 50 99066 726872 733719 10 55 13773 13905 100962 55 99046 719124 726044 5 60 13917 14054 100982 60 99026 711536 718529 0 82   172  

Canon Triangulorum.  8 sins. Tang. Secants   98 sins. Tangents Secants     5 14061 14202 101009 5 99006 704104 711170 55 10 24205 14350 101024 10 98985 696823 703962 50 15 24349 14499 101045 15 98965 689687 696899 45 20 14493 14647 101067 20 98944 682694 689979 40 25 14637 14796 101088 25 98922 675838 683196 35 30 14780 14945 101110 30 98901 669115 676546 30 35 14924 15093 101132 35 98879 662522 670026 25 40 15068 15242 101155 40 98858 656055 663632 20 45 15212 15391 101177 45 98836 649710 657361 15 50 15356 15540 101200 50 98813 643484 651208 10 55 15499 15689 101223 55 98791 637373 645170 5 60 15643 15838 101246 60 18768 631375 639245 0 81   171  

Canon Triangulorum.  9 sins. Tang. Secants   99 sins. Tangent Secants     5 15787 15987 101269 5 98745 625485 633429 55 10 15930 16136 101293 10 98722 619702 627719 50 15 16074 16286 101317 15 98699 614023 622112 45 20 16217 19435 101341 20 98676 608443 616606 40 25 16361 16584 101365 25 98652 602962 611198 35 30 16504 16734 101390 30 98628 597576 605885 30 35 16648 16883 101415 35 98604 592283 600665 25 40 16791 17033 101440 40 98580 587080 595536 20 45 16934 17182 101465 45 98555 581965 590494 15 50 17078 17332 101491 50 98530 576936 585539 10 55 17221 17482 101516 55 98505 571991 580667 5 60 17364 17632 101542 60 98480 567128 575877 0 80   170  

Canon Trangulorum.  10 sins. Tang. Secants   100 sins. Tangenrs Secants. M   5 17508 17782 101568 5 98455 562344 521166 55 10 17651 17932 101595 10 98429 557637 566533 50 15 17794 18082 101621 15 98404 553007 561975 45 20 17937 18233 101648 20 98478 148450 55749● 40 25 18080 18383 101675 25 98351 543966 553081 35 30 18223 18533 101703 30 98325 539551 548740 30 35 18366 18684 101730 35 98298 535206 544468 25 40 18509 18834 101758 40 98272 530927 540263 20 45 18652 18985 101786 45 98245 526715 536123 15 50 18795 19136 101814 50 98217 522566 532048 10 55 18938 19287 101842 55 98190 518480 528035 5 60 1908● 19438 10187 60 98162 514455 524084 0 79   169  

Canon Triangulorum.  11 sins. Tang. Secants   101 sins. Tangents Secants M   5 19223 19589 101900 5 98134 510490 520192 55 10 19366 19740 101929 10 98106 506583 516359 50 15 19509 19891 101959 15 98078 509733 512583 45 20 19651 2004● 101988 20 98050 498940 508862 40 25 19794 20183 102018 25 88021 495201 505197 35 30 19936 20345 102048 30 97992 491515 501585 30 35 20079 20496 102078 35 97963 487882 498025 25 40 202●1 20648 102109 40 97934 484300 494516 20 45 20364 20800 102140 45 97904 480764 491058 15 50 20506 20951 102171 50 97874 477285 487649 10 55 20648 21103 102202 55 97844 473850 484287 5 60 20791 21255 102234 60 97814 470463 480937 0 78   168  

Canon Triangulorum.  12 sins. Tang. Secants   102 sins. Tangent Secants     5 20933 21407 102265 5 97784 467121 477705 55 10 21075 21559 102279 10 97753 463825 474482 50 15 21217 21712 102329 15 97723 460572 471303 45 20 21359 22864 102362 20 97692 457362 468167 40 25 21501 22016 102395 25 97660 454196 465074 35 30 21643 22169 102427 30 97629 451070 462022 30 35 21785 22322 102461 35 97598 447986 459011 25 40 21927 22474 102494 40 97566 444941 456040 20 45 22069 22627 102528 45 97534 441936 453109 15 50 22211 22780 102561 50 97502 438969 450215 10 55 22353 22933 102596 55 97469 436040 447359 5 60 22495 23086 102630 60 17437 433147 444541 0 77   167  

Canon Triangulorum.  13 sins. Tang. Secants   103 sins Tangent Secants     5 22636 23240 102664 5 97404 430291 441758 55 10 22778 23393 102699 10 97371 427470 439011 50 15 22920 23546 102734 15 97337 424684 436299 45 20 23061 23700 102770 20 97364 421933 433621 40 25 23203 23854 102805 25 97270 419215 430977 35 30 23344 23007 102841 30 97236 416529 428365 30 35 23485 24161 102877 35 97202 413877 425786 25 40 23627 24315 102913 40 97168 411256 423239 20 45 23768 24469 102950 45 97134 408666 420723 15 50 23909 24624 102587 50 97099 406107 418237 10 55 24051 24778 103024 55 97064 403577 415782 5 60 24192 24932 103061 60 97029 401078 413356 0 76   160  

Canon Triangulorum.  14 sins. Tang. Secants   104 sins. Tangents Secants. M   5 24333 25087 103098 5 96994 398607 410959 55 10 24474 25242 103136 10 96958 396165 408591 50 15 24615 25396 103174 15 96923 393750 406250 45 20 24756 25551 103212 20 96887 391364 403938 40 25 24897 25706 103251 25 96851 389004 401652 35 30 35038 25861 103290 30 96814 386671 399392 30 35 25178 26016 103329 35 96778 384364 397159 25 40 25319 26172 103368 40 96741 382082 394952 20 45 25460 26327 103407 45 96704 379826 392769 15 50 25600 26483 103447 50 96667 377595 390612 10 55 25741 26639 103487 55 96630 375388 388479 5 60 25881 26794 103527 60 96592 373205 386370 0 75   165  

Canon Triangulorum.  15 sins. Tang. Secants   105 sins. Tangents Secants. M   5 26022 26950 103568 5 96554 371045 384284 55 10 26162 27106 103608 10 96516 368909 382222 50 15 26303 27263 103649 15 96478 366795 380183 45 20 26443 27419 103691 20 96440 364704 378165 40 25 26583 27575 103732 25 96401 362635 376171 35 30 26723 27732 103774 30 96363 360588 374197 30 35 27863 27889 103816 35 96324 358562 372245 25 40 27004 28045 103858 40 96284 356557 370315 20 45 27144 28202 103900 45 96245 354573 368404 15 50 27284 28359 103943 50 96245 352609 366515 10 55 27451 28517 103986 55 96166 356065 364645 5 60 27563 28674 104029 60 96126 348741 362795 0 74   164  

Canon Triangulorum.  16 sins Tang. Secants   106 sins Tang. Secants M   5 27703 28832 104073 5 96085 346836 360965 55 10 27843 28989 104117 10 96045 344951 359153 50 15 27982 29147 104161 15 96004 343084 357361 45 20 28122 29305 104205 20 95964 341236 355587 40 25 28262 29463 104250 25 959●3 339406 353831 35 30 28401 29621 104294 30 95881 337594 352093 30 35 28540 29779 104339 35 95840 335800 350373 25 40 28680 29938 104385 40 55798 334023 348671 20 45 28819 30096 104430 45 95757 332263 346985 15 50 28958 30255 104476 50 95715 330520 345317 10 55 29098 30414 104522 55 95672 328794 343665 5 60 29237 30573 104569 60 95630 327085 342030 0 73   103  

Canon Triangulorum.  17 sins Tang. Secants   107 sins Tang. Secants M   5 29376 30732 104615 5 95587 325391 340411 55 10 29515 30891 104662 10 95545 323714 338808 50 15 29654 31050 104709 15 95501 322052 337220 45 20 29793 31210 104757 20 95458 320406 335649 40 25 29931 131370 04804 25 95415 318775 334092 35 30 30070 31529 104852 30 95371 317159 332550 30 35 30209 31689 104901 35 95327 315558 331024 25 40 30347 31849 104949 40 95283 313971 329512 20 45 30486 32010 104998 45 95239 312399 328014 15 50 30624 32170 105047 50 95195 310842 326531 10 55 30763 32331 105096 55 95150 309298 325062 5 60 30901 32491 105146 60 95105 307768 323606 0 72   162  

Canon Triangulorum.  18 sins. Tang. Secants   108 sins. Tangents Secants. M   5 31039 32652 105196 5 95060 306252 322165 55 10 31178 32813 105246 10 95015 304749 320736 50 15 31316 32975 105296 15 94969 303259 319321 45 20 31454 33136 105347 20 94924 301783 317919 40 25 31592 33297 105398 25 94878 300319 316530 35 30 31730 33459 105449 30 94832 398868 315154 30 35 31868 33621 105500 35 94786 297430 313790 25 40 32006 33783 105552 40 94739 296004 312439 20 45 22143 33945 105604 45 94693 294590 311100 15 50 32281 34107 105656 50 94646 293188 309773 10 55 32419 34270 105709 55 94599 291799 308458 5 60 42556 34432 105762 60 94551 290421 307155 0 71   161  

Canon Triangulorum.  19 sins. Tang. Secants.   109 sins. Tangents Secants. M   5 32694 34595 105815 5 94504 289054 305863 55 10 32831 34758 105868 10 94456 287699 304583 50 15 32969 34921 105922 15 94408 286356 303314 45 20 33106 35084 105976 20 94360 285023 302056 40 25 33243 35248 106030 25 94312 283701 30081● 35 30 33380 35411 106084 30 94264 282391 299574 30 35 33517 35575 106139 35 94215 281091 298349 25 40 33654 35739 106194 40 94166 279801 297134 20 45 33791 35903 106250 45 94117 278523 295930 15 50 33928 36067 106350 50 94068 277254 294737 10 55 34065 36232 106361 55 94018 275996 293553 5 60 34202 36397 106417 60 93969 274747 292308 0 70   160  

Canon Triangulorum.  20 sins. Tang. Secants   110 sins. Tangents Secants M   5 34338 36561 106474 5 93919 273509 291217 55 10 34475 36726 106531 10 93869 272280 290063 50 15 34611 36891 107588 15 93819 271061 288919 45 20 34748 37057 106645 20 93768 269852 287785 40 25 34884 37222 106703 25 93718 268652 286660 35 30 35020 37388 106760 30 93667 267462 285545 30 35 35156 37554 106819 35 93616 266280 284438 25 40 35293 37720 106877 40 93564 265108 283341 20 45 35429 37886 106936 45 93513 263945 282253 15 50 35565 38053 106995 50 93461 262791 281174 10 55 35700 38219 107054 55 93410 261645 280104 5 60 35836 38386 107114 60 93358 260508 279042 0 69   159  

Canon Triangulorum.  21 sins. Tang. Secants   111 sins Tangent Secants M   5 35972 38553 107174 5 93305 259380 277989 55 10 36108 387●0 107234 10 93253 258266 276945 50 15 36243 38887 107295 15 93200 257149 275909 45 20 36379 39055 107856 20 93147 256046 274881 40 25 36514 39223 107417 25 93094 254951 273861 35 30 36650 39391 107478 30 93041 253864 272850 30 35 36785 39559 107540 35 92988 252785 271846 25 40 36920 39727 107602 40 92934 251715 270851 20 45 37055 39895 107664 45 92880 250651 269863 15 50 37190 40064 107727 50 92826 249596 268883 10 55 37325 40233 107790 55 92772 248548 267911 5 60 37460 40402 107853 60 92718 247508 266946 0 68   158  

Canon Triangulorum.  22 sins Tang. Secants   112 sins Tang. Secants. M   5 37595 40571 107917 5 92668 246475 265989 55 10 37730 40741 107980 10 92609 245450 265039 50 15 37864 40911 108044 15 92554 244432 264097 45 20 37999 41080 108109 20 92498 243421 263161 40 25 38133 41251 108174 25 92443 242418 262233 35 30 38268 41421 108239 30 92387 241421 261312 30 35 38402 41591 108304 35 92332 240431 260398 25 40 38536 41762 108370 40 92276 239448 259491 20 45 38671 41933 108436 45 92220 238472 258591 15 50 28805 42104 108502 50 92163 237503 257697 10 55 38939 42275 108569 55 92107 236541 256810 5 60 39073 42447 108636 60 92050 235585 255930 0 67   157  

Canon Triangulorum.  23 sins Tang. Secants   113 sins Tang. Secants M   5 39206 42619 108703 5 91993 234635 255056 55 10 39340 22791 108770 10 91936 233692 254189 50 15 39474 42963 108838 15 91879 232756 253328 45 20 39607 43135 108906 20 91821 231826 252474 40 25 39741 43308 108975 25 91763 230902 251626 35 30 39874 43481 109044 30 91706 229984 250784 30 35 40008 43654 109113 35 91647 229072 249948 25 40 40141 43827 109182 40 91589 228166 249118 20 45 40274 44001 109252 45 91531 227267 248295 15 50 40407 44174 109322 50 91472 226373 247477 10 55 40540 44348 109392 55 91413 225485 246665 5 60 40673 44522 109463 60 91354 224603 245859 0 66   156  

Canon Triangulorum.  24 sins. Tang. Secants   114 sins. Tangents Secants M   5 40806 44697 109534 5 91295 223727 245059 55 10 40939 44871 109606 10 91235 222856 244264 50 15 41071 45046 109677 15 91176 221991 243475 45 20 41204 45221 109749 20 91116 221132 242692 40 25 41336 45397 109822 25 91056 220278 241914 35 30 41469 45572 109894 30 90996 219429 241142 30 35 41601 45748 109967 35 90935 218586 240375 25 40 41733 45924 110041 40 90875 217749 239613 20 45 41865 46100 110114 45 90814 216916 238857 15 50 41998 46277 110188 50 90753 216089 238106 10 55 42129 46453 110263 55 90692 215267 237360 5 60 42261 46630 110337 60 90630 214459 236620 0 65   155  

Canon Triangulorum.  25 sins. Tang. Secants   115 sins Tangent Secants M   5 42393 46807 110412 5 90569 213638 235884 55 10 42525 46985 110488 10 90507 212832 235154 50 15 42656 47163 110563 15 90445 212030 224428 45 20 42788 47340 110639 20 90383 211233 233708 40 25 42919 47519 110716 25 90321 210441 23299● 35 30 43051 47697 110792 30 90258 209654 232282 30 35 43182 47876 110869 35 90195 208872 231576 25 40 43313 48055 110947 40 90132 208094 230875 20 45 43444 48234 111024 45 90069 207321 230178 15 50 43575 48413 111103 50 90006 206553 229486 10 55 43706 48593 111181 55 89943 205789 228799 5 60 43837 48773 111260 60 89879 205030 228117 0 64   154  

Canon Triangulorum.  26 sins Tang. Secants   116 sins Tang. Secants. M   5 43967 48953 111339 5 89815 204275 227439 55 10 44098 49133 111418 10 89751 203525 226765 50 15 44228 49314 111498 15 89687 202779 226096 45 20 44359 49495 111578 20 89622 202038 225432 40 25 44489 49676 111659 25 89558 201301 224771 35 30 44619 49858 111740 30 87493 200568 224115 30 35 44749 50039 111821 35 89428 199840 223464 25 40 44879 50221 111902 40 89363 199116 222816 20 45 45009 50404 111984 45 89297 198396 222173 15 50 45139 50586 112066 50 89232 197680 221534 10 55 45269 50769 112149 55 89166 196968 220899 5 60 45399 50952 112232 60 89100 196261 220268 0 63   153  

Canon Triangulorum.  27 sins Tang. Secants   117 sins Tang. Secants M   5 45528 51135 112315 5 89034 195557 219642 55 10 45658 51319 112399 10 88968 194857 219019 50 15 45787 51503 112483 15 88901 194162 218400 45 20 45916 51687 112568 20 88835 193470 217785 40 25 46045 51871 112653 25 88768 192782 217175 35 30 46174 52056 112738 30 88701 192098 216568 30 35 46303 52241 112823 35 88633 191417 215964 25 40 46432 52426 112909 40 88566 190741 215365 20 45 46571 52612 112995 45 88498 190068 214769 15 50 46690 52798 113082 50 88430 189399 214178 10 55 46818 52984 113169 55 88362 188734 213589 5 60 46947 53170 113257 60 88294 188072 213005 0 62   152  

Canon Triangulorum.  28 sins Tang. Secants   118 sins Tang. Secants M   5 47075 53357 113344 5 88226 187414 212424 55 10 47203 53544 113432 10 88157 186760 211847 50 15 47331 53731 113521 15 88089 186109 211273 45 20 47460 53919 113610 20 88020 185461 210703 40 25 47588 54107 113699 25 87951 184817 210136 35 30 47715 54295 113789 30 87881 184177 209573 30 35 47843 54484 113879 35 87812 183539 209014 25 40 47971 54672 113969 40 87742 182906 208457 20 45 48098 54861 114060 45 87602 182275 207905 15 50 48226 55051 114151 50 87602 181648 207555 10 55 48353 55240 114243 55 87532 181025 206809 5 60 48480 55430 114335 60 87461 180404 206266 0 61   151  

Canon Triangulorum.  29 sins Tang. Secants   119 sins Tang. Secants M   5 48608 55621 114427 5 87391 179787 205726 55 10 48735 55811 114520 10 87320 179173 205190 50 15 48862 56002 114613 15 87245 178562 204657 45 20 48988 56193 114707 20 87178 177955 204127 40 25 49115 56385 114801 25 87107 177350 203600 35 30 49242 56577 114895 30 87035 176749 204077 30 35 49368 56769 114990 35 86963 176151 202556 25 40 49495 56961 115085 40 86891 175555 202039 20 45 49621 57154 115180 45 86819 174963 201524 15 50 49747 57347 115276 50 86747 174374 201013 10 55 49873 57541 115373 55 86675 173788 200505 5 60 50000 57735 115450 60 86602 173205 200000 0 60   150  

Canon Triangulorum.  30 sins Tang. Secants   120 sins Tang. Secants M   5 50125 57929 115567 5 86529 172624 199497 55 10 50251 58123 115664 10 86456 172047 198998 50 15 50377 58318 115762 15 86383 171472 198501 45 20 50502 58513 115861 20 86310 170901 198008 40 25 50628 58708 115959 25 86236 170332 197517 35 30 50753 58904 116059 30 86162 169767 197029 30 35 50879 59100 116158 35 86089 169203 196544 25 40 51004 59296 116258 40 86014 168642 196062 20 45 51129 59493 116359 45 85940 168084 195582 15 50 51254 59690 116460 50 85868 167529 195105 10 55 51379 59888 116561 55 85791 166977 194631 5 60 51503 60086 116663 60 85716 166427 194160 0 59   149  

Canon Triangulorum.  31 sins Tang. Secants   121 sins Tang. Secants. M   5 51628 60284 116765 5 85641 165880 193691 55 10 51725 60482 116868 10 85566 165336 193225 50 15 51877 60681 116971 15 85491 164794 192762 45 20 52001 60880 117074 20 85415 164255 192301 40 25 52125 61086 117178 25 85339 163719 191843 35 30 52249 61280 117282 30 85264 163185 191388 30 35 52373 61480 117387 35 85187 162653 190935 25 40 52497 61680 117492 40 85111 162124 190484 20 45 52621 61881 117598 45 85035 161598 190036 15 50 52745 62083 117704 50 84958 161074 189591 10 55 52868 62284 117810 55 84881 160552 189148 5 60 52991 62486 117917 60 84804 160033 188707 0 58   148  

Canon Triangulorum.  32 sins Tang. Secants   122 sins Tang. Secants M   5 53115 62689 118025 5 84727 159516 188269 55 10 53238 62892 118133 10 84650 159002 187834 50 15 53361 63095 118241 15 84572 158490 187401 45 20 53484 63298 118350 20 84495 157980 186970 40 25 53607 63502 118459 25 84417 157473 186541 35 30 33729 63707 118568 30 84339 156968 186115 30 35 53852 63911 118679 35 84260 156465 185692 25 40 53975 64116 118789 40 84182 155965 185270 20 45 54097 64322 118900 45 84103 155467 184851 15 50 54219 64527 119012 50 84025 154971 184434 10 55 54341 64734 119123 55 83946 154477 184020 5 60 54463 64940 119236 60 83867 153986 183607 0 57   147  

Canon Triangulorum.  33 sins Tang. Secants   123 sins Tang. Secants M   5 54585 65147 119549 5 83787 153497 183197 55 10 54707 65355 119462 10 83708 153010 181789 50 15 54829 65562 119576 15 83628 152525 182384 45 20 54050 65771 119690 20 83548 152042 181980 40 25 55072 65979 119805 25 83468 151562 181579 35 30 55191 66188 119920 30 83388 151083 181180 30 35 55314 66397 120036 35 83308 150607 180783 25 40 55436 66607 120152 40 83227 150132 180388 20 45 55557 66817 120268 45 83146 149660 179995 15 50 55677 67028 120386 50 83066 149190 179604 10 55 55798 67239 120503 55 82985 148722 179215 5 60 55919 67450 120621 60 82903 148256 178829 0 56   146  

Canon Triangulorum.  34 sins Tang. Secants   124 sins Tang. Secants M   5 56039 67662 120740 5 82822 147791 178444 55 10 56160 67878 120859 10 82740 147329 178062 50 15 56280 68087 120979 15 82658 146869 177681 45 20 56400 68300 121099 20 82577 146411 177302 40 25 56520 68514 121219 25 82494 145955 176926 35 30 56640 68728 121320 30 82412 145500 176551 30 35 56760 68942 121461 35 82330 145048 176179 25 40 56880 99157 121584 40 82247 144598 175808 20 45 56999 69372 121706 45 82164 144149 175439 15 50 57119 99588 121829 50 82081 143702 175072 10 55 57238 69805 121953 55 81998 143257 174707 5 60 57357 70020 122077 60 81915 142814 174344 0 55   145  

Canon Triangulorum.  35 sins Tang. Secants   125 sins Tang. Secants M   5 57476 70237 122202 5 81831 142373 173983 55 10 57595 70455 122327 10 81748 141934 173624 50 15 57714 70673 122452 15 81664 141496 173266 45 20 57833 70891 122578 20 81580 141060 172910 40 25 57951 71110 122705 25 81495 140627 172557 35 30 58070 71329 122832 30 81411 140194 172205 30 35 58188 71548 122960 35 81327 139764 171854 25 40 58306 71769 123088 49 81242 139335 271506 20 45 59424 71989 123217 45 81157 138908 171159 15 50 58542 72210 123346 50 81070 138483 170814 10 55 58660 72432 123476 55 80987 138060 170471 5 60 58778 72654 123606 60 80901 137638 170130 0 54   144  

Canon Triangulorum.  36 sins Tang. Secants   126 sins Tang. Secants M   5 58896 72876 123737 5 80826 137218 169790 55 10 59013 73099 123869 10 80730 136799 169452 50 15 59130 73323 124001 15 80644 136382 169116 45 20 59248 73546 124133 20 80558 135967 168782 40 25 59365 73771 124266 25 80472 135554 168448 35 30 59482 73996 124400 30 80385 135142 168117 30 35 59599 74221 124534 35 80299 134731 167787 25 40 59715 74447 124669 40 80212 134323 167459 20 45 59832 74673 124804 45 80125 133916 167133 15 50 59948 74900 124940 50 80038 133510 166808 10 55 60065 75127 125076 55 79952 133106 166485 5 60 60181 75555 125213 60 79863 132704 166164 0 53   143  

Canon Triangulorum.  37 sins Tang. Secants   127 sins Tang. Secants M   5 60274 75583 125351 5 78711 132303 165844 55 10 60413 75812 125489 10 79688 131904 165462 50 15 60529 76041 125627 15 79600 131506 165208 45 20 60645 76271 125767 20 79512 131110 164893 40 25 60760 76501 125906 25 79423 130715 164580 35 30 60876 76632 126047 30 79335 130322 164267 30 35 60991 76964 126188 35 79246 129930 163957 25 40 61106 77195 126393 40 79157 129540 163648 20 45 61221 77428 126461 45 79068 129151 163340 15 50 61336 77614 126614 50 78979 128764 163034 10 55 61451 77894 126757 55 78890 128378 162730 5 60 61566 78128 126901 60 78801 127994 162426 0 52   142  

Canon Triangulorum.  38 sins. Tang. Secants   128 sins. Tangents Secants. M   5 61680 78363 127046 5 78711 127611 162125 55 10 61795 75812 127191 10 78621 127229 161825 50 15 61909 78833 127337 15 78531 126849 161526 45 20 62023 79069 127483 20 78441 126470 161229 40 25 62137 79306 127630 25 78351 126093 160933 35 30 62251 79543 127777 30 78260 125717 160638 30 35 62365 79781 127926 35 78170 125342 160345 25 40 62478 80019 128074 40 78079 124969 160054 20 45 62592 80258 128224 45 77988 124597 159763 15 50 62705 80497 128374 50 77898 124226 159475 10 55 62818 80737 128524 55 77806 123857 159187 5 60 62932 80978 128676 60 77714 123489 158901 0 51   141  

Canon Triangulorum.  39 sins. Tang. Secants   129 sins. Tangents Secants. M   5 63045 81219 128827 5 77622 123123 158617 55 10 63157 81461 128980 10 77521 122757 158333 50 15 63270 81703 129133 15 77439 122393 158051 45 20 63383 81946 129287 20 77347 122031 157770 40 25 63495 82189 129441 25 77254 121669 157491 35 30 63607 82433 129596 30 77762 121309 157213 30 35 63719 82678 129752 35 77669 120950 156936 25 40 63832 82972 129908 40 76977 120593 156661 20 45 63943 83169 130065 45 76884 120236 156397 15 50 64055 83415 130223 50 76791 119881 156114 10 55 64167 83662 130381 55 76697 119527 155842 5 60 64278 83909 130540 60 76604 119175 155572 0 50   140  

Canon Triangulorum.  40 sins. Tang. Secants   130 sins. Tangents Secants M   5 64390 84158 130700 5 76510 118823 155303 55 10 64501 84406 130860 10 76417 118473 155035 50 15 64612 84656 131021 15 76323 118124 154769 45 20 64723 84906 131183 20 76229 117777 154503 40 25 64834 85156 131345 25 76134 117430 154239 35 30 64944 85408 131508 30 76049 117084 153976 30 35 65055 85659 131672 35 75946 116740 153715 25 40 65165 85912 131836 40 75851 116397 153454 20 45 65275 86165 132001 45 75756 116055 153195 15 50 65386 86419 132167 50 75661 115714 152937 10 55 65496 86673 132334 55 75506 115375 152680 5 60 65605 86928 132501 60 75470 115036 152425 0 49   139  

Canon Triangulorum.  41 sins. Tang. Secants   131 sins Tang. Secants M   5 65715 87184 132669 5 75375 114699 152170 55 10 65815 87440 132837 10 75279 114363 151917 50 15 65934 87697 133007 15 75183 114028 151665 45 20 66043 87955 133177 20 75088 113694 151414 40 25 66153 88213 133347 25 74991 113361 151164 35 30 66262 88473 133519 30 74895 113029 150916 30 35 66370 88730 133691 35 74799 112698 150668 25 40 66479 88992 133864 40 74702 112369 150422 20 45 66588 89253 134037 45 74605 112040 150176 15 50 66696 89567 134212 50 74508 111713 149932 10 55 66804 89777 134387 55 74411 111386 149689 5 60 66913 90040 134563 60 74314 111061 149447 0 48   138  

Canon Triangulorum.  42 sins Tang. Secants.   132 sins Tang. Secants M   5 67021 90304 134739 5 74217 110736 149206 55 10 67128 90568 134917 10 74119 110413 148967 50 15 67239 90833 135095 15 74021 110091 148728 45 20 67344 91099 135274 20 73923 109770 148490 40 25 67451 91365 135453 25 73825 109450 148234 35 30 67559 91633 135634 30 73727 109130 148018 30 35 67666 91901 135815 35 73629 108812 147784 25 40 67773 92169 135997 40 73530 108495 147550 20 45 67880 92439 136179 45 73432 108179 147318 15 50 67986 92709 136363 50 73333 107864 147087 10 55 68093 92979 136547 55 73214 107550 146857 5 60 68199 93251 136732 60 73135 107236 146667 0 47   137  

Canon Triangulorum.  43 sins Tang. Secants   133 sins. Tang. Secants. M   5 68306 93523 136918 5 73036 106924 146399 55 10 68412 93796 137105 10 72936 106613 146217 50 15 68518 94070 137292 15 72837 106303 145946 45 20 68624 94345 137480 20 72737 105992 145721 40 25 68729 94620 137669 25 72637 105685 145497 35 30 68835 94896 137859 30 72537 105378 145273 30 35 68940 95173 138050 35 72437 105071 145051 25 40 69046 95450 138242 40 72336 104765 144830 20 45 69151 95729 138434 45 72236 104461 144610 15 50 69256 96008 138627 50 72135 104157 144391 10 55 69361 96288 138821 55 72034 103854 144172 5 60 69465 96568 139016 60 71933 103553 143955 0 46   136  

Canon Triangulorum.  44 sins. Tang. Secants   134 sins. Tangents Secants M   5 69570 96850 139212 5 71832 103252 143739 55 10 69674 97132 139408 10 71731 102952 143523 50 15 69790 97415 139605 15 71630 102652 143309 45 20 69883 97699 139884 20 71528 102354 143095 40 25 69987 97984 140003 25 71426 102057 142883 35 30 70090 98269 140203 30 71325 101760 142671 30 35 70194 98556 140404 35 71223 101465 142461 25 40 70298 98843 140605 40 71120 101170 141251 20 45 70401 99131 140808 45 71018 100876 142047 15 50 70504 99419 141011 50 70916 100583 141814 10 55 70607 99709 141216 55 70813 100291 141621 5 60 70710 10000 147427 60 70710 100000 141421 0 45   135  
I might here take occasion to teach you to work matters astronomical, but it was not my intent so to do in this volume, such that desire the same, may find some satisfaction therein, wrote by Master Blundeuil,& who so desireth it more ample, most also in such matters haue more ample tables, for these tables be onely drawn to be applied to instrumental observation as I haue said before, wherein we may find the just minute, as precisely as can be observed instrumentally, and what needeth more?    

THE EIGHTH book Of the Geodeticall staff:  
Whereby is performed most conclusions that may be wrought by the Globe or any Astrolabe, with the art of Horomancie, wherein you be taught to make horizontal, Vertical, and East and West Dials, with diverse others only by the staff, without any kind of arithmetic; to draw a Map of the world, according to the Longitude and Latitude of the places; to find how one place doth bear from another, by the points of the Mariners compass and such like: in all which you shall crave but the onely help of your staff.  


CHAP. I.  
principal notes and definitions for the better understanding of this book.  
FIRST, this kind of work consisteth in projecting the celestial circles into parallel and Ellipsicall lines.  
2 According unto perspective demonstrations, there can but the half of any sphere be laid down in plano.  
3 Def. All parallel lines be circles of latitude, declination, and circles of altitude and depression.  
4 Def. Lines Ellipsicall be meridians, Azimuths and circles of longitude.  
5 Def. All circles of latitude be parallel to the ecliptic in all astronomical operations, and are measured by Chords vpon the limb.  
6 Def. All circles of declination lie parallel unto the equinoctial, and are expressed by chords.  
7 Def. Circles of altitude and depression lie parallel to the horizon, and are expressed as other parallels be.  
8 Def. Circles of declination be counted vpon the limb from the equinoctial towards the poles.  
9 Def. Circles of latitude are counted from the zodiac vpon the limb.  
10 Def. Circles of altitude and depression are counted from the Horizon on the limb.  
11 Def. Take it general that all parallel lines are counted on the limb from that great circled they lie parallel unto towards the pole thereof.  
Lines Ellipsicall.  
12 Def. You must first understand that all kind of ellipticall lines by a certain reducement are measured by chords vpon the limb, like to parallels.  
13 Def. Ameridian is a great circled passing by the poles of the world, and cutting the equinoctial at right angles expressed by chords, and reduced to the limb by lines parallel to the 6 of clock meridian.  
14 Def. All Azimuths be great circles passing by the vertical point, and crossing the horizon at right angles, expressed by chords on the limb, and transferred thereunto by lines parallel to the east and west Azimuth.  
15 All circles of longitude be great circles passing by the poles of the zodiac, cutting the ecliptic at right angles, measured vpon the limb and transferred thereunts by lines parallel unto the axletree of the ecliptic.  
16 Def. All great circles be meridians, circles of longitude, vertical circles, Coluers, the equinoctial, ecliptake, &c.  
17 Def. All less circles be parallels.  
18 Def. Generally all great circles cross one the other vpon the center, and all other less circles never cross over the center.  
19 The center in this proiectment doth represent the equinoctial points of ♈ &c ♎, the circumference, the meridian, or 12 of clock line, but in some conclusions, the Solstitiall colour.  
20 Def. All times are reckoned from 12 of the clock at noon, to 12 at night, and are measured in the parallel of the sun( and so do Astronomers count it in their. Ephemerides) and counted from the axes of the equinoctial.  
21 Def. One degree of the equinoctial contains four minutes of time, and 15 degrees one hour, for which purpose behold the ensuing table.  

A Table of reduction of the parts of time, into the parts of the equinoctial ark.  parts of houres. the ark of the ecquino, circled.   parts of houres. the ark of the equator. min. gra. mi. min. gra. mi. 1 0 15 31 7 45 2 0 30 32 8 0 3 0 45 33 8 15 4 1 0 34 8 30 5 1 15 35 8 45 6 1 30 36 9 0 7 1 45 37 9 15 8 2 0 38 9 30 9 2 15 39 9 45 10 2 30 40 10 0 11 2 45 41 10 15 12 3 0 42 10 30 13 3 15 43 10 45 14 3 30 44 11 0 15 3 45 45 11 15 16 4 0 46 11 30 17 4 15 47 11 45 18 4 30 48 12 0 19 4 45 49 12 15 20 5 0 50 12 30 21 5 15 51 12 45 22 5 30 52 13 0 23 5 45 53 13 15 24 6 0 54 13 30 25 6 15 55 13 45 26 6 30 56 14 0 27 6 45 57 14 15 28 7 0 58 14 30 29 7 15 59 14 45 30 7 30 60 15 0 m./ 2 m./ 2  

A Table of reduction of the parts of the Equator circled into parts of the like time.  the ark of the equator circled. parts of time.   the ark of the equator parts of time. gra. ho. mi. gra. how. min. 1 0 4 31 2 4 2 0 8 32 2 8 3 0 12 33 2 12 4 0 16 34 2 16 5 0 20 35 2 20 6 0 24 36 2 24 7 0 28 37 2 28 8 0 32 38 2 32 9 0 36 39 2 36 10 0 40 40 2 40 11 0 44 41 2 44 12 0 48 42 2 48 13 0 52 43 2 52 14 0 56 44 2 56 15 1 0 45 3 0 16 1 4 46 3 4 17 1 8 47 3 8 18 1 12 48 3 12 19 1 16 49 3 16 20 1 20 50 3 20 21 1 24 51 3 24 22 1 28 52 3 28 23 1 32 53 3 32 24 1 36 54 3 36 25 1 40 55 3 40 26 1 44 56 3 44 27 1 48 67 3 48 28 1 52 58 3 52 29 1 56 59 3 56 30 2 0 60 4 0 m./ 2 m./ 2 m./ 2 m./ 2   

CHAP. II.  
To express any chord in the limb of a circled, and to protract any angle.  
THis Chapter, and the other fire ensuing, well understood, the rest of the book is but easy.  
To measure any thing in a circumference is no other thing but either having a chord given to express the quantity therof, or having an angle given to protract the like: for which purpose the 2 Chapter of the 6 book shall very fully instruct you in all points, therefore I will refer you thereunto for brevities sake.   

CHAP. III.  
Any parte of a diameter being given or required, to express the same in degrees.  
TO measure a diameter, is either for to express the parts given, or to give the part expressed.  
 
Any diameter in this work is a great circled, by the 17 def. because he crosseth the center; then to express the part given, work thus.  
Let A B C D E be an hemisphere, and let the proposition be to express A F part of the diameter B D, by the 2 def. this must be transserred to the circumference, thus: draw a line G H, to pass by the point F parallel to C E, then by the first Chap. measure G C or H E, so haue you your desire.  
2 To give the part expressed is, when there is any number of degrees expressed, to find the point whereon they fall, performed thus.  
I haue a number of degrees given as 30, and am required to show in what part of the diameter A D, they end.  
By the first Chapter, I protract a chord G C of 30 degrees, from whence I draw a line G H parallel to C D,& where that point cuts the diameter A B as at F, there is your desire; I conclude that A F contains 30 degrees  
And you must note that this work is general for the transserring of al kind of ellipsical lines, be they Meridians, circles of longitude &c. remembering that the parallel line G H, must always in Meridians be drawn parallel to the axletree line, as in the 13 Def. in azimuths to the east and west azimuth, as in the 14 def. and so of circles of Longitude, as in the 15 Def.   

CHAP. IIII.  
Any part of a parallel being given or required, to express the same.  
TO measure a parallel, is either to find what Ellipsscall line passeth by some point in any parallel, or some ellipsicall line being given, to find where he should cut the parallel: and to do either, you must first make the parallel the diameter of a circled, whereupon describe a semicircle, then work thus.  
First, let the parallel be G L, and let the prop. be to find what Ellipsicall line passeth by the point M, make G L the diameter of a circled, then on the point N describe the semicircle, G I L, next from that point M draw a parallel to F A as I M, then the portion of the circumference I O, is your desire, 30 degrees, by the 1 Chapter, and so of the parallel H P or any other.  
2 But if you haue the number of the ellipsical line given, and be required to find what point of the parallel he should pass by, work thus.  
Let the parallel be H P, and let the proposition be to find where the 30 Ellipsicall line shall pass by?  
Make H P the diameter of a circled, and then vpon O the center describe H F P the demicircle, then by the first Chap. protract a chord F K 30 degrees, whence draw a line parallel to F A as K R, which shall cut the parallel H P in R: I conclude that R is the point where as the 30 ellipsicall line should pass by, and so of G L, or any other parallel.   

CHAP. V.  
Any point being given in the hemisphere, to find what Ellipsicall line passeth by the same:  
THe premises well understood, this Chapter is easy. The point being assigned, you shal consider whether your question be to know what Meridian, what circled of longitude or Azimuth passeth there by, and then calling to mind the end of the third Chapter, work thus.  
Let the point taken in the hemispere A B C D E be R, 
The figure in the 4 Chapter serveth here.  and let the question be to find what Meridian passeth thereby, B D is the equinoctial, E C the axletree, draw therefore by the point R a line parallel to B D; then by the 4 Chapter, first difference, find what Ellipsicall line passeth by R, which you shall find 30. I conclude, the 30 Meridian doth pass by the point P, and so of any other.   

CHAP. VI.  
To draw a parallel by any number of degrees assigned, or a parallel being drawn, to find what number of degrees he contains.  
WHat parallel circles be, is shewed in the first Chap. which considered, work thus.  
1 Let the proposition be to draw a parallel of declination 23 degrees, 28 minutes distant North from the equinoctial, which let be the tropic of Cancer.  
Let A B C D E be the hemispers, B D the equinoctial, F C the axletree, then by the first Chapter I protract a chord B G, and L D 23 degrees, 28 minutes, from the equinoctial: I then conclude that G L is the parallel representing the tropic of Cancer, and so of any other.  
2 But if a parallel be drawn, and you would find what number of degrees it contains, do thus.  
Let the parallel be H P, then by the first Chapter, find the chord B H 50 degrees, your demand by the 10 def.  
  

CHAP. VII.  
To lay down a right obliqne sphere.  
AN obliqne sphere is such an one, whose North or South pole is elevated any thing at all above the Horizon, and is thus laid down.  
Let A B C D E be the hemisphere, then draw the diameter B D which call the equinoctial, then cross B D with a right line vpon the center A at right angles as C E, which call the axletree of the world, C the North pole, E the South; then draw a parallel to B D the equinoctial 23 degrees, 28 toward the North pole A, by the last Chap. which let be F K, call that the tropic of Cancer; do so vpon the other side the equinoctial as L M, call that the tropic of capricorn; next draw N O, 23 degrees, 28 minutes, from C, and parallel to B D, which call the circled articke, do so to B Q, so haue you the circled antactick: then the line L K is the Zodratke, so now haue you a right sphere; but of right spheres I note two kindes, that is a polar and an equinoctial, if you will make this an equinoctial, make C E the Horizon, and B D the equinoctial passing by the zenith; but if you will make it a polar, then make B D the Equinoctial the Horizon, and so you haue either of the right spheres, but to make the obliqne sphere, consider if the North or South pole be elevated, and how much, let C the North pole be elevated 52 degrees and minutes, therefore by the first Chap. protract the chord R C 52 degrees min. whence draw alone by the center A, to the circumference, so haue you the line R S the Horizon C, being elevated 52 degree min& this rule is general: well then the poles of the Horizon viz. the Zenith and Nadit be T w; and the poles of the ecliptic L K be N Q, for the poles of the ecliptic be always found in the circles, Articke and antarctic; well then the Meridian or 12 of clock line is the Peripher.  
Now shall follow some counclustion Astrononicall, which will be very easy if you understand what is said before, for all the work doth consist thereupon.   

CHAP. VIII.  
To take the altitude of the sun.  
I haue taught you how to perform this prope already in my second book, howbeit I shall not bring it out of season here to deliver an other way no less easy, and fa●●e betted for seamen: cause therefore the left leg to hang perpendicular, the center upwards( which you may does diners wires, as by hanging a plumb at the lower end) turn the ●nds of the legs fromwards the Sun, beauing up or putting down the right leg until the sun beams that pierceth through the slitte in the center-sight agree with the fidutiall edge of the right leg; note then the quantity of that angle by the second book Chapter 5, for that is the distance of the sun from the Zenith, therefore take that from 90 and the remainder is your desire.  
Example.  
The 20 day of September 1607, making observation at London as before, I found the angle vpon the legs to contain 64 42/ 60 degrees, which argued that the Sun was then 64 deg. 40 min. distant from our Zenith; take that therefore from 90, so haue you 25 11/ 62 deg. the true altitude of the sun at London vpon the to day of September, at 10 of the clock before noon.   

CHAP. IX.  
To find the suins declination for every day in the year, and the declination of any parte of the ecliptic.  
AS the latitudes in the heaven be counted from the ecliptic, so the declinations be numbered from the equator, therefore the declination is the ark of a great circled intercepted betwixt the Equator and the place of Eclinticke or star, thus had.  
By the 10, find the place of the sun in the Eclinticke, by which point draw a line parallel to the equinoctial, then the thord on the peripher betwixt that panalel& the Equinoctial is your desire.  
Example.  
 
B C D E is the hemisphere, B D the Equinoctial, F K the ecliptic 〈◇〉 the place of the Sun, viz. in the 20 degree of ♉ draw therefore by the point 〈◇〉 a line Y V, parallel to B D, the equinoctial; I conclude the ark V D, is the declination of the 20 degree of which by the 2, is 17 degrees 45 min. 40 sec.   

CHAP. X.  
To find any place of the sun in the ecliptic, or any point proposed.  
TO find the place of the Sun in the ecliptic, work thus. Let A B C D E be the hemisphere, F K the tropic of ♋, K L the ecliptic, and let the prop. be to find where the 20 degree of ♉ falleth.  
In this work you be to consider, how many degrees the sun is distant from the nearest Equinoctial point of ♈ or ♎ which can never exceed 90 degrees, and thereby whether the question be for North or South sins, as héere it falleth out: for by the 20 degree if ♉ a North sign, whose distance from the beginning of ♈ is 50 degrees, draw therefore by the 6 Chap, 
The figure in the third Chapter serveth here.  a line parallel to N Q the artrées of the Eclpticke 50 degrees distant, as O M, which shall intersect the North part of the ecliptic A K at X the very place of the 20 degree of ♉, and so of any other North sign, point or star.  
But for South signs, you must work vpon the other side the axletree of the ecliptic N Q, and then the work is all one with the former.  
Example.  
Let the question be to find the first min. of scorpion, of the two equinoctial points M is nearest to ♎, being 30 degrees, distant, therefore by the prescript, I draw R Z parallel to N Q the axletree of the ecliptic, 30 degrees distant, whose intersection with A L, the South moety of the zodiac is the beginning of M, as is plain in the figure.   

CHAP. XI.  
To find the altitude of the pole for any country whatsoever.  
THe altitude or elevation of the pole is a portion of a Maridian included betwixt the Horizon,& the pole elevated, had thus.  
Lay down a right sphere as if his poles were in the Zenith by the 7, then by the 6. draw a parallel according to the suins declination, Next having the Meridian altitude of the sun draw a line by the 6, distant from the parallel of the sun, according to the Meridians altitude, by the end of which numbers draw a line crossing the center, whose distance on the peripher from the pole, is your desire.  
Example.  
Let the hemisphere be A B C D E, then I draw a parallel according to the declination of the sun, the 10 of april, 11 40/ 60 degrees, then I count the Meridian Altitude of the sun 50 degrees, from F towards E, as to H; then did I draw a line from H to A, and so to I: I conclude the thord B I is the elenation 51 40/ 50 by the 2 Chap.   

CHAP. XII.  
To know what hour of the day it is, the sun shinng in any Region.  
To know what of the clock it is, is to find how many 15 degrees the sun is distant on his parallel from the Meridian, and so many hours, is the time of the day lacking or past the common 12 of clock, according as the question, fell before or after noon.  
Lay therefore down a sphere according to the elevation of your pole, and draw a parallel by the 6 according unto the declination of the sun then take the altitude of the Sun, and by the 7 and 10 def. of the first Chapter, draw an almicanter according to the altitude taken. Lastly, the Meridian by the 4 Chapter, passing by the intersection of the almicanter and parallel of declination, is your desire.  
Upon the 10 of April I made observation as followeth; First I laid down the hemisphere A B C D Eas before, then I draw F G a parallel by the 6, according unto the Suns declination 11 40/ 60 degrees, next by the 8, I observed the altitude of the sun 40 12/ 60 degrees, and by the 6 according to the 7 and 10 def. Chap. 1, I draw an almicanter I K according to the suins altitude taken, which intersects with F G the parallel of declination at L. Lastly I found by the 5. that the 30 Meridian passed by L, reckoned from F, then for that my observation was before noon, I may safely conclude that it was precisely 10 of the clock, for that 30 degrees contains two houres, as is plain in the table in the first chap. If my obsernation had been after noon, it had been just two of the clock, then the Azimuth cutin there, sheweth the cost: and if you know not whether your observation were before or after noon after your first taking the suins altitude, anon after observe it again if then the altitude be greater then it was before, then were your obsernations before noon, if less, afternoon.   

CHAP. XIII.  
To find the right asention of any portion of the ecliptic, together with the dogrees of the Equinoctial, that do ascend with cou●ry degree of the ecliptic in a right sphere.  
THe right ascention, is a point or ark of the Equator that doth ascend with any g●uen ark of the ecliptic, or any point or star in a right sphere, and found thus.  
Lay down a right sphere, so that the parts may be in the horizon, traverse what Meridian passeth by that point to the equinoctial, so haue you your desire.  
Example.  

The figure in the third Chapter serveth here.  Let A B C D E be the h●●●spheare, B D the equinoctial, L K the ecliptic,& let the proposition be for to find the right ascention of the ☉ degree ♉, by the 10 Chapter find the place of the ☉ deg. of ♉ in the ecliptic A L, which is marked with his character, and by that point in the ecliptic, draw therefore a line paras●el to O E as O LL. I conclude the chord O C is your demand, viz. 27 54/ 60 deg. the right ascention of the ☉ degree of Taurus, and this rule is general for all degrees or points not about 90 deg. distant-from 〈◇〉, which is three signs just.  
But you must note that the point given, may either contain just 90 degrees, which is three signs, or be more then 90, and yet less then 180, or more then 180, and yet less then 270 or more then 270, and ●●●●e then 360.  
If the who●e just 90 deg. containing 3 signs as the ☉ degree of ●dollar;, then is 90 the right ascention.  
2 If the ark be more then 90 as the ☉ degree of ☊ 120, and less then 180, then subtract the ark given from 180 viz. 120 from 180, so haue you 60 remaining, then seek the right ascention of the remainder( as of 60) as before: 60 contains 2 signs, therefore I seek the right ascention of ☉ deg. ♎, which is 2 signs from ♈, which I find 57 49/ 6● deg, which take from 180, the remainder is the right ascention of the given ark, viz. ●7 degr. 48 min. 48 see, from 180, so haue you 122 21/ 60 degr. and bette●, you desire.  
3. But if ●he ark comprehended betwixt ♈ and the point given, be more then 180, and yet less then 270, that is, if the ark be more then 6 signs and less then 9, as the ☉ deg. of ♏ 210 degrees or 7 signs, subtract 180 degrees from that ark the right ascension of the remainder taken from the beginning of ♎ and add to 180, so haue you your desire; as for the purpose 180 taken from 210 leaveth 30, which c●ntaineth one sign, whose right ascention by the former is 27 deg. 54 min. 20 sec. which added to 180, maketh 207 deg. 54 min. 20 sec. the right ascention of the 0 deg. of ♏.  
4. Lastly, if the ark given, comprehended betwi●t Aries and the point assigned be greater then 270 or 9 signs, as ●he 0 deg. of ● 300 deg. or 10 signs, then substrast that ark from 360, then the right ascention of the remainder taken from 360 leaveth your desire, as 300 taken from 360 leaveth 60, or two signs, whose right ascention 57 deg. 48 min 48 sec. taken from 360 leaveth 302 deg. 11 min. 12 sec. the right ascention of 0 deg. of Aquariust so that you see by finding the right ascention of any ark under 90 deg. as you be taught to do, you haue the right ascention of any other ark.  
And I would haue you note, that opposite signs, and signs alike distant from the equinoctial point, do ascend in a right sphere in equal time, as ♈ ♓ ♎ and ♍ so doth also ♉ ♌ ♏ ♒ and in like manner Gem. Can. Cap. Sagit.  
And for as much as every Meridian is a right Horizon to some one place or other, therefore the degree of the ecliptic coascending with any deg. of the equinoctial, in a right sphere must of necessity pass by the Meridian of any place together, and this is called the culminating of a star, &c. or celimeditation, so that you may gather hereby that the culminating of any point in the heauens is all one in substance with his right ascention.   

CHAP. XIIII.  
Of the difference of ascensions.  
THe difference of ascention is an ark of the Equator whereby the obliqne ascention or descension of the ecliptic doth differ from the right.  
Now to find the difference of ascensions, haning laid down a sphere according unto your Latitude, set in the ecliptic line, the deg. whose ascentionall difference you séeks, by which point draw a line parallel to the equinoctial; then note the intersection of that parallel with the Horizon, for the Meridian passing by that point is your desire, being reckoned from the axletree line.  
Example.  
Let A B C D E be an hemisphere, let the proposition be to find the ascentionall difference of the 29 degree of ♉, let M be the 26 of ♉, by which draw F G parallel to C D the equinoctial, let the Horizon be I H B, the pole elevated 52 degrees 2●/ 60 then I note the intersection of F G with I H, as at N. Lastly by the fourth I seek what Meridian passeth by N reckoned from B E, which I find 13 46/ 60 Meridian and better. I then conclude that the ascentionall difference of the 26 degree of ♉ in the 52 20/ 60 degree of latitude, is 13 degrees, 16 min. 26 sec. so that Taurus, because he is a sign obliqne ascending, I affirm that he riseth so much sooner in this obliqne sphere, then in a right sphere, as 13 deg. &c. cometh unto.  
And note further that the ascentionall difference from one quarter goeth for all the rest in the same latitude.   

CHAP. XV.  
Of the obliqne ascensions  
THe obliqne ascensions, be an ark or point of the Equator, which do coascend with any given ark of the ecliptic or with any star above the Horizon, and found thus. Consider if the signs be North or South, if South, add to their right ascention the difference of ascention; if North, take the difference of ascension from the right ascension, and the total of the one, or remainder of the other, is your desire.  
Example.  
The right ascension of the 4 deg. of ♊ is 62 degrees, by the 13 Chapter, the difference by the 14, is 29 47/ 60 deg. geminy is a North sign, therefore I subtract the difference viz 29 47/ 60 from 62, so haue you 32 13/ 60 deg. the obliqne ascension.  
Then for the South sign we will take the opposite deg. to the former as 4 of ♐, whose right ascension by the 13 is 242, to which I add 29 47/ 60 deg. the difference of ascensions by the 14, so haue you the obliqne ascension of the 4 deg. of Taurus, viz. 271 47/ 60 deg.  
These propositions be most excellent in all astronomical operations.   

CHAP. XVI.  
Of descensions both right and obliqne.  
unto the obliqne ascension of the opposite degree to the degree proposed, add 180, and so haue you your desire.  
Example.  
I would find the obliqne descension of the 20 deg. of$, therefore I add 180 unto 323 deg. 9 min. 34 sec. the obliqne ascension of the 20 deg. of Capt. so haue you 503 de. 9 mi. 34 sec. your desire, were it not that it doth exceed the whole circled 360, which for because it so falleth out, I must always in such a case subtract 360, as 360 deg. from 503 deg. 9 min. 34 sec. there remaineth 143 deg. 9 min 34 sec. your desire.   

CHAP. XVII.  
The longitude and latitude of any star proposed, to find his right and obliqne ascention together with his declination from the equinoctial, and to place any star in the hemisphere.  
THis matter is easily performed by your staff, understanding but what is said before, but to the matter.  
Here we shall haue occasion to use the circles of longitude remembered in the first Chap. def. 15, and circles of latitude as in the 9 def. first therefore by some tables find the true longitude and latitude of the star proposed, then in the hemisphere draw a line parallel to the ecliptic by the 6 according unto the latitude given: then by the 4 find where the circled of longitude 〈◇〉 the said parallel, for that is the stars place in the hemis●●●, the● work according unto the 12 def.& by the 4 Chapter, what Meridian passeth by that point, for that is the stars right ascention by the 13, and the parallel of declination cutting there also, is the declination by the 9, and according to the 6 and 8 def.  
Example.  
 
There is a faire star of the first magnitude placed in the North, whose name is Capella, Hircus, or Alhaoith, in English the goat: I find the longitude of this star to extend itself into the 1637 deg. of geminy, or to be distant from the vernal intersection of ♈ 76 37/ 69 deg. and that his latitude from the ecliptic is 22 2/ 4 deg. into the North. I draw therefore as I am wont my hemisphere A B C D E, wherein F G is the equinoctial M I the ecliptic, H Q the artrée of the ecliptic; then 1 I draw a line F P, parallel to the ecliptic M I, according to the latitude, v●z 22 deg. 30 min, then by the 4 I find where the 76 ●7/ 60 Elipticall line or circled of longitude ●●tteth the said parallel, as at P, which point I affirm is the true place of the star; then the distance of the parallel B F from F G, the equinoctial by the 6, is the declination viz. B F or P G, 45 ●/ 60 deg. and the denomination is North.  
Then the Meridian passing by that place of the star is his right astention, which by the 4 is. 72 ●1/ ●● Meridian.   

CHAP. XVIII.  
The difference of ascensions or the obliqne ascension of any point in the ecliptic proposed, or of any star, to find the latitude of the country.  
DRaw a parallel according to the declination of the point proposed, then by the 14 number the difference of ascensions proposed vpon the said parallel, fromwards the axletree of the world, by which point in the parallel draw a line to pass by the center touching the peripher, the chord then included betwixt the pole and the intersection of that line with the peripher is your desire.  
Example.  
having laid down a right sphere as A B C D E, making F G the equinoctial, L I the tropic of Cancer, 
The figure in the 17 Chapter serveth here.  let then the question be of the 0 deg, of$, whose difference of ascensions let be 34 13/ 60 deg, which I count by the 4 def. on, L I the tropic fromwards S towards L, which falleth at R. I conclude alone drawn from the circumference by R to A, and so to the peripher again, is the Semihorizon, as B R A C, then E being the pole elevated I affirm by the 2, that the chord B E limits the elevation, viz. 52 20/ 60 deg.  
But if the obliqne ascension be proposed you must subtract that from the right ascension, and so get the difference of ascensions.  
Example.  
The obliqne ascension of the 0 deg. of$ is proposed to be 55 47/ 60 deg. which take from the right ascension, which by the 13 is 90: so haue you 34 1●/ 60 deg. then work as before.   

CHAP. XIX.  
Of the amplitude of the rising and setting of the sun or any star or point in the heauens.  
THe amplitude of rising is an ark of the Horizon intercepted betwixt the rising of the Equinoctial& the true rising of the Ellipticke, or the sun or any star or point in the heauens  
For where as the sun and other stars, as most of them be without the Equator, and by their diuers cites do obtain diuers declination, and do describe diuers parallels, by the first mover, which do cut the Horizon in diuers places, then who doth not consider that there is also great variety of the amplitude of rising, which doth much vary in the stars and points of the ecliptic, by reason of the diverse inclination of the equinoctial to the Horizon, and the diuers places vpon the earth? but to the matter.  
You shal therefore lay down an obliqne sphere by the 7 according to the elevation, thē draw a parallel according to the declination of the star or point whose amplitude you seek; note thē the intersection of the parallel with the Horizon, for the Azimuth or distance of that point from the center is your desire.  
Example.  
 
The sun being in 0 deg of$ I do desire his amplitude in the 52 20/ 60 latitude, first I lay down a sphere according to my latitude, as A B C D E, F G being the equinoctial, O P the Tropical parallel, B the North pole, H I the horizon, and I the chord of elevation: so then I note where O P the tropic, and H I the Horizon intersect, as at Q. I conclude Q A is the amplitude then by the 3 Cha. and Chap. 1 def. 14. And for that the 40 40/ 60 Azimuth passeth by Q which is the true distance of Q from A, I conclude that the amplitude of ☉ being in the 0 deg of Cancer, was just 40 40/ 60 deg.   

CHAP. XX.  
The amplitude of rising of any star known, or of the sun with his place in the zodiac: to find the Latitude of the country.  
IMagine the amplitude of ☉ being in 0 deg of$ were given to be 40 40/ 60 deg.  
Let A B C D, be the hemisphere, O P a parallel drawn, according unto the declination of the 0 deg. of$ then do I protract a chord from R towards B according unto the amplitude as R B 40 40/ 60 deg, 
The figure last before serveth here.  then from B I draw a line B Q parallel to R A, which shal entersect with O P the tropic at G, I conclude a line drawn from A by Q to the circumference, as H A Q I, is the Horizon, so is B I the chord of elevation, which by the 2 is 52 20/ 60 deg.   

CHAP. XXI.  
To know the rising and setting of the sun, the length of the day and night, with the continuance of twilight.  
THe day, after vulgar capacity, is taken two-fold, civil and natural: the day civil is the space of time wherein one renolution of the equator with the adding of the motion of the sun, is full complete, and is always 24 hours.  
The natural day is a space of time, wherein the Sun is carried from the East, by the midst of heanen to the West, and is altered in every several latitude.  
Now the night is the space of time wherein the center of the sun is carried from the West by the Imum Coeli to the East, thus sound.  
having laid down your hemisphere, as you be accustomend according unto your elevation, draw a parallel according unto the declination of the sun for the day proposed, note then the intersection of that parallel with the Horizon, for the Meridian circled passing by is your desire, viz. the hour of sun rising, and the Azimuth passing by that point sheweth the cost.  
Example.  
I would know the rising of the sun, &c. in the 52 deg. 20 min. of latitude, the ☉ being in no degree$, lay down an obliqne sphere B D C E, according to the elevation: let B C be the Horizon, E the North pole, L I the tropic of$, the entersection of the tropic and the Horizon R: by the 4 I find the the 39 Meridian counted from S passeth by R, R is the 6 of clock Meridian, and 39 deg. by the first is 2 houres, 4 deg. therefore the sun rose 4 deg. before 4 in the morning, which is 16 mi. before 4: if you had sought a setting you must haue counted those deg. forward from 6, so had the setting been 8 deg. after 8 at night.  
Then if you add the hour of sun rising and setting together, they make always 12,& if you double the hour of sun setting you haue the length of the day, that taken from 24 leaveth the length of the night, as double 8 and 4 min. so haue you 16, 8 min. the length of the day, which taken from 24, there remaineth 7 56/ 60 hours, the length of night.  
Otherwise.  
By the 14, learn the difference of ascensions, which by the first reduce into time, which take from 6, if the sun haue declination toward the pole elevated, or add to 6, if the Sun haue declination from the pole elevated, the remainder of the one, or total of the other is your hour of ☉ rising.  
Then the hour of sun rising taken from 12 sheweth the hour of Sun setting, which doubled, sheweth the length of the day as before.  
Now for the continuance of twilight, draw a parallel 18 deg. under the Horizon, then the portion of the parallel of the Sun included betwixt that line of depression last drawn, and the Horizon is your desire, whereby you may gather that in summer we haue no ending of twilight, but all day; for that the circled of depression L T drawn 8 deg. under B C, the Horizon doth not reach L I the tropic of Cancer, whereas otherwise when the sun is in the equinoctial F G we haue a great twilight, for that the portion of the suins parallel A V is great, containing by the three above 30 deg. which is more then two houres; and so long is it then that twilight doth begin before Suune rising called commonly the break of day, and so long likewise doth it endure after sun set.  
So that you see in summer for a certain space we haue all day, and in other times a kind of twilight, which is nothing else but dis●unction of day and might, or lightness and darkness made by reason of the refulgent beams of the, Sun which cannot quiter be obscured until he be 18 degrees below the Horizon.   

CHAP. XXII.  
To know the Semidurnall arch of the sun or stars with the length of day and night, &c.  
LEt the question be to know the semidurnal arch of the 25 deg. of ♈, in the 52 deg. of latitude, lay down your obliqne hemisphere according to your latitude as you wont, which let be A B C E D, let C D be equinoctial, and let F G be a parallel drawn according to the declination of the 25 deg. of ♈, which let intersect with H I the Horizon at N: I then conclude that I N is the semidurnall arch of ☉ being in the 25 deg. of ♈, which by the 4 I find to be 102 deg. or 6 hours, 48 min. which doubled, so haue you 204 deg. or 13 howeres, 36 mi. the whole diurnal arch, which taken from 360 or 24 hours, so haue you 156 deg. or 10 hours, 24 min. the length of the night: then if you take the length of the night from 24, you leave the length of the day, and so get sun rising and setting, as in the last Chapter.  
Then for the semidiurnall ark of the stars, work in all respects with their parallel of declination as you did with the suins, and you haue your desire; and so shall you find the informed star under the belly of lo, to agree very near with the former.   

CHAP. XXIII.  
The quantity of the longest day of any country given, to find the elevation of the pole, and to distinguish the climates.  
THe Cosinographicall tables show that the island called Insula Zirisaea, containeth in latitude, ●r deg. and therfore by the 27, the longest day is 16 2/ 4 hours, but now imagine that you had but onely the length of the longest day given, and were thereby required to deliver the latitude.  
Draw therefore an hemisphere which let be A F D K E, then draw L I the tropic of Cancer, now the longest day is 16 2/ 4 hours, half that is 8 ¼ hours, the semidiurnall arch, which by the first contains 123 deg. and 3 min. therefore by the 4, I find where the 123 Meridian counted from I curteth the Tropical parallel as at R, or take 90 from 123, so haue you 33; then by the foresaid 4 Chap. find where the 33 Meridian counted from E passeth, by the said tropic of$, which will also fall out at R: I conclude by either of the ways, that a right line drawn from A by R to the circumference as to B, is the half semihorizon, and C B the whole semihorizon, and that the chord B E is the elevation of the pole viz. 52 deg.  
Now saith Gemma Frifius, if out of the longest day you take 12, and reduce the remainder into quarters of hours, to which adding one, the number or order of the parallel desixed is known.  
Example.  
In the island before, the longest day was 16 ●0/ 60 hours, from which take 12, so haue you 4 30/ 60 hours remaining; which containeth 18 quarters of an hour, to which I add one, so haue I 19 I then affirm that Insula Zirisaea lieth under the 19 parallel.  
Now having found the parallel, take thence 3 and half the remainder acquaints you the order of climate, as for the purpose, 3 from 19 there remaineth 16, half which is 8, which sheweth that they do inhabit in or under the 8 climate.   

CHAP. XXIIII.  
To know the Meridian altitude of the sun or any star.  
THe Meridian altitude of the Sun or any star, is the greatest altitude that either of them can haue,& is found thus.  
Lay down an hemisphere according unto the latitude, and then draw a parallel according to the declination of the sun or star proposed, the chord then that is intercepted betwixt the Horizon and the parallel counted vpon the limb, is your desire.  
Example.  
The sun being in the 0 degré, of$, I would know his Meridian altiude here at London.  
Let A B C E D, be your hemisphere, C D the equinoctial, O P a parallel drawn according to the declination of the 0 degree of s, H I the Horizon, then do I conclude that the chord O H is your desire, which by the second is found 62 deg.   

CHAP. XXV.  
To find the degree of Medium Coeli, or Culmination at any time proposed.  
The degrees of the ecliptic any time being in the Meridian, is called Gradus Medii Coeli, with some, Cor Coeli, but most general, Culmem Coeli, thus had.  
YOu must seek the right ascension of the degree proposed,& then get this distance from the Meridian line in degrees: then for the forenoon hours, take the distance from the right ascension, adding 360, if other wise substraction cannot be made, and then make substraction: but for afternoon hours, add the ascension to the distance, casting away 360. if the whole sum exceed 360; so haue you the right ascension of the culminating, then the deg. of the zodiac is found by the 13.  
Example.  
I would know the degree of culmination, the ☉ being in the 0 degree of$ at three a clock before noon, the right ascension by the 13 is 90 degrees; then I take 45 deg. out of 90, so haue I 45 degrees remaining, the right ascension of the degree of culmination: then by the 13 may you find the deg. of the zodiac answering thereunto, forit is but using that Chap. 'vice versa, so will it be near the 13 deg. of ♉.   

CHAP. XXVI.  
To know the height of the sun, every hour in the year above the Horizon without the sight thereof, as also to find the Azimuth in any region of the world.  
having laid down your hemisphere according unto your latitude, draw then a parallel according unto L the declination, and find a Meridian cutting the same according to the hour assigned; by which point of intersection, draw an almicanter parallel tooth Horizon, whose distance on the peripher from the Horizon is your desire, then the Azimuth passing by that point also, 
The figure in the 11 Chapter serveth here.  is the vertical circled.  
Example.  
Upon the 10 day of april 1607, I desired the altitude of the sun at London, just at ten of the clock before noon.  
First, I laid down the hemisphere A B C E D, according to the latitude, C D being the equinoctial, B E the North and South pole, H I the Horizon, R the Zenith, then is F G a parallel drawn according unto the suins declination 11 40/ 60; deg. next is L a point by the 4, where the 10 of clock hourly, or 60 Meridian counted from B, passeth by: then is I K an almicanter, drawn by the point L according to the 7 def. the ark of whose distance from the Horizon as I H, is the altitude of the sun, 46 12/ 90; deg, then the Azimuth by the 14 def. passing by L, show the Azimuth or vetical circled that the sun was in, reckoned from R. which by the 4. I found 47 or there abouts: whereby I might conclude that the Sun was 47 deg. distant from the East into the South.  
And by this means may you find the points of the compass, thus, the points of the compass be 32, which divide by 306, the quotientis 11 8/ 60; deg. which is one point; then divide 47 the Azimuth by 11 8/ 60; deg. the quotient doth yield the number of points from the East to be 4, which is Southeast.   

CHAP. XXVII.  
To find the Meridian and four quarters of the world by the Geodeticall staff.  
By the last Chap. you shall get the Azimuth of the sun, or by the 28; then shall you draw a line to point direct towards the sun, and then protract an angle according unto to the Azimuth, and that line represents the true Meridian.  
And you must note if you work according to the finding of the Azimuth in the last Chap. you must remember to note this Azimuth counted from the South.  
Example.  
 
By the last Chap. I found the sun was 47 deg. distant from the East, and therefore 43 deg. distant from the Merididian, I draw therefore a line A B to point just towards the sun, then I protract an angle of 4●. deg. B A C: I conclude the line A C lieth just under the Meridian.  
In like manner may you do with the legs without proing any angle.  
And you must note, if the sun were East from the Meridian, you must protract the foresaid angle on the right hand: if west on the left.   

CHAP. XXVIII.  
To find the Azimuth of the sun.  
COnsider if the sun be on the East or West side the Meridian, if on the East, apply a needle to the right leg: if on the West, to the left leg, in such fort that the Meridian line in the carded under the needle may lie parallel to the leg he is applied unto: then cause the needle to stand just over the Meridian in the card,& open the other leg, until the shadow of the sight placed therein agree with the fidutiall edge of the leg: then note the quantity of the angle, for the distance of the sun from the South, or Azimuth of the Sun?   

CHAP. XXIX.  
To know what hour the sun cometh to any Azimuth proposed.  
DRaw an hemisphere according unto your latitude, then note what hour line cutteth the proposed Azimuth on the parallel of declination, and that is your desire,  
Example.  
Let A B C E D, 
The figure in the 11 Chapter serveth here.  be an hemisphere, H I the Horizon, &c. F P a parallel drawn according to the declination 11 deg. 40 mi and let the question be to know when the sun cometh unto the 47 Azimuth: by the 4 find where the 47 Azimuth cutteth F G the parallel of declination, as at L: then by the said 4 see what Meridian circled passeth by L, reckoned from B E the 6 of clock Meridian, so shall you find the 60 Meridian; therefore by the first I conclude that the sun cometh to the 47 Azimuth according as is said, at the precise hour of 10 of the clock before noon: for 1● the number of degrees in our Horizon is in 60, 4 times; which is 4 hours,& that added to 6, hours maketh 10, as before.   

CHAP. XXX.  
Another way most easy and true to know the hour of the day.  
YOu shall observe by the 28 Chap. or some other way the Azimuth of the sun, and then lay down a ●●he are according to the latitude, and draw a parallel according unto the declination of the sun: note then where the Azimuth and the parallel of the sun intersect, for the Meridian passing by sheweth the hour desired.  
Example.  
Lay down your hemisphe are A B C E D, let H I be the Horizon, F G a parallel drawn according unto the declination 11 40/ 60; deg. let the Azimuth by the 28 Chap. be 43 deger. distant from the South which let cut F G at L: 
The figure in the 11 Chapter serveth here.  then by the 4 see what Meridian passeth by that point as by L, which you shall find the 60, reckoned from the 6 of clock line E; therfore it is 10 before noon, or 2 after noon, but because my observation was before noon, it is 10.  
I could enlarge this book greatly in matters. Astro●● uncall, as to teach you how to erect a figure to find the beginning quantity and duration of Eclipses, to observe the motion of the star●es, &c. would my occasions afford me so much time but since I am called away, I will onely teach you how to make some kind of dials, very easy and speedily, and so then conclude the book, with certain pleasant Cosmo graphicall propositions.   
Here ends the first part of the eighth book, called Planisphera.   

HOROMANCI, OR THE ART OF dialing By the Geodeticall staff, Containing the making of many, or rather, most Dials, without other help.  

CHAP. XXXI.  
To make an horizontal horoscope or dial in any obliqne Horizon, with the theorical ground thereof.  
TO make an horizontal dial is no other thing then to find how many be, of the Horizon is contained betwixt every 15 Meridian, so that every hour in the dial doth comprehend 15 Meridians, therefore if the Meridians in the heaven were visible to the eye, if you did but stand on the South part of the dial, and behold the North, you should see that the 12 of clock would just point to the North part of our common Meridian,& that 11 or one of the clock would di●●●●●●● your sight to the is Meridian into the East or West, and so 10 or 2 of clock to the 30, &c. whereupon you may gather, that the less the pose is elevated, the less the hours be towards noon, and the greater towards evening, because the Meridians answering to the midst of the day be nearer together, because their distance is taken near the pole, as concurring at a conunon center: and further distant in the evening, because they point or answer to the Meridians in the midst of the sphere, which are so much the greater by how much the pole is less elevated, by reason of the obliqne section of the Horizon. And as I distinguish the hours in any horizontal sun dial by the number of the Horizontal. degrees included betwixt enery 15 Meridian, so likewise it shall come all to one pass, if you take the number of Meridians included betwixt the section of every 15 degree of the Horizon: the onely difference is, that the axletree line or 6 of clock Meridian in this work, must be the 12 of clock line.  
Let A E C D be an hemisphear●, and let the proposition be to make an horizontal dial to the 53 degree of latitude, which may aptly enough be applied to the latitude of the town of Shrowsebury.  
Let C be the North pole, B the South, F G the Horizon, drawn according to the elevation 53 degas F C: let H be the ●enith, C B the 12 of clock Meridian,  
The first thing then that we be to do, is to find the distance betwixt every 15 deg. vpon the Horizon A F, counting from wards A, which by the 14 def. of the first, is performed by Azimuths, then it is easy by the 3 Chapter at 2 to find, where every 15 Azimuth cutteth, so shall you find A I to be 15, and there the 15 Azimuth cutteth A K to 30, A L 45, A M 60, A N 75, and A F 90: then are you to count what number of Meridians counted from A pass by éach of those points, which the 4 Chapter will easily perform, which limits the distance of one hour from another in deg. so shall you find the 12 5/ 60 Meridian to pass by I, the 24 43/ 60 by K, the 38 37/ 60 by L: the 54 3/ 60 by M, the 71 28/ 60 by N, and the 90 by F, as is plain by the pricked lines and the foresaid 4 Chapter: so then haue you found the one quarter of your dial, by which all the rest is made.  
The distance of each hour thus found, the labour then is no more but to describe a blind circled vpon your perfit, and then protract each hour by the first Chapter, according to the degree found, and then to put figures thereunto to signify 1, 2, 3, &c. of the clock, and solder a gnome according to the elevation, as hereafter.  
Example.  
Let A B C D be the perfit of brass for your dial, E the center, whereupon describe a blind circled F G H, and cross the same with two diameters at right augles, as F G H E: make E H the 12 of clock mark, and F G the 6 of clock marks: then protract the one quarter of the dial, as H F, thus: count the number of Meridians included betwixt F A, in the 1 figure or F S: viz. 12 5/ 60 deg. from H towards F in the ensuing figure, as H K, then draw the line E K for the 1 of clock line: next take the number of Meridians included betwixt A and K, or K R in the first figures, as 24 43/ 60 deg. and then protract the angle L E H 24 43/ 60 deg. so haue you E L the 2 of clock markernext note the number of Meridians included betwixt A and L in the first figure, or L and Q, viz. 38 ●7/ 60 degrees, and then from I protract H E M an angle of 38 37/ 60 degrees, so haue you E M the 3 of clock mark: do so with M A and N A, as with 54 3/ 60, and 71 28/ 60 degrees, so haue you E N and E O, the 4 and 5 of clock mark, the 6 of clock is F E.  
Now for the hour beyond 6 to 8, F P, and F O be equal, so is F N and F Q, then for the forenoon part of the dial, make it in all respects like unto the other, and the hour a like distant from 12 of clock, as 1 and 11, 2 and 10, 3 and 9, 4 and 8, &c. and by this means I hold you may make a dial as precisely as by any instrument whatsoever.  
Now for the gnome note the elevation as 53 degrees, according to which protract an angle of 53 degrees, as I E H: I conclude the triangle shadowed with lines as I E H, is the true pattern of the cock or gnome H E, being souldered fast on the Meridian at right angles with the perfit.  
I could here you teach another way to describe this kind of Dial, but by this instrumet more troublesone, tho haply more agreeing to the ground of Dials, as by the section of every 15 Meridian with the Horizon, as is remembered in the beginning of the Chap. notwithstanding any man of reasonable capacity may easily conceiue the making thereof; but I hold it less certain then this, which is most precise: but howsouer, 〈◇〉 intend not here to aggregate multiplicities.   

CHAP. XXXII.  
To make an horoscope dial to any wall, directly beholding the South, and standing perpendicular.  
SUch a South wall in any country is such an one, whose pole Zenith lieth or pointeth directly to the intersection of the Meridian with the Horizon, or we may term it to be an Horizon, but all is one: for the making thereof doth not much differ from the former.  
 
having therefore drawn your Horizon F G according to the elenation as in the last Chapter, cross the same with another line at right angles just over the center, as H I, then make I H the Horizon, and work in all respects as in the last Chap. for the finding out of the hours.  
Example.  
having made I H the Horizon, working as you were instructed in the last Chapter, you shall find that the 9 10/ 60 Meridian passeth by H, and so many deg. must the 11, or 1 of clock marks stand from 12: then proceeding you shall find the 19 ●/ 60 Meridian pass by K K, and so much must the 10 or 2 of clock line stand from 12: then by LL passeth the 31 2/ 60 Meridian, by 4 the 46 33/ 60 Meridian, by 5, 66 Meridian:& so the 6 of clock H endeth at 90, whereby you may also perce●ue, that in a direct South dial there can be placed but 12 hours, as 6 before noon, and 6 after noon: therefore having found the distance of your hour lines, fall to protract thus.  
Let A B C D be a board, or such like, to make your dial vpon; vpon the midst of A B, as E, erect a perpendieular E F,& then on L as a center, describe the semicircle A F B: make E F the 12 of clock line, then in the quadrant A F, protract by the first Chap. an angle of 9 10/ 60 deg. as F E G, so shall E G be the 〈◇〉 of clock line: do so to all the other hours found according as in the last Chap. so shall F H contain 19 ●/ 60, D F I 31 2/ 60 deg. F K 46 12/ 60 deg. F L 60 deg. and F A 90 deg. this side of your dial made, the other for the after noon is made in the same manner, then put figures thereunto according as in the demonstration.  
Now to make a gnomon or still for this dial, you must take 53 degrees the elevation of your pole out of 90, so haue you 37 degrees, and then from F protract an angle of 37 deg. as F E M, which is the height of the gonion; and the line E F must always be placed very perpendicularly vpon the wall, and so haue you finished your South dial: and note that in the former of the 2 figures, the chord C H, is always equal to the height of the cock in a South dial.  
Now the differences of placing this Dial from the former, is that the end of the cock, which pointed into the South in an horizontal, must lye in the Zenith here in this vertical or South dial, and hang perpendicular, as is said already.   

CHAP. XXXIII.  
To make a direct North dial or horoscope.  
I will not stand vpon circumstances in this place, for this dial is all one with the last, onely you must turn the side C D which was downwards in the South, to stand upward, E being the nadir and, F the Zenith: so then shall the cock behold the North pole, and M E lye parallel to the axletree line, which of necessity must be in al kind of Dials whatsoever: for the side E M doth always represent the axtrées line about which the Sun is carried,& you may omit the hour of 11, 10& 9 before noon, and so likewise of 12 and 3 after noon, because the sun cometh into the direct North wall, but onely before 6 in the morning, and after fix at evening.   

CHAP. XXXIIII.  
To make a dial to a direct East wall, called a vertical horoscope.  
LEt A B C D be an East wall, whereon draw a perpendicular, then note the elevation of your pole for the country where you make the dial viz. 53 degrees, which take from 90, so haue you 37 degrees the height of the equinoctial: then from the perpendicular E F draw an arch on the South side the same, and in any place of the perpendicular as G H: next from G protract an angle of 37 degrees as G I,& then draw the line E F which represents the axletree of the world, and doth lye parallel there with: so is also F E G the true patten of the cock for the South dial: but to the purpose, from any point in the line E F draw a perpendicular as K L: then see what breadth you will haue your dial, and accordingly draw another line parallel thereunto as M N, so is your dial prepared to reeiue declination.  
In any place of your dial towards the South end draw a line overthwart the dial perpendicular to K L, and extend that line beyond the Dial as O P, thē on the point P making P Q the diameter of a circled, describe a demicircle according as you see C Q R, then from Q, towards R protract an angle of 15 degrees as Q S, then draw a line from Q by S to cut the line K L, as at S; do so to every 15 degree, in the quadrant Q R extending the lines until they make section with K L, so shall Q F be 30 degrees, Q R 45 degrees, &c. and so shall the lines drawn by those number make section with the line K L at S F V w and L, then from S draw a line parallel to O Q for the 7 of clock line, do so from F for the 8 of clock line, and so likewise from V w and L, for the 9, 10, and 11 of clock lines: then for 4 and 5 before 6, make those hours equal to 8, and 7 after 6; then putting figures to hours as in the demonstration, the howr● lines be finished.  
Now for the cock of this dial, he must not be made like to a triangle, but as a parallellogram like to Q O Y Z, containing the just height of the semidiameter P Q, he must be placed in the 6 of clock line at right angles with the wall.   

CHAP. XXXV.  
Of an West dial.  
work in all respects, according unto the last Chap. onely this difference there is, that where you placed the 11, 10, 9, 8, 7, 6, 5, 4, of clock houres before noon, here must you set in their places 1. 2. 3. 4. 5. 6. 7. 8. of clock houres for the afternoon, because the Sun cometh not into the West until after 12.   

CHAP. XXXVI.  
To make a dial unto any right Horizon.  
WHat a right Horizon is, I haue already shewed in the 7 Chap. and that there be two kinds thereof, viz. Polar and Equinoctial: and in that Chapter I something disagree from other writers, in calling that an equinoctial sphere which is used for a sphere Polar, my reason is, because I hold this nomination more proper: not for that it cometh from the most worthiest part as that which is highest, which the Equinoctial is, for that he passeth by the Zenith, but for because the daies& nights there be always equal through the year, what declination soever the Sun haue:& this circled is called the Equinoctial for no other cause but for that the Sun being there doth make equal daies and nights through the world, onely that Horizon excepted which they term the equinoctial Horizon: for the sun being there in that circled, he beginneth either two depart out of sight, or remain in sight for half the year, so that in that Horizon, there is neither equal day or night at any time; and therefore the denomination of the sphere should seem unproper, but howsoever.  
For a right Horizon where the pole is in the Zenith, which I call a polar Horizon, the making of a dial thereunto is easy: for if you divide a circled by your staff into 24 equal partes, and put a style perpendicular therein, the labour is finished, onely adding figures to the hours, as the common order is.  
But for a dial where the equinoctial is in the Zenith, work thus.  
Let the piece of perfit whereon you would make this dial be A B C D, then make choice of what breadth he shal be as D E, then draw the line E F parallel to D C, and in the midst thereof draw a line squire wise to E F, as G H; make H I the diameter of a circled, and on H as a center describe a demicircle K I L, then take one of the quadrants as I L, and first protract an angle of 15 degrees a I H O, then of 30 as I H P, then of 45 deg. as I H Q, then of 60 degrees as I H R, lastly of 75 as I H S; so haue you denided I L into 6 equal parts: then keeping arule vpon H, draw a line by S, noting where it cuts the line E F as A F, then draw a line from H by R noting where it cutteth E F as be ore, as at M, do so to Q P O, and the line E F shall be cut at N Z O; then from F to the line B D C draw a line parallel to I G, and that is the 7 of clock in the morning: do so from M N Z O, so haue you the 8. 9. 10. and 11. of clock as in the figure, then the hours afternoon as 1. 2. 3. &c. be all one with the forenoon hours, as is plain.  
Now for the cock of this dial, he must stand over I G the 12 of clock line, containing the just height of the semidiameter I H, as you may well perceive by the cock in the dial, which is shadowed with lines: for your better understanding he must stand perpendicular vpon the perfit; and thus farrs of direct Dials.   

CHAP. XXXVII.  
To take the declination of any wall.  
A Declining wall is such a sur face that doth notdicectly behold the South or North, but bendeth either into the East or West, and is thus found.  
Draw a line perpendicular vpon the wall, and in that line stick a style to make right angles with the said wall, note when the shadow that the style yieldeth is in one line with the perpendicular, and at that instance take the height of the sun, and so get by the 26 or 28 Chapter his Azimuth, reckoned from the South, for that is the true declination of the wall.  
Otherwise.  
You shall first by some of the forenamed Chapters get the true Azimuth of the sun reckoned from the South, and consider whether it be East or West, then whether the wall be East or West declining from the South: first therefore in East declining walls open the legs unto a right angle, and apply the right leg( the equal diuisious upward) unto the said wall; then if the shadow of the left sight, agree with the fidutiall edge of the left leg, the Azimuth of the sun is the declination of the wall; but if the shadow do not agree, then open or close the left leg until the said shadow do agree, and note the quantity of the angle vpon the legs, and then work thus.  

1 If the angle exceed 90, sustract 90, and note the remainder.  
2 If the angle be less then 90, take the said angle 〈◇〉 of 90, and note the remainder. 

1 If the angle exceed 90, and the sun be East from the Meridian, then take the foresaid remainder out of the Suns Azimuth, the remainder is the declination.  
2 If the angle be less then 60, and the sun West from the Meridian, then take the Azimuth out of the foresaid remainder, and that which remaineth is your desire.  
3 If the angle be less then 90, and the sun East, then add the said remainder to the Azimuth, so haue you the declination: and these rules be general for any South wall declining into the East.     
For walls declining into the West.  
For West walls apply the left leg to the wall, and use the right as before the left.  
First then, if the angle be more then 90, and the Azimuth West, take the foresaid remainder out of the Azimuth.  
If the angle belesse then 90, and the sun East from the Meridian, take the Azimuth out of the remainder.  
But if the Azimuth be West, and the angle less then 90, add the said remainder to the Azimuth of the sun: so in all these ways haue you the declination of any South wall declining into the West.  
Note for East declining walls betimes in the morning, is good for West late near sun set.  
Example.  
I was to observe the declination of a wall, bending into the East, I therefore took the Azimuth of the sun, and found it East from the Meridian 16 deg. then did I apply the right leg unto the wall as before, and found the angle 110 deg. I take therefore 90 from 110, so is the remainder 20, which I take from 30, and that which remaineth is 10, the declination from the wall, by the first def. of East walls.  
Otherwise without the sun.  
Consider if the sun decline into the East or West.  
If East, apply an needle unto the left leg as in the 28 Chap. then apply the right leg unto the wall, opening the left until the needle stand just over the line in the bottom of the carded: then note the quantity of the angle vpon the legs for the declination.  
But if the wall decline West, apply the needle as before unto the right leg, then apply the left unto the wall, opening the right until the needle stand over his due place, the angle then vpon the legs is the true declination.   

CHAP. XXXVIII.  
To learn the Elenation and Angle of deflection, of any declining wall, by the declination.  
DRaw asemicircle A B C, then on the center D erect a perpendicular D B, then note the dcelination of the wall, which by the last Chapter was 10, therfore from B protract a chord B E of 10 deg.& from E to D draw a line E D: then note your latitude which taken from 90, the latitude is 53, which taken from 90 leaveth 37: therefore by the 6 draw a parallel to A C 37 degrees distant as F G, next draw a line so far from C as E is from B, as I D: now I say, the Ellipsicall line passing by H, counted from E D, is the angle of deflection, which by the 5 is about 9 degrees. I could not judge precisely for want of an instrment, when I wrote this Chap. then by the 3 Chap. the distance of H from D I is the elenation of the wall, as I K, which by the 6 you shall find near 36 deg. or thereabouts. I cannot speak precisely, because as I haue said I want my instrument.  
If you desire the ground of this work, you may read M. Blagraues 6 book, Chap. 16 of his worthy jewel: for A C is the Equinoctial, B D half the axletree line, E D the Zenith line, D I part of the Horizon.   

CHAP. XXXIX.  
To take the reclination or inclination of any wall.  
IF the wall recline, apply the left leg to lye parallel with the superficies of the reclining perfit, then heave up the right leg, until he stand perpendicular, which you may eastly try by help of a plumb: note the angle then vpon the legs, for the quantity of the reclination.  
The like must you observe for inclining walls, onely the right leg must hang right downward towards the ground, which before did stand perpendicular pointing to the Zenith.   

CHAP. XL.  
To make a dial to a South or North inclining or reclining flat.  
YOu must consider, whether the flat do recline or incline, and thereby whether it be North or South: if the flat incline or recline North, take the reclination from the elevation: but if it be Southwards, add the reclination to the elevation, so haue you the height of the pole above that flat: but it the addition exceed 90, subtract 90: the remainder taken from 90, leaveth the poles elevation, which had, by the 31 Chapter make a dial according to the elevation, which shall serve your flat.  
At this time I will speak no more of Dials, because I haue drawn them into a faire volume by themselves, keeping it for one of my mathematical Flowers, where they be performed diuers ways, as well instrumentally and Arithmetically, as by an easy kind of mechanical working: for here my intent was not to enter into the whole Art of dialing, but onely to show you some such few propositions, that the staff easily performed of himself.   

CHAP. XLI.  
certain Cosmographicall notes.  
I mind not here to make ample relation of Cosmography, but onely teach you one or two conclusions: for the better understanding whereof, it were fit that these notes were precisely considered,  
1 All places according as they be situate vpon the earth, haue both longitude and latitude, as the stars haue in heaven, I speak Cosmographically: and so is the earth distinguished into like circles as the heaven is.  
2 Latitude on the earth is no other thing then declination in the heauens, therefore by the 1. def. 6, all circles of latitude lye parallel to the equinoctial.  
3 Latitude is denominated by the ark of the Meridian or circled of longitude included betwixt the equinoctial and the place proposed, so that there is both North& South latitude.  
4 Def. North latitude is reckoned from the equinoctial towards the North pole, and South towards the South pole.  
5 Longitude on the earth, is not taken as Longitude in the heauens, according to the 15 def. of the 1. but longitude of the earth is the distance of any place East from the islands called Insulae Fortunatae, which are certain islands situated West from us at London.  
6 The Meridian passing by those Islands is the first deg. of longitude reckoned vpon the equinoctial into the East, so that there is both East and West longitude.  
7 East longitude continues until 180 deg. and West longitude, from 180 unto 360 complete.  
8 The number of Meridians included betwixt any two places reckoned on the equinoctial, is called the difference of longitude.  
9 Our late Cosmographers make the first degree of longitude to take beginning at the Islands of Azores, which is 5 demore West then the fortunate Islands, because then the compass is found to haue no variation.  
10 All towns either haue one longitude, and onely differ in latitude, or one latitude differing in longitude, or else differing both in longitude and latitude.  
11 The earth is also divided by the waters after 4 manners, either it is divided into an iceland, or a Peninsula, or an Isthmus, or a Continent.  
12 An island is a part of the earth that is enclosed round with waters, as Britannia, Rhodus, Scilia, America, &c.  
13 Peninsula called Chersonesus, is neither an absolute island nor yet a Continent, and is such a one that is included round with waters, onely some narrow or streight place remaining, such is Cymbrica Chersonesus, or England and Scotland.  
14 Isthmus, is a narrow piece of land included betwixt 2 seas, stretching itself so out, that you may pass thereby into the Peninsula, as an appendix, knitting the 2 Peninsulaes together, as Northumberland doth England and Scotland; so may you pass from Dania to Cymbrus.  
15 A Continent is called all the fixed or solid earth, which is neither island, Peninsula, Isthmus, but doth onely cohere and consist of itself, such is Bohemia, Saxonia &c.  
So thē the men inhabiting in these diuers parts the of earth, haue diuers names according to their shadows, as Amphiscij, Heteroscij, Periscij, and according to the situation, as Antaeci, Periaeci and Antipodes.  
16. Amphiscij, be those thata remain under the equinoctial, to whom the sun yieldeth 4 shadows.  
17 Heteroscij, be those that dwell under either of the temperate Zones, casting their shadows but one way at noontide always  
18 Periscij, be those that cast their shadows round about, and such inhabit under the poles, according to situation.  
19 Antaeci or Anticolae, be such that inhabit under one Metidian, and be a like distance from the equinoctial, that is, having equal latitude but on different sides the equinoctial, as if ours be North, theirs is South, and they haue paria tempora, but not pariter.  
20 Periaeci or Circumcolae, be they that remain all under 1 Meridian& in 1 latitude, with whom there is all things alike to us, as times, seasons, &c. onely this difference, when we haue noon, they haue midnight, when we haue night they haue day, our sun set is there sun rising &c.  
21 Antipodes be those that dwell feet to feet with us diametrically, so that if there be a plumb let fall through the center of the earth from us, it should pass to them.   

CHAP. XLII.  
To make a map of the would, and therein to describe any town according unto his t●●e longitude and latitude.  
THis matter is very easily performed by your staff, first you must repair to the tables of Apian, or some more late writer, and therein seek the true longitude and latitude of the town or city whose place you require, then draw a parallel unto the equinoctial according unto the latitude, and find where the Meridian or circled of longitude cutteth the said parallel for that is the place of the town or city.  
Example.  
I would place London in the hemisphere according to true longitude and latitude, by A●ians Tables I find the latitude of London to contain 52 ½ degrees, which is more then we allow, and the longitude to contain 13 1/ 3 degr.( which is less then we allow) I make A B C D an hemisphe are, B D the equinoctial, C the North pole: then I protract by the 6, a parallel E F 52 ½ degrees distant from A D: now I make the circumference to be the first Meridian according to the old writers, then do I find by the 4. where the 13 1/ 3 Meridian doth cut E F counted from the circumference, which falleth at E the true place of London, as you may perceive in the figure.  
 
In like manner may you see edinburgh, Venice, and Toletum, &c. placed.  
Now if you desire hereafter by your staff to seek the longitude of any place, the 5 Chapter will teach you, and for latitudes the 6 may suffice.  
By this instrument may you most easily find out the distance of places by a demicircle, according as master Blundiuell hath set it down, without the dividing of the same into 180 degr.   

CHAP. XLIII.  
How to know, how one place beareth from another, by the Geodeticall staff.  
THe places being given which you desire, to know how that one beareth from the other, you shall first take the place of your abode, which is the place you desire to know how the other beareth from it: then in your map find the next Meridian passing by the same, which had, let fall a perpendicular from the said place of your abode vpon the said nearest Meridian: then cross the perpendicular with a live at right angles, in the very place of your abode, which shall represent the Meridian of the said place, and the perpendicular the latitude: so the one end of the Meridian representeth the North, the other the South, and the perpendicular on the East side the Meridian the East, and the part on the other side the West: now you are to consider on which side this new found Meridian the other place standeth either East or West, and towards the same draw a demicircle, the one foot of your compass placed in the very intersection of the lines vpon the place of your abode; then vpon the center of this demicircle and the place desired lay a ruler, or extend a thread, which shall cut the demicircle in some point, the chord whereof included betwixt that point and the Meridian, doth show the number of degrees; and so by allowing 11 ¼ deg. to a rombe, you may easily find what wind the section made in the demicircle, the opposite wind thereunto is the wind that driveth you unto the desired place.  
And here you haue to note, whether the demicircle were made vpon the East or West side the new found Meridian, if on the East the partes of the compass be either S E or north-east, and the contrary for the West: if you desire to work this proposition by this staff, you may haue the wind distinguished vpon the legs among the chord divisions.  
Example.  
 
G H, is the Meridian in the Map, C the place of your abode, C I the line falling perpendicular vpon the Meridan, F the place sought how it beareth from C lying Westwards from the Meridian, N S, S w N, the semicircle drawn on the West side, F C a line drawn from the one place to the other A, a place where the line cutteth the semicircle A N, the chord measured 45 deg. distant from N the North, to the West: now A N, 45 deg. reduced into winds, contains 4 winds from the North, which is north-west, and so doth F bear from C; now the oppositewind to N w, is SE, which is the wind must drive you from C to F.  
I might here adjoin many pleasant& easy ways by this instrument to get the elevation, as by the declination, almicanter, and Azimuth, or by the declination hour, and almicanter, or by the declination and amplitude, or by the stars that never set, or by the stars that set, &c. but since I haue shewed some few ways to that purpose before, I will here leave to confounded the reader with varieties, presuming that what is said before, may give light sufficient to the performing of these and many other ways.    
Here ends the eight books of Geodetia.   

A brief Table of all the Chapters, in these eight books of the Geodeticall staff.  


The first book.  
Chap. 1 OF the framing, composing, and quantity of timber required to make the Geodeticall staff. Fol. 1  
Chap. 2 Of the deffinition and vocation of the staff and his principal partes. 3  
Chap. 3 Of the division of the upside of both the legs, together with the one side of the graduator. 5  
Chap. 4 To project the degree of a circled to 90 vpon the graduator, and unto 180, vpon the legs. 6  
Chap. 5 The order of projecting the Geometrical quadrant and hypsometrical scale vpon the graduator. 8  
Chap. 6 Of the staff his parts and application. 9  
Chap. 7 A needful proposition to be understood for the performing of the fourth proposition. 13  
Chap. 8 The order of projecting the chord of a circled vpon the lower side of the legs of the Geodetical staff. 14    


The Chapters contained in the second book.  
Chap. 1 THe manner of using the Geodeticall staff, with the things therein to be considered. 17  
Chap. 2 Of English measures, both old and such as be allowed by statute, with the comparing of them together &c. 19  
Chap. 3 Of the parts of geometrical mensuration. 22  
Chap. 4 Of Geodeticall angles, and how to use the staff in taking them. 25  
Chap. 5 The legs of the staff being opened to any angle to find the quantity therof in degrees. 27  
Chap. 6 To open the legs unto any angle proposed. 28  
Chap. 7 To take the height of the Sun, a comet, or any other thing seen in the heauens. 29  
Chap. 8 The altitude of the sun being given, to find the hour of the day easily. Ibid  
Chap. 9 To take the distance betwixt any stars unknown. 31  
Chap. 10 To find the hour of sun rising, &c. Ibid  
Chap. 11 To find the altitude of the North-pole, by the stars as never set. Ibid  
Chap. 12 To perform the last Chapter by the stars as set, or by the sun. Ibid  
Chap. 13 The latitude of any place given, to get the declination of the Sun, or any planet, star &c. seen in the heauens. Ibid  
Chap. 14 Of the Anomality in the situation of proportional triangles in the legs of the staff. 34  
Chap. 15 A castle, sort, &c. seen, to get the true distance thereof, without arithemeticall calculation, Ibid  
Chap 16 To take the height of any castle, Tower, or three, &c. 36  
Chap. 17 You standing vpon a tower, &c. to tell how far any thing seeneis from you 37  
Chap 18 The height of a fort, castle, &c. given, 38  
Chap. 19 To fetch his distance from you, by onely taking the angle of altitude. 39  
Chap. 20 You seeing two ships vpon the seas, and two castles vpon the land, &c. to fetch their distance without calculation. 40  
Chap. 21 A castle, &c. seen before you, so that by reason of impediments you cannot get his distance from you, as in the 15, how to perform it here at 2 stations, &c. 41  
Chap. 22 To take the altitude of any castle &c. though you cannot approach thereunto by reason of waters, as in the 16 Chapter. 42  
Chap. 23 To measure any valley or such like profundity 44  
Chap. 24 A Church or such like situate vpon a high hill, and you standing vpon a low hill, and a great valley betwixt, so that you cannot place the legs as before, &c. yet to find the height. Ibid  
Chap. 25 To take any profundity, and of another working by the staff then before. 45  
Chap. 26 Of the equation of angles in the Geodeticall staff. 47  
Chap. 27 Two numbers being given, or 2 lines assigned, to find the 3 in proportion. 48  
Chap 28 To perform the golden rule vpon the Geodeticall staff. 49  
Chap. 29 If a tower, army of men, &c. be seen before you, to fetch the distance thereof from you. 50  
Chap. 30 To take the height of any accessible thing standing perpendicular, 52  
Chap. 31 The height of a tower &c. given, to fetch the distance thereof from you, with the length of the scaling ladder. Ibid  
Chap. 32 Two ships seen vpon the seas, &c. to seek their true distance. 53  
Chap. 33 To find the distance of any two places, though you cannot come to take the angle of latitude as in the last Chapter. 54  
Chap. 34 To seek any vnapprochable altitude at two stations 56    


The Chapters contained in the third book.  
Chap. 1 TO place your staff perpendicular by art &c. 59.  
Chap. 2 To measure the distance of any mark seen before you in a right line. 60  
Chap. 3 To perform the last Chap. by a carpenters squire, represented by the staff. Ibid  
Chap. 4 To take the height of any thing by his shadow. 62  
Chap. 5 To perform the last Chap. with more ease. 66  
Chap. 6 To take any altitude without a shadow, by the visual beams. 67  
Chap. 7 To search out inaccessible heights. 68  
Chap. 8 To perform the 7 Chap. with more ease after Gem. Frisius. 70  
Chap. 9 To take any perpendicular height, by the onely help of the hollow staff. Ibid.  
Chap. 10 To fetch altitudes, by the visual beams reflected into a glass &c. 73  
Chap. 11 To fetch the altitude of a tower by a new way not spoken of heretofore. 74  
Chap. 12 To ferch heights after another way by their shadows, then was spoken of before. 75    


The Chapters contained in the fourth book.  
Chap. 1 TO take the horizontal distance of any mark from you without any calculation, or by calculation. 81  
Chap. 2 You standing vpon the top of a turret, to fetch the distance of any mark from you. 82  
Chap. 3 You standing vpon the ground, how to measure the altitud of any thing. 84  
Chap. 4 To measure inaccessible heights by the geometrical quadrant. 85  
Chap. 5 You being vpon the top of a high turret, how to measure an inferior altitude. 86  
Chap. 6 To work the 5 Chapter, versa 'vice. 88  
Chap. 7 To measure the height of mountaines. Ibid.  
Chap. 8 To take the altitude of a tower, &c. placed vpon the top of a high hill. 89  
Chap. 9 How lengths in heights be found. 90  
Chap. 10 To take the depth or profundity of any well, &c. 91  
Chap. 11 To perform the last Chap. by the equal parts of the legs. 92  
Chap. 12 To perform the 11 Chapter otherwise 93  
Chap. 13 To measure profundities by the hypsometrical scale. Ibid.  
Chap. 14 To fetch as well the length of the sides, as also the perpendicular depth of any valley. Ibid.    


The Chapters contained in the fift book.  
Chap. 1 OF the Iacobs staff, together with the comparing him with the Geodeticall staff, with the things requisite to be observed in the use therof 95  
Chap. 2 Towrke the rule of proportion by scale and compass. 96  
Chap. 3 To work the golden rule reversed, &c. by scale and compass. 100  
Chap. 4 How many ways the lacobs staff will perform dimensions, whereon he dependeth, and what he requireth. 101  
Chap. 5 By some known altitude to find out the unknown longitude at one station. 102  
Chap. 6 To measure lengths, standing, vpon the ground at one station. 103  
Chap. 7 By the known altitude to find the unknown longitude at one station. Ibid.  
Chap. 8 By the known altitude to seek any unknown longitude at two stations. 104  
Chap 9 By the known longitude to find the unknown altitude at one station. 106  
Chap. 10 By knowing the breadth of any pit, well, &c. to fetch the unknown profundity. 107  
Chap. 11 By the given longitude to find the unknown altitade. Ibid.  
Chap. 12 By the known part of some altitude, to find out the whole altitude itself. 108  
Chap. 13 To find the altitude of any thing by help of two stations. 109  
Chap. 14 To find the distance betwixt any two towns at two stations. 110    


The chapters contained in the sixth book  
Chap. 1 OF measuring ground,& what instruments are best for that purpose. Ibid  
Chap. 2 Of protracting angles, and finding out of the quantity of angles protracted. 111  
Chap. 3 To take the perfit of any Lordship &c. and to lay the same down according to the true symmetry therof. 112  
Chap. 4 To know if your perfit will close, and whether you haue wrought well or not. 116  
Chap. 5 To take the perfect perfit of a field at two stations,& to lay the limits and every angle therein down according to the true porportion thereof, &c. Ibid  
Chap. 6 To take the perfect perfit of a piece of ground by measuring round about the same. 120  
Chap. 7 Diuers ways of taking plattes by the Geodeticall staff. 123  
Chap. 8 To measure woodland grounds in such order as hath not been published &c. 124  
Chap. 9 Of errors daily practised &c. and of the reformation thereof. 128  
Chap. 10 To find the difference betwixt the horizontal& hypothenusal line, &c. 130  
Chap. 11 The making of a new quadrant, to perform the 9 or 10 Chap. 131  
Chap. 12 To seek altitudes by the new quadrant 133  
Chap. 13 The legs of the instrument being opened to any angle, to find the quantity, or to open them unto any quantity assigned, after another way then in the second book. 134  
Chap. 14 having taken the quantity of each angle in any field, to know if you haue wrought truly. 135  
Chap. 15 Of such things which are to be considered in the casting up of grounds, with the protracting of the square roots. 136   

The second parte of Geodetia, Chapter 16.  
Chap. 16 Of right angled triangles, &c. and to measure them. prop. 2. 143  
Chap. 17 Of acute angled triangles, &c. and to measure the contents thereof. 145  
18 Of an oxigonium, Isosceles, &c. and to measure the Area thereof. 146  
19 Of a Scalenum, oxigonium, and for to measure the contents therof. 147  
20 A general rule, to measure all kind of superficial triangles, and of obtuse angles in special. 148  
21 To find the length of the perpendicular in any triangle. 151  
22 Of Quadrangles, &c. and to measure them. 154  
23 Of multangled figures,& of finding the center thereof, &c. and to measure irregular grounds. 160  
24 To reduce perches into stature measure of acres, &c. 163  
25 How to measure all manner of land without arithmetic. 164  
26 To measure a trapez, rombus, &c. without arithmetic. 166  
27 The use of the Table for land measure. 172  
28 How to work when you haue part of perches, and the use of the Table for the purpose. 194  
29 From a place assigned in any tryangular piece of ground, to divide the same into two equal parts. 195  
30 To cut off any portion assigned in any triangle. Ibid  
31 In a rectangled or an oblong parallelogram, to cut off any quantity assigned. 199  
32 A Table for to divide grounds, and the use thereof. 208  
33 To know if water will be brought unto any place. 209  
34 To measure boards, glasses, pavements, &c. with the ground thereof. Ibid.  
35 To measure timber, ston, &c. with the ground of the work. 212  
36 Another kind of new working, in timber measure. 213  
37 Of slude measure, and the ground of the work: Ibid  
38 To seek altitudes after a a new way. 214  
39 To search inaccessible heights. 216  
40 To seek the distance of any place from you. 217  
41 To take the distance betwixt any two places. 218  
42 The use of the legs of the Geodeticall staff, as a scale. 219  
43 To divide a piece of ground as shal be assigned. 220  
44 The rare metamorphosing of all kind of figures both regular or irregular, containing 10 metamorphoses. 221  
45 Of the proportion of lines and numbers one to the other, containing propositions. 6. 233  
46 Of finding the proportion, of one figure, piece of ground or such like one to the other, with the finding of the diameter of circles or chords of regular figures the circumference being vnkowne, and to find the proportion of any chord to the circumference prop. 6. 237  
47 Of making of one figure, like and proportional to any other, of dividing grounds, Prob. 11. 240  
48 To measure mountaines, and valleys, with a further discourse of measuring grounds. 250  
49 The making of a compendious Table, for the legal parte of survey. 253  
50 certain Apophegmes, requisite for al Geodetors, &c. to understand. 256  
51 Of surveying of grounds, of making a new kind of particular, of apportionating lands, of ancient measure, and buying annuities. 259  
52 To reduce statute measure, into customary measure, or any kind of measure used in any particular place or country. 263  
53 Of the plaiting or casting up of one manor, &c. by diuers scales, as stature measure, and woodland measure, &c. 264  
54 In the surveying of a manor, how to join all the parcels in on plate as they lie in the whole manor. 265  
55 To alter the whole perfit of a manor, and every particular therein, easily and speedily. 266  
56 To take the plate out of one piece of paper, and to place the same truly in another piece according to the same proportion and fashion it had before, and that diuers ways. 267  
57 To garnish your plate. 268    


The Chapters contained in the seventh book.  
1 geometrical deffinitions, propositions, and consequences, of lines, angles, and triangles, &c. 269  
2 having given you some necessary geometrical instructions, for the better opening of your understanding, now shal follow axioms of Trygonometria itself, which chiefly is effected by the rule of proportion. The first axiom shall teach what protions shall be in the triangles or parts thereof, afterwards shall be declared, how those axioms may be applied to use. 272    


The Chapters contained in the eight book.  
1 principal notes and definitions for the better understanding of this Book. 325  
2 To express any chord in the limb of a circled, and to protract any angle. 328  
3 Any part of a diameter being given or required, to express the same in degrees. Ibid.  
4 Any part of a parallel being given or required, to express the same. 329  
5 Any point being given in the hemisphere, to find what Ellipsicall line passeth by the same. 331  
6 To draw a parallel by any number of degrees assigned, or a parallel being drawn, to find what number of de. he contains. Ibid  
7 To lay down a right obliqne sphere. 332  
8 To take the altitude of the sun. 334  
9 To find the suins declination for every day in the year, and the declination of any part of the ecliptic. Ibid.  
10 To find any place of the sun in the ecliptic, or any point proposed. 335  
11 To find the altitude of the pole for any country whatsoever. 336  
12 To know what hour of the day it is, the sun shining in any region. 337  
13 To find the right ascension of any ecliptic, together with the degrees of the equinoctial that do ascend, with every degree of the ecliptic in a right sphe are. 339  
14 Of the difference of ascensions. 340  
15 Of the obliqne ascensions 343  
16 Of the descentions both right and obliqne. Ibid.  
17 The longitude and latitude of any star proposed, to find his right and obliqne ascention, together with his declination from the equinoctial, and to place any star in the hemisphere. Ibid.  
18 The difference of ascensions, or the obliqne ascention of any point in the ecliptic proposed, or of any star, to find the latititude of any country. 344  
19 Of the amplitude of the rising and setting of the sun or any star, or point in the heauens. 345  
20 The amplitude of rising of any star known, or of the sun with his place in the zodiac, to find the latitude of the country. 346  
21 To know the rising and setting of the sun, the length of the day& night, with the continuance of twilight. 347  
22 To know the semidurnal arch of the Sun of stars, with the length of day and night, &c. 348  
23 The quantity of the longest day of any country given, to find the elevation of the pole, and to distinguish the climates. 350  
24 To know the Meridian altitude of the sun or any star. Ibid.  
25 To find the degrees of Medium Coeli, or Culmination at any time proposed, &c. 351  
26 To know the height of the sun every hour in the year above the Horizon, without the sight thereof, as also to find the azimuth in any region of the world 352  
27 To find the Meridian& four quarters of the world by the Geodeticall staff. 353  
28 To find the azimuth of the sun. 354  
29 To know what hour the sun cometh to any azimuth proposed. Ibid.  
30 Another way most easy& true, to know the hour of the day. 355  
31 To make an horizontal horoscope or dial in any obliqne Horizon, with the theorical ground thereof. 356  
32 To make an horoscope dial to any wall, directly beholding the South, and standing perpendicular. 360  
33 To make a direct North dial or horoscope. 362  
34 To make a dial to a direct East wall, called a vertical horoscope. Ibid.  
35 Of an West dial. 364  
36 To make a Dial unto any right Horizon. Ibid.  
37 To take the declination of any wall. 366  
38 To learn the elevation and Angle of deflection, of any declining wall, by the declination. 367  
39 To take the reclination or inclination of any wall. 368  
40 To make a Dial to a South or North inclining or reclining perfit. 369  
41 certain Cosmographicall notes. Ibid.  
42 To make a map of the world, and therein to describe any town according unto his true longitude& latitude. 371  
43 How to know how one place beareth from another by the Geodeticall staff●. 372    
FINIS: