long squares, that may be made of that whole line and one of
his partes with the square of the other part, shall be
equall to the square that is made of the whole line and the
saide first portion ioyned to him in lengthe as one whole
line.
_Example._
[Illustration]
The firste line is A.B, and is deuided by C. into two vnequall
partes as happeneth. The long square of yt, and his lesser
portion A.C, is foure times drawen, the first is E.G.M.K, the
seconde is K.M.Q.O, the third is H.K.R.S, and the fourthe is
K.L.S.T. And where as it appeareth that one of the little
squares (I meane K.L.P.O) is reckened twise, ones as parcell of
the second long square and agayne as parte of the thirde long
square, to auoide ambiguite, you may place one insteede of it,
an other square of equalitee, with it. that is to saye, D.E.K.H,
which was at no tyme accompting as parcell of any one of them,
and then haue you iiij. long squares distinctly made of the
whole line A.B, and his lesser portion A.C. And within them is
there a greate full square P.Q.T.V. whiche is the iust square of
B.C, beynge the greatter portion of the line A.B. And that those
fiue squares doo make iuste as muche as the whole square of that
longer line D.G, (whiche is as longe as A.B, and A.C. ioyned
togither) it may be iudged easyly by the eye, sith that one
greate square doth comprehend in it all the other fiue squares,
that is to say, foure long squares (as is before mencioned) and
one full square. which is the intent of the Theoreme.
_The xliij. Theoreme._
If a right line be deuided into ij. equal partes first, and
one of those parts again into other ij. parts, as chaunce
hapeneth, the square that is made of the last part of the
line so diuided, and the square of the residue of that whole
line, are double to the square of halfe that line, and to
the square of the middle portion of the same line.
_Example._
[Illustration]
The line to be deuided is A.B, and is parted in C. into two
equall partes, and then C.B, is deuided againe into two partes
in D, so that the meaninge of the Theoreme, is that the square
of D.B. which is the latter parte of the line, and the square of
A.D, which is the residue of the whole line. Those two squares,
I say, ar double to the square of one halfe of the line, and to
the square of C.D, which is the middle portion of those thre
diuisions. Which thing that you maye more easilye perceaue,
I haue drawen foure squares, whereof the greatest being marked
with E. is the square of A.D. The next, which is marked with G,
is the square of halfe the line, that is, of A.C, And the other
two little squares marked with F. and H, be both of one bignes,
by reason that I did diuide C.B. into two equall partes, so that
you amy take the square F, for the square of D.B, and the square
H, for the square of C.D. Now I thinke you doubt not, but that
the square E. and the square F, ar double so much as the square
G. and the square H, which thing the easyer is to be
vnderstande, bicause that the greate square hath in his side
iij. quarters of the firste line, which multiplied by itselfe
maketh nyne quarters, and the square F. containeth but one
quarter, so that bothe doo make tenne quarters.
Then G. contayneth iiij. quarters, seynge his side containeth
twoo, and H. containeth but one quarter, whiche both make
but fiue quarters, and that is but halfe of tenne.
Whereby you may easylye coniecture,
that the meanynge of the theoreme
is verified in the
figures of this
example.
_The xliiij. Theoreme._
If a right line be deuided into ij. partes equally, and an
other portion of a righte lyne annexed to that firste line,
the square of this whole line so compounded, and the square
of the portion that is annexed, ar doule as much as the
square of the halfe of the firste line, and the square of
the other halfe ioyned in one with the annexed portion, as
one whole line.
_Example._
[Illustration]
The line is A.B, and is diuided firste into twoo equal partes in
C, and then is there annexed to it an other portion whiche is
B.D. Now saith the Theoreme, that the square of A.D, and the
square of B.D, ar double to the square of A.C, and to the square
of C.D. The line A.B. containing four partes, then must needes
his halfe containe ij. partes of such partes I suppose B.D.
(which is the annexed line) to containe thre, so shal the hole
line comprehend vij. parts, and his square xlix. parts, where
vnto if you ad y^e square of the annexed lyne, whiche maketh
nyne, than those bothe doo yelde, lviij. whyche must be double
to the square of the halfe lyne with the annexed portion. The
halfe lyne by it selfe conteyneth but .ij. partes, and therfore
his square dooth make foure. The halfe lyne with the annexed
portion conteyneth fiue, and the square of it is .xxv, now put
foure to .xxv, and it maketh iust .xxix, the euen halfe of fifty
and eight, wherby appereth the truthe of the theoreme.
_The .xlv. theoreme._
In all triangles that haue a blunt angle, the square of the
side that lieth against the blunt angle, is greater than the
two squares of the other twoo sydes, by twise as muche as is
comprehended of the one of those .ij. sides (inclosyng the
blunt corner) and the portion of the same line, beyng drawen
foorth in lengthe, which lieth betwene the said blunt corner
and a perpendicular line lightyng on it, and drawen from one
of the sharpe angles of the foresayd triangle.
_Example._
For the declaration of this theoreme and the next also, whose
vse are wonderfull in the practise of Geometrie, and in
measuryng especially, it shall be nedefull to declare that euery
triangle that hath no ryght angle as those whyche are called (as
in the boke of practise is declared) sharp cornered triangles,
and blunt cornered triangles, yet may they be brought to haue a
ryght angle, eyther by partyng them into two lesser triangles,
or els by addyng an other triangle vnto them, whiche may be a
great helpe for the ayde of measuryng, as more largely shall be
sette foorthe in the boke of measuryng. But for this present
place, this forme wyll I vse, (whiche Theon also vseth) to adde
one triangle vnto an other, to bryng the blunt cornered triangle
into a ryght angled triangle, whereby the proportion of the
squares of the sides in suche a blunt cornered triangle may the
better bee knowen.
[Illustration]
Fyrst therfore I sette foorth the triangle A.B.C, whose corner
by C. is a blunt corner as you maye well iudge, than to make an
other triangle of yt with a ryght angle, I must drawe forth the
side B.C. vnto D, and from the sharp corner by A. I brynge a
plumbe lyne or perpendicular on D. And so is there nowe a newe
triangle A.B.D. whose angle by D. is a right angle. Nowe
accordyng to the meanyng of the Theoreme, I saie, that in the
first triangle A.B.C, because it hath a blunt corner at C, the
square of the line A.B. whiche lieth against the said blunte
corner, is more then the square of the line A.C, and also of the
lyne B.C, (whiche inclose the blunte corner) by as muche as will
amount twise of the line B.C, and that portion D.C. whiche lieth
betwene the blunt angle by C, and the perpendicular line A.D.
The square of the line A.B, is the great square marked with E.
The square of A.C, is the meane square marked with F. The square
of B.C, is the least square marked with G. And the long square
marked with K, is sette in steede of two squares made of B.C,
and C.D. For as the shorter side is the iuste lengthe of C.D, so
the other longer side is iust twise so longe as B.C, Wherfore I
saie now accordyng to the Theoreme, that the greatte square E,
is more then the other two squares F. and G, by the quantitee of
the longe square K, wherof I reserue the profe to a more
conuenient place, where I will also teache the reason howe to
fynde the lengthe of all suche perpendicular lynes, and also of
the line that is drawen betweene the blunte angle and the
perpendicular line, with sundrie other very pleasant
conclusions.
_The .xlvi. Theoreme._
In sharpe cornered triangles, the square of anie side that
lieth against a sharpe corner, is lesser then the two
squares of the other two sides, by as muche as is comprised
twise in the long square of that side, on whiche the
perpendicular line falleth, and the portion of that same
line, liyng betweene the perpendicular, and the foresaid
sharpe corner.
_Example._
[Illustration]
Fyrst I sette foorth the triangle A.B.C, and in yt I draw a
plumbe line from the angle C. vnto the line A.B, and it lighteth
in D. Nowe by the theoreme the square of B.C. is not so muche as
the square of the other two sydes, that of B.A. and of A.C. by
as muche as is twise conteyned in the long square made of A.B,
and A.D, A.B. beyng the line or syde on which the perpendicular
line falleth, and A.D. beeyng that portion of the same line
whiche doth lye betwene the perpendicular line, and the sayd
sharpe angle limitted, whiche angle is by A.
For declaration of the figures, the square marked with E. is the
square of B.C, whiche is the syde that lieth agaynst the sharpe
angle, the square marked with G. is the square of A.B, and the
square marked with F. is the square of A.C, and the two longe
squares marked with H.K, are made of the hole line A.B, and one
of his portions A.D. And truthe it is that the square E. is
lesser than the other two squares C. and F. by the quantitee of
those two long squares H. and K. Wherby you may consyder agayn,
an other proportion of equalitee, that is to saye, that the
square E. with the twoo longsquares H.K, are iuste equall to the
other twoo squares C. and F. And so maye you make, as it were an
other theoreme. _That in al sharpe cornered triangles, where a
perpendicular line is drawen frome one angle to the side that
lyeth againste it, the square of anye one side, with the ij.
longesquares made at that hole line, whereon the perpendicular
line doth lighte, and of that portion of it, which ioyneth to
that side whose square is all ready taken, those thre figures,
I say, are equall to the ij. squares, of the other ij. sides of
the triangle._ In whiche you muste vnderstand, that the side on
which the perpendiculare falleth, is thrise vsed, yet is his
square but ones mencioned, for twise he is taken for one side of
the two long squares. And as I haue thus made as it were an
other theoreme out of this fourty and sixe theoreme, so mighte I
out of it, and the other that goeth nexte before, make as manny
as woulde suffice for a whole booke, so that when they shall bee
applyed to practise, and consequently to expresse their
benefite, no manne that hathe not well wayde their wonderfull
commoditee, would credite the possibilitie of their wonderfull
vse, and large ayde in knowledge. But all this wyll I remitte to
a place conuenient.
_The xlvij. Theoreme._
If ij. points be marked in the circumference of a circle, and
a right line drawen frome the one to the other, that line
must needes fal within the circle.
_Example._
[Illustration]
The circle is A.B.C.D, the ij. poinctes are A.B, the righte line
that is drawenne frome the one to the other, is the line A.B,
which as you see, must needes lyghte within the circle. So if
you putte the pointes to be A.D, or D.C, or A.C, other B.C, or
B.D, in any of these cases you see, that the line that is drawen
from the one pricke to the other dothe euermore run within the
edge of the circle, els canne it be no right line. How be it,
that a croked line, especially being more croked then the
portion of the circumference, maye bee drawen from pointe to
pointe withoute the circle. But the theoreme speaketh only of
right lines, and not of croked lines.
_The xlviij. Theoreme._
If a righte line passinge by the centre of a circle, doo
crosse an other right line within the same circle, passinge
beside the centre, if he deuide the saide line into twoo
equal partes, then doo they make all their angles righte.
And contrarie waies, if they make all their angles righte,
then doth the longer line cutte the shorter in twoo partes.
_Example._
[Illustration]
The circle is A.B.C.D, the line that passeth by the centre, is
A.E.C, the line that goeth beside the centre is D.B. Nowe saye
I, that the line A.E.C, dothe cutte that other line D.B. into
twoo iuste partes, and therefore all their four angles ar righte
angles. And contrarye wayes, bicause all their angles are righte
angles, therfore it muste be true, that the greater cutteth the
lesser into two equal partes, accordinge as the Theoreme would.
_The xlix. Theoreme._
If twoo right lines drawen in a circle doo crosse one an
other, and doo not passe by the centre, euery of them dothe
not deuide the other into equall partions.
_Example._
[Illustration]
The circle is A.B.C.D, and the centre is E, the one line A.C,
and the other is B.D, which two lines crosse one an other, but
yet they go not by the centre, wherefore accordinge to the
woordes of the theoreme, eche of theim doth cuytte the other
into equall portions. For as you may easily iudge, A.C. hath one
portion longer and an other shorter, and so like wise B.D.
Howbeit, it is not so to be vnderstand, but one of them may be
deuided into ij. euen parts, but bothe to bee cutte equally in
the middle, is not possible, onles both passe through the centre,
therfore much rather when bothe go beside the centre, it can not
be that eche of theym shoulde be iustely parted into ij. euen
partes.
_The L. Theoreme._
If two circles crosse and cut one an other, then haue not
they both one centre.
_Example._
[Illustration]
This theoreme seemeth of it selfe so manifest, that it neadeth
nother demonstration nother declaracion. Yet for the plaine
vnderstanding of it, I haue sette forthe a figure here, where
ij. circles be drawen, so that one of them doth crosse the other
(as you see) in the pointes B. and G, and their centres appear
at the firste sighte to bee diuers. For the centre of the one is
F, and the centre of the other is E, which diffre as farre
asondre as the edges of the circles, where they bee most
distaunte in sonder.
_The Li. Theoreme._
If two circles be so drawen, that one of them do touche the
other, then haue they not one centre.
_Example._
[Illustration]
There are two circles made, as you see, the one is A.B.C, and
hath his centre by G, the other is B.D.E, and his centre is by
F, so that it is easy enough to perceaue that their centres doe
dyffer as muche a sonder, as the halfe diameter of the greater
circle is longer then the half diameter of the lesser circle. And
so must it needes be thought and said of all other circles in
lyke kinde.
_The .lij. theoreme._
If a certaine pointe be assigned in the diameter of a
circle, distant from the centre of the said circle, and from
that pointe diuerse lynes drawen to the edge and
circumference of the same circle, the longest line is that
whiche passeth by the centre, and the shortest is the
residew of the same line. And of al the other lines that is
euer the greatest, that is nighest to the line, which
passeth by the centre. And contrary waies, that is the
shortest, that is farthest from it. And amongest them all
there can be but onely .ij. equall together, and they must
nedes be so placed, that the shortest line shall be in the
iust middle betwixte them.
_Example._
[Illustration]
The circle is A.B.C.D.E.H, and his centre is F, the diameter is
A.E, in whiche diameter I haue taken a certain point distaunt
from the centre, and that pointe is G, from which I haue drawen
.iiij. lines to the circumference, beside the two partes of the
diameter, whiche maketh vp vi. lynes in all. Nowe for the
diuersitee in quantitie of these lynes, I saie accordyng to the
Theoreme, that the line whiche goeth by the centre is the
longest line, that is to saie, A.G, and the residewe of the same
diameter beeyng G.E, is the shortest lyne. And of all the other
that lyne is longest, that is neerest vnto that parte of the
diameter whiche gooeth by the centre, and that is shortest, that
is farthest distant from it, wherefore I saie, that G.B, is
longer then G.C, and therfore muche more longer then G.D, sith
G.C, also is longer then G.D, and by this maie you soone
perceiue, that it is not possible to drawe .ij. lynes on any one
side of the diameter, whiche might be equall in lengthe
together, but on the one side of the diameter maie you easylie
make one lyne equall to an other, on the other side of the same
diameter, as you see in this example G.H, to bee equall to G.D,
betweene whiche the lyne G.E, (as the shortest in all the
circle) doothe stande euen distaunte from eche of them, and it
is the precise knoweledge of their equalitee, if they be equally
distaunt from one halfe of the diameter. Where as contrary waies
if the one be neerer to any one halfe of the diameter then the
other is, it is not possible that they two may be equall in
lengthe, namely if they dooe ende bothe in the circumference of
the circle, and be bothe drawen from one poynte in the diameter,
so that the saide poynte be (as the Theoreme doeth suppose)
somewhat distaunt from the centre of the said circle. For if
they be drawen from the centre, then must they of necessitee be
all equall, howe many so euer they bee, as the definition of a
circle dooeth importe, withoute any regarde how neere so euer
they be to the diameter, or how distante from it. And here is to
be noted, that in this Theoreme, by neerenesse and distaunce is
vnderstand the nereness and distaunce of the extreeme partes of
those lynes where they touche the circumference. For at the
other end they do all meete and touche.
_The .liij. Theoreme._
If a pointe bee marked without a circle, and from it diuerse
lines drawen crosse the circle, to the circumference on the
other side, so that one of them passe by the centre, then
that line whiche passeth by the centre shall be the loongest
of them all that crosse the circle. And of the other lines
those are longest, that be nexte vnto it that passeth by the
centre. And those ar shortest, that be farthest distant from
it. But among those partes of those lines, whiche ende in
the outewarde circumference, that is most shortest, whiche
is parte of the line that passeth by the centre, and
amongeste the othere eche, of them, the nerer they are vnto
it, the shorter they are, and the farther from it, the
longer they be. And amongest them all there can not be more
then .ij. of any one length, and they two muste be on the two
contrarie sides of the shortest line.
_Example._
[Illustration]
Take the circle to be A.B.C, and the point assigned without it
to be D. Now say I, that if there be drawen sundrie lines from
D, and crosse the circle, endyng in the circumference on the
contrary side, as here you see, D.A, D.E, D.F, and D.B, then of
all these lines the longest must needes be D.A, which goeth by
the centre of the circle, and the nexte vnto it, that is D.E, is
the longest amongest the rest. And contrarie waies, D.B, is the
shorteste, because it is farthest distaunt from D.A. And so maie
you iudge of D.F, because it is nerer vnto D.A, then is D.B,
therefore is it longer then D.B. And likewaies because it is
farther of from D.A, then is D.E, therfore is it shorter then
D.E. Now for those partes of the lines whiche bee withoute the
circle (as you see) D.C, is the shortest. because it is the
parte of that line which passeth by the centre, And D.K, is next
to it in distance, and therefore also in shortnes, so D.G, is
farthest from it in distance, and therfore is the longest of
them. Now D.H, beyng nerer then D.G, is also shorter then it,
and beynge farther of, then D.K, is longer then it. So that for
this parte of the theoreme (as I think) you do plainly perceaue
the truthe thereof, so the residue hathe no difficulte. For
seing that the nearer any line is to D.C, (which ioyneth with
the diameter) the shorter it is and the farther of from it, the
longer it is. And seyng two lynes can not be of like distaunce
beinge bothe on one side, therefore if they shal be of one
lengthe, and consequently of one distaunce, they must needes bee
on contrary sides of the saide line D.C. And so appeareth the
meaning of the whole Theoreme.
And of this Theoreme dothe there folowe an other lyke. whiche
you maye calle other a theoreme by it selfe, or else a Corollary
vnto this laste theoreme, I passe not so muche for the name. But
his sentence is this: _when so euer any lynes be drawen frome
any pointe, withoute a circle, whether they crosse the circle,
or eande in the utter edge of his circumference, those two lines
that bee equally distaunt from the least line are equal
togither, and contrary waies, if they be equall togither, they
ar also equally distant from that least line._
For the declaracion of this proposition, it shall not need to
vse any other example, then that which is brought for the
explication of this laste theoreme, by whiche you may without
any teachinge easyly perceaue both the meanyng and also the
truth of this proposition.
_The Liiij. Theoreme._
If a point be set forthe in a circle, and from that pointe
vnto the circumference many lines drawen, of which more then
two are equal togither, then is that point the centre of
that circle.
_Example._
[Illustration]
The circle is A.B.C, and within it I haue sette fourth for an
example three prickes, which are D.E. and F, from euery one of
them I haue drawen (at the leaste) iiij. lines vnto the
circumference of the circle but frome D, I haue drawen more, yet
maye it appear readily vnto your eye, that of all the lines
whiche be drawen from E. and F, vnto the circumference, there
are but twoo equall, and more can not bee, for G.E. nor E.H.
hath none other equal to theim, nor canne not haue any beinge
drawen from the same point E. No more can L.F, or F.K, haue anye
line equall to either of theim, beinge drawen from the same
pointe F. And yet from either of those two poinctes are there
drawen twoo lines equall togither, as A.E, is equall to E.B, and
B.F, is equall to F.C, but there can no third line be drawen
equall to either of these two couples, and that is by reason
that they be drawen from a pointe distaunte from the centre of
the circle. But from D, althoughe there be seuen lines drawen,
to the circumference, yet all bee equall, bicause it is the
centre of the circle. And therefore if you drawe neuer so mannye
more from it vnto the circumference, all shall be equal, so that
this is the priuilege (as it were of the centre) and therfore no
other point can haue aboue two equal lines drawen from it vnto
the circumference. And from all pointes you maye drawe ij.
equall lines to the circumference of the circle, whether that
pointe be within the circle or without it.
_The lv. Theoreme._
No circle canne cut an other circle in more pointes then
two.
_Example._
[Illustration]
The first circle is A.B.F.E, the second circle is B.C.D.E, and
they crosse one an other in B. and in E, and in no more pointes.
Nother is it possible that they should, but other figures ther
be, which maye cutte a circle in foure partes, as you se in this
example. Where I haue set forthe one tunne forme, and one eye
forme, and eche of them cutteth euery of their two circles into
foure partes. But as they be irregulare formes, that is to saye,
suche formes as haue no precise measure nother proportion in
their draughte, so can there scarcely be made any certaine
theorem of them. But circles are regulare formes, that is to
say, such formes as haue in their protracture a iuste and
certaine proportion, so that certain and determinate truths may
be affirmed of them, sith they ar vniforme and vnchaungable.
_The lvi. Theoreme._
If two circles be so drawen, that the one be within the
other, and that they touche one an other: If a line bee
drawen by bothe their centres, and so forthe in lengthe,
that line shall runne to that pointe, where the circles do
touche.
_Example._
[Illustration]
The one circle, which is the greattest and vttermost is A.B.C,
the other circle that is y^e lesser, and is drawen within the
firste, is A.D.E. The centre of the greater circle is F, and the
centre of the lesser circle is G, the pointe where they touche
is A. And now you may see the truthe of the theoreme so
plainely, that it needeth no farther declaracion. For you maye
see, that drawinge a line from F. to G, and so forth in lengthe,
vntill it come to the circumference, it wyll lighte in the very
poincte A, where the circles touche one an other.
_The Lvij. Theoreme._
If two circles bee drawen so one withoute an other, that
their edges doo touche and a right line bee drawnenne frome
the centre of the one to the centre of the other, that line
shall passe by the place of their touching.
_Example._
[Illustration]
The firste circle is A.B.E, and his centre is K, The second
circle is D.B.C, and his centre is H, the point wher they do
touch is B. Nowe doo you se that the line K.H, whiche is drawen
from K, that is centre of the firste circle, vnto H, beyng
centre of the second circle, doth passe (as it must nedes by the
pointe B,) whiche is the verye poynte wher they do to touche
together.
_The .lviij. theoreme._
One circle can not touche an other in more pointes then one,
whether they touche within or without.
_Example._
[Illustration]
For the declaration of this Theoreme, I haue drawen iiij.
circles, the first is A.B.C, and his centre H. the second is
A.D.G, and his centre F. the third is L.M, and his centre K. the
.iiij. is D.G.L.M, and his centre E. Nowe as you perceiue the
second circle A.D.G, toucheth the first in the inner side, in so
much as it is drawen within the other, and yet it toucheth him
but in one point, that is to say in A, so lykewaies the third
circle L.M, is drawen without the firste circle and toucheth
hym, as you maie see, but in one place. And now as for the
.iiij. circle, it is drawen to declare the diuersitie betwene
touchyng and cuttyng, or crossyng. For one circle maie crosse
and cutte a great many other circles, yet can be not cutte any
one in more places then two, as the fiue and fiftie Theoreme
affirmeth.
_The .lix. Theoreme._
In euerie circle those lines are to be counted equall,
whiche are in lyke distaunce from the centre, And contrarie
waies they are in lyke distance from the centre, whiche be
equall.
_Example._
[Illustration]
In this figure you see firste the circle drawen, whiche is
A.B.C.D, and his centre is E. In this circle also there are
drawen two lines equally distaunt from the centre, for the line
A.B, and the line D.C, are iuste of one distaunce from the
centre, whiche is E, and therfore are they of one length. Again
thei are of one lengthe (as shall be proued in the boke of
profes) and therefore their distaunce from the centre is all
one.
_The lx. Theoreme._
In euerie circle the longest line is the diameter, and of
all the other lines, thei are still longest that be nexte
vnto the centre, and they be the shortest, that be farthest
distaunt from it.
_Example._
[Illustration]
In this circle A.B.C.D, I haue drawen first the diameter, whiche
is A.D, whiche passeth (as it must) by the centre E, Then haue I
drawen ij. other lines as M.N, whiche is neerer the centre, and
F.G, that is farther from the centre. The fourth line also on
the other side of the diameter, that is B.C, is neerer to the
centre then the line F.G, for it is of lyke distance as is the
lyne M.N. Nowe saie I, that A.D, beyng the diameter, is the
longest of all those lynes, and also of any other that maie be
drawen within that circle, And the other line M.N, is longer
then F.G. Also the line F.G, is shorter then the line B.C, for
because it is farther from the centre then is the lyne B.C. And
thus maie you iudge of al lines drawen in any circle, how to
know the proportion of their length, by the proportion of their
distance, and contrary waies, howe to discerne the proportion of
their distance by their lengthes, if you knowe the proportion of
their length. And to speake of it by the waie, it is a
maruaylouse thyng to consider, that a man maie knowe an exacte
proportion betwene two thynges, and yet can not name nor attayne
the precise quantitee of those two thynges, As for exaumple, If
two squares be sette foorthe, whereof the one containeth in it
fiue square feete, and the other contayneth fiue and fortie
foote, of like square feete, I am not able to tell, no nor yet
anye manne liuyng, what is the precyse measure of the sides of
any of those .ij. squares, and yet I can proue by vnfallible
reason, that their sides be in a triple proportion, that is to
saie, that the side of the greater square (whiche containeth
.xlv. foote) is three tymes so long iuste as the side of the
lesser square, that includeth but fiue foote. But this seemeth
to be spoken out of ceason in this place, therfore I will omitte
it now, reseruyng the exacter declaration therof to a more
conuenient place and time, and will procede with the residew of
the Theoremes appointed for this boke.
_The .lxi. Theoreme._
If a right line be drawen at any end of a diameter in
perpendicular forme, and do make a right angle with the
diameter, that right line shall light without the circle,
and yet so iointly knitte to it, that it is not possible to
draw any other right line betwene that saide line and the
circumference of the circle. And the angle that is made in
the semicircle is greater then any sharpe angle that may be
made of right lines, but the other angle without, is lesser
then any that can be made of right lines.
_Example._
[Illustration]
In this circle A.B.C, the diameter is A.C, the perpendicular
line, which maketh a right angle with the diameter, is C.A,
whiche line falleth without the circle, and yet ioyneth so
exactly vnto it, that it is not possible to draw an other right
line betwene the circumference of the circle and it, whiche
thyng is so plainly seene of the eye, that it needeth no farther
declaracion. For euery man wil easily consent, that betwene the
croked line A.F, (whiche is a parte of the circumference of the
circle) and A.E (which is the said perpendicular line) there can
none other line bee drawen in that place where they make the
angle. Nowe for the residue of the theoreme. The angle D.A.B,
which is made in the semicircle, is greater then anye sharpe
angle that may bee made of ryghte lines. and yet is it a sharpe
angle also, in as much as it is lesser then a right angle, which
is the angle E.A.D, and the residue of that right angle, which
lieth without the circle, that is to saye, E.A.B, is lesser then
any sharpe angle that can be made of right lines also. For as it
was before rehersed, there canne no right line be drawen to the
angle, betwene the circumference and the right line E.A. Then
must it needes folow, that there can be made no lesser angle of
righte lines. And againe, if ther canne be no lesser then the
one, then doth it sone appear, that there canne be no greater
then the other, for they twoo doo make the whole right angle, so
that if anye corner coulde be made greater then the one parte,
then shoulde the residue bee lesser then the other parte, so
that other bothe partes muste be false, or els bothe graunted to
be true.
_The lxij. Theoreme._
If a right line doo touche a circle, and an other right line
drawen frome the centre of the circle to the pointe where
they touche, that line whiche is drawenne frome the centre,
shall be a perpendicular line to the touch line.
_Example._
[Illustration]
The circle is A.B.C, and his centre is F. The touche line is
D.E, and the point wher they touch is C. Now by reason that a
right line is drawen frome the centre F. vnto C, which is the
point of the touche, therefore saith the theoreme, that the
sayde line F.C, muste needes bee a perpendicular line vnto the
touche line D.E.
_The lxiij. Theoreme._
If a righte line doo touche a circle, and an other right
line be drawen from the pointe of their touchinge, so that
it doo make righte corners with the touche line, then shal
the centre of the circle bee in that same line, so drawen.
_Example._
[Illustration]
The circle is A.B.C, and the centre of it is G. The touche line
is D.C.E, and the pointe where it toucheth, is C. Nowe it
appeareth manifest, that if a righte line be drawen from the
pointe where the touch line doth ioine with the circle, and that
the said lyne doo make righte corners with the touche line, then
muste it needes go by the centre of the circle, and then
consequently it must haue the sayde centre in him. For if the
saide line shoulde go beside the centre, as F.C. doth, then
dothe it not make righte angles with the touche line, which in
the theoreme is supposed.
_The lxiiij. Theoreme._
If an angle be made on the centre of a circle, and an other
angle made on the circumference of the same circle, and
their grounde line be one common portion of the
circumference, then is the angle on the centre twise so
great as the other angle on the circumference.
_Example._
[Illustration]
The circle is A.B.C.D, and his centre is E: the angle on the
centre is C.E.D, and the angle on the circumference is C.A.D t
their commen ground line, is C.F.D. Now say I that the angle
C.E.D, whiche is on the centre, is twise so greate as the angle
C.A.D, which is on the circumference.
_The lxv. Theoreme._
Those angles whiche be made in one cantle of a circle, must
needes be equal togither.
_Example._
Before I declare this theoreme by example, it shall bee
needefull to declare, what is it to be vnderstande by the wordes
in this theoreme. For the sentence canne not be knowen, onles
the uery meaning of the wordes be firste vnderstand. Therefore
when it speaketh of angles made in one cantle of a circle, it is
this to be vnderstand, that the angle muste touch the
circumference: and the lines that doo inclose that angle, muste
be drawen to the extremities of that line, which maketh the
cantle of the circle. So that if any angle do not touch the
circumference, or if the lines that inclose that angle, doo not
ende in the extremities of the corde line, but ende other in
some other part of the said corde, or in the circumference, or
that any one of them do so eande, then is not that angle
accompted to be drawen in the said cantle of the circle. And
this promised, nowe will I cumme to the meaninge of the
theoreme. I sette forthe a circle whiche is A.B.C.D, and his
centre E, in this circle I drawe a line D.C, whereby there ar
made two cantels, a more and a lesser. The lesser is D.E.C, and
the geater is D.A.B.C. In this greater cantle I drawe two
angles, the firste is D.A.C, and the second is D.B.C which two
angles by reason they are made bothe in one cantle of a circle
(that is the cantle D.A.B.C) therefore are they both equall. Now
doth there appere an other triangle, whose angle lighteth on the
centre of the circle, and that triangle is D.E.C, whose angle is
double to the other angles, as is declared in the lxiiij.
Theoreme, whiche maie stande well enough with this Theoreme, for
it is not made in this cantle of the circle, as the other are,
by reason that his angle doth not light in the circumference of
the circle, but on the centre of it.
[Illustration]
_The .lxvi. theoreme._
Euerie figure of foure sides, drawen in a circle, hath his
two contrarie angles equall vnto two right angles.
_Example._
[Illustration]
The circle is A.B.C.D, and the figure of foure sides in it, is
made of the sides B.C, and C.D, and D.A, and A.B. Now if you
take any two angles that be contrary, as the angle by A, and the
angle by C, I saie that those .ij. be equall to .ij. right
angles. Also if you take the angle by B, and the angle by D,
whiche two are also contray, those two angles are like waies
equall to two right angles. But if any man will take the angle
by A, with the angle by B, or D, they can not be accompted
contrary, no more is not the angle by C. estemed contray to the
angle by B, or yet to the angle by D, for they onely be
accompted _contrary angles_, whiche haue no one line common to
them bothe. Suche is the angle by A, in respect of the angle by
C, for there both lynes be distinct, where as the angle by A,
and the angle by D, haue one common line A.D, and therfore can
not be accompted contrary angles, So the angle by D, and the
angle by C, haue D.C, as a common line, and therefore be not
contrary angles. And this maie you iudge of the residewe, by
like reason.
_The lxvij. Theoreme._
Vpon one right lyne there can not be made two cantles of
circles, like and vnequall, and drawen towarde one parte.
_Example._
[Illustration]
Cantles of circles be then called like, when the angles that are
made in them be equall. But now for the Theoreme, let the right
line be A.E.C, on whiche I draw a cantle of a circle, whiche is
A.B.C. Now saieth the Theoreme, that it is not possible to draw
an other cantle of a circle, whiche shall be vnequall vnto this
first cantle, that is to say, other greatter or lesser then it,
and yet be lyke it also, that is to say, that the angle in the
one shall be equall to the angle in the other. For as in this
example you see a lesser cantle drawen also, that is A.D.C, so
if an angle were made in it, that angle would be greatter then
the angle made in the cantle A.B.C, and therfore can not they be
called lyke cantels, but and if any other cantle were made
greater then the first, then would the angle in it be lesser
then that in the firste, and so nother a lesser nother a greater
cantle can be made vpon one line with an other, but it will be
vnlike to it also.
_The .lxviij. Theoreme._
Lyke cantelles of circles made on equal righte lynes, are
equall together.
_Example._
What is ment by like cantles you haue heard before. and it is
easie to vnderstand, that suche figures a called equall, that be
of one bygnesse, so that the one is nother greater nother lesser
then the other. And in this kinde of comparison, they must so
agree, that if the one be layed on the other, they shall exactly
agree in all their boundes, so that nother shall excede other.
[Illustration]
Nowe for the example of the Theoreme, I haue set forthe diuers
varieties of cantles of circles, amongest which the first and
seconde are made vpon equall lines, and ar also both equall and
like. The third couple ar ioyned in one, and be nother equall,
nother like, but expressyng an absurde deformitee, whiche would
folowe if this Theoreme wer not true. And so in the fourth
couple you maie see, that because they are not equall cantles,
therfore can not they be like cantles, for necessarily it goeth
together, that all cantles of circles made vpon equall right
lines, if they be like they must be equall also.
_The lxix. Theoreme._
In equall circles, suche angles as be equall are made vpon
equall arch lines of the circumference, whether the angle
light on the circumference, or on the centre.
_Example._
[Illustration]
Firste I haue sette for an exaumple twoo equall circles, that is
A.B.C.D, whose centre is K, and the second circle E.F.G.H, and
his centre L, and in eche of them is there made two angles, one
on the circumference, and the other on the centre of eche
circle, and they be all made on two equall arche lines, that is
B.C.D. the one, and F.G.H. the other. Now saieth the Theoreme,
that if the angle B.A.D, be equall to the angle F.E.H, then are
they made in equall circles, and on equall arch lines of their
circumference. Also if the angle B.K.D, be equal to the angle
F.L.H, then be they made on the centres of equall circles, and
on equall arche lines, so that you muste compare those angles
together, whiche are made both on the centres, or both on the
circumference, and maie not conferre those angles, wherof one is
drawen on the circumference, and the other on the centre. For
euermore the angle on the centre in suche sorte shall be double
to the angle on the circumference, as is declared in the three
score and foure Theoreme.
_The .lxx. Theoreme._
In equall circles, those angles whiche bee made on equall
arche lynes, are euer equall together, whether they be made
on the centre, or on the circumference.
_Example._
This Theoreme doth but conuert the sentence of the last Theoreme
before, and therfore is to be vnderstande by the same examples,
for as that saith, that equall angles occupie equall archelynes,
so this saith, that equal arche lines causeth equal angles,
consideringe all other circumstances, as was taughte in the
laste theoreme before, so that this theoreme dooeth affirming
speake of the equalitie of those angles, of which the laste
theoreme spake conditionally. And where the laste theoreme spake
affirmatiuely of the arche lines, this theoreme speaketh
conditionally of them, as thus: If the arche line B.C.D. be
equall to the other arche line F.G.H, then is that angle B.A.D.
equall to the other angle F.E.H. Or els thus may you declare it
causally: Bicause the arche line B.C.D, is equal to the other
arche line F.G.H, therefore is the angle B.K.D. equall to the
angle F.L.H, consideringe that they are made on the centres of
equall circles. And so of the other angles, bicause those two
arche lines aforesaid ar equal, therfore the angle D.A.B, is
equall to the angle F.E.H, for as muche as they are made on
those equall arche lines, and also on the circumference of
equall circles. And thus these theoremes doo one declare an
other, and one verifie the other.
_The lxxi. Theoreme._
In equal circles, equall right lines beinge drawen, doo
cutte awaye equalle arche lines frome their circumferences,
so that the greater arche line of the one is equall to the
greater arche line of the other, and the lesser to the
lesser.
_Example._
[Illustration]
The circle A.B.C.D, is made equall to the circle E.F.G.H, and
the right line B.D. is equal to the righte line F.H, wherfore it
foloweth, that the ij. arche lines of the circle A.B.D, whiche
are cut from his circumference by the right line B.D, are equall
to two other arche lines of the circle E.F.H, being cutte frome
his circumference, by the right line F.H. that is to saye, that
the arche line B.A.D, beinge the greater arch line of the firste
circle, is equall to the arche line F.E.H, beynge the greater
arche line of the other circle. And so in like manner the lesser
arche line of the firste circle, beynge B.C.D, is equal to the
lesser arche line of the seconde circle, that is F.G.H.
_The lxxij. Theoreme._
In equall circles, vnder equall arche lines the right lines
that bee drawen are equall togither.
_Example._
This Theoreme is none other, but the conuersion of the laste
Theoreme beefore, and therefore needeth none other example. For
as that did declare the equalitie of the arche lines, by the
equalitie of the righte lines, so dothe this Theoreme declare
the equalnes of the right lines to ensue of the equalnes of the
arche lines, and therefore declareth that right lyne B.D, to be
equal to the other right line F.H, bicause they both are drawen
vnder equall arche lines, that is to saye, the one vnder B.A.D,
and thother vnder F.E.H, and those two arch lines are estimed
equall by the theoreme laste before, and shal be proued in the
booke of proofes.
_The lxxiij. Theoreme._
In euery circle, the angle that is made in the halfe circle,
is a iuste righte angle, and the angle that is made in a
cantle greater then the halfe circle, is lesser thanne a
righte angle, but that angle that is made in a cantle,
lesser then the halfe circle, is greatter then a right
angle. And moreouer the angle of the greater cantle is
greater then a righte angle and the angle of the lesser
cantle is lesser then a right angle.
_Example._
In this proposition, it shal be meete to note, that there is a
greate diuersite betwene an angle of a cantle, and an angle made
in a cantle, and also betwene the angle of a semicircle, and y^e
angle made in a semicircle. Also it is meet to note y^t al
angles that be made in y^e part of a circle, ar made other in a
semicircle, (which is the iuste half circle) or els in a cantle
of the circle, which cantle is other greater or lesser then the
semicircle is, as in this figure annexed you maye perceaue
euerye one of the thinges seuerallye.
[Illustration]
Firste the circle is, as you see, A.B.C.D, and his centre E, his
diameter is A.D, Then is ther a line drawen from A. to B, and so
forth vnto F, which is without the circle: and an other line
also frome B. to D, whiche maketh two cantles of the whole
circle. The greater cantle is D.A.B, and the lesser cantle is
B.C.D, In whiche lesser cantle also there are two lines that
make an angle, the one line is B.C, and the other line is C.D.
Now to showe the difference of an angle in a cantle, and an
angle of a cantle, first for an example I take the greter cantle
B.A.D, in which is but one angle made, and that is the angle by
A, which is made of a line A.B, and the line A.D, And this angle
is therfore called an angle in a cantle. But now the same cantle
hathe two other angles, which be called the angles of that
cantle, so the twoo angles made of the righte line D.B, and the
arche line D.A.B, are the twoo angles of this cantle, whereof
the one is by D, and the other is by B. Wher you must remembre,
that the angle by D. is made of the right line B.D, and the arche
line D.A. And this angle is diuided by an other right line
A.E.D, which in this case must be omitted as no line. Also the
angle by B. is made of the right line D.B, and of the arch line
.B.A, & although it be deuided with ij. other right lines, of
w^{ch} the one is the right line B.A, & thother the right line
B.E, yet in this case they ar not to be considered. And by this
may you perceaue also which be the angles of the lesser cantle,
the first of them is made of y^e right line B.D, & of y^e arch
line B.C, the second is made of the right line .D.B, & of the
arch line D.C. Then ar ther ij. other lines, w^{ch} deuide those
ij. corners, y^t is the line B.C, & the line C.D, w^{ch} ij.
lines do meet in the poynte C, and there make an angle, whiche
is called an angle made in that lesser cantle, but yet is not
any angle of that cantle. And so haue you heard the difference
betweene an angle in a cantle, and an angle of a cantle. And in
lyke sorte shall you iudg of the angle made in a semicircle,
whiche is distinct from the angles of the semicircle. For in this
figure, the angles of the semicircle are those angles which be
by A. and D, and be made of the right line A.D, beeyng the
diameter, and of the halfe circumference of the circle, but by
the angle made in the semicircle is that angle by B, whiche is
made of the righte line A.B, and that other right line B.D,
whiche as they mete in the circumference, and make an angle, so
they ende with their other extremities at the endes of the
diameter. These thynges premised, now saie I touchyng the
Theoreme, that euerye angle that is made in a semicircle, is a
right angle, and if it be made in any cantle of a circle, then
must it neds be other a blunt angle, or els a sharpe angle, and in
no wise a righte angle. For if the cantle wherein the angle is
made, be greater then the halfe circle, then is that angle a
sharpe angle. And generally the greater the cantle is, the lesser
is the angle comprised in that cantle: and contrary waies, the
lesser any cantle is, the greater is the angle that is made in
it. Wherfore it must nedes folowe, that the angle made in a
cantle lesse then a semicircle, must nedes be greater then a
right angle. So the angle by B, beyng made at the right line
A.B, and the righte line B.D, is a iuste righte angle, because
it is made in a semicircle. But the angle made by A, which is
made of the right line A.B, and of the right line A.D, is lesser
then a righte angle, and is named a sharpe angle, for as muche
as it is made in a cantle of a circle, greater then a
semicircle. And contrary waies, the angle by C, beyng made of
the righte line B.C, and of the right line C.D, is greater then
a right angle, and is named a blunte angle, because it is made
in a cantle of a circle, lesser then a semicircle. But now
touchyng the other angles of the cantles, I saie accordyng to
the Theoreme, that the .ij. angles of the greater cantle, which
are by B. and D, as is before declared, are greatter eche of
them then a right angle. And the angles of the lesser cantle,
whiche are by the same letters B, and D, but be on the other
side of the corde, are lesser eche of them then a right angle,
and be therfore sharpe corners.
_The lxxiiij. Theoreme._
If a right line do touche a circle, and from the pointe
where they touche, a righte lyne be drawen crosse the
circle, and deuide it, the angles that the saied lyne dooeth
make with the touche line, are equall to the angles whiche
are made in the cantles of the same circle, on the contrarie
sides of the lyne aforesaid.
_Example._
[Illustration]
The circle is A.B.C.D, and the touche line is E.F. The pointe of
the touchyng is D, from which point I suppose the line D.B, to
be drawen crosse the circle, and to diuide it into .ij. cantles,
wherof the greater is B.A.D, and the lesser is B.C.D, and in ech
of them an angle is drawen, for in the greater cantle the angle
is by A, and is made of the right lines B.A, and A.D, in the
lesser cantle the angle is by C, and is made of y^e right lines
B.C, and C.D. Now saith the Theoreme that the angle B.D.F, is
equall to the angle made in the cantle on the other side of the
said line, that is to saie, in the cantle B.A.D, so that the
angle B.D.F, is equall to the angle B.A.D, because the angle
B.D.F, is on the one side of the line B.D, (whiche is according
to the supposition of the Theoreme drawen crosse the circle) and
the angle B.A.D, is in the cantle on the other side. Likewaies
the angle B.D.E, beyng on the one side of the line B.D, must be
equall to the angle B.C.D, (that is the angle by C,) whiche is
made in the cantle on the other side of the right line B.D. The
profe of all these I do reserue, as I haue often saide, to a
conuenient boke, wherein they shall be all set at large.
_The .lxxv. Theoreme._
In any circle when .ij. right lines do crosse one an other,
the likeiamme that is made of the portions of the one line,
shall be equall to the lykeiamme made of the partes of the
other lyne.
[Illustration]
Because this Theoreme doth serue to many vses, and wold be wel
vnderstande, I haue set forth .ij. examples of it. In the
firste, the lines by their crossyng do make their portions
somewhat toward an equalitie. In the second the portions of the
lynes be very far from an equalitie, and yet in bothe these and
in all other y^e Theoreme is true. In the first example the
circle is A.B.C.D, in which thone line A.C, doth crosse thother
line B.D, in y^e point E. Now if you do make one likeiamme or
longsquare of D.E, & E.B, being y^e .ij. portions of the line
D.B, that longsquare shall be equall to the other longsquare
made of A.E, and E.C, beyng the portions of the other line A.C.
Lykewaies in the second example, the circle is F.G.H.K, in
whiche the line F.H, doth crosse the other line G.K, in the
pointe L. Wherfore if you make a lykeiamme or longsquare of the
two partes of the line F.H, that is to saye, of F.L, and L.H,
that longsquare will be equall to an other longsquare made of
the two partes of the line G.K. which partes are G.L, and L.K.
Those longsquares haue I set foorth vnder the circles containyng
their sides, that you maie somewhat whet your own wit in
practisyng this Theoreme, accordyng to the doctrine of the
nineteenth conclusion.
_The .lxxvi. Theoreme._
If a pointe be marked without a circle, and from that pointe
two right lines drawen to the circle, so that the one of
them doe runne crosse the circle, and the other doe touche
the circle onely, the long square that is made of that whole
lyne which crosseth the circle, and the portion of it, that
lyeth betwene the vtter circumference of the circle and the
pointe, shall be equall to the full square of the other
lyne, that onely toucheth the circle.
_Example._
The circle is D.B.C, and the pointe without the circle is A,
from whiche pointe there is drawen one line crosse the circle,
and that is A.D.C, and an other lyne is drawn from the said
pricke to the marge or edge of the circumference of the circle,
and doeth only touche it, that is the line A.B. And of that
first line A.D.C, you maie perceiue one part of it, whiche is
A.D, to lie without the circle, betweene the vtter circumference
of it, and the pointe assigned, whiche was A. Nowe concernyng
the meanyng of the Theoreme, if you make a longsquare of the
whole line A.C, and of that parte of it that lyeth betwene the
circumference and the point, (whiche is A.D,) that longesquare
shall be equall to the full square of the touche line A.B,
accordyng not onely as this figure sheweth, but also the saied
nyneteenth conclusion dooeth proue, if you lyste to examyne the
one by the other.
[Illustration]
_The lxxvii. Theoreme._
If a pointe be assigned without a circle, and from that
pointe .ij. right lynes be drawen to the circle, so that the
one doe crosse the circle, and the other dooe ende at the
circumference, and that the longsquare of the line which
crosseth the circle made with the portion of the same line
beyng without the circle betweene the vtter circumference
and the pointe assigned, doe equally agree with the iuste
square of that line that endeth at the circumference, then
is that lyne so endyng on the circumference a touche line
vnto that circle.
_Example._
In as muche as this Theoreme is nothyng els but the sentence of
the last Theoreme before conuerted, therfore it shall not be
nedefull to vse any other example then the same, for as in that
other Theoreme because the one line is a touche lyne, therfore
it maketh a square iust equal with the longsquare made of that
whole line, whiche crosseth the circle, and his portion liyng
without the same circle. So saith this Theoreme: that if the
iust square of the line that endeth on the circumference, be
equall to that longsquare whiche is made as for his longer sides
of the whole line, which commeth from the pointt assigned, and
crosseth the circle, and for his other shorter sides is made of
the portion of the same line, liyng betwene the circumference of
the circle and the pointe assigned, then is that line whiche
endeth on the circumference a right touche line, that is to
saie, yf the full square of the right line A.B, be equall to the
longsquare made of the whole line A.C, as one of his lines, and
of his portion A.D, as his other line, then must it nedes be,
that the lyne A.B, is a right touche lyne vnto the circle D.B.C.
And thus for this tyme I make an ende of the Theoremes.
+FINIS,+
_IMPRINTED at London in Poules
churcheyarde, at the signe of the Bra-
senserpent, by Reynold Wolfe._
Cum priuilegio ad imprimen-
dum solum.
+ANNO DOMINI+ .M.D.L.I.
[Illustration: Publisher's Device (brazen serpent, NVM. XXI.)]
* * * * *
* * * *
_Errors and Inconsistencies:_
Unless otherwise noted, spelling, punctuation and capitalization are
unchanged. Forms were regularized only where there was a very large
disparity between the expected form and the apparent errors (for
example, a thousand "A.B" against a dozen "A,B"), or a flagrant
misprint such as "cnt" for "cut".
The letters u and v follow the conventional "initial v, non-initial u"
pattern except in numbers (xv, iv). The lower-case j form occurs only
as the last digit of a number (ij, xxj); upper-case I and J share a
form, always read as I. Capital and lower-case "w" were often used
interchangeably. Words split across line breaks may or may not have a
hyphen.
_Language:_
The word "other" is used interchangeably with both "or" and "either";
similarly, "nother" is used in place of "nor" and "neither". The
expression "an other" is almost always written as two words.
The spellings "then(ne)" and "than(ne)" are used interchangeably;
"than(ne)" is rare. The spelling "liyng", both by itself and as the
end of a longer word, is used consistently.
_Illustrations:_
A number of illustrations contain errors such as unmarked or mislabeled
points ("circle B.C.D" where only C and D are labeled). Errors of this
type are identified in the illustrated HTML version, but not in the
present text-only file.
_Greek:_
All Greek is shown and transliterated as printed. Errors or anomalies
include missing, misplaced or incorrect diacritics; the non-final form
of sigma used at the end of words; and word-final #m# for #n#.
_Meaningful Errors and Anomalies:_
shal be clean extrirped and rooted out
[_spelling unchanged: intended form not certain_]
new and new causes to pray for your maiestie, perceiuyng
[_text "new and new causes" probably intentional_]
[Preface] And thus for this tyme I make an end...
[Body text] The definitions of the principles of GEOMETRY.
[_Pagination as shown by signature numbers demands another leaf
(two pages) between the end of the Preface and the beginning of
the body text. But no text is missing, and the facsimile has no
blank pages._]
Otherlesse then it as you se D
[_text unchanged: intended wording uncertain_]
THE .XXII. CONCLVSION. [.XXI.]
THE XXXVIII. CONCLVSION. [XXXVII.]
_The xxvi. Theoreme._
There is no xxv. (25) theorem.
_The .xxxix. theoreme._
The text of the Example is garbled, and does not fit the
illustration. Among other problems, points C and D seem to have
been switched, either in the text or in the illustration.
The circle is A.B.C.D, and his centre is E: the angle on the centre
is C.E.D, and the angle on the circumference is C.A.D t their commen
ground line, is C.F.D.
[cirle]
[_printing of "C.E.D" unclear: looks like "C.F.D" but center of
circle is E_]
[_lone "t" may be error for ampersand or other punctuation_]
[C.F.D,]
_Misprints:_
Aristotle had putte forthe certaine bookes [kookes]
And his father king Dauid ioyneth [Danid]
those thynges are done by negromancy. And hereof came it that
fryer Bakon was accompted so greate a negromancier
[_spelling unchanged_]
For vndoubtedly if they mean [vndoudtedly if the]
that Godde was alwaies workinge [alwaaies]
all the lines that be drawen to the circumference [circumfernece]
An other hath but one compassed [hatht]
whose sydes partlye are all equall as in A [eqnall]
as this A. doth partly expresse [A doth partly eppresse]
To make a threlike triangle on any lyne measurable. [or any]
and it shall cut the first line in two equall portions. [cnt]
then open I the compass as wide as .iiij. partes [compaas]
To make a plumbe lyne on any porcion of a circle [or any]
for euery triangle together an equal likeiamme
[_text has "to/gether" at line break: may be meant for two words_]
as you mai se by C and D, for ij. sides of both the trianngles ar
parallels.
[you maise ... triangls]
then will the square of that greater portion [portior]
that line I say is a touche line [touthe]
(that is to saie M. G,) [_error for "in G"?_]
one perpendicular from G. vnto the side side B.C [A.C]
then must I first draw a tuche lyne [wust]
anye of the two lynes contayninge the angle appointed. [nye of]
draw thence two lines, one to D [one to A]
which is made accordinge to the conclusion. [ancordinge]
you shal perceaue that there will be [peceaue]
a / more conueni/ent time.
[_text has "conui/ent" at line break_]
other in deciding some controuersy [decising]
a great deale the soner [somner]
Whiche example hath beene vsede [hat]
whan I wrote these first connclusions [cunclusions]
suche bokes as ar appoynted [as at]
Will I refuse. [Willl]
all right angles be equall [eqnall]
whiche thinge the better to perceaue [peceaue]
whiche I do only by examples declare [declae]
about the ground line, are equal togither [groud]
you shal take this triangle A.B.C. which hath a very blunt corner
[_word "this" illegible_] [veery blunt]
the one is a longe square A.B.E
[_printed as shown: expected form is "A.B.F.E"_]
though they be diuers in numbre. [numbhe]
proofe that G.H. being the ground line [groud]
and standing betwene one paire of parallels [an]
for thei ar the two y^e contrary sides [_word "y^e" superfluous?_]
they haue one ground line D.E. [on]
By the square of any lyne [sqnare]
squares that are made of the same line [sane]
The fyrst lyne propounded is A.B [propouned]
hath another longe square equall to hym
[_text has anomalous "a / nother" at line break_]
one of those parts again into other ij. parts [iuto]
which thing the easyer is to be vnderstande [eayser]
and blunt cornered triangles [couered]
the square marked with G. is the square of A.B [with C.]
And from all pointes you maye drawe ij. equall lines [poittes]
If two circles bee drawen so one withoute an other [and other]
drawen frome the centre of the circle to the pointe [tge]
which in the theoreme is supposed. [the .heoreme]
and therfore can not they be called lyke cantels [ban]
then would the angle in it be lesser [it]
being cutte frome his circumference, by the right line F.H.
[circumforence]
And in lyke sorte shall you iudg [And n lyke]
_Punctuation, Spacing, Capitalization:_
Phrase breaks where a comma is followed by a capital letter, or a
period by a lower-case letter, are not individually noted.
Number forms such as "those. ij. last" or "line. A.B." were silently
regularized to "those .ij. last" and "line .A.B." Missing sentence-final
periods at the end of a printed line were silently supplied.
_Initial u or medial v unchanged:_
ioyneth uertuous conuersacion
XXV. CONCLVSION.... the whole circle agreynge therevnto.
or eande in the utter edge of his circumference
onles the uery meaning of the wordes be firste vnderstand
Geometry teacheth the drawyng, Measuring and proporcion
[_capital M in original_]
And herof commeth that seconnde thing wherin al agree
[that seconnde . thing]
if they can with their wysedome ouercome all vyces. Of the firste of
those three sortes
[_text reads "... their rwysedome ... / ... th ee sortes ..."
on consecutive lines. The extraneous "r" is directly above the
missing or invisible "r"_]
#akroamatikoi#. [_final . missing_]
was he estemed for his wisdom. [_final . missing_]
wisedom is better then pretious stones . yea
[_punctuation unchanged_]
your Maiesties excellencye, [excellencye,,]
if his subiectes be riche in substaunce, [sub staunce]
new and new causes to pray for your maiestie, perceiuyng
[maiestie,perceiuyng]
can be ignorant thereof, in so much that [thereof. in]
those thynges are done by negromancy. [_spelling unchanged_]
thogh it be but smal, and thefore not notable. [_final . missing_]
but boweth any waye, such are called [waye. such]
not call it one croked lyne, but rather [cr oked]
Now to geue you example of triangles, [triangles,,]
a portion of a globe as the figur marked w^t A. [_final . missing_]
as is the figure G. other ij. sharp and one blunt
these examples .M. N, and O. where M. hath a right angle, N,
a blunte angle
[.M.N, and O where ... N, A, blunte]
as in this example because A.B, is drawen in length [A,B,]
the angle C, is called an vtter angle. [_final . missing_]
_diamondlike_, whose figur is noted with T. [_final . missing_]
and partlye vnequall, as in B, and they [in, B and]
as in this figure is some what expressed. [_final . missing_]
then draw a line from B, to F, & so I haue mine intent.
[_final . missing_]
then open I the compasse as wyde as A.C, [A,C,]
F, this space betwene D. and F. [betwene D and F.]
crosse that other first line in .ij. places. [_final . missing_]
The arch to be diuided ys A.D.C, the corde is A.B.C, [A,B.C,]
and ther set a mark. Then take a long line [mark Then]
a parallel must be drawen howe you shall doo it. [doo it,]
equal to A.B, as for example D.F. [D,F.]
accordyng to the fifth conclusion [accordyngt o]
(accordyng as the .viij. conclusion teacheth)
[_comma for close parenthesis_]
it hath one angle (that is B.A.E.) like to D
[_close parenthesis missing_]
therefore I wyl by example sette forth [ther efore]
thenne drawe I a line also from N. vnto C. [vnto C]
is equall vnto A.B, and so is K.L. [K.L,]
you mai se by C. and D, for ij. sides [by C and D,]
accordyng to the lengthe lo the peece that remaineth
[_error for "of the"?_]
[th e peece]
this line A.B, (w^{ch} was assigned vnto me) [A,B,]
& thus haue you attained y^e vse of this conclusion.
[_final . missing_]
Make in a table the like draught [A table]
vsing it as it wer a croked ruler. [_final . missing_]
the lines of diuision A.D, B.E [A. D. B. E]
whose two sides A.C. and B.C. are diuided [B,C.]
enclose those iij prickes. that centre as you se is D
[prickes that centre . as]
as you se B.A.D, and B.C.E. [B.C,E.]
at this tyme wyll shewe you [she,we]
as youre selfe mai easily gesse. [_final . missing_]
A.C. and B.D. are the two diameters [A.C. and B D.]
A.B.C.D. is the quadrate appointed [A,B.C.D.]
quadrate from angle to angle, as you se A.C. & B D. [A.C. & B D.]
A.B.C. is the circle, whiche I haue deuided [A,B.C.]
And from eche pricke ij. lines drawen [line sdrawen]
I would make a circle. Therfore I drawe [circle, Therfore]
of equall sides and equall angles. [angles,]
perceaue by the xxxvij. xxxviij. xxxix. and xl. conclusions
[xxxviij xxxix.]
that be nighest [that be . nighest]
learne the demonstrations by harte, (as somme
[_missing open parenthesis_]
sundrye woorkes partely ended, and partely to bee ended [And]
As for example A--------B. A. being the one pricke [A being]
that corner which is greatter then a right angle
[_s in "is" invisible_]
firste these two right lines A.B. and A.C. [A B. and A.C.]
ij. quantities, as A. and B, be equal to an other [as A and B,]
in an other place. In the mean season [in]
drawen forthe vnto D. and E. [vnto D and E.]
[The thirde Theoreme.] Example. [_final . missing_]
and yet the ij. lesser sides togither ar greater then it.
[_text has "... yet thr ... ar greate" at consecutive line-ends_]
the angle C. (whiche are the ij. angles contayned
[_missing open parenthesis_]
M.N. equall also to H.L. [H,L.]
therefore are A.C. and B.D. bothe equall [A,C.]
A.D.E, and D.E.B, which (as the xxvij.
[_missing open parenthesis_]
_The xxxi. theoreme._ [xxxi, theoreme.]
there is made a triangle B.C.G, and a lykeiamme [B.C G,]
fyll vp the sydes of the .ij. fyrste square lykeiammes [.ij fyrste]
equall to the .ij. squares of both the other sides.
[_final . missing_]
The ij. lines proposed ar A.B. and C.D [A B. and C.D]
for the square D.E.F.G. is equal to the two other partial squares
of D.H.K.G and H.E.F.K,
[D,E.F.G. ... D.H.K G]
_The xxxvij. Theoreme._ [xxxvij Theoreme.]
the square that is made of the whole line
[_first t in "that" invisible_]
(which is equall with D.G.) [D,G.]
Lette the diuided line bee A.B, and parted in C [A,B,]
the square of the whole lyne A.B, [A,B,]
as herafter I will declare in conuenient place. [_final . missing_]
two lesser squares beyng taken away,
[_close parenthesis for comma_]
the great square, and that is G.F.M.H. [G,F.M.H.]
two vnequall partes as happeneth. The long square [the]
_The .xlvi. Theoreme._ [The.xlvi.]
_The xlvij. Theoreme._ [xlvij Theoreme.]
and doo not passe by the centre [passeby]
For as you may easily iudge, A.C. hath one portion [A C.]
if they be equally distaunt from one halfe of the diameter
[_second l in "equally" invisible_]
then it, and beynge farther of [it. and]
the second circle is B.C.D.E, and they crosse [B.C.D,E,]
The second circle is D.B.C, and his centre is H [D,B.C,]
that is farther from the centre. The fourth [centre, The]
twise so great as the other angle on the circumference.
[_final . missing_]
The lesser is D.E.C, and the geater is D.A.B.C. [D.F.C,]
therefore are they both equall. [_final . missing_]
What is ment by like cantles you haue heard before [mentby]
by the equalitie of the righte lines, so dothe this Theoreme
[so do the this]
Also it is meet to note y^t al angles that be made [y^{t}al]
whiche maketh two cantles of the whole circle. [circle,]
which is made of a line A.B, and the line A.D, [A,B,]
I saie accordyng to the Theoreme, that the .ij. angles [.ij angles]
so that the angle B.D.F, is equall to the angle B.A.D [B D.F,]
make their portions somewhat toward an equalitie. [_final . missing_]
doth crosse thother line B.D, in y^e point E. [B D,]
whiche was A. Nowe concernyng the meanyng [No we]