<B CESCIE2B>
<Q E2 EX SCIO BLUNDEV>
<N TABLES>
<A BLUNDEVILE THOMAS>
<C E2>
<O 1570-1640>
<M X>
<K X>
<D ENGLISH>
<V PROSE>
<T SCIENCE OTHER>
<G X>
<F X>
<W WRITTEN>
<X MALE>
<Y X>
<H PROF>
<U PROF>
<E X>
<J X>
<I X>
<Z EXPOS>
<S SAMPLE X>


[^BLUNDEVILE.
A BRIEFE DESCRIPTION OF THE TABLES OF
THE THREE SPECIALL RIGHT LINES BELONGING TO
A CIRCLE, CALLED SIGNES, LINES TANGENT, 
AND LINES SECANT.
LONDON: JOHN WINDET, 1597.
PP. 48R.1   - 51V.8      (SAMPLE 1)
PP. 152R.15 - 157R.36    (SAMPLE 2)^]

<S SAMPLE 1>
<P 48R>
[}THE DESCRIPTION AND VSE OF THE 
TABLES OF SINES.}]

   Because there is no proportion, comparison,
or likenes betwixt a right line and a
crooked, the auntient Philosophers, as
(^Ptolomey^) and divers other, were much
troubled in seeking to know the measures
of a Circle or of any portion thereof by his
Diameter, and by knowing the Diameter
to finde out the length of any Chorde in a
circle, which is alwaies lesser then the Diameter it selfe,     #
and finding
that the more parts whereinto the Diameter was diuided,
the nearer they approched to the truth: Some of them therefore,
as (^Ptolomey^) , diuided the Diameter of a circle into a 120.  #
parts,
and the Semidiameter into 60. parts, and euery such part into
6'0. and euery minute in 60. seconds &c. And in like manner did
(^Arzahel^) , an auntient Arabian, who diuided the Diameter     #
into
300. partes and the Semidiameter into 150. and euery of those
parts into 6'0. and so forth as before, according to which      #
computation
they made their Tables: but because the working by those
Tables was very tedious and troublesome, by reason that it was
needfull continually to vse the art of numbring by              #
Astronomicall
fractions: therefore (^Georgius Purbachius^) , and (^Regio      #
Montanus^)
his Scholer to auoide that trouble of calculating by            #
Astronomicall
fractions, diuided the Diameter of a Circle into a farre
greater number of parts, and made such tables as are vsed at    #
this
present, the description and vse whereof both hereafter follow,
first of those that are set downe by (^Monte Regio^) in Folio,  #
and
then of those that were lately Corrected and made perfect by    #
(^Clauius^)
the Jesuite which are Printed in quarto.
<P 48V>
   And because that the way to find out the proportion which    #
any
chord hath to the whole Diameter, was very hard, therefore the
said (^Purbachius^) and (^Monte Regio^) hauing direction from   #
certaine
propositions of (^Euclyd^) as from the 47. proposition of his
first booke, and from the third proposition of his third        #
booke, and 
also from the 15. proposition of his fift booke, they made      #
choise of
the halfe chord and Semidiameter of the Circle, calling the     #
halfe
chord, (\Sinum rectum\) , and the Semediameter                  #
(\Sinum totum\) . And 
because that the proportion of any circumference to his         #
diameter
neuer changeth, how great or how little so euer the Circle be:  #
after
that they had calculated for one Circle, they made such tables
as might serue for all Circles, and though these Tables of      #
sines
doe suffice to worke thereby all manner of conclusions, as      #
well of
Astronomie, as of Geometrie, yet for more ease, our moderne
Geometricians haue of late inuented two other right lines       #
belonging
to a Circle called lines Tangent, and lines Secant, and
haue made like tables for them that were made for sines, and    #
both
tables, that is to say, as well of the sines, as of the lines   #
Tangent
and Secant, haue one selfe manner of working thereby, as shall
plainely appeare hereafter when wee come to describe the same.
But first we will beginne with the tables of sines, and         #
plainely define
euery terme or vocable of Art, belonging thereunto: The 
termes are these here following: (^An arch, a Chord^) ,         #
(\Sinus rectus
Sinus versus, Quadrans, Complementum\) , (^and^) (\sinus 
Complementi\) .

[}THE DEFINITIONS OF THE FORESAID TEARMES.}]

   An (^Arch^) is any part or portion of the circumference of   #
a cirle,
which in this practise doth not commonly extend beyond
180. degrees which is one halfe of the circumference of any     #
Circle
how great or small so euer it be, for euery Circle containeth   #
360.
degrees.
   A (^Chorde^) is a right line drawne from one end of the      #
(^Arch^) to 
the other end thereof, and note that all chordes are alwaies    #
lesser
then the Diameter it selfe, for that is the greatest Chorde in  #
anye
Circle.
   (\Sinus rectus\) is the one halfe of a (^Chord^) or string   #
of any Arke
<P 49R>
which is double to the Arke that is giuen or supposed, and      #
falleth
with right Angles vppon that Semidiameter which diuideth the
double Arke into two equall parts.
   (\Sinus versus\) that is to say turned the contrary way, is  #
a right
line, and that part of the Semidiameter, which is intercepted   #
betwixt
the beginning of the giuen Arke and the right Sine of the
same Arke, and this is also called in Latine (\Sagitta\) , in   #
English a
Shaft or Arrowe, for the Demonstratiue figure thereof hereafter
following, is not vnlike to the string of a bowe ready bent     #
hauing
a Shaft in the midst thereof.
   (\Quadrans\) is the fourth part of a Circle containing 90. 
degrees.
   (\Complementum arcus\) , is that portion of the Circle,      #
which
sheweth how much the giuen Arke is lesser then the Quadrant, if
the giuen Arke doe containe fewer degrees then the Quadrant,    #
but
if it containe more degrees then the Quadrant, then the         #
difference
betwixt the quarter of the Circle and the said arch, is the     #
complement
of the said giuen Arke.
   (\Sinus complementi\) , is the right Sine of that Arch       #
which is
the complement of the giuen Arke.
   (\Sinus totus\) , is the Semidiameter of the Circle, and is  #
the greatest 
Sine that may be in the Quadrant of a Circle, which according 
to the first tables of (^Monte Regio^) containeth 6/000/000.
and according to the last tables 10/000/000. parts, for the     #
more
parts that the totall Sine hath, the more true and exact shall  #
your
worke bee, notwithstanding sometime it shall suffice to         #
attribute
unto the totall Sine but 60/000. parts, which numbers           #
(^Appian^)
obserueth in teaching the way to finde out the distance of two  #
places
differing both in Longitude and Latitude by the Tables of
Sines, and some doe make the totall Sine to containe 100/000.
partes, as (^Wittikindus^) in his treatise of Dials, and        #
diuers other
doe the like. Also (^Clauius^) himselfe saith that in the       #
tables set
downe by him in quarto, you may sometime make the totall Sine
to be but 100/000, so as you cut off the two last figures on    #
the
right hand in euery Sine, but you shall better understand       #
euerye
thing here aboue mentioned, by the figure Demonstratiue heere
following.
<P 49V>
[}THE FIGURE DEMONSTRATIUE.}] [^FIGURE OMITTED^]

   In this figure 
you see first a whole
Circle drawne upon
the Centre (^E.^)
and marked with
the letters (^A.B.C
D.^) which Circle
by two crosse Diameters
marked
with the letters (^A.
C.^) and (^B.D.^) & passing
both through
the Centre (^E.^) is diuided
into fower 
Quadrantes or
quarters, the upper
Quadrante
whereof on the left hand is marked with the letters             #
(^A.B.E.^) in
which Quadrant, the right perpendicular line marked with the
letters (^F.H.^) betokeneth the right Sine of the giuen Arke    #
(^A.F.^)
which right Sine is the one halfe of the chord or string        #
(^F.G.^) and
the giuen Arke (^A.F.^) is the one halfe of the double Arke or  #
bowe
(^G.A.F.^) and (^A.H.^) is the Shaft called in Latine           #
(\Sinus versus\) :
Againe the letters (^F.B.^) doe shew the complement which       #
together
with the giuen Arke (^A.F.^) doe make the whole Quadrant        #
(^A.F.B.^)
which is diuided into 9. spaces, euery space containing 10.     #
degrees
whereby you may plainely perceiue that in this demonstration,
the giuen Arke (^A.F.^) is 50. degrees, and the complement      #
(^F.B.^) is
40. degrees, both which being added together doe make up the
whole Quadrant of 90. degrees, marked with the letters          #
(^A.F.B.^)
Now (\Sinus complementi\) is the crosse line marked with the    #
letters
(^F.K.^) the totall Sine which is the whole Semidiameter and
greatest right Sine, is marked with the letters (^B.E.^) But    #
because
it is not enough to know the signification of the things aboue  #
specified
to vse the foresaid Tables when neede is, vnlesse you know
<P 50R>
also how to find out those things in the said tables, I thinke  #
it good
therefore to shew you the order of the said tables by           #
describing the
same as followeth. 
   You haue then to vnderstand that the tables of (^Monte       #
Regio^)
printed in Folio, are contained in 18. Pages, and euery Page
containeth eleauen partitions, called collums, whereof the      #
first on
the left hand containeth 60. minutes, which are to be counted   #
from
head to foote, as they stand in order one right under another   #
in seuerall
places, proceeding from 1. to 60. The second collum containeth
Sines. The third containeth onely a portion or part of
one second, and from thence foorth proceeding towardes the      #
right
hand all the other collums doe containe in like manner Sines    #
and
the portion of one second. And right ouer the head of euery     #
Sine
(the first collum of Sines onely excepted, hauing nothing but a
Cypher ouer his head) are set downe the degrees of the whole
Quadrant called arches, in such order as from the first Page to
the last, there are in all 89. degrees, or arches, as by        #
perusing the
said tables you may plainely see. Now to find out in these      #
tables
the things aboue mentioned, you must doe as followeth.
   First to find out the right Sine of any giuen Arke, you must
seeke out the number of the said Arke in the front of the       #
tables, and
if the giuen Arke hath no minutes ioyned thereunto, then the    #
first
number of Sines right under the said Arke, is the right Sine
thereof. But if it hath any minutes ioyned thereunto, then you
must seeke out in that Page, where you found the giuen Arke,    #
the
number of the minutes in the first collum of the said Page, on  #
the
left hand, and right against those minutes on the right hand,   #
in
the square Angle right under the said arch, you shall find the  #
right
Sine. As for example, you would find out the right Sine of a    #
giuen 
Arke containing 8. degrees, and 2'0. heere hauing found out
in the front of the second Page the figure of 8. standing       #
right ouer
the eight collum seeke in the first collum on the left hand of  #
the said
Page, for 20. minutes, and right against the 20. minutes you    #
shal
find on the right hand in the common Angle or square 869593.
which is the right Sine of the foresaid giuen Arke, so as you   #
make
6/000/000. to be the totall Sine: but if you make 60/000. the   #
totall
Sine, then you must alwaies reiect the two last figures         #
standing 
on the right hand of the said right sine, & the rest of the     #
figures
shall be the right Sine.
<P 50V>
   Now to find out the complement, there is nothing to be done,
but onely to subtract the giuen Arke out of the whole Quadrant
which is 90. degrees, and the remainder shall be the            #
complement:
as in the former example by subtracting 8. degrees, 2'0. out    #
of 90
degrees, you shal find that there remaineth 81. degrees, 4'0.   #
which
is the complement of that arch. Againe to find out the Sine of  #
the
complement you must doe thus, seeke the complement in the       #
front 
of the tables of Sines, euen as you doe to find out any giuen   #
arke:
as in the former example, the complement being 81. degrees 4'0.
you must seeke 81. in the front of the 17. Page of the first    #
tables,
which being found, seeke out also the 4'0. in the first collum  #
of the
said Page on the left hand, and right against those 4'0. in     #
the common
Angle right under the Arke 81. you shall finde 5/936/649.
which number is the right Sine of the foresaid complement, so   #
as
you make 6/000/000. to be the totall Sine, for if 60/000. be    #
the
totall Sine, then you must reiect (as I said before) the two    #
last figures
on the right hand, and the number remaining shall bee the
right Sine of the foresaid complement, and therefore in         #
working 
by these tables, you must alwaies remember what number you
make the totall Sine to be.
   (\Sinus versus\) commeth seldome in vse, notwithstanding if  #
you
would know how to find it out, you neede to do no more but      #
subtract
(\Sinum complementi\) of the giuen Arke, out of the totall
Sine, and the remainder shall bee (\Sinus versus\) , as in the  #
former
example your (\Sinus complementi\) was 5/936/649. which being
subtracted out of the totall Sine 6/000/000. there remaineth
63/351. and that number is (\Sinus versus\) : for if you adde   #
this
remainder to the number which you subtracted, it will make up
the totall Sine 6/000/000. But there is one thing more          #
necessarie
to be knowne then this, because it commeth oftner in vse, and
that is upon some diuision made how to find out the Arke of any
quotient, which is to be done thus: Enter with the quotient     #
into
the body of the tables, and leaue not seeking amongst the       #
squares
of the Sines, vntill you haue found out the iust number of the  #
quotient
(if it be there) if not, you must take the number of that Sine
which is in value most nigh vnto it, whether it bee a little    #
more or
lesse, it maketh no matter, and hauing found that number,       #
looke in
the front of that collum, and you shall find the Arke of your   #
quotient,
<P 51R>
standing right ouer the head of that collum, and also the       #
mynutes
thereof in the first collum of the said Page on the left hand.
As for example, hauing diuided one number by another, I finde
the quotient to be 469/012. whereof I would know the arch, now
in seeking this quotient amongst the Sines, I cannot finde that
iust number, but I find in the first Page, and in the tenth     #
collum
469/015. which is the nighest number vnto it that I can see. In
the front of which collum I find the Arke to be 4. degrees,     #
and directly
against that Sine on the left hand, I find 2'9. belonging to
that arch, whereof that quotient is the (^Sinus^) , so as I     #
gather hereof
that the arch of the foresaid quotient is 4. degrees, 2'9. But  #
you
haue to note by the way that the number of your quotient must   #
neuer
be much lesse then 1745. for otherwise it is not to bee found   #
in
these tables, unlesse you make the totall Sine to bee but       #
60/000.
for then by reiecting the last two figures on the right hand,   #
as I
haue said before, the first right Sine of these tables shal be  #
no more
but 17. and by that account a very small quotient may be found  #
in 
these tables. And whatsoeuer hath beene said here touching the  #
order 
that is to be obserued in the first tables of                   #
(^Monte Regio^) , whose
totall Sine is 6/000/000. the like in all points is to be       #
obserued
in the last tables, whose totall Sine is 10/000/000. Thus much
touching the order of the foresaid tables of (^Monte Regio^)    #
Printed
in Folio: but for as much as those tables be not altogether     #
truely
Printed, and for that they haue beene lately corrected, and     #
made
more perfect by (^Clauius^) , who doth set downe the saide      #
Tables in
quarto and not in folio, whereby they are the more portable,    #
and
the more commodious, as well for that they are more truely      #
Printed,
as also for that the complement of euery Arke is set downe in
euery Page at the foote of euery collum, so as you need to      #
spend no
time in subtracting the Arke from 90. I thinke it good          #
therefore
to make a briefe description of those Tables, and the rather    #
for
that I haue requested the Printer to print the like here in     #
quarto,
and I doe worke all such conclusions as hereafter follow, by    #
the
said tables, the totall Sine whereof is 10/000/000. according   #
to
the last tables of (^Monte Regio^) . But for so much as some    #
may
haue already the tables of (^Monte Regio^) Printed in Folio,    #
not 
knowing perhaps the vse thereof, I will set downe two           #
conclusions
to bee wrought by those tables, and all the rest of the         #
conclusions
<P 51V>
are to be wrought by these tables which I haue here caused to
be Printed in quarto like to those of (^Clauius^) : and though  #
the two
conclusions next following, which are to shew the vse of the    #
foresaid
tables, may be wrought by the tables of Sines in what forme
so euer they be truely Printed in Folio, or in quarto, yet      #
because I
had appointed them to bee done by the Tables of (^Monte         #
Regio^) ,
Printed in folio before that euer I saw (^Clauius^) his booke,  #
I mind
not now to alter them but to let them stand still as they are. 

<S SAMPLE 2>
<P 152R>
[}OF THE HORIZON BOTH RIGHT AND OBLIQUE, MAKING THEREBY
THREE KINDS OF SPHEARES, THAT IS, THE RIGHT, THE PARALELL, AND  #
THE 
OBLIQUE SPHEARE.}]  

[}CAP. 17.}]

[}WHAT IS THE HORIZON?}]

   It is a great immooueable circle which deuideth
the upper Hemispheare, which is as
much to say, as the upper halfe of the world
which we see, from the nether Hemispheare
which wee see not, for standing in a plaine
field, or rather upon some high mountaine 
void of bushes and trees, and looking round
about, you shall see your selfe inuironed as
it were with a circle, and to be in the very midst or centre    #
thereof,
beneath or beyond which circle, your sight cannot passe, and    #
therfore
this circle in Greeke is called (^Horizon^) , and in Latine     #
(\Finitor\) ,
that is to say, that which determineth, limitteth or boundeth
the sight, the Poles of which circle are imagined to be two     #
points
in the firmament, whereof the one standeth right ouer your      #
heade,
called in Arabick (^Zenith^) : and the other directlie vnder    #
your
feete, called in the same tongue (^Nadir^) , that is to say     #
the pointe opposite,
and from point to point you must imagine that there goeth
a right line passing through the centre of the worlde, and also
<P 152V>
through your bodie both head and feet, which is called the      #
Arletree
of the Horizon, and you haue to understand that of Horizons     #
there
be 2. kinds, that is, right & oblique, making 3. kinds of       #
Sphears,
that is to say, the right Spheare, the paralel Spheare, and the
oblique Spheare.

[}WHEN IS THE HORIZON SAID TO BE RIGHT, AND THEREBY TO MAKE 
A RIGHT SPHEARE?}]

   It may be said to be right two manner of waies, first, when  #
the
Horizon passeth through both the Poles of the world, cutting    #
the
Equinoctiall with right angles, in which Spheare they that      #
dwell
haue their (^Zenith^) in the Equinoctiall, which passeth right  #
ouer
their heads, to whom the daies and nights are alwaies equal.    #
Secondly,
they are said to haue a right Horizon, & to dwell in a right
Spheare, to whom one of the Poles of the world is their         #
(^Zenith^) ,
and their Horizon is all one with the Equinoctiall, cutting     #
the Arletree
of the world in the very midst with right angles, and because
the Horizon & the Equinoctial are Paralels, this kind of        #
Spheare
is called a paralel Spheare, in which Sphear they that dwel     #
haue
6. moneths day, and 6. moneths night, as you may easily         #
perceiue
by placing the Spheare, so as one of the Poles may stand right  #
vp
in the midst of the Horizon, by meanes wherof you shal see 6.   #
signes
of the Zodiaque to be alwayes aboue the Horizon, and 6. signes
to be alwayes under the Horizon: Againe by placing the Spheare
so as both the Poles may lie vppon the Horizon, you shall see   #
the
shape of the first right Sphear, wherin the Horizon passeth     #
throgh
both the Poles of the world, and the Equinoctiall passeth       #
through
the Poles of the Horizon, which are the two points called       #
before the
(^Zenith^) and (^Nadir^) .

[}WHEN IS IT SAID TO BEE AN OBLIQUE HORIZON, AND THEREBY TO
MAKE AN OBLIQUE SPHEARE?}]

   When the Pole of the world is eleuated aboue the Horizon,    #
bee
it neuer so little, so as the Horizon doe cut the Equinoctiall  #
with
oblique angles, and looke how much the Pole of the world is     #
eleuated 
aboue your Horizon, so much is your (^Zenith^) distant from
the Equinoctiall, and the nigher that your Horizon approcheth   #
to
the Pole, the nigher your (^Zenith^) approcheth to the          #
Equinoctial.
Againe, looke how much the Equinoctiall is eleuated aboue your
Horizon, so much is your (^Zenith^) distant from the Pole, all  #
which
<P 153R>
things this figure here following doth plainely shew, whereby   #
you
may easily perceiue that the latitude, which is the distance    #
of your
(^Zenith^) from the Equinoctiall, is alwaies equall to the      #
altitude of
the Pole, which is the distance betwixt your Horizon & the      #
Pole,
as for example, knowing the latitude of (^Norwich^) to be 52,   #
degrees
lay the (^Zenith^) of this figure upon the 52. degrees,         #
reckoning from
the Equinoctiall towards the pole Arctique on your left hand,   #
and
looke what distance is betwixt the saide (^Zenith^) and the     #
Equinoctiall, 
the selfe same distance you shall find to be betwixt the        #
Horizon
and the foresaid Pole on your right hand, and you may doe the   #
like
upon the Spheare it selfe by raising the moouable Meridian      #
aboue
the Horizon at that altitude, so as the 52. degr. may be euen   #
with
the Horizon.
(^A Figure shewing the latitude of any place to bee equall to
the eleuation of the Pole.^) [^FIGURE OMITTED.^]
<P 153V>
[}WHAT OTHER VSES HATH THIS CIRCLE?}]
 
   In this circle are set downe the foure quarters of the       #
world, as 
East, West, North and South, and the rest of the winds: Againe,
this circle deuideth the artificiall day from the artificiall   #
night, for
all the while that the Sun is aboue the Horizon it is day, &    #
whilest
it is under the same it is night. And by this circle wee knowe  #
what
starres do continually appeare, and which are continually       #
hidden,
also what starres doe rise and goe downe. Againe, in taking     #
the eleuation 
of the Pole, this circle is chiefly to be considered, for when
we know how many degrees the Pole is raised aboue the Horizon,
then we haue the eleuation therof for that place. For to euery  #
seuerall
place, yea to euerye little moment of the earth in an oblique
Spheare, belongeth his proper Horizon and seuerall altitude of
the Pole, whereby it appeareth that the Horizons are infinite   #
and
without number.

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

   That is declared in the second booke of this Treatise,       #
wheras I
speake of the latitude and longitude of the earth, in the 8.    #
chapter.

[}OF THE MERIDIAN, AND OF THE VSES THEREOF.}]

[}CAP. 18.}]

[}WHAT IS THE MERIDIAN?}]

   It is a great immoouable circle passing through the
Poles of the worlde, and through the Poles of the 
Horizon.

[}WHY IS IT CALLED THE MERIDIAN?}]

   Because that when the sun rising aboue the Horizon in the    #
East,
commeth to touch this line with the Center of his body, then    #
it is
midday or noonetide to those, through whose (^Zenith^) that     #
Circle
passeth. And when the Sun after his going downe in the west     #
commeth 
to touch the selfe line againe in the point opposit, it is to   #
them
midnight, and note that diuers Cities, hauing diuers Latitudes,
that is to say, being distant one from another North and South  #
be
it neuer so far, may haue one selfe Meridian: but if they be    #
distant
one from another East & West, bee it neuer so little, then      #
they must
<P 154R>
needes haue diuers Meridians, and such distance betwixt the two
seuerall Meridians, is called the difference of Longitude       #
whereof
we shall speake hereafter more at large when we come to treate  #
of
the Longitude and Latitude of the earth, which something        #
differeth 
from the Longitude and Latitude of the starres of Planets,
whereof we haue already spoken in the 11. Chapter.

[}HOW MANY MERIDIANS BE THERE?}]

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

[}WHAT OTHER VSES HATH THIS CIRCLE?}]

   This circle deuideth the East part of the world from the     #
West
and also it sheweth both the North and South, for by turning
your face towardes the East, you shall finde the Sunne being in
that line at noonetide to bee on your right hand right South,   #
the
opposit part of which circle sheweth on your left hand the      #
North.
Also this Circle by reason that it passeth through both the     #
Poles
of the world, deuideth both the Equinoctiall and all his        #
Paralels
into two equall parts as well aboue the Horizon as under the    #
Horizon,
and by that meanes it deuideth the artificial day and           #
artificiall
night each of them into two parts, that is to say, into two     #
semidiurnall 
and into two seminocturnall parts. For betwixt that part
of the Horizon where the Sun riseth, mounting still untill he   #
come
to this Circle, which is at noonetide, is contayned the first   #
halfe of
the day, & the other halfe is from the same circle to the       #
going down
of the Sunne under the Horizon. And the first parte of the      #
night is
<P 154V>
the space betwixt the Suns going down and his comming againe
to the Meridian, which is at midnight, and from thence to the   #
time
of his rising is the other halfe of the night, and also the     #
Astronomers
take the beginning of their naturall day from this circle,
counting either from noontide to noontide, or else from         #
midnight to
midnight. Againe, this circle sheweth the right ascentions and  #
declinations
of the starres, and the highest altitude, otherwise called
the Meridian altitude of the Sun or of any star, or degree of   #
the Ecliptique,
or of any other point in the firmament, al which vses and
many others more you shal better understand hereafter, when wee
come to shew the vses of the globe as well terrestriall as      #
celestiall.

[}OF THE VERTICALL CIRCLES, AND VSES THEREOF}]

[}CAP. 13}] [^13 MISTAKENLY FOR 19^]

   But here you haue to note that though the most part of
Geographers doe set downe in their Spheares but 6. 
great circles, yet ther is another great circle called the
circle Verticall, which passeth right ouer our heades
through our (^Zenith^) , wheresoeuer we be vpon the land or     #
sea, crossing
our Horizon in 2. points opposite, and deuiding the same into 
two equall parts, and such kind of circles are called in        #
Arabick (^Azimuthes^) , 
whereof you may imagine that there be so many as ther
be rombes or winds in the Marriners compasse, which are in      #
number
32. yea, and if you will, you may make halfe so many as there
be degrees in the Horizon, which are in nu~ber 360. the halfe   #
whereof
is 180. If you be right under the Equinoctiall, and doe goe or
saile right East or West, then the Equinoctiall is your         #
Verticall
circle, and if you goe or saile right North or South, then the  #
Meridian
is your uerticall circle, which two circles notwithstanding
do alwaies keepe their names. But in sayling by any other       #
rombe,
that circle which is imagined to passe from the true East       #
pointe
right ouer your head unto the true West point, or which         #
crosseth
your Meridian in the (^Zenith^) point with right Sphericall     #
angles,
is most properly called the uerticall circle, and the learned   #
seamen
haue great respect to two speciall kinds of Verticall circles,  #
that
is, the Magneticall Meridian, and the (^Azimuth^) of the Sunne.
<P 155R>
[}WHAT MANNER OF VERTICALL CIRCLES BEE THOSE, AND WHERETO
SERUE THEY?}]

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

[}WHAT VSE IS THERE OF THE VERTICALL CIRCLES, OR 
AZIMUTHES?}]

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

[}OF CERTAINE CIRCLES CALLED (^ALMICANTERATHES^) .}]

   Since I haue spoken heere somewhat of the uerticall Circles 
called (^Azimuthes^) , it shall not be amisse to shew you also  #
that there
be other Circles to bee considered of in the Spheare as well    #
as in
the Astrolabe called (^Almicanterathes^) , that is to say,      #
Circles of
Altitude, which though they be not al great Circles, for euery  #
one
<P 155V>
is lesser then other proceeding fro~ the oblique Horizon of     #
any place 
to the Zenith of the said place, yet the first                  #
(^Almicanterath^) which
is the verie oblique Horizon it selfe, is a great Circle        #
deuiding the 
Spheare into two equall parts, and all the rest are lesser and  #
lesser,
untill you come to the verie Zenith, and are paralels to the    #
Horizon, 
euen as the Tropiques and the other lesser Circles are          #
paralels 
unto the Equinoctiall. And the Zenith in Sphericall bodies
is the Centre of them all, though it bee not so in Astrolabes,  #
in
there euerie (^Almicanterath^) is saine to haue his seuerall    #
Centre,
of which Circles there be in all 90. according to the number    #
of 90
degrees contained betwixt the oblique Horizon and the Zenith, 
and these Circles doe serue to shew the Altitude of the Sunne   #
or
Moone, or of any other starre fixed or wandring, being mounted  #
at
any time aboue the oblique Horizon, which is easie to bee       #
found by
any Quadrant, Crosse-staffe, or Astrolabe. But leauing to       #
speake
any further of these Circles, because they are not vsed to be   #
described
in Spheares but onely in Astrolabes, I will now treate of the 
foure lesser Circles before mentioned, which are commonly set
downe in euery Spheare or Globe.

[}OF THE FOURE LESSER CIRCLES, THAT IS TO SAY, THE CIRCLE       #
ARCTIQUE,
THE CIRCLE ANTARCTIQUE, THE TROPIQUE OF CANCER, AND THE         #
TROPIQUE
OF CAPRICORNE, AND ALSO OF THE FIUE ZONES, THAT IS TO SAY,
TWO COLD, TWO TEMPERATE, AND ONE EXTREMELY HOAT.}]
         
[}CAP. 20.}]

[}WHICH CALL YOU THE LESSER CIRCLES?}]

   They are those that doe not deuide the 
Spheare into two equall parts, as the great
Circles doe, and of such there bee foure, that
is the two Polar circles, and the two Tropiques,
that is to say, the Tropique of (^Cancer^) ,
and the Tropique of (^Capricorne^) , of 
which Polar circles the one is called Arctique, and the other   #
Antarctique,
and are made by the turning about of the two Poles of
the Zodiaque, which Poles being situated in the Colure of the
<P 156R>
Solstices are so farre distant from the Poles of the world, as  #
is the
greatest declination of the Sunne from the Equinoctiall, which  #
is
23. degrees, 2'8. as hath beene said before. 

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

   The Arctique Circle is that which is next to the North Pole,
and hath his name of this worde (\Arctos\) , which is the       #
great Beare
or Charles wayne, which are seuen stars placed next to this     #
Circle 
on the outside thereof, and it is otherwise called the          #
Septentrionall
Circle of this word (\Septentrio\) , which is as much to say as
seuen Oxen, signified by the seuen stars of the little Beare,   #
which
doe mooue slowly like Oxen, and are placed all within the sayde
Circle, and the bright starre that is in the tippe of the       #
tayle of the
sayde little Beare, is called of the Mariners the loade starre  #
or
North starre, whereby they sayle on the Sea, and the Centre of 
this Circle is the North Pole of the world which is not to be   #
seene
with mans eye.

[}WHAT IS THE ANTARCTIQUE CIRCLE?}]

   It is that which is next unto the South Pole, and it is so
called, because it is opposite or contrarie to the Circle 
Arctique.

[}NOW DESCRIBE THE TWO TROPIQUES.}]

   The Tropique of (^Cancer^) is a Circle imagined to bee       #
betwixt
the Equinoctiall and the Circles Arctique, which Circles the
Sun maketh when he entreth into the first degree of             #
(^Cancer^) , which
is about y=e= twelue or thirteenth day of June being then in    #
his greatest
declination from the Equinoctiall Northward, and nighest to 
our Zenith, being ascended to the highest point that he can     #
goe, at
which time the daies with us be at the longest, and the         #
nightes at
the shortest. And so from thence he declineth to the other      #
Tropique
called the Tropique of (^Capricorne^) , which is a Circle       #
imagined to
be betwixt the Equinoctiall and the Circle Antarctique, which   #
the
Sunne maketh when hee entreth into the first degree of          #
(^Capricorne^) ,
which is about the twelfth or thirteenth daye of December 
at which time hee is againe in his greatest declination from
the Equinoctiall Southwarde, and furthest from our Zenith:
whereby the dayes with us bee then at the shortest, and the     #
nights
<P 156V>
at the longest: And note that these two Circles are called      #
Tropiques
of this Greeke word (\Tropos\) , which is as much to say as a
conuersion or turning, for when the Sunne arriueth at any of    #
these 
two Circles, he turneth backe againe either ascending or        #
descending,
by reason of which foure Circles as well the firmament as 
the earth is deuided into fiue Zones, that is to say, two       #
colde, two
temperate, & one extremely hoat, otherwise called the burnt     #
Zone,
of which fiue Zones, the foresaid foure circles are the true    #
bounds.
For of the two cold Zones, the one lyeth betwixt the North pole
and the Circle Arctique, and the other lyeth betwixt the South
Pole and the Circle Antarctique, & of the two temperate Zones, 
the one lyeth betwixt the Circle Arctique, & the Tropique of    #
(^Cancer^) ,
and the other lyeth betwixt the Circle Antarctique, and the
Tropique of (^Capricorne^) , & the extreme hoat Zone lyeth      #
betwixt
the two Tropiques, in the middest of which two Tropiques, is    #
the 
Equinoctiall line, as you may see in this figure and also in    #
the
Spheare or Globe it selfe.
(^A figure shewing the fiue foresaid Zones.^) [^FIGURE          #
OMITTED^]
<P 157R>
   Of which Zones the auncient men were wont to say that three
were unhabitable, that is, the two colde, and the extreame      #
hoat, 
which experience sheweth in these latter daies, to be untrue,   #
as we
shall declare more at large when we come to treate of the       #
diuision
of the earth: Againe you haue to understand that euery one of   #
these
lesser Circles doth containe in length, 360. degrees as well as #
euery
one of the greater Circles, but the degrees are not of like     #
bignesse,
no more then the Circles themselues are like in compasse or
circuit, for the lesser the Circles are in circuit, the lesser  #
their degrees
must needes be.

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

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



