Chapter VI: Part 6

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.

_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.

* * * * *
* * * *

_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]

End of Project Gutenberg's The Path-Way to Knowledg, by Robert Record