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1

THE DESCRIPTION AND VSE OF THE
2TABLES OF SINES.

3
4    Because there is no proportion, comparison,
5or likenes betwixt a right line and a
6crooked, the auntient Philosophers, as
7Ptolomey and divers other, were much
8troubled in seeking to know the measures
9of a Circle or of any portion thereof by his
10Diameter, and by knowing the Diameter
11to finde out the length of any Chorde in a
12circle, which is alwaies lesser then the Diameter it selfe, and finding
13that the more parts whereinto the Diameter was diuided,
14the nearer they approched to the truth: Some of them therefore,
15as Ptolomey , diuided the Diameter of a circle into a 120. parts,
16and the Semidiameter into 60. parts, and euery such part into
176'0. and euery minute in 60. seconds &c. And in like manner did
18Arzahel , an auntient Arabian, who diuided the Diameter into
19300. partes and the Semidiameter into 150. and euery of those
20parts into 6'0. and so forth as before, according to which computation
21they made their Tables: but because the working by those
22Tables was very tedious and troublesome, by reason that it was
23needfull continually to vse the art of numbring by Astronomicall
24fractions: therefore Georgius Purbachius , and Regio Montanus
25his Scholer to auoide that trouble of calculating by Astronomicall
26fractions, diuided the Diameter of a Circle into a farre
27greater number of parts, and made such tables as are vsed at this
28present, the description and vse whereof both hereafter follow,
29first of those that are set downe by Monte Regio in Folio, and
30then of those that were lately Corrected and made perfect by Clauius
31the Jesuite which are Printed in quarto.

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32    And because that the way to find out the proportion which any
33chord hath to the whole Diameter, was very hard, therefore the
34said Purbachius and Monte Regio hauing direction from certaine
35propositions of Euclyd as from the 47. proposition of his
36first booke, and from the third proposition of his third booke, and
37also from the 15. proposition of his fift booke, they made choise of
38the halfe chord and Semidiameter of the Circle, calling the halfe
39chord, Sinum rectum , and the Semediameter Sinum totum . And
40because that the proportion of any circumference to his diameter
41neuer changeth, how great or how little so euer the Circle be: after
42that they had calculated for one Circle, they made such tables
43as might serue for all Circles, and though these Tables of sines
44doe suffice to worke thereby all manner of conclusions, as well of
45Astronomie, as of Geometrie, yet for more ease, our moderne
46Geometricians haue of late inuented two other right lines belonging
47to a Circle called lines Tangent, and lines Secant, and
48haue made like tables for them that were made for sines, and both
49tables, that is to say, as well of the sines, as of the lines Tangent
50and Secant, haue one selfe manner of working thereby, as shall
51plainely appeare hereafter when wee come to describe the same.
52But first we will beginne with the tables of sines, and plainely define
53euery terme or vocable of Art, belonging thereunto: The
54termes are these here following: An arch, a Chord , Sinus rectus
55Sinus versus, Quadrans, Complementum , and sinus
56Complementi .
57
58

THE DEFINITIONS OF THE FORESAID TEARMES.

59
60    An Arch is any part or portion of the circumference of a cirle,
61which in this practise doth not commonly extend beyond
62180. degrees which is one halfe of the circumference of any Circle
63how great or small so euer it be, for euery Circle containeth 360.
64degrees.
65    A Chorde is a right line drawne from one end of the Arch to
66the other end thereof, and note that all chordes are alwaies lesser
67then the Diameter it selfe, for that is the greatest Chorde in anye
68Circle.
69    Sinus rectus is the one halfe of a Chord or string of any Arke

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70which is double to the Arke that is giuen or supposed, and falleth
71with right Angles vppon that Semidiameter which diuideth the
72double Arke into two equall parts.
73    Sinus versus that is to say turned the contrary way, is a right
74line, and that part of the Semidiameter, which is intercepted betwixt
75the beginning of the giuen Arke and the right Sine of the
76same Arke, and this is also called in Latine Sagitta , in English a
77Shaft or Arrowe, for the Demonstratiue figure thereof hereafter
78following, is not vnlike to the string of a bowe ready bent hauing
79a Shaft in the midst thereof.
80    Quadrans is the fourth part of a Circle containing 90.
81degrees.
82    Complementum arcus , is that portion of the Circle, which
83sheweth how much the giuen Arke is lesser then the Quadrant, if
84the giuen Arke doe containe fewer degrees then the Quadrant, but
85if it containe more degrees then the Quadrant, then the difference
86betwixt the quarter of the Circle and the said arch, is the complement
87of the said giuen Arke.
88    Sinus complementi , is the right Sine of that Arch which is
89the complement of the giuen Arke.
90    Sinus totus , is the Semidiameter of the Circle, and is the greatest
91Sine that may be in the Quadrant of a Circle, which according
92to the first tables of Monte Regio containeth 6/000/000.
93and according to the last tables 10/000/000. parts, for the more
94parts that the totall Sine hath, the more true and exact shall your
95worke bee, notwithstanding sometime it shall suffice to attribute
96unto the totall Sine but 60/000. parts, which numbers Appian
97obserueth in teaching the way to finde out the distance of two places
98differing both in Longitude and Latitude by the Tables of
99Sines, and some doe make the totall Sine to containe 100/000.
100partes, as Wittikindus in his treatise of Dials, and diuers other
101doe the like. Also Clauius himselfe saith that in the tables set
102downe by him in quarto, you may sometime make the totall Sine
103to be but 100/000, so as you cut off the two last figures on the
104right hand in euery Sine, but you shall better understand euerye
105thing here aboue mentioned, by the figure Demonstratiue heere
106following.

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107

THE FIGURE DEMONSTRATIUE.

FIGURE OMITTED
108
109    In this figure
110you see first a whole
111Circle drawne upon
112the Centre E.
113and marked with
114the letters A.B.C
115D. which Circle
116by two crosse Diameters
117marked
118with the letters A.
119C. and B.D. & passing
120both through
121the Centre E. is diuided
122into fower
123Quadrantes or
124quarters, the upper
125Quadrante
126whereof on the left hand is marked with the letters A.B.E. in
127which Quadrant, the right perpendicular line marked with the
128letters F.H. betokeneth the right Sine of the giuen Arke A.F.
129which right Sine is the one halfe of the chord or string F.G. and
130the giuen Arke A.F. is the one halfe of the double Arke or bowe
131G.A.F. and A.H. is the Shaft called in Latine Sinus versus :
132Againe the letters F.B. doe shew the complement which together
133with the giuen Arke A.F. doe make the whole Quadrant A.F.B.
134which is diuided into 9. spaces, euery space containing 10. degrees
135whereby you may plainely perceiue that in this demonstration,
136the giuen Arke A.F. is 50. degrees, and the complement F.B. is
13740. degrees, both which being added together doe make up the
138whole Quadrant of 90. degrees, marked with the letters A.F.B.
139Now Sinus complementi is the crosse line marked with the letters
140F.K. the totall Sine which is the whole Semidiameter and
141greatest right Sine, is marked with the letters B.E. But because
142it is not enough to know the signification of the things aboue specified
143to vse the foresaid Tables when neede is, vnlesse you know

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144also how to find out those things in the said tables, I thinke it good
145therefore to shew you the order of the said tables by describing the
146same as followeth.
147    You haue then to vnderstand that the tables of Monte Regio
148printed in Folio, are contained in 18. Pages, and euery Page
149containeth eleauen partitions, called collums, whereof the first on
150the left hand containeth 60. minutes, which are to be counted from
151head to foote, as they stand in order one right under another in seuerall
152places, proceeding from 1. to 60. The second collum containeth
153Sines. The third containeth onely a portion or part of
154one second, and from thence foorth proceeding towardes the right
155hand all the other collums doe containe in like manner Sines and
156the portion of one second. And right ouer the head of euery Sine
157(the first collum of Sines onely excepted, hauing nothing but a
158Cypher ouer his head) are set downe the degrees of the whole
159Quadrant called arches, in such order as from the first Page to
160the last, there are in all 89. degrees, or arches, as by perusing the
161said tables you may plainely see. Now to find out in these tables
162the things aboue mentioned, you must doe as followeth.
163    First to find out the right Sine of any giuen Arke, you must
164seeke out the number of the said Arke in the front of the tables, and
165if the giuen Arke hath no minutes ioyned thereunto, then the first
166number of Sines right under the said Arke, is the right Sine
167thereof. But if it hath any minutes ioyned thereunto, then you
168must seeke out in that Page, where you found the giuen Arke, the
169number of the minutes in the first collum of the said Page, on the
170left hand, and right against those minutes on the right hand, in
171the square Angle right under the said arch, you shall find the right
172Sine. As for example, you would find out the right Sine of a giuen
173Arke containing 8. degrees, and 2'0. heere hauing found out
174in the front of the second Page the figure of 8. standing right ouer
175the eight collum seeke in the first collum on the left hand of the said
176Page, for 20. minutes, and right against the 20. minutes you shal
177find on the right hand in the common Angle or square 869593.
178which is the right Sine of the foresaid giuen Arke, so as you make
1796/000/000. to be the totall Sine: but if you make 60/000. the totall
180Sine, then you must alwaies reiect the two last figures standing
181on the right hand of the said right sine, & the rest of the figures
182shall be the right Sine.

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183    Now to find out the complement, there is nothing to be done,
184but onely to subtract the giuen Arke out of the whole Quadrant
185which is 90. degrees, and the remainder shall be the complement:
186as in the former example by subtracting 8. degrees, 2'0. out of 90
187degrees, you shal find that there remaineth 81. degrees, 4'0. which
188is the complement of that arch. Againe to find out the Sine of the
189complement you must doe thus, seeke the complement in the front
190of the tables of Sines, euen as you doe to find out any giuen arke:
191as in the former example, the complement being 81. degrees 4'0.
192you must seeke 81. in the front of the 17. Page of the first tables,
193which being found, seeke out also the 4'0. in the first collum of the
194said Page on the left hand, and right against those 4'0. in the common
195Angle right under the Arke 81. you shall finde 5/936/649.
196which number is the right Sine of the foresaid complement, so as
197you make 6/000/000. to be the totall Sine, for if 60/000. be the
198totall Sine, then you must reiect (as I said before) the two last figures
199on the right hand, and the number remaining shall bee the
200right Sine of the foresaid complement, and therefore in working
201by these tables, you must alwaies remember what number you
202make the totall Sine to be.
203    Sinus versus commeth seldome in vse, notwithstanding if you
204would know how to find it out, you neede to do no more but subtract
205Sinum complementi of the giuen Arke, out of the totall
206Sine, and the remainder shall bee Sinus versus , as in the former
207example your Sinus complementi was 5/936/649. which being
208subtracted out of the totall Sine 6/000/000. there remaineth
20963/351. and that number is Sinus versus : for if you adde this
210remainder to the number which you subtracted, it will make up
211the totall Sine 6/000/000. But there is one thing more necessarie
212to be knowne then this, because it commeth oftner in vse, and
213that is upon some diuision made how to find out the Arke of any
214quotient, which is to be done thus: Enter with the quotient into
215the body of the tables, and leaue not seeking amongst the squares
216of the Sines, vntill you haue found out the iust number of the quotient
217(if it be there) if not, you must take the number of that Sine
218which is in value most nigh vnto it, whether it bee a little more or
219lesse, it maketh no matter, and hauing found that number, looke in
220the front of that collum, and you shall find the Arke of your quotient,

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221standing right ouer the head of that collum, and also the mynutes
222thereof in the first collum of the said Page on the left hand.
223As for example, hauing diuided one number by another, I finde
224the quotient to be 469/012. whereof I would know the arch, now
225in seeking this quotient amongst the Sines, I cannot finde that
226iust number, but I find in the first Page, and in the tenth collum
227469/015. which is the nighest number vnto it that I can see. In
228the front of which collum I find the Arke to be 4. degrees, and directly
229against that Sine on the left hand, I find 2'9. belonging to
230that arch, whereof that quotient is the Sinus , so as I gather hereof
231that the arch of the foresaid quotient is 4. degrees, 2'9. But you
232haue to note by the way that the number of your quotient must neuer
233be much lesse then 1745. for otherwise it is not to bee found in
234these tables, unlesse you make the totall Sine to bee but 60/000.
235for then by reiecting the last two figures on the right hand, as I
236haue said before, the first right Sine of these tables shal be no more
237but 17. and by that account a very small quotient may be found in
238these tables. And whatsoeuer hath beene said here touching the order
239that is to be obserued in the first tables of Monte Regio , whose
240totall Sine is 6/000/000. the like in all points is to be obserued
241in the last tables, whose totall Sine is 10/000/000. Thus much
242touching the order of the foresaid tables of Monte Regio Printed
243in Folio: but for as much as those tables be not altogether truely
244Printed, and for that they haue beene lately corrected, and made
245more perfect by Clauius , who doth set downe the saide Tables in
246quarto and not in folio, whereby they are the more portable, and
247the more commodious, as well for that they are more truely Printed,
248as also for that the complement of euery Arke is set downe in
249euery Page at the foote of euery collum, so as you need to spend no
250time in subtracting the Arke from 90. I thinke it good therefore
251to make a briefe description of those Tables, and the rather for
252that I haue requested the Printer to print the like here in quarto,
253and I doe worke all such conclusions as hereafter follow, by the
254said tables, the totall Sine whereof is 10/000/000. according to
255the last tables of Monte Regio . But for so much as some may
256haue already the tables of Monte Regio Printed in Folio, not
257knowing perhaps the vse thereof, I will set downe two conclusions
258to bee wrought by those tables, and all the rest of the conclusions

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259are to be wrought by these tables which I haue here caused to
260be Printed in quarto like to those of Clauius : and though the two
261conclusions next following, which are to shew the vse of the foresaid
262tables, may be wrought by the tables of Sines in what forme
263so euer they be truely Printed in Folio, or in quarto, yet because I
264had appointed them to bee done by the Tables of Monte Regio ,
265Printed in folio before that euer I saw Clauius his booke, I mind
266not now to alter them but to let them stand still as they are.

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267

OF THE HORIZON BOTH RIGHT AND OBLIQUE, MAKING THEREBY
268THREE KINDS OF SPHEARES, THAT IS, THE RIGHT, THE PARALELL, AND THE
269OBLIQUE SPHEARE.

270
271

CAP. 17.

272
273

WHAT IS THE HORIZON?

274
275    It is a great immooueable circle which deuideth
276the upper Hemispheare, which is as
277much to say, as the upper halfe of the world
278which we see, from the nether Hemispheare
279which wee see not, for standing in a plaine
280field, or rather upon some high mountaine
281void of bushes and trees, and looking round
282about, you shall see your selfe inuironed as
283it were with a circle, and to be in the very midst or centre thereof,
284beneath or beyond which circle, your sight cannot passe, and therfore
285this circle in Greeke is called Horizon , and in Latine Finitor ,
286that is to say, that which determineth, limitteth or boundeth
287the sight, the Poles of which circle are imagined to be two points
288in the firmament, whereof the one standeth right ouer your heade,
289called in Arabick Zenith : and the other directlie vnder your
290feete, called in the same tongue Nadir , that is to say the pointe opposite,
291and from point to point you must imagine that there goeth
292a right line passing through the centre of the worlde, and also

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293through your bodie both head and feet, which is called the Arletree
294of the Horizon, and you haue to understand that of Horizons there
295be 2. kinds, that is, right & oblique, making 3. kinds of Sphears,
296that is to say, the right Spheare, the paralel Spheare, and the
297oblique Spheare.
298
299

WHEN IS THE HORIZON SAID TO BE RIGHT, AND THEREBY TO MAKE
300A RIGHT SPHEARE?

301
302    It may be said to be right two manner of waies, first, when the
303Horizon passeth through both the Poles of the world, cutting the
304Equinoctiall with right angles, in which Spheare they that dwell
305haue their Zenith in the Equinoctiall, which passeth right ouer
306their heads, to whom the daies and nights are alwaies equal. Secondly,
307they are said to haue a right Horizon, & to dwell in a right
308Spheare, to whom one of the Poles of the world is their Zenith ,
309and their Horizon is all one with the Equinoctiall, cutting the Arletree
310of the world in the very midst with right angles, and because
311the Horizon & the Equinoctial are Paralels, this kind of Spheare
312is called a paralel Spheare, in which Sphear they that dwel haue
3136. moneths day, and 6. moneths night, as you may easily perceiue
314by placing the Spheare, so as one of the Poles may stand right vp
315in the midst of the Horizon, by meanes wherof you shal see 6. signes
316of the Zodiaque to be alwayes aboue the Horizon, and 6. signes
317to be alwayes under the Horizon: Againe by placing the Spheare
318so as both the Poles may lie vppon the Horizon, you shall see the
319shape of the first right Sphear, wherin the Horizon passeth throgh
320both the Poles of the world, and the Equinoctiall passeth through
321the Poles of the Horizon, which are the two points called before the
322Zenith and Nadir .
323
324

WHEN IS IT SAID TO BEE AN OBLIQUE HORIZON, AND THEREBY TO
325MAKE AN OBLIQUE SPHEARE?

326
327    When the Pole of the world is eleuated aboue the Horizon, bee
328it neuer so little, so as the Horizon doe cut the Equinoctiall with
329oblique angles, and looke how much the Pole of the world is eleuated
330aboue your Horizon, so much is your Zenith distant from
331the Equinoctiall, and the nigher that your Horizon approcheth to
332the Pole, the nigher your Zenith approcheth to the Equinoctial.
333Againe, looke how much the Equinoctiall is eleuated aboue your
334Horizon, so much is your Zenith distant from the Pole, all which

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335things this figure here following doth plainely shew, whereby you
336may easily perceiue that the latitude, which is the distance of your
337Zenith from the Equinoctiall, is alwaies equall to the altitude of
338the Pole, which is the distance betwixt your Horizon & the Pole,
339as for example, knowing the latitude of Norwich to be 52, degrees
340lay the Zenith of this figure upon the 52. degrees, reckoning from
341the Equinoctiall towards the pole Arctique on your left hand, and
342looke what distance is betwixt the saide Zenith and the Equinoctiall,
343the selfe same distance you shall find to be betwixt the Horizon
344and the foresaid Pole on your right hand, and you may doe the like
345upon the Spheare it selfe by raising the moouable Meridian aboue
346the Horizon at that altitude, so as the 52. degr. may be euen with
347the Horizon.
348A Figure shewing the latitude of any place to bee equall to
349the eleuation of the Pole. FIGURE OMITTED.

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350

WHAT OTHER VSES HATH THIS CIRCLE?

351
352    In this circle are set downe the foure quarters of the world, as
353East, West, North and South, and the rest of the winds: Againe,
354this circle deuideth the artificiall day from the artificiall night, for
355all the while that the Sun is aboue the Horizon it is day, & whilest
356it is under the same it is night. And by this circle wee knowe what
357starres do continually appeare, and which are continually hidden,
358also what starres doe rise and goe downe. Againe, in taking the eleuation
359of the Pole, this circle is chiefly to be considered, for when
360we know how many degrees the Pole is raised aboue the Horizon,
361then we haue the eleuation therof for that place. For to euery seuerall
362place, yea to euerye little moment of the earth in an oblique
363Spheare, belongeth his proper Horizon and seuerall altitude of
364the Pole, whereby it appeareth that the Horizons are infinite and
365without number.
366
367

HOW SHAL I KNOW IN ANY PLACE, HAUING AN OBLIQUE HORIZON,
368HOW MUCH THE POLE IS ELEUATED ABOUE THE HORIZON?

369
370    That is declared in the second booke of this Treatise, wheras I
371speake of the latitude and longitude of the earth, in the 8. chapter.
372
373

OF THE MERIDIAN, AND OF THE VSES THEREOF.

374
375

CAP. 18.

376
377

WHAT IS THE MERIDIAN?

378
379    It is a great immoouable circle passing through the
380Poles of the worlde, and through the Poles of the
381Horizon.
382
383

WHY IS IT CALLED THE MERIDIAN?

384
385    Because that when the sun rising aboue the Horizon in the East,
386commeth to touch this line with the Center of his body, then it is
387midday or noonetide to those, through whose Zenith that Circle
388passeth. And when the Sun after his going downe in the west commeth
389to touch the selfe line againe in the point opposit, it is to them
390midnight, and note that diuers Cities, hauing diuers Latitudes,
391that is to say, being distant one from another North and South be
392it neuer so far, may haue one selfe Meridian: but if they be distant
393one from another East & West, bee it neuer so little, then they must

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394needes haue diuers Meridians, and such distance betwixt the two
395seuerall Meridians, is called the difference of Longitude whereof
396we shall speake hereafter more at large when we come to treate of
397the Longitude and Latitude of the earth, which something differeth
398from the Longitude and Latitude of the starres of Planets,
399whereof we haue already spoken in the 11. Chapter.
400
401

HOW MANY MERIDIANS BE THERE?

402
403    The Astronomers doe appoint for euery two degrees of the Equinoctiall
404a Meridian, so as they make in all 180. Albeit most
405commonly in the Spheare they set downe but one, which serueth
406for all by turning the body of the Spheare to it, which for y^e cause
407is called the mooueable Meridian. And in such Spheares as haue
408not a foote and a standing Horizon, there is no Meridian at al, but
409the two Colures are faine to supply their want, but all terrestriall
410Globes are commonly described with twelue Meridians, cutting
411the Equinoctiall in 24. points, and deuiding the same into 24. spaces,
412euery space containing 15. degrees, which is an houre, by
413meanes whereof we know how much sooner or latter it is noontide
414in any place, for it is noonetide sooner to those whose Meridian is
415more Eastward then to them whose Meridian is more Westward.
416And contrariwise the Eclipse of the Sun or Moone appeareth sooner
417to those whose Meridian is more Westward.
418
419

WHAT OTHER VSES HATH THIS CIRCLE?

420
421    This circle deuideth the East part of the world from the West
422and also it sheweth both the North and South, for by turning
423your face towardes the East, you shall finde the Sunne being in
424that line at noonetide to bee on your right hand right South, the
425opposit part of which circle sheweth on your left hand the North.
426Also this Circle by reason that it passeth through both the Poles
427of the world, deuideth both the Equinoctiall and all his Paralels
428into two equall parts as well aboue the Horizon as under the Horizon,
429and by that meanes it deuideth the artificial day and artificiall
430night each of them into two parts, that is to say, into two semidiurnall
431and into two seminocturnall parts. For betwixt that part
432of the Horizon where the Sun riseth, mounting still untill he come
433to this Circle, which is at noonetide, is contayned the first halfe of
434the day, & the other halfe is from the same circle to the going down
435of the Sunne under the Horizon. And the first parte of the night is

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436the space betwixt the Suns going down and his comming againe
437to the Meridian, which is at midnight, and from thence to the time
438of his rising is the other halfe of the night, and also the Astronomers
439take the beginning of their naturall day from this circle,
440counting either from noontide to noontide, or else from midnight to
441midnight. Againe, this circle sheweth the right ascentions and declinations
442of the starres, and the highest altitude, otherwise called
443the Meridian altitude of the Sun or of any star, or degree of the Ecliptique,
444or of any other point in the firmament, al which vses and
445many others more you shal better understand hereafter, when wee
446come to shew the vses of the globe as well terrestriall as celestiall.
447
448

OF THE VERTICALL CIRCLES, AND VSES THEREOF

449
450

CAP. 13

13 MISTAKENLY FOR 19
451
452    But here you haue to note that though the most part of
453Geographers doe set downe in their Spheares but 6.
454great circles, yet ther is another great circle called the
455circle Verticall, which passeth right ouer our heades
456through our Zenith , wheresoeuer we be vpon the land or sea, crossing
457our Horizon in 2. points opposite, and deuiding the same into
458two equall parts, and such kind of circles are called in Arabick Azimuthes ,
459whereof you may imagine that there be so many as ther
460be rombes or winds in the Marriners compasse, which are in number
46132. yea, and if you will, you may make halfe so many as there
462be degrees in the Horizon, which are in nu~ber 360. the halfe whereof
463is 180. If you be right under the Equinoctiall, and doe goe or
464saile right East or West, then the Equinoctiall is your Verticall
465circle, and if you goe or saile right North or South, then the Meridian
466is your uerticall circle, which two circles notwithstanding
467do alwaies keepe their names. But in sayling by any other rombe,
468that circle which is imagined to passe from the true East pointe
469right ouer your head unto the true West point, or which crosseth
470your Meridian in the Zenith point with right Sphericall angles,
471is most properly called the uerticall circle, and the learned seamen
472haue great respect to two speciall kinds of Verticall circles, that
473is, the Magneticall Meridian, and the Azimuth of the Sunne.

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474

WHAT MANNER OF VERTICALL CIRCLES BEE THOSE, AND WHERETO
475SERUE THEY?

476
477    M. Borrough in his discourse of the variation of the Compasse,
478defineth the Magneticall Meridian to bee a great Circle, which
479passeth through the Zenith and the Pole of the load stone called in
480Latine Magnes , and deuideth the Horizon into two equall parts,
481by crossing the same in two points opposite. Againe the Azimuth
482of the Sunne is a great Circle, passing through the Zenith and
483the Centre of the Sunne in what part of the heauen so euer he be,
484so as he be aboue the Horizon, which Circle deuideth the Horizon
485into two equall parts by crossing the same in two points opposite.
486And by helpe of these two Circles and a certaine instrument made
487of purpose to giue a true shadow, he teacheth to finde out the true
488Meridian of any place: And also to know how much any Mariners
489Compasse doth varie from the true North and South, in Northeasting
490or Northwesting, whereof I shall speake more at large
491hereafter in my treatise of Nauigation.
492
493

WHAT VSE IS THERE OF THE VERTICALL CIRCLES, OR
494AZIMUTHES?

495
496    The uerticall Circle sheweth what time the Sunne or any other
497starre rysing beyond the true East pointe, is passed that
498Sunne or saide starre, commeth to the true East or anye other
499rombe. Also in what Coast or part of heauen, the Sunne, Moone,
500or any other starre is at any time being mounted aboue the Horizon,
501as whether it bee Southeast or Northeast, or in any other
502rombe: Also by helpe of the uerticall Circle most properly so called,
503are the twelue houses of heauen set, according to Campanus
504and Gazula . And by helpe of these Circles you may also knowe
505how any place vppon the earth beareth one from another eyther
506Eastward of Westward, and so foorth, for euerie place hath his
507seuerall Azimuth aunswerable to the Horizon and Zenith of the
508saide place.
509
510

OF CERTAINE CIRCLES CALLED ALMICANTERATHES .

511
512    Since I haue spoken heere somewhat of the uerticall Circles
513called Azimuthes , it shall not be amisse to shew you also that there
514be other Circles to bee considered of in the Spheare as well as in
515the Astrolabe called Almicanterathes , that is to say, Circles of
516Altitude, which though they be not al great Circles, for euery one

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517is lesser then other proceeding fro~ the oblique Horizon of any place
518to the Zenith of the said place, yet the first Almicanterath which
519is the verie oblique Horizon it selfe, is a great Circle deuiding the
520Spheare into two equall parts, and all the rest are lesser and lesser,
521untill you come to the verie Zenith, and are paralels to the Horizon,
522euen as the Tropiques and the other lesser Circles are paralels
523unto the Equinoctiall. And the Zenith in Sphericall bodies
524is the Centre of them all, though it bee not so in Astrolabes, in
525there euerie Almicanterath is saine to haue his seuerall Centre,
526of which Circles there be in all 90. according to the number of 90
527degrees contained betwixt the oblique Horizon and the Zenith,
528and these Circles doe serue to shew the Altitude of the Sunne or
529Moone, or of any other starre fixed or wandring, being mounted at
530any time aboue the oblique Horizon, which is easie to bee found by
531any Quadrant, Crosse-staffe, or Astrolabe. But leauing to speake
532any further of these Circles, because they are not vsed to be described
533in Spheares but onely in Astrolabes, I will now treate of the
534foure lesser Circles before mentioned, which are commonly set
535downe in euery Spheare or Globe.
536
537

OF THE FOURE LESSER CIRCLES, THAT IS TO SAY, THE CIRCLE ARCTIQUE,
538THE CIRCLE ANTARCTIQUE, THE TROPIQUE OF CANCER, AND THE TROPIQUE
539OF CAPRICORNE, AND ALSO OF THE FIUE ZONES, THAT IS TO SAY,
540TWO COLD, TWO TEMPERATE, AND ONE EXTREMELY HOAT.

541
542

CAP. 20.

543
544

WHICH CALL YOU THE LESSER CIRCLES?

545
546    They are those that doe not deuide the
547Spheare into two equall parts, as the great
548Circles doe, and of such there bee foure, that
549is the two Polar circles, and the two Tropiques,
550that is to say, the Tropique of Cancer ,
551and the Tropique of Capricorne , of
552which Polar circles the one is called Arctique, and the other Antarctique,
553and are made by the turning about of the two Poles of
554the Zodiaque, which Poles being situated in the Colure of the

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555Solstices are so farre distant from the Poles of the world, as is the
556greatest declination of the Sunne from the Equinoctiall, which is
55723. degrees, 2'8. as hath beene said before.
558
559

WHICH IS THE ARCTIQUE CIRCLE, AND WHY IS IT SO CALLED?

560
561    The Arctique Circle is that which is next to the North Pole,
562and hath his name of this worde Arctos , which is the great Beare
563or Charles wayne, which are seuen stars placed next to this Circle
564on the outside thereof, and it is otherwise called the Septentrionall
565Circle of this word Septentrio , which is as much to say as
566seuen Oxen, signified by the seuen stars of the little Beare, which
567doe mooue slowly like Oxen, and are placed all within the sayde
568Circle, and the bright starre that is in the tippe of the tayle of the
569sayde little Beare, is called of the Mariners the loade starre or
570North starre, whereby they sayle on the Sea, and the Centre of
571this Circle is the North Pole of the world which is not to be seene
572with mans eye.
573
574

WHAT IS THE ANTARCTIQUE CIRCLE?

575
576    It is that which is next unto the South Pole, and it is so
577called, because it is opposite or contrarie to the Circle
578Arctique.
579
580

NOW DESCRIBE THE TWO TROPIQUES.

581
582    The Tropique of Cancer is a Circle imagined to bee betwixt
583the Equinoctiall and the Circles Arctique, which Circles the
584Sun maketh when he entreth into the first degree of Cancer , which
585is about y^e twelue or thirteenth day of June being then in his greatest
586declination from the Equinoctiall Northward, and nighest to
587our Zenith, being ascended to the highest point that he can goe, at
588which time the daies with us be at the longest, and the nightes at
589the shortest. And so from thence he declineth to the other Tropique
590called the Tropique of Capricorne , which is a Circle imagined to
591be betwixt the Equinoctiall and the Circle Antarctique, which the
592Sunne maketh when hee entreth into the first degree of Capricorne ,
593which is about the twelfth or thirteenth daye of December
594at which time hee is againe in his greatest declination from
595the Equinoctiall Southwarde, and furthest from our Zenith:
596whereby the dayes with us bee then at the shortest, and the nights

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597at the longest: And note that these two Circles are called Tropiques
598of this Greeke word Tropos , which is as much to say as a
599conuersion or turning, for when the Sunne arriueth at any of these
600two Circles, he turneth backe againe either ascending or descending,
601by reason of which foure Circles as well the firmament as
602the earth is deuided into fiue Zones, that is to say, two colde, two
603temperate, & one extremely hoat, otherwise called the burnt Zone,
604of which fiue Zones, the foresaid foure circles are the true bounds.
605For of the two cold Zones, the one lyeth betwixt the North pole
606and the Circle Arctique, and the other lyeth betwixt the South
607Pole and the Circle Antarctique, & of the two temperate Zones,
608the one lyeth betwixt the Circle Arctique, & the Tropique of Cancer ,
609and the other lyeth betwixt the Circle Antarctique, and the
610Tropique of Capricorne , & the extreme hoat Zone lyeth betwixt
611the two Tropiques, in the middest of which two Tropiques, is the
612Equinoctiall line, as you may see in this figure and also in the
613Spheare or Globe it selfe.
614A figure shewing the fiue foresaid Zones. FIGURE OMITTED

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615    Of which Zones the auncient men were wont to say that three
616were unhabitable, that is, the two colde, and the extreame hoat,
617which experience sheweth in these latter daies, to be untrue, as we
618shall declare more at large when we come to treate of the diuision
619of the earth: Againe you haue to understand that euery one of these
620lesser Circles doth containe in length, 360. degrees as well as euery
621one of the greater Circles, but the degrees are not of like bignesse,
622no more then the Circles themselues are like in compasse or
623circuit, for the lesser the Circles are in circuit, the lesser their degrees
624must needes be.
625
626

SITH EUERY OF THE LESSER CIRCLES DIFFER ONE FROM ANOTHER IN
627CIRCUIT, AND THEREBY THE DEGREES OF EUERY CIRCLE BE LESSER THEN
628OTHER, HOW SHALL I KNOW THE TRUE QUANTITIE OF EUERY DEGREE IN
629ECH CIRCLE, AND HOW MANYE MINUTES ARE REQUIRED IN EUERIE
630LESSER DEGREE PROPORTIONALLY TO ANSWERE ONE DEGREE OF THE
631EQUINOCTIALL.

632
633    For the better knowledge hereof, you must first imagine that
634there may bee as many Circles made from the Equinoctiall towards
635any of the Poles, as there be degrees of Latitude, which are
636in number 90. as hath beene said before: And the nigher that any
637circle is to the Equinoctiall, the greater it is in circuit, and the further
638from the Equinoctiall towards any of the Poles, the lesser in
639circuit, and therfore more or lesse minutes are requisite to answere
640to one degree of the Equinoctiall, as you may easily perceiue by
641this Table following, consisting of 6. collums, euery front or head
642whereof is noted with three great letters, D. M. S. signifying degrees,
643minutes and seconds, sixe times repeated, and in the beginning
644of the first collum on the left hand is set downe one degree,
645which is the first degree of 90. and nighest unto the Equinoctiall,
646right against which one degree is placed towards the right hand,
64759. minutes, and 59. seconds: and so proceeding from degree to degree
648successiuely, untill you come to 90. you shall finde how many
649minutes and seconds doe answere to one degree of the Equinoctiall,
650and this Table will also serue to shew the difference of miles in
651euery sundry clyme or paralell, whereof we shall speake hereafter
652when we come to treat of the earth.