<P_48R>

<heading>

THE DESCRIPTION AND VSE OF THE TABLES OF SINES . (BLUNDEV-E2-H,48R.3)

</heading>

Because there is no proportion , comparison , or likenes betwixt a
right line and a crooked , the auntient Philosophers , as <font>
Ptolomey </font> 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 ,
(BLUNDEV-E2-H,48R.5)

and finding that the more parts whereinto the Diameter was diuided ,
the nearer they approched to the truth : Some of them therefore , as
<font> Ptolomey </font> , diuided the Diameter of a circle into a parts
, and the Semidiameter into parts , and euery such part into and euery
minute in seconds &c. (BLUNDEV-E2-H,48R.6)

And in like manner did <font> Arzahel </font> , an auntient Arabian ,
who diuided the Diameter into partes and the Semidiameter into and
euery of those parts into and so forth as before , according to which
computation they made their Tables : (BLUNDEV-E2-H,48R.7)

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 <font> Georgius
Purbachius </font> , and <font> Regio Montanus </font> his Scholer to
auoide that trouble of calculating by Astronomicall fractions , diuided
the Diameter of a Circle into a farre greater number of parts ,
(BLUNDEV-E2-H,48R.8)

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 <font> Monte Regio </font> in Folio , and then of those that were
lately Corrected and made perfect by <font> Clauius </font> the Jesuite
which are Printed in quarto . (BLUNDEV-E2-H,48R.9)

<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 <font>
Purbachius </font> and <font> Monte Regio </font> hauing direction from
certaine propositions of <font> Euclyd </font> 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 . (BLUNDEV-E2-H,48V.11)

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 , (BLUNDEV-E2-H,48V.12)

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 , (BLUNDEV-E2-H,48V.13)

and haue made like tables for them that were made for sines ,
(BLUNDEV-E2-H,48V.14)

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
. (BLUNDEV-E2-H,48V.15)

But first we will beginne with the tables of sines , and plainely
define euery terme or vocable of Art , belonging thereunto :
(BLUNDEV-E2-H,48V.16)

The termes are these here following : <font> An arch , a Chord </font>
, Sinus rectus Sinus versus , Quadrans , Complementum , <font> and
</font> sinus Complementi . (BLUNDEV-E2-H,48V.17)

<heading>

THE DEFINITIONS OF THE FORESAID TEARMES . (BLUNDEV-E2-H,48V.19)

</heading>

An <font> Arch </font> is any part or portion of the circumference of a
$circle {TEXT:cirle} , which in this practise doth not commonly extend
beyond degrees which is one halfe of the circumference of any Circle
how great or small so euer it be , (BLUNDEV-E2-H,48V.21)

for euery Circle containeth degrees . (BLUNDEV-E2-H,48V.22)

A <font> Chorde </font> is a right line drawne from one end of the
<font> Arch </font> to the other end thereof , (BLUNDEV-E2-H,48V.23)

and note that all chordes are alwaies lesser then the Diameter it selfe
, (BLUNDEV-E2-H,48V.24)

for that is the greatest Chorde in anye Circle . (BLUNDEV-E2-H,48V.25)

Sinus rectus is the one halfe of a <font> Chord </font> 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 . (BLUNDEV-E2-H,49R.26)

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 ,
(BLUNDEV-E2-H,49R.27)

and this is also called in Latine Sagitta , in English a Shaft or
Arrowe , (BLUNDEV-E2-H,49R.28)

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 . (BLUNDEV-E2-H,49R.29)

Quadrans is the fourth part of a Circle containing degrees .
(BLUNDEV-E2-H,49R.30)

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 , (BLUNDEV-E2-H,49R.31)

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 . (BLUNDEV-E2-H,49R.32)

Sinus complementi , is the right Sine of that Arch which is the
complement of the giuen Arke . (BLUNDEV-E2-H,49R.33)

Sinus totus , is the Semidiameter of the Circle , (BLUNDEV-E2-H,49R.34)

and is the greatest Sine that may be in the Quadrant of a Circle ,
which according to the first tables of <font> Monte Regio </font>
containeth and according to the last tables parts ,
(BLUNDEV-E2-H,49R.35)

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 parts , which numbers <font> Appian
</font> obserueth in teaching the way to finde out the distance of two
places differing both-4 in Longitude and Latitude by the Tables of
Sines , (BLUNDEV-E2-H,49R.36)

and some doe make the totall Sine to containe partes , as <font>
Wittikindus </font> in his treatise of Dials , (BLUNDEV-E2-H,49R.37)

and diuers other doe the like . (BLUNDEV-E2-H,49R.38)

Also <font> Clauius </font> himselfe saith that in the tables set downe
by him in quarto , you may sometime make the totall Sine to be but , so
as you cut off the two last figures on the right hand in euery Sine ,
(BLUNDEV-E2-H,49R.39)

but you shall better understand euerye thing here aboue mentioned , by
the figure Demonstratiue heere following . (BLUNDEV-E2-H,49R.40)

<P_49V>

<heading>

THE FIGURE DEMONSTRATIUE . (BLUNDEV-E2-H,49V.43)

</heading>

In this figure {COM:figure_omitted} you see first a whole Circle drawne
upon the Centre <font> E. </font> and marked with the letters <font> A.
B. C D. </font> which Circle by two crosse Diameters marked with the
letters <font> A. C. </font> and <font> B. D. </font> & passing both
through the Centre <font> E. </font> is diuided into fower Quadrantes
or quarters , the upper Quadrante whereof on the left hand is marked
with the letters <font> A. B. E. </font> in which Quadrant , the right
perpendicular line marked with the letters <font> F. H. </font>
betokeneth the right Sine of the giuen Arke <font> A. F. </font> which
right Sine is the one halfe of the chord or string <font> F. G. </font>
(BLUNDEV-E2-H,49V.45)

and the giuen Arke <font> A. F. </font> is the one halfe of the double
Arke or bowe <font> G. A. F. </font> (BLUNDEV-E2-H,49V.46)

and <font> A. H. </font> is the Shaft called in Latine Sinus versus :
(BLUNDEV-E2-H,49V.47)

Againe the letters <font> F. B. </font> doe shew the complement which
together with the giuen Arke <font> A. F. </font> doe make the whole
Quadrant <font> A. F. B. </font> which is diuided into 9. spaces ,
euery space containing degrees whereby you may plainely perceiue that
in this demonstration , the giuen Arke <font> A. F. </font> is degrees
, and the complement <font> F. B. </font> is degrees , both which being
added together doe make up the whole Quadrant of degrees , marked with
the letters <font> A. F. B. </font> (BLUNDEV-E2-H,49V.48)

Now Sinus complementi is the crosse line marked with the letters <font>
F. K. </font> (BLUNDEV-E2-H,49V.49)

the totall Sine which is the whole Semidiameter and greatest right Sine
, is marked with the letters <font> B. E. </font> (BLUNDEV-E2-H,49V.50)

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 . (BLUNDEV-E2-H,50R.51)

You haue then to vnderstand that the tables of <font> Monte Regio
</font> printed in Folio , are contained in 18. Pages , and euery Page
containeth eleauen partitions , called collums , whereof the first on
the left hand containeth 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 (BLUNDEV-E2-H,50R.52)

The second collum containeth Sines . (BLUNDEV-E2-H,50R.53)

The third containeth onely a portion or part of one second ,
(BLUNDEV-E2-H,50R.54)

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
. (BLUNDEV-E2-H,50R.55)

And right ouer the head of euery Sine <paren> the first collum of Sines
onely excepted , hauing nothing but a Cypher ouer his head </paren> 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 .
(BLUNDEV-E2-H,50R.56)

Now to find out in these tables the things aboue mentioned , you must
doe as followeth . (BLUNDEV-E2-H,50R.57)

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 ,
(BLUNDEV-E2-H,50R.58)

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 .
(BLUNDEV-E2-H,50R.59)

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 ,
(BLUNDEV-E2-H,50R.60)

and right against those minutes on the right hand , in the square Angle
right under the said arch , you shall find the right Sine .
(BLUNDEV-E2-H,50R.61)

As for example , you would find out the right Sine of a giuen Arke
containing 8. degrees , and (BLUNDEV-E2-H,50R.62)

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 minutes , (BLUNDEV-E2-H,50R.63)

and right against the 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 to be the totall Sine :
(BLUNDEV-E2-H,50R.64)

but if you make the totall Sine , then you must alwaies reiect the two
last figures standing on the right hand of the said right sine ,
(BLUNDEV-E2-H,50R.65)

& the rest of the figures shall be the right Sine .
(BLUNDEV-E2-H,50R.66)

<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
degrees , (BLUNDEV-E2-H,50V.68)

and the remainder shall be the complement : (BLUNDEV-E2-H,50V.69)

as in the former example by subtracting 8. degrees , out of degrees ,
you shal find that there remaineth 81. degrees , which is the
complement of that arch . (BLUNDEV-E2-H,50V.70)

Againe to find out the Sine of the complement you must doe thus ,
(BLUNDEV-E2-H,50V.71)

seeke the complement in the front of the tables of Sines , euen as you
doe to find out any giuen arke : (BLUNDEV-E2-H,50V.72)

as in the former example , the complement being 81. degrees you must
seeke 81. in the front of the 17. Page of the first tables , which
being found , seeke out also the in the first collum of the said Page
on the left hand , (BLUNDEV-E2-H,50V.73)

and right against those 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 to be the totall Sine ,
(BLUNDEV-E2-H,50V.74)

for if be the totall Sine , then you must reiect <paren> as I said
before </paren> the two last figures on the right hand ,
(BLUNDEV-E2-H,50V.75)

and the number remaining shall bee the right Sine of the foresaid
complement , (BLUNDEV-E2-H,50V.76)

and therefore in working by these tables , you must alwaies remember
what number you make the totall Sine to be . (BLUNDEV-E2-H,50V.77)

Sinus versus commeth seldome in vse , (BLUNDEV-E2-H,50V.78)

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 , (BLUNDEV-E2-H,50V.79)

and the remainder shall bee Sinus versus , (BLUNDEV-E2-H,50V.80)

as in the former example your Sinus complementi was 5$$936$$649. which
being subtracted out of the totall Sine there remaineth 63$$351.
(BLUNDEV-E2-H,50V.81)

and that number is Sinus versus : (BLUNDEV-E2-H,50V.82)

for if you adde this remainder to the number which you subtracted , it
will make up the totall Sine (BLUNDEV-E2-H,50V.83)

But there is one thing more necessarie to be knowne then this , because
it commeth oftner in vse , (BLUNDEV-E2-H,50V.84)

and that is upon some diuision made how to find out the Arke of any
quotient , which is to be done thus : (BLUNDEV-E2-H,50V.85)

Enter with the quotient into the body of the tables ,
(BLUNDEV-E2-H,50V.86)

and leaue not seeking amongst the squares of the Sines , vntill you
haue found out the iust number of the quotient <paren> if it be there
</paren> (BLUNDEV-E2-H,50V.87)

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 , (BLUNDEV-E2-H,50V.88)

and hauing found that number , looke in the front of that collum ,
(BLUNDEV-E2-H,50V.89)

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 . (BLUNDEV-E2-H,51R.90)

As for example , hauing diuided one number by another , I finde the
quotient to be whereof I would know the arch , (BLUNDEV-E2-H,51R.91)

now in seeking this quotient amongst the Sines , I $can $not
{TEXT:cannot} finde that iust number , (BLUNDEV-E2-H,51R.92)

but I find in the first Page , and in the tenth collum 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 , (BLUNDEV-E2-H,51R.93)

and directly against that Sine on the left hand , I find 2'9. belonging
to that arch , whereof that quotient is the <font> Sinus </font> , so
as I gather hereof that the arch of the foresaid quotient is 4. degrees
, 2'9. (BLUNDEV-E2-H,51R.94)

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
(BLUNDEV-E2-H,51R.95)

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. (BLUNDEV-E2-H,51R.96)

and by that account a very small quotient may be found in these tables
. (BLUNDEV-E2-H,51R.97)

And whatsoeuer hath beene said here touching the order that is to be
obserued in the first tables of <font> Monte Regio </font> , whose
totall Sine is the like in all points is to be obserued in the last
tables , whose totall Sine is (BLUNDEV-E2-H,51R.98)

Thus much touching the order of the foresaid tables of <font> Monte
Regio </font> Printed in Folio : (BLUNDEV-E2-H,51R.99)

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
<font> Clauius </font> , 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 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 , (BLUNDEV-E2-H,51R.100)

and I doe worke all such conclusions as hereafter follow , by the said
tables , the totall Sine whereof is according to the last tables of
<font> Monte Regio </font> . (BLUNDEV-E2-H,51R.101)

But for so much as some may haue already the tables of <font> Monte
Regio </font> Printed in Folio , not knowing perhaps the vse thereof ,
I will set downe two conclusions to bee wrought by those tables ,
(BLUNDEV-E2-H,51R.102)

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 <font> Clauius </font> : (BLUNDEV-E2-H,51V.103)

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 <font> Monte
Regio </font> , Printed in folio before that euer I saw <font> Clauius
</font> his booke , I mind not now to alter them but to let them stand
still as they are . (BLUNDEV-E2-H,51V.104)

<P_152R>

<heading>

OF THE HORIZON BOTH RIGHT AND OBLIQUE , MAKING THEREBY THREE KINDS OF
SPHEARES , THAT IS , THE RIGHT , THE PARALELL , AND THE OBLIQUE SPHEARE
. (BLUNDEV-E2-H,152R.107)

CAP. 17 . (BLUNDEV-E2-H,152R.108)

WHAT IS THE HORIZON ? (BLUNDEV-E2-H,152R.109)

</heading>

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 ,
(BLUNDEV-E2-H,152R.111)

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 $can
$not {TEXT:cannot} passe , (BLUNDEV-E2-H,152R.112)

and therfore this circle in Greeke is called <font> Horizon </font> ,
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 <font> Zenith </font> : and
the other directlie vnder your feete , called in the same tongue <font>
Nadir </font> , that is to say the pointe opposite ,
(BLUNDEV-E2-H,152R.113)

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 , (BLUNDEV-E2-H,152V.114)

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 .
(BLUNDEV-E2-H,152V.115)

<heading>

WHEN IS THE HORIZON SAID TO BE RIGHT , AND THEREBY TO MAKE A RIGHT
SPHEARE ? (BLUNDEV-E2-H,152V.117)

</heading>

It may be said to be right two manner of waies ,
(BLUNDEV-E2-H,152V.119)

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 <font> Zenith </font> in the Equinoctiall , which
passeth right ouer their heads , to whom the daies and nights are
alwaies equal . (BLUNDEV-E2-H,152V.120)

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 <font>
Zenith </font> , (BLUNDEV-E2-H,152V.121)

and their Horizon is all one with the Equinoctiall , cutting the
Arletree of the world in the very midst with right angles ,
(BLUNDEV-E2-H,152V.122)

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 : (BLUNDEV-E2-H,152V.123)

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 <font> Zenith </font> and <font> Nadir
</font> . (BLUNDEV-E2-H,152V.124)

<heading>

WHEN IS IT SAID TO BEE AN OBLIQUE HORIZON , AND THEREBY TO MAKE AN
OBLIQUE SPHEARE ? (BLUNDEV-E2-H,152V.126)

</heading>

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 , (BLUNDEV-E2-H,152V.128)

so much is your <font> Zenith </font> distant from the Equinoctiall ,
(BLUNDEV-E2-H,152V.129)

and the nigher that your Horizon approcheth to the Pole , the nigher
your <font> Zenith </font> approcheth to the Equinoctial .
(BLUNDEV-E2-H,152V.130)

Againe , looke how much the Equinoctiall is eleuated aboue your Horizon
, (BLUNDEV-E2-H,152V.131)

so much is your <font> Zenith </font> distant from the Pole , all which
{COM:see_info_file_regarding_missing_page_153}
<P_154R_misnumbered_as_151R> things this figure here following doth
plainely shew , whereby you may easily perceiue that the latitude ,
which is the distance of your <font> Zenith </font> from the
Equinoctiall , is alwaies equall to the altitude of the Pole , which is
the distance betwixt your Horizon & the Pole ,
(BLUNDEV-E2-H,154R_misnumbered_as_151R.132)

as for example , knowing the latitude of <font> Norwich </font> to be
52 , degrees lay the <font> Zenith </font> of this figure upon the 52.
degrees , reckoning from the Equinoctiall towards the pole Arctique on
your left hand , (BLUNDEV-E2-H,154R_misnumbered_as_151R.133)

and looke what distance is betwixt the saide <font> Zenith </font> and
the Equinoctiall , (BLUNDEV-E2-H,154R_misnumbered_as_151R.134)

the selfe same distance you shall find to be betwixt the Horizon and
the foresaid Pole on your right hand ,
(BLUNDEV-E2-H,154R_misnumbered_as_151R.135)

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 .
(BLUNDEV-E2-H,154R_misnumbered_as_151R.136)

<font> A Figure shewing the latitude of any place to bee equall to the
eleuation of the Pole . </font> {COM:figure_omitted}
(BLUNDEV-E2-H,154R_misnumbered_as_151R.137)

<P_154V_misnumbered_as_151V>

<heading>

WHAT OTHER VSES HATH THIS CIRCLE ?
(BLUNDEV-E2-H,154V_misnumbered_as_151V.140)

</heading>

In this circle are set downe the foure quarters of the world , as East
, West , North and South , and the rest of the winds :
(BLUNDEV-E2-H,154V_misnumbered_as_151V.142)

Againe , this circle deuideth the artificiall day from the artificiall
night , (BLUNDEV-E2-H,154V_misnumbered_as_151V.143)

for all the while that the Sun is aboue the Horizon it is day ,
(BLUNDEV-E2-H,154V_misnumbered_as_151V.144)

& whilest it is under the same it is night .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.145)

And by this circle wee knowe what starres do continually appeare , and
which are continually hidden , also what starres doe rise and goe downe
. (BLUNDEV-E2-H,154V_misnumbered_as_151V.146)

Againe , in taking the eleuation of the Pole , this circle is chiefly
to be considered , (BLUNDEV-E2-H,154V_misnumbered_as_151V.147)

for when we know how many degrees the Pole is raised aboue the Horizon
, then we haue the eleuation therof for that place .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.148)

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 .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.149)

<heading>

HOW SHAL I KNOW IN ANY PLACE , HAUING AN OBLIQUE HORIZON , HOW MUCH THE
POLE IS ELEUATED ABOUE THE HORIZON ?
(BLUNDEV-E2-H,154V_misnumbered_as_151V.151)

</heading>

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 .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.153)

<heading>

OF THE MERIDIAN , AND OF THE VSES THEREOF .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.155)

CAP. 18 . (BLUNDEV-E2-H,154V_misnumbered_as_151V.156)

WHAT IS THE MERIDIAN ? (BLUNDEV-E2-H,154V_misnumbered_as_151V.157)

</heading>

It is a great immoouable circle passing through the Poles of the worlde
, and through the Poles of the Horizon .
(BLUNDEV-E2-H,154V_misnumbered_as_151V.159)

<heading>

WHY IS IT CALLED THE MERIDIAN ?
(BLUNDEV-E2-H,154V_misnumbered_as_151V.161)

</heading>

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 <font> Zenith </font> that
Circle passeth . (BLUNDEV-E2-H,154V_misnumbered_as_151V.163)

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 ,
(BLUNDEV-E2-H,154V_misnumbered_as_151V.164)

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 :
(BLUNDEV-E2-H,154V_misnumbered_as_151V.165)

but if they be distant one from another East & West , bee it neuer so
little , then they must <P_155R_misnumbered_as_152R> needes haue diuers
Meridians , (BLUNDEV-E2-H,155R_misnumbered_as_152R.166)

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 or {HELSINKI:of} Planets , whereof we haue already spoken in
the 11. Chapter . (BLUNDEV-E2-H,155R_misnumbered_as_152R.167)

<heading>

HOW MANY MERIDIANS BE THERE ?
(BLUNDEV-E2-H,155R_misnumbered_as_152R.169)

</heading>

The Astronomers doe appoint for euery two degrees of the Equinoctiall a
Meridian , so as they make in all
(BLUNDEV-E2-H,155R_misnumbered_as_152R.171)

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 .
(BLUNDEV-E2-H,155R_misnumbered_as_152R.172)

And in such Spheares as haue not a foote and a standing Horizon , there
is no Meridian at al , (BLUNDEV-E2-H,155R_misnumbered_as_152R.173)

but the two Colures are faine to supply their want ,
(BLUNDEV-E2-H,155R_misnumbered_as_152R.174)

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 , (BLUNDEV-E2-H,155R_misnumbered_as_152R.175)

for it is noonetide sooner to those whose Meridian is more Eastward
then to them whose Meridian is more Westward .
(BLUNDEV-E2-H,155R_misnumbered_as_152R.176)

And contrariwise the Eclipse of the Sun or Moone appeareth sooner to
those whose Meridian is more Westward .
(BLUNDEV-E2-H,155R_misnumbered_as_152R.177)

<heading>

WHAT OTHER VSES HATH THIS CIRCLE ?
(BLUNDEV-E2-H,155R_misnumbered_as_152R.179)

</heading>

This circle deuideth the East part of the world from the West
(BLUNDEV-E2-H,155R_misnumbered_as_152R.181)

and also it sheweth both the North and South ,
(BLUNDEV-E2-H,155R_misnumbered_as_152R.182)

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 .
(BLUNDEV-E2-H,155R_misnumbered_as_152R.183)

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 ,
(BLUNDEV-E2-H,155R_misnumbered_as_152R.184)

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 .
(BLUNDEV-E2-H,155R_misnumbered_as_152R.185)

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 ,
(BLUNDEV-E2-H,155R_misnumbered_as_152R.186)

& the other halfe is from the same circle to the going down of the
Sunne under the Horizon . (BLUNDEV-E2-H,155R_misnumbered_as_152R.187)

And the first parte of the night is <P_155V_misnumbered_as_152V> the
space betwixt the Suns going down and his comming againe to the
Meridian , which is at midnight ,
(BLUNDEV-E2-H,155V_misnumbered_as_152V.188)

and from thence to the time of his rising is the other halfe of the
night , (BLUNDEV-E2-H,155V_misnumbered_as_152V.189)

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 . (BLUNDEV-E2-H,155V_misnumbered_as_152V.190)

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 .
(BLUNDEV-E2-H,155V_misnumbered_as_152V.191)

<heading>

OF THE VERTICALL CIRCLES , AND VSES THEREOF
(BLUNDEV-E2-H,155V_misnumbered_as_152V.193)

CAP. $19 {TEXT:13} (BLUNDEV-E2-H,155V_misnumbered_as_152V.194)

</heading>

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 <font> Zenith </font> , wheresoeuer we be vpon the
land or sea , crossing our Horizon in 2. points opposite , and deuiding
the same into two equall parts ,
(BLUNDEV-E2-H,155V_misnumbered_as_152V.196)

and such kind of circles are called in Arabick <font> Azimuthes </font>
, whereof you may imagine that there be so many as ther be rombes or
winds in the Marriners compasse , which are in number 32.
(BLUNDEV-E2-H,155V_misnumbered_as_152V.197)

yea , and if you will , you may make halfe so many as there be degrees
in the Horizon , which are in nu~ber the halfe whereof is
(BLUNDEV-E2-H,155V_misnumbered_as_152V.198)

If you be right under the Equinoctiall , and doe goe or saile right
East or West , then the Equinoctiall is your Verticall circle ,
(BLUNDEV-E2-H,155V_misnumbered_as_152V.199)

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 . (BLUNDEV-E2-H,155V_misnumbered_as_152V.200)

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 <font> Zenith </font>
point with right Sphericall angles , is most properly called the
uerticall circle , (BLUNDEV-E2-H,155V_misnumbered_as_152V.201)

and the learned seamen haue great respect to two speciall kinds of
Verticall circles , that is , the Magneticall Meridian , and the <font>
Azimuth </font> of the Sunne .
(BLUNDEV-E2-H,155V_misnumbered_as_152V.202)

<P_156R_misnumbered_as_155R>

<heading>

WHAT MANNER OF VERTICALL CIRCLES BEE THOSE ,
(BLUNDEV-E2-H,156R_misnumbered_as_155R.205)

AND WHERETO SERUE THEY ? (BLUNDEV-E2-H,156R_misnumbered_as_155R.206)

</heading>

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 , (BLUNDEV-E2-H,156R_misnumbered_as_155R.208)

and deuideth the Horizon into two equall parts , by crossing the same
in two points opposite . (BLUNDEV-E2-H,156R_misnumbered_as_155R.209)

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 . (BLUNDEV-E2-H,156R_misnumbered_as_155R.210)

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 . (BLUNDEV-E2-H,156R_misnumbered_as_155R.211)

<heading>

WHAT VSE IS THERE OF THE VERTICALL CIRCLES , OR AZIMUTHES ?
(BLUNDEV-E2-H,156R_misnumbered_as_155R.213)

</heading>

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 .
(BLUNDEV-E2-H,156R_misnumbered_as_155R.215)

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 :
(BLUNDEV-E2-H,156R_misnumbered_as_155R.216)

Also by helpe of the uerticall Circle most properly so called , are the
twelue houses of heauen set , according to <font> Campanus </font> and
<font> Gazula </font> . (BLUNDEV-E2-H,156R_misnumbered_as_155R.217)

And by helpe of these Circles you may also knowe how any place vppon
the earth beareth one from another eyther Eastward or {HELSINKI:of}
Westward , and so foorth , (BLUNDEV-E2-H,156R_misnumbered_as_155R.218)

for euerie place hath his seuerall Azimuth aunswerable to the Horizon
and Zenith of the saide place .
(BLUNDEV-E2-H,156R_misnumbered_as_155R.219)

<heading>

OF CERTAINE CIRCLES CALLED <font> ALMICANTERATHES </font> .
(BLUNDEV-E2-H,156R_misnumbered_as_155R.221)

</heading>

Since I haue spoken heere somewhat of the uerticall Circles called
<font> Azimuthes </font> , 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 <font> Almicanterathes </font> , that is to say
, Circles of Altitude , which though they be not al great Circles , for
euery one <P_156V_misnumbered_as_155V> lesser then other proceeding
fro~ the oblique Horizon of any place to the Zenith of the said place ,
yet the first <font> Almicanterath </font> which is the verie oblique
Horizon it selfe , is a great Circle deuiding the Spheare into two
equall parts , (BLUNDEV-E2-H,156V_misnumbered_as_155V.223)

and all the rest are lesser and lesser , untill you come to the verie
Zenith , (BLUNDEV-E2-H,156V_misnumbered_as_155V.224)

and are paralels to the Horizon , euen as the Tropiques and the other
lesser Circles are paralels unto the Equinoctiall .
(BLUNDEV-E2-H,156V_misnumbered_as_155V.225)

And the Zenith in Sphericall bodies is the Centre of them all , though
it bee not so in Astrolabes ,
(BLUNDEV-E2-H,156V_misnumbered_as_155V.226)

in there euerie <font> Almicanterath </font> is saine to haue his
seuerall Centre , of which Circles there be in all according to the
number of degrees contained betwixt the oblique Horizon and the Zenith
, (BLUNDEV-E2-H,156V_misnumbered_as_155V.227)

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 .
(BLUNDEV-E2-H,156V_misnumbered_as_155V.228)

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 .
(BLUNDEV-E2-H,156V_misnumbered_as_155V.229)

<heading>

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 .
(BLUNDEV-E2-H,156V_misnumbered_as_155V.231)

CAP. . (BLUNDEV-E2-H,156V_misnumbered_as_155V.232)

WHICH CALL YOU THE LESSER CIRCLES ?
(BLUNDEV-E2-H,156V_misnumbered_as_155V.233)

</heading>

They are those that doe not deuide the Spheare into two equall parts ,
as the great Circles doe , (BLUNDEV-E2-H,156V_misnumbered_as_155V.235)

and of such there bee foure , that is the two Polar circles , and the
two Tropiques , that is to say , the Tropique of <font> Cancer </font>
, and the Tropique of <font> Capricorne </font> , of which Polar
circles the one is called Arctique , and the other Antarctique ,
(BLUNDEV-E2-H,156V_misnumbered_as_155V.236)

and are made by the turning about of the two Poles of the Zodiaque ,
which Poles being situated in the Colure of the
<P_157R_misnumbered_as_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 . (BLUNDEV-E2-H,157R_misnumbered_as_156R.237)

<heading>

WHICH IS THE ARCTIQUE CIRCLE ,
(BLUNDEV-E2-H,157R_misnumbered_as_156R.239)

AND WHY IS IT SO CALLED ? (BLUNDEV-E2-H,157R_misnumbered_as_156R.240)

</heading>

The Arctique Circle is that which is next to the North Pole ,
(BLUNDEV-E2-H,157R_misnumbered_as_156R.242)

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 , (BLUNDEV-E2-H,157R_misnumbered_as_156R.243)

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 ,
(BLUNDEV-E2-H,157R_misnumbered_as_156R.244)

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 ,
(BLUNDEV-E2-H,157R_misnumbered_as_156R.245)

and the Centre of this Circle is the North Pole of the world which is
not to be seene with mans eye .
(BLUNDEV-E2-H,157R_misnumbered_as_156R.246)

<heading>

WHAT IS THE ANTARCTIQUE CIRCLE ?
(BLUNDEV-E2-H,157R_misnumbered_as_156R.248)

</heading>

It is that which is next unto the South Pole ,
(BLUNDEV-E2-H,157R_misnumbered_as_156R.250)

and it is so called , because it is opposite or contrarie to the Circle
Arctique . (BLUNDEV-E2-H,157R_misnumbered_as_156R.251)

<heading>

NOW DESCRIBE THE TWO TROPIQUES .
(BLUNDEV-E2-H,157R_misnumbered_as_156R.253)

</heading>

The Tropique of <font> Cancer </font> 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 <font> Cancer
</font> , 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 . (BLUNDEV-E2-H,157R_misnumbered_as_156R.255)

And so from thence he declineth to the other Tropique called the
Tropique of <font> Capricorne </font> , 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 <font>
Capricorne </font> , 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_157V_misnumbered_as_156V> at the longest :
(BLUNDEV-E2-H,157V_misnumbered_as_156V.256)

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 ,
(BLUNDEV-E2-H,157V_misnumbered_as_156V.257)

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 .
(BLUNDEV-E2-H,157V_misnumbered_as_156V.258)

For of the two cold Zones , the one lyeth betwixt the North pole and
the Circle Arctique , (BLUNDEV-E2-H,157V_misnumbered_as_156V.259)

and the other lyeth betwixt the South Pole and the Circle Antarctique ,
(BLUNDEV-E2-H,157V_misnumbered_as_156V.260)

& of the two temperate Zones , the one lyeth betwixt the Circle
Arctique , & the Tropique of <font> Cancer </font> ,
(BLUNDEV-E2-H,157V_misnumbered_as_156V.261)

and the other lyeth betwixt the Circle Antarctique , and the Tropique
of <font> Capricorne </font> ,
(BLUNDEV-E2-H,157V_misnumbered_as_156V.262)

& 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 .
(BLUNDEV-E2-H,157V_misnumbered_as_156V.263)

<font> A figure shewing the fiue foresaid Zones . </font>
(BLUNDEV-E2-H,157V_misnumbered_as_156V.264)

{COM:figure_omitted}

<P_158R_misnumbered_as_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 : (BLUNDEV-E2-H,158R_misnumbered_as_157R.267)

Againe you haue to understand that euery one of these lesser Circles
doth containe in length , degrees as well as euery one of the greater
Circles , (BLUNDEV-E2-H,158R_misnumbered_as_157R.268)

but the degrees are not of like bignesse , no more then the Circles
themselues are like in compasse or circuit ,
(BLUNDEV-E2-H,158R_misnumbered_as_157R.269)

for the lesser the Circles are in circuit , the lesser their degrees
must needes be . (BLUNDEV-E2-H,158R_misnumbered_as_157R.270)

<heading>

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 .
(BLUNDEV-E2-H,158R_misnumbered_as_157R.272)

</heading>

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 as hath beene
said before : (BLUNDEV-E2-H,158R_misnumbered_as_157R.274)

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 ,
(BLUNDEV-E2-H,158R_misnumbered_as_157R.275)

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 ,
(BLUNDEV-E2-H,158R_misnumbered_as_157R.276)

and in the beginning of the first collum on the left hand is set downe
one degree , which is the first degree of and nighest unto the
Equinoctiall , right against which one degree is placed towards the
right hand , 59. minutes , and 59. seconds :
(BLUNDEV-E2-H,158R_misnumbered_as_157R.277)

and so proceeding from degree to degree successiuely , untill you come
to you shall finde how many minutes and seconds doe answere to one
degree of the Equinoctiall ,
(BLUNDEV-E2-H,158R_misnumbered_as_157R.278)

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 .
(BLUNDEV-E2-H,158R_misnumbered_as_157R.279)

