A MATHEMATICAL APENDIX, CONTAINING MANY PROPOSITIONS AND CONclusions mathematical: with necessary observations both for Mariners at Sea, and for Cherographers and Surveyors of Land; TOGETHER WITH AN EASY perspective mechanical way, to Delineat Sun dial's upon any Wall or Plane given, be it direct, inclining, declining, or reclyning, from the Horizon, or Meridian, in any Region or Place of known Latitude. With other things pleasant and profitable for the weal public, not heretofore extant in our vulgar: Partly colllected out of foreign modern writers, and partly invented and practised by the Author. Written by R. N. Gent. LONDON, Printed by R. B. for Roger jackson, and are to be sold at his shop in Fleet-strtet, 〈…〉 conduit, 1604. TO THE RIGHT HONOURABLE MY SINGULER GOOD LORD AND MAster, Sir EDWARD SEYMOUR Knight, Baron Beauchamp, Earl of Hertford, and his majesties Lieutenant within the Counties of Somerset, and Wiltess, and the City and County of Bristol. RIght honourable, since amongst the rudest Creatures (even the brute beasts devoid of Reason) Ingratitude, that foul Monster, is not once to be found: how hateful then the very remembrance thereof aught to be unto human Creatures; but especially to those that profess Christianity (in whom it hath pleased the sovereign Creator of all things, to make his chief Storehouse of all Reason in these terrestial parts) so much as to purpose, much more in action to prove ingrateful to any; but chiefly to those from whom especial favours have proceeded: who is it that will not absolutely censure? Wherefore (my honourable good Lord) because I would not seem more than brute, although I acknowledge my ability cannot extend unto the moiety of my desire: yet like the poor man, who (wanting better wherewith to express his loyal affection) presented the mighty Prince Alexander the great, with a cup of cool water: so offer I this small (and therefore unworthy) present, unto your honourable acceptance: my bounden duty ever challenging greater, doth nevertheless encourage me to begin with this little. Humbly praying your Lordship to shroud these my first fruits under the wings of your honourable patronage, against the envious storms of Zoilus, and his Critic associates; who are always more ready to carp and find fault with others, than any way able to do the like themselves. Herein promising to myself that the more learned in the Mathematics, (who I confess had been much fit to have handled this subject than myself) and those that desire the benefit of knowledge in the common wealth (for which, I protest, I chiefly undertook this work, in regard that scant any thing herein contained; is yet extant in our vulgar tongue) will if not embrace, yet at the least not utterly reject and contemn this my travail. The acceptance of the more learned and better sort I assume to myself, having ever noted such to favour the desire of good studies: And of those that love the common good I much presume, judging them by myself; having ever borne especial affection to such, as have showed their willing endeavours to profit the public weal with their best means. And so praying almighty God to bless your Lordship and my good Lady your honourable Countess, with long life, and all happy increase of honour and prosperity in this world, and in the world to come eternal felicity, I rest always Your honours most humble and faithful servant, ROBERT NORTON. THe Longitude (gentle Reader) for which this present Treatise was chief composed) is defined to be the shortest distance that can possibly be taken upon the Globe of the Earth between two Meridian's propounded, beginning to reckon the same (from west to East) at the Meridian of the Island Coruo, which our latest writers for divers good reasons account for the first Meridian: To find the longitude, is by some means to search how many degrees the Meridian of the place, where you make observation, is distant from some one Meridian given. The knowledge whereof being so necessary as that the Geographers and Navigators cannot with exactness perform what (in their Arts) necessity requireth, without the same: Therefore have many worthy men taken great pains, and with all ingeniousnes laboured to supply that defect, with the most exquisite means they could devise for the readiest finding thereof; who have set down many Propositions to that effect, and both learnedly and ingeniously left unto posterity such tokens of their love to the public good, as may justly challenge to merit infinite memory and commendation. As Ptolemy, by the time of an Eclipse in two several places being known, to find the longitude. Appian, the true places of the Moon and a fixed Star being given, to find the Longitude. Others by the Angle, that a fixed Star maketh at the moment of time she entereth into the first minute of Cancer or Capricornus, and the difference of time wherein such an angle appeareth, to find the Longitude. Some with portable watches, the true hour of sundry places being given, to find the Longitude. The time being given, and the way of the Ship, to find the Longitude. The point of the Compass & the way of the ship given, to find the longitude. The Latitude, and the way the ship maketh, to find the Longitude, etc. All which are found to be subjecteth unto inavoydable errors, or else clogged with such difficulties, or want of expedient execution; as that they cannot absolutely be concluded for perfect apt means to find the Longitude at Sea: the want of which, is the greatest imperfection in the Art of Navigation. Take here in good part (friendly Reader) these few propositions, not heretofore written in this our vulgar, containing partly collections which from sundry Authors in other Languages I have sought out, and augmented or abbreviated for thy better use; and partly practised observations, which (time & opportunity permitting) myself have experienced both on Sea & Land. Requesting, that with the friendly eye of judgement, thou wouldst consider them well, before thou divulge any unadvised censure against them: Not doubting but every one that shall deign a due perusing hereof may find some thing worthy his bestowed labour. And so, greying excuse if I seem to omit any necessary point in the premises (which I might soon do, in regard I much strived to avoid prolixity) do hearty wish thee thy lawful desire, and myself present to expound any thing that seems herein difficil; referring my labour to your courteous favours ●end. ROBERT NORTON. A Table containing the conten's of the Propositions of this Book. Proposition. 1. How to find the Longitudes of places, by the daily declination of the Sun. 2 How to find the Longitude by Arches of great Circles, which pass by the Centres of the fixed Stars, and Planets that yield no sensible paralax or difference of Aspect. A Corollary upon the same, showing to perform the same more exquisitely, by comparing the Planets one with an other. Another Corollary upon the same Consequence, to find the Longitude, by comparing the Sun and Moon being nigh Conjunction, and their Paralaxis abstracted. 3 A Comet, or the Moon appearing, how to find the paralax or difference of aspect thereof, two several ways; and thereby how far it is from the Centre or Circumference of the Earth. 4 The Paralax and distance from the Earth given, to find 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 Moon, appearing; and the difference of time from one Meridian to another, by comparing it with a fixed Star. A Corollary upon the same, showing more evident accomplishment thereof, by comparing it with the true moving of one of the Planets. 6 A Star m Heaven propounded, to find the Longitude without the difference of Time. 7 How to find the Longitude and Latitude of any place at once, without the difference of time, by observation of the Celestial Luminaries. 8 How to find the Longitude at all times from moment to moment, Mechanically: And withal to describe all the places of note in a Region or Country, on a Map or plat, according to their several distances and situations exactly. With the use of the Protractor. A Corollary upon the same, applying it to the Survey and plaiting of Land. Another Corollary upon the same Consequence, expressing a Mechanical means to find the Longitude at Sea at all times, and to keep a perfect Traverse, for a whole voyage. An Annotaion, showing an artificial devise how the Master or Pilot, at Sea, may much more exactly make observation of the Celestial Luminaries with any Instrument, than the ordinary manner can possibly admit, by reason of the heaving and setting of the Ship. The making of the Cosmodelite, an excellent Instrument for many Mathematical conclusions: As for the Longitude of places, to take any Altitude, Latitude, or distance in sight, for Survey and plaiting of Land, to make Sun dials, etc. invented by the Author. How by the Cosmodelite to delineate a Sun dial on any plane given, with great facility, & without arithmetical calculation. Master Robert Smith his invention to delineat a Sun dial otherwise. The making of an artificial engine, whereby with any strength given (be it never so little) to elevate and lift up any ponderous weight assigned. Brief Expositions of the Geometrical and astronomical terms mentioned in this Treatise. Finis Tabulae. The first Proposition. How to find the Longitudes of places, by the daily Declination of the Sun. THe Sun continually declining from the AEquator, according to the several points of the Ecliptic wherein he properly moveth, sometimes towards the South (being in the Meridional signs), and sometimes towards the North Pole (being in the Septentrional:) making his greatest declination on either side to be 23. degrees and 28. or thereabouts, doth, from the noon of one day unto the noon of the next day, so sensibly vary his declination on any one Meridian, as that you may easily find how much he declineth from Meridian to Meridian, on all the Meridian's that may be imagined upon the face of the whole earth. As for example, if from the noon of this day until the same moment to morrow, he shall be found to vary one minute of a degree in his declination upon one same Meridian: imagining then 60. several Meridian's on the Earth, equally distant one from another; it is most certain that he will make from one of those Meridian's unto the next 1/60 part of a minute (which is a second difference of declination: and so consequently more or less, as the same shall hap to be beyond or short of the said Meridian. Wherefore if you divide 360. the degrees of the Equator, by 60. the number of the supposed Meridian's, you shall find the Quotient to produce 6. degrees, the difference of the Longitude which one of them shall be from the next. This may be observed from day to day at all times of the year, be it that the Sun have greater or less declination than the said minute, under the assigned Meridian: which collected into a Table will be a ready means for to find the Longitude of any unknown Meridian. The practice to find the Longitude (the Table being made) is thus: Having exactly observed the declination of the Sun (by some perfect Mathematical Instrument) for the hour of Noon, then enter the said Table, seeking there the declination set down for that day: which being found, the difference of the declination will show you under what Meridian you observe and are, with the difference of Longitude: which difference being added or subducted (as reason will direct) to or from the former given Meridian, will yield the sought unknown Longitude. Note that the Table may be enlarged, not only to thirds and fowerths, but unto tenths: which will be much better for the more exact expressing of his slow declination. And thus you may find the Longitude by the daily declination of the Sun. The second Proposition. How to find the Longitudes by Arches of great Circles, which pass by the Centres of the fixed Stars, and such of the Planets as yield no sensible paralax or difference of aspect. FOrasmuch as great Circles, which pass by the Centres of all the Stars, do express in their Arches the several distances of all the Planets and fixed Stars, (considered by two and two): And for that the Planets in their proper movings do overgo the Stars of the Firmament; It must necessarily follow, that their said distances do continually vary, (either more or less) and the Angles also subtending them. Wherefore, if by Astronomical Tables you obtain the true place of one of the Planets (having no Paralax) and compare the same with one of the fixed Stars given, be it Septentrional or Meridional; seeking for every day, hour, and minute, what Angles the same Planet shall make with the said fixed Star, and of such Angles to make a Table; it will be an Artificial preparative to the finding of the Longitudes of places with facility. For all the Inhabitants of the Earth (according to Ptolemy) being in the Centre of the world in respect of the fixed Stars and higher Planets, it is certain that all such Stars and Planets will appear to them all to vary their angles continually; but chiefly when the Planet shall be direct: and that those Planets, which move most swiftly, do yield more evidently such variation than the slower ♄. and ♃. The Table being prepared: when you desire to find an unknown Longitude, Observe exactly by Instrument, what Angle the same Planet maketh with the said Star, 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 observation: and the difference of the Angles will give you the difference of the Longitude (if any be) between the place of observation sought, and the place given, for which the table was made. Corollary. I. Then it followeth, that by the same means, and with more evident variation of Angles, we may find the Longitudes, by comparing the said Planets one with another to find their Angles, especially one being direct, and the other Retrograde: & so to make like Tables of their proper movings. Corollary. II. Likewise if you take the continual true movings of the Sun and Moon, during the space of three, or four days, before and after their conjunction, and so find their true angles: and observe the Angle (by instrument) exactly, which they for one same moment, do make in the eye of the obseruour, you shall find the observed Angle greater than the Angle calculated, by their paralaxis: which being also known and subducted, the difference of the remaining angle (from the Angle calculated), is the difference of the Longitudes, from one Meridian to the other. The third Proposition. A Comet, or the Moon appearing, to find the Paralax or difference of aspect thereof, and thereby the distance of the same above 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 line, directly respecting a Celestial luminary; and of the line of the true moving; which proceedeth from the Centre of the Earth, passing by the Centre of the same Luminary, unto the Firmament; The semidiametre of the Earth, being the sole efficient cause of such Angles. Neither do such angles happen to every of the celestial Luminaries, save only to such as are nearest unto the Earth, as the ☽ ☿ ♀ and ☉; and amongst them also, those which are nearest, have most sensible paralax, or difference of Aspect: as the figure following doth plainly demonstrate. diagram wherein A representeth the Centre of the Earth, A B the Semidiametre, A F the vertical line, D the farthest or more remote Luminary, yielding smallest paralax, A E the line of his true place in the firmament, K the nearest Luminary, yielding greatest Paralax, A I the line of the true place thereof, B G the visual line, the Obseruor being in B. Whereby it appeareth, that the Angle E D G, or his vertical B D A, being the Paralax of the more remote Luminary, is less than the angle I K G, or his vertical B K A, the Paralax of the nearer Luminary: both which Paralaxes, are caused by the sensible quantity of the Semidiametre of the Earth, being compared with their Orbs. How to find the quantity of the Paralax. To find the quantity of the Paralax of the Moon, or Comet, for any hour or Meridian given, during the appearance thereof above the Horizon: you shall with the Cosmodelite (the Circle thereof being inclined according to the inclination of the Equator) take two several Observations (having some space of time between): at each of which, seeing the said Comet or Moon through the sights of the Index, you must carefully note the degrees intersered therewith, in the Limb of the said Circle. Then comparing the quantity of the arch contained between the two points (at the several observations) intersected in the said Limb, with the arch answering to the space of time between the two observations (being reduced into degrees and minutes of the Equinoctial) and the lesser arch substracted from the greater, the remainder is the Paralax or difference of aspect sought. Although some perhaps will say here, that the Planets, by reason of their excentricities and proper movings, do unequally move, in respect of the Primum Mobile: which although I confess, yet I say, that in the space of time, spent in the said Observation, the same is so insensibly small, as that it were absurd to make any scruple thereof. Another way to find the Paralax. First seek, out of the Astronomical Tables or otherwise, the true place, in the Zodiac, of the Luminary you will observe, for the time and Meridian assigned; then examine the true place of some known fixed star, or one of the Superior Planets, (appearing at that time above the Horizon given): thereby obtaining the Angle, subtending the Arch of the Distance from one of those Luminaryes to the other: Then take the visual distance (by some exact Mathematical Instrument) between 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, being near the Horizon, and having some Planet or notable fixed Star nigh unto him. And here you may note, that the Paralax is always greater, near the Horizon, then being more elevated: As also that the superior Planets ♄ ℞ and ♂ scant yield any sensible Paralax, by reason of the small quantity of the Earth's semidiametre, in respect of their Orbs. To find the distance of any Comet or Planet above the Earth. diagram diagram Then to find how far the said Luminary is distant from the Earth (having described the proportional triangle aforesaid) and having the measure of one of the sides thereof given, viz, the Semidiametre of the Earth AK, which according to most writers is 3436 4\11 miles, which measure I find to be 6. times and a half, contained in the side KC, and therefore conclude the same to be distant from the Centre of the Earth, 21477 2/11 miles; and from the surface of the Earth, 18041 1/11 miles. Thus much is sufficient, for the finding of the Paralax, and distance of any Celestial body (having Paralax) from the Centre or superficies of the Earth. The fourth Proposition. The Paralax, and distance of any Star from the Earth, given, to find the true angle it will make in the Centre of the Earth, with any Star that yields no Paralax. Having taken the Paralax, & so consequently (as in the precedent) the distance thereof from the Centre or the Earth, and observed the visual angle by Instrument, which such a star shall make, with any known star, that yields no Paralax; you shall describe, on a clean Paper, or Slate, an Angle equal to the visual angle so found: then draw a small Circle, representing the Globe of the Earth: and describe one Arch, concentric with the same Circle, & so many Semidiametres without the same, as the said Star or luminary (yielding Paralax) is distant from the Earth; and one other Arch as much further distant again without that, or more or less, as you please: which last described Arch shall represent unto you, part of the Firmament or Heaven of fixed stars. Then draw a strait line from the Centre, unto the said outmost arch: and let the place, where the same line intersecteth the middle Arch, be the place of the star, yielding Paralax: then express the distance between the two stars, 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 line, the same line may fall in the point of the angle, and make a square angle with the Semidiametre of the small Circle: Then the visual lines, and the lines of the true places of the said stars, being extended to the utmost arch; and a strait line drawn from the intersecting point (which the line of the true place of the Star that yields Paralax, maketh) in the utmost arch, unto the point of the visual Angle in the Circumference of the small Circle, it is certain, that that line, and the visual line, which respects the fixed star (or star yielding no Paralax) will (by this means) frame an angle in the Circumference of the small Circle (representing the Earth) equal to the angle, which the said two stars made, at the instant of observation, in the Centre of the Earth. The fift Proposition. To find the Longitude of any place, by a Comet, or the Moon, appearing, and the difference of time from one Meridian to another. FIrst the Obseruators, who may see the Moon or Comet, at one instant, above their several Orisons, must subduct the difference of Aspect, or Paralax, by the doctrine precedent, and so find the true place thereof, in the Firmament. Then observing, from Moment to Moment, what change of Angles it shall make, with a fixed Star given, in the Centre of the Earth, as in the preceding Proposition is taught: let them confer the computation of time, in which they found them to agree by equal angles, with the observations of one another: so will the difference of time, being reduced into degrees and minutes, express the difference of the Longitudes of their several Meridian's. The reason hereof is the roundness of the Earth, causing the different rising and setting of the Sun, unto the Inhabitants of the East, from those of the West. If then you make an exact Table of the true angles, which the Moon or Comet will from time to time make with a fixed Star given, under one certain Meridian known; and observe the same angle, under another Meridian: It cannot but happen at a time of the night, differing from the former, by hours, minutes, or both: which, being as is aforesaid reduced, will yield your desired effect. Thus much, for the finding of the Longitude, by the difference of time of the happening angles. Corollary. Then it followeth, that if you collect the proper movings of all the other Planets (out of Astronomical Tables) and compare their true movings, with the true moving of the Moon or Comet appearing, you shall find more speedy change of angles, than by comparing them with fixed stars; by reason of the swifter motions of the Planets, then of the fixed Stars. The sixth Proposition. A Star in heaven propounded, to find the Longitudes of places, without the difference of tyme. FOr performance of this proposition, the representation of Meridional planes only is necessarily required: wherefore, for your better understanding hereof, it will be here requisite to acquaint you, with some of the delineations and uses of the Cosmodelite (an Instrument, in this Book, hereafter described): which (by reason of the several bend of the Ears and Semicircle thereof) is very apt for the true representation as of any other Circle or Plane, that is, or can be imagined in the Heavens, so also of the Meridian's & Meridional Planes. For, having declined the Instrument by the Ears thereof, until the line of 6. be parallel to the Axis of the world, the extremes thereof directly respecting the Poles of the world; and then causing the great Circle to move circularly, upon the Centre of the Semicircle; you shall perceive the same Circle and the Plane thereof (in that motion) to represent so many Meridian's, and Meridional Planes, as can be imagined in the world. Therefore, if you first abstract the Paralax of the Moon or Star; and then extract, out of Astronomical Tables (being calculated for a Meridian given) what angle the same will make, with some of the other Planets or fixed Stars, with which the said Star or Moon shall culminate, or at once come to the assigned Meridian; And shall also carefully observe, 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 (contained between the Fiducial line, which intersecteth the Semicircle, and the end next to the same Semicircle) show 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 observation: which number of degrees is the true difference of their Longitudes, etc. So, having the Longitude of the one, that of the other can not be unknown. Thus may the Longitude be found, by the Moon or a Comet, and some other Star, given, without the difference of time. The Seventh Proposition. How to find the Longitude and Latitude, at once, by the Celestial bodies, without observing the difference of tyme. FOrasmuch as many observers, both on Land and Sea, may, at one instant time, see certain of the Planets in one same Constitution and position to respect one an other, being compared one to one, or one of them with a fixed Star: If therefore every one of them do observe the said stars (by the great Circles of several Cosmodelites inclined according to the inclination of the several planes, which they will at one time be seen to make, to the eyes of the several obseruators, on their several Orisons, it is certain, that each of them shall find the Inclination of the great Circle of his Cosmodelite (representing the plane of the two Stars) to differ from the inclination of all the rest, whether they be situate under divers parallels, or divers Meridian's. And then every of them applying, on a Globe, the particular angle, which his inclination made with the vertical line of his Horizon, it will be presently found what Longitude and Latitude their several places have; and consequently, the proportional distances from one place of observation, to another. The Eighth Proposition. How to find the Longitude Mechanically, & to describe all the notable places in a Region or Country, on a Map or Plate, according to their true situations and distances one from another. TO perform this Proposition, there must be first provided a kind of Horselitter (like to the figure following) having one wheel, of such greatness, as (the horses travailing, with the same) it may lightly trample on the face of the Earth, and easily turn about, always answering, in his motion, the swiftness or slowness of the horses pasing. This wheel must also move or turn about other wheels, so framed, as that with an Index, it may (after the ordinary manner of common watches) express when the said great wheel (so trampling) shall have justly measured a certain number of paces, perches, or furlongs, and their parts; in such sort, as that the Obseruator (in the Hoslitter) may readily find at any time by the same, how many paces, perches, or furlongs, the said Horselitter shall have passed over. There must be also a perfect Marine Compass, or else a large magnetical Needle and Fly, hanged in the same manner, with circles of brass which I hold better and more exact, by reason it is not so ponderous, upon the point of the Pin; and therefore must of necessity be more quick and ready to show the true respective point. All which, being fitly placed in the said Litter, and he being seated therein; when the horses begin to make way, let him note in his Marine Compass justly, upon what point of the Compass or parts, they make their direct coutses: Or if he shall use the Needle and Fly afore mentioned, then let him observe what point, or degree, the South point of the Needle shall cut (in the Limb of the said Fly), until he come to the next turning or angle; which must be diligently set down in a paper or Table, at every angle, together with the number of Pases, perches, or Furlongs found (from one angle to the next) by the said Index: always distinguishing each angle, with the measured Quantity between that and the last before, with several characters or Alphabetical notes; to avoid the mistaking of them, one for another. Thus much carefully performed, and the notes of the several direct courses from Angle to angle, with the measured quantity between each angle exactly set down; there is no more to be done abroad, but he may then, in his chair at home, draw with a protractor (divided in all points answerable to the divisions of the Marine Compass, or to the Fly under the Needle) by the notes aforenamed, the whole way, with every angle or turning, and all things and places of note, as Rivers, houses, gates, paths, styles, trees, & such like, by or through which the said Litter passed. diagram diagram The practice and use of the Protractor followeth. Having the notes of the true distance from Angle to Angle, and how they bear and are situate one from another, draw on the Parchment or paper, whereon the Plate shall be drawn, certain obscure strait lined parallels over the whole face thereof; each, two inches or thereabouts, distant one from an other. Then begin at such a place, as that the Parchment or Paper may each way be able to receive the whole description (which reason will show), and place the Protractor (by means of the divisions engraved on the ends of the long square thereof) upon, or parallel to, one of the describe obscure pararelles: then seek in the Limb for the point of the Compass, or degree, by which the second Angle is noted to bear or be situate from the first: there make a prick, and draw an infinite obsceure line, from the Centre of the Protractor, through the said prick; Then open a pair of Compasses, to the proportion of a Scale (answerable to the distance noted, from the first Angle to the second) and set one foot in the point, where the Centre of the Protractor was, and with the other foot (the Compasses unaltered) note a prick in the said obscure line; which will there represent the true place, distance and situation, of the second angle from the first. Then, placing the Centre of the Protractor upon the last described prick, do in all things as before, both for the point of the Compass or degree, and for the distance also, and consequently the like for all the other angles, until they be all described in the same, according to their true distance, situations, ways, & turnings, and by which the Horselitter passed, and for which the said notes were so taken. Note, that if there should be between any two of the angles some small crookedness or bending, the form thereof may be well represented on rue paper of notes, the same being 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 their situations and distances. Then, having the true Longitude and Latitude of any one place described in the said Plot or Map, it cannot be but the Longitudes and Latitudes of all other places therein will offer themselves, by reason of their known situations and distances. Thus have you the means to find the Longitudes mechanicallie: and how to make a Map or Chart of any Region, Country, Shire, and all the places of note within them, very exactly. The form of the Horselitter. It will be also necessary to place a scale of the same proportional even ●arts, were they miles, Furlongs or paces, by which you made your said Plot or Map, in the most convenient spare place: and if you will also, a Fly containing the points of the Compass duly situate will not be impertinent. A Corollary. Then it followeth by the same reason, using only a Chayne-lyne of 2.3. or 4. perch length, in the stead of the Horselitter, to measure the distances between the several angles, and an Instrument with sights (as the Cosmodelite herein mentioned, or such 〈…〉 a perfect: Magnetical needle, to expresie (upon a Fly placed underneath the same) the degrees or points of the Compass intersected when the said sights shall be directed unto or respect the several angles, everally; with which and the quantity of perches, contained between the several angles distinctly noted, and the same then protracted in all points according to the foremen instructions, you may truly describe any field, Manor, Park, forest wood, or Fortification, in a Chart, Map or Plate, very exactly. Corollary. II. Likewise to find the Longitude of any place at Sea, mechanically; there may be an engine made & joined to the side of the Ship, having a wheel finned (like the wheel of a water mil) and hanged in a Chamnel (made of boards) with Circles like the marine Compass, to the end it may always hang level in the said Channel, and so move by means of the water passing through the same; it being both at the entry and issue narrow, and wide in the midst to withstand the billows and waves, which would otherwise hinder the proportional turning of the said wheel: which may also with an Index in the manner of that of the Horselitter, express how much the Ship shall have made way, from the time of setting forth, alteration of the course, or from watch to watch: which together with true notes of the several courses by what point of the Copasse the Ship hath performed her way, and in all things (in this practice) imitating the foretaught operations for the Hoslitter, the true description of the course and quantity of Leagues, with the angles and turnings, that such a Ship shall make, from time to time in a whole voyage, shall by these means be truly set forth and kept, in the manner of a Traverse. And at any time applying the same on the Sea Chard or plat, the Longitude & Latitude of any place in the same will be presently found, together with the true place where the same Ship for that time is, or was at any time in the whole voyage; and how far distant from any Haven, Island, Sands, Rocks, or dangers described in the said Map or Seacard. Annotatio. How the Master or Pilot at Sea may, with any Instrument, much more exactly observe the Celestial luminaries, than the ordinary manner can admit. I Cannot here omit, though in digressing manner, to set down a means written by an excellent French Author, (from whom I confess I gathered many flowers to decorate this small posy,) whereby the Seamen may, notwithstanding the heaving & setting of the Ship, with much more exactness observe, with their Instrument, the Sun, Stars and any Celestial Luminary, than their ordinary and usual manner can permit, For by mine own practice I continually found it almost impossible (in a calm excepted) to make any exact observation, by reason of the violent shaking of the body, caused by the continual agitation of the Ship. First then in some convenient place in the Ship, there must be two small p●●●es of Timber, of 4. feet a piece in length, set upright 4. or 5. feet distant one from another, to support, at the upper ends of them, two great Iron circle's hanged like the brass circles of the marine Compass: within which Circles there may be a seat & Table so placed, as that the Obseruor (being entered therein) may hang steady & level, although the Ship itself do heave and set, and shall be able thereby more perfectly to observe any Celestial Luminary with exactness. And the whole frame may be so made, that it may be taken asunder, and set up and down at pleasure, to avoid encumbrance. The making of the Cosmodelite, an excellent Instrument for many Mathematical practices 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 Plate of any Country, Manor, field, fortification or Town; To delineate Sun dials upon all planes given, with great facility, invented by the Author. WHether you will have your instrument of Brass or fine grained Wood, you must frame a Circle of five or six Inches diameter at the least, within the Limb of which, you may describe & engrave certain Circles one within another, (for several uses): one of which Circles shall be divided into 360. equal parts or degr●●s, which shall serve for astronomical and 〈◊〉 geometrical observations; another of the said circles shall be divided into 24. equal parts, (and each of them again into subdivisions, as halves & quarters) which shall both serve of itself for an Equinoctial Dial, being inclined to the inclination of the Aequator; and also most readily to delineate all Sun dials, Mural and horizontal, as in the Proposition preceding is fully taught. Then shall you make an Index of the length of the whole Circles diameter, whose fiducial line may serve for each several inscribed Circle, and erect a sight near unto each end thereof. Then also make a Semicircle about three inches diameter, leaving some part more than a Semicircle above the diameter thereof, both for the better joining the same under the Circle, which must be done directly under one o● the Diametres, and for the better scope to elevate or embase the same; The limb of which Semicircle you shall divide into 180. equal parts or degrees, viz, each Quadrant into 90. There must be also a piece of the same substance so framed, as that the Semicircle may, as between two Cheeks or supporters, move up and down to any inclination, upon a pin which must pass through the Centre of the Semicircle and heads of the said Cheeks or supporters, and must have at the smaller end a Nut and screw to lock the same at pleasure; & the said Cheeks must have a fiducial edge, to note in the Limb of the Semicircle the elevation and inclination of the great C●●●e, when occasion shall require. Furthermore the lower end of the Cheeks must be joined to the staff (which must support the whole) as the two feet of a pair of Compasses, or rather as the legs of the Sector or Circumferental Scale are joined together, with three round planes to make one joint head: the two outmost of those planes we will call Ears, & do differ only from the head of a Sector in a pin which is to pass through the Centre carrying a double Index at one end thereof, and the other end must be fitted to a screw nut, to lock 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 Centre of the three round planes, the hole of the middle plane being square to the end, when the Instrument is inclined; that then this square hole may, by reason of the squareness of the ●iddle of the Pin, carry the said double Index with it proportionally Circular upon that ear whereon it moves: whose Limb, being likewise divided into degrees, will express 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 usual manner) a box with a perfect Magnetical needle therein, and a fly fixed in the bottom, exactly divided into degrees and Rombes, having the variation of the compass for the place of observation always noted therein. Thus is the Cosmodelite fitly prepared ●o perform your expectation in any Mathematical observation. The figures of each part, before mentioned, follow here demonstrated. diagram 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 joined directly under the diameter L M. The piece with two Cheeks is E, the upper pin and screw thereof, I: the lower end of the same piece, being the middle plane with a square hole at the Centre, D; the other two planes joined to the Staff F: the pin for the same and the screws thereof are G & H: the box with a Magnetical needle P, to be placed on the Centre and Index. All which joined together is represented by the figure at N. Here could I further add two other necessary parts unto the said Instrument: one whereof, for ●●e making of a chorographical description, on a ●●●p or Plate, of all the angles in sight, at one sole sta●●●●. The other, by two several stations, (without ●●ng the respecting lines by degrees, points of 〈◊〉 Compass, or parts of either; and without the 〈◊〉 of Protractor, or any arithmetical notations) ideally to describe on the said Instrument all the notable places, which at these two stations were to be seen: according to their true distances and situations, each from other. All which and many other, if I shall find these acceptable, I will hereafter either generally publish, or particularly explain to any true well willer of these practice. Also I might here apply the use of this Instrument to measure all visual distances within the Angle of sight, and how to carry a Rolling trench under Ground directly unto any place appointed to be undermined with infinite other necessary conclusions: But because the most of them have been 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 seem superfluous for me here to iterate the same. How by the help of the Cosmodelite to delineate a Sun Dial, on any Plane given, with great facility & without the use of Arithmetic. PLace the Cosmodelite so upon his foot, as that it may be near unto the Plane or wall whereon you desire to draw your Sun Dial, being se● so directly upright, by the help of a Plummet 〈◊〉 otherwise, that the planes of the Ears thereof be 〈◊〉 perpendicular to the Horizon: that done, lock fa●● the said Ears with the screw and nut. Then by the Magnetical needle or otherwise, place the line of 12 (in the Instruments great Circle) directly in the Meridian line 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 equinoctial is embased under 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 here representeth. Thus all parts of the Cosmodelite being fast locked, it is readily prepared to delineate a Sun Dial on any plane, against which it is placed as above 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 thread of 5. or six foot in length, and erect the same perpendicular to the plane of the same great Circle; extending it so far, as it may intersect the given Plane, and therein note a point or prick; which prick shall be both a Centre to the hour lines, & also the point from which the Style or Ostens●r for the same Dial must proceed: and the same line, so erected, doth also well demonstrate unto you the very fiducial edge of the Ostensor or style: These things performed, you have now no more to do for the setting ●●rth of the hour pricks in the said Plane, but to 〈◊〉 the said three● strait, upon each of the hour ●es engraved on the Instrument, and to extend ●●e same forth so far, at each time, as it may touch ●nd make so many seu●rall pricks as there will fall hour lines (in that manner) on the said Plane: as for such hours whereon the thread being applied, and infinitely extended, will not yet any where intersect the said Plane, it is certain that those hours cannot be found on that Plane, by the shadow of the Sun. All the pricks being so set down, on the Plane, that will fall thereon; you must then draw a strait line from the first described prick, to each of the other pricks, and those shall be the hour lines, for the same Dial, truly described. Then an Ostensor or Style having his fiducial edge and being placed according to the former instruction, and the figures for each hour set down at the end of each of them, to distinguish them one from another: you shall so have describe an exact Sun Dial, Perspectivelie. Note that if, in the erecting of the thread perpendicular to the plane of the great Circle, it happen to be parallel to the plane (on which you have described your Sun dial as aforesaid) and so will no where intersect the same plane; Then must the Ostensor or Style be also parallel, and of such distance from the plane given as the perpendicular erected thread extended is. Thus may you by the help of the Cosmodelite delineate a Sun Dial, Direct, Inclining, Declining, or Reclyning, on any Plane, or Horizon given. Master Robert Smyth his intention to delineate a Sun Dialll on any Plane in any one Parallel given, with a Instrument of sleight charge. FIrst draw a Quadrant, and divide the Arch thereof into 90. equal parts or degrees, and extend a strait line from the Centre or square angle of the same, unto the degree in the arch answering the Latitude of the parallel given; prolonlonging the same line, beyond the Arch or limb, some 6. or 8. inches, or more or less at pleasure. Also extend the Base line of the Quadrant as far as you think fit (for the bigness of your instrument) and at the end of the same line erect a perpendicular which may intersect the first extended line. So shall the said perpendicular, the Base, and the line drawn by the degree of Latitude, describe unto you a rectangled triangle. Then draw a Circle (whose Dyametre may be about half the length of the Base line of the Tryangle aforesaid) and divide the Circumference of the said circle into 24. equal parts: in the Centre of which Circle you shall fix a thread of 4. or 5. feet in length: The said Tryangle and Circle being then cut out of some thin board of fine grained wood, and the circle let on, upon the side of the triangle which subtendeth the square angle, so far as that the centre of the circle may touch or exactly join to the said line, ●nd the plane of the circle stand square to the said ●yne, and the divisions being distinguished Arithmetically from 12. to 12. as appeareth in the figure following: The said Instrument is ready to perform the promised effect in manner following. diagram The use and practise of the said Instrument. Place and fasten your triangle upon or near unto the Plane on which you purpose to draw the Sundiall, in such manner as that the side E F may be perpendicular unto the Horizon, and the line E A directly in the Meridian line of the place: so shall the line A F be found to represent the Axis of the world; and the circles circumference to be exactly in the Plane of the equinoctial. Then have you no more to do, but apply the thread on so many of the 24. divisions severally, as, being infinitely extended, it note in the plane given, so many touch points as there can be hours found by the suns shadow upon the said plane: And drawing strait lines from the point in the plane, (which the line A F of the triangle respects) unto each of the said touch points, such lines shall be the true hour lines to express the exact hour of the day, by the shadow of an Ostensor or Style; which must be placed to carry the true form of the said line A F of the same triangle upon the said plane given: Applying also the Arithmetical characters, correspondent to those of the circle, upon the hour lines severally, you shall so have delineated a perfect Sundiall upon the Plane given. The making of an artificial Engine, whereby with a small strength given, you may lift up any ponderous weight assigned WHen Hieron (King of Sicilia) had builded a a Ship of such admirable greatness (to present unto Ptolemy, King of Egypt) as that all the inhabitants of Syracusia, were not able by any means they could practise, to launch it into the water: Archimedes, that excellent Mathematician, caused this engine to be framed; whereby the King himself, did with one hand lift up the said mighty vessel from the Earth, and set it into the Sea: which although many, aswell Historiographers, as others, have in their works mentioned; yet none have hitherto set down the perfect framing thereof, save only the learned Besson; who ha●● thus left it unto posterity. The Tripaston (for so 〈◊〉 nameth it) may be composed, of as many wheel and screws (to turn those wheels) as you wi●● but of four of each at the least. The first screw must have a handle to turn about like the handle of a handmyll or grinding stone; and so fitted, as being turned round, it may also turn the first wheel, by means of the obliqne swellings of the same screw, falling between the Teeth or Clogs of the said first wheel: which Teeth must be so many in number as may be proportional to the strength you would multiply by the same. The axle-tree of the first wheel must have upon the same a second screw, which may in like manner, and proportion, turn a second wheel, and that second a third; which third a fourth, and so infinitely at pleasure. Now if the first screw (by the handle) be turned about 20. times, to the turning of the first wheel once: I affirm that the said first wheel will lift up as much poised or burden, as the strength of 20. men will extend unto, having a cord fastened to the same, and to the axle-tree of the said first wheel, and a man to turn about the first screw by the handle thereof. The second wheel having the like proportion in motion to the first as the first hath to the handle, I conclude that the second wheel will in like sort raise up 20 times 20 men's strength, which is 400 men's strength. The third wheel 20 times 400 men's strength, which is 8000. The fourth 20 times 8000 which is 160000 men's strength, and so forth infinitely; A thing which to many will seem incredible: but who so will duly put the same in practice shall find it fully able to perform the promised effect: whereby it appeareth that it was not without reason, that Archimedes affirmed that he could move the whole Globe of the Earth out of her place, if he had any firm place in the Air, that could support his said Engine; and thereupon made this Problem, Datum pondus datis veribus movere. But here may some make question, if the slowness of this Engine cannot by some Artificial means be hastened: to which I answer it may, by taking away one or two of the last wheels screws and Axis, and in their places so use common Pulleys, whereof vitrvuius writeth in his tenth book and third Chapter: which Pappus in his Annotations upon the Mechanics of Archimedes affirmeth to have also infinite force, with great celerity. Thus much may suffice for the framing of this Engine, whose benefit may be extended to infinite necessary uses: Only I will here demonstrate in the figure following the form of the said Screwes and wheels. diagram diagram Brief Expositions of the Geometrical and Astronomical terms mentioned in this Treatise. A line is a length without breadth or deepness. line A Superficies or Surface hath only length and breadth without deepness. Surface A plane is equally flat, contained within lines, Plane and doth not bulk out or shrink in at any place: and is said to be represented, when a like figure hath an absolute like situation and constitution. An Angle is the concourse of two or more several lines in one same point: Angle And is given when the degrees of the subtending arch thereof is known. A right or square angle, Right angle is when two lines fall square one upon another, making all the angles framed thereby equal. A Sharp or acute angle, Sharp angle is any angle that is less than a square angle. A Blunt or Obtuse angle, Blunt angle is any angle that is greater than a right or square angle. A Triangle is a Figure of three Corners or angles: Triangle And is given, when the quantity of all the Angles and sides are known. A Circle is a round Figure, Circle made by the turning of a line upon a point fixed. The Circumference of a Circle is the outmost edge or limb of the Circle, Circumference being in all places equidistant from the aforesaid fixed point. Any part of a Circumference is an Arch; An arch is given, Arch when the degrees contained therein are known. The Centre is a point in the midst of a Circle, Centre Globe, or Sphere. The dyametre of a Circle is the longest strait line that can be drawn within a Circle, diameter and it passeth through the Centre from side to side: Semidiametre The half thereof is the Semidiametre. A great Circle is that which divideth the world into two equal parts. Great Circle The edge or Limb thereof containing 360. equal parts or degrees. Degree A Degree is therefore 1/360 part of a Circle. The Aequator or equinoctial is a great Circle, Equinoctial. girding the world in the midst between the two Poles. The Zodiac is a great Circle broad and slopewise situate, zodiac bearing the 12 Signs. In the midst of which Circle is a line called the Ecliptic, Ecliptic from which the Sun never swerveth. The Meridian is a great Circle passing through the Zenith and Poles of the world, Meridian being always permanent, though the Sphere be moved. The Horizon is a great Circle, Horizon dividing the world (according to sense) into 2 equal parts; viz, the Superior seen, or Diurnal Hemisphere; and the inferior unseen, or Nocturnal Hemisphere. Azimuthes, Azimuthes or Circles vertical, are great Circles, and pass through the Zenith intersecting the Horizon with right angles. Almicanterathes or Circles of altitude, Almicantares. are Circles parallel to the Horizon: and are greatest, being nearest the Horizon; and least, being nearest the Zenith. The Axis, or axle-tree of the world, axle-tree is a line supposed to pass through the Centre of the Earth: the extremes or ends of which line are the Poles of the world: viz, the North end the Pole artic, 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 have North Latitude; and between the Equinoctial and the South Pole have South Latitude. The Longitude of the Earth is as the Circuit of the Equator in the Heavens. Longitude of places. And is divided into 360 even parts or degrees. Any two places, being less than 180 degrees distant, have one same Longitude, if they be under one same Meridian; Otherwise they have different Longitude. Any two places having like Latitude (being both North, or both South Latitude) are in one same Parallel. The vertical point or Zenith is a point in Heaven directly over our heads, Zenith. and is the Centre or Pole of the Horizon. The opposite point is the Nadire 〈…〉 Nadir. The Paralax, or difference of Asp●●● 〈◊〉 a Comet Planet or other Luminary, Paralax. is the angle 〈…〉 intersection of the Line of the true Place 〈…〉 place thereof reckoned in the Firmament. FINIS. Faults escaped, in the original Copy, itself. Page 10. in the line of the figure A C write G at the upper end of the arch D, and at the star * write Q. And page 11. line 10, for, as the arch E C doth at E, read as the arch B C doth at B; and in l. 15 for point E read point B; and l. 26 for side A C on C, read A Q on Q: And 1. 27, for side A C at the point A, read side A Q at the point D. And page 12. l. 1 for the point A of the line A C, read the point 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 read from Q. l. 4. for point E read point D, l. 6 for angle A K C read A K Q.