To make a figure of fifteene equall sides and angles in any
circle appointed.
This rule is generall, that how many sides the figure shall
haue, that shall be drawen in any circle, into so many partes
iustely muste the circles bee deuided. And therefore it is the
more easier woorke commonly, to drawe a figure in a circle, then
to make a circle in an other figure. Now therefore to end this
conclusion, deuide the circle firste into fiue partes, and
then eche of them into three partes againe: Or els
first deuide it into three partes, and then ech
of them into fiue other partes, as you
list, and canne most readilye.
Then draw lines betwene
euery two prickes
that be nighest
togither, and
ther wil appear rightly drawen the figure, of fiftene sides,
and angles equall. And so do with any other figure
of what numbre of sides so euer it bee.
+FINIS.+
THE SECOND BOOKE
+OF THE PRINCIPLES+
_of Geometry, containing certaine_
_Theoremes, whiche may be cal-_
led Approued truthes. And be as
it were the moste certaine
groundes, wheron the
practike conclusions
of Geometry ar
founded.
[Leaf]
Whervnto are annexed certaine declarations by
examples, for the right vnderstanding of the
same, to the ende that the simple reader
might not iustly complain of hardnes
or obscuritee, and for the same
cause ar the demonstra-
tions and iust profes
omitted, vntill a
more conueni-
ent time.
1551.
If truthe maie trie it selfe,
By Reasons prudent skyll,
If reason maie preuayle by right,
And rule the rage of will,
I dare the triall byde,
For truthe that I pretende.
And though some lyst at me repine,
Iuste truthe shall me defende.
THE PREFACE VNTO
the Theoremes.
I Doubt not gentle reader, but as my argument is straunge and
vnacquainted with the vulgare toungue, so shall I of many men be
straungly talked of, and as straungly iudged. Some men will saye
peraduenture, I mighte haue better imployed my tyme in some
pleasaunte historye, comprisinge matter of chiualrye. Some other
wolde more haue preised my trauaile, if I hadde spente the like
time in some morall matter, other in deciding some controuersy
of religion. And yet some men (as I iudg) will not mislike this
kind of mater, but then will they wishe that I had vsed a more
certaine order in placinge bothe the Propositions and Theoremes,
and also a more exacter proofe of eche of theim bothe, by
demonstrations mathematicall. Some also will mislike my
shortenes and simple plainesse, as other of other affections
diuersely shall espye somwhat that they shall thinke blame
worthy, and shal misse somewhat, that thei wold with to haue
bene here vsed, so that euerie manne shall giue his verdicte of
me according to his phantasie, vnto whome ioinctly, I make this
my firste answere: that as they ar many and in opinions verie
diuers, so were it scarse possible to please them all with anie
one argumente, of what kinde so euer it were. And for my seconde
aunswere, I saye thus. That if annye one argumente mighte please
them all, then should thei be thankfull vnto me for this kind of
matter. For nother is there anie matter more straunge in the
englishe tungue, then this whereof neuer booke was written
before now, in that tungue, and therefore oughte to delite all
them, that desire to vnderstand strange matters, as most men
commonlie doo. And againe the practise is so pleasaunt in
vsinge, and so profitable in appliynge, that who so euer dothe
delite in anie of bothe, ought not of right to mislike this
arte. And if any manne shall like the arte welle for it selfe,
but shall mislyke the fourme that I haue vsed in teachyng of it,
to hym I shall saie, Firste, that I dooe wishe with hym that
some other man, whiche coulde better haue doone it, hadde shewed
his good will, and vsed his diligence in suche sorte, that I
myght haue bene therby occasioned iustely to haue left of my
laboure, or after my trauaile to haue suppressed my bookes. But
sithe no manne hath yet attempted the like, as far as I canne
learne, I truste all suche as bee not exercised in the studie of
Geometrye, shall finde greate ease and furtheraunce by this
simple, plaine, and easie forme of writinge. And shall perceaue
the exacte woorkes of Theon, and others that write on Euclide,
a great deale the soner, by this blunte delineacion afore hande
to them taughte. For I dare presuppose of them, that thing which
I haue sette in my selfe, and haue marked in others, that is to
saye, that it is not easie for a man that shall trauaile in a
straunge arte, to vnderstand at the beginninge bothe the thing
that is taught and also the iuste reason whie it is so. And by
experience of teachinge I haue tried it to bee true, for whenne
I haue taughte the proposition, as it is imported in meaninge,
and annexed the demonstration with all, I didde perceaue that it
was a greate trouble and a painefull vexacion of mynde to the
learner, to comprehend bothe those thinges at ones. And therfore
did I proue firste to make them to vnderstande the sence of the
propositions, and then afterward did they conceaue the
demonstrations muche soner, when they hadde the sentence of the
propositions first ingrafted in their mindes. This thinge caused
me in bothe these bookes to omitte the demonstrations, and to
vse onlye a plaine forme of declaration, which might best serue
for the firste introduction. Whiche example hath beene vsed by
other learned menne before nowe, for not only Georgius Ioachimus
Rheticus, but also Boetius that wittye clarke did set forth some
whole books of Euclide, without any demonstration or any other
declaration at al. But & if I shal hereafter perceaue that it
maie be a thankefull trauaile to sette foorth the propositions
of geometrie with demonstrations, I will not refuse to dooe it,
and that with sundry varietees of demonstrations, bothe
pleasaunt and profitable also. And then will I in like maner
prepare to sette foorth the other bookes, whiche now are lefte
vnprinted, by occasion not so muche of the charges in cuttyng of
the figures, as for other iuste hynderances, whiche I truste
hereafter shall bee remedied. In the meane season if any man
muse why I haue sette the Conclusions beefore the Teoremes,
seynge many of the Theoremes seeme to include the cause of some
of the conclusions, and therfore oughte to haue gone before
them, as the cause goeth before the effecte. Here vnto I saie,
that although the cause doo go beefore the effect in order of
nature, yet in order of teachyng the effect must be fyrst
declared, and than the cause therof shewed, for so that men best
vnderstand things First to lerne that such thinges ar to be
wrought, and secondarily what thei ar, and what thei do import,
and than thirdly what is the cause therof. An other cause why y^t
the theoremes be put after the conclusions is this, whan I wrote
these first connclusions (which was .iiiij. yeres passed)
I thought not then to haue added any theoremes, but next vnto
y^e conclusions to haue taught the order how to haue applied them
to work, for drawing of plottes & such like vses. But afterward
considering the great commoditie y^t thei serue for, and the light
that thei do geue to all sortes of practise geometricall, besyde
other more notable benefites, whiche shall be declared more
specially in a place conuenient, I thoughte beste to geue you
some taste of theym, and the pleasaunt contemplation of suche
geometrical propositions, which might serue diuerselye in other
bookes for the demonstrations and proofes of all Geometricall
woorkes. And in theim, as well as in the propositions, I haue
drawen in the Linearie examples many tymes more lynes, than be
spoken of in the explication of them, whiche is doone to this
intent, that yf any manne lyst to learne the demonstrations by
harte, (as somme learned men haue iudged beste to doo) those
same men should find the Linearye exaumples to serue for this
purpose, and to wante no thyng needefull to the iuste proofe,
whereby this booke may bee wel approued to be more complete then
many men wolde suppose it.
And thus for this tyme I wyll make an ende without any larger
declaration of the commoditiees of this arte, or any farther
answeryng to that may bee obiected agaynst my handelyng of it,
wyllyng them that myslike it, not to medle with it: and vnto
those that will not disdaine the studie of it, I promise all
suche aide as I shall be able to shewe for their farther
procedyng both in the same, and in all other commoditees that
thereof maie ensue. And for their incouragement I haue here
annexed the names and brefe argumentes of suche bookes, as I
intende (God willyng) shortly to sette forth, if I shall
perceaue that my paynes maie profyte other, as my desyre is.
+The brefe argumentes of suche bokes as ar appoynted shortly
to be set forth by the author herof.+
THE seconde part of Arithmetike, teachyng the workyng by
fractions, with extraction of rootes both square and cubike: And
declaryng the rule of allegation, with sundrye plesaunt
exaumples in metalles and other thynges. Also the rule of false
position, with dyuers examples not onely vulgar, but some
appertaynyng to the rule of Algeber, applied vnto quantitees
partly rationall, and partly surde.
THE arte of Measuryng by the quadrate geometricall, and the
disorders committed by vsyng the same, not only reueled but
reformed also (as muche as to the instrument pertayneth) by the
deuise of a new quadrate newely inuented by the author hereof.
THE arte of measuryng by the astronomers staffe, and by the
astronomers ryng, and the form of makyng them both.
THE arte of makyng of Dials, bothe for the daie and the nyght,
with certayn new formes of fixed dialles for the moon and other
for the sterres, whiche may bee sette in glasse windowes to
serue by daie and by night. And howe you may by those dialles
knowe in what degree of the Zodiake not only the sonne, but also
the moone is. And how many howrs old she is. And also by the
same dial to know whether any eclipse shall be that moneth, of
the sonne or of the moone.
The makyng and vse of an instrument, wherby you maye not onely
measure the distance at ones of all places that you can see
togyther, howe muche eche one is from you, and euery one from
other, but also therby to drawe the plotte of any countreie that
you shall come in, as iustely as maie be, by mannes diligence
and labour.
THE vse bothe of the Globe and the Sphere, and therin also of
the arte of Nauigation, and what instrumentes serue beste
thervnto, and of the trew latitude and longitude of regions and
townes.
Euclides woorkes in foore partes, with diuers demonstrations
Arithmeticall and Geometricall or Linearie. The fyrst parte of
platte formes. The second of numbres and quantitees surde or
irrationall. The third of bodies and solide formes. The fourthe
of perspectiue, and other thynges thereto annexed.
BESIDE these I haue other sundrye woorkes partely ended, and
partely to bee ended, Of the peregrination of man, and the
originall of al nations, The state of tymes, and mutations of
realmes, The image of a perfect common welth, with diuers other
woorkes in naturall sciences, Of the wonderfull workes and
effectes in beastes, plantes, and minerals, of whiche at this
tyme, I will omitte the argumentes, beecause thei doo appertaine
littel to this arte, and handle other matters in an other sorte.
To haue, or leaue,
Nowe maie you chuse,
No paine to please,
Will I refuse.
The Theoremes of Geometry, before
_WHICHE ARE SET FORTHE_
_certaine grauntable requestes_
_which serue for demonstrations_
Mathematicall.
[Sidenote: I.]
That from any pricke to one other, there may be drawen a
right line.
As for example A--------B. A. being the one pricke, and B. the
other, you maye drawe betwene them from the one to the other,
that is to say, frome A. vnto B, and from B. to A.
[Sidenote: II.]
That any right line of measurable length may be drawen forth
longer, and straight.
[Illustration]
Example of A.B, which as it is a line of measurable lengthe, so
may it be drawen forth farther, as for example vnto C, and that
in true streightenes without crokinge.
[Sidenote: III.]
[Illustration]
That vpon any centre, there may be made a circle of anye
quantitee that a man wyll.
Let the centre be set to be A, what shal hinder a man to drawe a
circle aboute it, of what quantitee that he lusteth, as you se
the forme here: other bygger or lesse, as it shall lyke him to
doo:
That all right angles be equall eche to other.
[Illustration]
Set for an example A. and B, of which two though A. seme the
greatter angle to some men of small experience, it happeneth
only bicause that the lines aboute A, are longer then the lines
about B, as you may proue by drawing them longer, for so that B.
seme the greater angle yf you make his lines longer then the
lines that make the angle A. And to proue it by demonstration,
I say thus. If any ij. right corners be not equal, then one
right corner is greater then an other, but that corner which is
greatter then a right angle, is a blunt corner (by his
definition) so must one corner be both a right corner and a
blunt corner also, which is not possible: And againe: the lesser
right corner must be a sharpe corner, by his definition, bicause
it is lesse then a right angle. which thing is impossible.
Therefore I conclude that all right angles be equall.
Yf one right line do crosse two other right lines, and make
ij. inner corners of one side lesser then ij. righte corners,
it is certaine, that if those two lines be drawen forth
right on that side that the sharpe inner corners be, they
wil at length mete togither, and crosse on an other.
[Illustration]
The ij. lines beinge as A.B. and C.D, and the third line
crossing them as dooth heere E.F, making ij inner cornes (as ar
G.H.) lesser then two right corners, sith ech of them is lesse
then a right corner, as your eyes maye iudge, then say I, if
those ij. lines A.B. and C.D. be drawen in lengthe on that side
that G. and H. are, the will at length meet and crosse one an
other.
Two right lines make no platte forme.
[Illustration]
A platte forme, as you harde before, hath bothe length and
bredthe, and is inclosed with lines as with his boundes, but ij.
right lines cannot inclose al the bondes of any platte forme.
Take for an example firste these two right lines A.B. and A.C.
whiche meete togither in A, but yet cannot be called a platte
forme, bicause there is no bond from B. to C, but if you will
drawe a line betwene them twoo, that is frome B. to C, then will
it be a platte forme, that is to say, a triangle, but then are
there iij. lines, and not only ij. Likewise may you say of D.E.
and F.G, whiche doo make a platte forme, nother yet can they
make any without helpe of two lines more, whereof the one must
be drawen from D. to F, and the other frome E. to G, and then
will it be a longe rquare. So then of two right lines can bee
made no platte forme. But of ij. croked lines be made a platte
forme, as you se in the eye form. And also of one right line, &
one croked line, maye a platte fourme bee made, as the
semicircle F. doothe sette forth.
Certayn common sentences manifest to
sence, and acknowledged of all men.
_The firste common sentence._
What so euer things be equal to one other thinge, those same
bee equall betwene them selues.
[Illustration]
Examples therof you may take both in greatnes and also in
numbre. First (though it pertaine not proprely to geometry, but
to helpe the vnderstandinge of the rules, whiche may bee wrought
by bothe artes) thus may you perceaue. If the summe of monnye in
my purse, and the mony in your purse be equall eche of them to
the mony that any other man hathe, then must needes your mony
and mine be equall togyther. Likewise, if anye ij. quantities,
as A. and B, be equal to an other, as vnto C, then muste nedes A.
and B. be equall eche to other, as A. equall to B, and B. equall
to A, whiche thinge the better to perceaue, tourne these
quantities into numbre, so shall A. and B. make sixteene, and C.
as many. As you may perceaue by multipliyng the numbre of their
sides togither.
_The seconde common sentence._
And if you adde equall portions to thinges that be equall,
what so amounteth of them shall be equall.
Example, Yf you and I haue like summes of mony, and then receaue
eche of vs like summes more, then our summes wil be like styll.
Also if A. and B. (as in the former example) bee equall, then by
adding an equal portion to them both, as to ech of them, the
quarter of A. (that is foure) they will be equall still.
_The thirde common sentence._
And if you abate euen portions from things that are equal,
those partes that remain shall be equall also.
This you may perceaue by the last example. For that that was
added there, is subtracted heere. and so the one doothe approue
the other.
_The fourth common sentence._
If you abate equalle partes from vnequal thinges, the
remainers shall be vnequall.
As bicause that a hundreth and eight and forty be vnequal if I
take tenne from them both, there will remaine nynetye and eight
and thirty, which are also vnequall. and likewise in quantities
it is to be iudged.
_The fifte common sentence._
When euen portions are added to vnequalle thinges, those
that amounte shalbe vnequall.
So if you adde twenty to fifty, and lyke ways to nynty, you
shall make seuenty and a hundred and ten whiche are no lesse
vnequall, than were fifty and nynty.
_The syxt common sentence._
If two thinges be double to any other, those same two
thinges are equal togither.
[Illustration]
Bicause A. and B. are eche of them double to C, therefore must
A. and B. nedes be equall togither. For as v. times viij. maketh
xl. which is double to iiij. times v, that is xx so iiij. times
x, likewise is double to xx. (for it maketh fortie) and
therefore muste neades be equall to forty.
_The seuenth common sentence._
If any two thinges be the halfes of one other thing, then
are thei .ij. equall togither.
So are D. and C. in the laste example equal togyther, bicause
they are eche of them the halfe of A. other of B, as their
numbre declareth.
_The eyght common sentence._
If any one quantitee be laide on an other, and thei agree,
so that the one excedeth not the other, then are they equall
togither.
[Illustration]
As if this figure A.B.C, be layed on that other D.E.F, so that
A. be layed to D, B. to E, and C. to F, you shall see them agre
in sides exactlye and the one not to excede the other, for the
line A.B. is equall to D.E, and the third lyne C.A, is equall to
F.D so that eueryside in the one is equall to some one side of
the other. Wherfore it is playne, that the two triangles are
equall togither.
_The nynth common sentence._
Euery whole thing is greater than any of his partes.
This sentence nedeth none example. For the thyng is more playner
then any declaration, yet considering that other common sentence
that foloweth nexte that.
_The tenthe common sentence._
Euery whole thinge is equall to all his partes taken
togither.
[Illustration]
[Illustration]
It shall be mete to expresse both w^t one example, for of thys
last sentence many men at the first hearing do make a doubt.
Therfore as in this example of the circle deuided into sundry
partes it doeth appere that no parte can be so great as the
whole circle, (accordyng to the meanyng of the eight sentence)
so yet it is certain, that all those eight partes together be
equall vnto the whole circle. And this is the meanyng of that
common sentence (whiche many vse, and fewe do rightly
vnderstand) that is, that _All the partes of any thing are
nothing els, but the whole_. And contrary waies: _The whole is
nothing els, but all his partes taken togither_. whiche saiynges
some haue vnderstand to meane thus: that all the partes are of
the same kind that the whole thyng is: but that that meanyng is
false, it doth plainly appere by this figure A.B, whose partes
A. and B, are triangles, and the whole figure is a square, and
so are they not of one kind. But and if they applie it to the
matter or substance of thinges (as some do) then it is most
false, for euery compound thyng is made of partes of diuerse
matter and substance. Take for example a man, a house, a boke,
and all other compound thinges. Some vnderstand it thus, that
the partes all together can make none other forme, but that that
the whole doth shewe, whiche is also false, for I maie make fiue
hundred diuerse figures of the partes of some one figure, as you
shall better perceiue in the third boke. And in the meane season
take for an example this square figure following A.B.C.D, w^{ch}
is deuided but in two parts, and yet (as you se) I haue made
fiue figures more beside the firste, with onely diuerse ioynyng
of those two partes. But of this shall I speake more largely in
an other place. In the mean season content your self with these
principles, whiche are certain of the chiefe groundes wheron all
demonstrations mathematical are fourmed, of which though the
moste parte seeme so plaine, that no childe doth doubte of them,
thinke not therfore that the art vnto whiche they serue, is
simple, other childishe, but rather consider, howe certayne the
profes of that arte is, y^t hath for his groundes soche playne
truthes, & as I may say, suche vndowbtfull and sensible
principles, And this is the cause why all learned menne dooth
approue the certenty of geometry, and consequently of the other
artes mathematical, which haue the grounds (as Arithmeticke,
musike and astronomy) aboue all other artes and sciences, that
be vsed amongest men. Thus muche haue I sayd of the first
principles, and now will I go on with the theoremes, whiche I do
only by examples declare, minding to reserue the proofes to a
peculiar boke which I will then set forth, when I perceaue this
to be thankfully taken of the readers of it.
[Illustration]
The theoremes of Geometry brieflye
declared by shorte examples.
_The firste Theoreme._
When .ij. triangles be so drawen, that the one of them hath
ij. sides equal to ij sides of the other triangle, and that
the angles enclosed with those sides, bee equal also in
bothe triangles, then is the thirde side likewise equall in
them. And the whole triangles be of one greatnes, and euery
angle in the one equall to his matche angle in the other,
I meane those angles that be inclosed with like sides.
_Example._
[Illustration]
This triangle A.B.C. hath ij. sides (that is to say) C.A. and
C.B, equal to ij. sides of the other triangle F.G.H, for A.C. is
equall to F.G, and B.C. is equall to G.H. And also the angle C.
contayned beetweene F.G, and G.H, for both of them answere to
the eight parte of a circle. Therfore doth it remayne that A.B.
whiche is the thirde lyne in the firste triangle, doth agre in
lengthe with F.H, w^{ch} is the third line in y^e second triangle
& y^e hole triangle. A.B.C. must nedes be equal to y^e hole
triangle F.G.H. And euery corner equall to his match, that is to
say, A. equall to F, B. to H, and C. to G, for those bee called
match corners, which are inclosed with like sides, other els do
lye against like sides.
_The second Theoreme._
In twileke triangles the ij. corners that be about the ground
line, are equal togither. And if the sides that be equal, be
drawen out in length then wil the corners that are vnder the
ground line, be equal also togither.
_Example_
[Illustration]
A.B.C. is a twileke triangle, for the one side A.C, is equal to
the other side B.C. And therfore I saye that the inner corners
A. and B, which are about the ground lines, (that is A.B.) be
equall togither. And farther if C.A. and C.B. bee drawen forthe
vnto D. and E. as you se that I haue drawen them, then saye I
that the two vtter angles vnder A. and B, are equal also
togither: as the theorem said. The profe wherof, as of al the
rest, shal apeare in Euclide, whome I intende to set foorth in
english with sondry new additions, if I may perceaue that it
wilbe thankfully taken.
_The thirde Theoreme._
If in annye triangle there bee twoo angles equall togither,
then shall the sides, that lie against those angles, be
equal also.
[Illustration]
_Example._
This triangle A.B.C. hath two corners equal eche to other, that
is A. and B, as I do by supposition limite, wherfore it foloweth
that the side A.C, is equal to that other side B.C, for the side
A.C, lieth againste the angle B, and the side B.C, lieth against
the angle A.
_The fourth Theoreme._
When two lines are drawen from the endes of anie one line,
and meet in anie pointe, it is not possible to draw two
other lines of like lengthe ech to his match that shal begin
at the same pointes, and end in anie other pointe then the
twoo first did.
_Example._
[Illustration]
The first line is A.B, on which I haue erected two other lines
A.C, and B.C, that meete in the pricke C, wherefore I say, it is
not possible to draw ij. other lines from A. and B. which shal
mete in one point (as you se A.D. and B.D. mete in D.) but that
the match lines shalbe vnequal, I mean by _match lines_, the two
lines on one side, that is the ij. on the right hand, or the ij.
on the lefte hand, for as you se in this example A.D. is longer
then A.C, and B.C. is longer then B.D. And it is not possible,
that A.C. and A.D. shall bee of one lengthe, if B.D. and B.C.
bee like longe. For if one couple of matche lines be equall (as
the same example A.E. is equall to A.C. in length) then must
B.E. needes be vnequall to B.C. as you see, it is here shorter.
_The fifte Theoreme._
If two triangles haue there ij. sides equal one to an other,
and their ground lines equal also, then shall their corners,
whiche are contained betwene like sides, be equall one to
the other.
_Example._
[Illustration]
Because these two triangles A.B.C, and D.E.F. haue two sides
equall one to an other. For A.C. is equall to D.F, and B.C. is
equall to E.F, and again their ground lines A.B. and D.E. are
lyke in length, therfore is eche angle of the one triangle
equall to ech angle of the other, comparyng together those
angles that are contained within lyke sides, so is A. equall
to D, B. to E, and C. to F, for they are contayned within like
sides, as before is said.
_The sixt Theoreme._
When any right line standeth on an other, the ij. angles
that thei make, other are both right angles, or els equall
to .ij. righte angles.
_Example._
[Illustration]
A.B. is a right line, and on it there doth light another right
line, drawen from C. perpendicularly on it, therefore saie I,
that the .ij. angles that thei do make, are .ij. right angles as
maie be iudged by the definition of a right angle. But in the
second part of the example, where A.B. beyng still the right
line, on which D. standeth in slope wayes, the two angles that
be made of them are not righte angles, but yet they are equall
to two righte angles, for so muche as the one is to greate, more
then a righte angle, so muche iuste is the other to little, so
that bothe togither are equall to two right angles, as you maye
perceiue.
_The seuenth Theoreme._
If .ij. lines be drawen to any one pricke in an other lyne,
and those .ij. lines do make with the fyrst lyne, two right
angles, other suche as be equall to two right angles, and
that towarde one hande, than those two lines doo make one
streyght lyne.
_Example._
[Illustration]
A.B. is a streyght lyne, on which there doth lyght two other
lines one frome D, and the other frome C, but considerynge that
they meete in one pricke E, and that the angles on one hand be
equal to two right corners (as the laste theoreme dothe declare)
therfore maye D.E. and E.C. be counted for one ryght lyne.
_The eight Theoreme._
When two lines do cut one an other crosseways they do make
their matche angles equall.
[Illustration]
_Example._
What matche angles are, I haue tolde you in the definitions of
the termes. And here A, and B. are matche corners in this
example, as are also C. and D, so that the corner A, is equall
to B, and the angle C, is equall to D.
_The nynth Theoreme._
Whan so euer in any triangle the line of one side is drawen
forthe in lengthe, that vtter angle is greater than any of
the two inner corners, that ioyne not with it.
_Example._
[Illustration]
The triangle A.D.C hathe hys grounde lyne A.C. drawen forthe in
lengthe vnto B, so that the vtter corner that it maketh at C, is
greater then any of the two inner corners that lye againste it,
and ioyne not wyth it, whyche are A. and D, for they both are
lesser then a ryght angle, and be sharpe angles, but C. is a
blonte angle, and therfore greater then a ryght angle.
_The tenth Theoreme._
In euery triangle any .ij. corners, how so euer you take
them, ar lesse then ij. right corners.
_Example._
[Illustration]
In the firste triangle E, whiche is a threlyke, and therfore
hath all his angles sharpe, take anie twoo corners that you
will, and you shall perceiue that they be lesser then ij. right
corners, for in euery triangle that hath all sharpe corners (as
you see it to be in this example) euery corner is lesse then a
right corner. And therfore also euery two corners must nedes be
lesse then two right corners. Furthermore in that other triangle
marked with M, whiche hath .ij. sharpe corners and one right,
any .ij. of them also are lesse then two right angles. For
though you take the right corner for one, yet the other whiche
is a sharpe corner, is lesse then a right corner. And so it is
true in all kindes of triangles, as you maie perceiue more
plainly by the .xxij. Theoreme.
_The .xi. Theoreme._
In euery triangle, the greattest side lieth against the
greattest angle.
_Example._
[Illustration]
As in this triangle A.B.C, the greattest angle is C. And A.B.
(whiche is the side that lieth against it) is the greatest and
longest side. And contrary waies, as A.C. is the shortest side,
so B. (whiche is the angle liyng against it) is the smallest and
sharpest angle, for this doth folow also, that is the longest
side lyeth against the greatest angle, so it that foloweth
_The twelft Theoreme._
In euery triangle the greattest angle lieth against the
longest side.
For these ij. theoremes are one in truthe.
_The thirtenth theoreme._
In euerie triangle anie ij. sides togither how so euer you
take them, are longer then the thirde.
[Illustration]
For example you shal take this triangle A.B.C. which hath a very
blunt corner, and therfore one of his sides greater a good deale
then any of the other, and yet the ij. lesser sides togither ar
greater then it. And if it bee so in a blunte angeled triangle,
it must nedes be true in all other, for there is no other kinde
of triangles that hathe the one side so greate aboue the other
sids, as thei y^t haue blunt corners.
_The fourtenth theoreme._
If there be drawen from the endes of anie side of a triangle
.ij. lines metinge within the triangle, those two lines
shall be lesse then the other twoo sides of the triangle,
but yet the corner that thei make, shall bee greater then
that corner of the triangle, whiche standeth ouer it.
_Example._
[Illustration]
A.B.C. is a triangle. on whose ground line A.B. there is drawen
ij. lines, from the ij. endes of it, I say from A. and B, and
they meete within the triangle in the pointe D, wherfore I say,
that as those two lynes A.D. and B.D, are lesser then A.C. and
B.C, so the angle D, is greatter then the angle C, which is the
angle against it.
_The fiftenth Theoreme._
If a triangle haue two sides equall to the two sides of an
other triangle, but yet the angle that is contained betwene
those sides, greater then the like angle in the other
triangle, then is his grounde line greater then the grounde
line of the other triangle.
[Illustration]
_Example._
A.B.C. is a triangle, whose sides A.C. and B.C, are equall to
E.D. and D.F, the two sides of the triangle D.E.F, but bicause
the angle in D, is greatter then the angle C. (whiche are the
ij. angles contayned betwene the equal lynes) therfore muste the
ground line E.F. nedes bee greatter thenne the grounde line A.B,
as you se plainely.
[Illustration]
_The xvi. Theoreme._
If a triangle haue twoo sides equalle to the two sides of an
other triangle, but yet hathe a longer ground line then that
other triangle, then is his angle that lieth betwene the
equall sides, greater then the like corner in the other
triangle.
_Example._
This Theoreme is nothing els, but the sentence of the last
Theoreme turned backward, and therfore nedeth none other profe
nother declaration, then the other example.
_The seuententh Theoreme._
If two triangles be such sort, that two angles of the one be
equal to ij. angles of the other, and that one side of the
one be equal to on side of the other, whether that side do
adioyne to one of the equall corners, or els lye againste
one of them, then shall the other twoo sides of those
triangles bee equalle togither, and the thirde corner also
shall be equall in those two triangles.
_Example._
[Illustration]
Bicause that A.B.C, the one triangle hath two corners A. and B,
equal to D.E, that are twoo corners of the other triangle.
D.E.F. and that they haue one side in theym bothe equall, that
is A.B, which is equall to D.E, therefore shall both the other
ij. sides be equall one to an other, as A.C. and B.C. equall to
D.F. and E.F, and also the thirde angle in them both shal be
equall, that is, the angle C. shal be equall to the angle F.
_The eightenth Theoreme._
When on ij. right lines ther is drawen a third right line
crosse waies, and maketh .ij. matche corners of the one line
equall to the like twoo matche corners of the other line,
then ar those two lines gemmow lines, or paralleles.
_Example._
[Illustration]
The .ij. fyrst lynes are A.B. and C.D, the thyrd lyne that
crosseth them is E.F. And bycause that E.F. maketh ij. matche
angles with A.B, equall to .ij. other lyke matche angles on C.D,
(that is to say E.G, equall to K.F, and M.N. equall also to
H.L.) therfore are those ij. lynes A.B. and C.D. gemow lynes,
vnderstand here by _lyke matche corners_, those that go one way
as doth E.G, and K.F, lyke ways N.M, and H.L, for as E.G. and
H.L, other N.M. and K.F. go not one waie, so be not they lyke
match corners.
_The nyntenth Theoreme._
When on two right lines there is drawen a thirde right line
crossewaies, and maketh the ij. ouer corners towarde one
hande equall togither, then ar those .ij. lines paralleles.
And in like maner if two inner corners toward one hande, be
equall to .ii. right angles.
_Example._
As the Theoreme dothe speake of .ij. ouer angles, so muste you
vnderstande also of .ij. nether angles, for the iudgement is
lyke in bothe. Take for example the figure of the last theoreme,
where A.B, and C.D, be called paralleles also, bicause E. and K,
(whiche are .ij. ouer corners) are equall, and lykewaies L.
and M. And so are in lyke maner the nether corners N. and H, and
G. and F. Nowe to the seconde parte of the theoreme, those .ij.
lynes A.B. and C.D, shall be called paralleles, because the ij.
inner corners. As for example those two that bee toward the
right hande (that is G. and L.) are equall (by the fyrst parte
of this nyntenth theoreme) therfore muste G. and L. be equall to
two ryght angles.
_The xx. Theoreme._
When a right line is drawen crosse ouer .ij. right gemow
lines, it maketh .ij. matche corners of the one line, equall
to two matche corners of the other line, and also bothe ouer
corners of one hande equall togither, and bothe nether
corners like waies, and more ouer two inner corners, and two
vtter corners also towarde one hande, equall to two right
angles.
_Example._
Bycause A.B. and C.D, (in the laste figure) are paralleles,
therefore the two matche corners of the one lyne, as E.G. be
equall vnto the .ij. matche corners of the other line, that is
K.F, and lykewaies M.N, equall to H.L. And also E. and K. bothe
ouer corners of the lefte hande equall togyther, and so are M.
and L, the two ouer corners on the ryghte hande, in lyke maner
N. and H, the two nether corners on the lefte hande, equall eche
to other, and G. and F. the two nether angles on the right hande
equall togither.
[Sec.] Farthermore yet G. and L. the .ij. inner angles on the right
hande bee equall to two right angles, and so are M. and F. the
.ij. vtter angles on the same hande, in lyke manner shall you
say of N. and K. the two inner corners on the left hand. and of
E. and H. the two vtter corners on the same hande. And thus you
see the agreable sentence of these .iii. theoremes to tende to
this purpose, to declare by the angles how to iudge paralleles,
and contrary waies howe you may by paralleles iudge the
proportion of the angles.
_The xxi. Theoreme._
What so euer lines be paralleles to any other line, those
same be paralleles togither.
_Example._
[Illustration]
A.B. is a gemow line, or a parallele vnto C.D. And E.F,
lykewaies is a parallele vnto C.D. Wherfore it foloweth, that
A.B. must nedes bee a parallele vnto E.F.
_The .xxij. theoreme._
In euery triangle, when any side is drawen forth in length,
the vtter angle is equall to the ij. inner angles that lie
againste it. And all iij. inner angles of any triangle are
equall to ij. right angles.
[Illustration]
_Example._
The triangle beeyng A.D.E. and the syde A.E. drawen foorthe vnto
B, there is made an vtter corner, whiche is C, and this vtter
corner C, is equall to bother the inner corners that lye agaynst
it, whyche are A. and D. And all thre inner corners, that is to
say, A.D. and E, are equall to two ryght corners, whereof it
foloweth, _that all the three corners of any one triangle are
equall to all the three corners of euerye other triangle_. For
what so euer thynges are equalle to anny one thyrde thynge,
those same are equalle togitther, by the fyrste common sentence,
so that bycause all the .iij. angles of euery triangle are
equall to two ryghte angles, and all ryghte angles bee equall
togyther (by the fourth request) therfore must it nedes folow,
that all the thre corners of euery triangle (accomptyng them
togyther) are equall to iij. corners of any triangle, taken all
togyther.
_The .xxiii. theoreme._
When any ij. right lines doth touche and couple .ij. other
righte lines, whiche are equall in length and paralleles,
and if those .ij. lines bee drawen towarde one hande, then
are thei also equall together, and paralleles.
_Example._
[Illustration]
A.B. and C.D. are ij. ryght lynes and paralleles and equall in
length, and they ar touched and ioyned togither by ij. other
lynes A.C. and B.D, this beyng so, and A.C. and B.D. beyng
drawen towarde one syde (that is to saye, bothe towarde the
lefte hande) therefore are A.C. and B.D. bothe equall and also
paralleles.
_The .xxiiij. theoreme._
In any likeiamme the two contrary sides ar equall togither,
and so are eche .ij. contrary angles, and the bias line that
is drawen in it, dothe diuide it into two equall portions.
_Example._
[Illustration]
Here ar two likeiammes ioyned togither, the one is a longe
square A.B.E, and the other is a losengelike D.C.E.F. which ij.
likeiammes ar proued equall togither, bycause they haue one
ground line, that is, F.E, And are made betwene one payre of
gemow lines, I meane A.D. and E.H. By this Theoreme may you know
the arte of the righte measuringe of likeiammes, as in my booke
of measuring I wil more plainly declare.
_The xxvi. Theoreme._
All likeiammes that haue equal grounde lines and are drawen
betwene one paire of paralleles, are equal togither.
_Example._
Fyrste you muste marke the difference betwene this Theoreme and
the laste, for the laste Theoreme presupposed to the diuers
likeiammes one ground line common to them, but this theoreme
doth presuppose a diuers ground line for euery likeiamme, only
meaning them to be equal in length, though they be diuers in
numbre. As for example. In the last figure ther are two
parallels, A.D. and E.H, and betwene them are drawen thre
likeiammes, the firste is, A.B.E.F, the second is E.C.D.F, and
the thirde is C.G.H.D. The firste and the seconde haue one
ground line, (that is E.F.) and therfore in so muche as they are
betwene one paire of paralleles, they are equall accordinge to
the fiue and twentye Theoreme, but the thirde likeiamme that is
C.G.H.D. hathe his grounde line G.H, seuerall frome the other,
but yet equall vnto it. wherefore the third likeiam is equall to
the other two firste likeiammes. And for a proofe that G.H.
being the ground or ground line of the third likeiamme, is equal
to E.F, whiche is the ground line to both the other likeiams,
that may be thus declared, G.H. is equall to C.D, seynge they
are the contrary sides of one likeiamme (by the foure and twenty
theoreme) and so are C.D. and E.F. by the same theoreme.
Therfore seynge both those ground lines E.F. and G.H, are equall
to one thirde line (that is C.D.) they must nedes bee equall
togyther by the firste common sentence.
_The xxvii. Theoreme._
All triangles hauinge one grounde lyne, and standing betwene
one paire of parallels, ar equall togither.
_Example._
[Illustration]
A.B. and C.F. are twoo gemowe lines, betweene which there be
made two triangles, A.D.E. and D.E.B, so that D.E, is the common
ground line to them bothe. wherfore it doth folow, that those
two triangles A.D.E. and D.E.B. are equall eche to other.
_The xxviij. Theoreme._
All triangles that haue like long ground lines, and bee made
betweene one paire of gemow lines, are equall togither.
_Example._
Example of this Theoreme you may see in the last figure, where
as sixe triangles made betwene those two gemowe lines A.B. and
C.F, the first triangle is A.C.D, the seconde is A.D.E, the
thirde is A.D.B, the fourth is A.B.E, the fifte is D.E.B, and
the sixte is B.E.F, of which sixe triangles, A.D.E. and D.E.B.
are equall, bicause they haue one common grounde line. And so
likewise A.B.E. and A.B.D, whose commen grounde line is A.B, but
A.C.D. is equal to B.E.F, being both betwene one couple of
parallels, not bicause thei haue one ground line, but bicause
they haue their ground lines equall, for C.D. is equall to E.F,
as you may declare thus. C.D, is equall to A.B. (by the foure
and twenty Theoreme) for thei are two contrary sides of one
lykeiamme. A.C.D.B, and E.F by the same theoreme, is equall to
A.B, for thei ar the two y^e contrary sides of the likeiamme,
A.E.F.B, wherfore C.D. must needes be equall to E.F. like wise
the triangle A.C.D, is equal to A.B.E, bicause they ar made
betwene one paire of parallels and haue their groundlines like,
I meane C.D. and A.B. Againe A.D.E, is equal to eche of them
both, for his ground line D.E, is equall to A.B, inso muche as
they are the contrary sides of one likeiamme, that is the long
square A.B.D.E. And thus may you proue the equalnes of all the
reste.
_The xxix. Theoreme._
Al equal triangles that are made on one grounde line, and
rise one waye, must needes be betwene one paire of
parallels.
_Example._
Take for example A.D.E, and D.E.B, which (as the xxvij.
conclusion dooth proue) are equall togither, and as you see,
they haue one ground line D.E. And againe they rise towarde one
side, that is to say, vpwarde toward the line A.B, wherfore they
must needes be inclosed betweene one paire of parallels, which
are heere in this example A.B. and D.E.
_The thirty Theoreme._
Equal triangles that haue their ground lines equal, and be
drawen toward one side, ar made betwene one paire of
paralleles.
_Example._
The example that declared the last theoreme, maye well serue to
the declaracion of this also. For those ij. theoremes do diffre
but in this one pointe, that the laste theoreme meaneth of
triangles, that haue one ground line common to them both, and
this theoreme dothe presuppose the grounde lines to bee diuers,
but yet of one length, as A.C.D, and B.E.F, as they are ij.
equall triangles approued, by the eighte and twentye Theorem, so
in the same Theorem it is declared, y^t their ground lines are
equall togither, that is C.D, and E.F, now this beeynge true,
and considering that they are made towarde one side, it
foloweth, that they are made betwene one paire of parallels when
I saye, drawen towarde one side, I meane that the triangles must
be drawen other both vpward frome one parallel, other els both
downward, for if the one be drawen vpward and the other
downward, then are they drawen betwene two paire of parallels,
presupposinge one to bee drawen by their ground line, and then
do they ryse toward contrary sides.
_The xxxi. theoreme._
If a likeiamme haue one ground line with a triangle, and be
drawen betwene one paire of paralleles, then shall the
likeiamme be double to the triangle.
_Example._
[Illustration]
A.H. and B.G. are .ij. gemow lines, betwene which there is made
a triangle B.C.G, and a lykeiamme, A.B.G.C, whiche haue a
grounde lyne, that is to saye, B.G. Therfore doth it folow that
the lyke iamme A.B.G.C. is double to the triangle B.C.G. For
euery halfe of that lykeiamme is equall to the triangle, I meane
A.B.F.E. other F.E.C.G. as you may coniecture by the .xi.
conclusion geometrical.
And as this Theoreme dothe speake of a triangle and likeiamme
that haue one groundelyne, so is it true also, yf theyr
groundelynes bee equall, though they bee dyuers, so that thei be
made betwene one payre of paralleles. And hereof may you
perceaue the reason, why in measuryng the platte of a triangle,
you must multiply the perpendicular lyne by halfe the grounde
lyne, or els the hole grounde lyne by halfe the perpendicular,
for by any of these bothe waies is there made a lykeiamme equall
to halfe suche a one as shulde be made on the same hole grounde
lyne with the triangle, and betweene one payre of paralleles.
Therfore as that lykeiamme is double to the triangle, so the
halfe of it, must needes be equall to the triangle. Compare the
.xi. conclusion with this theoreme.
_The .xxxij. Theoreme._
In all likeiammes where there are more than one made aboute
one bias line, the fill squares of euery of them must nedes
be equall.
[Illustration]
_Example._
Fyrst before I declare the examples, it shal be mete to shew the
true vnderstandyng of this theorem. [Sidenote: _Bias lyne._]
Therfore by the _Bias line_, I meane that lyne, whiche in any
square figure dooth runne from corner to corner. And euery
square which is diuided by that bias line into equall halues
from corner to corner (that is to say, into .ij. equall
triangles) those be counted _to stande aboute one bias line_,
and the other squares, whiche touche that bias line, with one of
their corners onely, those doo I call _Fyll squares_, [Sidenote:
_Fyll squares._] accordyng to the greke name, which is
_anapleromata_, [Sidenote: #anaple:ro:mata#] and called in latin
_supplementa_, bycause that they make one generall square,
includyng and enclosyng the other diuers squares, as in this
example H.C.E.N. is one square likeiamme, and L.M.G.C. is an
other, whiche bothe are made aboute one bias line, that is N.M,
than K.L.H.C. and C.E.F.G. are .ij. fyll squares, for they doo
fyll vp the sydes of the .ij. fyrste square lykeiammes, in suche
sorte, that all them foure is made one greate generall square
K.M.F.N.
Nowe to the sentence of the theoreme, I say, that the .ij. fill
squares, H.K.L.C. and C.E.F.G. are both equall togither, (as it
shall bee declared in the booke of proofes) bicause they are the
fill squares of two likeiammes made aboute one bias line, as the
exaumple sheweth. Conferre the twelfthe conclusion with this
theoreme.
_The xxxiij. Theoreme._
In all right anguled triangles, the square of that side
whiche lieth against the right angle, is equall to the .ij.
squares of both the other sides.
_Example._
[Illustration]
A.B.C. is a triangle, hauing a ryght angle in B. Wherfore it
foloweth, that the square of A.C, (whiche is the side that lyeth
agaynst the right angle) shall be as muche as the two squares of
A.B. and B.C. which are the other .ij. sides.
[Sec.] By the square of any lyne, you muste vnderstande a figure made
iuste square, hauyng all his iiij. sydes equall to that line,
whereof it is the square, so is A.C.F, the square of A.C.
Lykewais A.B.D. is the square of A.B. And B.C.E. is the square
of B.C. Now by the numbre of the diuisions in eche of these
squares, may you perceaue not onely what the square of any line
is called, but also that the theoreme is true, and expressed
playnly bothe by lines and numbre. For as you see, the greatter
square (that is A.C.F.) hath fiue diuisions on eche syde, all
equall togyther, and those in the whole square are twenty and
fiue. Nowe in the left square, whiche is A.B.D. there are but
.iij. of those diuisions in one syde, and that yeldeth nyne in
the whole. So lykeways you see in the meane square A.C.E. in
euery syde .iiij. partes, whiche in the whole amount vnto
sixtene. Nowe adde togyther all the partes of the two lesser
squares, that is to saye, sixtene and nyne, and you perceyue
that they make twenty and fiue, whyche is an equall numbre to
the summe of the greatter square.
By this theoreme you may vnderstand a redy way to know the syde
of any ryght anguled triangle that is vnknowen, so that you
knowe the lengthe of any two sydes of it. For by tournynge the
two sydes certayne into theyr squares, and so addynge them
togyther, other subtractynge the one from the other (accordyng
as in the vse of these theoremes I haue sette foorthe) and then
fyndynge the roote of the square that remayneth, which roote
(I meane the syde of the square) is the iuste length of the
vnknowen syde, whyche is sought for. But this appertaineth to
the thyrde booke, and therefore I wyll speake no more of it at
this tyme.
_The xxxiiij. Theoreme._
If so be it, that in any triangle, the square of the one
syde be equall to the .ij. squares of the other .ij. sides,
than must nedes that corner be a right corner, which is
conteined betwene those two lesser sydes.
_Example._
As in the figure of the laste Theoreme, bicause A.C, made in
square, is asmuch as the square of A.B, and also as the square
of B.C. ioyned bothe togyther, therefore the angle that is
inclosed betwene those .ij. lesser lynes, A.B. and B.C. (that is
to say) the angle B. whiche lieth against the line A.C, must
nedes be a ryght angle. This theoreme dothe so depende of the
truthe of the laste, that whan you perceaue the truthe of the
one, you can not iustly doubt of the others truthe, for they
conteine one sentence, contrary waies pronounced.
_The .xxxv. theoreme._
If there be set forth .ij. right lines, and one of them
parted into sundry partes, how many or few so euer they be,
the square that is made of those ij. right lines proposed,
is equal to all the squares, that are made of the vndiuided
line, and euery parte of the diuided line.
[Illustration]
_Example._
The ij. lines proposed ar A.B. and C.D, and the lyne A.B. is
deuided into thre partes by E. and F. Now saith this theoreme,
that the square that is made of those two whole lines A.B. and
C.D, so that the line A.B. standeth for the length of the square,
and the other line C.D. for the bredth of the same. That square
(I say) wil be equall to all the squares that be made, of the
vndiueded lyne (which is C.D.) and euery portion of the diuided
line. And to declare that particularly, Fyrst I make an other
line G.K, equall to the line .C.D, and the line G.H. to be equal
to the line A.B, and to bee diuided into iij. like partes, so
that G.M. is equall to A.E, and M.N. equal to E.F, and then
muste N.H. nedes remaine equall to F.B. Then of those ij. lines
G.K, vndeuided, and G.H. which is deuided, I make a square, that
is G.H.K.L, In which square if I drawe crosse lines frome one
side to the other, according to the diuisions of the line G.H,
then will it appear plaine, that the theoreme doth affirme. For
the first square G.M.O.K, must needes be equal to the square of
the line C.D, and the first portion of the diuided line, which is
A.E, for bicause their sides are equall. And so the seconde
square that is M.N.P.O, shall be equall to the square of C.D,
and the second part of A.B, that is E.F. Also the third square
which is N.H.L.P, must of necessitee be equal to the square of
C.D, and F.B, bicause those lines be so coupeled that euery
couple are equall in the seuerall figures. And so shal you not
only in this example, but in all other finde it true, that if
one line be deuided into sondry partes, and an other line whole
and vndeuided, matched with him in a square, that square which
is made of these two whole lines, is as muche iuste and equally,
as all the seuerall squares, whiche bee made of the whole line
vndiuided, and euery part seuerally of the diuided line.
_The xxxvi. Theoreme._
If a right line be parted into ij. partes, as chaunce may
happe, the square that is made of the whole line, is equall
to bothe the squares that are made of the same line, and the
twoo partes of it seuerally.
_Example._
[Illustration]
The line propounded beyng A.B. and deuided, as chaunce
happeneth, in C. into ij. vnequall partes, I say that the square
made of the hole line A.B, is equal to the two squares made of
the same line with the twoo partes of itselfe, as with A.C, and
with C.B, 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, but that the greater
square is equall to the square of the whole line A.B, and the
partiall squares equall to the squares of the second partes of
the same line ioyned with the whole line, your eye may iudg
without muche declaracion, so that I shall not neede to make
more exposition therof, but that you may examine it, as you did
in the laste Theoreme.
_The xxxvij. Theoreme._
If a right line be deuided by chaunce, as it maye happen,
the square that is made of the whole line, and one of the
partes of it which soeuer it be, shal be equall to that
square that is made of the ij. partes ioyned togither, and
to an other square made of that part, which was before
ioyned with the whole line.
_Example._
[Illustration]
The line A.B. is deuided in C. into twoo partes, though not
equally, of which two partes for an example I take the first,
that is A.C, and of it I make one side of a square, as for
example D.G. accomptinge those two lines to be equall, the other
side of the square is D.E, whiche is equall to the whole line
A.B.
Now may it appeare, to your eye, that the great square made of
the whole line A.B, and of one of his partes that is A.C, (which
is equall with D.G.) is equal to two partiall squares, whereof
the one is made of the saide greatter portion A.C, in as muche
as not only D.G, beynge one of his sides, but also D.H. beinge
the other side, are eche of them equall to A.C. The second
square is H.E.F.K, in which the one side H.E, is equal to C.B,
being the lesser parte of the line, A.B, and E.F. is equall to
A.C. which is the greater parte of the same line. So that those
two squares D.H.K.G and H.E.F.K, bee bothe of them no more then
the greate square D.E.F.G, accordinge to the wordes of the
Theoreme afore saide.
_The xxxviij. Theoreme._
If a righte line be deuided by chaunce, into partes, the
square that is made of that whole line, is equall to both
the squares that ar made of eche parte of the line, and
moreouer to two squares made of the one portion of the
diuided line ioyned with the other in square.
[Illustration]
_Example._
Lette the diuided line bee A.B, and parted in C, into twoo
partes: Nowe saithe the Theoreme, that the square of the whole
lyne A.B, is as mouche iuste as the square of A.C, and the
square of C.B, eche by it selfe, and more ouer by as muche
twise, as A.C. and C.B. ioyned in one square will make. For as
you se, the great square D.E.F.G, conteyneth in hym foure lesser
squares, of whiche the first and the greatest is N.M.F.K, and is
equall to the square of the lyne A.C. The second square is the
lest of them all, that is D.H.L.N, and it is equall to the
square of the line C.B. Then are there two other longe squares
both of one bygnes, that is H.E.N.M. and L.N.G.K, eche of them
both hauyng .ij. sides equall to A.C, the longer parte of the
diuided line, and there other two sides equall to C.B, beeyng he
shorter parte of the said line A.B.
So is that greatest square, beeyng made of the hole lyne A.B,
equal to the ij. squares of eche of his partes seuerally, and
more by as muche iust as .ij. longe squares, made of the longer
portion of the diuided lyne ioyned in square with the shorter
parte of the same diuided line, as the theoreme wold. And as
here I haue put an example of a lyne diuided into .ij. partes,
so the theoreme is true of all diuided lines, of what number so
euer the partes be, foure, fyue, or syxe. etc.
This theoreme hath great vse, not only in geometrie, but also in
arithmetike, as herafter I will declare in conuenient place.
_The .xxxix. theoreme._
If a right line be deuided into two equall partes, and one
of these .ij. partes diuided agayn into two other partes, as
happeneth the longe square that is made of the thyrd or
later part of that diuided line, with the residue of the
same line, and the square of the mydlemoste parte, are bothe
togither equall to the square of halfe the firste line.
_Example._
[Illustration]
The line A.B. is diuided into ij. equal partes in C, and that
parte C.B. is diuided agayne as hapneth in D. Wherfore saith the
Theorem that the long square made of D.B. and A.D, with the
square of C.D. (which is the mydle portion) shall bothe be
equall to the square of half the lyne A.B, that is to saye, to
the square of A.C, or els of C.D, which make all one. The long
square F.G.N.O. whiche is the longe square that the theoreme
speaketh of, is made of .ij. long squares, wherof the fyrst is
F.G.M.K, and the seconde is K.N.O.M. The square of the myddle
portion is L.M.O.P. and the square of the halfe of the fyrste
lyne is E.K.Q.L. Nowe by the theoreme, that longe square
F.G.M.O, with the iuste square L.M.O.P, muste bee equall to the
greate square E.K.Q.L, whyche thynge bycause it seemeth somewhat
difficult to vnderstande, althoughe I intende not here to make
demonstrations of the Theoremes, bycause it is appoynted to be
done in the newe edition of Euclide, yet I wyll shew you brefely
how the equalitee of the partes doth stande. And fyrst I say,
that where the comparyson of equalitee is made betweene the
greate square (whiche is made of halfe the line A.B.) and two
other, where of the fyrst is the longe square F.G.N.O, and the
second is the full square L.M.O.P, which is one portion of the
great square all redye, and so is that longe square K.N.M.O,
beynge a parcell also of the longe square F.G.N.O, Wherfore as
those two partes are common to bothe partes compared in
equalitee, and therfore beynge bothe abated from eche parte, if
the reste of bothe the other partes bee equall, than were those
whole partes equall before: Nowe the reste of the great square,
those two lesser squares beyng taken away, is that longe square
E.N.P.Q, whyche is equall to the long square F.G.K.M, beyng the
rest of the other parte. And that they two be equall, theyr
sydes doo declare. For the longest lynes that is F.K and E.Q are
equall, and so are the shorter lynes, F.G, and E.N, and so
appereth the truthe of the Theoreme.
_The .xl. theoreme._
If a right line be diuided into .ij. euen partes, and an
other right line annexed to one ende of that line, so that
it make one righte line with the firste. The longe square
that is made of this whole line so augmented, and the
portion that is added, with the square of halfe the right
line, shall be equall to the square of that line, whiche is
compounded of halfe the firste line, and the parte newly
added.
_Example._
[Illustration]
The fyrst lyne propounded is A.B, and it is diuided into ij.
equall partes in C, and an other ryght lyne, I meane B.D annexed
to one ende of the fyrste lyne.
Nowe say I, that the long square A.D.M.K, is made of the whole
lyne so augmented, that is A.D, and the portion annexed, y^t is
D.M, for D.M is equall to B.D, wherfore y^t long square A.D.M.K,
with the square of halfe the first line, that is E.G.H.L, is
equall to the great square E.F.D.C. whiche square is made of the
line C.D. that is to saie, of a line compounded of halfe the
first line, beyng C.B, and the portion annexed, that is B.D. And
it is easyly perceaued, if you consyder that the longe square
A.C.L.K. (whiche onely is lefte out of the great square) hath
another longe square equall to hym, and to supply his steede in
the great square, and that is G.F.M.H. For their sydes be of
lyke lines in length.
_The xli. Theoreme._
If a right line bee diuided by chaunce, the square of the
same whole line, and the square of one of his partes are
iuste equall to the long square of the whole line, and the
sayde parte twise taken, and more ouer to the square of the
other parte of the sayd line.
_Example._
[Illustration]
A.B. is the line diuided in C. And D.E.F.G, is the square of the
whole line, D.H.K.M. is the square of the lesser portion (whyche
I take for an example) and therfore must bee twise reckened.
Nowe I saye that those ij. squares are equall to two longe
squares of the whole line A.B, and his sayd portion A.C, and
also to the square of the other portion of the sayd first line,
whiche portion is C.B, and his square K.N.F.L. In this theoreme
there is no difficultie, if you consyder that the litle square
D.H.K.M. is .iiij. tymes reckened, that is to say, fyrst of all
as a parte of the greatest square, whiche is D.E.F.G. Secondly
he is rekned by him selfe. Thirdely he is accompted as parcell
of the long square D.E.N.M, And fourthly he is taken as a part
of the other long square D.H.L.G, so that in as muche as he is
twise reckened in one part of the comparison of equalitee, and
twise also in the second parte, there can rise none occasion of
errour or doubtfulnes therby.
_The xlij. Theoreme._