A MATHEMATICALL APENDIX, CONTAINING MA­NY PROPOSITIONS AND CON­clusions mathematicall: with necessary obser­uations both for Mariners at Sea, and for Chero­graphers and Surueyors of Land;

TOGETHER WITH AN EASIE perspectiue mechanicall way, to Delineat Sunne dyalls vpon any Wall or Plane giuen, be it direct, in­clyning, declyning, or reclyning, from the Hori­zon, or Meridian, in any Region or Place of knovvne Latitude.

With other things pleasant and profitable for the weale publick, not heretofore extant in our vulgar: Partly coll­lected out of Foreigne moderne writers, and part­lie inuented and practised by the Author.

Written by R. N. Gent.

LONDON, Printed by R. B. for Roger Iackson, and are to be sold at his shoppe in Fleet-strtet, [...] [...] duit, 1604.

TO THE RIGHT HO­NORABLE MY SINGV­LER GOOD LORD AND MA­ster, Sir EDVVARD SEYMOVR Knight, Baron Beauchamp, Earle of Hertford, and his Maiesties Liuetenant within the Counties of Somerset, and Wiltess, and the City and County of Bristoll.

RIght honorable, since amōgst the rudest Crea­tures (euen the brute beasts deuoide of Rea­son) Ingratitude, that foule Monster, is not once to be found: how hatefull then the very remembrance thereof ought to be vnto humane Creatures; but especially to those that professe Christianitie (in whome it hath pleased the so­uereigne Creator of all things, to make his chiefe Storehouse of all Reason in these Terrestiall parts) so much as to purpose, much more in action [Page]to proue ingratefull to any; but chiefely to those from whom especiall fauours haue proceeded: who is it that will not absolutely censure? Wherefore (my honorable good Lord) because I would not seeme more then brute, although J acknowledge my abilitie cannot extend vnto the moytie of my desire: yet like the poore man, who (wanting better wherewith to expresse his loyall affection) presen­ted the mightie Prince Alexander the great, with a cup of coole water: so offer I this small (and therefore vnworthy) present, vnto your hono­rable acceptance: my bounden duety euer chal­lenging greater, doth neuerthelesse encourage me to beginne with this little. Humbly praying your Lordship to shroud these my first fruits vnder the wings of your honorable patronage, against the enuious stormes of Zoilus, and his Crittick associates; who are alwayes more ready to carpe and find fault with others, then any way able to doe the like themselues. Herein promising to my selfe that the more learned in the Mathema­ticks, (who I confesse had bene much fitter to haue handled this subiect then my selfe) and those that desire the benefite of knowledge in the common [Page]wealth (for which, I protest, I chiefely vnder­tooke this work, in regard that scant any thing herein contained; is yet extant in our vulgar tongue) will if not imbrace, yet at the least not vt­terly reiect and contemne this my trauaile. The acceptance of the more learned and better sort I assume to my selfe, hauing euer noted such to fauor the desire of good studies: And of those that loue the common good I much presume, iudging them by my selfe; hauing euer borne especiall affection to such, as haue shewed their willing endeauours to profit the publique weale with their best meanes. And so praying almighty God to blesse your Lord­ship and my good Lady your honorable Coun­tesse, with long life, and all happie encrease of honour and prosperity in this world, and in the world to come eternall fe­licity, I rest alwayes

Your honors most humble and faithfull seruant, ROBERT NORTON.

THe Longitude (gentle Rea­der) for which this present Treatise was chiefly compo­sed) is defined to be the shor­test distance that can possibly be taken vpon the Globe of the Earth be­tweene two Meridians propounded, begin­ning to reckon the same (from west to East) at the Meridian of the Iland Coruo, which our latest writers for diuers good reasons account for the first Meridian: To find the longitude, is by some meanes to search how many degrees the Meridian of the place, where you make obseruatiō, is distant from some one Meridian giuen. The knowledge whereof being so necessary as that the Geo­graphers and Nauigators cannot with exact­nesse performe what (in their Arts) necessity requireth, without the same: Therefore haue many worthy men taken great paynes, and with al ingeniousnes labored to supply that defect, with the most exquisite meanes they could deuise for the readiest finding therof; who haue set downe many Propositions to [Page]that effect, and both learnedly and ingeni­ously left vnto posteritie such tokens of their loue to the publique good, as may iustly challenge to merit infinite memorie and commendation. As Ptolome, by the time of an Eclipse in two seuerall places beeing knowne, to find the longitude. Appian, the true places of the Moone and a fixed Starre being giuen, to find the Longitude. Others by the Angle, that a fixed Star maketh at the moment of time shee entreth into the first minute of Cancer or Capricornus, and the dif­ference of tyme wherein such an angle ap­peareth, to find the Longitude. Some with portable watches, the true hower of sundry places beeing giuen, to find the Longitude. The tyme beeing giuen, and the way of the Ship, to find the Longitude. The poynt of the Compasse & the way of the ship giuē, to find the longitude. The Latitude, and the way the ship maketh, to find the Longitude, &c. All which are found to be subiected vn­to inauoydable errors, or els clogged with such difficulties, or want of expedient exe­cution; as that they cannot absolutely bee [Page]concluded for perfect apt meanes to finde the Longitude at Sea: the want of which, is the greatest imperfection in the Art of Naui­gatiō. Take here in good part (friendly Rea­der) these few propositions, not heretofore written in this our vulgar, cōtaining partly collections which from sundry Authors in other Languages I haue sought out, and augmented or abreuiated for thy better vse; and partly practised obseruations, which (time & opportunity permitting) my selfe haue experienced both on Sea & Land. Re­questing, that with the friendly eye of Iudgement, thou wouldest consider them well, be­fore thou divulge any vnaduised censure a­gainst them: Not doubting but euery one that shall deign a due perusing hereof may find som thing worthy his bestowed labor. And so, graying excuse if I seem to omit a­ny necessary poynt in the premisses (which I might soon do, in regard I much striued to auoid prolixity) doe heartily wish thee thy lawfull desire, and my selfe present to ex­pound any thing that seemes herin difficil; referring my labor to your curteous fauors [...]end.

ROBERT NORTON.

A Table containing the conten's of the Propositions of this Booke.

Proposition. 1.

How to finde the Longitudes of places, by the dayly decli­nation of the Sunne.

2 How to find the Longitude by Arches of great Cir­cles, which passe by the Centres of the fixed Starres, and Planets that yeeld no sensible paralax or difference of Aspect.

A Corollarie vpon the same, shewing to performe the same more exquisitely, by comparing the Planets one with an other.

Another Corollary vpon the same Consequence, to find the Longitude, by comparing the Sunne and Moon being nigh Coniunction, and their Paralaxis abstrac­ted.

3 A Comet, or the Moone appearing, how to finde the paralax or difference of aspect thereof, two seuerall wayes; and thereby how farre it is from the Centre or Circumference of the Earth.

4 The Paralax and distance from the Earth giuen, to finde what Angle it will make in the Centre of the Earth with any other Star that hath no Paralax.

5 To find the Longitude by a Comet, or the Moone, appearing; and the difference of tyme from one Me­ridian to another, by comparing it with a fixed Star.

A Corollarie vpon the same, shewing more euident accomplishment thereof, by comparing it with the true [Page]moouing of one of the Planets.

6 A Starre m Heauen propounded, to finde the Longitude without the difference of Tyme.

7 How to finde the Longitude and Latitude of anie place at once, without the difference of time, by obseruati­on of the Coelestiall Luminaries.

8 How to finde the Longitude at all times from mo­ment to moment, Mechanically: And withall to describe all the places of note in a Region or Country, on a Map or platte, according to their seuerall distances and situations exactlie. With the vse of the Protrac­tor.

A Corollary vpon the same, applying it to the Sur­uey and platting of Land.

Another Corollary vpon the same Consequence, expressing a Mechanicall meanes to find the Longitude at Sea at all times, and to keep a perfect Trauerse, for a whole voyage.

An Annotaion, shewing an artificiall deuise how the Master or Pylot, at Sea, may much more exactly make obseruation of the Coelestiall Luminaries with any In­strument, then the ordinarie manner can possiblie admitte, by reason of the heauing and setting of the Ship.

The making of the Cosmodelite, an excellent Instru­ment for many Mathematicall conclusions: As for the Longitude of places, to take any Altitude, Latitude, or distance in sight, for Suruey and platting of Land, to make Sun dialls, &c. inuented by the Author.

How by the Cosmodelite to delineate a Sun diall on [Page]any plane giuen, with great facility, & without arithme­ticall calculation.

Master Robert Smith his inuention to delineat a Sun diall otherwise.

The making of an artificiall engine, whereby with any strength giuen (be it neuer so little) to eleuate and lift vp any ponderous weight assigned.

Brief Expositions of the Geometricall and Aftro­nomicall tearms mentioned in this Treatise.

Finis Tabulae.

The first Proposition. How to find the Longitudes of places, by the daylie Decli­nation of the Sun.

THe Sunne continually declyning from the AEquator, according to the seuerall poynts of the Ecliptick wherein hee properly mooueth, sometimes towards the South (be­ing in the Meridionall signes), and sometimes towards the North Pole (beeing in the Septentrionall:) making his greatest declination on eyther side to be 23. degrees and 28. or therea­bouts, doth, from the noone of one day vnto the noone of the next day, so sensibly vary his declina­tion on any one Meridian, as that you may easilie find how much hee declyneth from Meridian to Meridian, on all the Meridians that may be imagi­ned vpon the face of the whole earth. As for exam­ple, if from the noone of this day vntill the same moment to morrow, he shall be found to vary one minute of a degree in his declination vpon one same Meridian: imagining then 60. seuerall Me­ridians on the Earth, equally distant one from ano­ther; it is most certaine that he will make from one [Page 2]of those Meridians vnto the next 1/60 part of a mi­nute (which is a second difference of declination: and so consequently more or lesse, as the same shal happē to be beyond or short of the said Meridian. Wherefore if you diuide 360. the degrees of the E­quator, by 60. the number of the supposed Meridi­ans, you shall finde the Quotient to produce 6. de­grees, the difference of the Longitude which one of them shall be from the next. This may be obser­ued from day to day at all times of the yeare, bee it that the Sunne haue greater or lesse declination then the saide minute, vnder the assigned Meridian: which collected into a Table will be a ready means for to find the Longitude of any vnknowne Meri­dian. The practise to find the Longitude (the Table being made) is thus: Hauing exactly obserued the declination of the Sunne (by some perfect Mathe­maticall Instrument) for the hower of Noone, then enter the sayd Table, seeking there the declination set downe for that day: which beeing found, the difference of the declination will shew you vnder what Meridian you obserue and are, with the diffe­rence of Longitude: which difference being added or subducted (as reason will direct) to or from the former giuen Meridian, will yeelde the sought vn­knowne Longitude.

Note that the Table may be enlarged, not one­lie to thirds and fowerths, but vnto tenths: which will be much better for the more exact expressing of his slowe declination. And thus you may finde [Page 3]the Longitude by the daylie declination of the Sunne.

The second Proposition. How to find the Longitudes by Arches of great Circles, which passe by the Centres of the fixed Starres, and such of the Planets as yeelde no sensible paralax or difference of aspect.

FOrasmuch as great Circles, which passe by the Centres of all the Stars, doe expresse in theyr Arches the seuerall distances of all the Pla­nets and fixed Stars, (considered by two and two): And for that the Planets in their proper mouings do ouergoe the Stars of the Firmament; It must ne­cessarily follow, that their said distances do conti­nually vary, (either more or lesse) and the Angles also subtending them. Wherefore, if by Astrono­micall Tables you obtaine the true place of one of the Planets (hauing no Paralax) and compare the same with one of the fixed Stars giuen, be it Septē ­trionall or Meridional; seeking for euery day, how­er, and minute, what Angles the same Planet shall make with the said fixed Star, and of such Angles to make a Table; it wil be an Artificiall preparatiue to the finding of the Longitudes of places with fa­cility. For all the Inhabitants of the Earth (accor­ding to Ptolomy) being in the Centre of the worlde in respect of the fixed Stars and higher Planets, it is certaine that all such Stars and Planets will ap­peare to them all to vary theyr angles continually; [Page 4]but chiefely when the Planet shall bee direct: and that those Planets, which mooue most swiftlie, do yeelde more euidently such variation then the slower ♄. and ♃. The Table beeing prepared: when you desire to finde an vnknowne Longitude, Ob­serue exactly by Instrument, what Angle the same Planet maketh with the saide Starre, for which the said Table was made: which had, seek then in your said Table for the Angle calculated for that mo­ment of time, in which you make your obseruati­on: and the difference of the Angles will giue you the difference of the Longitude (if any bee) be­tween the place of obseruation sought, and the place giuen, for which the table was made.

Corollary. I.

Then it followeth, that by the same meanes, and with more euident variation of Angles, wee may find the Longitudes, by comparing the saide Pla­nets one with another to find their Angles, especi­ally one being direct, and the other Retrograde: & so to make lyke Tables of theyr proper moo­uings.

Corollarie. II.

Likewise if you take the continuall true moo­uings of the Sunne and Moone, during the space of three, or fower dayes, before and after theyr coniunction, and so finde theyr true angles: and obserue the Angle (by instrument) exactly, which [Page 5]they for one same moment, doe make in the eye of the obseruour, you shall finde the obserued Angle greater then the Angle calculated, by theyr para­laxis: which being also knowne and subducted, the difference of the remayning angle (from the Angle calculated), is the difference of the Longitudes, from one Meridian to the other.

The third Proposition. A Comet, or the Moone appearing, to find the Paralax or difference of aspect thereof, and thereby the distance of the same aboue the Centre or Circumference of the Earth.

Wh [...] Par [...] is.THe Paralax or difference of Aspect (accor­ding to the Astronomers) is an angle, made of the concurrence of the visual lyne, directly respecting a Coelestiall luminary; and of the lyne of the true mouing; which proceedeth from the Centre of the Earth, passing by the Centre of the same Luminarie, vnto the Firmament; The se­midiametre of the Earth, beeing the sole efficient cause of such Angles. Neyther do such angles hap­pen to euery of the coelestial Luminaries, saue one­ly to such as are neerest vnto the Earth, as the ☽ ☿ ♀ and ☉; and amongst them also, those which are neerest, haue most sensible paralax, or difference of Aspect: as the figure following doth plainely de­monstrate.

[Page 6]

[diagram]

wherein A representeth the Centre of the Earth, A B the Semidiametre, A F the verticall lyne, D the farthest or more remote Luminarie, yeelding smal­lest paralax, A E the lyne of his true place in the fir­mament, K the neerest Luminary, yeelding greatest Paralax, A I the lyne of the true place thereof, B G the visuall lyne, the Obseruor being in B. Whereby it appeareth, that the Angle E D G, or his verticall B D A, being the Paralax of the more remote Lu­minary, is lesse then the angle I K G, or his verticall B K A, the Paralax of the neerer Luminary: both which Paralaxes, are caused by the sēsible quātity of [Page 7]the Semidiametre of the Earth, beeing compared with their Orbs.

How to find the quantitie of the Paralax.

To finde the quantitie of the Paralax of the Moone, or Comet, for any hower or Meridian gi­uen, during the appearance thereof aboue the Ho­rizon: you shall with the Cosmodelite (the Circle thereof being inclyned according to the inclinati­on of the Equator) take two seuerall Obseruati­ons (hauing some space of tyme betweene): at each of which, seeing the said Comet or Moone through the sights of the Index, you must carefully note the degrees intersered therewith, in the Limbe of the said Circle. Then comparing the quantity of the arch contayned betweene the two poyntes (at the seuerall obseruations) intersected in the said Limb, with the arch answering to the space of tyme be­tweene the two obseruations (beeing reduced into degrees and minutes of the Equinoctiall) and the lesser arch substracted frō the greater, the remayn­der is the Paralax or difference of aspect sought. Although some perhaps will say here, that the Pla­nets, by reason of their excentricities and proper mouings, do vnequally mooue, in respect of the Primum Mobile: which although I confesse, yet I say, that in the space of tyme, spent in the said Ob­seruation, the same is so insensibly small, as that it were absurd to make any scruple thereof.

Another way to find the Paralax.

[Page 8]First seeke, out of the Astronomicall Tables or otherwise, the true place, in the Zodiack, of the Lu­minary you will obserue, for the tyme and Meridi­an assigned; then examine the true place of some knowne fixed star, or one of the Superior Planets, (appearing at that tyme aboue the Horizon giuen): thereby obtayning the Angle, subtending the Arch of the Distance from one of those Lumina­ryes to the other: Then take the visuall distance (by some exact Mathematicall Instrument) betweene the said two Luminaries: which had, and compared with the former so calculated, the difference is the Paralax of the said lower Planet or Luminary, for that instant. The like may be performed by the Sun, at his rising or setting, beeing neere the Horizon, and hauing some Planet or notable fixed Star nigh vnto him. And here you may note, that the Paralax is alwayes greater, neere the Horizon, then beeing more eleuated: As also that the superior Planets ♄ ℞ and ♂ scant yeelde any sensible Paralax, by reason of the small quantitie of the Earths semidi­ametre, in respect of their Orbs.

To finde the distance of any Comet or Planet aboue the Earth.

The distance of any Comet or Planet aboue the Earth, is found by the Doctryne of Tryangles; viz, two angles and one side being giuen, to find the o­ther angle, and the other two sides. Herein the two angles giuen are these; The one, the Paralax found: the other, the Altitude aboue the Horizon taken [Page 9]by Instrument, beeing added to 90: The side giuen, is the Semidiametre of the Earth. For the better explanation hereof, behold the figure following:

[diagram]

Wherein, the Paralax or angle EDG, contayneth 8. degrees. The angle DBH is the altitude founde, conteyning 31. degrees: which being added to 90, make 121. degrees, the second angle: Now these two angles added together, make 129: which solo­ducted from 180 (the degrees of two right angles, alwayes equall to the three angles of any Tryan­gle) the remaynder is 51, the degrees of the third angle sought, represented here by the angle BAD.

[Page 10]Then to find the two vnknowne sides (by any two of those angles, and one of the sides knowne) be­cause euery one is not capable of the Rules giuen by the learned Astronomers, who haue written hereof, I will here set downe an easie meanes (as it were Mechanicall) to be vnderstood and practiced with facility, as followeth: Hauing any Circle ex­actly

[diagram]

[Page 11]diuided into 360. degrees, or euen partes, thereby to finde, readilie, the angle subtending e­uerie Arche: or contrarywise, to finde any arch subtending any angle giuen; First drawe an infi­nite lyne, which may represent the side giuen, as the lyne AB; and set your Compasses (opened to the Semidiametre of the Circle) vpon A the vpper end of the same lyne, and draw an arch (with the other foote) of such greatnesse, as may crosse the sayd lyne, as the arch FC doth at F; then open your Compasses, vnto so many degrees of the lymbe of your said Circle, as one of your giuen angles con­teyneth: as, according to the former example, to 121. degrees, and set one foote of your Compasse so opened, in the intersected point F, and extend the said quantity in the said arch, which will fall at C. Then drawe a strait line from the Centre of the arch A, to the sayd poynt C: and that lyne will re­present a second side of the Tryangle desired. Then againe, for the third side, you are to worke in all things as before: as, first to open your Compasses vnto your diuided Circles Semidiametre, and set one foote vpon one of the sides already described, and draw an arch that may intersect the said side to­wards the alreadie described angle: as, vpon the side AC, on C, set the Compasses, and draw the arch GD, which intersecteth the side AC at the poynt A: and then opening the Compasses, to so many de­grees of the diuided Circle, as the other angle gi­uen conteyneth, viz. 8. degrees, carrying the same to [Page 12]the poynt A of the lyne AC, and expresse the same in the arch AD. which will fall in E. Then draw a straite lyne from C, the Centre of the last described arch, vnto the said point E: which being continued, will cut the lyne AB in the poynt K; and both giue you the third Angle AKC containing 51 degrees, and also at once represent vnto you, all the angles and sides proportionallie of the Try­angle, made by the Centre of the Earth, the Coe­lestiall Luminary, and the Centre of the Eye of the Obseruatour; each respecting other, with strayte lines.

Then to finde how farre the said Luminarie is distant from the Earth (hauing described the pro­portionall Tryangle aforesaide) and hauing the measure of one of the sides thereof giuen, viz, the Semidiametre of the Earth AK, which according to most writers is 3436 4\11 myles, which measure I find to be 6. tymes and a halfe, contained in the side KC, and therefore conclude the same to be distant from the Centre of the Earth, 21477 2/11 myles; and from the surface of the Earth, 18041 1/11 myles. Thus much is sufficient, for the finding of the Pa­ralax, and distance of any Coelestiall body (hauing Paralax) from the Centre or superficies of the Earth.

The fourth Proposition. The Paralax, and distance of any Star from the Earth, giuen, to find the true angle it wil make in the Centre of the Earth, with any Star that yeelds no Paralax.

HAuing taken the Paralax, & so consequent­ly (as in the precedent) the distance thereof from the Centre or the Earth, and obserued the visuall angle by Instrument, which such a star shall make, with any knowne star, that yeelds no Paralax; you shall describe, on a cleane Paper, or Slate, an Angle equall to the visuall angle so found: then draw a small Circle, representing the Globe of the Earth: and describe one Arch, con­centrick with the same Circle, & so many Semidi­ametres without the same, as the said Star or lumi­narie (yeelding Paralax) is distant from the Earth; and one other Arch as much further distant again without that, or more or lesse, as you please: which last described Arch shall represent vnto you, part of the Firmament or Heauen of fixed starres. Then drawe a straite lyne from the Centre, vnto the said outmost arch: and let the place, where the same lyne intersecteth the middle Arch, be the place of the star, yeelding Paralax: then expresse the distāce betweene the two starres, with an angle in such a part of the circumference of the small circle, as the said angle, being parted in two with an obscure lyne, the same lyne may fall in the poynt of the [Page 14]angle, and make a square angle with the Semidia­metre of the small Circle: Then the visuall lynes, and the lynes of the true places of the said starres, beeing extended to the vtmost arch; and a straite lyne drawne from the intersecting poynt (which the lyne of the true place of the Star that yeelds Pa­ralax, maketh) in the vtmost arch, vnto the poynt of the visuall Angle in the Circumference of the small Circle, it is certaine, that that lyne, and the visuall lyne, which respects the fixed starre (or starre yeelding no Paralax) will (by this meanes) frame an angle in the Circumference of the small Circle (representing the Earth) equall to the angle, which the saide two stars made, at the instant of obseruati­on, in the Centre of the Earth.

The fift Proposition. To find the Longitude of any place, by a Comet, or the Moone, appearing, and the difference of tyme from one Meridian to another.

FIrst the Obseruators, who may see the Moone or Comet, at one instant, aboue theyr seuerall Horizons, must subduct the difference of As­pect, or Paralax, by the doctrine precedent, and so find the true place thereof, in the Firmament. Then obseruing, from Moment to Moment, what change of Angles it shall make, with a fixed Starre giuen, in the Centre of the Earth, as in the preceding Proposition is taught: let them conferre the com­putation [Page 15]of tyme, in which they found them to a­gree by equal angles, with the obseruations of one another: so will the difference of tyme, beeing re­duced into degrees and minutes, expresse the dif­ference of the Longitudes of theyr seuerall Meri­dians. The reason hereof is the roundnesse of the Earth, causing the different rising and setting of the Sunne, vnto the Inhabitants of the East, from those of the West. If then you make an exact Table of the true angles, which the Moone or Comet will from time to time make with a fixed Starre giuen, vnder one certaine Meridian knowne; and obserue the same angle, vnder another Meridian: It cannot but happen at a time of the night, differing from the former, by howers, minutes, or both: which, be­ing as is aforesaide reduced, will yeeld your desired effect. Thus much, for the finding of the Longi­tude, by the difference of time of the happening angles.

Corollarie.

Then it followeth, that if you collect the proper moouings of al the other Planets (out of Astrono­micall Tables) and compare theyr true moouings, with the true moouing of the Moone or Comet appearing, you shall finde more speedy change of angles, thē by comparing them with fixed stars; by reason of the swifter motions of the Planets, then of the fixed Starres.

The sixt Proposition. A Star in heauen propounded, to find the Longitudes of places, without the difference of tyme.

FOr performance of this proposition, the re­presentation of Meridionall planes onely is necessarily required: wherefore, for your better vnderstanding hereof, it will be here requisite to acquaint you, with some of the delineations and vses of the Cosmodelite (an Instrument, in this Book, hereafter described): which (by reason of the seue­rall bendings of the Eares and Semicircle therof) is very apt for the true representation as of any o­ther Circle or Plane, that is, or can be imagined in the Heauens, so also of the Meridians & Meridional Planes. For, hauing declyned the Instrument by the Eares thereof, vntill the lyne of 6. be paralell to the Axis of the world, the extreames thereof directlie respecting the Poles of the world; and then causing the great Circle to mooue circularly, vpon the Centre of the Semicircle; you shall perceiue the same Circle and the Plane thereof (in that motion) to represent so many Meridians, and Meridionall Planes, as can be imagined in the world. Therfore, if you first abstract the Paralax of the Moone or Starre; and then extract, out of Astronomicall Ta­bles (being calculated for a Meridian giuen) what angle the same will make, with some of the other Planets or fixed Starres, with which the saide Star or Moone shall culminate, or at once come to the [Page 17]assigned Meridian; And shal also carefully obserue, till you find (by the great Circle and Index of the Cosmodelite) that they agree in angle, with the angle calculated for them: Then shall the number of the degrees (contayned betweene the Fiduciall lyne, which intersecteth the Semicircle, and the end next to the same Semicircle) shew the true num­ber of degrees (and thereby, the distance) between the Meridian for which the Calculation of the Ta­ble was made, and the Meridian of obseruation: which number of degrees is the true difference of their Longitudes, &c. So, hauing the Longitude of the one, that of the other can not be vnknown. Thus may the Longitude be found, by the Moone or a Comet, and some other Star, giuen, without the difference of time.

The Seuenth Proposition. How to find the Longitude and Latitude, at once, by the Coelestiall bodies, without obseruing the difference of tyme.

FOrasmuch as many Obseruors, both on Land and Sea, may, at one instāt time, see certain of the Planets in one same Constitution and position to respect one an other, beeing compared one to one, or one of them with a fixed Starre: If therefore euery one of them doe obserue the saide starres (by the great Circles of seuerall Cosmodelites inclyned according to the inclyna­tion [Page 18]of the seuerall planes, which they will at one tyme be seene to make, to the eyes of the seuerall obseruators, on theyr seuerall Horizons, it is cer­taine, that each of them shall finde the Inclination of the great Circle of his Cosmodelite (representing the plane of the two Starres) to differ from the in­clination of all the rest, whether they be situate vn­der diuers Paralelles, or diuers Meridians. And then euery of them applying, on a Globe, the par­ticular angle, which his inclination made with the verticall lyne of his Horizon, it will be presentlie found what Longitude and Latitude theyr seuerall places haue; and consequently, the proportionall distances from one place of obseruation, to ano­ther.

The Eighth Proposition. How to find the Longitude Mechanically, & to describe al the notable places in a Region or Country, on a Map or Plat, according to theyr true situations and distan­ces one from another.

TO performe this Proposition, there must bee first prouided a kynd of Horse-litter (lyke to the figure following) hauing one wheele, of such greatnesse, as (the horses trauayling, with the same) it may lightlie trample on the face of the Earth, and easilie turne about, alwayes answering, in his motion, the swiftnesse or slownesse of the horses pasing. This wheele must also mooue or turn about other wheels, so framed, as that with an [Page 19]Index, it may (after the ordinary manner of com­mon watches) expresse when the said great wheele (so trampling) shall haue iustly measured a certain number of pases, pearches, or furlongs, and theyr parts; in such sorte, as that the Obseruator (in the Hoslitter) may readily find at any tyme by the same, how many pases, pearches, or furlongs, the saide Horselitter shall haue passed ouer. There must bee also a perfect Marine Compasse, or else a large magneticall Needle and Fly, hanged in the same manner, with circles of brasse which I hold better and more exact, by reason it is not so ponderous, vpon the poynt of the Pinne; and therefore must of necessity be more quick and ready to shew the true respectiue poynt. All which, beeing fitly placed in the saide Litter, and he being seated therein; when the horses begin to make way, let him note in his Maryne Compasse iustly, vpon what poynt of the Compasse or parts, they make theyr direct coutse: Or if hee shall vse the Needle and Flye afore men­tioned, then let him obserue what poynt, or de­gree, the South poynt of the Needle shall cut (in the Lymbe of the said Flye), vntill hee come to the next turning or angle; which must bee diligently set downe in a paper or Table, at euery angle, to­gether with the number of Pases, Pearches, or Furlongs found (from one angle to the next) by the said Index: alwayes distinguishing each angle, with the measured Quantitie betweene that and the last before, with seueral characters or Alphabe­ticall [Page 20]notes; to auoyde the mistaking of them, one for another. Thus much carefully performed, and the notes of the seuerall direct courses from Angle to angle, with the measured quantity between each angle exactly set down; there is no more to be don abroad, but hee may then, in his chayre at home, draw with a protractor (diuided in al poynts answe­rable to the diuisions of the Marine Compasse, or to the Flye vnder the Needle) by the notes afore­named, the whole way, with euery angle or turning, and all things and places of note, as Riuers, houses, gates, paths, styles, trees, & such lyke, by or through which the said Litter passed.

[diagram]
[...]

[Page]

[diagram]

The practise and vse of the Protractor followeth.

Hauing the notes of the true distance from Angle to Angle, and how they beare and are situate one from another, draw on the Parchement or paper, whereon the Plat shall be drawne, certaine obscure straite lyned Paralelles ouer the whole face there­of; each, two inches or thereabouts, distant one from an other. Then begin at such a place, as that the Parchement or Paper may each way be able to receiue the whole description (which reason will shew), and place the Protractor (by meanes of the diuisions engraued on the ends of the long square thereof) vpon, or paralell to, one of the descrybed obscure pararelles: then seeke in the Lymbe for the poynt of the Compasse, or degree, by which the se­cond Angle is noted to bear or be situate from the first: there make a prick, and drawe an infinite ob­sceure lyne, from the Centre of the Protractor, through the said prick; Then open a paire of Com­passes, to the proportion of a Scale (answerable to the distance noted, from the first Angle to the se­cond) and set one foote in the poynt, where the Centre of the Protractor was, and with the other foote (the Compasses vnaltered) note a prick in the saide obscure lyne; which will there represent the true place, distance and situation, of the second angle from the first. Then, placing the Centre of the Protractor vpon the last described prick, do in al things as before, both for the poynt of the Com­passe or degree, and for the distance also, and con­sequently [Page 22]the lyke for all the other angles, vntill they bee all described in the same, according to theyr true distance, situations, wayes, & turnings, and by which the Horselitter passed, and for which the saide notes were so taken. Note, that if there should bee betweene any two of the angles some small crookednesse or bending, the forme thereof may be well represented on rue paper of notes, the same beeing present to the Obseruators eye. Like­wise if there be any House, Gate, style, tree, or any other thing fit to be noted; the Obseruator, making remembrances in his said notes, may easily with his Protractor make description of them, answerable to theyr situations and distances. Then, hauing the true Longitude and Latitude of anie one place described in the sayde Plot or Mappe, it cannot bee but the Longitudes and Latitudes of all other places therein will offer them selues, by reason of theyr knowne situations and distan­ces. Thus haue you the meanes to finde the Longitudes mechanicallie: and how to make a Mappe or Chart of any Region, Countrey, Shiere, and all the places of note within them, verie ex­actlie.

The forme of the Horselitter.

[Page 23]

[figure]

It will bee also necessarie to place a scale of the same proportionall euen [...]arts, were they myles, Furlongs or pases, by which you made your saide Plot or Map, in the most conuenient spare place: and if you wil also, a Flye containing the poynts of the Cōpasse duely situate will not be impertinent.

A Corollary.

Then it followeth by the same reason, vsing on­ly a Chayne-lyne of 2.3. or 4. pearch length, in the stead of the Horselitter, to measure the distances betweene the seuerall angles, and an Instrumente with sights (as the Cosmodelite herein mentioned, or such [...] a perfect: Magneticall needle, to expresie (vpon a Flye placed vnderneath the same) the degrees or poynts of the Compasse intersected [Page 24]when the said sights shall bee directed vnto or re­spect the seuerall angles, euerally; with which and the quantity of pearches, conteyned betweene the seuerall angles distinctly noted, and the same then protracted in all poynts according to the formen instructions, you may truely describe any field, Mannour, Park, forrest wood, or Fortification, in a Chart, Map or Plat, very exactly.

Corollarie. II.

Likewise to find the Longitude of any place at Sea, mechanically; there may be an engine made & ioyned to the side of the Ship, hauing a wheele fin­ned (lyke the wheel of a water mil) and hanged in a Chānel (made of boards) with Circles lyke the ma­rine Compasse, to the end it may alwayes hang le­uell in the said Channell, and so moue by meanes of the water passing through the same; it being both at the entry and issue narrow, and wide in the midst to withstand the billowes and waues, which would otherwise hinder the proportionall turning of the said wheel: which may also with an Index in the mā ­ner of that of the Horselitter, expresse how much the Ship shall haue made way, from the time of setting forth, alteratiō of the course, or from watch to watch: which together with true notes of the seuerall courses by what poynt of the C̄opasse the Ship hath performed her way, and in all things (in this practice) imitating the fore-taught operations for the Hoslitter, the true descriptiō of the course and quantity of Leagues, with the angles and tur­nings, [Page 25]that such a Ship shall make, fr̄o time to time in a whole voyage, shal by these means be truly set forth and kept, in the maner of a Trauerse. And at any time applying the same on the Sea Chard or plat, the Longitude & Latitude of any place in the same will be presētly found, together with the true place where the same Ship for that tyme is, or was at any tyme in the whole voyage; and how farre di­stant from any Hauen, Iland, Sands, Rocks, or dan­gers described in the said Map or Seacard.

Annotatio. How the Master or Pylot at Sea may, with any Instru­ment, much more exactly obserue the Coelestial lumina­ries, then the ordinary manner can admitte.

I Cannot here omit, though in digressing māner, to set down a means written by an excellēt Frēch Author, (from whome I confesse I gathered manie flowers to decorate this small posie,) whereby the Seamē may, notwithstāding the heauing & setting of the Ship, with much more exactnesse obserue, with their Instrument, the Sun, Stars and any Coe­lestiall Luminary, then theyr ordinary and vsuall manner can permit, For by mine owne practise I continually found it almost impossible (in a calme excepted) to make any exact obseruation, by reason of the violent shaking of the body, caused by the continuall agitation of the Ship. First then in some conuenient place in the Shippe, there must be two small p [...]es of Tymber, of 4. feet a peece in length, set vpright 4. or 5. feete distant one from another, to [Page 26]support, at the vpper ends of them, two great Iron circles hanged like the brasse circles of the marine Compasse: within which Circles ther may be a seat & Table so placed, as that the Obseruor (beeing entered therin) may hang steady & leauel, although the Ship it selfe do heaue and set, and shall be able thereby more perfectly to obserue any Coelestiall Luminary with exactnesse. And the whole frame may be so made, that it may be taken asunder, and set vp and down at pleasure, to auoide incōbrance.

The making of the Cosmodelite, an excellent Instru­ment for many Mathematical practises in Astronomy, Geometry, and Cosmography, viz, for the finding of the Longitude of any place; To take any Altitude Latitude or distance within the Angle of sight; To make a Map or Plat of any Country, Mannor, field, fortification or Towne; To delineate Sun dialls vpon all planes giuen, with great facility, inuented by the Author.

WHether you will haue your instrument of Brasse or fine grained Woode, you must frame a Circle of fiue or six Inches Dia­metre at the least, within the Lymbe of which, you may describe & engraue certain Circles one with­in another, (for seuerall vses): one of which Circles shall be diuided into 360. equall parts or degr [...]s, which shall serue for astronomicall and [...] [...]e­ometricall obseruations; another of the said circles shall be diuided into 24. equall parts, (and eache of [Page 27]them again into subdiuisions, as halves & quarters) which shal both serue of it selfe for an Equinoctial Diall, being inclyned to the inclination of the Ae­quator; and also most readily to delineate all Sun Dialls, Murall and Horizontall, as in the Proposi­tion preceding is fully taught. Then shal you make an Index of the length of the whole Circles Dia­metre, whose fiduciall lyne may serue for each seue­rall inscribed Circle, and crect a sight neere vnto each end thereof. Then also make a Semicircle a­bout three inches Diametre, leauing some part more then a Semicircle aboue the Diametre there­of, both for the better ioyning the same vnder the Circle, which must be done directlie vnder one o [...] the Diametres, and for the better scope to eleua [...] or embase the same; The lymbe of which Se [...] ­circle you shall diuide into 180. equall partes or degrees, viz, each Quadrant into 90. There must be al­so a peece of the same substance so framed, as that the Semicircle may, as betweene two Cheekes or supporters, mooue vp and downe to any inclinati­on, vpon a pin which must passe through the Cen­tre of the Semicircle and heads of the said Cheeks or supporters, and must haue at the smaller end a Nut and screw to locke the same at pleasure; & the saide Cheeks must haue a fiduciall edge, to note in the Lymbe of the Semicircle the eleuation and inclination of the great C [...]e, when occasion shall require. Furthermore the lower end of the Cheeks must bee ioyned to the staffe (which must support [Page 28]the whole) as the two feete of a payre of Compas­ses, or rather as the legs of the Sector or Circum­ferental Scale are ioyned together, with three roūd planes to make one ioynt head: the two outmost of those planes we wil cal Eares, & do differ only frō the head of a Sector in a pinne which is to passe through the Centre carrying a double Index at one end thereof, and the other end must bee fitted to a screw nut, to locke the same fast when you please: also the middle part of the same pin must be square; yet round towards each end, fitted to the three holes in the Cētre of the three round planes, the hole of the middle plane beeing square to the end, when the Instrument is inclyned; that then this [...]quare hole may, by reason of the squarenesse of the [...]iddle of the Pin, carry the said double Index with it proportionally Circular vpon that eare whereon it mooues: whose Lymbe, being likewise diuided in­to degrees, will expresse there the true quantity of such inclination. And it is also very necessary to place, on the Centre of the great Circle and Index thereof (after the vsuall maner) a box with a perfect Magneticall needle therein, and a fly fixed in the bottom, exactly diuided into degrees and Rombes, hauing the variation of the compasse for the place of obseruation alwayes noted therein. Thus is the Cosmodelite fitly prepared [...]o performe your expec­tation in any Mathematicall obseruation. The fi­gures of each part, before mentioned, follow heere demonstrated.

[Page 29]

[diagram]

[Page 30]The great Circle is A, the Index thereof C, and the sights for the same Index K K: the Semicircle is B, which is to be ioyned directly vnder the Diame­tre L M. The peece with two Cheeks is E, the vpper pin and screw thereof, I: the lower end of the same peece, beeing the middle plane with a square hole at the Centre, D; the other two planes ioyned to the Staffe F: the pinne for the same and the screwes therof are G & H: the box with a Magnetical needle P, to be placed on the Centre and Index. All which ioyned together is represented by the figure at N.

Here could I further adde two other necessarie parts vnto the saide Instrument: one whereof, for [...]e making of a Cherographicall description, on a [...]p or Plat, of al the angles in sight, at one sole sta­ [...]. The other, by two seuerall stations, (without [...]ng the respecting lynes by degrees, poynts of [...] Compasse, or parts of eyther; and without the [...] of Protractor, or any arithmeticall notations) [...]eally to describe on the saide Instrument all the notable places, which at these two stations were to be seene: according to their true distances and si­tuations, each from other. All which and many o­ther, if I shall find these acceptable, I wil hereafter eyther generally publish, or particularly explane to any true well willer of these practise. Also I might heere apply the vse of this Instrument to measure all visuall distances within the Angle of sight, and how to cary a Rowling trench vnder Grounde di­rectly vnto any place appoynted to be vndermined [Page 31]with infinite other necessary conclusions: But be­cause the most of them haue beene already taught how to be performed by other instruments (not so general as this) I therefore presume, the ingenious practiser will with so much ease apply them here­unto as it might seeme superfluous for me here to iterate the same.

How by the helpe of the Cosmodelite to delineate a Sunne Diall, on any Plane giuen, with great facility & without the vse of Arithmetick.

PLace the Cosmodelite so vpon his foote, as that it may be neere vnto the Plane or wall where­on you desire to draw your Sun Dial, being se [...] so directly vpright, by the helpe of a Plummet [...] otherwise, that the planes of the Eares thereof be [...] perpendicular to the Horizon: that done, locke fa [...] the saide Eares with the screw and nut. Then by the Magneticall needle or otherwyse, place the lyne of 12 (in the Instruments great Circle) directly in the Meridian lyne of that place: and (by the Semicircle and degrees thereof) embase the North part of the same great Circle, so many degrees as the North part of the Aequinoctiall is embased vnder the Horizon, and so shall the plane of the great Circle be found to be in the true plane of the Aequator of the world, which it heere representeth. Thus all parts of the Cosmodelite being fast locked, it is readi­ly prepared to delineate a Sun Diall on any plane, [Page 32]against which it is placed as aboue said. To practise this Proposition you shall first (the Cosmodelite pla­ced according to the former directions) fasten in the Centre of the great Circle, a threed of 5. or six foote in length, and erect the same perpendicular to the plane of the same great Circle; extending it so farre, as it may intersect the giuen Plane, and therein note a poynt or prick; which prick shall be both a Cētre to the hower lynes, & also the poynt from which the Style or Ostens [...]r for the same Diall must proceede: and the same lyne, so erected, doth also well demonstrate vnto you the verie fiduciall edge of the Ostensor or style: These things perfor­med, you haue now no more to doe for the setting [...]rth of the hower pricks in the said Plane, but to [...] the said three [...] straite, vpon eache of the hower [...]es engraued on the Instrument, and to extend [...]e same forth so far, at each tyme, as it may touch [...]nd make so many seu [...]rall pricks as there will fall hower lynes (in that maner) on the said Plane: as for such howers whereon the threed beeing applyed, and infinitely extended, will not yet any where in­tersect the said Plane, it is certaine that those ho­wers cannot be found on that Plane, by the shadow of the Sun. All the pricks beeing so set downe, on the Plane, that will fall thereon; you must then draw a straite lyne from the first described pricke, to eache of the other pricks, and those shall be the hower lynes, for the same Diall, truely described. Then an Ostensor or Style hauing his fiducial edge [Page 33]and being placed according to the former instruc­tion, and the figures for eache hower set downe at the end of each of them, to distinguish them one from another: you shall so haue descrybed an ex­act Sunne Dyall, Perspectiuelie. Note that if, in the erecting of the threed perpendicular to the plane of the great Circle, it happen to bee para­lell to the plane (on which you haue described your Sunne dyall as aforesayde) and so will no where intersect the same plane; Then must the Ostensor or Style be also paralel, and of such distance from the plane giuen as the perpendicular erected threed extended is. Thus may you by the helpe of the Cosmodelite delineate a Sunne Diall, Direct, In­clyning, Declyning, or Reclyning, on any Plane, or H [...] rizon giuen.

Master Robert Smyth his intention to delineate a Sun Dialll on any Plane in any one Paralel giuen, with a Instrument of sleight charge.

FIrst draw a Quadrant, and diuide the Arch thereof into 90. equall partes or degrees, and extend a straite lyne from the Centre or square angle of the same, vnto the degree in the arch an­swering the Latitude of the paralell giuen; prolon­longing the same lyne, beyond the Arch or lymbe, some 6. or 8. inches, or more or lesse at pleasure. Al­so extend the Base lyne of the Quadrant as farre as you thinke fit (for the bignesse of your instrument) [Page 34]and at the end of the same lyne erect a perpendicu­lar which may intersect the first extended lyne. So shall the sayde perpendicular, the Base, and the line drawne by the degree of Latitude, describe vnto you a rectangled Tryangle. Then drawe a Circle (whose Dyametre may be about halfe the length of the Base lyne of the Tryangle aforesaide) and di­uide the Circumference of the said circle into 24. equall partes: in the Centre of which Circle you shall fix a threed of 4. or 5. feete in length: The said Tryangle and Circle being then cut out of some thin boord of fine grained woode, and the circle let on, vpon the side of the Tryangle which subtendeth the square angle, so farre as that the centre of the circle may touch or exactlie ioyne to the said line, [...]nd the plane of the circle stand square to the said [...]yne, and the diuisions being distinguished Arith­metically from 12. to 12. as appeareth in the figure following: The sayde Instrument is readie to performe the promised effecte in manner fol­lowing.

[Page 35]

[diagram]

[Page 36] The vse and practise of the said Instrument.

Place and fasten your Tryangle vpon or neere vnto the Plane on which you purpose to drawe the Sundiall, in such manner as that the syde E F may be perpendicular vnto the Horizon, and the lyne E A directly in the Meridian lyne of the place: so shall the lyne A F be found to represent the Axis of the world; and the circles circumference to bee exactly in the Plane of the Aequinoctiall. Then haue you no more to doe, but apply the threed on so many of the 24. diuisions seuerally, as, being in­finitely extended, it note in the plane giuen, so ma­ny touch poynts as there can be howers found by the Sunnes shadow vpon the saide plane: And drawing straite lynes from the poynt in the plane, (which the lyne A F of the Tryangle respects) vnto eache of the saide touch poynts, such lynes shall be the true hower lynes to expresse the exact hower of the day, by the shadowe of an Ostensor or Style; which must bee placed to carry the true forme of the said lyne A F of the same Tryangle vpon the said plane giuen: Applying also the Arithmeticall characters, correspondēt to those of the circle, vp­pon the hower lynes seuerallie, you shall so haue delineated a perfect Sundiall vpon the Plane giuen.

The making of an artificiall Engine, whereby with a small strength giuen, you may lift vp any ponderous weight assigned

WHen Hieron (King of Sicilia) had builded a a Ship of such admirable greatnesse (to present vnto Ptolome, King of Aegypt) as that al the inhabitāts of Syracusia, were not able by any meanes they could practice, to launch it into the water: Archimedes, that excellent Mathematici­an, caused this engine to be framed; whereby the King himselfe, did with one hand lift vp the sayde mightie vessell from the Earth, and set it into the Sea: which although many, aswell Historiogra­phers, as others, haue in theyr works mentioned; yet none haue hitherto set downe the perfect fra­ming thereof, saue only the learned Besson; who ha [...] thus left it vnto posterity. The Tripaston (for so [...] nameth it) may be composed, of as many wheele and screwes (to turne those wheeles) as you wi [...] but of fower of each at the least. The first screw must haue a handle to turne about lyke the handle of a handmyll or grynding stone; and so fitted, as being turned round, it may also turne the first wheele, by meanes of the oblique swellings of the same screw, falling betweene the Teeth or Cogs of the saide first wheele: which Teeth must be so ma­ny in nūber as may be proportionall to the strēgth you would multiplie by the same. The Axletree of the first wheele must haue vpon the same a second screw, which may in like manner, and proportion, turne a second wheele, and that second a thirde; which third a fowerth, and so infinitely at pleasure. [Page 38]Now if the first screw (by the handle) be turned a­bout 20. tymes, to the turning of the first wheele once: I affirme that the sayde first wheele will lift vp as much poyse or burthen, as the strength of 20. men will extend vnto, hauing a cord fastened to the same, and to the Axletree of the saide first wheele, and a man to turne about the first screw by the handle thereof. The second wheele hauing the lyke proportion in motion to the first as the first hath to the handle, I conclude that the second wheele will in lyke sort rayse vp 20 tymes 20 mens strength, which is 400 mens strength. The third wheele 20 tymes 400 mens strēgth, which is 8000. The fowerth 20 tymes 8000 which is 160000 mens strength, and so foorth infinitely; A thing which to many will seeme incredible: but who so will duely put the same in practice shall find it ful­ly able to performe the promised effect: whereby it appeareth that it was not without reason, that Archimedes affirmed that he could moue the whole Globe of the Earth out of her place, if he had any firme place in the Ayre, that could support his said Engine; and therevpon made this Probleme, Da­tum pondus datis veribus mouere. But here may some make questiō, if the slowenesse of this Engin cānot by some Artificiall meanes be hastened: to which I answere it may, by taking away one or two of the last wheeles screwes and Axis, and in their places so vse common Pullies, whereof Vitruuius wryteth in his tenth booke and third Chapter: which Pappus [Page 39]in his Annotations vpon the Mechanicks of Ar­chimedes affirmeth to haue also infinite force, with great celerity. Thus much may suffice for the fra­ming of this Engine, whose benefite may be ex­tended to infinite necessary vses: Onely I will here demonstrate in the figure following the forme of the said Screwes and wheeles.

Briefe Expositions of the Geometricall and Astronomicall tearmes mentioned in this Treatise.

  • A lyne is a length without breadth or deepnesse.
    Lyne
  • A Superficies or Surface hath onely length and bredth without deepenesse.
    Surface
  • A plane is equally flat, contained within lynes,
    Plane
    and doth not bulke out or shrinke in at any place: and is saide to be represented, when a lyke figure hath an absolute lyke situation and constitution.
  • An Angle is the concourse of two or moe seuerall lynes in one same poynt:
    Angle
    And is giuen when the degrees of the subtending arch thereof is knowne.
  • A right or square angle,
    Right angle
    is when two lynes fall square one vpon another, making all the angles framed thereby equall.
  • A Sharpe or acute angle,
    Sharpe angle
    is any angle that is lesse then a square angle.
  • A Blunt or Obtuse angle,
    Blunt angle
    is any angle that is greater then a right or square angle.
  • A Triangle is a Figure of three Corners or angles:
    Triangle
    And is giuen, when the quantity of all the Angles and sides are knowne.
  • A Circle is a round Figure,
    Circle
    made by the turning of a lyne vpon a poynt fixed.
  • The Circumference of a Circle is the outmost edge or lymbe of the Circle,
    Circumfe­rence
    being in all places equidistant from the aforesaid fixed poynt. Any part of a Circumferēce is an Arch; An arch is giuen,
    Arch
    when the degrees contai­ned [Page]therein are knowne.
  • The Centre is a poynt in the midst of a Circle,
    Centre
    Globe, or Spheare.
  • The dyametre of a Circle is the longest straite lyne that can be drawne within a Circle,
    Diametre
    and it passeth through the Centre from side to side:
    Semidia­metre
    The halfe there­of is the Semidiametre.
  • A great Circle is that which diuideth the world into two equall parts.
    Great
    Circle
    The edge or Lymbe thereof containing 360. equall parts or degrees.
    Degree
    A Degree is therefore 1/360 part of a Circle.
  • The Aequator or Aequinoctiall is a great Circle,
    Equinoc­tiall.
    girding the world in the midst between the two Poles.
  • The Zodiack is a great Circle broad and slopewise situ­ate,
    Zodiack
    bearing the 12 Signes. In the midst of which Cir­cle is a lyne called the Ecliptick,
    Ecliptick
    from which the Sun neuer swarueth.
  • The Meridian is a great Circle passing through the Ze­nith and Poles of the world,
    Meridian
    being alwayes permanent, though the Sphere be moued.
  • The Horizon is a great Circle,
    Horizon
    diuiding the world (accor­ding to sēse) into 2 equal parts; viz, the Superior seen, or Diurnall Hemisphere; and the inferior vnseene, or Nocturnall Hemisphere.
  • Azimuthes,
    Azimu­thes
    or Circles verticall, are great Circles, and passe through the Zenith intersecting the Horizon with right angles.
  • Almicanterathes or Circles of altitude,
    Almican­tares.
    are Circles para­lell to the Horizon: and are greatest, being neerest the Horizon; and least, being neerest the Zenith.
  • [Page]The Axis, or axletree of the world,
    Axletree
    is a lyne supposed to passe through the Centre of the Earth: the extreames or ends of which lyne are the Poles of the world: viz, the North end the Pole artick, and the South end the aniartick.
  • There is North Latitude, and South Latitude of places:
    Latitude of places.
    For all places between the Equinoctial and the North pole haue North Latitude; and between the Equinoc­tiall and the South Pole haue South Latitude.
  • The Longitude of the Earth is as the Circuit of the E­quator in the Heauens.
    Longitude of places.
    And is diuided into 360 euen parts or degrees.
  • Any two places, being lesse then 180 degrees distant, haue one same Longitude, if they be vnder one same Meri­dian; Otherwyse they haue different Longitude.
  • Any two places hauing lyke Latitude (being both North, or both South Latitude) are in one same Paralell. The verticall poynt or Zenith is a poynt in Heauen di­rectly ouer our heads,
    Zenith.
    and is the Centre or Pole of the Horizon.
  • The Oposite poynt is the Nadire [...]
    Nadir.
  • The Paralax, or difference of Asp [...] [...] a Comet Planet or other Luminary,
    Paralax.
    is the angle [...] inter­section of the Lyne of the true Place [...] place thereof reckoned in the Firmament.
FINIS.

Faultes escaped, in the originall Copie, it selfe.

Page 10. in the lyne of the figure A C write G at the vpper end of the arch D, and at the star * write Q. And page 11. lyne 10, for, as the arch E C doth at E, reade as the arch B C doth at B; and in l. 15 for poynt E reade poynt B; and l. 26 for side A C on C, reade A Q on Q: And 1. 27, for side A C at the poynt A, reade side A Q at the poynt D. And page 12. l. 1 for the poynt A of the line A C, reade the poynt G of the line A Q, And l. 2 for arch A D which will fall in E, read arch G D which will fall in D; l. 3 for from C reade from Q. l. 4. for poynt E reade poynt D, l. 6 for angle A K C reade A K Q.

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