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Thiſ inſtrument or any other for the Mathema
The making, description, and vse of a small portable Instrument for ye Pocket (or according to any Magnitude) in forme of a mixt Trapezia thus Called a Horizontall Quadrant.
Composed and prodused soly for the benefit and vse of such which are studious of Mathematicall Practice Written and delivered by Delamain student and Teacher of the Mathematickes.
Attribuit nullo praescripto tempore vitae vsuram nobis ingenij
London printed for Richard Hawkins and are to be ſold at his ſhop in Chancery lane neere Sarjants Inne 1632.
YOur ſingular knowledge in all excellent, and ſolid Literature, and your ever HeLearning, are not unknowne unto the world; And amongſt
Vniverſitie, you tooke no little affection, to the Mathematicall Arts, as by your Lo. owne Manu-ſcripts and exBookes in your Lo. great Librarie I have often ſeene; Beſides, not onely by mine owne ſundry conferences with your Lo. but alſo by the relation of others of more mature judgement I have bin amply informed in theſe your L. more aged yeares not onely of your continued love to theſe Arts, but alſo that your knowledge in them far exceedes many of the Nobilitie of this kingdome. Now my L. when I cauTractat to be made for your Lo. laſt Summer (I meane your Lo. Horizontall Quadrant) I had not then any intention ſo ſoone to have written publikely upon it; But, haInſtrument
propoſitions then ſlightly compiled, (farre exceeding the Inſtrumentall way in this nature, that eyther Nobilitie, Gentrie, or others are now acquainkingdome, for a recreative Inſtrument, as well for the copious uſe thereof, as its great facilitie, and expedition in operation) your Lo. then incouraging me to the publiſhMoneths after I conſidered thereon: and drew it up into a Body, and thus acPatronage (to be ſheltred under the wings of your Lo. clemency againſt all calumInſtrument before it came
Quadrant that ever was made common in this kind: accept therefore favourably I beſeech your Lo. this ſmall mite of my labours, as from the hands of one of your pooreſt ſervants (yet true affectionate) who ſhall alwayes acknowledge your Lo. Nobleneſſe towards him, and ever reſt
BEfore I ſhew the Projection, deſcribe the particulars, and deliver the uſes of this Horizontall Quadrant, it will not bee impertinent, for the ſaReader to underInstruments, and whence it might be drawne, and projected.
Now the ingenious aptneſſe for Inuention, and accomodating of things in a faire and expedite courſe for Mathematicall Practices, of that late profeſſor of Aſtronomy Mr. Gunter is not unTranſcripts in that of more ſolid matter, but ſuch of vulgar Practices he hath publikely made manifeſt for the uſe of all ſuch as affect thoſe Studies. In which worke maScheme, or Diagramme of the fourth Projection in
Booke of the Sector Page the 64. & 65. wch acConcaue hemiſphere (but being in truth a naturall Projection of the viſible Hemiſphere, that is, one Moytie of the Globe, projected on a Plaine) which Diagram and Projection is now chalenged by my Reverend good friend Master Oughtred, and it ſhould ſeeme that Master Gunter had the Original of it from his labours, & invention, who compoſed and made the ſame ſo, for more then thirty yeares paſt, as appeares by his owne Writings, & Manuſcripts upon that Protection ſhewTractat upon my Horizontall Quadrant, whoſe exMathematicall Learning may evidently confirme it: which Projection the ſaid Master Oughtred gave to the late Biſhop of Wincheſter, Doctor Bilſon, for more then 20. yeares paſt, and to ſome others of very good quality.
And it may alſo, by a Letter from that moſt famous and admired Geometer, Master Henry Brigges unto Master Oughtred dated from Greſham College Iune, the 2. Anno. 1618. be collected that the ſaid Maſter Gunter had the firſt overture of that fourth Projection, from the ſaid Maſter Oughtred, in which letter are theſe words: Master Gunter doth here ſend you the Print of an Horizontall Diall of his drawing after your Inſtrument; And afterwards the ſaid Projection was alſo preſented by Master Gunter to many Noble Perſonages, and in particular to
Right Honourable the Earle of Bridgewater, cauſing it to be cut in Braſſe, in ſuch a forme aTractat, ſome uſes of which Diall are extant, viz. the 2. 18. 21. and 34. Pro. of the Index or Table following.
Now having conſidered diverſe Pocket Inſtruments (that many men are practiſed in) & looked into ſundry Projections, amongſt which that of Gemmafriſius (there drawne in the Booke of the Sector) is of admirable uſe, yet making a more ſerious quaere, & contemplating more intenſively upon that Diagram, drawne and ſpecified in that 64. and 65. Pages of the Sector (aforeſaid) I found it farre to exceede all others in the Multiplicity, and excellencie of performance.
If I ſhould adde unto it a Kalender of time, and an Index graduated with an Axis, and Perpendiculars to be erected vpon it at pleaſure: & referring only the Trapeziall forme, it ſhould be fitted farre to exceede any portable Instrument for the Pocket, ever yet produced in reſpect of the general uſes of it: in reſolving ſuch ordinary Propoſitions which are practiced in Astronomie, upGlobe, Spheare, Hemiſpheare, Quadrants of all ſorts, Aſtrolabe of Friſius, Blagraue, and others for facility, expedition, or certaintie, (like Magnitudes conſidered) for in theſe Inſtruments for ſeverall times, and ſeverall Propoſitions, there muſt be diverſe rectifications of the parts belonging to theſe Instruments, and
Horizontall Quadrant, the former redifications are avoided, Contemplation & the eye being onIndex, the aptneſſe, & fitneſſe of the parts, and lines ſo naturally projected, or deſcribed as they are upon the plaine of the Instrument (being a part of the Horizon the Parallels Meridians, & Verticall Circles, that are contained or may be deſLatitude ſufficiently neceſſary) induGlobe, what to ſpeake, and what to anſwere in a Propoſition without farther direction: And having had this Horizontall Quadrant for many yeares paſt, as a Pocket Instrument, diverſe about this Kingdome being imPocket Instruments as are now uſed, here, or in forraigne parts: I was willing at laſt after I had given order for the making of fower of theſe Instruments in Silver for ſeverall Noble Perſonages, to disburTranſcribing the uſes of the Inſtrument, and Tables for the making of it, to ſatiſMathematicall Practices alſo participate of it.
Now, what I have delivered vpon the accommodating of the Instrument thus, the making thereof, with the uſes that I
Tractat upon it folInferiour aſſiſtant, but to mine owne Induſtry, ſearch and labour, and that 64. 65. and 66. Pages of the Booke: of the Sector before ſpecified in which is onely ſhewne the 2. 3. 19. 22. 25. and 30. Propoſitions of the Index, or Table following, as uſes of the ſaid Projection.
But I have extended them to many more, and abundantly, and plentifully ſupplied the obſcuritie of that Scheme, or Diagramme there drawne (as for a generall good) in the uſe of this Horizantal Quadrant. I deliver therfore firſt the making of it, firſt by the Sector (ſomewhat different from that of Maſter Gunters) ſecondly by Geometrie, and laſtly I ſhew a third way, how it may be Proiected and made by my Mathematicall Ring, and by Numbers, which I have Calculated and accommodated to that end in Tables, for more exactneſſe. Part of the generall ſcope, and uſe of which Inſtrument I deliuer in the Index, or Table following.
Viz of the
Horizon.
Line of Shadowes.
Kalender.
Parallels.
Aequator.
Eclipticke.
Houre-lines.
Index.
MVch more I might have laid open upInſtrument, as the maHorizontall, direct, declining, cilindrical, & Ring dyalls, the diſtance of the houres, ſubſtiler & ſtiles hight, Stoflerius Astrolabe, Master Gunters Quadrant, with many other Inſtruments, now uſed, but let theſe be ſufficient for the preſent; the ingenious, may eaſily adde vnto that which I have delivered, & therefore I ſhew firſt how to project the Instruments, then the Deſcription, and laſtly how theſe uſes, are compendiouſly con
FIrſt, according to any Semidiameter as Z N. or Z S. deſcribe a Circle repreſenting the Horizon, and draw the line S N. for the Meridian: Divide the halfe Meridian Z N. and Z S. into 90. gr. according to the Tangents of halfe their Arkes, by the helpe of the Sines on the edge of the Sector: or the ſemidiameters may be diviProjection, thus. Conſider what parallels you would deſcribe, and how much they are diſtant from the Zenith in their interſections, in the Meridian both towards the South and North of the Zenith (for every parallell in an oblique Spheare, in his interſection with the Meridian, is farther from the Zenith in one part, than in an other) Then if the ſemidiameter Z N. be placed over in the ſigne Complement of halfe that diſtance, from the Zenith, the parallell Sine of the former halfe taken from the Sector: ſhall from ſhew the interſection in the Meridian with that parallell.
Now if the ſemidiameter Z. S. bee placed in the Sine Complement of A. viz. 76. gr. and then the parallell Sine of A taken, viz. 14. gr. it will reach from Z. to ♋. the inTropicke of ♋ with the South part of the Meridian, but if the ſemidiameter Z. S. be placed over in the Sine Complement. B. viz. 37. gr. 30. m. & then the paB. viz. 52. gr. 30. m. being taken it will reach from Z to V. the other interſection of the Tropicke of 69 with the Meridian below the Pole, the Middle betweene this V. and 69. will bee at 1. which is the Center of that Tropicke: In like manner may be found the interſections, and Centers of the other parallels with the Meridians, and ſo may be deſcribed.
Secondly, ſeeing the Lat. is 51. gr. 30. take the SemidiameZ S. and fit it over in the Sine Complement of it, viz. 38. gr. 30. then the parallell ſine of 51. gr. 30. m. will reach from Z. to T. the center of the houre of 6. E P W. but if the Radius Z S. be fitted over in the Sine Complement of halfe 38. gr. 30. viz. 70. gr. 45. m. and the parallell ſine of halfe 38. gr. 30. viz. 19. gr. 15. m. be taken, it will reach from Z. to P. the Pole, then upon T. erect a perpenP T. viz. 2. T. 10. which ſhall ſerve for the finding of the Centers of the Meridians, or houre CirPole P. now ſeeing that T P. is the neereſt diſtance in the right line 2. 10. unto P. the right line P. T. ſhall be Radius, to a Circle, and the line 2. T. 10. ſhal be a Tangent line to that Circle. Now the Radius of a Circle being knowne, the Tangent of any Angle, or Arke, may be alſo knowne, according to the Naturall projection and congruity of lines, but becauſe in this firſt direction we would apply it ſoly to the Sector: the center of the Meridians or houres may be had by the helpe of the Sines thereon thus.
Conſider what houre, or Meridian circle from the houre of 6. viz. E, P, W. you would diſcribe, for then if the Radius P, T. be fitted over in the Sine complement of it and the parallell Sine of the houre Angle Taken, it will ſhew from T, in the line 2. 10. the center of that Meridian, or houre circle: ſo if the houre circle of 5. or 7. were
P, the Pole is 15. gr. fit the Radius or ſemediameter T, P. over in the Sine complement of it viz. 75. gr. for then the Sine perallell of 15. gr. being taken will reach from T, to 5. and from T, to 7, the Center of the houre circles of 5 and 7. If therefore one foote of the Cumpaſſes be placed in 5. and then extended to P. the Pole, you may deſcribe the houre Circle of 5. and placed in 7. you may draw the houre Circle of 7. and ſo may be deſcribed the reſt of the Meridians. and houre Circles.
Thirdly, to deſcribe the Eclipticke, conſider the diſtanZenith Z. and the Tropicks of ♑. and 69. according to the former Lat. of 51. gr. 30. which will be Z, ♑ 75. gr. and Z. ♋. 28. gr. then take the ſemideameter Z, S. and fit it over in the Sine of thoſe Arkes, then the parSines of the Complements of thoſe Arkes will ſhew from Z. the diſtances of the Centers of theſe TroEclipticke, will be neare the Pole P, viz. at ♑. and the Center of the Northerne ſemicircle of the Eclipticke will be below the Pole at ♋. Therefore placing one foote of the Cumpaſſes in ♋. below the Pole, and extending the other foote to ♋. above the Pole you may deſcribe the ſemicircle E, ♋. W. and placing one foote in ♑. neare the Pole, you may deſcribe the ſemicircle, E, ♑, W.
Now for the dividing of the Eclipticke; this Mr. Gunter delivers ſo obſtruſly in his 66, page of the Sector. That if a man had not more fundamentall Mathematicall DoEclipticke, Equator and Meridians, viz, ♈ B ♓. or ♈ B ♉: ♈ R ♒. or ♈ R ♊, &c. and get the right aſcention of theſe Arkes of the Eclipticke hat you intend to divide, ſo ♈, B. is the right aſcention of the Arkes ♈, ♉, and ♈, ♓. and ♈, R. is the Right aſention of Arkes ♈, ♊, and ♈, ♒, from which ground the Table S. is calculated according to the Arkes in the Eclipticke in the Table R. Now to finde the Centers of thoſe Meridians which may divide the Eclipticke according to the
Right aſcenſion here calculated anſwerable to
FIrſt, having diſcribed a Cirrle at pleaſure as before, E. S. W. N. draw a line to paſſe by the center as S. N. and croſſe it at right Angles, with the line E. W. in Z. then let the ſemcircles W S E and W N E. bee divided from W. each of them into 180. gr. or rather upon E. wee may diſQuadrant at pleaſure, as C D. and augmenting it unto ω, divide the Quadrant D C. from D. into the uſuall diviſions of a Quadrant, and ſo from D. unto ω, inſert or protract the ſame diviſions, then having conſidered as beLatitude of the place, and diſtance of the parallels from Z. the Zenith, towards S. the South, and alſo towards N. the North, in the Meridian as in the former Table there is ſpecified. Account the diſtance of the parallels from the Zenith towards S. the South, in the ſemicircle W S E. but thoſe towards N the North, in the ſemicircle W N E. from W. ſo ſuppoſing the Latitude as before to bee 51. gr. 30.Zenith & the TroSouth, is 28. gr. which account from W. to F. but rather halfe of it from D. to F. then conſider the diſtance betweene the Zenith and the other part beyond the Pole, viz. 105. gr. number this from W. to G. but rather halfe uf it from D. to G. and laying a ruE. F. and E. G. the Meridian A B. may be interV. the middle, betweene which will be at 1. the Center of the Tropicke: in like manner the diAequator from the Zenith towards the South is 51. gr 30. reckon it from W. to H. or halfe of it from D. to H. but the diſtance of the Aequater from the Zenith towards the North beyond the Pole is 128. gr. 30. m. which I account from W. to I. or halfe of it from D. to I, then laying a ruler upon E. H. and E. I. the Meridian A B. may be interſected in Q and Y. the halfe diſtance beQ and Y. will be at 2. the Center of the Aequater: In like manner may the Meridian A B. bee divided into any of the reſt of the diviſions, and the parallels alſo diſE. then may you ſoftly move the ruler from D. towards ω. and as it paſſet by the degrees according to the Colume B. of the Tables following, beginnig at the bottome, ſo the edge of the ruler ſhall ſhew the interſections that the parallels of deMeridian Z. S. then move the Ruler ſoftly along from D. toC. as it paſſeth by the degrees in the Colume G. beRuler ſhall inMeridian A. B. in the Centers of thoſe paral
Secondly, account the Latitude from D to M. and halfe D to R. and laying a
E M. and E R. the Meridian, S N. ſhall be inT. and P. P. repreſenting the Pole of the world,T the center of the houre of 6. then unto the line T P. upon the point T. erect a perpendicular 2. 10. and according to the ſemidiameter P T. deſcribe a ſemicirT ς. divide the Quadrants T α and T ς. from T. each of them into 90. gr. then lay a ruler upon P. and the ſeveQuadrants, T α. and T ς. interT. 10. in the houre points, 2. 3. 4. 5. 7. 8. 9. 10. &c. then placing the Compaſſes in T. and extending the other foote to P. you may diſcribe the houre Circle of 6. but placing it in 5. and extended to P. you may diſcribe the houre Circle of 5. the ſame extent placed in 7. will diſP. as it paſſeth by the degrees of the houres in the Quadrants from T. ſo the edge of the Ruler ſhall interſect the line 2. 10. in the Centers of thoſe houres from T.
Thirdly, to deſcribe the Eclipticke, conſider the Altitude of each Tropicke above the Horizon, according to the Latitude given, which was 51. gr. 30. m. So the Altitude of ♑. is 15. gr. and that of 69. is 62. gr. In the Quadrant D C. acD. viz. D y and D δ. lay a ruler upon E. and thoſe ſeverall points, ſo may the Meridian S. N. be interſected in the points ♑. and 69. which are the Centers of the ſemicircles of the Eclipticke, therefore placing one foote of the Compaſſes in 69. below the Pole, and extending the other foote to 69. above the Pole, you may diſcribe the Northeren ſemicircle E. 69. W. and plaS. you may deſcribe the Southerne ſemicircle E ♑ W. thoſe ſemicircles of the Eclipticke may bee divided Geometrically, without the helpe of the Table of right aſcenſion, but for more expeQuadrants T α. and T ς. account the degrees of the right aſcenſion for ſuch diEclipticke as you intend to have, ſuppoſe the beginning of ♉ ♓ or ♍ ♏ the diſtances of the beginning of any of theſe ſignes, fromithe Equinoctiall points are
R. ſo right againſt it under S. is 27. gr. 54. m. this account from T. towards α. and ς. and laying a ruler upon P. and thoſe degrees interſect the Tangent line 2. 10. in 30. and then placing one foote of the Compaſſes in 30. towards 10. and extending the other foote to P. you may interſect the Eclipticke in the beginning of ♉. ♓. & ſo in ♍ ♏. the points required: In the likemanner you may divide the other part of the Eclipticke. So the Centers of the deP. and then to move thereon, Now as it paſI.O. in tha Tables folT. in the ſeveral Quadrants: ſo the Ruler ſhall Interſect the line 2. to in the Centers of thoſe Arkes anK N. the degrees of the Ecliptick.
LEt a Circle bee diſcribed according to any capacity, as before, and croſſed with diametrall lines at right Angles
E Z W. and S Z N. then take the ſemidiameter E Z. and divide it into 10. parts, & ſupdivide each of thoſe parts into 10. or 100. (according to the capacity of the ſcale) as A. or more accurately according to the ſame Radius make a diagonall ſeale, then conſider the diſtance of the parallels from the Zenith according to the Latitude you intend, as admit 51. gr. 30. m. to be the Latitude as before. Take the halfe of thoſe diſtances (according to the firſt directions) by which is made the Colume B. which are the halfe diTropickes and the Zenith, then move the Tangent of 45. gr. unto the parts of the Radius or ſemidiameter, viz. 10000. in the Circle of Numbers, ſo right againſt the Tangent of any one of thoſe degrees in the Colume B.
C. (or they may be extracted out of the Tables of naturall Tangents.) Further if we conſider the diſtance betweene the Zenith and the other interſections of thoſe parallels, with the Meridian beyond the Pole, and take halfe of thoſe diſtances wee may make the Colume D. then moving the moveable ſoftly along, as the Tangent of any degree of the Colume D. in the moveable, paſſeth by the parts of the ſcall, viz. 10000. in the fixed (on the Circle of Numbers) ſo the Tangent of 45. gr. in the moveable, ſhall point out in the Circle of Numbers, the diſtance betweene Z. and thoſe parallels beyond the Pole. From theſe directions are calcuE. or they may bee alſo taken from the Table of naturall Tagents as before: The Numbers of the Colume C and E. ſerve to finde the diſtanparallels in the Meridian froZ. & to diſcribe thoſe parallels, note that at the bottome of the Columes C. & E. are the Numbers, 2493. & 13032. take 2493. from the diagonall ſcale, and protract it from Z. towards S. viz. at ♋. then take from the ſame ſcale alſo, 13032. and protract it from Z. to ♋. below N. divide the ſpace betweene ♋. and ♋. into two equall parts which will be at 1. neere P. ſo have you the Center of the TroCompaſſes therefore from 1. to ♋. then may you deſcribe that Tropicke, viz. δ. ♋. ζ. In like
parallels, but for more eaſe we may take halfe of the differences of the NumC and the Colume E. and ſo may we have the Colume F. and then wi h greater expedition wee may protract the Centers of theſe parallels, from Z. for if 5269. which is at the bottome of the Colume F. (& betweene the former two Numbers) be taken from the ſcale, and protracted from Z. it will reach unto 1. the CenF. is the diſtance of the Centers from Z. of his oppoſite Number in the Colume B. or A. by which Columes C. and F. you may deſcribe all the parallels, betweene the Tropicks from degree to degree.
But for more exactneſſe it were conveni
Secondly, move the Tangent of the Latitude in the moveable viz. 51 gr. 30. m. unto the former part of the Radius or ſcale. viz. 10000. in the Circle of Numbers in the fixed,
A. and protracted from Z. to T. it ſhall be the Center of the houre of 6. upon T. erect a perpendiT. 10. ſerving for the Centers of the other houres: then move the Tangent of 45: gr. to the parts of the ſcale, viz. 10000 in the Circle of Numbers, and conſider the diZenith and the Pole, viz. 38. gr. 30. m. the Tangent of halfe of it in the movable doth point out in the Circle of Numbers 34921. which taken alſo froZ. will reach to P. the Pole, through which all the houres muſt be drawne, and the Centers of which houres in the line 2.10. from T. may be had thus: which two numbers 125717. & 34921. I place over the Columes H. and I.
According to the diſtance P.T. make a ſcale B. (or rather a diagonall ſcale) to containe 10000. parts, then move the Tangent of 45. gr. to the parts of this ſcale in the Circle of Numbers, viz. 10000. ſo every degree in the mouable aTangents unto 45. gr. doth point out in the Circle of Numbers, the diſtances of the Centers of thoſe deT. in the line 2. 10. by which the Colume H. is made, then moving the moveable ſoftly along as the Tangent of any degree in the movable above 45. gr. paſſeth by the parts of the ſcale B. viz. 10000. in the Circle of Numbers, ſo the Tangent of 45. gr. in the moveable, paſſeth by the diſtance of the Centers of thoſe degrees from T. in the Circle of Numbers in the fixed, above 10000. by which is made up the reſt of the Colume H. viz. I. by helpe of which Colume H. and I. the houres may be thus drawne.
Marke, what Numbers are againſt the houres in the Colume H and I. for if thoſe Numbers be taken from the ſcale B. and protracted from T. in the line 2.10. they ſhall be the Centers of thoſe houres: ſo in the Colume H. againſt the houre of 7. or 5. is 2679, which take from the ſcale B. and protract it from T. to 7. and from T. to 5. in the line 2. 10. then placing one foote of the Compaſſes in 7. and exP. deſcribe the houre of 7. and one foote of the Compaſſes at the ſame extent being plaT. out of the Colume H. I. the Centers of the other houres with their intermediats, and ſo al
But here note, that it were convenient to finde the Inhoure lines (and their intermediate deHorizon as before was delivered of the inDeclination with the Horizon, and it may be drawne from my Ring thus.
Now to deſcribe the Ecliptick, conſider as before the hight of the Tropicks above the Horizon, in the Latitude given, viz. 51. gr, 30. m. ſo ♑ will be 15. above the Horizon, and ♋ will be 62. gr. high. Then move the Tangent of 45.
A. in the Circle of Numbers in the fixed. viz. 10000. ſo right againſt the Tangent of 15. gr. in the moveable is 2679. the diſtance of the Center of the Southerne ſemicircle of the Eclipticke from Z. which I place in the Colume over M. and againſt ♑. then move the moveable ſoftly along untill the Tangent of 62 gr. bee right againſt 10000.Circle of Numbers, ſo the TanCircle of Numbers in the fixed: The diſtance of the CenEclipticke, from Z. which I place in the Colume over P. againſt ♋. if theſe numbers be taken from the ſcale A. and protracted from Z. they will reach from Z. to ♑. and from Z. to ♋, and ſo placing one foote of the Compaſſes in ♑. neere the Pole, and extending the other foote to ♑. neere S. you may deEclipticke E, ♑. W. and placing one foote of the Compaſſes in ♋. below the Pole, and extending the other foote to ♋. above the Pole: you may deſcribe the Northerne part of the Eclipticke E, ♋, W. and thoſe ſemicircles of the Eclipticke may be di
Move the Tangent of 45. unto the Sine of 66. and 30. ſo right againſt the Tangent of the degrees of the Sunnes Longitude in the Eclipticke in the moveable,right aſcention in the fixed, or they may be had by reſolving of a Triangle, in which there will be 9Ring they are given at one rectification, and onely by a glance of the eye: for proportionals either in Sines or Tangents are had by the Ring, with the ſame expedition that Numbers are had, As by the uſe of the Circles of Sines and Tangents upon the projection of this Ring, in diverſe parAppendix upon the uſe of the Ring; and ſo according to the former Conſtruction is made the Columes L. and O. for 45. being brought to 66. gr. 30. m. as before, right againſt 10. gr. in the moveable, is 9. gr. 11. m. in the fixed, againſt 20. gr. in the moveable, is 18. gr. 28. m. in the fixed, and ſo of the reſt. Then move the Tangent of 45. gr. to the parts of the ſcale B. viz.
L. in the moveable are the diſtances of the Centers of thoſe Arks, from T. in the Circle of Numbers in the fixed, and ſo is made the Colume M. & if you move the moveable ſoftly along as the Tangent of any degree in the Colume O. paſſeth by 10000 the parts of the ſcale B. ſo the Tangent of 45. in the moveT. in the Circle of Numbers in the fixed, by which is made the Colume P. or they may bee had from the Table of naturall Tangents.
Then by the ſcale B. protract the Numbers,
M. and P. from T. in the line 2. 10. for they ſhall bee the Centers of thoſe degrees of the Eclipticke, which are oppoſite unto them, viz. in the Columes K. and N. ſo if I would interſect the Eclipticke, in the beginning of ♉. ♓. ♍. or ♏. each being diſtant from ♈. 30. gr. which I ſeeke in the Colume. K. and finde right againſt it in the Colume M. 5294. which I take from the ſcale B. and protract it from T. to 30. in the line 2. 10. Now placing one foote of the Compaſſes in 30. next 10. and extending the other foote to P. the Eclipticke may be interſected in the points of ♉. and ♓. and placed in 30. towards 2. the ſame extent will Interſect the Eclipticke in ♍. and ♏. In like manner may the Centers of the reſt of the degrees of the Eclipticke be protracted in the line, 2. 10. from T. out of the Columes M. and P. and ſo all the Eclipticke divided from degree, to degree: but this may be otherwiſe done.
Beſides that which is delivered touching the drawing of the Parallels, Eclipticke, and Houre lines, there remaines yet how to put on the Callender, to graduate the Index, and to draw, and divide the line of Shadowes.
This may be eaſily done from the Table R. Calculated, and accommodated to that purpoſe for the yeare 1640, and may ſufficiently ſerve for many yeares after, without any ſenſible error.
Having divided the Quadrants,
E.S. and E.N. (as before into the uſuall degrees of a Quadrant,) lay a ruler upon
Z. and account the degrees from the point E. in the Quadrant towards N. and S. out of the Table R. according to the ſeverall Columes of the Table R. and InQuadrants,
Horizon of the Inſtrument from E. be divided into the uſuall dayes of the Month, which is the Callender and the beginning of theſe diviſions, may be at the 10. of March, and ſo going on to the 11. of Iune, and then againe to begin from the 10. of March, and go on unto the 10. of December, and theſe dayes may bee noted upon the inſide of the Horizontall Arke with ſhort lines from E. as before, and at every Month may bee placed a repreſentative letter for that Month, and every 10. and 5. day of every Month, may bee noted with a ſmall ſtroke ſomewhat longer then the reſt, to helpe the memory the readier to number. In like manner may the reſt of the dayes of the Calender be interſected in the out ſide of the Horizontall Arke, toLimbe, beginning at the 13. of September, and ſo going on to the 11. of Iune, then againe from the 13. of September, and going on unto the 10. of December, and theſe Months may bee alſo noted with ſignificant letters, appropriate to each Month, and each 10. and 15. day of the Month, may be alſo denoted as before, with a ſtroke ſome
Let the Index be equall to the ſemidiameter, Z. E. and then may it bee divided out of the Table Q. by the helpe of the ſcale A. beginning at the Center: the Index being diuided, and placed on the Center of the Inſtrument at Z. it ſhall helpe to put on, and divide the line of ſhadowes. as followeth.
Lay the edge of the Index to A. in the Limbe which is neere the 10. of December, and move it to any degree in
Table S. and account the like degree in the Index, and then make a marke upon the plaine of the Inſtrument where that degree toucheth, and ſo goe on from point to point, untill the whole line bee deſcribed and divided, acTable S. This line might bee placed beCalender and the Limbe, or in a Quadrant, &c. But I have cauſed it to bee deſcribed as is ſeene upon the Scheme againſt Page the 1. for expedition and conveni
Let A. B. C. D. be a plaine, divide the length A. B. within halfe an Inch of the higher end, and an Inch of the lower end, in to 3. equall parts which ſuppoſe the line M.N. then diM. N. ſhall bee diZ. Then take 4. of theſe parts for Radius, and on Z. deArke, A. E. B. and upon Z. erect a perM. N. to cut the Arke A. B. in E. now from E. to A. protract 40. gr. and
E. to B. protract 50. gr. ſo the Angle E. Z. M. ſhall be 90. and alſo A.Z.B. ſhall be 90. Now having made a ſcale of Z.E. like to the ſcale A. according to the former directions) then out of the Colume C. and by helpe of the ſaid ſcale A. from Z. you may protract Z. ♋. 2493. Z.Q. 4823. and Z. ♑ 7673. and from the Colume F. y u may protract the diſtances of the Centers of thoſe interZ. viz. Z. 1.5269. Z. 2.7949. and Z. 3.16217. and ſo placing the Compaſſe in theſe Centers, you may deſcribe the Equator, and both Tropicks. But if Z. M, and Z. N. be divided according to the ſcale A. then from Z. you may account the interſections of the parallels, and diſtanScheme or Trapeziall forme of the Inſtrument, B. A. ♑. ♋. and may bee finiſhed according to that againſt Page the 1. by the Tables and directions here calculated, and delivered to that end.
Now to augment the Inſtrument to any proportion aſTropickes were ſuppoſed to bee 10. Inches, the Radius might be found out, or if the Radius were 4. foote, (which is according to mine owne Inſtrument:) what diſtance might there bee betweene the Tropickes: the proportion would be as 516. to 1000. ſo the breadth to the ſcale, or as 1000. to 516. ſo the ſcale to the breadth: therefore by the Ring, bring 516. in the moveRadius in the fixed, is the diſtance betweene the Tropickes in the Moveable, or againſt the diſtance aſſigned for the Tropickes in the Moveable, is the meaſure of the Radius or Scale, in the fixed: So if ♑. ♋. be allotted to be 10. Inches, for the diſtance betweene the Tropickes, the Scale, or Radius, of the Inſtrument ſhould be 19 4/10. fere: but if the ſcale or Radius were 4. foote, or 48. Inches, then the diTropickes of ♑, and ♋. will be neere 24 77/100. Inches. Thus for the making of the Inſtrument, the deſcription of which followeth.
THe forme of this Inſtrument is like a mixt Trapezia, as appeares againſt Page. 1. where of two ſides are right, and the other two ſides are Circular, which falleth out to bee ſo from the nature of the Projection, and that part which I have thought moſt convenient for uſe, and is fully ſufficient for that which I have deliuered upon it; and may be made of any plaine Materiall, but fitteſt in Braſſe, or Silver: the ſeverall parts of which Inſtrument are five, viz. the Backe, the Face, the Sights, the Index, and the diFace.
Firſt, the Backe of the Inſtrument, is a part of Gemma Friſius projection, whoſe particular deſcription and admi
Secondly, the face of the Inſtrument, is that upon which Index, and ſights are placed on.
Thirdly, the Sights are the ſmall peeces of Braſſe in Face of the Inſtrument; one of which Sights is neere the Center of the Inſtrument, and the other is neere the Circumference thereof.
Fourthly, the Index, is the movable peece of braſſe, Center, upon which alſo two other ſights may be placed, the edge of this Index is divided and noted thus. 10. 20. 30. 40. 50. 60. 70. 80. 90. which are called the degrees of the Index, and there is adjoyned unto it three ſmall plates to be rectified as occaſion requires, one of which is called the Axis, and the other two are perpendicu
Fiftly, the lines deſcribed on the Face of the Quadrant are ſixfold.
Viz.
The Limbe and its Parallels.
The Kalender and its diviſions.
The Aequator and its Parallels.
The Eclipticke and its diviſions.
The Houre lines and their intermediates.
The line of Shadowes and its diviſions.
Firſt, the Limbe is the outmoſt Circle, which is divided into 140. gr. and noted at every 10. degree thus, 10. 20. 30. 40. 50. 60. 70. 80. 90. and each of theſe degrees is divided into parts according to the Capacity of a degree in the Inſtrument.
Secondly, the next parallell line to the Limbe is the Horizon or Kalender, which is noted with letters thus, I.A. S.O. N.D. I.F. M.A. M.I. of which I. in the firſt place ſtands for Iuly, A. for Auguſt, S. for September, O. for October. &c. then againe on the inſide, I. ſtands for Ianuary, F. for February, &c. each letter repreſenting its Month, and each of thoſe Months is divided into dayes by ſmall ſhort lines, whereof the 10. and 15. day of every Month is ſignified by Numbers, or elſe by a line ſomewhat longer then any of the others, to helpe the memory the readier to Number, and for more promptneſſe of finding the day of the Month, in the Kalender as occaſion requireth.
Thirdly, The Equater is that line that meeteth with the tenth of March, and the thirteenth of September in the Kalender, and is divided into degrees, and numbred thus, 10. 20. 30. 40. 50. 60. 70. 80. 90. and the parallels to the Equator are theſe lines which are on each ſide of it, every 5. degree of which being noted thus, 5. 10. 15. 20. the outmoſt of thoſe Parallels on each ſide of the Equator are the two Tropickes that which is neareſt the Center, is calTropicke of ♋, and that which is fartheſt of, is called the Tropicke of ♑, thoſe two Tropicks, the Kalender, and the houre of 12. comprehend the whole Projectin: and here note farther that theſe parallels are called
Moneth, as well as the parallels of the Sunnes Declination, according as they ſhall be vſed, and farther below the Tropicke of Cancer is a graduation of the common houres of a Horizontall Dyall: ſome vſe of whſch is ſhowne, by pro. 36.
Fourthly, the Eclipticke on the inſtrument is repreſenEclipticke which croſſeth the former parallels, and meeteth with the Equator, in the Horizon or Kalender, in the former 10. of March, and 13. of September: that Quarter which is towards the Center of the inſtrument, ſerves for the Northerne ſemicircle of the Eclipticke, and that which is farther from the Center ſerves for the Southerne ſemicircle of the Eclipticke; & each of theſe ſemicircles is divided into the Signes of the Zodiacke, & charactered accordingly thus, ♈. ♉. ♊. ♋. ♌. ♍. ♎. ♏. ♐. ♑. ♒. ♓. Of which the firſt 6. Signes are called Northerne ſignes and are in the Northerne ſemicircle, & the other 6. Southerne ſignes & are in the Southerne Semicircle. And each of thoſe ſignes is divided into 30. gr. and if the Inſtrument be large, each of theſe degrees may be divided into 6. or 12. diviſions more, So every diviſion ſhall ac
Fiftly, the houre lines are thoſe that croſſe the Aequator
Tropicke of ♋ with numerall Characters thus IIII. V. VI. VII. VIII. IX. X. XI. XII. And are the forenoone houre notes: thoſe hourelines ſerve alſo for the afterArithmeticall figures, for the houres in the afternoone thus, 1. 2. 3. 4. 5. 6. 7. 8. each of thoſe houres is divided into 3. parts, each part being 20. minuts: and each of thoſe parts is ſubdiHoure circles with their intermediates are alſo called Meridians or degrees of meaſure, and are Numbred by tens in the Aequator, from the meeting of the Aequator with the Eclipticke, as before
Sixtly, the line of ſhadowes is that which makes a ſphaericall Equilaterall Triangle upon the plaine of the Inſtrument, the baſis of which is the Horizon, or Kalender and one of whoſe legges is below the Tropicke of ♋ and the other croſſeth the Tropicke and parallels, and meeteth with the Kalender neere in the 10. of December: both of thoſe equall ſides are called the line of ſhadowes, and are divided alike into 10. vnequall diviſions, and each of thoſe diviſions againe is divided into 10. other diviſions, and againe each of them into other 10. (if the Inſtrument be large.) The firſt Capitall 10. diviſions are noted with Arithmeticall figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. of which 1. is at the very meeting of the two lines, not farre from the Center, which ſignifieth Equall: the figure of 2. Double: the figure of 3. Triple, the figure of 4. Quadruple: the fiQuintuple, &c, Thus for the making, and deſcription of the Inſtrument, the uſe of it now followeth.
OF which ſome have relation to the obſerSunne, others without obſervation, or ſight of the Sunne.
The Vſes of theſe which are knowne without ſeeing the Sunne, are 30. of the ſaid Index or Table, as followeth, viz. the 2. 9. 10. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 32. 33. 35. 37. 38. 39. 40. 41. and 42. of which 13. of them will be ſhowne onely by knowing the day of the Month, viz. the 18. 33. 21. 19. 25. 24. 35. 2. 38. 37. 20. 41. and 42. as followeth.
Seeke the day of the Month in the Kalender,
and the houre line that meeteth therewith, ſheweth the time of Sunne riſing, or ſetting.
So if the day of the Month were the 13. of October, the parallell that meeteth therewith is the houre, viz. 7. of the clocke, at which time the Sun riſeth: the ſame houre is noted alſo with 5. which is ye time of Sun ſetting that day, this dou
Marke what Meridian meeteth with the day of the Month in the Kalender: as ſuppoſe the day to be the former 13. of October, which is the houre line of 7. and 5. (as before) and account the Numbers of Meridians to the houre of 6. ſo have you 15. gr. or an houre, which is the difference of Aſcention for the 13. day of October re
Marke what parallell of Declination meeteth with the day of the Month in the Kalender, and account how many degrees it is from the Equinoctiall, ſo have you the Sunnes Declination for that day.
So if the day were the laſt of Auguſt, the path. from the Equator, and ſomuch is the Sunnes declination, that day, viz. 5. gr. North declination.
For the firſt, note that the dayes betweene the 10th of December and the 11th. of Iune, are dayes of Increaſe, and the reſt are dayes of DeKalender, is the day of increaſe, which dayes are equall one to the other. So the 19. day of May, is againſt the 4. of Iuly, at which time the Sunne riſeth and ſetteth alike without ſenſible
Admit the day to be the 18th. of February and according to the firſt pro. finde the time of Sunne riſing, which is at 40. m. after 6. of the clocke for that day, and the ſetting 20. m, after 5. then marke what day of the Month the houre line of 20 m, after 5. in the forenoone, meeteth with the Kalender, which will be the 23. of Auguſt, ſo the 18th. day of February the Sun did ſet at the ſame houre that it did riſe, the 23. day of Auguſt.
Note where the parallel of the day of the Month croſſeth the Eclipticke, that is the Sunnes
place. So the former parallell of the 13th. of October meeteth with the Eclipticke in the beginning of ♏, and ♓, but which of theſe is the Sunnes place, the quarter of the yeare may eaſily tell you, viz. ♏ which is the Sunnes place or the degree in the Eclipticke for that day.
Conſider what Meridean meeteth with the Sunnes place in the Eclipticke for the day given, and marke the number of Meridians in the Equator (for the Meridians are numbred in the Equator, as is ſayd before in the deſcription) ſo have you the Sunnes right Aſcention: but here note that the degrees in the Eclipticke are numbred forward and backeward in the Eclipticke unto 360. gr. upon this Inſtrument: ſo are the right Aſcentions of thoſe degrees alſo numbred forward and backward in the Aequator: for the right aſcenEclipticke, is that degree of the Aequator which is oppoſite unto it, (the ſucceſſion of the ſignes conſidered) ſo if the Sun were in the beginning of ♏, the right Aſcention is neere 208. degrees, for the Meridian that paſEquator from ♈, and is within 6. m, of 28. gr. Now the right Aſcention of ♋, is 90. gr. and the beginning the of ♎, is 180. gr. and from the beginning of ♎, to the beginning ♏, is within 6. m, of 28. gr. as before, all is which put togeAſcention of the Sun the 13th. day of October.
Note that the difference of Aſcention, is the difference alwayes betweene the right Aſcention of the Sun, and the oblique Aſcention thereof: therAſcention known by the laſt diAſcention by the ſecond direction, the oblique Aſcention is eaſily had, by Addition, or ſubſtraction thus. If the Sun be in a Southerne ſigne then the oblique Aſcention, is greaAſcention, by ſo much as the difference of Aſcention comes to: but if the Sun be in a Northerne ſigne, the oblique Aſcention is ſo much leſſe: which difference of Aſcention as bePro: for the ſaid 13th. of October was 15. gr. this ad unto the right Aſcention of the beSuns oblique Aſcention for the beginning of ♏, on the 13th. day of October; but if the Sun had beene in the beginning of ♉, the oblique aſcention would have beene onely neere 13. gr. viz. 12. gr. 54. m.
Here note that this Propoſition holds in uſe
Suns being in the Northerne ſignes that is from the 10th. of March to the 13th. of September: therefore lay the Index to the Eaſt, or Aequinoctiall point noted with E. or 40. & 50. in the Limbe: ſo have you inſtantly at once without farther rectification both the Altitude and houre of the Suns comming East or Weſt, aHorizon for all or any of the dayes aMoneth meeting with the edge of the Index gives the Suns Altitude in the Index, and the Meridian meeting therewith ſhewes the houre.
So if it were the ſecond of May, or the 22. of Iuly, the parallel belonging to thoſe daies meetes with the Index neere about, 23. gr. 17. m, and there alſo meetes with that point, the houre line of 7, and 5. which ſheweth that when the Sun is 23. gr. 17. m, high either upon the ſecond of May or the 22. of Iuly; then the Sun will be due Eaſt or Weſt, and that will happen to be at 7 of the clocke in the forenoone, and 5. of the clocke in the af
Lay the Index to the day of the Moneth, for the
Limbe of the Instrument ſhall ſhew the Amplitude required.
So if the day were the 13th. of October the numEaſt, noted with 40. 50. unto the Index is 18. gr. 40. m. which is the Suns Amplitude for the given day, viz. the 13th. day of October.
Lay the Index unto the houre of 12, and where the parallel of the day of the Moneth meeteth yt therewith ſhal be the Suns Meridionall Altitude.
So if it were the 13th. day of October, as before, the parallell, for that day is 11. gr. and a halfe from the Equator South: this croſſeth the Index in 27. gr. which is the Sunnes Meridionall Altitude that day. Now for the Sunnes depreſſion at midnight, here is to be noted, that any degree of the Eclipticke is at any time ſo much below the Horizon, as his oppoſite degree in the Eclipticke, is above the Horizon at the ſame time.
Therefore where the contrary parallell of the Sunne viz. 11. gr. and a halfe North, meeteth with
Index in the houre of 12. that ſhall bee the Sunnes Meridionall depreſſion at midnight, the ſaid 13th. day of October.
This propoſition, hath reference to the Sunnes depreſſion under the Horizon, for it is ſaid to bee day breake or twi-light to end, when the Sun is, 18. gr. under the Horizon: therefore the Con
Account 18. gr. on the Index then move the Index untill that degree meete with the Contrarie parallel of Declination for the day given, ſo the Meridian or Houre-line that meeteth therewith ſhall bee the houre of day breake required.
So if the day were the 10. of Aprill, the paralDeclination for that day is North II. gr. and a halfe which I ſeeke out one the other ſide of the Aequator viz. II. gr. and a halfe South Declination, and Marke where the 18th. gr. of the Index meeteth therewith, for there alſo is the houre of day breake viz. with in 20. m. of 3. in the Morning, and 20. m. paſt 9. for the end of twi-light the ſayd 10th. of Aprill, alſo the Index in the Horizon at that Inſtant
Sun under the Horizon viz. neere 48 gr. 10. m. to the North of the Eaſt: but if the day had beene the 13th of October the houre of daybreake had beene 2 minuts before 5. and twi-light would have ended 2. m. af
By the firſt Conſtruction, for the dayes given finde the time of Sun riſing, and by the former 13th Conſtruction the houre and time of day breake belonging to thoſe dayes: then compare the time betweene the Sun riſing, and day breake of the one, with that of the other, ſo the difference of thoſe two, ſhall bee the difference of time re
So the difference of time betweene day breake and Sunne riſing the 10th of December is neere a
houre longer then that of the 10th of March; but more then an houre and halfe longer betweene day breake, and Sun riſing the 10th of May, then the 10th of March.
Seeing the declinations of Plaines, or Walls, are acEaſt or Weſt in the Horizon, as the ſunnes Amplitude is the numLimbe of the Inſtrument. Now admit the Declination of a Plaine or Wall, to be 22. gr. the oppeIndex being ſet therehoure the Sunne will come upon the Plaine, for any day in the yeare; for where the parallell of the day of the Moneth croſſeth the Index amongſt the hourelines, (which Index repreſents the Plaine) that is the houre of the Suns comming upon the Plaine and the degrees in the Index gives the Sunnes Altitude.
So if the Sunne were in the Tropicke of ♋ the Tropicke meeteth with the Index almoſt within 5 m. of 9 in the Morning, and at that time the Sunne commeth upon the Plaine, and there the Tropicke cuts alſo the Index in 45. gr.
ſuns Altitude at that time that ye
Sun wil glance or begin to ſhine upoPlaine.
As for the time of the ſuns continuance on the Plaine (as is ſpecified in the Index or Table) acDeclination, on the other ſide of the Eaſt point, and lay the Index thereto, ſo the edge of it in the Tropicke of ♋, will point out at what houre the Sun goes of the Plaine viz. at 6. of the clock & 38. m, neere, if the declination were Weſt, (as here it is ſuppoſed) which added to the time of the ſuns coPlaine, makes 9. houres 33. m, & ſo long the ſun ſhines on the Plaine.
Let a Plaine decline from the Eaſt point toSouth 22. gr. account this in the Limbe
from the houre of 12. and lay the Index thereto, ſo the parallell that croſſeth the Index, doth ſhew the Sunnes Altitude, and the Meridian meeting therewith, gives the houre, at which time the Sun will be oppoſite to the Plaine; ſo have you at one inſtant for every day in the yeare, at what houre and Altitude, the Sunne will bee oppoſite to the Plaine.
Thus touching the reſolution of the former 13 uſes of the aforeſaid Table, or Index which had reMonth, there are 13. other uſes of the foreſaid Index, or Table viz. the 10. 12. 32. 39. 40. 23. 17. 16. 26. 27. 28. 29. 30. Which have no dependance upon the ſight of the Sun, of which the 6 firſt are reſolued, only by knowing the day of the Month, and the other 7. are as followeth.
As if it were required the 20 of Aprill,
at what houre of the day, and how high the Sun muſt be either in the forenoone or afternoone, that the ſhaman or any Altitude, ſhall be equall unto his height double, triple, quadruple Quintuple &c.
Lay the Index unto the numbers in the line of ſhadowes viz. to 1. 2. 3. 4. 5. &c. and whereſoever any of thoſe diviſions in the line of ſhadowes meete with the Index amongſt the degrees; there it ſheweth what height the Sun muſt have, to make the ſhadowes equall, double, triple &c. to the Altitude.
So laying the Index upon 1 in the line of ſhadowes, it meeteth with 45. gr. in the Index: & ſo high the Sun muſt be to make the ſhadow of a man or any thing equall to his height upon an Horizontall plaine: then move the Index to and fro, unIndex meete with the paMonth, viz the 20. of April, ſo the houre line that meeteth therewith, is the houre of the day that the ſhadow of a Man: or other Altitudes, will be equall to his heigth or Altitude, viz neere 10. of the Clocke in the forenoone, or 2. of the Clocke, in the afternoone.
If the houre be knowne, or ſuppoſed, move the
Index until it meete with the houre in the parallel of the day of the Month, ſo the interſection of that parallel with the Index is the ſuns Altitude, and the edge of the Index, in the Limbe, wil ſhew the Suns Azimuth, then move the Index until the degree of Altitude interſect the line of ſhadowes, ſo ſhall you have the proportion of ſhadowes, to their bodies required.
So if on the 11th. of Aprill at, 7. of the Clocke in the forenoone, (if the Sun ſhine,) it were required what proportion the ſhadow of a man ſhall beare to his height, or the ſhadow of an Altitude to the Altitude, the parallel that belongeth to the given day is neere 12. gr. Marke where this parallel meeteth with the given houre of 7. and bring the Index to it; ſo have you the Suns height at that houre viz 18. gr. 26. m, and the edge of the Index in the Limbe of the Inſtrument, ſhall give the Azimuth viz 4. gr. from the Eaſt: then move the Index, untill the degree of the Suns Altitude
ſhadowes which will be in 3, which ſheweth that at 7. of the Clock in the forenone the ſaid 11th. of Aprill, the ſhadow of a man, or the ſhadow of an Altitude, ſhall be Triple to his height: the like will be at 5. of the Clocke in the afternoone, for equall diſtances of the Sun from the Meridian the ſame day, withAltitudes of the ſun, and equall Altitude of the ſun doth proſhadowes upon Horizontall Plaines
Secondly, if the Poſition or Azimuth of the Sun be knowne or ſuppoſed, which admit 4 gr. from the Eaſt towards the South.
Lay the Index unto it in the limbe, & marke what degree in the Index the parallel meeteth with, which is with 18. gr. 26. m, ſo have you the ſuns Altitude in the Index: then move the Index until ye degree meete with the Line of ſhadowes; ſo have you the proportion of ſhadowes required at that inſtant, viz Triple as before.
Thirdly, if the Suns height be knowne or ſupIndex, and moove the Index until that degree meete with the line of ſhadowes; ſo where it interline of the ſhadowes, there you have the proportion of ſhadowes to their bodies at that intriple as beth. of Aprill if the houre be 7. or the Altitude 18. g. 26. m, or the Azimuth 4. gr. from the Eaſt toward the South, the proportion of ſhadowes to their bodies will be Triple.
For the firſt account the Suns Altitude in the
Index, and move it to and fro untill that degree meete with the parallel of the day of the Month: ſo the Meridian that paſſeth by that point, ſhal be the houre required.
Thus if the day were the tenth of March,
the Sun being that day in the Aequinoctial, & the Altitude ſuppoſed to be 32. gr. 37. m, this I ſeeke out upon the Index and move the Index till that degree meete with the Aequator; ſo the Meridian or houre Circle that paſſeth thereby is the houre viz. 10. of the Clocke in the forenoone or 2. of the Clocke in the afternoone, and if you move the Index ſoftly along as the degrees of the Suns Altitude in the Index interſect the Aequator (and ſo of any parallel:) ſo the Meridian that meeteth therehoure of the day agreeable to that Altitude.
As ſuppoſe it were 36. gr. 35. m, from the
South. Move the Index in the Limbe unto this Azimuth known or ſuppoſed; ſo where the Index croſſeth the parallel for the day given, there the Meridian that meeteth therewith, ſhewes the houre of the day viz 10. of the Clocke in the forenoone or 2 in the afternoone as before. And if you move the Index ſoftly along, as the Index paſAzimuth in the Limbe: ſo the edge of the Index ſhall interſect the parallel of declination for the day of the Month, in the houre of the day agreeable to that Azimuth: by which propoſition and the laſt, Glaſſes may be eaſily placed to burne according to the Suns Azimuth, or houre aſſigned.
By the 11th Conſtruction it is ſaid that any deEclipticke, is aſmuch belowe the Horizon at any time, as his oppoſite degree is above the Horizon at the ſame time: therefore if the Index be layed to the like parallel, on the contrary ſide of the Aequator, that meeteth with the given
houre the interſection in the Index ſhall ſhew you the degree of the Suns depreſsion under the Horizon at that houre.
So if at 10 of the Clocke at night the ſaid 13th of October it were required to finde the Suns deHorizon,
conſider the declination, or the ſuns parallel for that day, which is 11. gr. and a halfe South, which declination I ſeeke in the other ſide of the Aequator, and marke where it meeteth with the houre of 10. unto which I lay the Index, ſo the edge thereof in the Limbe ſheweth the ſuns Azimuth to be nere 42 gr. 30. m, from the South, and the parallels interſection that meeteth with the Index, gives the Suns depreſſion, viz. neere 43. gr. and ſo much is the Sun below the Horizon, and in that poſition the 13th of October at 10. of the Clocke at night.
But if it were required at what houre of the Night the Sun would touch the verticall Circle of Eaſt and Weſt under the Horizon.
Lay the Index to the point of Eaſt and marke where about the Contrary palleIndex, for there you have both the houre and the degree of the ſuns depreſſion.
So the day being as before the 13th of October,
& the declination ſouth 11. gr. and a halfe, this acNorth declinations & it meeteth with the Index in 38 m, paſt 6. the houre of the Suns being Weſt, and with all the ſuns depreſſion, at the ſame time is neere 14. gr. and 50. m.
Conſider the declination for the day, and move the Index to and fro untill the degree of the ſuns depreſſion in the Index, meeteth with the like paAequator, ſo the houre that meeteth therwith is the houre of the day to our Antipodes.
So if on the 20th. of Aprill, we ſhould ſuppoſe the ſun to be 13. gr. under the Horizon, & deſire to know the houre to our Antipodes, the parallel of declention for that day is 15. gr. North, Now in the Index account 13. degrees and move it to and fro untill the ſaid thirteenth degree in the Index meete with the 15th. parallel of South declination, ſo the Meridian that meeteth therewith is the houre of the day to our Antipodes, within 2 m, of 9. at night.
Move the Index, to & fro untill the edge of the
Index meete with the parallel belonging to that day, in the ſame Number of degrees that the end of the Index in the Limbe from the point of Eaſt doth; ſo have you the degree of the Suns Azimuth, and Altitude equall the one to the other, and the Meridian meeting with the Index in the parallel of the given day, ſheweth at what houre that Azimuth, and Altitude will be equall.
So admit the ſun to be in the Tropicke of ♋, the Index being moved to and fro untill there be like degrees in the Index, and in the Limbe, which will be neere 16. gr. 45. m, and there the houre that meeteth therewith is 12. m, after 6. in the forenoon, at which houre the eleventh of Iune, the Suns Azimuth, and Altitude, will be equall viz. neere 16. gr. 45. m, as before.
Account 7. gr. and a halfe amongſt the Meridians
from the given day in the Kallender, and note the day of the Month againſt it, then number the dayes betweene that day and the given day, and you haue the anſwer.
So if the day were the laſt of February,
or the firſt of March, conſider the Suns ſetting that day by the Inſtrument, which is 40. m, paſt 5. this doubled
houres 20. m, then from the laſt of February account 7. gr. and a halfe and it will point out the fifteenth of March at which time the Sun ſeteth at 10. m, paſt 6 which doubled makes 12 houres 20. m, ſo the fifteenth of March, the length of the day is an houre longer then it was the firſt of March; and the difference of time only but 15. dayes, but if the number of daies were accounted to or from the Suns entring into the Tropicall points, it will be more then 35. dayes before the day will be an houre longer or ſhorter.
So if from the tenth of Iune we ſhould account 7. gr. and a halfe amongſt the Meridians from that Meridian that meeteth with the tenth of Iune, it would fall out at the 16th. day of Iuly, at which time the day will be an houre ſhorter then it was the tenth of Iune, and the intervall of time more then twiſe as much as the former viz. 39. dayes.
This propoſition at the firſt ſeemes as a Paradox, yet by this Inſtrument may eaſily be reſolued, and ſo conſequently from Mathematicall principles demonſtrated, not onely the inequalitie of equall Months, but alſo the inequalitie of Naturall dayes.
Now a day naturall according to the generall definition is one revolution of the Aequator or primum mobile, that is from ſun riſing to ſun riſing: or it is the time wherein the ſun paſſeth by the Meridian, and commeth to the Meridian againe, commonly taken for 24. houres: but be cauſe that in that intervale of time the ſun paſſing from the Meridian and commeth to the Meridian againe, the Sun moves according to his Naturall motion (ſecundum antiquiorum traditionem) neere a degree more or leſſe; therefore the Naturall day ſhall be ſome what longer or ſhorter then 24. houres, viz. by ſo much as the difference of right aſcention of that degree of the Eclipticke comes to that the ſun is in, and ſeeing the degrees of the Eclipticke amongſt themſelues have not the ſame difference of right Aſcention that the other degrees have, (notwithſtanding the degrees of the Eclipticke amongſt themſelves being equall the one to the other) the ſuns motion ender thoſe deſun will move more or leſſe untill the ſun can touch the Meridian, which is the limit or tearme of the ſuns diurnall revolution as before: this difference and inequalitie of time in naturall dayes may by calculation be given from day to day, but beInſtrument of this nature can be ſeene, but by a number of dayes, compared with another number of dayes it will evidently appeare.
So,
if it were required how much the Month
of December is longer then the Month of March, in the firſt of which months the ſuns motion is quicker, being about the Perigeum then at other times, now both of which months have equal num
If the degree given be in the Northerne part of the Ecliptike, the oblique Aſcention is leſſe then the right Aſcention vel contra. Get therefore firſt the right Aſcention of the point given by the ſixt Pro. and the difference of Aſcention by the 2. Pro. for
Aſcention gives the deAequinoctiall in the Horizon, but if the given degree had beene in a Southren ſigne, the difference of Aſcention muſt be added to the right Aſcention, ſo have you the degree of the Aequator in the Horizon.
This is but the Conuerſe of the former, onely conſider the correſpondent quarters of the Aequinoctiall to theſe of the Eclipticke.
Seeke the degree of the Aequator in the Horizon,
Circle to it) then the degree of the Eclipticke oppoſite to the remainder, is the Anſwer, but note that if the reQuarter of the Ecliptike, if the remainder be betweene 180. and 270. then it reſpects the 3 quarter of the Eclipticke, if the remainder be betweene 90. and 180. it hath refe
This, is onely but the converſe of the former, & is thus performed firſt, ſeek the right Aſcentio ̄
of the given degree of Medium Coeli, & adde thereAſcention, & note the ſuns place oppoſit therto for the difference of Aſcention of this laſt degree being ſubtracted from the former degree of the Aequator in the Horizon, if it be a degree of the Southren ſignes (otherwiſe Adde) gives the degree of the Eclipticke in the Horizon demanded.
Firſt, note the right Aſcention for the day giAequator in the Meridian, at 12. of the Clocke, unto which degree adde 90. ſo have you the degree of the Aequator in the Horizon at 12. of the Clocke. Then conſider how many houres the given houres wants of 12. or is paſt 12. which conuerted into meaſure and accounted Eaſtward, or Weſtward, according to the houre given from the former points of the Aequator in the Horizon at 12. will give the degree of the Aequator in the Horizon at the houre propoſed, then by the 23. Pro. I ſeeke out the degree of the Eclipticke in the Horizo ̄
anſwerable to the degree of the Aequator ſo have you the degree Aſcendant, from which account 90. gr. or 3 ſignes, ſo have you the degree of the Nonageſſima point in the Horizon, but if you reckon 6. ſignes from the Aſcendant, you have the deſendant degree of the Eclipticke in the Weſt of the Horizon.
According to the laſt Pro. ſinde the degree
Aſcendant, and the Nonageſſima degree, then by the 24. Pro. finde what degree of the Eclipticke is in the Meridian, Anſwerable to the degree of the Eclipticke in the Horizon, ſo ſhall you know on which ſide of the Meridian the Nonageſſima degree is, & how far from the Meridian, then if yt
Index be layed upon the houre of 12, where the parallel of the Nonageſſima degree croſſeth it, that ſhould be the height of it, if it were in the Meridian; account therfore from the Meridian or houre of 12. in the Aequator, the number of degrees betweene the Nonageſſima degree, and the degree of the Eclipticke in the Meridian, & marke where that Meridian meeteth with the parallel of the of the Nonageſſima degree, lay the Index thereto,
Altitude of the Nonageſſima degree in the Index, and the Azimuth in the Horizon, or Limbe of the Inſtrument.
There are yet the 48. 49. 50. 9. 13. 14. and 15th. uſes of the ſaid Index or Table, which haue no relation to the ſuns ſight or obſerTractat; as for theſe uſes of the Inſtrument which depend upon the Suns ſight, or obſervatith.
Lift up the edge of the Inſtrument to the eye, ſo that the ſight which is at the Limbe or Circumference of the Quadrant be next the eye, and the Index to hang perpendicular and to play eaſiQuadrant up and downe untill you may through both ſights ſee the Center or midle of the Sun, or ſtarre: ſo the Index in the Limbe ſhall fall upon the degrees of the Sun or ſtarres Altitude above the Horizon at that time. Or without looking at the ſun, the Altitude thereof may be thus found: hould the Quadrant
that the Index may hang perpeverticall as before, then move about the Inſtrument untill the edge of it be oppoſite to the body of the Sun. Now ſuppoſing the Inſtrument to hang thus upon his Center, ſoftly lift up the edge thereof which is towards the Sun, untill you ſee the beames of the ſun paſſe through both ſights, then the Index in the Limbe ſhall give the ſuns Altitude as before.
By the laſt Pro. obſerue or take the ſuns Altitude and account it on the Index, then ſeeke for the parallel of the day of the Month for the day preIndex untill that degree of Altitude in the edge of the Index meete with the parallel of the day, ſo the Meridian that meeteth with that degree of Altitude in the Index, ſhall be the houre of the day required, & the edge of ye
Index in the Limbe of the Inſtrument, ſhall likewiſe ſhew the Suns Azimuth belonging to that houre.
So if upon the laſt of August the Suns Altitude in the forenoone ſhould be obſerued and found to be 30. gr. & a halfe, ſeeke this Altitude out upon the Index & move the Index untill the degree of Altitude meete with the parallel for the day of the Month given, viz. the fift parallel from the Aequator Northward ſo the houreline that meeteth alſo with the 30. gr. & a halfe in the Index, is the houre viz. neere 9. & that ſhall be the houre of the day at that inſtant, & the edge of the Index in the Limbe cutteth neere 35. gr. and 30. m, from the point of Eaſt, towards the South, and ſo much is the Suns Azimuth at that time.
According to the 27. Pro. firſt obſerue the
Suns Altitude above the Horizon, and by the laſt Conſtruction finde the Suns Azimuth agreeable to that Altitude: let the Index and reſt at that deIndex, then houlding the plaine or face of the Quadrant parallel to the Horizon, move the Inſtrument Circular, untill the ſhadow of the ſaid perpendicular fall by the ſide of the Index, and ſo the houerline of 12, or the edge of the Inſtrument which is parallel unto it (which is the North and ſouth edge of the Inſtrument) ſhall repreſent the Meridian line, and pointeth out the North and South in the Horizon of the world by the termes thereof, and the other ſtraight edge of the Inſtrument which is perpendicular unto that edge is the (Eaſt and Weſt edge of the Inſtrument) and denoteth or ſheweth the line of Eaſt, and Weſt in the Horizon, of the world. But this may be more accurately done if you place the backe
Inſtrument downe upon an Horizontall plaine, and the edge of the Index being at the deſuns Azimuth obſerved, and the perIndex as beInſtrument as it ſo lyeth untill the ſhadow of the perpendicular fall by the ſide of the Index, ſo the Meridian of the Inſtrument, ſhall be in the Meridian of the World, and every point and degree in the Limbe of the Inſtrument ſhall point out, and be oppoſite, and reHorizon of the world.
But here note that this Conſtruction ſerves onMeridian line may be thus. Having found the ſuns Azimuth as before, lay the Index upon the houre line of 12. and erect the perpendicular at the end thereof, and move the Inſtrument aCircular, untill the ſhadow of the ſaid perIndex: for then if the edge of the Index be moved unto the ſuns Azimuth before known, the edge of the Index ſhall repreſent the Meridian line, & 90. gr. farther ſhall be the point of Eaſt, and the Center of the Inſtrument the point of Weſt, therefore if upon the plaine that the Inſtrument lies upon, you make a marke at the edge of the Index which is in the Meridian as before, and another marke right unCenter and ſo place the North and South edge of the Inſtrument unto theſe two points: then
Horizon. or Limbe of the Inſtrument, ſhall point out as before his oppoſite or Horizon of the world.
By the laſt Pro. finde out or draw the Meridian line, and place the North and South edge of the Inſtrument unto it: if the Building or Place ly in the Eaſterne ſemicircle of the world (but if it ly in the Weſterne ſemicircle, then let the Eaſt & Weſt edge of the Inſtrument be placed upon the Meridian line) ſo the eye being over the Center of the Inſtrument, and behoulding the place, let the Index be moved untill it be alſo with the viſual line obIndex, from the Cardinall points of the Inſtrument in the Limbe, viz. from the Eaſt or Weſt, North or South, ſhall ſhew the bearing of that place from you, in reſpect of the Cardinall points of the world in the Horizon: but if two ſights be placed at the Index (which is according to the deſcription thereof) then may you obIndex by leting the Inſtrument reſt, and moving the Index to and fro untill you ſee the obiect, ſo the edge of the Index in the limbe, ſhall point out the
Poſition of the place from you in deEaſt, Weſt, North, or South, & acCompaſſe which the place, or obiect beares from you.
The Meridian line being drawne firſt upon a plaine according to the former directions, conNorth, and South edge of the Inſtrument be placed unto the Meridian line, but if it be in the afternoone, then ſet the edge of Eaſt, & Weſt of the Inſtrument, unto the Meridian line, and let the Inſtrument reſt there, then erect the perpendicular at the end of the Index, & move the Index about untill the ſhadow of the perpenIndex, ſo the edge of the Index will amongſt the degrees in the Limbe ſhew the Suns Azimuth at that time, and where the edge of the Index meeteth with the parallel of the day of the Month, that is the houre of the day at that time. But if the Axis be rectified then there is no neede of a Meridian line to be drawne, for this Inſtrument will with great facilitie finde out his owne Meridian, by moving it to and fro untill
Center of the Inſtrument, interſect the ſame houre in the Parallel of the day of the Moneth, that the Axis doth amongſt the Common houres ▪
ſo that houre ſhall be the houre of the day for that inſtant, and the ſhadow of the ſaid perpendicular, cutting the Limbe, or extended unto it, doth there ſhew the Suns Azimuth, and ſo the Meridian of the Inſtrument at that poſition, ſhal be in the Meridian of the world required.
By the twentie nineth Pro. vpon an even Plaine parallel to the Horizon draw the Meridian line, & place the North & South line of the Card directly over the ſaid Meridian line, ſo the Number of deNeedle cutteth in the Card from the North and South line of the Card, that ſhall be the variation of the Needle required; otherwiſe it may be found thus: Neere unto the Center of the Index, upon the Index may a ſmall Broſse pinne be ſo placed that it may be erected perpendicular to the Center of the Inſtrument and halfe an inch aNeedle by placed upon this pinne, then lay the Eaſt, and Weſt edge of the Inſtrument to the Meridian line, & when the Needle reſteth, move the Index, untill the edge of it be diNeedle ſo the edge of the Index; in the Limbe of the Inſtrument, ſhall
Needles variation required.
First, draw the Meridian line upon ſome plaine by helpe of the 38. Construction, then erect the prependicular at the end of the Index, and place the North and South edge of the Inſtrument, to the Meridian line ſo drawne upon the plaine, and move alſo the Index untill the edge thereof touch the houre of 12. let the Inſtrument reſt at this poſition, then marke diligently about noone or 12. of the Clocke when the ſhadow of the perpendicular doth fall by the edge of the Index, for then the ſun is in the Meridian, at which time according to the 27. Pro. obſerue or take the ſuns height (which is his Meridian Altitude, for that day) and by the 3. Pro. finde the Suns declination agreeable to that day, and adde it to the Suns Meridionall Altitude obſerve (if it be South declination, otherwiſe ſubtract it from the former Meridionall Altitude,) ſo have you the height of the Aequinoctial above the Horizon, that taken from 90. gives the depreſſion of the South Pole under the Horizon, which is alwayes equall to the elevation of the North Pole above the Horizon.
So if upon the tenth of Aprill,
the Meridian Altitude ſhould be found to be 50. gr. the Declination belonging to that day by the 3. Pro. is 11. gr. and a halfe North, which being ſubtracted (acAequinoctiall above the Horizon: & that taken from 90. leaves 51. gr. 30. m the depreſſion of the South Pole under the Horizon: or the elevation of the North Pole above the Horizon, for the height of the Aequinoctiall knowne, the Complement thereof is alwayes the Latitude of the place, or height of the Pole: and here note generally that the height of the Pole and Aequinoctiall together, doe alwayes make a Quadrant or 90. gr. therefore the height of one of them being knowne, the height of the other is alſo knowne, and further here note that if the ſun have North Declination, the ſun is ſo much higher then the Aequinoctiall at none that day, by ſo much as his Declination, cometh to, but if the Sun have South Declination, then the Sun is lower then the Aequinoctiall that day at noone, by ſo much as his Declination cometh to, by which you may eaſily gether when to adde, or ſubtract the ſuns Declination to, or from the ſuns Meridianall Altitude to get the height of Aequator, which knowne the Poles height cannot be unknowne.
Marke where the parallel for the day of the Month meeteth with the given houre, and bring the edge of the Index thereto, ſo the degree that the edge of the Index cutteth in the Limbe of the Inſtrument that ſhal be the Suns Azimuth, and the degree that the houre cutteth in the Index, that ſhall be the Suns Altitude required.
So, if upon the tenth of December at nine of the Clocke in the Morning, the Suns Azimuth and Altitude were required, marke firſt where the Tropick of Capricorne (which is the parallel, for that day given) meeteth with the given houre of nine, and bring the Index thereto, ſo the edge of it in the Limbe pointeth out neere 40. gr. and a halfe, & ſo much is the Suns Azimuth, from the South, at nine of the Clocke in the forenoone, the ſaid tenth of December, and the houre line meeting with the Index, ſheweth neere 5. gr. 25. m. ſo much is the ſuns Altitude at that time; now if you move the Index ſoftly along, as the edge of it paſſeth by any houre for any day of the yeare, ſo the edge of the Index in the Limbe of the Inſtrument ſheweth the ſuns Azimuth, and the interIndex ſhall ſhew the Suns Altitude belonging to that houre.
It is uſually noted by ſome, that when the ſhaAſtronomie (and may be eaſily contraSuns paſſages by the Meridians and verticall Circles of the Heavens, for by how much greater the proSuns approchment is unto the Zenith, or verticall point, by ſo much the more ſhall the houre or time be various in one and the ſame Azimuth.
So in the laſt Pro. the Azimuth of the Sun the tenth of December, at nine of the Clocke in the forenoone, was found to be 40. gr. and a halfe, and the Suns diſtance from the Zenith, at that time was neere 84. gr. 35. m, Now admitte the Suns diſtance from the Zenith the tenth of Iune were but 32. gr. 35. m, the Sunne being in the ſame Azimuth, the houre would be halfe an houre paſt 10. For the Index being layed to the houre of 9. in the Tropicke of ♑. (which is the Suns parallel, for the ſaid tenth of December,) and it cutteth the
Suns Motion in the inequalitie of time, and ſo the complement of the former 32. gr 35. m, in the Index, meeteth with the Tropicke of ♋, (which is the Suns parallel for the tenth of Iune) in halfe an houre paſt 10. ſo that it evidentDecember, denoting the houre of the day to be 9. of the Clocke, the ſame ſhadow the tenth of Iune, ſhall repreſent halfe an houre paſt 10. ſo the error ſhall be an houre and a halfe: but if you move the Index unto the houre of 9. belonging to the tenth of Iune, the Index ſhall point you out in the Limbe neere 68. gr. of Azimuth for that houre, which at 9. of the Clocke the tenth of December, was but 40. gr. & an halfe, ſo the difference of Azimuth in one and the ſame houre, ſhall be 27. gr. and a halfe, & the time as be
At any appearance of the Sun by the 27. Pro. take the Suns Altitude, then place the North and South edge of the Inſtrument unto the Meridian line formerly, drawne (if in the foreEast, and West, edge of Instrument to the Meridian-line,
Index, then moove the Index to and fro untill the ſhadow of the prependicular fall by the ſide of the Index, ſo the parallel that meeteth with the degree of the Suns obſerued Altitude, in the edge of the Index, parallel in the Kalender that ſhall ſhew the day of the Month required.
So if, upon a certaine day in the yeare the ſuns Altitude were obſerved and found to be 36. gr. having placed the edge of the Inſtrument to the Meridian line, and rectified the Index, then move the Index, untill the ſhadow of the prependicular fall by the edge of the Inſtrument, let the Inſtrument reſt at this poſition, and account the former 36. gr. upon the Index, which degree meeteth with the houre in the Aequator, and alſo that inKalender, in the tenth of March, & the thirteenth of September, but which of theſe dayes is the day of the Month, the next dayes obſeruation of the Sun upon the ſame houre will helpe you, for if the ſuns Altitude beſound to be greater then the day of the month inquired after it was the tenth of (March, becauſe the ſun from the tenth of December unto the eleventh of Iune, doth every day at one & the ſame houre, aſcend,) but if the Suns Altitude be found to be leſſe then the former dayes obſervation ſpecified was, then the day required, was the thirteenth of September, becauſe that from the eleventh of Iune, unto the tenth of December, the ſuns Altitude every day doth ſenſibly diminiſh at one and the ſame houre.
But here is to be noted that if there be no Meridionall line, then the prependicular over the Center and the Axis of the Index being erected, place downe the backe of the Inſtrument upon an Horizontall plaine, and move the Inſtrument to and fro, untill the ſhadow of the Axis meete with the ſame houre below the Tropicke, amongſt the comInſtrument meeteth with on the face of the Inſtrument, for then the parallel that croſſeth or meeteth with the ſhadow of the prependicular, and the houre, will in the Kalender ſhew the day of the Month required, and ſo then the Meridian of the Inſtrument ſhall be in the Meridian of the world, and every point or degree in the Horizon of the Inſtrument, it ſhall point out his like, or oppoſite degree in the Horizon of the world.
Or otherwiſe it may be done thus, take the Suns Altitude, then immediatly by ſome Watch, clock, or Sun-dyall, learne the houre of the day, and move the Index to and fro, untill the Suns Altitude in the Index, meete with the former houre, ſo the parallel that meeteth therewith, ſhall ſhew the day of the Month in the Kalender required, then having the day of the Month, you have the Quarter of the yeare, for from the tenth of March unto the eleventh of Iune, is the Spring quarter, from the eleventh of Iune, to the thirteenth of September, is the Summer quarter, from the thirteenth of September, to the tenth of December is the Autummuall
quarter, and from the tenth of December, unto the tenth of March, is the Winter Quarter.
Firſt, let the thing that caſteth the ſhadow, or ſomething equall in length unto it, be divided into ten equall parts, and each of thoſe parts ſubſcale,) then meaſcale, and marke how often the ſhadow is longer then the ſcale, and the Decimall part if there be any, ſo have you the proportion betweene the ſhadow, and that which
ſhadow, and then is it reſolved accor- to the conuerſe of the fifteenth Pro.,
Admit ſome one: upon the 12. of February, or on the ninth of October, houlding a ſtaffe prepenA B, or ſuppoſing it to be part of the Coyne of a Houſe, or edge of a Window or ſuch like ſhould caſt a ſhadow, as B, C, which being noted, or drawne and having divided the ſtaffe, or thing as before, and ſhould then meaſure the ſhadow, as B, C, by the ſaid ſtaffe or ſcale, and finde it to be contained therein three times, and 6, parts or 6, decimals, the porportion of the Gnomon, or ſcale, A B, to the ſhadow B C, would be as 1 to 3. and 6/10.
Move therefore the Index to and fro, untill the edge of it meere with 3. and 6/10, in the line of ſhadowes; ſo have you the degree of the Suns Altitude at that inſtant in the Index, viz. 15. gr. and ½ then ſeeke out the parallel for the 12. of February, or the ninth of October (the day given) which is neere the tenth degree from the Aequator South, move the Index, until the former 15. gr.
Index, meeteth with the ſaid parallel for the day, ſo have you the houre belonging to that time, which will be neere 42 m, paſt 8. in the Morning, or 18. m, paſt 3 in the after noone, and the edge of the Index in the Limbe of the Inſtrument ſheweth the ſuns Azimuth alſo at that inEaſt toSouth.
Now for the Meridionall line, this may be done at any time after, if the Azimuth be not forgotten: for if the Center of the Inſtrument be layed downe upon any part of the ſhadow B C, and ſo the Inſtrument to be mooved upon his Center untill the ſaid ſhadow B C, formerly drawne, cut the edge of the Limbe, in the aforeſaid Azimuth of 39. gr. 30. m, then the Meridian of the Inſtrument ſhall be in the Meridian of the world, and if that ſhadow were from a Window, or Building, the poſiInſtrument, ſhall denote the poſition of the Window or Building.
By the 27. Pro. take the height of the Sun, and by the 28. Pro. finde the Suns Azimuth for the
Altitude, ſo the Azimuth thus found ſhall be the declination of the Plaine required: for the decliPlaine, is accounted from the points of Eaſt, Weſt, North or South, in the Horizon, as the Suns Azimuth is: therefore whatſoever Plaine is in the plaine of any virticall Circle, that Plaine is as far from any of the Cardinal points of the Horizon, as the ſun is at that time, & ſo the Sun, being in that virticall Circle, ſhall nePlaine: and therefore looke what the Suns Azimuth is at that inſtant, ſuch ſhall be the Declination of the Plaine re
Thus for the Conſtruction of the aforeſaid 13. uſes which did depend upon the ſuns obſervation, the 48. 49. 50. 9. 13. 14. and 15th. uſes of the Index or Table againſt Page the firſt, ſhould have followInſtrument unto the reIndex or Table, which have reference to night obſervation, upon ſuch Starres which are, or may be placed on the face of the Instrument, betweene the two Tropickes, or under the Tropicke of Cancer, according to there Declinations, and right Aſcentions: which are theſe following.
Much may be ſaid upon the uſes of theſe starres, but for brevitie I onely delivere theſe foure ex
1. First, for any night of the yeare, to find at what houre, and Altitude any of the ſaid starres will be in the Meridian, (that ſo they may be known.)
2. To know at any day, at what houre any of theſe starres riſeth, or ſetteth, with their time of continuance above the Horizon, and in what part of the Hemiſpheare, they may be ſeene with their Azimuth, and Altitude at any houre.
Thirdly, to finde in any night at what part of the Horizon, any of the aforeſaid starres riſeth, or ſetteth, and at what houre, and Altitude they will be due East, or Weſt.
4. Fourthly, upon the ſight or apparance of any of the ſaid Starres, to finde the Azimuth thereof: and the houre of the night.
By the ſixth Pro: finde the Suns rigth Aſcention for the day given, which conuerted into time by allowing for every 15. degrees an houre, and for every degree 4. m, then ſubſtract this right Aſcention of the Sun, from the starres right Aſcention, ſo the remainder or difference of time, ſhall ſhew how many houres the ſtarres ſhall come later to the Meridian then the Sun: but if the ſubtractiAnſwere,
ſo, if upon the ſixth of February, it were required to find at what houre any of the aforeſaid starres will be in the Meridian, or due South, firſt therefore by the ſaid ſixth Pro. I find the ſuns right Aſcention for the day given viz, 330. gr. which containes three nineties or
houres, and ſo the whole 270. gr. makes 18. houres, and the other 60. gr. at 15. gr. to an houre makes 4. houres more all which put together makes 22. houres: ſo the right Aſcention of the Sun the ſixth of February, is neere 330. gr. as before, or 22. houres
For ſeeing that 22. houres the Sunnes right Aſcention, is greater then the right Aſcention of any of the starres afore ſpecified, ſubſtract this 22, houres from 24. houres, reſt 2. houres, which added to the right Aſcention of each Starre before delivered, you have the houre of the Stars coming to the Meridian: hence you may gather which of thoſe starres, are out of obſervation for that time, viz. Alae Pegaſi, Pri. ♈, and Aquila, which come to the Meridian in the day time: but if the day given had been the 26th of Iuly, the right Aſcention of the Sunne, that day is neere 135. gr. or 9. houres.
For the right Aſcention of the Sun being but 9. houres take it from the right aſcention of Cor. hidra which is 9. houres 10. m, reſt 10. m, which ſhewCor. Hydra comes to the Meridian 10. m, later then the Sun that day, that is, 10. m, after 12 and ſo the reſt, whoſe rightaſcention is greater then the Suns. But for theſe ſtarres, whoſe right Aſcention is leſſe then the ſaid 9. houres, ſubtract this 9 houre from 24. houres, reſt 15. houre (or rahoures) this adde unto the right aſcention of any of the aforeſaid ſtarres, as ſuppoſe Canis Minor makes 22. houres 20. m, which ſheweth that Canis minor, wil come to the Meridian. 22. houres 20, m, later that day then the ſun: therefore this, 22. houres and 20. m, being conſidered according to an hourCanis Minor will come to the Meridian at 10. of the clocke and 20. m: of the next day (the right aſcention of the Internall of
houres unto the right aſcention of theſe Starres, whoſe right aſcentions are leſſer then the Suns, ſo have you the Meridionall houre re
Hence may be gathered that Alae Pegaſi, Pri. ♈ and Aquila, are onely for obſervation that night, the other ſtarres are out of obſervation, and will come to the Meridian, in the day time.
Laſtly, to finde the Meridionall Altitude of any of theſe starres, lay the edge of the Index unto the houre line, of 12, ſo the parallel of the starres declination that croſſeth the edge of the Index, ſhall there ſhew you in the Index, the Meridionall Altitude of the starre required.
By the laſt direction finde the houre of the stars being in the Meridian, then marke what houre the parallel of the declination of any starre interHorizon or Kalender, ſo have you the houre or time of the starres riſing or ſetting, and the number of houres, from that point of the stars riſing in the Horizon, unto the Meridian being doubled, gives the countinuance of the starres aHorizon, required.
So if upon the 6th. of February, it were dehoure Oculus ♉. would aſcend, & how long it would continue above the Horizon.
By the laſt propoſition, get the houre of the starres being in the Meridian, which is at 6. of the Clocke and 15. minuts at night, and marke the Number of houres betweene the Meridian, and that point where the parallel of Oculus ♉, meeteth with the Kalender, which is 7. houres 24, minuts, this doubled makes 14. houres 48. m, and ſo long will Oculus ♉, be above the Horizon.
But if from the ſaid 7, houres and 24. m, the ſaid 6, houres 15. m, be taken, there will reſt 1. houre 9. m, and ſo much before 12. of the clocke at noone, doth Oculus ♉ riſe, that is 51. m, after 10, of the Clocke, and ſo conſequently if the ſaid 7. houres and 24. m, be added unto the houre of the starres being in the Meridian, viz. 6. of the Clocke and 15. m, as before, the ſaid starre will ſet at 39. m, paſt 1, in the Morning.
Laſtly, if at any houre betweene the riſing of the starre, and the ſetting thereof, it be required at what Poſition and Altitude the ſtarre is in. It is thus done.
Account to the given houre, from the houre of the ſtarre riſing, ſetting, or being in the Meridian,
(in the parallel of the ſtarres declination) and lay the Index thereto, ſo the edge of it in the Limbe of the Inſtrument, ſhall ſhew the ſtarres Azimuth or Poſition, and where the parallel of the ſtarres Declination croſſeth the edge of the Index, that ſhall be the ſtarres Altitude, at that houre.
So if on the ſaid 6th. of February, at 11. of the Clocke at night, it were required in what Poſition,
Azimuth Oculus ♉, was in, and alſo how high above the Horizon: I make, or ſuppoſe the houre of 12. to be the aforeſaid 6. of the Clocke and 15. m, (for at that houre as before Oculus ♉ was in the Mridian) and from thence in the ſtars parallel of Declination, I account untill I come unto 11. of the Clocke, viz. that is 4. houres, and 45. m, from 12. and lay the Index thereto, ſo the edge of the Index in the Limbe, pointeth out 4. gr: 24. m, and ſo farre Oculus ♉, is diſtant from the Weſt at 11. of the Clocke at night, and the parallel of the starres Declination meeteth with the Index in 24. gr, neere, which is the ſtarres Altitude, at that houre required.
For the firſt, Marke where the parallel of the ſtarres declination croſſeth the Horizon, or Kalender, Lay the edge of the Index hereto, ſo the
Index, and the point of Eaſt or Weſt, upon the limbe of the Inſtrument, ſheweth the diſtance of the ſtarres riſing from the East or Weſt.
So if it were required in what part of the Horizon Oculus ♉ riſeth, marke where the parallel of the ſtars Declination croſſeth the Horizon, and lay the edge of the Index thereto, ſo it cutteth the Limbe of the Inſtrument from the Eaſt neere 26. gr. and ſo farre Oculus ♉, riſeth from the East towards the North.
For the ſecond to finde the time of a starres coming Eaſt, or weſt.
By the 40th. Pro. conſider at what houre the star
is in the Meridian, then lay the edge of the Index to the point of Eaſt and West, and account in the parallel of the ſtars Declination the number of houres betweene the edge of the Index, and the houre of 12. which being taken from the houre of the stars being in the Meridian, gives the houre of the ſtars coming East, but added unto the houre of the ſtars being in the Meridian, ſhewes the houre of the ſtars being West.
So if it were demaunded at what houre,
upon the 6th. of February, Cor ♌, would be due Eaſt or Weſt, and what Altitude the ſtarre ſhould then have. Firſt, lay the edge of the Index, to the point of Eaſt and West, & whereſoever the parallel of the ſtarres declination croſſeth the edge of the Index that ſhall be the ſtarres Altitude, viz. neere 17. gr 45. m, then account the number of houres
Declination betweene the edge of the Index, and the houre of 12. which is neere 5. houres and 12. m, which taken from the houre of the ſtarres being in the Meridian,) which by the 40th. Pro. was at 11. of the clocke & 48. m, at night) reſts 6 houres, and 36. m: but if the ſaid 5. houres and 12. m, be added unto the ſaid 11. houres and 48. m, it makes 17. houres, from which 12. being taken leaves 5, houres. So upon the 6th. of February, Cor ♌ ſhall be due Eaſt, at 36. m, paſt 6, at night, and due Weſt, at 5, of the Clocke in the Morning, and the Stars Altitude, being either Eaſt or West, is neere 17. gr. 45. m, as was re
By the 40th. Pro. for the day given finde the
houre of the starres coming to the Merid
then by the 27. Pro. take the starres heighIndex, theIndex untill the degree of the ſtarre
Index, meete with the parallDeclination, ſo the edge of the Azimuth, and the Meridian that meeteth with the degree of the Altitude, in the Index ſhall ſhew you the houre that the ſtarre wants to be in the Meridian, or is paſt the Meridian, which added, or ſubtracted from the houre of the ſtarres being in the Meridian, gives the houre of the night required.
So if the day were the 26th. of Iuly,
and if Aquila, ſhould be obſerved to be on the Weſt of the Meridian, 29. gr. 20. m, high above the Horizon, this I ſeeke out upon the Index, and move the Index to and fro untill the ſaid, 29. gr. 20. m, meete with the parallel of Declination, of Aquila, ſo the edge of the Index, in the Limbe doth point out the ſtarres Azimuth from the South, viz. 63. gr. 12. m, and the Meridian that meeteth with the aforeſaid degree of Altitude, is the time of the ſtarres diſtance from the Meridian, viz. neere 3. houres and 28. m, this added unto the houre of Aquilas being in the Meridian, which by the 40th. Pro. was at 10. of the Clocke & 32. m, at night, makes 14. houres, or 2 of the Clocke, in the Morning, ſo if Aquila were obſerved the 26th. of Iuly, to be 29. gr. 20. m, high to the Weſt of the Meridian, then the Poſition or Azimuth of that ſtarre from the Meridian, was 63. gr. 12. m, and the houre at that inſtant, was at 2. of the Clocke in the Morning.
Thus touching the reſolution of the aforeſaid 44. 45. 46. and 47th. Pro. of the aforeſaid Index or Table, which did belong to Aſtronomicall obth. uſes of the Index or Table, which are onely proper to Geometricall Practices, viz.
to ſhew
How to meaſure the Quantitie of an Angle, or to take the diſtance of two Starres.
How to meaſure diſtances and bredthes.
How to take the Circuit of a figure, or the ſurueigh of a Place.
The inclination of a Plaine, or to Place a Plaine Horizontall.
Whether an Altitude be in the Point of libration, or above, or below the levell of the eye, and how much.
How much the hight of an Altitude is above the eye, which is acceſſable, or in acceſſable.
How to meaſure any Part of an Altitude, which is not approachable.
Let the Iſtrument be placed upon ſome Rest,
which may be ſo accommodated that the Inſtrument, may be elevated, depreſſed, or be placed Horizontall, as occaſion requires, then erect the ſights of the Index, & place the edge of the Index upon the houre line of 12. the Index ſo placed looke through the ſights thereof and moving the Inſtrument upon his Reſt to and fro, untill you ſee the marke or Starre, that makes the Angle or diſtance required. Then ſcrew faſt the Inſtrument to the ſocket, and move about the Index, untill through the ſights thereof you ſee the other marke or ſtar, ſo the number of degrees betweene the edge of the Index and the houre of 12. in the Limbe of the Inſtrument ſhall be the meaſure of ye Angle, or diſtance of the two Stars ſought for.
Let E and D be two markes or Starres,
and let the Angle E A D, or diſtance E D, be required. The Inſtrument, A B M, being placed upon his Reſt G H I K, obſerue one of the markes or ſtarres as D, through the ſights (admit A B,) ſo the viſuall line ſhall be A B D, then having made faſt the
Inſtrument, move the Index A B, untill through the ſights of it you ſee the other marke or ſtarre, E, which ſuppoſe to be in the viſuall line, A C E, ſo the Arke of the Limbe of the Inſtrument B C, ſhall be the diſtance betweene the two Stars, ED, or the meaſure of the Angle E A D,
required. Now Infinit are the uſes of knowing the Quantitie of Angles, in the Copious and vaſt Body of Mathematicall Practices, therefore from the multitude of Examples that might be raiſed upon them, or extracted from them, I will onely inſtance for brevitie, upon theſe two plaine, and ordinary ones following.
Let G F L H,
repreſent part of the Perimeter of a Fort, and let it be required, that ſtanding at ſome convenient place without Musket ſhot, as admit at Y, the diſtance betweene the points of the Bulwarke viz. F & L, as alſo the meaſure of the face of either of the Bulwarks viz. F E, or M L, with the length of the Cortaine D N, and all the diſtances from Y, viz. Y F, Y E, Y M, and Y L, were required.
Or ſuppoſe O P Q R, were 4 places, whoſe ſeO to P, then from P to Q, and from Q to R, & alY viz. Y O, Y P, Y Q and Y R, were demanded. The Conſtruction and reſolution upon either of theſe is alike, therefore we will inſtance upon the latter.
Place the Inſtrument upon his Reſt, at Y, and the edge of the Index, upon the houre of 12. then looſights of the Index, upon ſome marke taken at pleaſure in the field, which, admit to be S, then obſerve the firſt marke R, ſo have you the Angle S Y R, which ſuppoſe 52. gr. 30. m, then looke to Q, ſo have you the Angle SYQ, which let be 63. gr. 15. m. then looke to P, ſo haSYP, admit to be 74. gr. 15. m, and laſtly looke to O, ſo have you the Angle
S Y O, 123. gr. 15. m: for theſe Angles are taken with great facillitie, when once the Inſtrument is rectified as in the firſt direction is ſpecified, for you neede not but move the Index, Circular from obiect to obiect, ſo the Arkes of the Limbe of the Inſtrument as before, from the houre of 12, unto the edge of the Index, ſhall ſhew the meaſure of the ſeuerall Angles obſerved.
Thus at Y, place up a Marke, and in the viſuall line, Y S, and meaſure a certaine diſtance at pleaſure, as admit to S, and ſuppoſe it were found to be 900. foote (or 300. yardes) then placing the Inſtrument at S, upon his Reſt, and laying the edge of the Index to the houre of 12. I move the Inſtrument about, untill through the ſights of the Index I may ſee the marke which was ſet up at my, laſt station, then make faſt the Inſtrument, and obſerve O, ſo have you the Angle Y S O, which ſuppoſe to be 26. gr. 50. m, then looke to P, ſo have you the Angle Y S P, which let be 55. gr. 50. m, then looke to Q, ſo have you the Angle Y S Q, which admit to be 60. gr. 15. m, laſtly obſerve R, ſo have you the Angle Y S R, 87. gr. In like maner may you obſerve the Angles at the Fort from the ſtations Y, and S, formerly ſpecified all which obſervations may be placed downe in Tables, as here under appeares, which may be called the Tables of obſerved Angles.
Now touching the reſolution of the point, there is a triple way of operation, viz. either Arithmeticall, Inſtrumentall, or Geometricall, each of which being ſufficiently facill, to ſuch which are verſed in the documents of Mathematicall Practices, but the later becauſe it is more vulgar, and eaſieſt to be apprehended, I will inſtance here upon: which is that part of Geometrie, commonPretraction, a thing ſo common that alGeometrie, can performe it according to the ordinary way they are inſtructed in. But to facilitate that kind of practice, I aduiſe ſuch as affect this kinde of Practice to uſe the Protracter, which I uſe, which is a plaine then ſector, having a ſmal hole at the Center, whoſe two legges from the Center are made with a ſharpe edge, but ſo that they lie flat upon the Paper, & each of them to be divided into 100 or 1000 diuiſions with a Quadrantall Arke or more divided and faſtned alſo at the end of one of the leggs, but ſo that the Quadrantall,
Arke have alſo an edge to lie flat upon the Paper, and to ſlide in at the other leg, ſo ſhall it be acInſtrument to finde the Quantitie of Angles, in a Plot, or to proAngles, for ſervice, as followeth.
So to ſearch out the diſtance of O P Q R,
the one from the other, or all the diſtances from Y as was required: by the helpe of the former Angles of obſervations and Protracter, it may be done thus.
Let ABCD,
be a plaine to be raiſed as Fortifiers have it, or a field to be ploted as ſurveyers account it, or a Figure whoſe Perimeter is required, as Geometritiās treate of it. The Plaine Table,
may be held beſt for this ſervice, as ſuch would have it, whoſe learning is altogether verſed therein. But any Instrument ſhall be able to doe this Service, that can but accuratly take or meaſure any Angle, (not
Angles with greateſt ConveAngles of conAngles to fabricate the whole worke: for here eſpecially is to be noted, that the more Angles that are obſerued in any practiſe, by way of Circumſcribing a Field, or Campaigne, the greater, and more evident ſhall the error be in the Concluſion.
So in the Figure A B C D,
there is eleven Angles and as many ſides, now if at every Angle, an obſervation ſhould be made, it would be more. ſubiect to error (as before) then if leſſe Angles were obſerved, therfore in this Diagramme fewer Angles of obſervation may be fully ſufficient to raiſe that Plaine, Take the Plot of the Field, or give the Perimeter of that Figure, Therefore ſupAngles of Conſequence to be, A, B, C, D, Q, the worke may be then thus.
Place the Inſtrument, upon his reſt at A, and obſerue the Angle Q AB, which ſuppoſe 32. gr. 10 m, then meaſure Q A, with a Decimall Chaine (or ſuch like,) which ſuppoſe to be 5. Chaines, note this in a peece of paper, then take the Inſtrument up, and meaſure the line, A B, but firſt onely A E, which ſuppoſe to be 11. Chaines and 60 Linkes, which is written down thus. 11. 60. Then meaſure the diſtance from E to the Angle F, which admit to be 2. 20, Laſtly, go one with the meaſure A B, which ſuppoſe to be 15. 00, the Angle of obſeruation, and meaſures thus taken may be noted downe one againſt another, as in the Table following, then place the Inſtrument
upon his reſt at B, and obſerue the Angle: ABC, which note downe alſo and meaſure the diſtance, B G, and G H, and then going on with G B, to G I, and marke and meaſure B I, and then meaſure alſo I K, and ſo goe on with B C, which meaſures are all placed down as appeares in the Table of Angles
Q, and ſo all the Angles and meaſures will be according to the Table here under ſpecified.
But it had beene fully ſufficient (by helpe of this Protractor) to have plotted the aforeſaid Plaine, by knowing the former Meaſures, and two Angles of obſeruation only.
For the first, Set the Eaſt and West, edge of the
Instrument unto the Plaine, then if the edge of the Index in the Limbe of the Instrument, cut the point of Eaſt or West, the Plaine is verticall, and doth not Incline, but if the Index fall from the points, looke how many degrees it is from the points of East or West in the Limbe of the Inſtrument ſo much is that Inclination of the Plaine.
For the Second, to Elevate a Plaine, to an Angle aſſigned.
This is onely the ſame with the former, but may be applyed to ſeverall uſes, as to trie the mount, or to mount a Peece of Ordinance at any Randon: or to place Burning glaſſes (or others) at ſeverall Angles, to receive each others Reflexon, and that the point of concurſe, or inflammation in ſuch Glaſſes may be in the Radius or beame of the Sunne, or that the point of inflamation, the reImage, or the extenſive Elumination may be proiected to a point aſſigned.
For the Third, to rectifie a Plaine Horizontall.
Place the North and South edge of the Instrument, unto the under face of the Plaine, and then marke if the edge of the Index, cut the points of Eaſt or West in the Limbe of the Instrument, for then the Plaine is Horizontall, but if it ſwarne from that point, then it is not Horizontal, but the Plaine is to be raiſed, or depreſſed, untill by ſeverall tryalls in ſundry parts of the Plaine, you ſee the edge of the Index fall upon the points of Eaſt or West, for then ſhall it be truly Horizontall: OtherPlaine Horizontall, by operating upon the upper face of it, if you ſet a
Cube upon the plaine, and then placing the East and West edge of the Instrument unto the ſide of the Cube, for then the obſeruation will be as the former, and therefore, accommodated & conclu
Let C B and X, be three ſeverall obiects, and let their ſeverall ſituations be required.
Firſt, let the Instrument hang upon a reſt perIndex may be verticall, and play eaſily by the ſide of the Inſtrument, then looking through the ſights of it, lift the Instrument up and downe, untill you ſee your marke, which ſuppoſe firſt C, and admit the Index ſhould cut 5. in the line of ſhadowes, which ſheweth that C, is higher then the eye by the 5th. part of the diſtance of the baſſis of the obiect, from the eye, ſuppoſed at A.
Secondly, if through the ſights of the Instrument you ſee the ſecond obiect, B and the Index falling upon no part in the line of ſhadowes, then it ſhewB, is levell with the eye, for if in any obſervation the Index fall betweene the beginning of the line of ſhadowes, (which is neere the begining of December) and the ſight next
Index fall beyond the beginning of the line of ſhadowes, then the obiect is lower then the eye.
Thirdly, if you obſerve the obiect X, (the eye being at A,) then if in obſerving the Marke X, through the ſights of the Instrument, the Index ſhall fall beyond the beginning of the line of ſhadowes, that is from the Kalender number the degrees, in the Limbe from the edge of the Index unto that point, and account the ſame backLimbe that is oppoſit to the beginning of the line of ſhadowes, and lay the edge of the Index unto it, then ſuppoſe the Index, in the line of ſhadowes interſect 8. which ſheweth that the point X is lower then the levell of the eye, by the eight part of the diſtance from you to the marke. Now if the diſtance ſhould be 100. foote, then the point X, ſhall be below the Horizontall line, or line of levell A B, 12. foote and ½ which is the ⅛ part of 100. the diſtance be
Let, B C be an Altitude and the eye at,
A diBaſis of B, 100. foote.
If through the ſights of the Inſtrument the ſummet of the Altitude B C, viz. C, be ſeene, and the Index falling upon 5, in the line of ſhadowes,
it argueth the Altitude B C, to be the 5th. part of the diſtance, viz. of A. B. which is 20. foot. Or let the Altitude of G, be ſought out, whoſe Baſis cannot be ſeene Admit, the firſt ſtation be made at A and ſeing the ſummet of the Altitude G, the Index ſhould cut 3 in the line of ſhadowes, it ArguBaſis of the Altitude, is triple to the Altitude, then if I ſhould go neerer to the Altitude, viz. at D, and ſhould obſerue the ſummet or top of the Altitude G, and that the Index ſhould fall upon 1. in the line of ſhadowes, then it ſheweth that the diſtance from D, to the Baſis of the Altitude is equall to the Altitude. Now ſuppoſe that betweene D and A were 80. foote it ſhould ſeeme that the Altitude obſerued ſhould be 40. foote, for if at D. the diſtance to the Altitude be equall to the Altitude, & the diſtance from A, to the Altitude, be Triple to the Altitude, then the diſtance from D to the Altitude is the ⅓ of the diſtance AR, & ſo AD, ſhall be double to DR, therefore halfe the diAD, viz. 40. foot is the Altitude required.
Let G H, a part of an Altitude be required to be meaſured.
Firſt, ſearch out the height G R, as before 40. foote, then admit ſtanding at A and looking to H, through the ſights, the Index ſhould cut 4, which ſhewes the diſtance from A, to be QuaAltitude of H R, and if comming neerer the Altitude 80. foote, viz. at D, I ſhould obſerve H againe, through the ſights of the Inſtrument, and finde the Index to cut 1, and ⅓, in the line of the ſhadowes, then the diſtance from D, to the Altitude H R, viz. D R, ſhould containe the Altitude H R, once, and a third part of the Altitude, now ſeeing that D R, is 1 and ⅓, therefore H R, ſhall be 1, but the obſervation at A ſhewed the diſtance from A, to the Altitude H R, to be Quadruple, and ſeeing that D R, is 1, and 1 part of 3, therefore A D, muſt be 2, and 2 parts of 3, which makes A R, the whole diſtance to be 4, or Quadruple to H R, but if A B, 2, and 2 parts be 80. foote then D R, being 1, and 1, part ſhall be 40 foote, and if D R, 1, and ⅓, be 40. foote, then H R, (which was 1 ſhould be but 30. foote, & ſo conH R,) taken from G R, there ſhall reG H, 10. foote, the meaſure of the part of the Altitude required. In like manner might we applie the Inſtrument to the meaſuring of Bredths and diſtances: but that which is delivered may ſerue for the preſent, and as fully ſufficient for the Ingenious.
I might have Annexed unto this Tractat the demonſtraProiection, which might have ſatisfied thoſe which are more learned, but to ſhew them it would be impertinent, ſeeing the thing lies ſo obuious: for others, it would not be reſpected or regarded, ſeeing the making, and practicall uſe of the Inſtrument, principally & Totally they looke after, which I have plentifully delivered. Now by way of Compariſon it is ſaid in the deſcription of Maſter Gunters Quadrant, that if a Quadrant were made (as he there relateth) unto a foote ſemidiameter, it ſhould ſhew the Azimuth unto a degree, & the houre unto a minute. It is moſt probable that if this Horlzontal Quadrant have the ſame ſemidiameter, it ſhal ſhew the houre unto half a minut, and the Azimuth unto 3 m. And if in this Tractat I have beene too obſcure (which I have avoyded as much as poſReader to excuſe me. I confeſſe I might more Methodically have digeſted it, and more aHorizontall Quadrant, then I have done: I have made way for him, and vnuailed the ſubject, to helpe his ſight.
Motus Perpetuus Solis Diſtinguit Tempora VIGILATE QVAERE VT IVENIa