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+
+The Project Gutenberg EBook of The Way To Geometry, by Peter Ramus
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Way To Geometry
+
+Author: Peter Ramus
+
+Translator: William Bedwell
+
+Release Date: October 2, 2008 [EBook #26752]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE WAY TO GEOMETRY ***
+
+
+
+
+Produced by Jonathan Ingram, Keith Edkins and the Online
+Distributed Proofreading Team at https://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+<table border="0" cellpadding="10" style="background-color: #ccccff;">
+<tr>
+<td style="width:25%; vertical-align:top">
+Transcriber's note:
+</td>
+<td>
+<p>Several few typographical errors have been corrected. They
+appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the
+explanation will appear when the mouse pointer is moved over the marked
+passage. Cases which could conceivably be genuine variable orthography have been retained as printed.
+Many corrections are actually part of the arguments: no guarantee is given that all such errors
+in the text have been found and corrected.
+</p><p>
+In the original, 20 pages were printed with out-of-sequence page numbers, the numbers as printed being repeated in the index. These
+have been retained as printed but marked with an asterisk to distinguish them from the in-sequence pages which
+have the same number, thus 249*
+</p><p>
+Mixed fractions have been consistently transcribed as e.g. 5.5/96 although the period is not always present in the printed text (this
+avoids possible confusion of a hyphen, as 5-5/96, with a minus sign). The plus and minus signs in the printed text are apparently
+indistinguishable dashes - they have been transcribed as '+' and '-' as the context requires.
+</p></td>
+</tr>
+</table>
+
+<h3><i>VIA REGIA</i></h3>
+
+<p class="cenhead"><span class="sc">Ad</span></p>
+
+<h3>GEOMETRIAM.</h3>
+
+<hr class="full" />
+
+<h3>THE WAY</h3>
+
+<p class="cenhead">TO</p>
+
+<h1>GEOMETRY.</h1>
+
+<h2>Being necessary and usefull,</h2>
+
+<p class="cenhead">For</p>
+
+<p class="cenhead"><i>Astronomers.
+Engineres.
+Geographers.
+Architecks.
+Land-meaters.
+Carpenters.
+Sea-men.
+Paynters.
+Carvers, &amp;c.</i></p>
+
+<hr class="full" />
+
+<p class="cenhead">Written in Latine by <span class="sc">Peter Ramvs</span>, and now
+Translated and much enlarged by the Learned
+M<sup>r</sup>. <span class="sc">William Bedwell.</span></p>
+
+<hr class="full" />
+
+<p class="cenhead"><i>LONDON</i>,<br />
+Printed by <i>Thomas Cotes</i>, And are to be sold by<br />
+<i>Michael Sparke</i>; at the blew Bible in<br />
+Greene Arbour, 1636.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/156.png"><img style="width:100%" src="images/156.png"
+      alt="Surveying." title="Surveying." /></a>
+  </div>
+  <div class="figcenter" style="width:30%;">
+      <a href="images/149.png"><img style="width:100%" src="images/149.png"
+      alt="Surveying." title="Surveying." /></a>
+  </div>
+<h2>TO THE</h2>
+
+<h1>WORSHIPFVL</h1>
+
+<h2><span class="sc">M. Iohn Greaves</span>, <i>Professor</i> of</h2>
+
+<h3><i>Geometry</i> in <i>Gresham Colledge</i> London;</h3>
+
+<p class="cenhead"><i>All happinesse</i>.</p>
+
+  <p><i>SIR</i>,</p>
+
+  <p><i>Your acquaintance with the Author before his death was not long,
+  which I have oft heard you say, you counted your great unhappinesse, but
+  within a short time after, you knew not well whether to count your selfe
+  more happie in that you once knew him, or unhappy in that upon your
+  acquaintance you so suddenly lost him. This his worke then being to come
+  forth to the censorious eye of the world, and as the manner usually is to
+  have some Patronage, I have thought good to dedicate it to your selfe;
+  and that for these two reasons especially</i>.</p>
+
+  <p><i>First, in respect of the sympathy betwixt it, and your studies;
+  Laboures of this nature being usually offered to such persons whose
+  profession is that way setled</i>.</p>
+
+  <p><i>Secondly, for the great love and respect you alwayes shewed to the
+  Author, being indeed a man that would deserve no lesse, humble, void of
+  pride, ever ready to impart his knowledge to others in what kind soever,
+  loving and affecting those that affected learning</i>.</p>
+
+  <p><i>For these respects then, I offer to you this Worke of your so much
+  honoured friend. I my selfe also (as it is no lesse my duty) for his sake
+  striving to make you hereby some part of a requitall, least I should be
+  found guilty of ingratitude, which is a solecisme in manners, if having
+  so fit an opportunity, I should not expresse to the world some Testimonie
+  of love to you, who so much loved him. I desire then (good Sir) your kind
+  acceptance of it, you knowing so well the ability of the Author, and
+  being also able to judge of a Worke of this nature, and in that respect
+  the better able to defend it from the furie of envious Detractours, of
+  which there are not few. Thus with my best wishes to you, as to my much
+  respected friend, I rest</i>.</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>Yours to be commanded in</p>
+      <p>any thing that he is able.</p>
+    </div>
+
+    <div class="stanza">
+      <p><span class="sc">Iohn Clerke.</span></p>
+    </div>
+  </div>
+<h3><i>To the Reader.</i></h3>
+
+  <p>Friendly Reader, that which is here set forth to thy view, is a
+  Translation out of <i>Ramus</i>. Formerly indeed Translated by one
+  M<sup>r</sup>. <i>Thomas Hood</i>, but never before set forth with the
+  Demonstrations and Diagrammes, which being cut before the Authors death,
+  and the Worke it selfe finished, the Coppie I having in mine hands, never
+  had thought for the promulgation of it, but that it should have died with
+  its Author, considering no small prejudice usually attends the printing
+  of dead mens Workes, and wee see the times, the world is now all eare and
+  tongue, the most given with the <i>Athenians</i>, to little else than to
+  heare and tell newes: And if <i>Apelles</i> that skilfull Artist alwayes
+  found somewhat to be amended in those Pictures which he had most
+  curiously drawne; surely much in this Worke might have beene amended if
+  the Authour had lived to refine it, but in that it was onely the first
+  draught, and that he was prevented by death of a second view, though
+  perused by others before the Presse; I was ever unwilling to the
+  publication, but that I was often and much solicited with iteration of
+  strong importunity, and so in the end over-ruled: perswading me from time
+  to time unto it, and that it being finished by the Authour, it was farre
+  better to be published, though with some errours and escapes, than to be
+  onely moths-meat, and so utterly lost. I would have thee, Courteous
+  Reader know, that it is no conceit of the worth of the thing that I
+  should expose the name and credit of the Authour to a publike censure;
+  yet I durst be bold to say, had he lived to have fitted it, and corrected
+  the Presse, the worke would have pointed out the workeman. For I may say,
+  without vaine ostentation, he was a man of worth and note, and there was
+  not that kinde of learning in which he had not some knowledge, but
+  especially for the Easterne tongues, those deepe and profound Studies, in
+  the judgement of the learned, which knew him well, he hath not left his
+  fellow behind him; as his Workes also in Manuscript now extant in the
+  publike Library of the famous Vniversity of Cambridge; do testifie no
+  lesse; for him then being so grave and learned a Divine to meddle with a
+  worke of this nature, he gives thee a reason in his owne following
+  Preface for his principall end and intent of taking this Worke in hand,
+  was not for the deepe and Iudiciall, but for the shallowest skull, the
+  good and profit of the simpler sort, who as it was in the Latine, were
+  able to get little or no benifite from it. Therefore considering the
+  worth of the Authour, and his intent in the Worke. Reade it favourably,
+  and if the faults be not too great, cover them with the mantle of love,
+  and judge charitably offences unwillingly committed, and doe according to
+  the termes of equitie, as thou wouldest be done unto, but it is a common
+  saying, as <i>Printers</i> get Copies for their profit, so Readers often
+  buy and reade for their pleasure; and there is no worke so exactly done
+  that can escape the malevolous disposition of some detracting spirits, to
+  whom I say, as one well, <i>Facilius est unicuivis nostrum aliena curios�
+  observare: quam proproia negotia rect� agere</i>. It is a great deale
+  more easie to carpe at other mens doings, than to give better of his
+  owne. And as <i>Arist</i>. <span title="to pasin aresai duscherestaton esti" class="grk"
+  >&tau;&#x1F79; &pi;&#x1F71;&sigma;&iota;&nu;
+  &#x1F00;&rho;&#x1F73;&sigma;&alpha;&iota;
+  &delta;&upsilon;&sigma;&chi;&epsilon;&rho;&#x1F73;&sigma;&tau;&alpha;&tau;&#x1F79;&nu;
+  &#x1F10;&sigma;&tau;&iota;</span>; <i>omnibus placere difficilimum
+  est</i>. But wherefore, Gentle Reader, should I make any doubt of thy
+  curtesie, and favourable acceptance; for surely there can be nothing more
+  contrary to equitie, than to speake evill of those that have taken paines
+  to doe good, a Pagan would hardly doe this, much lesse I hope any good
+  Christian. Read then, and if by reading, thou reapest any profit, I have
+  my desire, if not, the fault shall be thine owne, reading haply more to
+  judge and censure, than for any good and benefit which otherwise may be
+  received from it; let but the same mind towards thine owne good possesse
+  thee in reading it, as did the Author in writing it, and there shall be
+  no neede to doubt of thy profit by it.</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p><i>Thine in the common</i></p>
+      <p><i>bond of love</i>,</p>
+    </div>
+
+    <div class="stanza">
+      <p><span class="sc">Iohn Clerke</span>.</p>
+    </div>
+  </div>
+<hr class="full" />
+
+<h2>The Authors Preface.</h2>
+
+  <p><i>Two things, I feare me, will here be objected against me: The one
+  concerneth my selfe, directly: The other mine Author, and the worke I
+  have taken in hand the translating of him. Concerning my selfe, I
+  suppose, some will aske, Why I being a Divine; should meddle or busie my
+  selfe with these prophane studies? </i>Geometry<i> may no way further
+  Divinity, and therefore is no fit study for a Divine? This objection
+  seemeth to smell of Brownisme, that is, of a ranke peevish humour
+  overflowing the stomach of some, whereby they are caused to loath all
+  manner of solid learning, yea of true Divinity it selfe, and therefore it
+  doth not deserve an answer: And this we in our Title before signified.
+  For we have not taken this paines for Turkes and others, who by the lawes
+  of their profession are bound to abandon all manner of learning. But if
+  any man shall propose it, as a question, with a desire of satisfaction,
+  we are ready to answer him to the best of our abilitie. First, that
+  </i>Theologia vera est ars artium &amp; scientia scientiarum<i>, Divinity
+  is the Art of Arts, and Science of Sciences; or Divinity is the Mistresse
+  upon which all Arts and Sciences are to attend as servants and
+  handmaides. And why then not </i>Geometry?<i> But in what place she
+  should follow her, I dare not say: For I am no herald, and therefore I
+  meddle not with precedencie: But if I were, she should be none of the
+  hindermost of her traine</i>.</p>
+
+  <p><i>The Oratour saith, and very truly doubtlesse, That, </i>Omnes
+  artes, qu� ad humanitat&#x113; pertinent, habent commune quoddam
+  vinculum, &amp; cognatione quadam inter se continentur<i>. All Arts which
+  pertaine unto humanity, they have a certaine common bond, and are knit
+  together by a kinde of affinity. If then any Arts and Sciences may be
+  thought necessary attendants upon this great Lady; Then surely
+  </i>Geometry<i> amongst the rest must needes be one: For otherwise her
+  traine will be but loose and shattered</i>.</p>
+
+  <p>Plato <i>saith</i> <span title="ton theon akei ge�metrein" class="grk"
+  >&tau;&#x1F78;&nu; &theta;&epsilon;&#x1F78;&nu;
+  &#x1F00;&kappa;&epsilon;&#x1F76;
+  &gamma;&epsilon;&omega;&mu;&epsilon;&tau;&rho;&epsilon;&#x1FD6;&nu;</span>,
+  <i>That God doth alwayes worke by</i> Geometry, <i>that is, as the
+  wiseman doth interprete it,</i> <span class="sc">Sap. XI. 21.</span>
+  Omnia in mensura &amp; numero &amp; pondere disponere. <i>Dispose all
+  things by measure, and number, and weight: Or, as the learned
+  </i>Plutarch<i> speaketh; He adorneth and layeth out all the parts of the
+  world according to rate, proportion, and similitude. Now who, I pray you,
+  understandeth what these termes meane, but he which hath some meane skill
+  in </i>Geometry?<i> Therefore none but such an one, may be able to
+  declare and teach these things unto others</i>.</p>
+
+  <p><i>How many things are there in holy Scripture which may not well be
+  understood without some meane skill in</i> Geometry? <i>The Fabricke and
+  bignesse of</i> Noah's <i>Arke: The Sciagraphy of the Temple set out
+  by</i> Ezechiel, <i>Who may understand, but he that is skilfull in these
+  Arts? I speake not of many and sundry words both in the New and Old
+  Testaments, whose genuine and proper signification is merely
+  Geometricall: And cannot well be conceived but of a Geometer</i>.</p>
+
+  <p><i>And here, that I may speake it without offence, I would have it
+  observed, how many men, much magnified for learning, not onely in their
+  speeches, which alwayes are not premeditated, but even in their writings,
+  exposed to the view and censure of all men, doe often
+  </i>paralogizein<i>, speake much, and little to the purpose. This they
+  could not so easily and often doe, if they had beene but meanely
+  practised in these kinde of studies. Wherefore that Epigramme which was
+  used to be written over their Philosophy Schoole doores,</i> <span
+  title="oudeis age�metr�tos eisit�" class="grk"
+  >&omicron;&#x1F50;&delta;&#x1FC6;&iota;&sigmaf;
+  &#x1F00;&gamma;&epsilon;&omega;&mu;&#x1F73;&tau;&rho;&eta;&tau;&omicron;&sigmaf;
+  &epsilon;&#x1F34;&sigma;&iota;&tau;&omega;</span>, <i>No man ignorant
+  of</i> Geometry <i>come within these doores: Now written over our
+  Divinitie Schooles. And if any man shall thinke this an hard sentence,
+  let him heare what Saint </i>Augustine<i> saith in the same case,</i>
+  Nemo ad divinarum humanarumq; rerum cognitionem accedat, nisi prius
+  annumerandi artem addiscat: <i>Let no man come neither within the
+  Divinity nor Philosophy Schooles, except he have first learned
+  Arithmeticke. Now that the one of these Arts cannot be learned without
+  the other;</i> Euclide <i>our great Master, who made but one of both,
+  hath sufficiently demonstrated</i>.</p>
+
+  <p><i>If I should alledge the like practise of famous Divines, greatly
+  admired for their great skill in this profession, as</i> T. Peckham
+  <i>Arch-Bishop of Canterbury,</i> Maurolycus <i>Bishop of</i> Messana
+  <i>in</i> Sicilia, Cusanus <i>Cardinall of</i> Rome, <i>and many others,
+  before indifferent judges, I am sure I should not be condemned. Who doth
+  not greatly magnifie the grave</i> Seb. Munster, <i>the nimble</i> Ph.
+  Melanchthon, <i>and the noble</i> Bernardino Baldo <i>Abbot of</i>
+  Guastill, <i>and the painefull</i> Barth. Pitiscus <i>of</i> Grunberg,
+  <i>for their knowledge and paines in these Arts and Sciences? And thus
+  much shall at this time suffice, to have spoken unto the first Question:
+  If any shall require further satisfaction, those I referre unto the
+  forenamed Authors, whose authority peradventure may more prevaile with
+  them, then my reasons may</i>.</p>
+
+  <p><i>The next is concerning mine Author, and the worke in hand</i>
+  Geometry, <i>it must needs be confest we are beholden to</i> Euclides
+  <i>Elements for: And he that would be rich in that profession, may have,
+  if he be not covetous, his fill there, if he will labour hard, and take
+  paines for it, it is true. But in what time thinke yau, may a man learne
+  all</i> Euclide, <i>and so by him be made skilfull in this Art? By
+  himselfe I know not whether ever or never: And with the helpe of another,
+  although very expert, I will not promise him that hee shall attaine to
+  perfection in many yeares</i>.</p>
+
+  <p>Hippocrates <i>the Prince of Physicians hath, as they say, in his
+  workes laid out the whole Art of Physicke; but I marvell how long a man
+  should study him alone, and read him over and over, before he should be a
+  good Physician? I feare mee all the friends that he hath, and neighbours
+  round about him, yea, and himselfe too, would all die before he should be
+  able to hele them, or per adventure ere he should be able to know what
+  they ail'd; and after 30, or 40. yeeres of such his study, I would be
+  very loath to commit my selfe unto him. How much therefore are the
+  students of this noble Science beholding unto those men, who by their
+  industry, practise, and painefull travells, have shewed them a ready and
+  certaine way through this wildernesse?</i></p>
+
+  <p><i>The Elements of</i> Euclide <i>they do containe generally the whole
+  art of</i> Geometry: <i>But if you will offer to travell thorow them
+  alone, you shall finde them, I will warrant you, Elements indeed: for
+  there you may walke through the spacious Aire, and over the great and
+  wide sea, and in and about the vaste and arid wildernesse many a day and
+  night, before you shall know where you are. This</i> Ramus, <i>my Authour
+  in reading him found to be true; and confesseth himselfe often to have
+  beene at a stand: Often to have lost himselfe: Often to have hitte upon a
+  rocke, when he had thought he had touch'd land</i>.</p>
+
+  <p><i>Least therefore other men, in this journey doe not likewise loose
+  themselves, for the benefit and safety, I meane, of others he hath
+  prick'd them out a charde or chack'd out a way, which if thou shalt
+  please to follow, it shall lead thee to thy wayes end, as directly, and
+  in as short time, as conveniently may be. Yet in what time I cannot
+  warrant thee: For all mens capacity, especially in these Arts, is not
+  alike: All are not a like painefull, industrious, or diligent: All are
+  not of the same ability of body, to be able to continue or sit at it: Or
+  all not so free from other imployments or businesse calling them from
+  their study, as some others are. For know this for certaine, Thou shalt
+  here make no great progresse, except thou doe make it as it were a
+  continued labour, Here you must observe that rule of the great
+  Painter</i>, Nulla dies sine linea, <i>Let no day passe over your head,
+  in which you draw not some diagram or figure or other</i>.</p>
+
+  <p><i>One other thing let me also advise thee of, how capable soever thou
+  art, refuse not, if thou maist have it, the helpe of a teacher; For
+  except thou be another</i> Hippocrates <i>or</i> Forcatelus, <i>wh&#x14D;
+  our Authour mentioneth, thou canst not in these Arts and Sciences attaine
+  unto any great perfection without infinite patience and great losse of
+  most precious time, For they are therefore called <span
+  title="Math�matikoi" class="grk"
+  >&Mu;&alpha;&theta;&eta;&mu;&alpha;&tau;&iota;&kappa;&#x1F79;&iota;</span>,
+  Mathematicks, that is, doctrinal or disciplinary Arts, because they are
+  not to be attained unto by our owne information and industry; but by the
+  helpe and instruction of others</i>.</p>
+
+  <p><i>This Worke gentle Reader, was in part above 30. yeares since
+  published by M.</i> Thomas Hood, <i>a learned man, and loving friend of
+  mine, who teaching these Arts, in the Staplers Chappell in Leadenhall
+  London, for the benefit of his Schollers and Auditory, did set out the
+  Elements apart by themselves. The whole at large, with the Diagrammes,
+  and Demonstrations, hee promised, as appeareth in the Preface to that his
+  Worke, at his convenient leysure to send out shortly, after them. This
+  for ought we know or can learne, is not by him or any other performed:
+  And yet are those alone, without these of small use or none to a learner,
+  where a teacher is not alwayes at hand. Wherefore we are bold being
+  (encouraged thereunto by some private friends, and especially by the
+  learned M.</i> H. Brigges, <i>professour of</i> Geometry <i>in the famous
+  Vniversity of</i> Oxford) <i>to publish this of ours long since finished
+  and ended</i>.</p>
+
+  <p><i>The usuall termes, whether Latine or Greeke, commonly used by
+  the</i> Geometers, <i>we have set downe and expressed in English, as well
+  as we could, as others, writing of this argument in our language, have
+  done before us. These termes, I doubt not, may by some in English
+  otherwise be expressed, but how harsh those termes, may unto
+  Mathematicall eares, at the first appeare, I will not say; and use in
+  short time will make these familiar, and as pleasing to the eare as those
+  possibly may be</i>.</p>
+
+  <p><i>Our Authour, in the declaration of the Elements hath many passages,
+  which in our judgement doe not make so much for the understanding of the
+  matter in hand, as for the defence of the method here used, against</i>
+  Aristotle, Euclide, Proclus, <i>and others, which we have therfore wholly
+  omitted. Some other things, which in our opinion, might in some respect
+  illustrate any particular in this businesse, we have here and there
+  inserted. Out of the learned</i> Finkius's Geometria Rotundi, <i>Wee have
+  added to the fifth Booke certaine Propositions with their Consectaries
+  out of</i> Ptolomi's <i>Almagest. The painfull and diligent</i> Rod.
+  Snellius <i>out of the Lectures and Annotations of B.</i> Salignacus, I.
+  Tho. Freigius, <i>and others, hath illustrated and altered here and there
+  some few things</i>.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/018.png"><img style="width:100%" src="images/018.png"
+      alt="Decorative spacer." title="Decorative spacer." /></a>
+  </div>
+  <div class="figcenter" style="width:40%;">
+      <a href="images/019.png"><img style="width:100%" src="images/019.png"
+      alt="Decorative spacer." title="Decorative spacer." /></a>
+  </div>
+<h3>The Contents.</h3>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>Booke I.  <i>Of a Magnitude.</i>   Page <a href="#page1">1</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Booke II.  <i>Of a Line.</i>        p. <a href="#page13">13</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book III.  <i>Of an Angle.</i>      p. <a href="#page21">21</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book IV.  <i>Of a Figure</i>.       p. <a href="#page32">32</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book V.  <i>Of Lines and Angles in a plaine Surface.</i>      p. <a href="#page51">51</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book VI.  <i>Of a Triangle.</i>     p. <a href="#page83">83</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book VII.  <i>The comparison of Triangles.</i>      p. <a href="#page94">94</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book VIII.  <i>Of the diverse kinds of Triangles.</i>      p. <a href="#page106">106</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book IX. <i>Of the measuring of right lines by like right-angled Triangles.</i>  p. <a href="#page113">113</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book X.  <i>Of a Triangulate and Parallelogramme.</i>      p. <a href="#page136">136</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XI.  <i>Of a Right-angle.</i>       p. <a href="#page148">148</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XII.  <i>Of a Quadrate.</i>      p. <a href="#page152">152</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XIII.  <i>Of an Oblong.</i>      p. <a href="#page167">167</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XIV.  <i>Of a right line proportionally cut: And of other Quadrangles, and Multangles.</i>       p. <a href="#page174">174</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XV.  <i>Of the Lines in a Circle.</i>       p. <a href="#pageastx201">201</a>*</p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XVI.  <i>Of the Segments of a Circle.</i>       p. <a href="#page201">201</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XVII.  <i>Of the Adscription of a Circle and Triangle.</i>      p. <a href="#page215">215</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XVIII.  <i>Of the adscription of a Triangulate.</i>      p. <a href="#page221">221</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XIX.  <i>Of the measuring of ordinate Multangle, and of a Circle.</i>       p. <a href="#pageastx252">252</a>*</p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XX.  <i>Of a Bossed surface.</i>       p. <a href="#pageastx257">257</a>*</p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXI. <i>Of Lines and Surfaces in solids.</i>      p. <a href="#page242">242</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXII.  <i>Of a Pyramis.</i>       p. <a href="#page249">249</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXIII.  <i>Of a Prisma.</i>       p. <a href="#page256">256</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXIV.  <i>Of a Cube.</i>       p. <a href="#page264">264</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXV.  <i>Of mingled ordinate Polyedra's.</i>       p. <a href="#page271">271</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXVI.   <i>Of a Spheare.</i>       p. <a href="#page284">284</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Book XXVII.  <i>Of the Cone and Cylinder.</i>        p. <a href="#page290">290</a></p>
+    </div>
+  </div>
+<hr class="full" />
+
+<p><!-- Page 1 --><span class="pagenum"><a name="page1"></a>[1]</span></p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/021.png"><img style="width:100%" src="images/021.png"
+      alt="Decorative spacer." title="Decorative spacer." /></a>
+  </div>
+<h1><i>VIA REGIA AD
+GEOMETRIAM.</i></h1>
+
+<h2>THE FIRST BOOKE OF
+<i>Peter Ramus's</i> Geometry,
+<i>Which is of a Magnitude.</i></h2>
+
+<hr class="full" />
+
+  <p><a name="1_e_j"></a> 1. <i>Geometry is the Art of measuring
+  well</i>.</p>
+
+  <p>The end or scope of Geometry is to measure well: Therefore it is
+  defined of the end, as generally all other Arts are. <i>To measure
+  well</i> therefore is to consider the nature and affections of every
+  thing that is to be measured: To compare such like things one with
+  another: And to understand their reason and proportion and similitude.
+  For all that is to measure well, whether it bee that by Congruency and
+  application of some assigned measure: Or by Multiplication of the termes
+  or bounds: Or by Division of the product made by multiplication: Or by
+  any other way whatsoever the affection of the thing to be measured be
+  considered.</p>
+
+  <p>But this end of Geometry will appeare much more beautifull and
+  glorious in the use and geometricall workes and <!-- Page 2 --><span
+  class="pagenum"><a name="page2"></a>[2]</span>practise then by precepts,
+  when thou shalt observe Astronomers, Geographers, Land-meaters, Sea-men,
+  Enginers, Architects, Carpenters, Painters, and Carvers, in the
+  description and measuring of the Starres, Countries, Lands, Engins, Seas,
+  Buildings, Pictures, and Statues or Images to use the helpe of no other
+  art but of Geometry. Wherefore here the name of this art commeth farre
+  short of the thing meant by it. (For <i>Geometria</i>, made of <i>G�</i>,
+  which in the Greeke language signifieth the Earth; and <i>M�tron</i>, a
+  measure, importeth no more, but as one would say <i>Land-measuring</i>.
+  And <i>Geometra</i>, is but <i>Agrimensor</i>, A land-meter: or as
+  <i>Tully</i> calleth him <i>Decempedator</i>, a Pole-man: or as
+  <i>Plautus</i>, <i>Finitor</i>, a Marke-man.) when as this Art teacheth
+  not only how to measure the Land or the Earth, but the Water, and the
+  Aire, yea and the whole World too, and in it all Bodies, Surfaces, Lines,
+  and whatsoever else is to bee measured.</p>
+
+  <p>Now <i>a Measure</i>, as <i>Aristotle</i> doth determine it, in every
+  thing to be measured, is some small thing conceived and set out by the
+  measurer; and of the Geometers it is called <i>Mensura famosa</i>, a
+  knowne measure. Which kinde of measures, were at first, as
+  <i>Vitruvius</i> and <i>Herodo</i> teache us, taken from mans body:
+  whereupon <i>Protagoras</i> sayd, <i>That man was the measure of all
+  things</i>, which speech of his, Saint <i>Iohn</i>, <i>Apoc.</i> 21. 17.
+  doth seeme to approve. True it is, that beside those, there are some
+  other sorts of measures, especially greater ones, taken from other
+  things, yet all of them generally made and defined by those. And because
+  the stature and bignesse of men is greater in some places, then it is
+  ordinarily in others, therefore the measures taken from them are greater
+  in some countries, then they are in others. Behold here a catalogue, and
+  description of such as are commonly either used amongst us, or some times
+  mentioned in our stories and other bookes translated into our English
+  tongue.</p>
+
+  <p><i>Granum hordei</i>, a Barley corne, like as a wheat corne in
+  weights, is no kinde of measure, but is <i>quiddam minimum <!-- Page 3
+  --><span class="pagenum"><a name="page3"></a>[3]</span>in mensura</i>,
+  some least thing in a measure, whereof it is, as it were, made, and
+  whereby it is rectified.</p>
+
+  <p><i>Digitus</i>, a Finger breadth, conteineth 2. barly cornes length,
+  or foure layd side to side:</p>
+
+  <p><i>Pollex</i>, a Thumbe breadth; called otherwise <i>Vncia</i>, an
+  ynch, 3. barley cornes in length:</p>
+
+  <p><i>Palmus</i>, or <i>Palmus minor</i>, an Handbreadth, 4. fingers, or
+  3. ynches.</p>
+
+  <p><i>Spithama</i>, or <i>Palmus major</i>, a Span, 3. hands breadth, or
+  9. ynches.</p>
+
+  <p><i>Cubitus</i>, a Cubit, halfe a yard, from the elbow to the top of
+  the middle finger, 6. hands breadth, or two spannes.</p>
+
+  <p><i>Ulna</i>, from the top of the shoulder or arme-hole, to the top of
+  the middle finger. It is two folde; A yard and an Elne. <i>A yard</i>,
+  containeth 2. cubites, or 3. foote: <i>An Elne</i>, one yard and a
+  quarter, or 2. cubites and �.</p>
+
+  <p><i>Pes</i>, a Foot, 4. hands breadth, or twelve ynches.</p>
+
+  <p><i>Gradus</i>, or <i>Passus minor</i>, a Steppe, two foote and an
+  halfe.</p>
+
+  <p><i>Passus</i>, or <i>Passus major</i>, a Stride, two steppes, or five
+  foote.</p>
+
+  <p><i>Pertica</i>, a Pertch, Pole, Rod or Lugge, 5. yardes and an
+  halfe.</p>
+
+  <p><i>Stadium</i>, a Furlong; after the Romans, 125. pases: the English,
+  40. rod.</p>
+
+  <p><i>Milliare</i>, or <i>Milliarium</i>, that is <i>mille passus</i>,
+  1000. passes, or 8. furlongs.</p>
+
+  <p><i>Leuca</i>, a League, 2. miles: used by the French, spaniards, and
+  seamen.</p>
+
+  <p><i>Parasanga</i>, about 4. miles: a Persian, &amp; common Dutch mile;
+  30. furlongs.</p>
+
+  <p><i>Sch&oelig;nos</i>, 40. furlongs: an Egyptian, or swedland mile.</p>
+
+  <p>Now for a confirmation of that which hath beene saide, heare the words
+  of the Statute.</p>
+
+  <p><i>It is ordained, That 3. graines of Barley, dry and round, do make
+  an</i> Ynch: 12. <i>ynches do make a</i> Foote: 3. <i>foote do make a</i>
+  <!-- Page 4 --><span class="pagenum"><a name="page4"></a>[4]</span>Yard:
+  <i>5. yardes and � doe make a Perch: And 40. perches in length, and 4. in
+  breadth, doe make an Aker: 33. Edwar. 1. De terris mensurandis: &amp; De
+  compositione ulnarum &amp; Perticarum</i>.</p>
+
+  <p>Item, <i>Bee it enacted by the authority aforesaid; That a</i> Mile
+  <i>shall be taken and reckoned in this manner, and no otherwise; That is
+  to say,</i> a Mile <i>to containe 8. furlongs: And every</i> Furlong
+  <i>to containe 40. lugges or poles: And every</i> Lugge <i>or</i> Pole
+  <i>to containe 16. foote and �. 25. Eliza.</i> An Act for restraint of
+  new building, &amp;c.</p>
+
+  <p>These, as I said, are according to diverse countries, where they are
+  used, much different one from another: which difference, in my judgment;
+  ariseth especially out of the difference of the Foote, by which generally
+  they are all made, whether they be greater of lesser. For the Hand being
+  as before hath beene taught, the fourth part of the foot whether greater
+  or lesser: And the Ynch, the third part of the hand, whether greater or
+  lesser.</p>
+
+  <p><i>Item</i>, the Yard, containing 3. foote, whether greater or lesser:
+  And the Rodde 5. yardes and �, whether greater or lesser, and so forth of
+  the rest; It must needes follow, that the Foote beeing in some places
+  greater then it is in other some, these measures, the Hand, I meane, the
+  Ynch, the Yard, the Rod, must needes be greater or lesser in some places
+  then they are in other. Of this diversity therefore, and difference of
+  the foot, in forreine countries, as farre as mine intelligence will
+  informe me, because the place doth invite me, I will here adde these few
+  lines following. For of the rest, because they are of more speciall use,
+  I will God willing, as just occasion shall be administred, speake more
+  plentifully hereafter.</p>
+
+  <p>Of this argument divers men have written somewhat, more or lesse: But
+  none to my knowledge, more copiously and curiously, then <i>Iames
+  Capell</i>, a Frenchman, and the learned <i>Willebrand</i>,
+  <i>Snellius</i>, of <i>Leiden</i> in Holland, for they have compared, and
+  that very diligently, many and sundry kinds of these measures one with
+  another. The first as you may <!-- Page 5 --><span class="pagenum"><a
+  name="page5"></a>[5]</span>see in his treatise <i>De mensuris
+  intervallorum</i> describeth these eleven following: of which the
+  greatest is <i>Pes Babylonius</i>, the Babylonian foote; the least,
+  <i>Pes Toletanus</i>, the foote used about <i>Toledo</i> in Spaine: And
+  the meane betweene both, <i>Pes Atticus</i>, that used about
+  <i>Athens</i> in Greece. For they are one unto another as 20. 15. and 12.
+  are one unto another. Therefore if the Spanish foote, being the least, be
+  devided into 12. ynches, and every inch againe into 10. partes, and so
+  the whole foote into 120. the <i>Atticke</i> foote shall containe of
+  those parts 150. and the <i>Babylonian</i>, 200. To this <i>Atticke</i>
+  foote, of all other, doth ours come the neerest: For our <i>English</i>
+  foote comprehendeth almost 152. such parts.</p>
+
+  <p>The other, to witt the learned <i>Snellius</i>, in his <i>Eratosthenes
+  Batavus</i>, a booke which hee hath written of the true quantity of the
+  compasse of the Earth, describeth many more, and that after a farre more
+  exact and curious manner.</p>
+
+  <p>Here observe, that besides those by us here set downe, there are
+  certaine others by him mentioned, which as hee writeth are found wholly
+  to agree with some one or other of these. For <i>Rheinlandicus</i>, that
+  of <i>Rheinland</i> or <i>Leiden</i>, which hee maketh his base, is all
+  one with <i>Romanus</i>, the <i>Italian</i> or <i>Roman</i> foote.
+  <i>Lovaniensis</i>, that of <i>Lovane</i>, with that of <i>Antwerpe</i>:
+  <i>Bremensis</i>, that of <i>Breme</i> in <i>Germany</i>, with that of
+  <i>Hafnia</i>, in <i>Denmarke</i>. Onely his <i>Pes Arabicus</i>, the
+  <i>Arabian</i> foote, or that mentioned in <i>Abulfada</i>, and
+  <i>Nubiensis</i>: the Geographers I have overpassed, because hee dareth
+  not, for certeine, affirme what it was.</p>
+
+  <div class="figcenter" style="width:60%;">
+      <a href="images/025.png"><img style="width:100%" src="images/025.png"
+      alt="Scale marked Digitus and Palmus." title="Scale marked Digitus and Palmus." /></a>
+  </div>
+<p><!-- Page 6 --><span class="pagenum"><a name="page6"></a>[6]</span></p>
+
+  <p>Looke of what parts <i>Pes Tolitanus</i>, the spanish foote, or that
+  of <i>Toledo</i> in Spaine, conteineth 120. of such is the
+  <i>Pes</i>.</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p><i>Heidelbergicus</i>, that of Heidelberg, 137.</p>
+      <p><i>Hetruscus</i>, that of Tuscan, in Italie, 138.</p>
+      <p><i>Sedanensis</i>, of Sedan in France, 139.</p>
+      <p><i>Romanus</i>, that of Rome in Italy, 144.</p>
+      <p><i>Atticus</i>, of Athens in Greece, 150.</p>
+      <p><i>Anglicus</i>, of England, 152.</p>
+      <p><i>Parisinus</i>, of Paris in France, 160.</p>
+      <p><i>Syriacus</i>, of Syria, 166.</p>
+      <p><i>�gyptiacus</i>, of Egypt, 171.</p>
+      <p><i>Hebraicus</i>, that of Iud�a, 180.</p>
+      <p><i>Babylonius</i>, that of Babylon, 200.</p>
+    </div>
+  </div>
+  <p>Looke of what parts <i>Pes Romanus</i>, the foote of Rome, (which is
+  all one with the foote of <i>Rheinland</i>) is 1000. of such parts is the
+  foote of</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p><i>Toledo</i>, in Spaine, 864.</p>
+      <p><i>Mechlin</i>, in Brabant, 890.</p>
+      <p><i>Strausburgh</i>, in Germany, 891.</p>
+      <p><i>Amsterdam</i>, in Holland, 904.</p>
+      <p><i>Antwerpe</i>, in Brabant, 909.</p>
+      <p><i>Bavaria</i>, in Germany, 924.</p>
+      <p><i>Coppen-haun</i>, in Denmarke, 934.</p>
+      <p><i>Goes</i>, in Zeland, 954.</p>
+      <p><i>Middleburge</i>, in Zeland, 960.</p>
+      <p><i>London</i>, in England, 968.</p>
+      <p><i>Noremberge</i>, in Germany, 974.</p>
+      <p><i>Ziriczee</i>, in Zeland, 980.</p>
+      <p>The ancient <i>Greeke</i>, 1042.</p>
+      <p><i>Dort</i>, in Holland, 1050.</p>
+      <p><i>Paris</i>, in France, 1055.</p>
+      <p><i>Briel</i>, in Holland, 1060.</p>
+      <p><i>Venice</i>, in Italy, 1101.</p>
+      <p><i>Babylon</i>, in Chald�a, 1172.</p>
+      <p><i>Alexandria</i>, in Egypt, 1200.</p>
+      <p><i>Antioch</i>, in Syria, 1360.</p>
+    </div>
+  </div>
+  <p>Of all other therefore our English foote commeth neerest unto that
+  used by the Greekes: And the learned Master <i>Ro. Hues</i>, was not much
+  amisse, who in his booke or Treatise <i>De Globis</i>, thus writeth of it
+  <i>Pedem nostrum Angli cum Gr�corum pedi �qualem invenimus, comparatione
+  facta <!-- Page 7 --><span class="pagenum"><a
+  name="page7"></a>[7]</span>cum Gr�corum pede, quem Agricola &amp; alij ex
+  antiquis monumentis tradiderunt</i>.</p>
+
+  <p>Now by any one of these knowne and compared with ours, to all English
+  men well knowne the rest may easily be proportioned out.</p>
+
+  <p><a name="2_e_j"></a> 2. <i>The thing proposed to bee measured is a
+  Magnitude</i>.</p>
+
+  <p><i>Magnitudo</i>, a Magnitude or Bignesse is the subject about which
+  Geometry is busied. For every Art hath a proper subject about which it
+  doth employ al his rules and precepts: And by this especially they doe
+  differ one from another. So the subject of Grammar was speech; of
+  Logicke, reason; of Arithmeticke, numbers; and so now of Geometry it is a
+  magnitude, all whose kindes, differences and affections, are hereafter to
+  be declared.</p>
+
+  <p><a name="3_e_j"></a> 3. <i>A Magnitude is a continuall
+  quantity</i>.</p>
+
+  <p>A Magnitude is <i>quantitas continua</i>, a continued, or continuall
+  quantity. A number is <i>quantitas discreta</i>, a disjoined quantity: As
+  one, two, three, foure; doe consist of one, two, three, foure unities,
+  which are disjoyned and severed parts: whereas the parts of a Line,
+  Surface, and Body are contained and continued without any manner of
+  disjunction, separation, or distinction at all, as by and by shall better
+  and more plainely appeare. Therefore a Magnitude is here understood to be
+  that whereby every thing to be measured is said to bee great: As a Line
+  from hence is said to be long, a Surface broade, a Body solid: Wherefore
+  Length, Breadth, and solidity are Magnitudes.</p>
+
+  <p><a name="4_e_j"></a> 4. <i>That is</i> continuum, <i>continuall, whose
+  parts are contained or held together by some common bound</i>.</p>
+
+  <p>This definition of it selfe is somewhat obscure, and to be <!-- Page 8
+  --><span class="pagenum"><a name="page8"></a>[8]</span>understand onely
+  in a geometricall sense: And it dependeth especially of the common
+  bounde. For the parts (which here are so called) are nothing in the
+  whole, but in a <i>potentia</i> or powre: Neither indeede may the whole
+  magnitude bee conceived, but as it is compact of his parts, which
+  notwithstanding wee may in all places assume or take as conteined and
+  continued with a common bound, which Aristotle nameth a <i>Common
+  limit</i>; but <i>Euclide</i> a <i>Common section</i>, as in a line, is a
+  Point, in a surface, a Line: in a body, a Surface.</p>
+
+  <p><a name="5_e_j"></a> 5. <i>A bound is the outmost of a
+  Magnitude</i>.</p>
+
+  <p><i>Terminus</i>, a Terme, or Bound is here understood to bee that
+  which doth either bound, limite, or end <i>actu</i>, in deede; as in the
+  beginning and end of a magnitude: Or <i>potentia</i>, in powre or
+  ability, as when it is the common bound of the continuall magnitude.
+  Neither is the Bound a parte of the bounded magnitude: For the thing
+  bounding is one thing, and the thing bounded is another: For the Bound is
+  one distance, dimension, or degree, inferiour to the thing bounded: A
+  Point is the bound of a line, and it is lesse then a line by one degree,
+  because it cannot bee divided, which a line may. A Line is the bound of a
+  surface, and it is also lesse then a surface by one distance or
+  dimension, because it is only length, wheras a surface hath both length
+  and breadth. A Surface is the bound of a body, and it is lesse likewise
+  then it is by one dimension, because it is onely length and breadth,
+  whereas as a body hath both length, breadth, and thickenesse.</p>
+
+  <p>Now every Magnitude <i>actu</i>, in deede, is terminate, bounded and
+  finite, yet the geometer doth desire some time to have an infinite line
+  granted him, but no otherwise infinite or farther to bee drawane out then
+  may serve his turne. <!-- Page 9 --><span class="pagenum"><a
+  name="page9"></a>[9]</span></p>
+
+  <p><a name="6_e_j"></a> 6. <i>A Magnitude is both infinitely made, and
+  continued, and cut or divided by those things wherewith it is
+  bounded</i>.</p>
+
+  <p>A line, a surface, and a body are made gemetrically by the motion of a
+  point, line, and surface: Item, they are conteined, continued, and cut or
+  divided by a point, line, and surface. But a Line is bounded by a point:
+  a surface, by a line: And a Body by a surface, as afterward by their
+  severall kindes shall be understood.</p>
+
+  <p>Now that all magnitudes are cut or divided by the same wherewith they
+  are bounded, is conceived out of the definition of <i>Continuum</i>, e.
+  4. For if the common band to containe and couple together the parts of a
+  Line, surface, &amp; Body, be a Point, Line, and Surface, it must needes
+  bee that a section or division shall be made by those common bandes: And
+  that to bee dissolved which they did containe and knitt together.</p>
+
+  <p><a name="7_e_j"></a> 7. <i>A point is an undivisible signe in a
+  magnitude</i>.</p>
+
+  <p>A Point, as here it is defined, is not naturall and to bee perceived
+  by sense; Because sense onely perceiveth that which is a body; And if
+  there be any thing lesse then other to be perceived by sense, that is
+  called a Point. Wherefore a Point is no Magnitude: But it is onely that
+  which in a Magnitude is conceived and imagined to bee undivisible. And
+  although it be voide of all bignesse or Magnitude, yet is it the
+  beginning of all magnitudes, the beginning I meane <i>potenti�</i>, in
+  powre.</p>
+
+  <p><a name="8_e_j"></a> 8. <i>Magnitudes commensurable, are those which
+  one and the same measure doth measure: Contrariwise, Magnitudes
+  incommensurable are those, which the same measure cannot measure.</i> 1,
+  2. d. X.</p>
+
+  <p>Magnitudes compared betweene themselves in respect of numbers have
+  Symmetry or commensurability, and <!-- Page 10 --><span
+  class="pagenum"><a name="page10"></a>[10]</span>Reason or rationality: Of
+  themselves, Congruity and Adscription. But the measure of a magnitude is
+  onely by supposition, and at the discretion of the Geometer, to take as
+  pleaseth him, whether an ynch, an hand breadth, foote, or any other thing
+  whatsoever, for a measure. Therefore two magnitudes, the one a foote
+  long, the other two foote long, are commensurable; because the magnitude
+  of one foote doth measure them both, the first once, the second twice.
+  But some magnitudes there are which have no common measure, as the
+  Diagony of a quadrate and his side, 116. p. X. <i>actu</i>, in deede, are
+  <i>Asymmetra</i>, incommensurable: And yet they are <i>potenti�</i>, by
+  power, <i>symmetra</i>, commensurable, to witt by their quadrates: For
+  the quadrate of the diagony is double to the quadrate of the side.</p>
+
+  <p><a name="9_e_j"></a> 9. <i>Rationall Magnitudes are those whose reason
+  may bee expressed by a number of the measure given. Contrariwise they are
+  irrationalls.</i> 5. d. X.</p>
+
+  <p><i>Ratio</i>, Reason, Rate, or Rationality, what it is our Authour
+  (and likewise <i>Salignacus</i>) have taught us in the first Chapter of
+  the second booke of their Arithmetickes: Thither therefore I referre
+  thee.</p>
+
+  <p><i>Data mensura</i>, a Measure given or assigned, is of <i>Euclide</i>
+  called <i>Rhet�</i>, that is spoken, (or which may be uttered) definite,
+  certaine, to witt which may bee expressed by some number, which is no
+  other then that, which as we said, was called <i>mensura famosa</i>, a
+  knowne or famous measure.</p>
+
+  <p>Therefore Irrationall magnitudes, on the contrary, are understood to
+  be such whose reason or rate may not bee expressed by a number or a
+  measure assigned: As the side of the side of a quadrate of 20. foote unto
+  a magnitude of two foote; of which kinde of magnitudes, thirteene sorts
+  are mentioned in the tenth booke of <i>Euclides Elements</i>: such are
+  the segments of a right line proportionally cutte, unto the whole line.
+  The Diameter in a circle is rationall: <!-- Page 11 --><span
+  class="pagenum"><a name="page11"></a>[11]</span>But it is irrationall
+  unto the side of an inscribed quinquangle: The Diagony of an Icosahedron
+  and Dodecahedron is irrationall unto the side.</p>
+
+  <p><a name="10_e_j"></a> 10. <i>Congruall or agreeable magnitudes are
+  those, whose parts beeing applyed or laid one upon another doe fill an
+  equall place</i>.</p>
+
+  <p><i>Symmetria</i>, Symmetry or Commensurability and Rate were from
+  numbers: The next affections of Magnitudes are altogether
+  geometricall.</p>
+
+  <p><i>Congruentia</i>, Congruency, Agreeablenesse is of two magnitudes,
+  when the first parts of the one doe agree to the first parts of the
+  other, the meane to the meane, the extreames or ends to the ends, and
+  lastly the parts of the one, in all respects to the parts, of the other:
+  so Lines are congruall or agreeable, when the bounding, points of the
+  one, applyed to the bounding points of the other, and the whole lengths
+  to the whole lengthes, doe occupie or fill the same place. So Surfaces
+  doe agree, when the bounding lines, with the bounding lines: And the
+  plots bounded, with the plots bounded doe occupie the same place. Now
+  bodies if they do agree, they do seeme only to agree by their surfaces.
+  And by this kind of congruency do we measure the bodies of all both
+  liquid and dry things, to witt, by filling an equall place. Thus also doe
+  the moniers judge the monies and coines to be equall, by the equall
+  weight of the plates in filling up of an equall place. But here note,
+  that there is nothing that is onely a line, or a surface onely, that is
+  naturall and sensible to the touch, but whatsoever is naturall, and thus
+  to be discerned is corporeall.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="11_e_j"></a> 11. <i>Congruall or agreeable Magnitudes are
+  equall.</i> 8. <i>ax. j</i>.</p>
+
+  <p>A lesser right line may agree to a part of a greater, but to so much
+  of it, it is equall, with how much it doth agree: <!-- Page 12 --><span
+  class="pagenum"><a name="page12"></a>[12]</span>Neither is that axiome
+  reciprocall or to be converted: For neither in deede are Congruity and
+  Equality reciprocall or convertible. For a Triangle may bee equall to a
+  Parallelogramme, yet it cannot in all points agree to it: And so to a
+  Circle there is sometimes sought an equall quadrate, <span
+  class="correction" title="text reads `althoughin congruall'">although
+  incongruall</span> or not agreeing with it: Because those things which
+  are of the like kinde doe onely agree.</p>
+
+  <p><a name="12_e_j"></a> 12. <i>Magnitudes are described betweene
+  themselves, one with another, when the bounds of the one are bounded
+  within the boundes of the other: That which is within, is called the
+  inscript: and that which is without, the Circumscript</i>.</p>
+
+  <p>Now followeth Adscription, whose kindes are Inscription and
+  Circumscription; That is when one figure is written or made within
+  another: This when it is written or made about another figure.</p>
+
+  <p><i>Homogenea</i>, Homogenealls or figures of the same kinde onely
+  betweene themselves <i>rectitermina</i>, or right bounded, are properly
+  adscribed betweene themselves, and with a round. Notwithstanding, at the
+  15. booke of <i>Euclides Elements</i> Heterogenea, Heterogenealls or
+  figures of divers kindes are also adscribed, to witt the five ordinate
+  plaine bodies betweene themselves: And a right line is inscribed within a
+  periphery and a triangle.</p>
+
+  <p>But the use of adscription of a rectilineall and circle, shall
+  hereafter manifest singular and notable mysteries by the reason and
+  meanes of adscripts; which adscription shall be the key whereby a way is
+  opened unto that most excellent doctrine taught by the subtenses or
+  inscripts of a circle as <i>Ptolomey</i> speakes, or Sines, as the latter
+  writers call them.</p>
+
+<hr class="full" />
+
+<p><!-- Page 13 --><span class="pagenum"><a name="page13"></a>[13]</span></p>
+
+<h2>The second Booke of <i>Geometry</i>.
+<i>Of a Line.</i></h2>
+
+  <p><a name="1_e_ij"></a> 1. <i>A Magnitude is either a Line or a
+  Lineate</i>.</p>
+
+  <p>The Common affections of a magnitude are hitherto declared: The
+  <i>Species</i> or kindes doe follow: for other then this division our
+  authour could not then meete withall.</p>
+
+  <div class="figright" style="width:25%;">
+      <a href="images/033.png"><img style="width:100%" src="images/033.png"
+      alt="Lines." title="Lines." /></a>
+  </div>
+  <p><a name="2_e_ij"></a> 2. <i>A Line is a Magnitude onely long</i>.</p>
+
+  <p>As are <i>ae.</i> <i>io.</i> and <i>uy.</i> such a like Magnitude is
+  conceived in the measuring of waies, or distance of one place from
+  another: And by the difference of a lightsome place from a darke:
+  <i>Euclide</i> at the 2 <i>d j.</i> defineth a line to be a length void
+  of breadth: And indeede length is the proper difference of a line, as
+  breadth is of a face, and solidity of a body.</p>
+
+  <p><a name="3_e_ij"></a> 3. <i>The bound of a line is a point</i>.</p>
+
+  <p><i>Euclide</i> at the 3. <i>d j.</i> saith that the extremities or
+  ends of a line are points. Now seeing that a Periphery or an hoope line
+  hath neither beginning nor ending, it seemeth not to bee bounded with
+  points: But when it is described or made it beginneth at a point, and it
+  endeth at a pointe. Wherefore a Point is the bound of a line, sometime
+  <i>actu</i>, in deed, as in a right line: sometime <i>potenti�</i>, in a
+  possibility, as in a perfect periphery. Yea in very deede, as before was
+  taught in the definition of <i>continuum</i>, 4 <i>e.</i> all lines,
+  whether they bee right lines, or crooked, are contained or continued with
+  points. But a line is made by the <!-- Page 14 --><span
+  class="pagenum"><a name="page14"></a>[14]</span>motion of a point. For
+  every magnitude generally is made by a geometricall motion, as was even
+  now taught, and it shall afterward by the severall kindes appeare, how by
+  one motion whole figures are made: How by a conversion, a Circle,
+  Spheare, Cone, and Cylinder: How by multiplication of the base and
+  heighth, rightangled parallelogrammes are made.</p>
+
+  <p><a name="4_e_ij"></a> 4. <i>A Line is either Right or Crooked</i>.</p>
+
+  <p>This division is taken out of the 4 d j. of <i>Euclide</i>, where
+  rectitude or straightnes is attributed to a line, as if from it both
+  surfaces and bodies were to have it. And even so the rectitude of a solid
+  figure, here-after shall be understood by a right line perpendicular from
+  the toppe unto the center of the base. Wherefore rectitude is propper
+  unto a line: And therefore also obliquity or crookednesse, from whence a
+  surface is judged to be right or oblique, and a body right or
+  oblique.</p>
+
+  <p><a name="5_e_ij"></a> 5. <i>A right line is that which lyeth equally
+  betweene his owne bounds: A crooked line lieth contrariwise.</i> 4. <i>d.
+  j</i>.</p>
+
+  <p>Now a line lyeth equally betweene his owne bounds, when it is not here
+  lower, nor there higher: But is equall to the space comprehended betweene
+  the two bounds or ends: As here <i>ae.</i> is, so hee that maketh
+  <i>rectum iter</i>, a journey in a straight line, commonly he is said to
+  treade so much ground, as he needes must, and no more: He goeth
+  <i>obliquum iter</i>, a crooked way, which goeth more then he needeth, as
+  <i>Proclus</i> saith.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/034.png"><img style="width:100%" src="images/034.png"
+      alt="Straight Line." title="Straight Line." /></a>
+  </div>
+<p><!-- Page 15 --><span class="pagenum"><a name="page15"></a>[15]</span></p>
+
+  <p><a name="6_e_ij"></a> 6. <i>A right line is the shortest betweene the
+  same bounds</i>.</p>
+
+  <p><i>Linea recta</i>, a straight or right line is that, as <i>Plato</i>
+  defineth it, whose middle points do hinder us from seeing both the
+  extremes at once; As in the eclipse of the Sunne, if a right line should
+  be drawne from the Sunne, by the Moone, unto our eye, the body of the
+  Moone beeing in the midst, would hinder our sight, and would take away
+  the sight of the Sunne from us: which is taken from the Opticks, in which
+  we are taught, that we see by straight beames or rayes. Therfore to lye
+  equally betweene the boundes, that is by an equall distance: to bee the
+  shortest betweene the same bounds; And that the middest doth hinder the
+  sight of the extremes, is all one.</p>
+
+  <p><a name="7_e_ij"></a> 7. <i>A crooked line is touch'd of a right or
+  crooked line, when they both doe so meete, that being continued or drawne
+  out farther they doe not cut one another</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/035.png"><img style="width:100%" src="images/035.png"
+      alt="Crooked Lines." title="Crooked Lines." /></a>
+  </div>
+  <p><i>Tactus</i>, Touching is propper to a crooked line, compared either
+  with a right line or crooked, as is manifest out of the 2. and 3.
+  <i>d</i> 3. A right line is said to touch a circle, which touching the
+  circle and drawne out farther, doth not cut the circle, 2 <i>d</i> 3. as
+  here <i>ae</i>, the right line toucheth the periphery <i>iou</i>. And
+  <i>ae</i>. doth touch the helix or spirall. <!-- Page 16 --><span
+  class="pagenum"><a name="page16"></a>[16]</span>Circles are said to touch
+  one another, when touching they doe not cutte one another, 3. <i>d</i> 3.
+  as here the periphery doth <i>aej.</i> doth touch the periphery
+  <i>ouy</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="8_e_ij"></a> 8. <i>Touching is but in one point onely. � 13.
+  p</i> 3.</p>
+
+  <p>This Consectary is immediatly conceived out of the definition; for
+  otherwise it were a cutting, not touching. So <i>Aristotle</i> in his
+  <i>Mechanickes</i> saith; That a round is easiliest mou'd and most swift;
+  Because it is least touch't of the plaine underneath it.</p>
+
+  <p><a name="9_e_ij"></a> 9. <i>A crooked line is either a Periphery or an
+  Helix.</i> This also is such a division, as our Authour could then hitte
+  on.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/036a.png"><img style="width:100%" src="images/036a.png"
+      alt="Periphery." title="Periphery." /></a>
+  </div>
+  <p><a name="10_e_ij"></a> 10. <i>A Periphery is a crooked line, which is
+  equally distant from the middest of the space comprehended</i>.</p>
+
+  <p><i>Peripheria</i>, a Periphery, or Circumference, as <i>eio.</i> doth
+  stand equally distant from <i>a</i>, the middest of the space enclosed or
+  conteined within it.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="11_e_ij"></a> 11. <i>A Periphery is made by the turning about
+  of a line, the one end thereof standing still, and the other drawing the
+  line</i>.</p>
+
+  <div class="figleft" style="width:15%;">
+      <a href="images/036b.png"><img style="width:100%" src="images/036b.png"
+      alt="Generation of Periphery." title="Generation of Periphery." /></a>
+  </div>
+  <p>As in <i>eio.</i> let the point <i>a</i> stand still: And let the line
+  <i>ao</i>, be turned about, so that the point <i>o</i> doe make a race,
+  and it shall make the periphery <i>eoi</i>. Out of this fabricke doth
+  <i>Euclide</i>, at the 15. d. j. frame the definition of a Periphery: And
+  so doth hee afterwarde define a Cone, a Spheare, and a Cylinder. <!--
+  Page 17 --><span class="pagenum"><a name="page17"></a>[17]</span></p>
+
+  <p>Now the line that is turned about, may in a plaine, bee either a right
+  line or a crooked line: In a sphericall it is onely a crooked line; But
+  in a conicall or Cylindraceall it may bee a right line, as is the side of
+  a Cone and Cylinder. Therefore in the conversion or turning about of a
+  line making a periphery, there is considered onely the distance; yea two
+  points, one in the center, the other in the toppe, which therefore
+  Aristotle nameth <i>Rotundi principia</i>, the principles or beginnings
+  of a round.</p>
+
+  <p><a name="12_e_ij"></a> 12. <i>An Helix is a crooked line which is
+  unequally distant from the middest of the space, howsoever
+  inclosed</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/037.png"><img style="width:100%" src="images/037.png"
+      alt="Examples of Helix." title="Examples of Helix." /></a>
+  </div>
+  <p><i>H�c tortuosa linea</i>, This crankled line is of <i>Proclus</i>
+  called <i>Helicoides</i>. But it may also be called <i>Helix</i>, a twist
+  or wreath: The <i>Greekes</i> by this word do commonly either understand
+  one of the kindes of Ivie which windeth it selfe about trees &amp; other
+  plants; or the strings of the vine, whereby it catcheth hold and twisteth
+  it selfe about such things as are set for it to clime or run upon.
+  Therfore it should properly signifie the spirall line. But as it is here
+  taken it hath divers kindes; As is the <i>Arithmetica</i> which is
+  Archimede'es Helix, as the <i>Conchois</i>, Cockleshell-like: as is the
+  <i>Cittois</i>, Iuylike: The <i>Tetragonisousa</i>, the Circle squaring
+  line, to witt that by whose meanes a circle may be brought into a square:
+  The Admirable line, found out by <i>Menelaus</i>: The Conicall
+  <i>Ellipsis</i>, the <i>Hyperbole</i>, the <i>Parabole</i>, such as these
+  are, they attribute to <!-- Page 18 --><span class="pagenum"><a
+  name="page18"></a>[18]</span><i>Menechmus</i>: All these
+  <i>Apollonius</i> hath comprised in eight Bookes; but being mingled
+  lines, and so not easie to bee all reckoned up and expressed,
+  <i>Euclide</i> hath wholly omitted them, saith <i>Proclus</i>, at the 9.
+  <i>p. j</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/038a.png"><img style="width:100%" src="images/038a.png"
+      alt="Mingled Lines." title="Mingled Lines." /></a>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/038b.png"><img style="width:100%" src="images/038b.png"
+      alt="Perpendicular Lines." title="Perpendicular Lines." /></a>
+  </div>
+  <p><a name="13_e_ij"></a> 13. <i>Lines are right one unto another,
+  whereof the one falling upon the other, lyeth equally: Contrariwise they
+  are oblique. � 10. d j</i>.</p>
+
+  <p>Hitherto straightnesse and crookednesse have beene the affections of
+  one sole line onely: The affections of two lines compared one with
+  another are <i>Perpendiculum</i>, Perpendicularity and
+  <i>Parallelismus</i>, Parallell equality; Which affections are common
+  both to right and crooked lines. Perpendicularity is first generally
+  defined thus:</p>
+
+  <p>Lines are right betweene themselves, that is, perpendicular one unto
+  another, when the one of them lighting upon the other, standeth upright
+  and inclineth or leaneth neither way. So two right lines in a plaine may
+  bee perpendicular; as are <i>ae.</i> and <i>io.</i> so two peripheries
+  upon a sphearicall may be perpendiculars, when the one of them falling
+  upon the other, standeth indifferently betweene, and doth not incline or
+  leane either way. So a right line may be <!-- Page 19 --><span
+  class="pagenum"><a name="page19"></a>[19]</span>perpendicular unto a
+  periphery, if falling upon it, it doe reele neither way, but doe ly
+  indifferently betweene either side. And in deede in all respects lines
+  right betweene themselves, and perpendicular lines are one and the same.
+  And from the perpendicularity of lines, the perpendicularity of surfaces
+  is taken, as hereafter shall appeare. Of the perpendicularity of bodies,
+  <i>Euclide</i> speaketh not one word in his <i>Elements</i>, &amp; yet a
+  body is judged to be right, that is, plumme or perpendicular unto another
+  body, by a perpendicular line.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/039.png"><img style="width:100%" src="images/039.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p><a name="14_e_ij"></a> 14. <i>If a right line be perpendicular unto a
+  right line, it is from the same bound, and on the same side, one onely. �
+  13. p. xj</i>.</p>
+
+  <p>Or, there can no more fall from the same point, and on the same side
+  but that one. This consectary followeth immediately upon the former: For
+  if there should any more fall unto the same point and on the same side,
+  one must needes reele, and would not ly indifferently betweene the parts
+  cut: as here thou seest in the right line <i>ae. io. eu</i>.</p>
+
+  <p><a name="15_e_ij"></a> 15. <i>Parallell lines they are, which are
+  everywhere equally distant. � 35. d j</i>.</p>
+
+  <p><i>Parallelismus</i>, Parallell-equality doth now follow: And this
+  also is common to crooked lines and right lines: As <!-- Page 20 --><span
+  class="pagenum"><a name="page20"></a>[20]</span>heere thou seest in these
+  examples following.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:30%;">
+      <a href="images/040a.png"><img style="width:100%" src="images/040a.png"
+      alt="Parallel Lines." title="Parallel Lines." /></a>
+  </div>
+  <p>Parallell-equality is derived from perpendicularity, and is of neere
+  affinity to it. Therefore Posidonius did define it by a common
+  perpendicle or plum-line: yea and in deed our definition intimateth
+  asmuch. Parallell-equality of bodies is no where mentioned in <i>Euclides
+  Elements</i>: and yet they may also bee parallells, and are often used in
+  the Optickes, Mechanickes, Painting and Architecture.</p>
+
+  <div class="figright" style="width:10%;">
+      <a href="images/040b.png"><img style="width:100%" src="images/040b.png"
+      alt="Three Parallel Lines." title="Three Parallel Lines." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="16_e_ij"></a> 16. <i>Lines which are parallell to one and the
+  same line, are also parallell one to another</i>.</p>
+
+  <p>This element is specially propounded and spoken of right lines onely,
+  and is demonstrated at the 30. <i>p. j.</i> But by an addition of equall
+  distances, an equall distance is knowne, as here.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<p><!-- Page 21 --><span class="pagenum"><a name="page21"></a>[21]</span></p>
+
+<h2>The third Booke of <i>Geometry</i>.
+Of an Angle.</h2>
+
+  <p><a name="1_e_iij"></a> 1. <i>A lineate is a Magnitude more then
+  long</i>.</p>
+
+  <p>A New forme of doctrine hath forced our Authour to use oft times new
+  words, especially in dividing, that the logicall lawes and rules of more
+  perfect division by a dichotomy, that is into two kindes, might bee held
+  and observed. Therefore a Magnitude was divided into two kindes, to witt
+  into a Line and a Lineate: And a Lineate is made the <i>genus</i> of a
+  surface and a Body. Hitherto a Line, which of all bignesses is the first
+  and most simple, hath been described: Now followeth a Lineate, the other
+  kinde of magnitude opposed as you see to a line, followeth next in order.
+  <i>Lineatum</i> therefore a Lineate, or <i>Lineamentum</i>, a Lineament,
+  (as by the authority of our Authour himselfe, the learned <i>Bernhard
+  Salignacus</i>, who was his Scholler, hath corrected it) is that
+  Magnitude in which there are lines: Or which is made of lines, or as our
+  Authour here, which is more then long: Therefore lines may be drawne in a
+  surface, which is the proper soile or plots of lines; They may also be
+  drawne in a body, as the Diameter in a Prisma: the axis in a spheare; and
+  generally all lines falling from aloft: And therfore <i>Proclus</i>
+  maketh some plaine, other solid lines. So Conicall lines, as the
+  Ellipsis, Hyperbole, and Parabole, are called solid lines because they do
+  arise from the cutting of a body.</p>
+
+  <p><a name="2_e_iij"></a> 2. <i>To a Lineate belongeth an Angle and a
+  Figure</i>.</p>
+
+  <p>The common affections of a Magnitude were to be bounded, cutt, jointly
+  measured, and adscribed: Then of a line to be right, crooked, touch'd,
+  turn'd about, and <!-- Page 22 --><span class="pagenum"><a
+  name="page22"></a>[22]</span>wreathed: All which are in a lineate by
+  meanes of a line. Now the common affections of a Lineate are to bee
+  Angled and Figured. And surely an Angle and a figure in all Geometricall
+  businesses doe fill almost both sides of the leafe. And therefore both of
+  them are diligently to be considered.</p>
+
+  <p><a name="3_e_iij"></a> 3. <i>An Angle is a lineate in the common
+  section of the bounds</i>.</p>
+
+  <p>So <i>Angulus Superficiarius</i>, a superficiall Angle, is a surface
+  consisting in the common section of two lines: So <i>angulus solidus</i>,
+  a solid angle, in the common section of three surfaces at the least.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/042.png"><img style="width:100%" src="images/042.png"
+      alt="Angles." title="Angles." /></a>
+  </div>
+  <p>[But the learned B. <i>Salignacus</i> hath observed, that all angles
+  doe not consist in the common section of the bounds, Because the touching
+  of circles, either one another, or a rectilineal surface doth make an
+  angle without any cutting of the bounds: And therefore he defineth it
+  thus: <i>Angulus est terminorum inter se invicem inclinantium concursus:
+  An angle is the meeting of bounds, one leaning towards another.</i>] So
+  is <i>aei.</i> a superficiall angle: [And such also are the angles
+  <i>ouy.</i> and <i>bcd.</i>] so is the angle <i>o.</i> a solid angle, to
+  witt comprehended of the three surfaces <i><span class="correction"
+  title="text reads `aei.'">aoi.</span></i> <i>ioe.</i> and <i>aoe.</i>
+  Neither may a surface, of 2. dimensions, be bounded with <!-- Page 23
+  --><span class="pagenum"><a name="page23"></a>[23]</span>one right line:
+  Nor a body, of three dimensions, bee bounded with two, at lest beeing
+  plaine surfaces.</p>
+
+  <p><a name="4_e_iij"></a> 4. <i>The shankes of an angle are the bounds
+  compreding the angle</i>.</p>
+
+  <p><i>Sc�le</i> or <i>Crura</i>, the Shankes, Legges, H. are the bounds
+  insisting or standing upon the base of the angle, which in the Isosceles
+  only or Equicrurall triangle are so named of <i>Euclide</i>, otherwise he
+  nameth them <i>Latera</i>, sides. So in the examples aforesaid,
+  <i>ea.</i> and <i>ei.</i> are the shankes of the superficiary angle
+  <i>e</i>; And so are the three surfaces <i><span class="correction"
+  title="text reads `aei.'">aoi.</span></i> <i>ieo.</i> and <i>aeo.</i> the
+  shankes of the said angle <i>o</i>. Therefore the shankes making the
+  angle are either Lines or Surfaces: And the lineates formed or made into
+  Angles, are either Surfaces or Bodies.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/044.png"><img style="width:100%" src="images/044.png"
+      alt="Lunular Angles, etc." title="Lunular Angles, etc." /></a>
+  </div>
+  <p><a name="5_e_iij"></a> 5. <i>Angles homogeneall, are angles of the
+  same kinde, both in respect of their shankes, as also in the maner of
+  meeting of the same:</i> [<i>Heterogeneall, are those which differ one
+  from another in one, or both these conditions.</i>]</p>
+
+  <p>Therefore this <i>Homogenia</i>, or similitude of angles is twofolde,
+  the first is of shanks; the other is of the manner of meeting of the
+  shankes: so rectilineall right angles, are angles homogeneall betweene
+  themselves. But right-lined right angles, and oblique-lined right angles
+  between themselves, are heterogenealls. So are neither all obtusangles
+  compared to all obtusangles: Nor all acutangles, to all acutangles,
+  homogenealls, except both these conditions doe concurre, to witt the
+  similitude both of shanke and manner of meeting. <i>Lunularis</i>, a
+  Lunular, or Moonlike corner angle is homogeneall to a <i>Systroides</i>
+  and <i>Pelecoides</i>, Hatchet formelike, in shankes: For each of these
+  are comprehended of <!-- Page 24 --><span class="pagenum"><a
+  name="page24"></a>[24]</span>peripheries: The Lunular of one convexe; the
+  other concave; as <i><span class="correction" title="there was no letter u on the printed diagram, inserted in place which fits the text"
+  >iue</span></i>. The Systroides of both convex, as <i>iao</i>. The
+  Pelecoides of both concave, as <i>eau</i>. And yet a lunular, in respect
+  of the meeting of the shankes is both to the Systroides and Pelecoides
+  heterogeneall: And therefore it is absolutely heterogeneall to it.</p>
+
+  <p><a name="6_e_iij"></a> 6. <i>Angels congruall in shankes are
+  equall</i>.</p>
+
+  <p>This is drawne out of the <a href="#10_e_j">10. e j</a>. For if twice
+  two shanks doe agree, they are not foure, but two shankes, neither are
+  they two equall angles, but one angle. And this is that which
+  <i>Proclus</i> speaketh of, at the 4. p j. when hee saith, that a right
+  lined angle is equall to a right lined angle, when one of the shankes of
+  the one put upon one of the shankes of the other, the other two doe
+  agree: when that other shanke fall without, the angle of the out-falling
+  shanke is the greater: when it falleth within, it is lesser: For there is
+  comprehendeth; here it is comprehended.</p>
+
+  <p>Notwithstanding although congruall or agreeable angles be equall: yet
+  are not congruity and equality reciprocall or convertible: For a Lunular
+  may bee equall to a right <!-- Page 25 --><span class="pagenum"><a
+  name="page25"></a>[25]</span>lined right angle, as here thou seest: For
+  the angles of equall semicircles <i>ieo.</i> and <i>aeu.</i> are equall,
+  as application doth shew. The angle <i>aeo.</i> is common both to the
+  right angle <i>aei.</i> and to the lunar <i>aueo.</i> Let therefore the
+  equall angle <i>aeo.</i> bee added to both: the right angle <i>aei.</i>
+  shall be equall to the Lunular <i>aueo</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/045.png"><img style="width:100%" src="images/045.png"
+      alt="Lunulars equal to right lined angles." title="Lunulars equal to right lined angles." /></a>
+  </div>
+  <p>The same Lunular also may bee equall to an obtusangle and Acutangle,
+  as the same argument will demonstrate.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="7_e_iij"></a> 7. <i>If an angle being equicrurall to an other
+  angle, be also equall to it in base, it is equall: And if an angle having
+  equall shankes with another, bee equall to it in the angle, it is also
+  equall to it in the base. �</i> 8. &amp; 4. <i>p j</i>.</p>
+
+  <p>For such angles shall be congruall or agreeable in shanks, and also
+  congruall in bases. <i>Angulus isosceles</i>, or <i>Angulus
+  �quicrurus</i>, is a triangle having equall shankes unto another. <!--
+  Page 26 --><span class="pagenum"><a name="page26"></a>[26]</span></p>
+
+  <div class="figright" style="width:10%;">
+      <a href="images/046a.png"><img style="width:100%" src="images/046a.png"
+      alt="Angles." title="Angles." /></a>
+  </div>
+  <p><a name="8_e_iij"></a> 8. <i>And if an angle equall in base to
+  another, be also equall to it in shankes, it is equall to it</i>.</p>
+
+  <p>For the congruency is the same: And yet if equall angles bee equall in
+  base, they are not by and by equicrurall, as in the angles of the same
+  section will appeare, as here. And so of two equalities, the first is
+  reciprocall: The second is not. [And therefore is this Consectary, by the
+  learned B. <i>Salignacus</i>, justly, according to the judgement of the
+  worthy Rud. <i>Snellius</i>, here cancelled; or quite put out: For angles
+  may be equall, although they bee unequall in shankes or in bases, as
+  here, the angle <i>a.</i> is not greater then the angle <i>o</i>,
+  although the angle <i>o</i> have both greater shankes and greater base
+  then the angle <i>a</i>.]</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_iij"></a> 9. <i>If an angle equicrurall to another angle,
+  be greater then it in base, it is greater: And if it be greater, it is
+  greater in base: �</i> 52 &amp; 24. <i>p j</i>.</p>
+
+  <p>As here thou seest; [The angles <i><span class="correction"
+  title="there was no letter e on the printed diagram, inserted in place which fits the text"
+  >eai.</span></i> and <i>uoy.</i> are equicrurall, that is their shankes
+  are equall one to another; But the base <i>ei</i> is greater then the
+  base <i>uy</i>: Therefore the angle <i>eai</i>, is greater then the angle
+  <i>uoy</i>. And contrary wise, they being equicrurall, and the angle
+  <i>eai.</i> being greater then the angle <i>uoy.</i> The base <i>ei.</i>
+  must needes be greater then the base <i>uy</i>.]</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/046b.png"><img style="width:100%" src="images/046b.png"
+      alt="Equicrurall Angles." title="Equicrurall Angles." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 27 --><span class="pagenum"><a name="page27"></a>[27]</span></p>
+
+  <div class="figright" style="width:10%;">
+      <a href="images/047.png"><img style="width:100%" src="images/047.png"
+      alt="Inscribed Angle." title="Inscribed Angle." /></a>
+  </div>
+  <p><a name="10_e_iij"></a> 10. <i>If an angle equall in base, be lesse in
+  the inner shankes, it is greater</i>.</p>
+
+  <p>Or as the learned Master <i>T. Hood</i> doth paraphrastically
+  translate it. <i>If being equall in the base, it bee lesser in the feete
+  (the feete being conteined within the feete of the other angle) it is the
+  greater angle.</i> [That is, if one angle enscribed within another angle,
+  be equall in base, the angle of the inscribed shall be greater then the
+  angle of the circumscribed.]</p>
+
+  <p>As here the angle <i><span class="correction" title="text interchanges aei and aoi."
+  >aoi.</span></i> within the angle <i>aei.</i> And the bases are equall,
+  to witt one and the same; Therefore <i>aoi.</i> the inner angle is
+  greater then <i>aei.</i> the outter angle. <i>Inner</i> is added of
+  necessity: For otherwise there will, in the section or cutting one of
+  another, appeare a manifest errour. All these consectaries are drawne out
+  of that same axiome of congruity, to witt out of the <a
+  href="#10_e_j">10. e j</a>. as <i>Proclus</i> doth plainely affirme and
+  teach: It seemeth saith hee, that the equalities of shankes and bases,
+  doth cause the equality of the verticall angles. For neither, if the
+  bases be equall, doth the equality of the shankes leave the same or
+  equall angles: But if the base bee lesser, the angle decreaseth: If
+  greater, it increaseth. Neither if the bases bee equall, and the shankes
+  unequall, doth the angle remaine the same: But when they are made lesse,
+  it is increased: when they are made greater, it is diminished: For the
+  contrary falleth out to the angles and shankes of the angles. For if thou
+  shalt imagine the shankes to be in the same base thrust downeward, thou
+  makest them lesse, but their angle greater: but if thou do againe
+  conceive them to be pul'd up higher, thou makest them greater, but their
+  angle lesser. For looke how much more neere they come one to another, so
+  much farther off is the toppe removed from the base: wherefore you may
+  boldly affirme, that the same <!-- Page 28 --><span class="pagenum"><a
+  name="page28"></a>[28]</span>base and equall shankes, doe define the
+  equality of Angels. This <i>Poclus</i>,</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="11_e_iij"></a> 11. <i>If unto the shankes of an angle given,
+  homogeneall shankes, from a point assigned, bee made equall upon an
+  equall base, they shall comprehend an angle equall to the angle given. �
+  23. p j.</i> &amp; 26. <i>p xj</i>.</p>
+
+  <p>[This consectary teacheth how unto a point given, to make an angle
+  equall to an Angle given. To the effecting and doing of each three things
+  are required; First, that the shankes be homogeneall, that is in each
+  place, either straight or crooked: Secondly, that the shankes bee made
+  equall, that is of like or equall bignesse: Thirdly, that the bases be
+  equall: which three conditions if they doe meete, it must needes be that
+  both the angles shall bee equall: but if one of them be wanting, of
+  necessity againe they must be unequall.]</p>
+
+  <p>This shall hereafter be declared and made plaine by many and sundry
+  practises: and therefore here we bring no example of it.</p>
+
+  <p><a name="12_e_iij"></a> 12. <i>An angle is either right or
+  oblique</i>.</p>
+
+  <p>Thus much of the Affections of an angle; the division into his kindes
+  followeth. An angle is either Right or Oblique: as afore, at the 4 <i>e
+  ij.</i> a line was right or straight, and oblique or crooked. <!-- Page
+  29 --><span class="pagenum"><a name="page29"></a>[29]</span></p>
+
+  <div class="figright" style="width:10%;">
+      <a href="images/049a.png"><img style="width:100%" src="images/049a.png"
+      alt="Right Angle." title="Right Angle." /></a>
+  </div>
+  <p><a name="13_e_iij"></a> 13. <i>A right angle is an angle whose shankes
+  are right (that is perpendicular) one unto another: An Oblique angle is
+  contrary to this</i>.</p>
+
+  <p>As here the angle <i>aio.</i> is a right angle, as is also <i>oie.</i>
+  because the shanke <i>oi.</i> is right, that is, perpendicular to
+  <i>ae.</i> [The instrument wherby they doe make triall which is a right
+  angle, and which is oblique, that is greater or lesser then a right
+  angle, is the square which carpenters and joyners do ordinarily use: For
+  lengthes are tried, saith <i>Vitruvius</i>, by the Rular and Line:
+  Heighths, by the Perpendicular or Plumbe: And Angles, by the <span
+  class="correction" title="text reads `spuare'">square</span>.]
+  Contrariwise, an Oblique angle it is, when the one shanke standeth so
+  upon another, that it inclineth, or leaneth more to one side, then it
+  doth to the other: And one angle on the one side, is greater then that on
+  the other.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="14_e_iij"></a> 14. <i>All straight-shanked right angles are
+  equall</i>.</p>
+
+  <p>[That is, they are alike, and agreeable, or they doe fill the same
+  place; as here are <i>aio.</i> and <i>eio.</i> And yet againe on the
+  contrary: All straight shanked equall angles, are not right-angles.]</p>
+
+  <p>The axiomes of the equality of angles were three, as even now wee
+  heard, one generall, and two Consectaries: Here moreover is there one
+  speciall one of the equality of Right angles.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/049b.png"><img style="width:100%" src="images/049b.png"
+      alt="Semicircular Right Angles." title="Semicircular Right Angles." /></a>
+  </div>
+  <p>Angles therfore homogeneall and recticrurall, that is whose shankes
+  are right, as are right lines, as plaine surfaces (For let us so take the
+  word) are equall right <!-- Page 30 --><span class="pagenum"><a
+  name="page30"></a>[30]</span>angles. So are the above written
+  rectilineall right angles equall: so are plaine solid right angles, as in
+  a cube, equall. The axiome may therefore generally be spoken of solid
+  angles, so they be recticruralls: Because all semicircular right angles
+  are not equall to all semicircular right angles: As here, when the
+  diameter is continued it is perpendicular, and maketh twice two angles,
+  within and without, the outter equall betweene themselves, and inner
+  equall betweene themselves: But the outer unequall to the inner: And the
+  angle of a greater semicircle is greater, then the angle of a lesser.
+  Neither is this affection any way reciprocall, That all equall angles
+  should bee right angles. For oblique angles may bee equall betweene
+  themselves: And an oblique angle may bee made equall to a right angle, as
+  a Lunular to a rectilineall right angle, as was manifest, at the <a
+  href="#6_e_iij">6 e</a>.</p>
+
+  <p>The definition of an oblique is understood by the obliquity of the
+  shankes: whereupon also it appeareth; That an oblique angle is unequall
+  to an homogeneall right angle: Neither indeed may oblique angles be made
+  equall by any lawe or rule: Because obliquity may infinitly bee both
+  increased and diminished.</p>
+
+  <p><a name="15_e_iij"></a> 15. <i>An oblique angle is either Obtuse or
+  Acute</i>.</p>
+
+  <p>One difference of Obliquity wee had before at the <a href="#9_e_ij">9
+  e ij</a>. in a line, to witt of a periphery and an helix; Here there is
+  another dichotomy of it into obtuse and acute: which difference is proper
+  to angles, from whence it is translated or conferred upon other things
+  and metaphorically used, as <i>Ingenium obtusum, acutum</i>; A dull, and
+  quicke witte, and such like. <!-- Page 31 --><span class="pagenum"><a
+  name="page31"></a>[31]</span></p>
+
+  <div class="figright" style="width:10%;">
+      <a href="images/051b.png"><img style="width:100%" src="images/051b.png"
+      alt="Acute Angle." title="Acute Angle." /></a>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/051a.png"><img style="width:100%" src="images/051a.png"
+      alt="Obtuse Angle." title="Obtuse Angle." /></a>
+  </div>
+  <p><a name="16_e_iij"></a> 16. <i>An obtuse angle is an oblique angle
+  greater then a right angle. 11. d j</i>.</p>
+
+  <p><i>Obtusus</i>, Blunt or Dull; As here <i>aei.</i> In the definition
+  the <i>genus</i> of both <i>Species</i> or kinds is to bee understood:
+  For a right lined right angle is greater then a sphearicall right angle,
+  and yet it is not an obtuse or blunt angle: And this greater inequality
+  may infinitely be increased.</p>
+
+  <p><a name="17_e_iij"></a> 17. <i>An acutangle is an oblique angle lesser
+  then a right angle. 12. d j</i>.</p>
+
+  <p><i>Acutus</i>, Sharpe, Keene, as here <i>aei.</i> is. Here againe the
+  same <i>genus</i> is to bee understood: because every angle which is
+  lesse then any right angle is not an acute or sharp angle. For a
+  semicircle and sphericall right angle, is lesse then a rectilineall right
+  angle, and yet it is not an acute angle.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<p><!-- Page 32 --><span class="pagenum"><a name="page32"></a>[32]</span></p>
+
+<h2>The fourth Booke, which is
+of a Figure.</h2>
+
+  <p><a name="1_e_iiij"></a> 1. <i>A figure is a lineate bounded on all
+  parts</i>.</p>
+
+  <p>So the triangle <i>aei.</i> is a figure; Because it is a plaine
+  bounded on all parts with three sides. So a circle is a figure: Because
+  it is a plaine every way bounded with one periphery.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/052a.png"><img style="width:100%" src="images/052a.png"
+      alt="Figures." title="Figures." /></a>
+  </div>
+  <p><a name="2_e_iiij"></a> 2. <i>The center is the middle point in a
+  figure</i>.</p>
+
+  <p>In some part of a figure the Center, Perimeter, Radius, Diameter and
+  Altitude are to be considered. The Center therefore is a point in the
+  midst of the figure; so in the triangle, quadrate, and circle, the center
+  is, <i>aei</i>.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/052b.png"><img style="width:100%" src="images/052b.png"
+      alt="Centers of Triangle, Quadrate, And Circle." title="Centers of Triangle, Quadrate, And Circle." /></a>
+  </div>
+<p><!-- Page 33 --><span class="pagenum"><a name="page33"></a>[33]</span></p>
+
+  <p><i>Centrum gravitatis</i>, the center of weight, in every plaine
+  magnitude is said to bee that, by the which it is handled or held up
+  parallell to the horizon: Or it is that point whereby the weight being
+  suspended doth rest, when it is caried. Therefore if any plate should in
+  all places be alike heavie, the center of magnitude and weight would be
+  one and the same.</p>
+
+  <p><a name="3_e_iiij"></a> 3. <i>The perimeter is the compasse of the
+  figure</i>.</p>
+
+  <p>Or, the perimeter is that which incloseth the figure. This definition
+  is nothing else but the interpretation of the Greeke word. Therefore the
+  perimeter of a Triangle is one line made or compounded of three lines. So
+  the perimeter of the triangle <i>a</i>, is <i>eio.</i> So the perimeter
+  of the circle <i>a</i> is a periphery, as in <i>eio.</i> So the perimeter
+  of a Cube is a surface, compounded of sixe surfaces: And the perimeter of
+  a spheare is one whole sph�ricall surface, as hereafter shall
+  appeare.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/053.png"><img style="width:100%" src="images/053.png"
+      alt="Perimeters." title="Perimeters." /></a>
+  </div>
+  <p><a name="4_e_iiij"></a> 4. <i>The Radius is a right line drawne from
+  the center to the perimeter</i>.</p>
+
+  <p><i>Radius</i>, the Ray, Beame, or Spoake, as of the sunne, and <!--
+  Page 34 --><span class="pagenum"><a name="page34"></a>[34]</span>cart
+  wheele: As in the figures under written are <i>ae</i>, <i>ai</i>,
+  <i>ao</i>. It is here taken for any distance from the center, whether
+  they be equall or unequall.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/054a.png"><img style="width:100%" src="images/054a.png"
+      alt="Examples of Radius." title="Examples of Radius." /></a>
+  </div>
+  <p><a name="5_e_iiij"></a> 5. <i>The Diameter is a right line inscribed
+  within the figure by his center</i>.</p>
+
+  <p>As in the figure underwritten are <i>ae</i>, <i>ai</i>, <i>ao</i>. It
+  is called the <i>Diagonius</i>, when it passeth from corner to corner. In
+  solids it is called the <i>Axis</i>, as hereafter we shall heare.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/054b.png"><img style="width:100%" src="images/054b.png"
+      alt="Examples of Diameter." title="Examples of Diameter." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="6_e_iiij"></a> 6. <i>The diameters in the same figure are
+  infinite</i>.</p>
+
+  <p>Although of an infinite number of unequall lines that be only the
+  diameter, which passeth by or through the center <!-- Page 35 --><span
+  class="pagenum"><a name="page35"></a>[35]</span>notwithstanding by the
+  center there may be divers and sundry. In a circle the thing is most
+  apparent: as in the Astrolabe the index may be put up and downe by all
+  the points of the periphery. So in a speare and all rounds the thing is
+  more easie to be conceived, where the diameters are equall: yet
+  notwithstanding in other figures the thing is the same. Because the
+  diameter is a right line inscribed by the center, whether from corner to
+  corner, or side to side, the matter skilleth not. Therefore that there
+  are in the same figure infinite diameters, it issueth out of the
+  difinition of a diameter.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="7_e_iiij"></a> 7. <i>The center of the figure is in the
+  diameter</i>.</p>
+
+  <p>As here thou seest <i>a, e, i</i> this ariseth out of the definition
+  of the diameter. For because the diameter is inscribed into the figure by
+  the center: Therefore the Center of the figure must needes be in the
+  diameter thereof: This is by <i>Archimedes</i> assumed especially at the
+  9, 10, 11, and 13 <i>Theoreme</i> of his <i>Isorropicks</i>, or
+  <i>�quiponderants</i>.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/055.png"><img style="width:100%" src="images/055.png"
+      alt="Centers in Diameters." title="Centers in Diameters." /></a>
+  </div>
+  <p>This consectary, saith the learned Rod. Snellius, is as it were a
+  kinde of invention of the center. For where the diameters doe meete and
+  cutt one another, there must the center needes bee. The cause of this is
+  for that in every figure <!-- Page 36 --><span class="pagenum"><a
+  name="page36"></a>[36]</span>there is but one center only: And all the
+  diameters, as before was said, must needes passe by that center.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="8_e_iiij"></a> 8. <i>It is in the meeting of the
+  diameters</i>.</p>
+
+  <p>As in the examples following. This also followeth out of the same
+  definition of the diameter. For seeing that every diameter passeth by the
+  center: The center must needes be common to all the diameters: and
+  therefore it must also needs be in the meeting of them: Otherwise there
+  should be divers centers of one and the same figure. This also doth the
+  same <i>Archimedes</i> propound in the same words in the 8. and 12
+  theoremes of the same booke, speaking of Parallelogrammes and
+  Triangles.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/056.png"><img style="width:100%" src="images/056.png"
+      alt="Meeting of Diameters." title="Meeting of Diameters." /></a>
+  </div>
+  <p><a name="9_e_iiij"></a> 9. <i>The Altitude is a perpendicular line
+  falling from the toppe of the figure to the base</i>.</p>
+
+  <p><i>Altitudo</i>, the altitude, or heigth, or the depth: [For that, as
+  hereafter shall bee taught, is but <i>Altitudo versa</i>, an heighth <!--
+  Page 37 --><span class="pagenum"><a name="page37"></a>[37]</span>with the
+  heeles upward.] As in the figures following are <i>ae</i>, <i>io</i>,
+  <i>uy</i>, or <i>sr</i>. Neither is it any matter whether the base be the
+  same with the figure, or be continued or drawne out longer, as in a blunt
+  angled triangle, when the base is at the blunt corner, as here in the
+  triangle, <i>aei</i>, is <i>ao</i>.</p>
+
+  <div class="figcenter" style="width:65%;">
+      <a href="images/057.png"><img style="width:100%" src="images/057.png"
+      alt="Altitudes." title="Altitudes." /></a>
+  </div>
+  <p><a name="10_e_iiij"></a> 10. <i>An ordinate figure, is a figure whose
+  bounds are equall and angles equall</i>.</p>
+
+  <p>In plaines the Equilater triangle is onely an ordinate figure, the
+  rest are all inordinate: In quadrangles, the Quadrate is ordinate, all
+  other of that sort are inordinate: In every sort of Multangles, or many
+  cornered figures one may be an ordinate. In crooked lined figures the
+  Circle is ordinate, because it is conteined with equall bounds, (one
+  bound alwaies equall to it selfe being taken for infinite many,) because
+  it is equiangled, seeing (although in deede there be in it no angle) the
+  inclination notwithstanding is every where alike and equall, and as it
+  were the angle of the perphery be alwaies alike unto it selfe: whereupon
+  of Plato and Plutarch a circle is said to be <i>Polygonia</i>, a
+  multangle; and of Aristotle <i>Holegonia</i>, a totangle, nothing else
+  but one whole angle. In mingled-lined figures there is nothing that is
+  ordinate: In <!-- Page 38 --><span class="pagenum"><a
+  name="page38"></a>[38]</span>solid bodies, and pyramids the Tetrahedrum
+  is ordinate: Of Prismas, the Cube: of Polyhedrum's, three onely are
+  ordinate, the octahedrum, the Dodecahedrum, and the Icosahedrum. In
+  oblique-lined bodies, the spheare is concluded to be ordinate, by the
+  same argument that a circle was made to bee ordinate.</p>
+
+  <p><a name="11_e_iiij"></a> 11. <i>A prime or first figure, is a figure
+  which cannot be divided into any other figures more simple then it
+  selfe</i>.</p>
+
+  <p>So in plaines the triangle is a prime figure, because it cannot be
+  divided into any other more simple figure although it may be cut many
+  waies: And in solids, the Pyramis is a first figure: Because it cannot be
+  divided into a more simple solid figure, although it may be divided into
+  an infinite sort of other figures: Of the Triangle all plaines are made;
+  as of a Pyramis all bodies or solids are compounded; such are <i>aei.</i>
+  and <i>aeio</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/058.png"><img style="width:100%" src="images/058.png"
+      alt="Prime Figures." title="Prime Figures." /></a>
+  </div>
+  <p><a name="12_e_iiij"></a> 12. <i>A rationall figure is that which is
+  comprehended of a base and height rationall betweene themselves</i>.</p>
+
+  <p>So <i>Euclide</i>, at the 1. d. ij. saith, that a rightangled
+  parallelogramme is comprehended of two right lines perpendicular one to
+  another, videlicet one multiplied by the other. For Geometricall
+  comprehension is sometimes as it were in numbers a multiplication:
+  Therefore if yee shall grant the base and height to bee rationalls
+  betweene themselves, <!-- Page 39 --><span class="pagenum"><a
+  name="page39"></a>[39]</span>that their reason I meane may be expressed
+  by a number of the assigned measure, then the numbers of their sides
+  being multiplyed one by another, the bignesse of the figure shall be
+  expressed. Therefore a Rationall figure is made by the multiplying of two
+  rationall sides betweene themselves.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="13_e_iiij"></a> 13. <i>The number of a rationall figure, is
+  called a Figurate number: And the numbers of which it is made, the Sides
+  of the figurate</i>.</p>
+
+  <p>As if a Right angled parallelogramme be comprehended of the base
+  foure, and the height three, the Rationall made shall be 12. which wee
+  here call the figurate: and 4. and 3. of which it was made, we name
+  sides.</p>
+
+  <p><a name="14_e_iiij"></a> 14. <i>Isoperimetrall figures, are figures of
+  equall perimeter</i>.</p>
+
+  <p>This is nothing else but an interpretation of the Greeke word; So a
+  triangle of 16. foote about, is a isoperimeter to a triangle 16. foote
+  about, to a quadrate 16. foote about, and to a circle 16. foote
+  about.</p>
+
+  <p><a name="15_e_iiij"></a> 15. <i>Of isoperimetralls homogenealls that
+  which is most ordinate, is greatest: Of ordinate isoperimetralls
+  heterogenealls, that is greatest, which hath most bounds</i>.</p>
+
+  <p>So an equilater triangle shall bee greater then an isoperimeter
+  inequilater triangle; and an equicrurall, greater then an unequicrurall:
+  so in quadrangles, the quadrate is greater then that which is not a
+  quadrate: so an oblong more ordinate, is greater then an oblong lesse
+  ordinate. So of those figures which are heterogeneall ordinates, the
+  quadrate is greater then the Triangle: And the Circle, then the Quadrate.
+  <!-- Page 40 --><span class="pagenum"><a
+  name="page40"></a>[40]</span></p>
+
+  <p><a name="16_e_iiij"></a> 16. <i>If prime figures be of equall height,
+  they are in reason one unto another, as their bases are: And
+  contrariwise</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/060.png"><img style="width:100%" src="images/060.png"
+      alt="Areas." title="Areas." /></a>
+  </div>
+  <p>The proportion of first figures is here twofold; the first is direct
+  in those which are of equall height. In Arithmeticke we learned; That if
+  one number doe multiply many numbers, the products shall be proportionall
+  unto the numbers, multiplyed. From hence in rationall figures the content
+  of those which are of equall height is to bee expressed by a number. As
+  in two right angled parallelogrammes, let 4. the same height, multiply 2.
+  and 3. the bases: The products 8. and 12. the parallelogrammes made, are
+  directly proportionall unto the bases 2. and 3. Therefore as 2. is unto
+  3. so is 8. unto 12. The same shall afterward appeare in right Prismes
+  and Cylinders. In plaines, Parallelogramms are the doubles of triangles:
+  In solids, Prismes are the triples of pyramides: Cylinders, the triples
+  of Cones. The converse of this element is plaine out of the former also:
+  First figures if they be in reason one to another as their bases are,
+  then are they of equall height, to witt when their products are
+  proportionall unto the multiplyed, the same number did multiply them.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="17_e_iiij"></a> 17. <i>If prime figures of equall heighth
+  have also equall bases, they are equall</i>.</p>
+
+  <p>[The reason is, because then those two figures compared, have equall
+  sides, which doe make them equall betweene themselves; For the parts of
+  the one applyed or laid unto the parts of the other, doe fill an equall
+  place, as was taught at the <a href="#10_e_j">10. e. j</a>. <i>Sn.</i>]
+  So Triangles, so Parallelogrammes, and so other figures proposed are
+  equalled upon an equall base. <!-- Page 41 --><span class="pagenum"><a
+  name="page41"></a>[41]</span></p>
+
+  <p><a name="18_e_iiij"></a> 18. <i>If prime figures be reciprocall in
+  base and height, they are equall: And contrariwise</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/061a.png"><img style="width:100%" src="images/061a.png"
+      alt="Equal Parallelogrammes." title="Equal Parallelogrammes." /></a>
+  </div>
+  <p>The second kind of proportion of first figures is reciprocall. This
+  kinde of proportion rationall and expressible by a number, is not to be
+  had in first figures themselves: but in those that are equally manifold
+  to them, as was taught even now in direct proportion: As for example, Let
+  these two right angled parallelogrammes, unequall in bases and heighths
+  3, 8, 4, 6, be as heere thou seest: The proportion reciprocall is thus,
+  As 3 the base of the one, is unto 4, the base of the other: so is 6. the
+  height of the one is to 8. the height of the other: And the
+  parallelogrammes are equall, viz. 24. and 24. Againe, let two solids of
+  unequall bases &amp; heights (for here also the base is taken for the
+  length and heighth) be 12, 2, 3, <span class="correction" title="text reads `0', cf. the diagram"
+  >6</span>, 3, 4. The solids themselves shall be 72. and 72, as here thou
+  seest; and the proportion of the bases and heights likewise is
+  reciprocall: For as 24, is unto 18, so is 4, unto 3. The cause is out of
+  the golden rule of proportion in Arithmeticke: Because twice two sides
+  are <!-- Page 42 --><span class="pagenum"><a
+  name="page42"></a>[42]</span>proportionall: Therefore the plots made of
+  them shall be equall. And againe, by the same rule, because the plots are
+  equall: Therefore the bounds are proportionall; which is the converse of
+  this present element.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/061b.png"><img style="width:100%" src="images/061b.png"
+      alt="Equal Solids." title="Equal Solids." /></a>
+  </div>
+  <p><a name="19_e_iiij"></a> 19. <i>Like figures are equiangled figures,
+  and proportionall in the shankes of the equall angles</i>.</p>
+
+  <p>First like figures are defined, then are they compared one with
+  another, similitude of figures is not onely of prime figures, and of such
+  as are compounded of prime figures, but generally of all other
+  whatsoever. This similitude consisteth in two things, to witt in the
+  equality of their angles, and proportion of their shankes.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="20_e_iiij"></a> 20. <i>Like figures have answerable bounds
+  subtended against their equall angles: and equall if they themselves be
+  equall</i>.</p>
+
+  <p>Or thus, They have their termes subtended to the equall angles
+  correspondently proportionall: And equall if the figures themselves be
+  equall; H. This is a consectary out of the former definition.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="21_e_iiij"></a> 21. <i>Like figures are situate alike, when
+  the proportionall bounds doe answer one another in like
+  situation</i>.</p>
+
+  <p>The second consectary is of situation and place. And this like
+  situation is then said to be when the upper parts of the one figure doe
+  agree with the upper parts of the other, the lower, with the lower, and
+  so the other differences of places. <i>Sn</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 43 --><span class="pagenum"><a name="page43"></a>[43]</span></p>
+
+  <p><a name="22_e_iiij"></a> 22. <i>Those figures that are like unto the
+  same, are like betweene themselves</i>.</p>
+
+  <p>This third consectary is manifest out of the definition of like
+  figures. For the similitude of two figures doth conclude both the same
+  equality in angles and proportion of sides betweene themselves.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="23_e_iiij"></a> 23. <i>If unto the parts of a figure given,
+  like parts and alike situate, be placed upon a bound given, a like figure
+  and likely situate unto the figure given, shall bee made
+  accordingly</i>.</p>
+
+  <p>This fourth consectary teacheth out of the said definition, the
+  fabricke and manner of making of a figure alike and likely situate unto a
+  figure given. <i>Sn</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/063.png"><img style="width:100%" src="images/063.png"
+      alt="Like Figures." title="Like Figures." /></a>
+  </div>
+  <p><a name="24_e_iiij"></a> 24. <i>Like figures have a reason of their
+  homologallor correspondent sides equally manifold unto their dimensions:
+  and a meane proportionall lesse by one</i>.</p>
+
+  <p>Plaine figures have but two dimensions, to witt Length, and Breadth:
+  And therefore they have but a doubled reason of their homologall sides.
+  Solids have three dimensions, videl. Length, Breadth, &amp; thicknesse:
+  therefore they shall have a treabled reason of their homologall or
+  correspondent sides. In 8. and 18. the two plaines given, first the
+  angles are equall: secondly, their homolegall side 2. and 4. and 3. and
+  6. are proportionall. Therefore the reason of 8. the first figure, unto
+  18. the <!-- Page 44 --><span class="pagenum"><a
+  name="page44"></a>[44]</span>second, is as the reason is of 2. unto 3.
+  doubled. But the reason of 2. unto 3. doubled, by the 3. chap. ij. of
+  Arithmeticke, is of 4. to 9. (for 2/3 2/3 is 4/9.) Therefore the reason
+  of 8. unto 18, that is, of the first figure unto the second, is of 4.
+  unto 9. In Triangles, which are the halfes of rightangled
+  parallelogrammes, there is the same truth, and yet by it selfe not
+  rationall and to be expressed by numbers.</p>
+
+  <p>Said numbers are alike in the trebled reason of their homologall
+  sides; As for example, 60. and 480. are like solids; and the solids also
+  comprehended in those numbers are like-solids, as here thou seest:
+  Because their sides, 4. 3. 5. and 8. 6. 10. are proportionall betweene
+  themselves. But the reason of 60. to 480. is the reason of 4. to 8.
+  trebled, thus 4/8 4/8 4/8 = 64/512; that is of 1. unto 8. or
+  <i>octupla</i>, which you shall finde in the dividing of 480. by 60.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:30%;">
+      <a href="images/064.png"><img style="width:100%" src="images/064.png"
+      alt="Like Solids." title="Like Solids." /></a>
+  </div>
+  <p>Thus farre of the first part of this element: The second, that like
+  figurs have a meane, proportional lesse by one, then are their
+  dimensions, shall be declared by few words. For plaines having but two
+  dimensions, have but one meane proportionall, solids having three
+  dimensions, have two meane proportionalls. The cause is onely
+  Arithmeticall, as afore. For where the bounds are but 4. as they are in
+  two plaines, there can be found no more but one meane proportionall, as
+  in the former example of 8. and 18. where the homologall or correspondent
+  sides are 2. 3. and 4. 6.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+<table class="nobctr">
+<tr><td class="spac">2</td><td class="spac">3</td><td class="spac">4</td><td class="spac">6</td></tr>
+<tr><td class="spac">&nbsp;</td><td class="spac">3</td><td class="spac">4</td><td class="spac">&nbsp;</td></tr>
+<tr><td class="spac">8</td><td class="spac" colspan="2" align="center">12</td><td class="spac">18</td></tr>
+</table>
+
+<p><!-- Page 45 --><span class="pagenum"><a name="page45"></a>[45]</span></p>
+
+  <p>Againe by the same rule, where the bounds are 6. as they are in two
+  solids, there may bee found no more but two meane proportionalls: as in
+  the former solids 30. and 240. where the homologall or correspondent
+  sides are 2. 4. 3. 6. 5. 10.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+<table class="nobctr">
+<tr><td class="spac">2</td><td class="spac">4</td><td class="spac">3</td><td class="spac">6</td><td class="spac">5</td><td class="spac">10</td></tr>
+<tr><td class="spac">&nbsp;</td><td class="spac">4</td><td class="spac">3</td><td class="spac">&nbsp;</td></tr>
+<tr><td class="spac">6</td><td class="spac" colspan="2" align="center">12</td><td class="spac">24</td></tr>
+<tr><td class="spac" colspan="3">&nbsp;</td><td class="spac">24</td><td class="spac">5</td></tr>
+<tr><td class="spac">&nbsp;</td><td class="spac">30</td><td class="spac">60</td><td class="spac">120</td><td class="spac" colspan="2" align="center">240</td></tr>
+</table>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/065a.png"><img style="width:100%" src="images/065a.png"
+      alt="Creation of Like Figures." title="Creation of Like Figures." /></a>
+  </div>
+  <div class="figcenter" style="width:45%;">
+      <a href="images/065b.png"><img style="width:100%" src="images/065b.png"
+      alt="Creation of Like Solids." title="Creation of Like Solids." /></a>
+  </div>
+<p><!-- Page 46 --><span class="pagenum"><a name="page46"></a>[46]</span></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="25_e_iiij"></a> 25. <i>If right lines be continually
+  proportionall, more by one then are the dimensions of like figures
+  likelily situate unto the first and second, it shall be as the first
+  right line is unto the last, so the first figure shall be unto the
+  second: And contrariwise</i>.</p>
+
+  <p>Out of the similitude of figures two consectaries doe arise, in part
+  only, as is their axiome, rationall and expressable by numbers. If three
+  right lines be continually proportionall, it shall be as the first is
+  unto the third: So the <span class="correction" title="text reads `rectineall'"
+  >rectilineall</span> figure made upon the first, shall be unto the
+  rectilineall figure made upon the second, alike and likelily situate.
+  This may in some part be conceived and understood by numbers. As for
+  example, Let the lines given, be 2. foot, 4. foote, and 8 foote. And upon
+  the first and second, let there be made like figures, of 6. foote and 24.
+  foote; So I meane, that 2. and 4. be the bases of them. Here as 2. the
+  first line, is unto 8. the third line: So is 6. the first figure, unto
+  24. the second figure, as here thou seest.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/066.png"><img style="width:100%" src="images/066.png"
+      alt="Proportionall Lines and Figures." title="Proportionall Lines and Figures." /></a>
+  </div>
+  <p>Againe, let foure lines continually proportionall, be 1. 2. 4. 8. And
+  let there bee two like solids made upon the first and second: upon the
+  first, of the sides 1. 3. and 2. <span class="correction" title="text reads `lee'"
+  >let</span> it be 6. Upon the second, of the sides 2. 6. and 4. let it be
+  48. As the first right line 1. is unto the fourth 8. So is the figure 6.
+  unto the second 48. as is manifest by division. The examples are thus.
+  <!-- Page 47 --><span class="pagenum"><a
+  name="page47"></a>[47]</span></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/067a.png"><img style="width:100%" src="images/067a.png"
+      alt="Proportionall Lines and Solids." title="Proportionall Lines and Solids." /></a>
+  </div>
+  <p>Moreover by this Consectary a way is laid open leading unto the reason
+  of doubling, treabling, or after any manner way whatsoever assigned
+  increasing of a figure given. For as the first right line shall be unto
+  the last: so shall the first figure be unto the second.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="26_e_iiij"></a> 26. <i>If foure right lines bee proportionall
+  betweene themselves: Like figures likelily situate upon them, shall be
+  also proportionall betweene themselves: And contrariwise, out of the 22.
+  p vj. and 37. p xj</i>.</p>
+
+  <p>The proportion may also here in part bee expressed by numbers: And yet
+  a continuall is not required, as it was in the former.</p>
+
+<p><!-- Page 48 --><span class="pagenum"><a name="page48"></a>[48]</span></p>
+
+  <p>In Plaines let the first example be, as followeth.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/067b.png"><img style="width:100%" src="images/067b.png"
+      alt="Two Pairs of Figures." title="Two Pairs of Figures." /></a>
+  </div>
+  <p>The cause of proportionall figures, for that twice two figures have
+  the same reason doubled.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/068.png"><img style="width:100%" src="images/068.png"
+      alt="Two Pairs of Solids." title="Two Pairs of Solids." /></a>
+  </div>
+  <p>In Solids let this bee the second example. And yet here the figures
+  are not proportionall unto the right lines, as before figures of equall
+  heighth were unto their bases, but they themselves are proportionall one
+  to another. And yet are they not proportionall in the same kinde of
+  proportion.</p>
+
+  <p>The cause also is here the same, that was before: To witt, because
+  twice two figures have the same reason trebled.</p>
+
+  <p><a name="27_e_iiij"></a> 27. <i>Figures filling a place, are those
+  which being any way set about the same point, doe leave no voide
+  roome</i>.</p>
+
+  <p>This was the definition of the ancient Geometers, as appeareth out of
+  <i>Simplicius</i>, in his commentaries upon the 8. chapter of
+  <i>Aristotle's</i> iij. booke of Heaven: which kinde of figures
+  <i>Aristotle</i> in the same place deemeth to bee onely ordinate, and yet
+  not all of that kind. But only three among the Plaines, to witt a
+  Triangle, a Quadrate, and a Sexangle: amongst Solids, two; the Pyramis,
+  and the Cube. But if the filling of a place bee judged by right angles,
+  4. in a Plaine, and 8. in a Solid, the Oblong of plaines, and the <!--
+  Page 49 --><span class="pagenum"><a
+  name="page49"></a>[49]</span>Octahedrum of Solids shall (as shall appeare
+  in their places) fill a place; And yet is not this Geometrie of
+  <i>Aristotle</i> accurate enough. But right angles doe determine this
+  sentence, and so doth <i>Euclide</i> out of the angles demonstrate, That
+  there are onely five ordinate solids; And so doth <i>Potamon</i> the
+  Geometer, as <i>Simplicus</i> testifieth, demonstrate the question of
+  figures filling a place. Lastly, if figures, by laying of their corners
+  together, doe make in a Plaine 4. right angles, or in a Solid 8. they doe
+  fill a place.</p>
+
+  <p>Of this probleme the ancient geometers have written, as we heard even
+  now: And of the latter writers, <i>Regiomontanus</i> is said to have
+  written accurately; And of this argument <i>Maucolycus</i> hath promised
+  a treatise, neither of which as yet it hath beene our good hap to
+  see.</p>
+
+  <p><i>Neither of these are figures of this nature, as in their due places
+  shall be proved and demonstrated</i>.</p>
+
+  <p><a name="28_e_iiij"></a> 28. <i>A round figure is that, all whose
+  raies are equall</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/069.png"><img style="width:100%" src="images/069.png"
+      alt="Round Objects." title="Round Objects." /></a>
+  </div>
+  <p>Such in plaines shall the Circle be, in Solids the Globe or Spheare.
+  Now this figure, the Round, I meane, of all Isoperimeters is the
+  greatest, as appeared before at the <a href="#15_e_iiij">15. e</a>. For
+  which cause <i>Plato</i>, in his <i>Tim�us</i> or his Dialogue of the
+  World said; That this figure is of all other the greatest. And therefore
+  God, saith he, did make the world of a <!-- Page 50 --><span
+  class="pagenum"><a name="page50"></a>[50]</span>sphearicall forme, that
+  within his compasse it might the better containe all things: And
+  <i>Aristotle</i>, in his Mechanicall problems, saith; That this figure is
+  the beginning, principle, and cause of all miracles. But those miracles
+  shall in their time God willing, be manifested and showne.</p>
+
+  <p><i>Rotundum</i>, a Roundle, let it be here used for <i>Rotunda
+  figura</i>, a round figure. And in deede <i>Thomas Finkius</i> or
+  <i>Finche</i>, as we would call him, a learned <i>Dane</i>, sequestring
+  this argument from the rest of the body of Geometry, hath intituled that
+  his worke <i>De Geometria rotundi</i>, Of the Geometry of the Round or
+  roundle.</p>
+
+  <p><a name="29_e_iiij"></a> 29. <i>The diameters of a roundle are cut in
+  two by equall raies</i>.</p>
+
+  <p>The reason is, because the halfes of the diameters, are the raies. Or
+  because the diameter is nothing else but a doubled ray: Therefore if thou
+  shalt cut off from the diameter so much, as is the radius or ray, it
+  followeth that so much shall still remaine, as thou hast cutte of, to
+  witt one ray, which is the other halfe of the diameter. <i>Sn</i>.</p>
+
+  <p>And here observe, That <i>Bisecare</i>, doth here, and in other places
+  following, signifie to cutte a thing into two equall parts or portions;
+  And so <i>Bisegmentum</i>, to be one such portion; And <i>Bisectio</i>,
+  such a like cutting or division.</p>
+
+  <p><a name="30_e_iiij"></a> 30. <i>Rounds of equall diameters are equall.
+  Out of the 1. d. iij</i>.</p>
+
+  <p>Circles and Spheares are equall, which have equall diameters. For the
+  raies, which doe measure the space betweene the Center and Perimeter, are
+  equall, of which, being doubled, the Diameter doth consist.
+  <i>Sn</i>.</p>
+
+<hr class="full" />
+
+<p><!-- Page 51 --><span class="pagenum"><a name="page51"></a>[51]</span></p>
+
+<h2>The fifth Booke, of <i>Ramus</i> his
+Geometry,
+which is of Lines and
+Angles in a plaine Surface.</h2>
+
+  <p><a name="1_e_v"></a> 1. <i>A lineate is either a Surface or a
+  Body</i>.</p>
+
+  <p><i>Lineatum</i>, (or <i>Lineamentum</i>) a magnitude made of lines, as
+  was defined at <a href="#1_e_iij">1. e. iij</a>. is here divided into two
+  kindes: which is easily conceived out of the said definition there, in
+  which a line is excluded, and a Surface &amp; a body are comprehended.
+  And from hence arose the division of the arte Metriall into Geometry, of
+  a surface, and Stereometry, of a body, after which maner <i>Plato</i> in
+  his vij. booke of his Common-wealth, and <i>Aristotle</i> in the 7.
+  chapter of the first booke of his <i>Posteriorums</i>, doe distinguish
+  betweene Geometry and Stereometry: And yet the name of Geometry is used
+  to signifie the whole arte of measuring in generall.</p>
+
+  <p><a name="2_e_v"></a> 2. <i>A Surface is a lineate only broade. 5. d
+  j</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/071.png"><img style="width:100%" src="images/071.png"
+      alt="Surfaces." title="Surfaces." /></a>
+  </div>
+  <p>As here <i>aeio.</i> and <i>uysr.</i> The definition of a Surface doth
+  comprehend the distance or dimension of a line, to <!-- Page 52 --><span
+  class="pagenum"><a name="page52"></a>[52]</span>witt Length: But it
+  addeth another distance, that is Breadth. <span class="correction"
+  title="text reads `Therefoce'">Therefore</span> a Surface is defined by
+  some, as <i>Proclus</i> saith, to be a magnitude of two dimensions. But
+  two doe not so specially and so properly define it. Therefore a Surface
+  is better defined, to bee a magnitude onely long and broad. Such, saith
+  <i>Apollonius</i>, are the shadowes upon the earth, which doe farre and
+  wide cover the ground and champion fields, and doe not enter into the
+  earth, nor have any manner of thicknesse at all.</p>
+
+  <p><i>Epiphania</i>, the Greeke word, which importeth onely the outter
+  appearance of a thing, is here more significant, because of a Magnitude
+  there is nothing visible or to bee seene, but the surface.</p>
+
+  <p><a name="3_e_v"></a> 3. <i>The bound of a surface is a line. 6. d
+  j</i>.</p>
+
+  <p>The matter in Plaines is manifest. For a three cornered surface is
+  bounded with 3. lines: A foure cornered surface, with foure lines, and so
+  forth: A Circle is bounded with one line. But in a Sphearicall surface
+  the matter is not so plaine: For it being whole, seemeth not to be
+  bounded with a line. Yet if the manner of making of a Sphearicall
+  surface, by the conversi&#x14D; or turning about of a semiperiphery, the
+  beginning of it, as also the end, shalbe a line, to wit a semiperiphery:
+  And as a point doth not only <i>actu</i>, or indeede bound and end a
+  line: But is <i>potentia</i>, or in power, the middest of it: So also a
+  line boundeth a Surface <i>actu</i>, and an innumerable company of lines
+  may be taken or supposed to be throughout the whole surface. A Surface
+  therefore is made by the motion of a line, as a Line was made by the
+  motion of a point.</p>
+
+  <p><a name="4_e_v"></a> 4. <i>A Surface is either Plaine or
+  Bowed</i>.</p>
+
+  <p>The difference of a Surface, doth answer to the difference of a Line,
+  in straightnesse and obliquity or crookednesse.</p>
+
+  <p><i>Obliquum</i>, oblique, there signified crooked; Not right or
+  straight: Here, uneven or bowed, either upward or downeward. <i>Sn.</i>
+  <!-- Page 53 --><span class="pagenum"><a
+  name="page53"></a>[53]</span></p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/073a.png"><img style="width:100%" src="images/073a.png"
+      alt="Plaine Surface." title="Plaine Surface." /></a>
+  </div>
+  <p><a name="5_e_v"></a> 5. <i>A plaine surface is a surface, which lyeth
+  equally betweene his bounds, out of the 7. d j</i>.</p>
+
+  <p>As here thou seest in <i>aeio.</i> That therefore a Right line doth
+  looke two contrary waies, a Plaine surface doth looke all about every
+  way, that a plaine surface should, of all surfaces within the same
+  bounds, be the shortest: And that the middest thereof should hinder the
+  sight of the extreames. Lastly, it is equall to the dimension betweene
+  the lines: It may also by one right line every way applyed be tryed, as
+  <i>Proclus</i> at this place doth intimate.</p>
+
+  <p><i>Planum</i>, a Plaine, is taken and used for a plaine surface: as
+  before <i>Rotundum</i>, a Round, was used for a round figure.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="6_e_v"></a> 6. <i>From a point unto a point we may, in a
+  plaine surface, draw a right line, 1 and 2. post. j</i>.</p>
+
+  <p>Three things are from the former ground begg'd: The first is of a
+  Right line. A right line and a periphery were in the ij. booke defined:
+  But the fabricke or making of them both, is here said to bee properly in
+  a plaine.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/073b.png"><img style="width:100%" src="images/073b.png"
+      alt="Right Line." title="Right Line." /></a>
+  </div>
+  <p>The fabricke or construction of a right line is the 1. petition. And
+  justly is it required that it may bee done onely upon a plaine: For in
+  any other surface it were in vaine to aske it. For neither may wee
+  possibly in a sphericall betweene two points draw a right line: Neither
+  may wee possibly in a Conicall and Cylindraceall betweene any two points
+  assigned draw a right line. For from the toppe <!-- Page 54 --><span
+  class="pagenum"><a name="page54"></a>[54]</span>unto the base that in
+  these is only possible: And then is it the bounde of the plaine which
+  cutteth the Cone and Cylinder. Therefore, as I said, of a right plaine it
+  may onely justly bee demanded: That from any point assigned, unto any
+  point assigned, a right line may be drawne, as here from <i>a</i> unto
+  <i>e</i>.</p>
+
+  <p>Now the Geometricall instrument for the drawing of a right plaine is
+  called <i>Amussis</i>, &amp; by <i>Petolemey</i>, in the 2. chapter of
+  his first booke of his Musicke, <i>Regula</i>, a Rular, such as heere
+  thou seest.</p>
+
+  <div class="figcenter" style="width:60%;">
+      <a href="images/074.png"><img style="width:100%" src="images/074.png"
+      alt="Rular." title="Rular." /></a>
+  </div>
+  <p>And from a point unto a point is this justly demanded to be done, not
+  unto points; For neither doe all points fall in a right line: But many
+  doe fall out to be in a crooked line. And in a Spheare, a Cone &amp;
+  Cylinder, a Ruler may be applyed, but it must be a sphearicall, Conicall,
+  or Cylindraceall. But by the example of a right line doth
+  <i>Vitellio</i>, 2 <i>p j.</i> demaund that betweene two lines a surface
+  may be extended: And so may it seeme in the Elements, of many figures
+  both plaine and solids, by <i>Euclide</i> to be demanded; That a figure
+  may be described, at the 7. and 8. e ij. Item that a figure may be made
+  vp, at the 8. 14. 16. 23. 28. p. vj. which are of Plaines. Item at the
+  25. 31. 33. 34. 36. 49. p. xj. which are of Solids. Yet notwithstanding a
+  plaine surface, and a plaine body doe measure their rectitude by a right
+  line, so that <i>jus postulandi</i>, this right of begging to have a
+  thing granted may seeme primarily to bee in a right plaine line.</p>
+
+  <p>Now the <i>Continuation</i> of a right line is nothing else, but the
+  drawing out farther of a line now drawne, and that from a point unto a
+  point, as we may continue the right line <i>ae.</i> unto <i>i.</i>
+  wherefore the first and second Petitions of <i><span class="correction"
+  title="text reads `Euclde'">Euclide</span></i> do agree in one.</p>
+
+<p><!-- Page 55 --><span class="pagenum"><a name="page55"></a>[55]</span></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/075.png"><img style="width:100%" src="images/075.png"
+      alt="Operations with Lines." title="Operations with Lines." /></a>
+  </div>
+  <p><a name="7_e_v"></a> 7. <i>To set at a point assigned a Right line
+  equall to another right line given: And from a greater, to cut off a part
+  equall to a lesser. 2. and 3. p j</i>.</p>
+
+  <p>As let the Right line given be <i>ae.</i> And to <i>i.</i> a point
+  assigned, grant that <i>io.</i> equall to the same <i>ae.</i> may bee
+  set. Item, in the second example, let <i>ae.</i> bee greater then
+  <i>io.</i> And let there <span class="correction" title="text reads `he'"
+  >be</span> cut off from the same <i>ae.</i> by applying of a rular made
+  equall to <i>io.</i> <span class="correction" title="text reads `the the' over two lines"
+  >the</span> lesser, portion <i>au.</i> as here. For if any man shall
+  thinke that this ought only to be don in the minde, hee also, as it were,
+  beares a ruler in his minde, that he may doe it by the helpe of the
+  ruler. Neither is the fabricke in deede, or making of one right equal to
+  another: And the cutting off from greater Right line, a portion equall to
+  a lesser, any whit harder, then it was, having a point and a distance
+  given, to describe a circle: Then having a Triangle, Parallelogramme, and
+  semicircle given, to describe or make a Cone, Cylinder, and spheare, all
+  which notwithstanding <i>Euclide</i> did account as principles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="8_e_v"></a> 8. <i>One right line, or two cutting one another,
+  are in the same plaine, out of the 1. and 2. p xj</i>.</p>
+
+  <p>One Right line may bee the common section of two plaines: yet all or
+  the whole in the same plaine is one: And all the whole is in the same
+  other: And so the whole is the same plaine. Two Right lines cutting one
+  another, may bee in two plaines cutting one of another; But then a plaine
+  may be drawne by them: Therefore both <!-- Page 56 --><span
+  class="pagenum"><a name="page56"></a>[56]</span>of them shall be in the
+  same plaine. And this plaine is geometrically to be conceived: Because
+  the same plaine is not alwaies made the ground whereupon one oblique
+  line, or two cutting one another are drawne, when a periphery is in a
+  sphearicall: Neither may all peripheries cutting one another be possibly
+  in one plaine.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_v"></a> 9. <i>With a right line given to describe a
+  peripherie</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/076.png"><img style="width:100%" src="images/076.png"
+      alt="To Describe a Peripherie." title="To Describe a Peripherie." /></a>
+  </div>
+  <p>This fabricke or construction is taken out of the 3. Petition which is
+  thus. Having a center and a distance given to describe, make, or draw a
+  circle. But here the terme or end of a circle is onely sought, which is
+  better drawne out of the definition of a periphery, at the <a
+  href="#10_e_ij">10. e ij</a>. And in a plaine onely may that conversion
+  or turning about of a right line bee made: Not in a sphearicall, not in a
+  Conicall, not in a Cylindraceall, except it be in top, where
+  notwithstanding a periphery may bee described. Therefore before (to witt
+  at the said <a href="#10_e_ij">10. e ij</a>.) was taught the generall
+  fabricke or making of a Periphery: Here we are informed how to discribe a
+  Plaine periphery, as here.</p>
+
+  <p>Now as the Rular was the instrument invented and used for the drawing
+  of a right line: so also may the same <i>Rular</i>, used after another
+  manner, be the instrument to describe or draw a periphery withall. And
+  indeed such is that instrument used by the Coopers (and other like
+  artists) for the rounding of their bottomes of their tubs, heads of
+  barrells and otherlike vessells: But the <i>Compasses</i>, whether
+  straight shanked or bow-legg'd, such as here thou seest, it skilleth not,
+  are for al purposes and practises, in this case the best and readiest.
+  And in deed the Compasses, of all <!-- Page 57 --><span
+  class="pagenum"><a name="page57"></a>[57]</span>geometricall instruments,
+  are the most excellent, and by whose help famous Geometers have taught:
+  That all the problems of geometry may bee wrought and performed: And
+  there is a booke extant, set out by <i>John Baptist</i>, an Italian,
+  teaching, How by one opening of the Compasses all the problems of
+  <i>Euclide</i> may be resolved: And <i>Jeronymus Cardanus</i>, a famous
+  Mathematician, in the 15. booke of his Subtilties, writeth, that there
+  was by the helpe of the Compasses a demonstration of all things
+  demonstrated by <i>Euclide</i>, found out by him and one
+  <i>Ferrarius</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/077.png"><img style="width:100%" src="images/077.png"
+      alt="Types of Compasses." title="Types of Compasses." /></a>
+  </div>
+  <p><i>Talus</i>, the nephew of <i>D�dalus</i> by his sister, is said in
+  the viij. booke of <i>Ovids Metamorphosis</i>, to have beene the
+  inventour of this instrument: For there he thus writeth of him and this
+  matter:&mdash;<i>Et ex uno duo ferrea brachia nodo: Iunxit, ut �quali
+  spatio distantibus ipsis: Altera pars staret, pars altera duceret
+  orbem</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therfore</p>
+    </div>
+  </div>
+  <p><a name="10_e_v"></a> 10. <i>The raies of the same, or of an equall
+  periphery, are equall</i>.</p>
+
+  <p>The reason is, because the same right line is every where converted or
+  turned about. But here by the Ray of the periphery, must bee understood
+  the Ray the figure contained within the periphery. <!-- Page 58 --><span
+  class="pagenum"><a name="page58"></a>[58]</span></p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/078.png"><img style="width:100%" src="images/078.png"
+      alt="Bisecting an Angle." title="Bisecting an Angle." /></a>
+  </div>
+  <p><a name="11_e_v"></a> 11. <i>If two equall peripheries, from the ends
+  of equall shankes of an assigned rectilineall angle, doe meete before it,
+  a right line drawne from the meeting of them unto the toppe or point of
+  the angle, shall cut it into two equall parts. 9. p j</i>.</p>
+
+  <p>Hitherto we have spoken of plaine lines: Their affection followeth,
+  and first in the Bisection or dividing of an Angle into two equall
+  parts.</p>
+
+  <p>Let the right lined Angle to bee divided into two equall parts bee
+  <i>eai.</i> whose equall shankes let them be <i>ae.</i> and <i>ai.</i>
+  (or if they be unequall, let them be made equall, by the <a
+  href="#7_e_v">7 e</a>.) Then two equall peripheries from the ends
+  <i>e</i> and <i>i.</i> meet before the Angle in <i>o</i>. Lastly, draw a
+  line from <i>o.</i> unto <i>a</i>. I say the angle given is divided into
+  two equall parts. For by drawing the right lines <i>oe.</i> and
+  <i>oi.</i> the angles <i>oae.</i> and <i>oai.</i> equicrurall, by the
+  grant, and by their common side <i>ao.</i> are equall in base <i>eo.</i>
+  and <i>io.</i> by the <a href="#10_e_v">10 e</a> (Because they are the
+  raies of equall peripheries.) Therefore by the <a href="#7_e_iij">7. e
+  iij</a>. the angles <i>oae.</i> and <i>oai.</i> are equall: And therefore
+  the Angle <i>eai.</i> is equally divided into two parts.</p>
+
+  <p><a name="12_e_v"></a> 12. <i>If two equall peripheries from the ends
+  of a right line given, doe meete on each side of the same, a right line
+  drawne from those meetings, shall divide the right line given into two
+  equall parts. 10. p j</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/079a.png"><img style="width:100%" src="images/079a.png"
+      alt="Bisecting a Line." title="Bisecting a Line." /></a>
+  </div>
+  <p>Let the right line given bee <i>ae.</i> And let two equall peripheries
+  from the ends <i>a.</i> and <i>e.</i> meete in <i>i.</i> and <i>o.</i>
+  Then from those meetings let the right line <i>io.</i> be drawne. I say,
+  That <i>ae.</i> is divided into two equall parts, by the said line thus
+  <!-- Page 59 --><span class="pagenum"><a
+  name="page59"></a>[59]</span>drawne. For by drawing the raies of the
+  equall peripheries <i>ia.</i> and <i>ie.</i> the said <i>io.</i> doth cut
+  the angle <i>aie.</i> into two equall parts, by the <a href="#11_e_v">11.
+  e</a>. Therefore the angles <i>aiu.</i> and <i>uie.</i> being equall and
+  equicrurall (seeing the shankes are the raies of equall peripheries, by
+  the grant.) have equall bases <i>au.</i> and <i>ue.</i> by the <a
+  href="#7_e_iij">7. e iij</a>. Wherefore seeing the parts <i>au.</i> and
+  <i>ue.</i> are equall, <i>ae.</i> the assigned right line is divided into
+  two equall portions.</p>
+
+  <div class="figright" style="width:8%;">
+      <a href="images/080b.png"><img style="width:100%" src="images/080b.png"
+      alt="Plumbe-rule." title="Plumbe-rule." /></a>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/079b.png"><img style="width:100%" src="images/079b.png"
+      alt="Perpendicular Lines." title="Perpendicular Lines." /></a>
+  </div>
+  <p><a name="13_e_v"></a> 13. <i>If a right line doe stand perpendicular
+  upon another right line, it maketh on each side right angles: And
+  contrary wise</i>.</p>
+
+  <p>A right line standeth upon a right line, which cutteth, and is not cut
+  againe. And the <i>Angles on each side</i>, are they which the falling
+  line maketh with that underneath it, as is manifest out of
+  <i>Proclus</i>, at the 15. pj. of <i>Euclide</i>; As here <i>ae.</i> the
+  line cut: and <i>io.</i> the insisting line, let them be perpendicular;
+  The angles on each side, to witt <i>aio.</i> and <i>eio.</i> shall bee
+  right angles, by the <a href="#13_e_iij">13. e iij</a>.</p>
+
+  <p>The <i>Rular</i>, for the making of straight lines on a plaine, was
+  the first Geometricall instrument: The <i>Compasses</i>, for the
+  describing of a Circle, was the second: The <i>Norma</i> or <i>Square</i>
+  for the true erecting of a right line in the same plaine upon another
+  right line, and then of a surface and body, upon a surface or body, is
+  the third. The figure therefore is thus.</p>
+
+  <p>Now <i>Perpendicul&#x16B;</i>, an instrument with a line &amp; a
+  plummet of leade appendant upon it, used of Architects, Carpenters, and
+  Masons, is meerely physicall: because heavie things <!-- Page 60 --><span
+  class="pagenum"><a name="page60"></a>[60]</span>naturally by their weight
+  are in straight lines carried perpendicularly downeward. This instrument
+  is of two sorts: The first, which they call a Plumbe-rule, is for the
+  trying of an erect perpendicular, as whether a columne, pillar, or any
+  other kinde of building bee right, that is plumbe unto the plaine of the
+  horizont &amp; doth not leane or reele any way. The second is for the
+  trying or examining of a plaine or floore, whether it doe lye parallell
+  to the horizont or not. Therefore when the line from the right angle,
+  doth fall upon the middle of the base; it shall shew that the length is
+  equally poysed. The Latines call it <i>Libra</i>, or <i>Libella</i>, a
+  ballance: of the <i>Italians Livello</i>, and vel <i>Archipendolo</i>,
+  <i>Achildulo</i>: of the <i>French</i>, <i>Nivelle</i>, or <i>Niueau</i>:
+  of us a <i>Levill</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/080a.png"><img style="width:100%" src="images/080a.png"
+      alt="Square and Level." title="Square and Level." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/081b.png"><img style="width:100%" src="images/081b.png"
+      alt="Figure for demonstration 14 converse." title="Figure for demonstration 14 converse." /></a>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/081a.png"><img style="width:100%" src="images/081a.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p><a name="14_e_v"></a> 14. <i>If a right line do stand upon a right
+  line, it maketh the angles on each side equall to two right angles: and
+  contrariwise out of the 13. and</i> 14. <i>p j</i>.</p>
+
+  <p>For two such angles doe occupy or fill the same place that two right
+  angles doe: Therefore <!-- Page 61 --><span class="pagenum"><a
+  name="page61"></a>[61]</span>they are equall to them by the 11. e j. If
+  the insisting line be perpendicular unto that underneath it, it then
+  shall make 2. right angles, by the <a href="#13_e_v">13. e</a>. If it bee
+  not perpendicular, &amp; do make two oblique angles, as here <i>aio.</i>
+  and <i>oie.</i> are yet shall they occupy the same place that two right
+  angles doe: And therefore they are equall to two right angles, by the
+  same.</p>
+
+  <p>The converse is forced by an argument <i>ab impossibli</i>, or <i>ab
+  absurdo</i>, from the absurdity which otherwise would follow of it: For
+  the part must otherwise needes bee equall to the whole. Let therefore the
+  insisting or standing line which maketh two angles <i>aeo.</i> and
+  <i>aeu.</i> on each side equall to two right angles, be <i>ae.</i> I say
+  that <i>oe.</i> and <i>ei.</i> are but one right line. Otherwise let
+  <i>oe.</i> bee continued unto <i>u.</i> by the <a href="#6_e_v">6. e</a>.
+  Now by the <a href="#14_e_v">14. e.</a> or next former element,
+  <i>aeo.</i> &amp; <i>aeu.</i> are equall to two right angles; To which
+  also <i>oea.</i> &amp; <i>aei.</i> are equall by the grant: Let
+  <i>aeo.</i> the common angle be taken away: then shall there be left
+  <i>aeu.</i> equall to <i>aei.</i> the part to the whole, which is absurd
+  and impossible. Herehence is it certaine that the two right lines
+  <i>oe,</i> and <i>ei,</i> are in deede but one continuall right line.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/082a.png"><img style="width:100%" src="images/082a.png"
+      alt="Right Lines Cutting." title="Right Lines Cutting." /></a>
+  </div>
+  <p><a name="15_e_v"></a> 15. <i>If two right lines doe cut one another,
+  they doe make the angles at the top equall and all equall to foure right
+  angles. 15. p j</i>.</p>
+
+  <p><i>Anguli ad verticem</i>, Angles at the top or head, are called
+  Verticall angles which have their toppes meeting in the same point. The
+  Demonstration is: Because the lines cutting one another, are either
+  perpendiculars, and then all <!-- Page 62 --><span class="pagenum"><a
+  name="page62"></a>[62]</span>right angles are equall as heere: Or else
+  they are oblique, and then also are the verticalls equall, as are
+  <i>aui</i>, and <i>oue</i>: And againe, <i>auo</i>, and <i>iue</i>. Now
+  <i>aui</i>, and <i>oue</i>, are equall, because by the <a
+  href="#14_e_v">14. e.</a> with <i>auo</i>, the common angle, they are
+  equall to two right angles: And therefore they are equall betweene
+  themselves. Wherefore <i>auo</i>, the said common angle beeing taken
+  away, they are equall one to another.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="16_e_v"></a> 16. <i>If two right lines cut with one right
+  line, doe make the inner angles on the same side greater then two right
+  angles, those on the other side against them shall be lesser then two
+  right angles</i>.</p>
+
+  <p><span class="correction" title="text reads `A'">As</span> here, if
+  <i>auy</i>, and <i>uyi</i>, bee greater then two right angles <i>euy</i>,
+  and <i>uyo</i>, shall bee lesser then two right angles.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/082b.png"><img style="width:100%" src="images/082b.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p><a name="17_e_v"></a> 17. <i>If from a point assigned of an infinite
+  right line given, two equall parts be on each side cut off: and then from
+  the points of those sections two equall circles doe meete, a right line
+  drawne from their meeting unto the point assigned, shall bee
+  perpendicular unto the line given. 11. p j</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/083a.png"><img style="width:100%" src="images/083a.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+  <p>As let <i>a</i>, be the point assigned of the infinite line given: and
+  from that on each side, by the <a href="#7_e_v">7. e.</a> cut off equall
+  <!-- Page 63 --><span class="pagenum"><a
+  name="page63"></a>[63]</span>portions <i>ae</i>, and <i>ai</i>, Then let
+  two equall peripheries from the points <i>e</i>, and <i>i</i>, meete, as
+  in <i>o</i>, I say that a right line drawne from <i>o</i>, the point of
+  the meeting of the peripheries. unto <i>a.</i> the point given, shalbe
+  perpendicular upon the line given. For drawing the right lines <i>oe</i>,
+  &amp; <i>oi</i>, the two angles <i>eao</i>, and <i>iao</i>, on each side,
+  equicrurall by the construction of equall segments on each side, and
+  <i>oa</i>, the common side, are equall in base by the <a href="#9_e_v">9.
+  e</a>. And therefore the angles themselves shall be equall, by the <a
+  href="#7_e_iij">7. e iij</a>. and therefore againe, seeing that
+  <i>ao</i>, doth lie equall betweene the parts <i>ea</i>, and <i>ia</i>,
+  it is by the <a href="#13_e_ij">13. e ij</a>. perpendicular upon it.</p>
+
+  <p><a name="18_e_v"></a> 18. <i>If a part of an infinite right line, bee
+  by a periphery for a point given without, cut off a right line from the
+  said point, cutting in two the said part, shall bee perpendicular upon
+  the line given. 12. p j</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/083b.png"><img style="width:100%" src="images/083b.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p>Of an infinite right line given, let the part cut off by a periphery
+  of an externall center be <i>ae</i>: And then let <i>io</i>, cut the said
+  part into two parts by the <a href="#12_e_v">12. e</a>. I say that
+  <i>io</i> is perpendicular unto the said infinite right line. For it
+  standeth upright, and maketh <i>aoi</i>, and <i>eoi</i>, equall angles,
+  for the same cause, whereby the next former perpendicular was
+  demonstrated.</p>
+
+  <p><a name="19_e_v"></a> 19. <i>If two right lines drawne at length in
+  the same plaine doe never meete, they are parallells. � 35. d j.</i></p>
+
+  <div class="figleft" style="width:13%;">
+      <a href="images/084a.png"><img style="width:100%" src="images/084a.png"
+      alt="Parallel lines." title="Parallel lines." /></a>
+  </div>
+  <p>Thus much of the Perpendicularity of plaine right lines:
+  <i>Parallelissmus</i>, or their parallell equality doth follow. <!-- Page
+  64 --><span class="pagenum"><a name="page64"></a>[64]</span><i>Euclid</i>
+  did justly require these lines so drawne to be granted paralels: for then
+  shall they be alwayes equally distant, as here <i>ae.</i> and
+  <i>io</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="20_e_v"></a> 20. <i>If an infinite right line doe cut one of
+  the infinite right parallell lines, it shall also cut the other</i>.</p>
+
+  <p>As in the same example <i>uy.</i> cutting <i>ae.</i> it shall also cut
+  <i>io.</i> Otherwise, if it should not cut it, it should be parallell
+  unto it, by the <a href="#18_e_v">18 e</a>. And that against the
+  grant.</p>
+
+  <p><a name="21_e_v"></a> 21. <i>If right lines cut with a right line be
+  pararellells, they doe make the inner angles on the same side equall to
+  two right angles: And also the alterne angles equall betweene themselves:
+  And the outter, to the inner opposite to it: And contrariwise,</i> 29,
+  28, 27. p 1.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/084b.png"><img style="width:100%" src="images/084b.png"
+      alt="Perpendicular to Parallel lines." title="Perpendicular to Parallel lines." /></a>
+  </div>
+  <p>The paralillesme, or parallell-equality of right lines cut with a
+  right line, concludeth a threefold equality of angles: And the same is
+  againe of each of them concluded. Therefore in this one element there are
+  sixe things taught; all which are manifest if a perpendicular, doe fall
+  <!-- Page 65 --><span class="pagenum"><a
+  name="page65"></a>[65]</span>upon two parallell lines. The first sort of
+  angles are in their owne words plainely enough expressed. But the word
+  <i>Alternum</i>, alterne [or <i>alternate</i>, <i>H.</i>] here, as
+  <i>Proclus</i> saith, signifieth situation, which in Arithmeticke
+  signified proportion, when the antecedent was compared to the consequent;
+  notwithstanding the metaphor answereth fitly. For as an acute angle is
+  unto his successively following obtuse; So on the other part is the acute
+  unto his successively following obtuse: Therefore alternly, As the acute
+  unto the acute: so is the obtuse, unto the obtuse. But the outter and
+  inner are opposite, of the which the one is without the parallels; the
+  other is within on the same part not successively; but upon the same
+  right line the third from the outer.</p>
+
+  <p>The cause of this threefold propriety is from the perpendicular or
+  plumb-line, which falling upon the parallells breedeth and discovereth
+  all this variety: As here they are right angles which are the inner on
+  the same part or side: Item, the alterne angles: Item the inner and the
+  outter: And therefore they are equall, both, I meane, the two inner to
+  two right angles: and the alterne angles between themselvs: And the
+  outter to the inner opposite to it.</p>
+
+  <p>If so be that the cutting line be oblique, that is, fall not upon them
+  plumbe or perpendicularly, the same shall on the contrary befall the
+  parallels. For by that same obliquation or slanting, the right lines
+  remaining and the angles unaltered, in like manner both one of the inner,
+  to wit, <i>euy</i>, is made obtuse, the other, to wit, <i>uyo</i>, is
+  made acute: And the alterne angles are made acute and obtuse: As also the
+  outter and inner opposite are likewise made acute and obtuse.</p>
+
+  <p>If any man shall notwithstanding say, That the inner angles are
+  unequall to two right angles: By the same argument may he say (saith
+  <i>Ptolome</i> in <i>Proclus</i>) That on each side they be both greater
+  than two right angles, and also lesser: As in the parallel right lines
+  <i>ae</i> and <i>io</i>, cut with <!-- Page 66 --><span
+  class="pagenum"><a name="page66"></a>[66]</span>the right line <i>uy</i>,
+  if thou shalt say that <i>auy</i> and <i>iyu</i>, are greater then two
+  right angles, the angles on the other side, by the <a href="#16_e_v">16
+  e</a>, shall be lesser then two right angles, which selfesame
+  notwithstanding are also, by the gainesayers graunt, greater then two
+  right angles, which is impossible.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/086a.png"><img style="width:100%" src="images/086a.png"
+      alt="Line crossing Parallel lines." title="Line crossing Parallel lines." /></a>
+  </div>
+  <p>The same impossibility shall be concluded, if they shall be sayd, to
+  be lesser than two right angles.</p>
+
+  <p>The second and third parts may be concluded out of the first. The
+  second is thus: Twise two angles are equall to two right angles
+  <i>oyu</i>, and <i>euy</i>, by the former part: Item, <i>auy</i>, and
+  <i>euy</i>, by the <a href="#14_e_v">14 e</a>. Therefore they are equall
+  betweene themselves. Now from the equall, Take away <i>euy</i>, the
+  common angle, And the remainders, the alterne angles, at <i>u</i>, and
+  <i>y</i> shall be least equall.</p>
+
+  <p>The third is thus: The angles <i>euy</i>, and <i>oys</i>, are equall
+  to the same <i>uyi</i>, by the second propriety, and by the <a
+  href="#15_e_v">15 e</a>. Therefore they are equall betweene
+  themselves.</p>
+
+  <p>The converse of the first is here also the more manifest by that light
+  of the common perpendicular, And if any man shall thinke, That although
+  the two inner angles be equall to two right angles, yet the right may
+  meete, as if those equall angles were right angles, as here; it must
+  needes be that two right lines divided by a common perpendicular, should
+  both leane, the one this way, the other that way, or at least one of
+  them, contrary to the <a href="#13_e_ij">13 e ij</a>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/086b.png"><img style="width:100%" src="images/086b.png"
+      alt="Converse cases." title="Converse cases." /></a>
+  </div>
+  <p>If they be oblique angles, as here, the lines one slanting or <!--
+  Page 67 --><span class="pagenum"><a
+  name="page67"></a>[67]</span>obliquely crossing one another, the angles
+  on one side will grow lesse, on the other side greater. Therefore they
+  would not be equall to two right angles, against the graunt.</p>
+
+  <p>From hence the second and third parts may be concluded. The second is
+  thus: The alterne angles at <i>u</i> and <i>y</i>, are equall to the
+  foresayd inner angles, by the 14 e: Because both of them are equall to
+  the two right angles: And so by the first part the second is
+  concluded.</p>
+
+  <p>The third is therefore by the second demonstrated, because the outter
+  <i>oys</i>, is equall to the verticall or opposite angle at the top, by
+  the <a href="#15_e_v">15 e</a>. Therefore seeing the outter and inner
+  opposite are equall, the alterne also are equall.</p>
+
+  <p>Wherefore as <i>Parallelismus</i>, parallell-equality argueth a
+  three-fold equality of angles: So the threefold equality of angles doth
+  argue the same parallel-equality.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <p><a name="22_e_v"></a> 22. <i>If right lines knit together with a right
+  line, doe make the inner angles on the same side lesser than two right
+  Angles, they being on that side drawne out at length, will meete</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/087.png"><img style="width:100%" src="images/087.png"
+      alt="Lines that will meet." title="Lines that will meet." /></a>
+  </div>
+  <p>As here <i>ae</i>, and <i>io</i>, knit together with <i>eo</i>, doe
+  make two angles <i>aeo</i>, and <i>ioe</i>, lesser than two right angles:
+  They shall therefore, I say, meete if they be continued out that wayward.
+  The assumption and complexion is out of the <a href="#21_e_v">21 e</a>,
+  of right lines in the same plaine. If right lines cut with a right line
+  be parallels, they doe make the inner angles on the same part equall to
+  two right-angles. Therefore if they doe not make them equall, but lesser,
+  they shall not be parallel, but shall meete.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 68 --><span class="pagenum"><a name="page68"></a>[68]</span></p>
+
+  <p><a name="23_e_v"></a> 23. <i>A right line knitting together parallell
+  right lines, is in the same plaine with them.</i> 7 <i>p xj</i>.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/088a.png"><img style="width:100%" src="images/088a.png"
+      alt="Line knitting together parallell right lines." title="Line knitting together parallell right lines." /></a>
+  </div>
+  <p>As here <i>uy</i>, knitting or joyning together the two parallels
+  <i>ae</i>, and <i>io</i>, is in the same plaine with them as is manifest
+  by the <a href="#8_e_v">8 e</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/088b.png"><img style="width:100%" src="images/088b.png"
+      alt="Alternate angles." title="Alternate angles." /></a>
+  </div>
+  <p><a name="24_e_v"></a> 24. <i>If a right line from a point given doe
+  with a right line given make an angle, the other shanke of the angle
+  equalled and alterne to the angle made, shall be parallell unto the
+  assigned right line.</i> 31 <i>p j</i>.</p>
+
+  <p>As let the assigned right line be <i>ae</i>: And the point given, let
+  it be <i>i</i>. From which the right line, making with the assigned
+  <i>ae</i>, the angle, <i>ioe</i>, let it be <i>io</i>: To the which at
+  <i>i</i>, let the alterne angle <i>oiu,</i> be made equall: The right
+  line <i>ui</i>, which is the other shanke, is parallel to the assigned
+  <i>ae</i>.</p>
+
+  <p>An angle, I confesse, may bee made equall by the first propriety: And
+  so indeed commonly the Architects and Carpenters doe make it, by erecting
+  of a perpendicular. It may also againe in like manner be made by the
+  outter angle: Any man may at his pleasure use which hee shall thinke
+  good: But that here taught we take to be the best.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/089a.png"><img style="width:100%" src="images/089a.png"
+      alt="Parallel shanks." title="Parallel shanks." /></a>
+  </div>
+  <p><a name="25_e_v"></a> 25. <i>The angles of shanks alternly parallell,
+  are equall.</i> Or Thus, <i>The angles whose alternate feete are
+  parallells, are equall. H</i>.</p>
+
+  <p>This consectary is drawne out of the third property of <!-- Page 69
+  --><span class="pagenum"><a name="page69"></a>[69]</span>the <a
+  href="#21_e_v">21 e</a>. The thing manifest in the example following, by
+  drawing out, or continuing the other shanke of the inner angle. But
+  <i>Lazarus Schonerus</i> it seemeth doth thinke the adverbe
+  <i>alterne</i>, (<i>alternely</i> or <i>alternately</i>) to be more then
+  needeth: And therefore he delivereth it thus: The angles of parallel
+  shankes are equall.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="26_e_v"></a> 26. <i>If parallels doe bound parallels, the
+  opposite lines are equall �</i> 34 <i>p. j.</i> Or thus: <i>If parallels
+  doe inclose parallels, the opposite parallels are equall. H</i>.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/089b.png"><img style="width:100%" src="images/089b.png"
+      alt="Parallels bounding parallels." title="Parallels bounding parallels." /></a>
+  </div>
+  <p>Otherwise they should not be parallell. This is understood by the
+  perpendiculars, knitting them together, which by the definition are
+  equall betweene two parallells: And if of perpendiculars they bee made
+  oblique, they shall notwithstanding remaine equall, onely the corners
+  will be changed.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="27_e_v"></a> 27. <i>If right lines doe joyntly bound on the
+  same side equall and parallell lines, they are also equall and
+  parallell</i>.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/090b.png"><img style="width:100%" src="images/090b.png"
+      alt="Not parallel bounds." title="Not parallel bounds." /></a>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/090a.png"><img style="width:100%" src="images/090a.png"
+      alt="Parallel bounds." title="Parallel bounds." /></a>
+  </div>
+  <p>This element might have beene concluded out of the next precedent: But
+  it may also be learned out of those <!-- Page 70 --><span
+  class="pagenum"><a name="page70"></a>[70]</span>which went before. As let
+  <i>ae</i>, and <i>io</i>, equall parallels be bounded joyntly of
+  <i>ai</i>, and <i>eo</i>: and let <i>ei</i> be drawn. Here because the
+  right line <i>ei</i> falleth upon the parallels <i>ae</i>, and <i>io</i>,
+  the alterne angles <i>aei</i> and <i>eio</i>, are equall, by the <a
+  href="#21_e_v">21 e</a>. And they are equall in shankes <i>ae</i>, and
+  <i>io</i>, by the grants, and <i>ei</i>, is the common shanke: Therefore
+  they are also equall in base <i>ai</i>, and <i>eo</i>, by the <a
+  href="#7_e_iij">7 e iij</a>. This is the first: Then by <a
+  href="#21_e_v">21 e</a>, the alterne angles <i>eia</i>, and <i>ieo</i>,
+  are equall betweene themselves: And those are made by <i>ai</i> and
+  <i>eo</i>, cut by the right line <i>ei</i>: Therefore they are parallell;
+  which was the second.</p>
+
+  <p>On the same part or side it is sayd, least any man might understand
+  right lines knit together by opposite bounds as here.</p>
+
+  <p><a name="28_e_v"></a> 28. <i>If right lines be cut joyntly by many
+  parallell right lines, the segments betweene those lines shall bee
+  proportionall one to another, out of the</i> 2 <i>p vj and</i> 17 <i>p
+  xj</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/090c.png"><img style="width:100%" src="images/090c.png"
+      alt="First case: perpendiculars." title="First case: perpendiculars." /></a>
+  </div>
+  <p>Thus much of the Perpendicle, and parallell equality of plaine right
+  lines: Their Proportion is the last thing to be considered of them.</p>
+
+  <p>The truth of this element dependeth upon the nature of the parallells:
+  And that throughout all kindes of equality and inequality, both greater
+  and lesser. For if the lines thus cut be perpendiculars, the portions
+  <!-- Page 71 --><span class="pagenum"><a
+  name="page71"></a>[71]</span>intercepted betweene the two parallels shall
+  be equall: for common perpendiculars doe make parallell equality, as
+  before hath beene taught, and here thou seest.</p>
+
+  <p>If the lines cut be not parallels, but doe leane one toward another,
+  the portions cut or intercepted betweene them will not be equall, yet
+  shall they be proportionall one to another. And looke how much greater
+  the line thus cut is: so much greater shall the intersegments or portions
+  intercepted be. And contrariwise, Looke how much lesse: so much lesser
+  shall they be.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/091.png"><img style="width:100%" src="images/091.png"
+      alt="Cases of non-perpendiculars." title="Cases of non-perpendiculars." /></a>
+  </div>
+  <p>The third parallell in the toppe is not expressed, yet must it be
+  understood.</p>
+
+  <p>This element is very fruitfull: For from hence doe arise and issue,
+  First the manner of cutting a line according to any rate or proportion
+  assigned: And then the invention or way to finde out both the third and
+  fourth proportionalls.</p>
+
+  <p><a name="29_e_v"></a> 29. <i>If a right line making an angle with
+  another right line, be cut according to any reason [or proportion]
+  assigned, parallels drawne from the ends of the segments, unto the end of
+  the sayd right line given and unto some contingent point in the same,
+  shall cut the line given according to the reason given</i>.</p>
+
+  <p><i>Schoner</i> hath altered this Consectary, and delivereth it <!--
+  Page 72 --><span class="pagenum"><a name="page72"></a>[72]</span>thus:
+  <i>If a right <span class="correction" title="word missing in text"
+  >line</span> making an angle with a right line given, and knit unto it
+  with a base, be cut according to any rate assigned, a parallell to the
+  base from the ends of the segments, shall cut the line given according to
+  the rate assigned.</i> 9 and 10 p vj.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/092a.png"><img style="width:100%" src="images/092a.png"
+      alt="Division into two parts." title="Division into two parts." /></a>
+  </div>
+  <p><i>Punctum contingens</i>, A contingent point, that is falling or
+  lighting in some place at al adventurs, not given or assigned.</p>
+
+  <p>This is a marvelous generall consectary, serving indifferently for any
+  manner of section of a right line, whether it be to be cut into two
+  parts, or three parts, or into as many <span class="correction"
+  title="text reads `patts'">parts</span>, as you shall thinke good, or
+  generally after what manner of way soever thou shalt command or desire a
+  line to be cut or divided.</p>
+
+  <p>Let the assigned Right line to be cut into two equall parts be
+  <i>ae</i>. And the right line making an angle with it, let it be the
+  infinite right line <i>ai.</i> Let <i>ao</i>, one portion thereof be cut
+  off. And then by the <a href="#7_e_v">7 e</a>, let <i>oi</i>, another
+  part thereof be taken equall to it. And lastly, by the <a
+  href="#24_e_v">24 e</a>, draw parallels from the points <i>i</i>, and
+  <i>o</i>, unto <i>e</i>, the end of the line given, and to <i>u</i>; a
+  contingent point therein. Now the third parallell is understood by the
+  point <i>a</i>, neither is it necessary that it should be expressed.
+  Therefore the line <i>ae</i>, by the <a href="#28_e_v">28</a>, is cut
+  into two equall portions: And as <i>ao</i>, is to <i>oi</i>: So is
+  <i>au</i>, to <i>ue</i>. But <i>ao</i>, and <i>oi</i>, are halfe parts.
+  Therefore <i>au</i>, and <i>ue</i>, are also halfe parts.</p>
+
+  <p>And here also is the <a href="#12_e_v">12 e</a> comprehended, although
+  not in the same kinde of argument, yet in effect the same. But that
+  argument was indeed shorter, although this be more generall.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/092b.png"><img style="width:100%" src="images/092b.png"
+      alt="Division into three parts." title="Division into three parts." /></a>
+  </div>
+  <p>Now let <i>ae</i> be cut into three parts, of which the first let it
+  bee <!-- Page 73 --><span class="pagenum"><a
+  name="page73"></a>[73]</span>the halfe of the second: And the second, the
+  halfe of the third: And the conterminall or right line making an angle
+  with the sayd assigned line, let it be cut one part <i>ao</i>: Then
+  double this in <i>ou</i>: Lastly let <i>ui</i> be taken double to
+  <i>ou</i>, and let the whole diagramme be made up with three parallels
+  <i>ie</i>, <i>uy</i>, and <i>os</i>, The fourth parallell in the toppe,
+  as afore-sayd, shall be understood. Therefore that section which was made
+  in the conterminall line, by the <a href="#28_e_v">28 e</a>, shall be in
+  the assigned line: Because the segments or portions intercepted are
+  betweene the parallels.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="30_e_v"></a> 30. <i>If two right lines given, making an
+  angle, be continued, the first equally to the second, the second
+  infinitly, parallels drawne from the ends of the first continuation, unto
+  the beginning of the second, and some contingent point in the same, shall
+  intercept betweene them the third proportionall. 11. p vj</i>.</p>
+
+  <div class="figleft" style="width:15%;">
+      <a href="images/093.png"><img style="width:100%" src="images/093.png"
+      alt="Third proportional." title="Third proportional." /></a>
+  </div>
+  <p>Let the right lines given, making an angle, be <i>ae</i>, and
+  <i>ai</i>: and <i>ae</i>, the first, let it be continued equally to the
+  same <i>ai</i>, and the same <i>ai</i>, let it be drawne out infinitly:
+  Then the parallels <i>ei</i>, and <i>ou</i>, drawne from the ends of the
+  first continuation, unto <i>i</i>, the beginning of the second: and
+  <i>u</i>, a contingent point in the second, doe cut off <i>iu</i>, the
+  third proportionall sought. For by the <a href="#28_e_v">28 e</a>, as
+  <i>ae</i>, is unto <i>eo</i>, so is <i>ai</i>, unto <i>iu</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+  <p><a name="31_e_v"></a> 31. <i>If of three right lines given, the first
+  and the third making an angle be continued, the first equally to the
+  second, and the third infinitly; parallels drawne <!-- Page 74 --><span
+  class="pagenum"><a name="page74"></a>[74]</span>from the ends of the
+  first continuation, unto the beginning of the second, and some contingent
+  point, the same shall intercept betweene them the fourth proportionall.
+  12. p vj</i>.</p>
+
+  <p>Let the lines given be these: The first <i>ae</i>, the second
+  <i>ei</i>, the third <i>ao</i>, and let the whole diagramme be made up
+  according to the prescript of the consectary. Here by <a
+  href="#28_e_v">28. e</a>, as <i>ae</i>, is to <i>ei</i> so is <i>ao</i>,
+  to <i>ou</i>. Thus farre <i>Ramus</i>.</p>
+
+  <p><i>Lazarus Schonerus</i>, who, about some 25. yeares since, did revise
+  and augment this worke of our Authour, hath not onely altered the forme
+  of these two next precedent consectaries: but he hath also changed their
+  order, and that which is here the second, is in his edition the third:
+  and the third here, is in him the second. And to the former declaration
+  of them, hee addeth these words: From hence, having three lines given, is
+  the invention of the fourth proportionall; and out of that, having two
+  lines given, ariseth the invention of the third proportionall.</p>
+
+  <p>2 <i>Having three right lines given, if the first and the third making
+  an angle, and knit together with a base, be continued, the first equally
+  to the second; the third infinitly; a parallel from the end of the
+  second, unto the continuation of the third, shall intercept the fourth
+  proportionall. 12. p vj</i>.</p>
+
+  <p>The Diagramme, and demonstration is the same with our <a
+  href="#31_e_v">31. e</a> or 3 c of <i>Ramus</i>.</p>
+
+  <p>3 <i>If two right lines given making an angle, and knit together with
+  a base, be continued, the first equally to the second, the second
+  infinitly; a parallell to the base from the end of the first continuation
+  unto the second, shall intercept the third proportionall. 11. p
+  vj</i>.</p>
+
+  <p>The Diagramme here also, and demonstration is in all <!-- Page 75
+  --><span class="pagenum"><a name="page75"></a>[75]</span>respects the
+  same with our <a href="#30_e_v">30 e</a>, or 2 c of <i>Ramus</i>.</p>
+
+  <p>Thus farre <i>Ramus</i>: And here by the judgement of the learned
+  <i>Finkius</i>, two elements of <i>Ptolomey</i> are to be adjoyned.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/095.png"><img style="width:100%" src="images/095.png"
+      alt="Parallels proportional to segments." title="Parallels proportional to segments." /></a>
+  </div>
+  <p><a name="32_e_v"></a> 32 <i>If two right lines cutting one another, be
+  againe cut with many parallels, the parallels are proportionall unto
+  their next segments</i>.</p>
+
+  <p>It is a consectary out of the <a href="#28_e_v">28 e</a>. For let the
+  right lines <i>ae.</i> and <i>ai</i>, cut one another at <i>a</i>, and
+  let two parallell lines <i>uo</i>, and <i>ei</i>, cut them; I say, as
+  <i>au</i>, is to <i>uo</i>, so <i>ae</i>, is to <i>ei</i>. For from the
+  end <i>i</i>, let <i>is</i>, be erected parallell to <i>ae</i>, and let
+  <i>uo</i>, be drawne out untill it doe meete with it. Then from the end
+  <i>s</i>, let <i>sy</i>, be made parallell to <i>ai</i>: and lastly, let
+  <i>ea</i>, be drawne out, untill it doe meete with it. Here now
+  <i>ay</i>, shall be equall to the right line <i>is</i>, that is, by the
+  <a href="#26_e_v">26. e</a>, to <i>ue</i>: and at length, by the <a
+  href="#28_e_v">28. e</a>, as <i>ua</i>, is to <i>uo</i>; so is <i>ay</i>,
+  that is, <i>ue</i>, to <i>os</i>. Therefore, by composition or addition
+  of <span class="correction" title="text reads `ptoportions'"
+  >proportions</span>, as <i>ua</i>, is unto <i>uo</i>, so <i>ua</i>, and
+  <i>ue</i>, shall be unto <i>uo</i>, and <i>os</i>, that is, <i>ei</i>, by
+  the <a href="#27_e_v">27. e</a>.</p>
+
+  <p>The same <span class="correction" title="text reads `demonstation'"
+  >demonstration</span> shall serve, if the lines do crosse one another, or
+  doe vertically cut one another, as in the same diagramme appeareth. For
+  if the assigned <i>ai</i>, and <i>us</i>, doe cut one another vertically
+  in <i>o</i>, let them be cut with the parallels <i>au</i>, and <i>si</i>:
+  the precedent fabricke or figure being made up, it shall be by <a
+  href="#28_e_v">28. e.</a> as <i>au</i>, is unto <i>ao</i>, the segment
+  next unto it: so <i>ay</i>, that is, <i>is</i>, shall be unto <i>oi</i>,
+  his next segment.</p>
+
+  <p>The <a href="#28_e_v">28. e</a> teacheth how to finde out the third
+  and fourth proportionall: This affordeth us a meanes how to find out <!--
+  Page 76 --><span class="pagenum"><a name="page76"></a>[76]</span>the
+  continually meane proportionall single or double.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p><span class="correction" title="text reads `Thefore'">Therefore</span></p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/096.png"><img style="width:100%" src="images/096.png"
+      alt="Squire." title="Squire." /></a>
+  </div>
+  <p><a name="33_e_v"></a> 33. <i>If two right lines given be continued
+  into one, a perpendicular from the point of continuation unto the angle
+  of the squire, including the continued line with the continuation, is the
+  meane proportionall betweene the two right lines given</i>.</p>
+
+  <p>A squire (<i>Norma</i>, <i>Gnomon</i>, or <i>Canon</i>) is an
+  instrument consisting of two shankes, including a right angle. Of this we
+  heard before at the <a href="#13_e_v">13. e</a>. By the meanes of this a
+  meane proportionall unto two lines given is easily found: whereupon it
+  may also be called a <i>Mesolabium</i>, or <i>Mesographus simplex</i>, or
+  single meane finder.</p>
+
+  <p>Let the two right lines given, be <i>ae</i>, and <i>ei</i>. The meane
+  proportional between these two is desired. For the finding of which, let
+  it be granted that as <i>ae</i>, is to <i>eo</i>, so <i>eo</i>, is to
+  <i>ei</i>: therefore let <i>ae</i>, be continued or drawne out unto
+  <i>i</i>, so that <i>ei</i>, be equall to the other given. Then from
+  <i>e</i>, the point of the continuation, let <i>eo</i>, an infinite
+  perpendicular be erected. Now about this perpendicular, up and downe,
+  this way and that way, let the squire <i>ao</i>, be moved, so that with
+  his angle it may comprehend at <i>eo</i>, and with his shanks it may
+  include the whole right line <i>ai</i>. I say that <i>eo</i>, the segment
+  of the perpendicular, is the meane proportionall between <!-- Page 77
+  --><span class="pagenum"><a name="page77"></a>[77]</span><i>ae</i>, and
+  <i>ei</i>, the two lines given. For let <i>ea</i>, be continued or drawne
+  out into <i>u</i>, so that the continuation <i>au</i>, be equall unto
+  <i>eo</i>: and unto <i>a</i>, the point of the continuation, let the
+  angle <i>uas</i>, be made equall, and equicrurall to the angle
+  <i>oei</i>, that is, let the shanke <i>as</i>, be made equall to the
+  shanke <i>ei</i>. Wherefore knitting <i>u</i>, and <i>s</i>, together,
+  the right lines <i>us</i>, and <i>oi</i>, shall be equall; and the angles
+  <i>eoi</i>, <i>aus</i>, by the <a href="#7_e_iij">7. e iij</a>. And by
+  the <a href="#21_e_v">21. e</a>, the lines <i>sa</i>, and <i>oe</i>, are
+  parallell: and the angle <i>sao</i>, is equall to the angle <i>aoe</i>.
+  But the angles <i>sae</i>, and <i>aoi</i>, are right angles by the
+  Fabricke and by the grant; and therefore they are equall, by the <a
+  href="#14_e_iij">14. e iij</a>. Wherefore the other angles <i>oae</i>,
+  and <i>eoi</i>, that is, <i>sua</i>, are equall. And therefore by the <a
+  href="#21_e_v">21. e.</a> <i>us</i>, and <i>ao</i> are parallell; and
+  <i>us</i>, and <i>eo</i>, continued shall meete, as here in <i>y</i>: and
+  by the <a href="#26_e_v">26. e.</a> <i>oy</i>, and <i>as</i> are equall.
+  Now, by the <a href="#32_e_v">32. e.</a> as <i>ue</i>, is to <i>ua</i>,
+  so is <i>ey</i>, to <i>as</i>. Therefore by subduction or subtraction of
+  proportions, as <i>ea</i>, is to <i>ua</i>, so is <i>eo</i>, that is,
+  <i>ua</i>, to <i>oy</i>, that is <i>as</i>.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/097.png"><img style="width:100%" src="images/097.png"
+      alt="Figure for demonstration 33." title="Figure for demonstration 33." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:25%;">
+      <a href="images/098.png"><img style="width:100%" src="images/098.png"
+      alt="Plato's Mesographus in use." title="Plato's Mesographus in use." /></a>
+  </div>
+  <p><a name="34_e_v"></a> 34 <i>If two assigned right lines joyned
+  together by their ends rightanglewise, be continued vertically; a square
+  falling with one of his shankes, and another to it parallell and moveable
+  upon the ends of the assigned, with the angles upon the continued lines,
+  shall cut betweene them from the continued two meanes continually
+  proportionall to the assigned</i>.</p>
+
+  <p>The former consectary was of a single mesolabium; this is of a double,
+  whose use in making of solids, to this or that bignesse desired is
+  notable.</p>
+
+  <p>Let the two lines assigned be <i>ae</i>, and <i>ei</i>; and let there
+  be two meane right lines, continually proportionall betweene them sought,
+  to wit, that may be as <i>ae</i>, is unto <!-- Page 78 --><span
+  class="pagenum"><a name="page78"></a>[78]</span>one of the lines found;
+  so the same may be unto the second line found. And as that is unto this,
+  so this may be unto <i>ei</i>. Let therefore <i>ae</i>, and <i>ei</i>, be
+  joyned rightanglewise by their ends at <i>e</i>; and let them be infinite
+  continued, but vertically, that is, from that their meeting from the
+  lines ward, from <i>ei</i>, towards <i>u</i>, but <i>ae</i>, towards
+  <i>o</i>. Now for the rest, the construction; it was <i>Plato's
+  Mesographus</i>; to wit, a squire with the opposits parallell. One of his
+  sides <i>au</i>, moueable, or to be done up and downe, by an hollow
+  riglet in the side adjoyning. Therefore thou shalt make thee a
+  Mesographus, if unto the squire thou doe adde one moveable side, but so
+  that how so ever it be moved, it be still parallell unto the opposite
+  side [which is nothing else, but as it were a double squire, if this
+  squire be applied unto it; and indeed what is done by this instrument,
+  may also be done by two squires, as hereafter shall be shewed.] And so
+  long and oft must the moveable side be moved up and downe, untill with
+  the opposite side it containe or touch the ends of the assigned, but the
+  angles must fall precisely upon the continued lines: The right lines from
+  the point of the continuation, unto the corners of the squire, are the
+  two meane proportionals sought.</p>
+
+  <p>As if of the Mesographus <i>auoi</i>, the moveable side be <i>au</i>;
+  <!-- Page 79 --><span class="pagenum"><a
+  name="page79"></a>[79]</span>thus thou shalt move up and downe, untill
+  the angles <i>u</i>, and <i>o</i>, doe hit just upon the infinite lines;
+  and joyntly at the same instant <i>ua</i>, and <i>oi</i>, may touch the
+  ends of the assigned <i>a</i>, and <i>i</i>. By the former consectary it
+  shall be as <i>ei</i>, is to <i>eo</i>, so <i>eo</i>, shall be unto
+  <i>eu</i>: and as <i>eo</i>, is to <i>eu</i>, so shall <i>eu</i>, be unto
+  <i>ea</i>.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/099a.png"><img style="width:100%" src="images/099a.png"
+      alt="Plato's Mesographus." title="Plato's Mesographus." /></a>
+  </div>
+  <p>And thus wee have the composition and use, both of the single and
+  double Mesolabium.</p>
+
+  <p><a name="35_e_v"></a> 35. <i>If of foure right lines, two doe make an
+  angle, the other reflected or turned backe upon themselves, from the ends
+  of these, doe cut the former; the reason of the one unto his owne
+  segment, or of the segments betweene themselves, is made of the reason of
+  the so joyntly bounded, that the first of the makers be joyntly bounded
+  with the beginning of the antecedent made; the second of this consequent
+  joyntly bounded with the end; doe end in the end of the consequent
+  made</i>.</p>
+
+  <p><i>Ptolomey</i> hath two speciall examples of this <i>Theorem</i>: to
+  those <i>Theon</i> addeth other foure.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/099b.png"><img style="width:100%" src="images/099b.png"
+      alt="Figure for demonstration 35." title="Figure for demonstration 35." /></a>
+  </div>
+  <p>Let therefore the two right lines be <i>ae</i>, and <i>ai</i>: and
+  from the ends of these other two reflected, be <i>iu</i>, and <i>eo</i>,
+  cutting themselves in <i>y</i>; and the two former in <i>u</i>, and
+  <i>o</i>. The reason of the particular right lines made shall be as <!--
+  Page 80 --><span class="pagenum"><a name="page80"></a>[80]</span>the
+  draught following doth manifest. In which the antecedents of the makers
+  are in the upper place: the consequents are set under neathe their owne
+  antecedents.</p>
+
+<table class="nobctr">
+<tr><td colspan="4" align="center">The I. is <i>Ptolemeys</i> and <i>Theons</i> I.</td></tr>
+
+<tr><td colspan="2" align="center"><i>The makers:</i></td><td colspan="2" align="center"><i>The reason made.</i></td></tr>
+<tr><td align="center"><i>iu</i>,</td><td align="center"><i>ye</i>.</td></tr>
+<tr><td align="center"><i>uy</i>,</td><td align="center"><i>eo</i>,</td><td align="center"><i>ia</i>,</td><td align="center"><i>ao</i>.</td></tr>
+
+<tr><td colspan="4" align="center" class="tpb">The II. is <i>Theons</i> VI.</td></tr>
+
+<tr><td align="center"><i>au</i>,</td><td align="center"><i>ey</i>.</td></tr>
+<tr><td align="center"><i>ue</i>,</td><td align="center"><i>yo</i>,</td><td align="center"><i>ai</i>,</td><td align="center"><i>io</i>.</td></tr>
+
+<tr><td colspan="4" align="center" class="tpb">The III. is <i>Theons</i> III.</td></tr>
+
+<tr><td align="center"><i>ea</i>,</td><td align="center"><i>ui</i>.</td></tr>
+<tr><td align="center"><i>au</i>,</td><td align="center"><i>iy</i>,</td><td align="center"><i>eo</i>,</td><td align="center"><i>oy</i>.</td></tr>
+
+<tr><td colspan="4" align="center" class="tpb">The IIII. is <i>Theons</i> II.</td></tr>
+
+<tr><td align="center"><i>oa</i>,</td><td align="center"><i>iu</i>.</td></tr>
+<tr><td align="center"><i>ai</i>,</td><td align="center"><i>uy</i>,</td><td align="center"><i>oe</i>,</td><td align="center"><i>ey</i>.</td></tr>
+
+<tr><td colspan="4" align="center" class="tpb">The V. is <i>Ptolemys</i>, II. <i>Theons</i> IIII.</td></tr>
+
+<tr><td align="center"><i>iy</i>,</td><td align="center"><i>ue</i>.</td></tr>
+<tr><td align="center"><i>yu</i>,</td><td align="center"><i>ea</i>,</td><td align="center"><i>io</i>,</td><td align="center"><i>ao</i>.</td></tr>
+
+<tr><td colspan="4" align="center" class="tpb">The VI. is <i>Theons</i> V.</td></tr>
+
+<tr><td align="center"><i>eu</i>,</td><td align="center"><i>ai</i>.</td></tr>
+<tr><td align="center"><i>ua</i>,</td><td align="center"><i>io</i>,</td><td align="center"><i>ey</i>,</td><td align="center"><i>yo</i>.</td></tr>
+
+</table>
+
+  <p>The businesse is the same in the two other, whether you doe crosse the
+  bounds or invert them.</p>
+
+  <p>Here for demonstrations sake we crave no more, but that from the
+  beginning of an antecedent made a parallell be drawne to the second
+  consequent of the makers, unto one of the assigned infinitely continued:
+  then the multiplied proportions shall be,</p>
+
+  <p>The Antecedent, the Consequent; the Antecedent, the <!-- Page 81
+  --><span class="pagenum"><a name="page81"></a>[81]</span>Consequent of
+  the second of the makers; every way the reason or rate is of
+  Equallity.</p>
+
+  <p>The Antecedent; the Consequent of the first of the makers; the
+  Parallell; the Antecedent of the second of the makers, by the <a
+  href="#32_e_v">32. e</a>. Therefore by multiplication of proportions, the
+  reason of the Parallell, unto the Consequent of the second of the makers,
+  that is, by the fabricke or construction, and the <a href="#32_e_v">32.
+  e.</a> the reason of the Antecedent of the Product, unto the Consequent,
+  is made of the reason, &amp;c. after the manner above written.</p>
+
+  <div class="figright" style="width:25%;">
+      <a href="images/101.png"><img style="width:100%" src="images/101.png"
+      alt="Figure for several demonstrations in 35." title="Figure for several demonstrations in 35." /></a>
+  </div>
+  <p>For examples sake, let the first speciall example be demonstrated. I
+  say therefore, that the reason of <i>ia</i>, unto <i>ao</i>, is made of
+  the reason of <i>iu</i>, unto <i>uy</i>, multiplied by the reason of
+  <i>ye</i>, unto <i>eo</i>. For from the beginning of the Antecedent of
+  the product, to wit, from the point <i>i</i>, let a line be drawne
+  parallell to the right line <i>ey</i>, which shall meete with <i>ae</i>,
+  continued or drawne out infinitely in <i>n</i>. Therefore, by the <a
+  href="#32_e_v">32. e</a>, as <i>ia</i>, is to <i>ao</i>: so is the
+  parallell drawne to <i>eo</i>, the Consequent of the second of the
+  makers. Therefore now the multiplied proportions are thus <i>iu</i>,
+  <i>uy</i>, <i>in</i>, <i>ey</i>, by the 32. e: <i>ye</i>, <i>eo</i>,
+  <i>ey</i>, <i>eo</i>. Therefore as the product of <i>iu</i>, by
+  <i>ye</i>, is unto the product of <i>uy</i>, by <i>eo</i>: So <i>in</i>,
+  is to <i>eo</i>, that is, <i>ia</i>, to <i>ao</i>.</p>
+
+  <p>So let the second of <i>Ptolemy</i> to be taught, which in our <!--
+  Page 82 --><span class="pagenum"><a name="page82"></a>[82]</span>Table
+  aforegoing is the fifth. I say therefore that the reason of <i>io</i>,
+  unto <i>oa</i>; is made of the reason of <i>iy</i>, unto <i>yu</i>, and
+  the reason of <i>ue</i>, unto <i>ea</i>. For now againe, from the
+  beginning of the Antecedent of the Product <i>i</i>, let a line be drawne
+  parallell unto <i>ea</i>, the Consequent of the second of the Makers,
+  which shall meete with <i>eo</i>, drawne out at length, in <i>n</i>:
+  therefore, by the <a href="#32_e_v">32. e.</a> as <i>io</i>, is to
+  <i>ao</i>; so is <i><span class="correction" title="text reads `in'"
+  >en</span></i>, unto <i>ea</i>. Therefore now again the multiplied
+  proportions are thus:</p>
+
+<table class="nobctr">
+<tr><td class="spac"><i>ue</i>,</td><td class="spac"><i>ea</i>,</td><td class="spac"><i>ue</i>,</td><td class="spac"><i>ea</i>.</td></tr>
+<tr><td class="spac"><i>iy</i>,</td><td class="spac"><i>yu</i>,</td><td class="spac"><i><span class="correction" title="text reads `in'">en</span></i>,</td><td class="spac"><i>ue</i>;</td></tr>
+</table>
+
+  <p>by the <a href="#32_e_v">32. e</a>. Therefore, by multiplication of
+  proportions, the reason of <i><span class="correction" title="text reads `in'"
+  >en</span></i>, unto <i>ea</i>, that is, of <i>io</i>, unto <i>oa</i>, is
+  made of the reason of <i>iy</i>, unto <i>yu</i>, by the reason of
+  <i>ue</i>, unto <i>ea</i>.</p>
+
+  <p>It shall not be amisse to teach the same in the examples of
+  <i>Theon</i>. Let us take therefore the reason of the Reflex, unto the
+  Segment; And of the segments betweene themselves; to wit, the 4. and 6.
+  examples of our foresaid draught: I say therefore, that the reason of
+  <i>oe</i>, unto <i>ey</i>, is made of the reason <i>oa</i>, unto
+  <i>ai</i>, by the reason of <i>iu</i>, unto <i>uy</i>. For from the end
+  <i>o</i>, to wit, from the beginning of the Antecedent of the product,
+  let the right line <i>no</i>, be drawne parallell to <i>uy</i>. It shall
+  be by the <a href="#32_e_v">32. e.</a> as <i>oe</i>, is to <i>ey</i>: so
+  the parallell <i>no</i>, shall be to <i>uy</i>: but the reason of
+  <i>no</i>, unto <i>uy</i>, is made of the reason of <i>oa</i>, unto
+  <i>ai</i>, and of <i>iu</i>, unto <i>uy</i>: for the multiplied
+  proportions are,</p>
+
+<table class="nobctr">
+<tr><td class="spac"><i>iu</i>,</td><td class="spac"><i>uy</i>,</td><td class="spac"><i>iu</i>,</td><td class="spac"><i>uy</i>.</td></tr>
+<tr><td class="spac"><i>oa</i>,</td><td class="spac"><i>ai</i>,</td><td class="spac"><i><span class="correction" title="text reads `oe'">on</span></i>,</td><td class="spac"><i>iu</i>.</td></tr>
+</table>
+
+  <p>by the <a href="#32_e_v">32. e.</a></p>
+
+  <p>Againe, I say, that the reason of <i>ey</i>, unto <i>yo</i>, is
+  compounded of the reason of <i>eu</i>, unto <i>ua</i>, and of <i>ai</i>,
+  unto <i>io</i>.</p>
+
+  <p><i>Theon</i> here draweth a parallell from <i>o</i>, unto <i>ui</i>.
+  By the generall fabricke it may be drawne out of <i>e</i>, unto <i><span
+  class="correction" title="text reads `oi'">ui</span></i>.</p>
+
+  <p>It shall be therefore as <i>ey</i>, is unto <i>yo</i>, so <i>en</i>,
+  shall be unto <i>oi</i>. Now the proportions multiplied are,</p>
+
+<table class="nobctr">
+<tr><td class="spac"><i>ai</i>,</td><td class="spac"><i>io</i>,</td><td class="spac"><i>ai</i>,</td><td class="spac"><i>io</i>.</td></tr>
+<tr><td class="spac"><i>eu</i>,</td><td class="spac"><i>ua</i>,</td><td class="spac"><i>en</i>,</td><td class="spac"><i><span class="correction" title="text reads `ay'">ai</span></i>.</td></tr>
+</table>
+
+  <p>by the <a href="#32_e_v">32. e.</a></p>
+
+  <p>Therefore the reason of <i>en</i>, unto <i>io </i>, that is of
+  <i>ey</i>, unto <!-- Page 83 --><span class="pagenum"><a
+  name="page83"></a>[83]</span><i>yo</i>, shall be made of the foresaid
+  reasons.</p>
+
+  <p>Of the segments of divers right lines, the <i>Arabians</i> have much
+  under the name of <i>The rule of sixe quantities.</i> And the
+  <i>Theoremes</i> of <i>Althindus</i>, concerning this matter, are in many
+  mens hands. And <i>Regiomontanus</i> in his <i>Algorithmus</i>: and
+  <i>Maurolycus</i> upon the 1 p iij. of <i>Menelaus</i>, doe make mention
+  of them; but they containe nothing, which may not, by any man skillfull
+  in Arithmeticke, be performed by the multiplication of proportions. For
+  all those wayes of theirs are no more but speciall examples of that kinde
+  of multiplication.</p>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i>, the sixt Booke, of
+a Triangle.</h2>
+
+  <p><a name="1_e_vj"></a> 1. <i>Like plaines have a double reason of their
+  homologall sides, and one proportionall meane, out of 20 p vj. and xj.
+  and 18. p viij</i>.</p>
+
+  <p>Or thus; Like plaines have the proportion of their correspondent
+  proportionall sides doubled, &amp; one meane proportionall: Hitherto wee
+  have spoken of plaine lines and their affections: Plaine figures and
+  their kindes doe follow in the next place. And first, there is premised a
+  common corollary drawne out of the <a href="#24_e_iiij">24. e. iiij</a>.
+  because in plaines there are but two dimensions.</p>
+
+  <p><a name="2_e_vj"></a> 2. <i>A plaine surface is either rectilineall or
+  obliquelineall,</i> [<i>or rightlined, or crookedlined. H.</i>]</p>
+
+  <p>Straightnesse, and crookednesse, was the difference of lines at the <a
+  href="#4_e_ij">4. e. ij</a>. From thence is it here repeated and
+  attributed to a surface, which is geometrically made of lines. That made
+  of right lines, is rectileniall: that which is made of crooked lines, is
+  Obliquilineall. <!-- Page 84 --><span class="pagenum"><a
+  name="page84"></a>[84]</span></p>
+
+  <p><a name="3_e_vj"></a> 3. <i>A rectilineall surface, is that which is
+  comprehended of right lines</i>.</p>
+
+  <p>A plaine rightlined surface is that which is on all sides inclosed and
+  comprehended with right lines. And yet they are not alwayes right
+  betweene themselves, but such lines as doe lie equally betweene their
+  owne bounds, and without comparison are all and every one of them right
+  lines.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/106.png"><img style="width:100%" src="images/106.png"
+      alt="Polygons on rectilineall surface." title="Polygons on rectilineall surface." /></a>
+  </div>
+  <p><a name="4_e_vj"></a> 4. <i>A rightilineall doth make all his angles
+  equall to right angles; the inner ones generally to paires from two
+  forward: the outter always to foure</i>.</p>
+
+  <p>Or thus: A right lined plaine maketh his angles equall unto right
+  angles: Namely the inward angles generally, are equall unto the even
+  numbers from two forward, but the outward angles are equall but to 4.
+  right angles. <i>H</i>.</p>
+
+  <p>The first kinde I meane of rectilineals, that is a triangle doth make
+  all his inner angles equall to two right angles, that is, to a binary,
+  the first even number of right angles: the second, that is a quadrangle,
+  to the second even number, that is, to a quaternary or foure: The third,
+  that is, a Pentangle, of quinqueangle to the third, that is a senary of
+  right angles, or 6. and so farre forth as thou seest in this
+  Arithmeticall progression of even numbers,</p>
+
+<table class="nobctr">
+<tr><td class="spac">2.</td><td class="spac">4.</td><td class="spac">6.</td><td class="spac">8.</td><td class="spac">10.</td><td class="spac">12.</td></tr>
+<tr><td class="spac">3.</td><td class="spac">4.</td><td class="spac">5.</td><td class="spac">6.</td><td class="spac">7.</td><td class="spac">8.</td></tr>
+</table>
+
+<p><!-- Page 85 --><span class="pagenum"><a name="page85"></a>[85]</span></p>
+
+  <p>Notwithstanding the outter angles, every side continued and drawne
+  out, are alwayes equall to a quaternary of right angles, that is to
+  foure. The former part being granted (for that is not yet demonstrated)
+  the latter is from thence concluded: For of the inner angles, that of the
+  outter, is easily proved. For the three angles of a triangle are equall
+  to two right angles. The foure of a quadrangle to foure: of a
+  quinquangle, to sixe: of a sexe angle, to eight: Of septangle, to tenne,
+  and so forth, <span class="correction" title="text reads `from'"
+  >form</span> a binarie by even numbers: Whereupon, by the <a
+  href="#14_e_v">14. e. V</a>. a perpetuall quaternary of the outer angles
+  is concluded.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/107.png"><img style="width:100%" src="images/107.png"
+      alt="Outer angles." title="Outer angles." /></a>
+  </div>
+  <p><a name="5_e_vj"></a> 5. <i>A rectilineall is either a Triangle or a
+  Triangulate</i>.</p>
+
+  <p>As before of a line was made a lineate: so here in like manner of a
+  triangle is made a triangulate.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/108a.png"><img style="width:100%" src="images/108a.png"
+      alt="Triangle." title="Triangle." /></a>
+  </div>
+  <p><a name="6_e_vj"></a> 6. <i>A triangle is a rectilineall figure
+  comprehended of three rightlines. 21. d j</i>.</p>
+
+  <p>As here <i>aei.</i> A triangular figure is of <i>Euclide</i> defined
+  from the three sides; whereupon also it might be called
+  <i>Trilaterum</i>, that is three sided, of the cause: rather than
+  <i>Trianglum</i>, three cornered, of the effect; especially seeing that
+  three angles, and three sides <!-- Page 86 --><span class="pagenum"><a
+  name="page86"></a>[86]</span>are not reciprocall or to be converted. For
+  a triangle may have foure sides, as is <i>Acidoides</i>, or
+  <i>Cuspidatum</i>, the barbed forme, which <i>Zonodorus</i> called
+  <i>C&oelig;logonion</i>, or <i>Cavangulum</i>, an hollow cornered figure.
+  It may also have both five, and sixe sides, as here thou seest. The name
+  therefore of <i>Trilaterum</i> would more fully and fitly expresse the
+  thing named: But use hath received and entertained the name of a triangle
+  for a trilater: And therefore let it be still retained, but in that same
+  sense:</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/108b.png"><img style="width:100%" src="images/108b.png"
+      alt="Not triangles." title="Not triangles." /></a>
+  </div>
+  <p><a name="7_e_vj"></a> 7. <i>A triangle is the prime figure of
+  rectilineals</i>.</p>
+
+  <p>A triangle or threesided figure is the prime or most simple figure of
+  all rectilineals. For amongst rectilineall figures there is none of two
+  sides: For two right lines cannot inclose a figure. What is meant by a
+  prime figure, was taught at the <a href="#7_e_iiij"><span
+  class="correction" title="text reads `7. e. iiij'">11. e.
+  iiij</span></a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/109a.png"><img style="width:100%" src="images/109a.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p><a name="8_e_vj"></a> 8. <i>If an infinite right line doe cut the
+  angle of a triangle, it doth also cut the base of the same: Vitell.</i>
+  29. <i>t j</i>.</p>
+
+  <p><a name="9_e_vj"></a> 9. <i>Any two sides of a triangle are greater
+  than the other</i>.</p>
+
+  <p>Thus much of the difinition of a triangle; the reason or <!-- Page 87
+  --><span class="pagenum"><a name="page87"></a>[87]</span>rate in the
+  sides and angles of a triangle doth follow. The reason of the sides is
+  first.</p>
+
+  <p>Let the triangle be <i>aei</i>; I say, the side <i>ai</i>, is shorter,
+  than the two sides <i>ae</i>, and <i>ei</i>, because by the <a
+  href="#6_e_ij">6. e ij</a>, a right line is betweene the same bounds the
+  shortest.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><br style="clear : both" /></p>
+  <div class="figright" style="width:25%;">
+      <a href="images/109b.png"><img style="width:100%" src="images/109b.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p><a name="10_e_vj"></a> 10. <i>If of three right lines given, any two
+  of them be greater than the other, and peripheries described upon the
+  ends of the one, at the distances of the other two, shall meete, the
+  rayes from that meeting unto the said ends, shall make a triangle of the
+  lines given</i>.</p>
+
+  <p>Let it be desired that a triangle be made of these three lines,
+  <i>aei</i>, given, any two of them being greater than the other: First
+  let there be drawne an infinite right; From this let there be cut off
+  continually three portions, to wit, <i>ou</i>, <i>uy</i>, and <i>ys</i>,
+  equall to <i>ae</i>, and <i>i</i>, the three lines given. Then upon the
+  ends <i>y</i>, and <i>u</i>, at the distances <i>ou</i>, and <i>ys</i>;
+  let two peripheries meet in the point <i>r</i>. The rayes from that
+  meeting unto the said ends, <i>u</i>, and <i>y</i>, shall make the
+  triangle <i>ury</i>: for those rayes shall be equall to the right lines
+  given, by the <a href="#10_e_v">10. e v</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="11_e_vj"></a> 11. <i>If two equall peripheries, from the ends
+  of a right line given, and at his distance, doe meete, lines <!-- Page 88
+  --><span class="pagenum"><a name="page88"></a>[88]</span>drawne from the
+  meeting, unto the said ends, shall make an equilater triangle upon the
+  line given. 1 p. j</i>.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/110a.png"><img style="width:100%" src="images/110a.png"
+      alt="Figure for demonstration 11." title="Figure for demonstration 11." /></a>
+  </div>
+  <p>As here upon <i><span class="correction" title="text reads `ue'"
+  >ae</span></i>, there is made the equilater triangle, <i>aei</i>; And in
+  like manner may be framed the construction of an equicrurall triangle, by
+  a common ray, unequall unto the line given; and of a scalen or various
+  triangle, by three diverse raies; all which are set out here in this one
+  figure. But these specialls are contained in the generall probleme:
+  neither doe they declare or manifest unto us any new point of
+  Geometry.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/110b.png"><img style="width:100%" src="images/110b.png"
+      alt="Proportions in a triangle." title="Proportions in a triangle." /></a>
+  </div>
+  <p><a name="12_e_vj"></a> 12. <i>If a right line in a triangle be
+  parallell to the base, it doth cut the shankes proportionally: And
+  contrariwise. 2 p vj</i>.</p>
+
+  <p>Such therefore was the reason or rate of the sides in one triangle;
+  the proportion of the sides followeth.</p>
+
+  <p>As here in the triangle <i>aei</i>, let <i>ou</i>, be parallell to the
+  base; and let a third parallel be understood to be in the toppe <i>a</i>;
+  therefore, by the <a href="#28_e_v">28. e. v</a>. the intersegments are
+  proportionall.</p>
+
+  <p>The converse is forced out of <!-- Page 89 --><span class="pagenum"><a
+  name="page89"></a>[89]</span>the antecedent: because otherwise the whole
+  should be lesse than the part. For if <i>ou</i>, be not parallell to the
+  base <i>ei</i>, then <i>yu</i>, is: Here by the grant, and by the
+  antecedent, seeing <i>ao</i>, <i>oe</i>, <i>ay</i>, <i>ye</i>, are
+  proportionall: and the first <i>ao</i>, is lesser than <i>ay</i>, the
+  third: <i>oe</i>, the second must be lesser than <i>ye</i>, the fourth,
+  that is the whole then the part.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figright" style="width:20%;">
+      <a href="images/111b.png"><img style="width:100%" src="images/111b.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/111a.png"><img style="width:100%" src="images/111a.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><a name="13_e_vj"></a> 13. <i>The three angles of a triangle, are
+  equall to two right angles. 32. p j.</i></p>
+
+  <p>Hitherto therefore is declared the comparison in the sides of a
+  triangle. Now is declared the reason or rate in the angles, which joyntly
+  taken are equall to two right angles.</p>
+
+  <p>The truth of this proposition, saith <i>Proclus</i>, according to
+  common notions, appeareth by two perpendiculars erected upon the ends of
+  the base: for looke how much by the leaning of the inclination, is taken
+  from two right angles at the base, so much is assumed or taken in at the
+  toppe, and so by that requitall the equality of two right angles is made;
+  as in the triangle <i>aei</i>, let, by the <a href="#24_e_v">24. e v</a>,
+  <i>ou</i>, be parallell against <i>ie</i>. Here three particular angles,
+  <i>iao</i>, <i>iae</i>, <i>eau</i>, are equall to two right lines; by the
+  <a href="#14_e_v">14. e v</a>. But the inner angles are equall to the
+  same three: For first, <i>eai</i>, is equall to it selfe: Then the other
+  two are equall to their alterne angles, by the <a href="#24_e_v">24. e
+  v</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="14_e_vj"></a> 14. <i>Any two angles of a triangle are lesse
+  than two right angles</i>.</p>
+
+  <p>For if three angles be equall to two right angles, then <!-- Page 90
+  --><span class="pagenum"><a name="page90"></a>[90]</span>are two lesser
+  than two right angles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:18%;">
+      <a href="images/112a.png"><img style="width:100%" src="images/112a.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p><a name="15_e_vj"></a> 15. <i>The one side of any triangle being
+  continued or drawne out, the outter angle shall be equal to the two inner
+  opposite angles</i>.</p>
+
+  <p>This is the rate of the inner angles in one and the same triangle: The
+  rate of the outter with the inner opposite angles doth followe. As in the
+  triangle <i>aei</i>, let the side <i>ei</i>, be continued or drawne out
+  unto <i>o</i>; the two angles on each side <i>aio</i> and <i>aie</i>, are
+  by the <a href="#14_e_v">14 e v</a>. equall to two right angles: and the
+  three inner angles, are by the <a href="#13_e_vj">13. e.</a> equall also
+  to two right angles; take away <i>aie</i>, the common angle, and the
+  outter angle <i>aio</i>, shall be left equall to the other two inner and
+  opposite angles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="16_e_vj"></a> 16. <i>The said outter angle is greater than
+  either of the inner opposite angles. 16. p j</i>.</p>
+
+  <p>This is a consectary following necessarily upon the next former
+  consectary.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/112b.png"><img style="width:100%" src="images/112b.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+  <p><a name="17_e_vj"></a> 17. <i>If a triangle be equicrurall, the angles
+  at the base are equall: and contrariwise, 5. and 6. p. j</i>.</p>
+
+  <p>The antecedent is apparent by the <a href="#7_e_iij">7. e iij</a>. The
+  converse is apparent by an impossibilitie, which otherwise must needs
+  follow. For if any one shanke be greater than the other, as <i>ae</i>:
+  Then by the <a href="#7_e_v">7. e v</a>. let <i>oe</i>, be cut off <span
+  class="correction" title="unclear grammar, he means `equall to the other (ai)'"
+  >equall to it</span>: and let <i>oi</i>, be drawne: then by <a
+  href="#7_e_iij">7. e iij</a>. the base <i>oi</i>, must <!-- Page 91
+  --><span class="pagenum"><a name="page91"></a>[91]</span>be equal to the
+  base <i>ae</i>; but the base <i>oi</i>, is lesser than <i>ae</i>. For by
+  the <a href="#9_e_vj">9. e</a>, <i>ia</i>; and <i>ao</i>, (to which
+  <i>ae</i>, is equall, seeing that <i>oe</i>, is supposed to be equall to
+  the same <i>ai</i>: and <i><span class="correction" title="text reads `ae'"
+  >ao</span></i>, is common to both) are greater than the said <i>oi</i>;
+  therefore the same, <i>oi</i>, must be equall to the same <i>ae</i>, and
+  lesser than the same, which is impossible. This was first found out by
+  <i>Thales Milesius</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="18_e_vj"></a> 18. <i>If the equall shankes of a triangle be
+  continued or drawne out, the angles under the base shall be equall
+  betweene themselves</i>.</p>
+
+  <p>For the angles <i>aei</i>, and <i>ieo</i>: Item <i>aie</i>, and
+  <i>eiu</i>, are equall to two right angles, by the <a href="#14_e_v">14.
+  e v</a>. Therefore they are equall betweene themselves: wherefore if you
+  shall take away the inner angles, equall betweene themselves, you shall
+  leave the outter equall one to another.</p>
+
+  <div class="figcenter" style="width:18%;">
+      <a href="images/113a.png"><img style="width:100%" src="images/113a.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:18%;">
+      <a href="images/113b.png"><img style="width:100%" src="images/113b.png"
+      alt="Equilater." title="Equilater." /></a>
+  </div>
+  <p><a name="19_e_vj"></a> 19. <i>If a triangle be an equilater, it is
+  also an equiangle: And contrariwise</i>.</p>
+
+  <p>It is a consectary out of the condition of an equicrurall triangle of
+  two, both shankes and angles, as in the example <i>aei</i>, shall be
+  demonstrated.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="20_e_vj"></a> 20. <i>The angle of an equilater triangle doth
+  <!-- Page 92 --><span class="pagenum"><a
+  name="page92"></a>[92]</span>countervaile two third parts of a right
+  angle. Regio. 23. p j</i>.</p>
+
+  <p>For seeing that 3. angles are equall to 2. 1. must needs be equall to
+  2/3.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="21_e_vj"></a> 21. <i>Sixe equilater triangles doe fill a
+  place</i>.</p>
+
+  <p>As here. For 2/3. of a right angle sixe lines added together doe make
+  12/3. that is foure right angles; but foure right angles doe fill a place
+  by the <a href="#27_e_iiij">27. e. iiij</a>.</p>
+
+  <div class="figcenter" style="width:18%;">
+      <a href="images/114a.png"><img style="width:100%" src="images/114a.png"
+      alt="Figure for demonstration 21." title="Figure for demonstration 21." /></a>
+  </div>
+  <p><a name="22_e_vj"></a> 22. <i>The greatest side of a triangle
+  subtendeth the greatest angle; and the greatest angle is subtended of the
+  greatest side. 19. and 18. p j</i>.</p>
+
+  <div class="figright" style="width:12%;">
+      <a href="images/114b.png"><img style="width:100%" src="images/114b.png"
+      alt="Figure for demonstration 22." title="Figure for demonstration 22." /></a>
+  </div>
+  <p><i>Subtendere</i>, to draw or straine out something under another; and
+  in this place it signifieth nothing else but to make a line or such like,
+  the base of an angle, arch, or such like. And <i>subtendi</i>, is to
+  become or made the base of an angle, arch, of a circle, or such like: As
+  here, let <i>ai</i>, be a greater side than <i>ae</i>, I say the angle at
+  <i>e</i>, shall be greater than that at <i>i</i>. For let there be cut
+  off from <i>ai</i>, a portion equall to <i>ae</i>,; and let that be
+  <i>io</i>: then the angle <i>aei</i>, equicrurall to the angle
+  <i>oie</i>, shall be greater in base, by the grant. Therefore the angle
+  shall be greater, by the <a href="#9_e_iij">9 e iij</a>.</p>
+
+  <p>The converse is manifest by the same figure: As let the angle
+  <i>aei</i>, be greater than the angle <i>aie</i>. Therefore by the same,
+  <a href="#9_e_iij">9 e iij</a>. it is greater in base. For what is there
+  spoken <!-- Page 93 --><span class="pagenum"><a
+  name="page93"></a>[93]</span>of angles in generall, are here assumed
+  specially of the angles in a triangle.</p>
+
+  <p><a name="23_e_vj"></a> 23. <i>If a right line in a triangle, doe cut
+  the angle in two equall parts, it shall cut the base according to the
+  reason of the shankes; and contrariwise. 3. p vj</i>.</p>
+
+  <p>The mingled proportion of the sides and angles doth now remaine to be
+  handled in the last place.</p>
+
+  <p>Let the triangle be <i>aei</i>; and let the angle <i><span
+  class="correction" title="text reads `aei'">eai</span></i>, be cut into
+  two equall parts, by the right line <i>ao</i>: I say, as <i>ea</i>, is
+  unto <i>ai</i>, so <i>eo</i>, is unto <i>oi</i>. For at the angle
+  <i>i</i>, let the parallell <i>iu</i>, by the <a href="#24_e_v">24. e
+  v</a>. be erected against <i>ao</i>; and continue or draw out <i>ea</i>,
+  infinitly; and it shall by the <a href="#20_e_v">20. e v</a>. cut the
+  same <i>iu</i>, in some place or other. Let it therefore cut it in
+  <i>u</i>. Here, by the <a href="#28_e_v">28. e v</a>. as <i>ea</i>, is to
+  <i>au</i>, so is <i>eo</i>, to <i>oi</i>. But <i>au</i>, is equall to
+  <i>ai</i>, by the <a href="#17_e_vj">17. e</a>. For the angle <i>uia</i>,
+  is equall to the alterne angle <i>oai</i>, by the <a href="#21_e_v">21. e
+  v</a>. And by the grant it is equall to <i>oae</i>, his equall: And by
+  the <a href="#21_e_v">21. e v</a>. it is equall to the inner angle
+  <i>aui</i>; and by that which is concluded it is equall to <i>uia</i>,
+  his equall. Therefore by the <a href="#17_e_vj">17. e</a>, <i>au</i>, and
+  <i>ai</i>, are equall. Therefore as <i>ea</i>, is unto <i>ai</i>, so is
+  <i>eo</i>, unto <i>oi</i>.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/115.png"><img style="width:100%" src="images/115.png"
+      alt="Figure for demonstration 23." title="Figure for demonstration 23." /></a>
+  </div>
+  <p>The Converse likewise is demonstrated in the same figure. For as
+  <i>ea</i>, is to <i>ai</i>; so is <i>eo</i>, to <i>oi</i>: And so is
+  <i>ea</i>, to <i>au</i>, by the 12 e: therefore <i>ai</i>, and <i>au</i>,
+  are equall, Item the angles <i>eao</i>, and <i>oai</i>, are equall to the
+  angles at <i>u</i>, and <i>i</i>, by the <a href="#21_e_v">21. e v</a>.
+  which are equall betweene themselves by the <a href="#17_e_vj">17.
+  e</a>.</p>
+
+<hr class="full" />
+
+<p><!-- Page 94 --><span class="pagenum"><a name="page94"></a>[94]</span></p>
+
+<h2>Of Geometry, the seventh Booke,
+Of the comparison of Triangles.</h2>
+
+  <p><a name="1_e_vij"></a> 1. <i>Equilater triangles are equiangles. 8. p.
+  j</i>.</p>
+
+  <p>Thus forre of the Geometry, or affections and reason of one triangle;
+  the comparison of two triangles one with another doth follow. And first
+  of their rate or reason, out of their sides and angles: Whereupon
+  triangles betweene themselves are said to be equilaters and equiangles.
+  First out of the equality of the sides, is drawne also the equalitie of
+  the angles.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/116a.png"><img style="width:100%" src="images/116a.png"
+      alt="Equilater triangles." title="Equilater triangles." /></a>
+  </div>
+  <p>Triangles therefore are here jointly called equilaters, whose sides
+  are severally equall, the first to the first, the second, to the second,
+  the third to the third; although every severall triangle be
+  inequilaterall. Therefore the equality of the sides doth argue the
+  equality of the angles, by the <a href="#7_e_iij">7. e iij</a>. As
+  here.</p>
+
+  <p><a name="2_e_vij"></a> 2. <i>If two triangles be equall in angles,
+  either the two equicrurals, or two of equall either shanke, or base of
+  two angles, they are equilaters, 4. and 26. p j</i>.</p>
+
+  <p><span class="correction" title="text reads `Oh'">Or</span> thus; If
+  two triangles be equall in their angles, either <!-- Page 95 --><span
+  class="pagenum"><a name="page95"></a>[95]</span>in two angles contained
+  under equall feet, or in two angles, whose side or base of both is
+  equall, those angles are equilater. <i>H</i>.</p>
+
+  <p>This element hath three parts, or it doth conclude two triangles to be
+  equilaters three wayes. 1. The first part is apparent thus: Let the two
+  triangles be <i>aei</i>, and <i>ouy</i>; because the equall angles at
+  <i>a</i>, and <i>o</i>, are equicrurall, therefore they are equall in
+  base, by the <a href="#7_e_iij">7. e iij</a>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/116b.png"><img style="width:100%" src="images/116b.png"
+      alt="Equilater triangles: equicrural equal angles." title="Equilater triangles: equicrural equal angles." /></a>
+  </div>
+  <p>2 The second thus: Let the said two triangles <i>aei</i>, and
+  <i>ouy</i>, be equall in two angles a peece, at <i>e</i>, and <i>i</i>,
+  and at <i>u</i>, and <i>y</i>. And let them be equall in the shanke
+  <i>ei</i>, to <i>uy</i>. I say, they are equilaters. For if the side
+  <i>ae</i>, (for examples sake) be greater than the side <i>ou</i>, let
+  <i>es</i>, be cut off equall unto it; and draw the right line <i>is</i>.
+  Here by the antecedent, the triangles <i>sei</i>, and <i>ouy</i>, shall
+  be equiangles, and the angles <i>sie</i>, shall be equall to the angle
+  <i>oyu</i>, to which <!-- Page 96 --><span class="pagenum"><a
+  name="page96"></a>[96]</span>also the whole angle <i>aie</i>, is equall,
+  by the grant. Therefore the whole and the part are equall, which is
+  impossible. Wherefore the side <i>ae</i>, is not unequall but equall to
+  the side <i>ou</i>: And by the antecedent or former part, the triangles
+  <i>aei</i>, and <i>ouy</i>, being equicrurall, are equall, at the angle
+  of the shanks: Therefore also they are equall in their bases <i>ai</i>,
+  and <i>oy</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/117.png"><img style="width:100%" src="images/117.png"
+      alt="Equilater triangles: two equal angles and equal shank." title="Equilater triangles: two equal angles and equal shank." /></a>
+  </div>
+  <p>3 The third part is thus forced: In the triangles <i>aei</i>, and
+  <i>ouy</i>, let the angles at <i>e</i>, and <i>i</i>, and <i>u</i>, and
+  <i>y</i>, be equall, as afore: And <i>ae</i>, the base of the angle at
+  <i>i</i>, be equall to <i>ou</i>, the base of angle at <i>y</i>: I say
+  that the two triangles given are equilaters. For if the side <i>ei</i>,
+  be greater than the side <i>uy</i>, let <i>es</i>, be cut off equall to
+  it, and draw the right line <i>as</i>. Therefore by the antecedent, the
+  two triangles, <i>aes</i>, and <i>ouy</i>, equall in the angle of their
+  equall shankes are equiangle: And the angle <i>ase</i>, is equall to the
+  angle <i>oyu</i>, which is equall by the grant unto the angle <i>aie</i>.
+  Therefore <i>ase</i>, is equall to <i>aie</i>, the outter to the inner,
+  contrary to the <a href="#15_e_vj">15. e. vj</a>. Therefore the base
+  <i>ei</i>, is not unequall to the base <i>uy</i>, but equall. And
+  therefore as above was said, the two triangles <i>aei</i>, and
+  <i>ouy</i>, equall in the angle of their equall shankes, are
+  equilaters.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/116c.png"><img style="width:100%" src="images/116c.png"
+      alt="Equilater triangles: two equal angles and equal base (diagram moved to correct position: it was printed on the previous page)." title="Equilater triangles: two equal angles and equal base (diagram moved to correct position: it was printed on the previous page)." /></a>
+  </div>
+  <div class="figright" style="width:22%;">
+      <a href="images/118a.png"><img style="width:100%" src="images/118a.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p><a name="3_e_vij"></a> 3. <i>Triangles are equall in their three
+  angles</i>.</p>
+
+  <p>The reason is, because the three angles in any triangle are <!-- Page
+  97 --><span class="pagenum"><a name="page97"></a>[97]</span>equall to two
+  right angles, by the <a href="#13_e_vj">13. e vj</a>. As here, the
+  greatest triangle, all his corners joyntly taken, is equall to the
+  least.</p>
+
+  <p>And yet notwithstanding it is not therefore to be thought to be
+  equiangle to it: For Triangles are then equiangles, when the severall
+  angles of the one, are equall to the severall angles of the other: Not
+  when all joyntly are equall to all.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="4_e_vij"></a> 4. <i>If two angles of two triangles given be
+  equall, the other also are equall</i>.</p>
+
+  <p>All the three angles, are equall betweene themselves, by the <a
+  href="#3_e_vij">3 e</a>. Therefore if from equall you take away equall,
+  those which shall remaine shall be equall.</p>
+
+  <p><a name="5_e_vij"></a> 5. <i>If a right triangle equicrurall to a
+  triangle be greater in base, it is greater in angle: And contrariwise.
+  25. and 24. p j</i>.</p>
+
+  <p>Thus farre of the reason or rate of equality, in the sides and angles
+  of triangles: The reason of inequality, taken out of the common and
+  generall inequality of angles, doth <!-- Page 98 --><span
+  class="pagenum"><a name="page98"></a>[98]</span>follow. The first is
+  manifest, by the <a href="#9_e_iij">9. e iij</a>. as here thou seest in
+  <i>aei</i> and <i>ouy</i>.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/118b.png"><img style="width:100%" src="images/118b.png"
+      alt="Figure for demonstrations 5 and 6." title="Figure for demonstrations 5 and 6." /></a>
+  </div>
+  <p><a name="6_e_vij"></a> 6. <i>If a triangle placed upon the same base,
+  with another triangle, be lesser in the inner shankes, it is greater in
+  the angle of the shankes</i>.</p>
+
+  <p>This is a consectary drawne also out of the <a href="#10_e_iij">10 e
+  iij</a>. As here in the triangle <i>aei</i>, and <i>aoi</i>, within it
+  and upon the same base. Or thus: If a triangle placed upon the same base
+  with another triangle, be lesse then the other triangle, in regard of his
+  feet, (those feete being conteined within the feete of the other
+  triangle) in regard of the angle conteined under those feete, it is
+  greater: <i>H</i>.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/119.png"><img style="width:100%" src="images/119.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p><a name="7_e_vij"></a> 7. <i>Triangles of equall heighth, are one to
+  another as their bases are one to another</i>.</p>
+
+  <p>Thus farre of the Reason or rate of triangles: The proportion of
+  triangles doth follow; And first of a right line with the bases. It is a
+  consectary out of the <a href="#16_e_iiij">16 e iiij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="8_e_vij"></a> 8. <i>Upon an equall base, they are
+  equall</i>.</p>
+
+  <p>This was a generall consectary at the <a href="#16_e_iiij">16. e
+  iiij</a>: From whence <i>Archimedes</i> concluded, If a triangle of
+  equall heighth with many other triangles, have his base equall to the
+  bases of them all, it is equall to them all: as here thou seest
+  <i>aei</i> to be equall to the triangles <i>aeo</i>, <i>uoy</i>,
+  <i>syr</i>, <i>lrm</i>, <i>nmi</i>. Here hence also thou mayst conclude,
+  that <i>Equilater</i> triangles are equall: Because they are of equall
+  heighth, and upon the same base.</p>
+
+<p><!-- Page 99 --><span class="pagenum"><a name="page99"></a>[99]</span></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:23%;">
+      <a href="images/120a.png"><img style="width:100%" src="images/120a.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p><a name="9_e_vij"></a> 9. <i>If a right line drawne from the toppe of
+  a triangle, doe cut the base into two equall parts, it doth also cut the
+  triangle into two equall parts: and it is the diameter of the
+  triangle</i>.</p>
+
+  <p>As here thou seest: For the bisegments, or two equall portions thus
+  cut are two triangles of equall heighth <span class="correction"
+  title="text reads `that that'">that</span> is to say, they have one toppe
+  common to both, within the same parallels) and upon equall bases:
+  Therefore they are equall: And that right line shall be the diameter of
+  the triangle, by the <a href="#5_e_iiij">5 e iiij</a>, because it passeth
+  by the center.</p>
+
+  <p><a name="10_e_vij"></a> 10. <i>If a right line be drawne from the
+  toppe of a triangle, unto a point given in the base (so it be not in the
+  middest of it) and a parallell be drawne from the middest of the base
+  unto the side, a right line drawne from the toppe of the sayd parallell
+  unto the sayd point, shall cut the triangle into two equall
+  parts</i>.</p>
+
+  <p>Let the triangle given be <i>aei</i>: And let <i>ao</i>, cut the base
+  <i>ei</i>, in <i>o</i> unequally: And let <i>uy</i> be parallell from
+  <i>u</i>, the middest of <span class="correction" title="text reads `y' in italics"
+  >the</span> base, unto the sayd <span class="correction" title="text reads `ei' in italics"
+  ><i>ao</i></span>. I say that <i>yo</i> shall divide the triangle into
+  two equall portions. For let <i>au</i> be knit together with a right
+  line: That line, by the <a href="#9_e_vij">9 e</a>, shall divide the
+  triangle into two equall parts. Now the two triangles <i>ayu</i>, and
+  <i>you</i>, are equall by the 8 e; because they are of equall height, and
+  upon the same base.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/120b.png"><img style="width:100%" src="images/120b.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+<p><!-- Page 100 --><span class="pagenum"><a name="page100"></a>[100]</span></p>
+
+  <p>Take away <i>ysu</i>, the common triangle; And you shall leave
+  <i>asy</i>, and <i>osu</i>, equall betweene themselves: The common right
+  lined figure <i>ysui</i>, let it be added to both the sayd equall
+  triangles: And then <i>oyi</i>, shall be equall to <i>aui</i>, the halfe
+  part; And therefore <i>aeoy</i>, the other right lined figure, shall be
+  the halfe of the triangle given.</p>
+
+  <p><a name="11_e_vij"></a> 11. <i>If equiangled triangles be reciprocall
+  in the shankes of the equall angle, they are equall: And
+  contrariwise.</i> 15. <i>p. vj.</i> Or thus, <i>as the learned Mr.</i>
+  Brigges <i>hath conceived it: If two triangles, having one angle, are
+  reciprocall, &amp;c</i>.</p>
+
+  <div class="figcenter" style="width:48%;">
+      <a href="images/121.png"><img style="width:100%" src="images/121.png"
+      alt="Figure for demonstration 11. Labels u and i have been interchanged to match the text." title="Figure for demonstration 11. Labels u and i have been interchanged to match the text." /></a>
+  </div>
+  <p>Direct proportion in triangles, is such as hath in the former beene
+  taught: Reciprocall proportion followeth. It is a consectary drawne out
+  of the <a href="#18_e_iiij">18 e iiij</a>; which is manifest, as oft as
+  the equall angle is a right angle: For then those shankes, [comprehending
+  the equall angles,] are the heights and the bases; As here thou seest in
+  the severed triangles. Notwithstanding in obliquangle triangles, although
+  the shankes are not the heights, the cause of the truth hereof is the
+  same. Yet if any man shall desire a demonstration of it, it is thus: Let
+  therefore the diagramme or figure bee in the triangles <i>aei</i>, and
+  <i>aou</i>: And the angles <i>oau</i>, and <i>eai</i>, let them be
+  equall: And as <i>ua</i> is to <i>ae</i>, so let <i>ia</i> be unto
+  <i>ao</i>: I say that the triangles <i>aou</i>, and <i>eai</i>, are
+  equall. For <i>eo</i> being knit together with a right line, <i>uao</i>
+  is unto <i>oae</i>, as <i>ua</i> is unto <i>ae</i>, by the 7 e: <!-- Page
+  101 --><span class="pagenum"><a name="page101"></a>[101]</span>And
+  <i>ia</i>, unto <i>ao</i>, by the grant, is as <i>eai</i> is unto
+  <i>eao</i>. Therefore <i>uao</i>, and <i>eai</i>, are unto <i>eao</i>
+  proportionall: And therefore they are equall one to another.</p>
+
+  <p>The converse, is concluded by the same sorites, but by saying all
+  backward. For <i>ua</i> unto <i>ae</i> is, as <i>uao</i> is unto
+  <i>oae</i>, by the 7 e: And as <i>eai</i>, by the grant: Because they are
+  equall: And as <i>ia</i> is unto <i>ao</i>, by the same, Wherefore
+  <i>ua</i> is unto <i>ae</i>, as <i>ia</i> is unto <i>ao</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/122.png"><img style="width:100%" src="images/122.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p><a name="12_e_vij"></a> 12. <i>If two triangles be equiangles, they
+  are proportionall in shankes: And contrariwise: 4 and 5.</i> <i>p.
+  vj</i>.</p>
+
+  <p>The comparison both of the rate and proportion of triangles hath in
+  the former beene taught: Their similitude remaineth for the last place.
+  Which similitude of theirs consisteth indeed of the reason, or rate of
+  their angles and proportion of the shankes. Therefore for just cause was
+  the reason of the angles set first: Because from thence not onely their
+  reason, but also their latter proportion is gathered. Let <i>aei</i> and
+  <i>iou</i>, be two triangles equiangled: And let them be set upon the
+  same line <i>eiu</i>, meeting or touching one another in the common point
+  <i>i</i>. Then, seeing that the angles at <i>e</i> and <i>i</i>, are
+  granted to bee equall, the lines <i>oi</i>, and <i>ae</i>, are parallel,
+  by the <a href="#21_e_v">21 e v</a>. Therefore by the <a
+  href="#22_e_v">22 e v</a> <i>uo</i> and <i>ea</i>, being continued, shall
+  meete. Item, The right lines <i>ai</i>, and <i>yu</i>, by the <a
+  href="#21_e_v">21 e v</a>, are parallel, because the angle <i>aie</i> is
+  equall to <i>oui</i>, the inner opposite to it. Therefore seeing that
+  <i>ai</i> is parallell to the base <i>yu</i>, by the <a href="#21_e_v">21
+  e v</a>, <i>ea</i> shall be to <i>ay</i>, that is, by the <a
+  href="#26_e_v">26 e v</a>, to <i>io</i>, as <i>ei</i> is to <i>iu</i>:
+  And alternly, or crosse wayes, <i>ea</i> shall be to <i>ei</i>, as
+  <i>io</i> is to <i>iu</i>. This is the first proportion. Item, <!-- Page
+  102 --><span class="pagenum"><a name="page102"></a>[102]</span>seeing
+  that <i>io</i> is parallell to the base <i>ye</i>; <i>yo</i>, that is, by
+  the <a href="#26_e_v">26 e v</a>, <i>ai</i> shall bee unto <i>ou</i>, as
+  <i>ei</i>, is unto <i>iu</i>: And crosse wise, as <i>ai</i> is unto
+  <i>ie</i>, so is <i>ou</i> unto <i>ui</i>. This is the second
+  proposition. Lastly, equiordinately: <i>ae</i> is to <i>ai</i>, as
+  <i>oi</i> is to <i>ou</i>: wherefore if triangles be equiangled, they are
+  proportional in shankes.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/123.png"><img style="width:100%" src="images/123.png"
+      alt="Figure for the converse." title="Figure for the converse." /></a>
+  </div>
+  <p>This converse is thus demonstrated. Let there be two triangles
+  <i>aei</i>, and <i>ouy</i>, proportionall in shankes: And as <i>ae</i> is
+  to <i>ei</i>; so let <i>ou</i>, be to <i>uy</i>: And as <i>ai</i> is to
+  <i>ie</i>; so let <i>oy</i> bee to <i>yu</i>. Then at the points <i>u</i>
+  and <i>y</i>, let angles be made by the <a href="#11_e_iij">11 e iij</a>.
+  equall to the angles at <i>e</i> and <i>i</i>, and let the triangle
+  <i>uys</i>, be made: for the other angles at <i>a</i> and <i>s</i>, shall
+  be equall by the <a href="#4_e_vij">4 e</a>. And the triangle <i>yus</i>,
+  shall be equiangled to the assigned <i>aei</i>. And by the antecedent, it
+  shall be proportionall to it in shankes. Thus are two triangles
+  <i>ouy</i>, by the grant; and <i>uys</i>, by the construction,
+  proportionall in shanks to the same triangle <i>aei</i>: And as
+  <i>ae</i>, is to <i>ei</i>, so is <i>ou</i>, to <i>uy</i>; so is
+  <i>su</i>, to <i>uy</i>. Therefore seeing <i>ou</i> and <i>su</i>, are
+  proportionall to the same <i>yu</i>, they are equall; Item, as <i>ai</i>
+  is to <i>ie</i>: so is <i>oy</i> unto <i>yu</i>: so also is <i>sy</i>
+  unto <i>yu</i>. Therefore <i>oy</i> and <i>sy</i>, seeing they are
+  proportionall to the same <i>yu</i>, are equall. (<i>yu</i> is the common
+  side.) The triangle therefore <i>ouy</i>, is equilater unto the triangle
+  <i>syu</i>. And by the <a href="#1_e_vij">1 e</a>, it is to it equiangle:
+  And therefore it is equiangled to the triangle <i>aei</i>, which was to
+  be prooved. This was generally before taught at the <a
+  href="#20_e_iiij">20 e iiij</a>, of homologall sides subtending equall
+  angles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore,</p>
+    </div>
+  </div>
+  <div class="figleft" style="width:18%;">
+      <a href="images/124a.png"><img style="width:100%" src="images/124a.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><a name="13_e_vij"></a> 13. <i>If a right line in a triangle be
+  parallell to the base, it doth cut off from it a triangle equiangle to
+  the whole, but lesse in base.</i> <!-- Page 103 --><span
+  class="pagenum"><a name="page103"></a>[103]</span></p>
+
+  <p>As in the triangle <i>aei</i>, the right line <i>ou</i>, doth cut off
+  the triangle <i>aou</i>, equiangle, by the <a href="#21_e_v">21 e v</a>,
+  to the whole <i>aei</i>; But the base <i>ou</i>, is lesse than the base
+  <i>ei</i>, as appeareth by the <a href="#21_e_v">21 e</a>, and by the
+  alternation of the sides.</p>
+
+  <div class="figright" style="width:28%;">
+      <a href="images/124b.png"><img style="width:100%" src="images/124b.png"
+      alt="Equiangles." title="Equiangles." /></a>
+  </div>
+  <p><a name="14_e_vij"></a> 14. <i>If two trangles be proportionall in the
+  shankes of the equall angle, they are equiangles: 6 p vj</i>.</p>
+
+  <p>Let therefore the triangles given be <i>aei</i>, and <i>ouy</i>,
+  equall in their angles <i>a</i> and <i>o</i>: And in their shankes let
+  <i>ea</i>, be unto <i>ai</i>, as <i>ou</i> is to <i>oy</i>: And by the <a
+  href="#11_e_iij">11 e iij</a>, let the angles <i>soy</i>, and <i>oys</i>,
+  be equall to the angles <i>eai</i>, and <i>eia</i>: The other at <i>s</i>
+  and <i>e</i>, shall be equall, by the <a href="#4_e_vij">4 e</a>. Here
+  thou seest that the triangle <i>aei</i>, is equiangle unto <i>oys</i>.
+  Now, by the <a href="#12_e_vij">12 e.</a> as <i>ea</i> is to <i>ai</i>:
+  so is <i>so</i> to <i>oy</i>: and therefore, by the grant, so is
+  <i>uo</i> to <i>oy</i>. Therefore seeing that <i>uo</i>, and <i>os</i>,
+  are proportionall to <i>oy</i>, they are both equall. Lastly, if the
+  common shanke <i>oy</i> bee added to both the shankes <i>ou</i>, and
+  <i>oy</i>, are equall to the shankes <i>so</i> and <i>oy</i>. [But by the
+  construction the angles <i>oys</i> and <i>aie</i> are equall. And, by the
+  <a href="#4_e_vij">4 e</a>, the other at <i>s</i> and <i>e</i> are
+  equall. Therefore the first triangle <i>aei</i>, is made equiangled to
+  the third. Now seeing the second triangle <i>uoy</i> is to the third
+  <i>soy</i>, equall in the shanks of the equall angle, it is to the same
+  equilater, and by the <a href="#1_e_vij">1 e</a>, equiangled:
+  <i>Shon.</i>] Wherefore the second triangle <i>ouy</i> shall likewise be
+  equiangled to <i>osy</i>, the third: And therefore if <!-- Page 104
+  --><span class="pagenum"><a name="page104"></a>[104]</span>two triangles
+  proportionall in shankes be equall in the angle of their shankes, they
+  are equiangles.</p>
+
+  <p><a name="15_e_vij"></a> 15. <i>If triangles proportionall in shankes,
+  and alternly parallell, doe make an angle betweene them, their bases are
+  but one right line continued. 32 p. vj</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/125.png"><img style="width:100%" src="images/125.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>Or thus: If being proportionall in their feet, and alternately
+  parallels, they make an angle in the midst betweene them, they have their
+  bases continued in a right line: <i>H</i>.</p>
+
+  <p>The cause is out of the <a href="#14_e_v">14 e v</a>. For they shall
+  make on each side, with the falling line <i>ai</i>, two angles equall to
+  two right angles.</p>
+
+  <p>Let the triangles <i>aei</i> and <i>oiu</i>, be proportionall in
+  shanks: As <i>ae</i> is to <i>ai</i>, so let <i>io</i> be to <i>ou</i>:
+  And let <i>ea</i> bee parallel to <i>io</i>: And <i>ai</i> to <i>ou</i>:
+  Item, let them make the angle <i>aio</i>, betweene them, to wit, betweene
+  their middle shankes <i>ai</i>, and <i>oi</i>, I say their bases
+  <i>ei</i>, and <i>iu</i>, are but one right line continued. For seeing
+  that by the grant <i>ae</i>, and <i>oi</i>, are parallels: Item <i>ai</i>
+  and <i>uo</i>, the right line <i>ai</i> and <i>oi</i>, shall make, by the
+  <a href="#21_e_v">21 e v</a>, the angles at <i>a</i>, and <i>o</i>,
+  equall to the alterne angle <i>aio</i>: And therefore they are equall
+  betweene themselves: And then, by the <a href="#14_e_vij">14 e</a>, the
+  triangles given are equiangles: Therefore the angle <i>oui</i>, is equall
+  to the angle <i>aie</i>: Wherfore the three angles <i>oiu</i>,
+  <i>oia</i>, and <i>aie</i>, by the <a href="#3_e_vij">3 e</a>, are equall
+  to the three angles of the triangle <i>eai</i>, which are equall by the
+  <a href="#13_e_vj">13 e vj</a>. Unto two right angles: And therefore they
+  themselves also are equall to two right angles. Wherefore, by the <a
+  href="#14_e_v">14 e v</a>, <i>ei</i>, and <i>iu</i>, are one right line
+  continued. <!-- Page 105 --><span class="pagenum"><a
+  name="page105"></a>[105]</span></p>
+
+  <p><a name="16_e_vij"></a> 16. <i>If two triangles have one angle equall,
+  another proportionall in shankes, the third homogeneall, they are
+  equiangles. 7. p. vj</i>.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/126.png"><img style="width:100%" src="images/126.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p>Let <i>aei</i>, and <i>ouy</i>, the triangles given be equall in their
+  angles <i>a</i>, and <i>o</i>: and proportionall in the shankes of the
+  angles <i>e</i>, and <i>u</i>: and their other angles, at <i>i</i>, and
+  <i>y</i>, homogeneall, that is, let them be both, either acute, or
+  obtuse, or right angles. But first let them be acute, I say, the other at
+  <i>e</i>, &amp; <i>u</i>, are equall. Otherwise let <i>aes</i>, by the <a
+  href="#11_e_iij">11 e iij.</a> be made equall to the same <i>ouy</i>;
+  Then have you them by the <a href="#4_e_vij">4 e</a>, equiangles; and the
+  angles <i>ase</i>, shall be equall to the angle <i>oyu</i>; and both are
+  acute angles: and by the <a href="#12_e_vij">12. e</a>, <i>aes</i>, and
+  <i>ouy</i>, are proportionall in sides: and as <i>ae</i>, is to
+  <i>es</i>; so shall <i>ou</i>, be to <i>uy</i>, that is, by the grant, so
+  shall <i>ae</i>, be to <i>ei</i>. Therefore because the same <i>ea</i>,
+  hath unto two, to wit, <i>es</i>, and <i>ei</i>, the same reason, the
+  said <i>es</i>, and <i>ei</i>, are equall one to another: And therefore,
+  by the <a href="#17_e_vj">17. e. vj.</a> the angles at the base in
+  <i>s</i> and <i>i</i>, are equall. Therefore both of them are acute
+  angles: And in like manner <i>ase</i>, is an acute angle, contrary to the
+  <a href="#14_e_v">14. e v</a>. The same will fall out altogether like to
+  both the other, being either obtuse or right angles. The last part of a
+  right angle is manifest by the <a href="#4_e_vij">4 e</a> of this
+  Booke.</p>
+
+<hr class="full" />
+
+<p><!-- Page 106 --><span class="pagenum"><a name="page106"></a>[106]</span></p>
+
+<h2>Of Geometry the eight Booke,
+of the diverse kindes of Triangles.</h2>
+
+  <p><a name="1_e_viij"></a> 1. <i>A triangle is either right angled, or
+  obliquangled</i>.</p>
+
+  <p>The division of a triangle, taken from the angles, out of their common
+  differences, I meane, doth now follow. But here first a speciall
+  division, and that of great moment, as hereafter shall be in quadrangles
+  and prismes.</p>
+
+  <p><a name="2_e_viij"></a> 2. <i>A right angled triangle is that which
+  hath one right angle: An obliquangled is that which hath none. 27. d
+  j</i>.</p>
+
+  <p>A right angled triangle in Geometry is of speciall use and force; and
+  of the best Mathematicians it is called <i>Magister matheseos</i>, the
+  master of the Mathematickes.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/127.png"><img style="width:100%" src="images/127.png"
+      alt="Right angled triangle." title="Right angled triangle." /></a>
+  </div>
+  <p><a name="3_e_viij"></a> 3. <i>If two perpendicular lines be knit
+  together, they shall make a right angled triangle</i>.</p>
+
+  <p>As here in <i>aei</i>. This construction and manner of making of a
+  right angled triangle, is drawne out of the definition of a right angle.
+  For right lines perpendicular are the makers of a right angle, as is
+  manifest by the <a href="#13_e_iij">13. e iij</a>.</p>
+
+  <p><a name="4_e_viij"></a> 4. <i>If the angle of a triangle at the
+  base</i>, <i>be a right <!-- Page 107 --><span class="pagenum"><a
+  name="page107"></a>[107]</span>angle, a perpendicular from the toppe
+  shall be the other shanke:</i> [<i>and contrariwise Schon.</i>]</p>
+
+  <p>As is manifest in the same example.</p>
+
+  <p><a name="5_e_viij"></a> 5. <i>If a right angled triangle be
+  equicrurall, each of the angles at the base is the halfe of a right
+  angle: And contrariwise</i>.</p>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/128a.png"><img style="width:100%" src="images/128a.png"
+      alt="Equicrurall right angled triangle." title="Equicrurall right angled triangle." /></a>
+  </div>
+  <p>As in the triangle <i>aei</i>: For they are both equall to one right
+  angle, by the <a href="#13_e_vj">13. e. vj</a>. And betweene themselves,
+  by the <a href="#17_e_vj">17. e. vj</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="6_e_viij"></a> 6. <i>If one angle of a triangle be equall to
+  the other two, it is a right angle</i> [<i>And contrariwise
+  Schon.</i>]</p>
+
+  <p>Because it is equall to the halfe of two right angles, by the <a
+  href="#13_e_vj">13. e vj</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="7_e_viij"></a> 7. <i>If a right line from the toppe of a
+  triangle cutting the base into two equall parts be equall to the
+  bisegment, or halfe of the base, the angle at the toppe is a right
+  angle:</i> [<i>And contrariwise Schon.</i>]</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/128b.png"><img style="width:100%" src="images/128b.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>As in the triangle <i>aei</i>, the right line <i>ao</i>, cutting the
+  base <i>ei</i>, in <i>o</i>, into two equall parts, is equall to
+  <i>eo</i>, or <i>oi</i>, the halfe of the base maketh two equicrural
+  triangles; and the severall angles at the top equall to the angles at the
+  ends, <i>viz.</i> <i>e</i>, and <i>i</i>, by the <a href="#17_e_vj">17.
+  e. vj</a>. Therefore the angle at the toppe <!-- Page 108 --><span
+  class="pagenum"><a name="page108"></a>[108]</span>is equall to the other
+  two: wherefore by the <a href="#6_e_viij">6 e</a>, it is a right
+  angle.</p>
+
+  <p><a name="8_e_viij"></a> 8. <i>A perpendicular in a triangle from the
+  right angle to the base, doth cut it into two triangles, like unto the
+  whole and betweene themselves, 8. p vj.</i> [<i>And contrariwise
+  Schon.</i>]</p>
+
+  <p>As in the triangle <i>aei</i>, the perpendicular <i>ao</i>, doth cut
+  the triangles <i>aoe</i>, and <i>aoi</i>, like unto the whole <i>aei</i>,
+  because they are equiangles to it; seeing that the right angle on each
+  side is one, and another common in <i>i</i>, and <i>e</i>: Therefore the
+  other is equall to the remainder, by <a href="#4_e_vij">4. e vij</a>.
+  Wherefore the particular triangles are equiangles to the whole: As
+  proportionall in the shankes of the equall angles, by the <a
+  href="#12_e_vij">12. e vij</a>. But that they are like betweene
+  themselves it is manifest by the <a href="#22_e_iiij">22. e iiij</a>.</p>
+
+  <div class="figcenter" style="width:28%;">
+      <a href="images/129.png"><img style="width:100%" src="images/129.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="9_e_viij"></a> 9. <i>The perpendicular is the meane
+  proportionall betweene the segments or portions of the base</i>.</p>
+
+  <p>As in the said example, as <i>io</i>, is to <i>oa</i>: so is
+  <i>oa</i>, to <i>oe</i>, because the shankes of equall angles are
+  proportionall, by the <a href="#8_e_viij">8. e</a>. From hence was
+  <i>Platoes</i> Mesographus invented.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="10_e_viij"></a> 10. <i>Either of the shankes is proportionall
+  betweene the base, and the segment of the base next adjoyning</i>.</p>
+
+  <p>For as <i>ei</i>, is unto <i>ia</i>, in the whole triangle, so is
+  <i>ai</i>, to <i>io</i>, in the greater. For so they are homologall
+  sides, which <!-- Page 109 --><span class="pagenum"><a
+  name="page109"></a>[109]</span>doe subtend equall angles, by the <a
+  href="#23_e_iiij">23. e. iiij</a>. Item, as <i>ie</i>, is to <i>ea</i>;
+  in the whole triangle, so is <i>ae</i>, to <i>eo</i>, in the lesser
+  triangle.</p>
+
+  <p>Either of the shankes is proportionall betweene the summe, and the
+  difference of the base and the other shanke. And contrariwise. If one
+  side be proportionall betweene the summe and the difference of the
+  others, the triangle given is a rectangle. M. <i>H.</i>
+  <i>Brigges</i>.</p>
+
+  <p>This is a consectary arising likewise out of the <a href="#4_e_viij">4
+  e.</a> of very great use.</p>
+
+  <p>In the triangle <i>ead</i>, the shanke <i>ad</i>, 12. is the meane
+  proportionall betweene <i>bd</i>, 18. (the summe of the base <i>ae</i>,
+  13. and the shanke <i>ed</i>, 5.) and 8. the difference of the said base
+  and shanke: For if thou shalt draw the right lines <i>ba</i>, and
+  <i>ac</i>, the angle <i>bac</i>, shall be by the <a href="#6_e_viij">6.
+  e</a>, a rectangle; (because it is equall to the angles at <i>b</i>, and
+  <i>c</i>, seeing that the triangles <i>bea</i>, and <i>eac</i>, are
+  equicrurall.) And by the <a href="#9_e_viij">9 e</a>, <i>bd</i>,
+  <i>da</i>, and <i>dc</i>, are continually proportionall.</p>
+
+  <p><i>If a quadrate of a number, given for the first shanke, be divided
+  of another, the halfe of the difference of the divisour, and quotient
+  shall be the other shanke, and the halfe of the summe shall be the
+  base.</i> Or thus, <i>The side of divided number doubled, and the
+  difference of the divisour and quotient, shall be the two shankes, and
+  the summe of them shall be the base</i>.</p>
+
+  <p>Let the number given for the first shanke be 4. And let 8. divide 16.
+  the quadrate of 4. by 2. The halfe of 8 - 2, that is 3. shall be the
+  other shanke: And the halfe of 8 <span class="correction" title="Note: modern + and - signs have been substituted as required for the printed symbols, - - - or - - with no apparent distinction."
+  >+</span> 2, that is 5. shall be the base.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><i>If any one number shall divide the quadrate of another, the side of
+  the divided, and the halfe of the difference of the divisour and the
+  quotient, shall be the two shankes of a rectangled triangle, and the
+  halfe of the summe of them shall be the base thereof</i>.</p>
+
+  <p>Let the two numbers given be 4. and 6. The square of <!-- Page 110
+  --><span class="pagenum"><a name="page110"></a>[110]</span>6. let it be
+  36. and the quotient of 36. by 4. be 9: And the side <span
+  class="correction" title="text reads `it'">is</span> 6. for the one
+  shanke. Now 9 - 4. that is, 5. is the difference of the divisour and
+  quotient, whose halfe 2.�, is the other shanke. And 9 + 4. that is 13. is
+  the summe the said devisour and quotient, whose halfe 6.�, is the
+  base.</p>
+
+  <p>Againe let 4. and 8. be given. The quadrate of 8. is 64. And the <span
+  class="correction" title="text reads `quient'. It is the quotient of 64. by 4."
+  >quotient</span> of 64 is 16. and the side of 64. is 8. for the one
+  shanke. The halfe 16 - 4. that is 6. is the other shanke. And the halfe
+  of 16 + 4. that is 10, is the base.</p>
+
+  <p><a name="11_e_viij"></a> 11. <i>If the base of a triangle doe subtend
+  a right-angle, the rectilineall fitted to it, shall be equall to the like
+  rectilinealls in like manner fitted to the shankes thereof: And
+  contrariwise, out of the 31. p. vj</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/131.png"><img style="width:100%" src="images/131.png"
+      alt="Figure for demonstration 11. (Transcriber: should s be the rightangle in eis?)" title="Figure for demonstration 11. (Transcriber: should s be the rightangle in eis?)" /></a>
+  </div>
+  <p>Or thus: If the base of a triangle doe subtend a right angle, the
+  right lined figure made upon the base, is equall to the right lined
+  figures like, and in like manner situate upon the feete: <i>H</i>.</p>
+
+  <p>Let the right angled triangle be <i>aei</i>: and let there be also the
+  triangles <i>eau</i>, and <i>aiy</i>, and to them upon the base of the
+  said right angle, by the <a href="#23_e_iiij">23 e iiij</a>. let the
+  triangle <i>ies</i>, be made like, and in like manner situate. I say,
+  that <i>eis</i>, is equall joyntly to <i>eau</i>, and <i>aiy</i>. Let
+  <i>ao</i>, a perpendicular fall from the right angle <i>a</i>, to the
+  base <i>ei</i>: This by the <i>ioe</i>, doth yeeld us twise three
+  proportionals, to wit, <i>ie</i>, <i>ea</i>, <i>eo</i>: Item, <i>ei</i>,
+  <i>ia</i>, <i>io</i>: Therefore, by the <a href="#25_e_iiij">25. e.
+  iiij</a>, as <i>ie</i>, is to <i>eo</i>: so is the triangle <i>ies</i>,
+  to the triangle <i>eau</i>; And as <i>ei</i>, is to <i>oi</i>, so is the
+  triangle <i>eis</i>, to the triangle <i>aiy</i>: But <i>ei</i>, is equall
+  to <i>eo</i>, and <i>oi</i>, the whole, to wit, to his parts. Wherefore
+  by the second composition in <!-- Page 111 --><span class="pagenum"><a
+  name="page111"></a>[111]</span>Arithmeticke (9. c. ij.) the triangle
+  <i>eis</i>, is equall to the triangles <i>eau</i>, and <i>iay</i>.</p>
+
+  <p>The Converse is thus proved: Let the triangle be <i>aei</i>: And let
+  the perpendicular <i>eo</i>, be erected upon <i>ae</i>, equall to
+  <i>ei</i>: And draw a right line from <i>o</i> to <i>a</i>: Here by the
+  former, the rectilinealls situate at <i>oe</i>, and <i>ea</i>, that is by
+  the construction, at <i>ae</i>, and <i>ie</i>, are equall to the
+  rightilineall at <i>ao</i>, made alike and situate alike: And by the
+  graunt they are equall, to the rectilineall at <i>ai</i>, made alike and
+  situated alike. Therefore seeing the like rectilineals at <i>ao</i>, and
+  <i>ai</i>, are equall; they have by the <a href="#20_e_iiij">20 e
+  iiij</a>, their homologall sides equall: And the two triangles are
+  equiliters: And by the <a href="#1_e_vij">1 e vij</a>, equiangles. But
+  <i>aeo</i>, is a right angle, by the construction: And <i>aei</i>, is
+  proved to be equall to the same <i>aeo</i>: Therefore, by the <a
+  href="#13_e_v">13 e v</a>. <i>aei</i>, also is a right angle.</p>
+
+  <p><a name="12_e_viij"></a> 12. <i>An obliquangled triangle is either
+  Obtusangled or Acutangled</i>.</p>
+
+  <p>The division of an obliquangled triangle is taken from the speciall
+  differences of an oblique angle. For at the 15 e iij, we were taught that
+  an oblique angle was either obtuse or acute: Therefore an obliquangled
+  triangle is an obtuseangle, and an Acutangle.</p>
+
+  <p><a name="13_e_viij"></a> 13. <i>An obtusangle is that triangle which
+  hath one blunt corner.</i> 28.<i>d i</i>.</p>
+
+  <p>There can be but one right angle in a triangle, by the <a
+  href="#2_e_viij">2 e</a>. Therefore also in it there can be but one blunt
+  angle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="14_e_viij"></a> 14. <i>If the obtuse or blunt angle be at the
+  base of the triangle given, a perpendicular drawne from the toppe <!--
+  Page 112 --><span class="pagenum"><a name="page112"></a>[112]</span>of
+  the triangle, shall fall without the figure: And contrarywise</i>.</p>
+
+  <p>As here in <i>aei</i>, the perpendicular <i>io</i>, falleth without:
+  This is manifest by the <a href="#4_e_viij">4 e</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="15_e_viij"></a> 15. <i>If one angle of a triangle be greater
+  than both the other two, it is an obtuse angle: And contrariwise</i>.</p>
+
+  <p>This is plaine by the <a href="#6_e_viij">6 e</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:22%;">
+      <a href="images/133.png"><img style="width:100%" src="images/133.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p><a name="16_e_viij"></a> 16. <i>If a right line drawne from the toppe
+  of the triangle cutting the base into two equall parts, be lesse than one
+  of those halfes, the angle at the toppe is a blunt-angle. And
+  contrariwise</i>.</p>
+
+  <p>As in <i>aei</i>, the perpendicular <i>eo</i>, cutting the base
+  <i>ai</i> into two equall parts <i>ao</i>, and <i>oi</i>: And the said
+  <i>eo</i> is lesse than either <i>ao</i>, or <i>oi</i>: Therefore the
+  angle <i>aei</i>, is a blunt angle by the <a href="#7_e_viij">7
+  e</a>.</p>
+
+  <p><a name="17_e_viij"></a> 17. <i>An acutangled triangle is that which
+  hath all the angles acute. 29 d j</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="18_e_viij"></a> 18. <i>A perpendicular drawne from the top
+  falleth <span class="correction" title="text reads `without'"
+  >within</span> the figure: And contrariwise</i>.</p>
+
+  <p>As in <i>aei</i>, the perpendicular <i>ao</i> falleth <span
+  class="correction" title="text reads `without'">within</span> as is
+  plaine by the <a href="#4_e_viij">4 e</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 113 --><span class="pagenum"><a name="page113"></a>[113]</span></p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/134.png"><img style="width:100%" src="images/134.png"
+      alt="Figure for demonstration 20." title="Figure for demonstration 20." /></a>
+  </div>
+  <p><a name="19_e_viij"></a> 19. <i>If any one angle of triangle be lesse
+  then the other two, it is acute: And contrariwise</i>.</p>
+
+  <p>As is manifest by the <a href="#6_e_viij">6 e</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="20_e_viij"></a> 20. <i>If a right line drawne from the toppe
+  of the triangle; cutting the base into two equall parts, be <span
+  class="correction" title="text reads `lesse'">greater</span> than either
+  of those portions, the angle at the toppe is an acute angle: And
+  contrariwise</i>.</p>
+
+  <p>As in <i>aei</i>, let <i>ao</i> cutting the base <i>ei</i> into two
+  equall parts, be <span class="correction" title="text reads `lesse'"
+  >greater</span> than any one of those parts, the angle at the toppe is an
+  acuteangle, as <span class="correction" title="text reads `apreareth'"
+  >appeareth</span> by the <a href="#7_e_viij">7 e</a>.</p>
+
+  <p><br style="clear :both" /></p>
+<hr class="full" />
+
+<h2>The ninth Booke, of <i>P. Ramus</i>
+Geometry, which intreateth of
+the measuring of right lines
+by like right-angled
+<i>triangles</i>.</h2>
+
+  <p>The Geometry of like right-angled triangles, amongst many other uses
+  that it hath, it doth especially afford us the geod�sy or measuring of
+  right lines: And that mastery, which before (at the <a href="#2_e_viij">2
+  e viij</a>) attributed the right angled triangles, shall here be found to
+  be a true mastery indeed. <!-- Page 114 --><span class="pagenum"><a
+  name="page114"></a>[114]</span>For it shall containe the geodesy of right
+  lines; and afterward the geodesy of plaines and solides, by the measuring
+  of their sides, which are right lines.</p>
+
+  <p><a name="1_e_ix"></a> 1. <i>For the measuring of right lines; we will
+  use the</i> Iacobs <i>staffe, which is a squire of unequall
+  shankes</i>.</p>
+
+  <p><i>Radius</i>, commonly called <i>Baculus Iacob</i>, <i>Iacobs</i>
+  staffe, as if it had been long since invented and practised by that holy
+  Patriarke, is a very auncient instrument, and of all other Geometricall
+  instruments, commonly used, the best and fittest for this use.
+  <i>Archimedes</i> in his book of the Number of the sand, seemeth to
+  mention some such thing: And <i>Hipparchus</i>, with an instrument not
+  much unlike this, boldly attempted an haynous matter in the sight of God,
+  as <i>Pliny</i> thinketh, namely to deliver unto posterity the number of
+  the starres, and to assigne or fixe them in their true places by the
+  <i>Norma</i>, the squire or <i>Iacobs</i> staffe. And indeed true it is
+  that the Radius is not onely used for the measuring of the earth and
+  land: But especially for the defining or limiting of the starres in their
+  places and order: And for the describing and setting out of all the
+  regions and waies of the heavenly city. Yea and <i>Virgill</i> the famous
+  Poet, in his 3 <i>Ecloge</i>, <i>Ecquis fuit alter</i>, <i>Descripsit
+  radio totum</i>, <i>qui gentibus orbem?</i> and againe afterward in the 6
+  of his <i>Eneiades</i>, hath noted both these uses. <i>C&oelig;liqu�
+  meatus.</i> <i>Describent radio &amp; surgentia sidera dicent.</i> Long
+  after this the <i>Iewes</i> and <i>Arabians</i>, as <i>Rabbi Levi</i>;
+  But in these latter daies, the <i>Germaines</i> especially, as
+  <i>Regiomontanus</i>; <i>Werner</i>, <i>Schoner</i>, and <i>Appian</i>
+  have grac'd it: But above all other the learned <i>Gemma Phrisius</i> in
+  a severall worke of that argument onely, hath illustrated and taught the
+  use of it plainely and fully.</p>
+
+  <p>The <i>Iacobs</i> staffe therefore according to his owne, and those
+  Geometricall parts, shall here be described (The <!-- Page 115 --><span
+  class="pagenum"><a name="page115"></a>[115]</span>astronomicall
+  distribution wee reserve to his time and place.) And that done, the use
+  of it shall be shewed in the measuring of lines.</p>
+
+  <p>This instrument, at the discretion of the measurer may be greater or
+  lesser. For the quantity of the same can no otherwayes be determined.</p>
+
+  <p><a name="2_e_ix"></a> 2. <i>The shankes of the staffe are the Index
+  and the Transome</i>.</p>
+
+  <div class="figcenter" style="width:80%;">
+      <a href="images/136a.png"><img style="width:100%" src="images/136a.png"
+      alt="Iacobs staffe: Index." title="Iacobs staffe: Index." /></a>
+  </div>
+  <div class="figcenter" style="width:40%;">
+      <a href="images/136b.png"><img style="width:100%" src="images/136b.png"
+      alt="Iacobs staffe: Transome." title="Iacobs staffe: Transome." /></a>
+  </div>
+  <p>The principall parts of this instrument are two, the <i>Index</i>, or
+  <i>Staffe</i>, which is the greater or longer part: and the
+  <i>Transversarium</i>, or Transome, and is the lesser and shorter.</p>
+
+  <p><a name="3_e_ix"></a> 3. <i>The Index is the double and one tenth part
+  of the transome</i>.</p>
+
+  <p>Or thus: The Index is to the transversary double and 1/10 part
+  thereof. <i>H.</i> As here thou seest.</p>
+
+  <p><a name="4_e_ix"></a> 4. <i>The Transome is that which rideth upon the
+  Index, and is to be slid higher or lower at pleasure</i>.</p>
+
+  <p>Or, The transversary is to be moved upon the Index, sometimes higher,
+  sometimes lower: <i>H.</i> This proportion in defining and making of the
+  shankes of the instrument is perpetually to be observed: as if the
+  transome be 10. parts, the <!-- Page 116 --><span class="pagenum"><a
+  name="page116"></a>[116]</span>Index must be 21. If that be 189. this
+  shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill
+  what the numbers be, so this be their proportion. More than this, That
+  the greater the numbers be, that is the lesser that the divisions be, the
+  better will it be in the use. And because the Index must beare, and the
+  transome is to be borne; let the index be thicker, and the transome the
+  thinner.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/137.png"><img style="width:100%" src="images/137.png"
+      alt="Parts of Iacobs staffe." title="Parts of Iacobs staffe." /></a>
+  </div>
+  <p>But of what matter each part of the staffe be made, whether of brasse
+  or wood it skilleth not, so it be firme, and will not cast or warpe.
+  Notwithstanding, the transome will more conveniently be moved up and
+  downe by brasen pipes, both by it selfe, and upon the Index higher or
+  lower right angle wise, so touching one another, that the alterne mouth
+  of the one may touch the side of the other. The thrid pipe is to be moved
+  or slid up and downe, from one end of the transome to the other; and
+  therefore it may be called the <i>Cursor</i>. The fourth and fifth pipes,
+  fixed and immoveable, are set upon the ends of the transome, are <!--
+  Page 117 --><span class="pagenum"><a name="page117"></a>[117]</span>unto
+  the third and second of equall height with finnes, to restraine when
+  neede is, the opticke line, and as it were, with certaine points to
+  define it in the transome.</p>
+
+  <p>The three first pipes may, as occasion shall require, be fastened or
+  staied with brasen scrues. With these pipes therefore the transome may be
+  made as great, as need shall require, as here thou seest.</p>
+
+  <p>The fabricke or manner of making the instrument hath hitherto beene
+  taught, the use thereof followeth: unto which in generall is required:
+  First, a just distance. For the sight is not infinite. Secondly, that one
+  eye be closed: For the optick faculty conveighed from both the eyes into
+  one, doth aime more certainely; and the instrument is more fitly applied
+  and set to the cheeke bone, then to any other place. For here the eye is
+  as it were the center of the circle, into which the transome is
+  inscribed. Thirdly, the hands must be steady; for if they shake, the
+  proportion of the Geodesy must needes be troubled and uncertaine. Lastly,
+  the place of the station is from the midst of the foote.</p>
+
+  <p><a name="5_e_ix"></a> 5. <i>If the sight doe passe from the beginning
+  of one shanke, it passeth by the end of the other: And the one shanke is
+  perpendicular unto the magnitude to be measured, the other
+  parallell</i>.</p>
+
+  <p>These common and generall things are premised. That the sight is from
+  the beginning of the Index by the end of the transome; Or contrariwise,
+  From the beginning of the transome, unto the end of the Index. And that
+  the Index is right, that is, perpendicular to the line to be measured,
+  the transome parallell. Or contrariwise. Now the perpendicularity of the
+  Index, in measurings of lengthts, may be tried by a plummet of lead
+  appendent; But in heights and breadths, the eye must be trusted; although
+  a little varying of the plummet can make no sensible errour. <!-- Page
+  118 --><span class="pagenum"><a name="page118"></a>[118]</span>By the end
+  of the transome, understand that which is made by the line visuall,
+  whether it be the outmost finne, or the Cursour in any other place
+  whatsoever.</p>
+
+  <p><a name="6_e_ix"></a> 6. <i>Length and Altitude have a threefold
+  measure; The first and second kinde of measure require but one distance,
+  and that by granting a dimension of one of them, for the third
+  proportionall: The third two distances, and such onely is the dimension
+  of Latitude</i>.</p>
+
+  <p>Geodesy of right lines is two fold; of one distance, or of two.
+  Geodesy of one distance is when the measurer for the finding of the
+  desired dimension doth not change his place of standing. Geodesy of two
+  distances is when the measurer by reason of some impediment lying in the
+  way betweene him and the magnitude to be measured, is constrained to
+  change his place, and make a double standing.</p>
+
+  <p>Here observe, That length and heighth, may be joyntly measured both
+  with one, and with a double station: But breadth may not be measured
+  otherwise than with two.</p>
+
+  <p><a name="7_e_ix"></a> 7. <i>If the sight be from the beginning of the
+  Index right or plumbe unto the length, and unto the farther end of the
+  same, as the segment of the Index is, unto the segment of the transome,
+  so is the heighth of the measurer unto the length</i>.</p>
+
+  <p>Let therefore the segment of the Index, from the toppe, I meane, unto
+  the transome be 6. parts. The segment of the transome, to wit, from the
+  Index unto the opticke line be 18. The Index, which here is the heighth
+  of the measurer, 4. foote: The length, by the rule of three, shall be 12.
+  foote. The figure is thus, for as <i>ae</i>, is to <i>ei</i>, so is
+  <i>ao</i>, <!-- Page 119 --><span class="pagenum"><a
+  name="page119"></a>[119]</span>unto <i>ou</i>, by the <a
+  href="#12_e_vij">12. e vij</a>. For they are like triangles. For
+  <i>aei</i>, and <i>aou</i>, are right angles: And that which is at
+  <i>a</i> is common to them both: Wherefore the remainder is equall to the
+  remainder, by the <a href="#4_e_vij">4. e vij</a>.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/140a.png"><img style="width:100%" src="images/140a.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+<p><!-- Page 120 --><span class="pagenum"><a name="page120"></a>[120]</span></p>
+
+  <p>The same manner of measuring shall be used from an higher place; as
+  out of <i>y</i>, the segment of the Index is 5. parts; the segment of the
+  transome 6: and then the height be 10 foote: the same Length shall be
+  found to bee 12 foote.</p>
+
+  <p>Neither is it any matter at all, whether the length in a plaine or
+  levell underneath: Or in an ascent or descent of a mountaine, as in the
+  figure under written.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/140b.png"><img style="width:100%" src="images/140b.png"
+      alt="Using Iacobs staffe on an ascent." title="Using Iacobs staffe on an ascent." /></a>
+  </div>
+  <p>Thus mayest thou measure the breadths of Rivers, Valleys, and Ditches.
+  For the Length is alwayes after this manner, so that one may measure the
+  distance of shippes on the Sea, as also <i>Thales Milesius</i>, in
+  <i>Proclus</i> at the 26 p j, did measure them. An example thou hast
+  here.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/141.png"><img style="width:100%" src="images/141.png"
+      alt="Using Iacobs staffe on a ship." title="Using Iacobs staffe on a ship." /></a>
+  </div>
+  <p>Hereafter in the measuring of Longitude and Altitude, sight is unto
+  the toppe of the heighth. Which here I doe now forewarne thee of, least
+  afterward it should in vaine be reitered often.</p>
+
+  <p>The second manner of measuring a Length is thus: <!-- Page 121
+  --><span class="pagenum"><a name="page121"></a>[121]</span></p>
+
+  <p><a name="8_e_ix"></a> 8. <i>If the sight be from the beginning of the
+  index parallell to the length to be measured, as the segment of the
+  transome is, unto the segment of the index, so shall the heighth given be
+  to the length</i>.</p>
+
+  <p>As if the segment of the Transome be 120 parts: the height given 400
+  foote: The segment of the Index 210 parts: The length, by the golden rule
+  shall be 700 foote. The figure is thus. And the demonstration is like
+  unto the former; or indeed more easier. For the triangles are equiangles,
+  as afore. Therefore as <i>ou</i> is to <i>ua</i>: so is <i>ei</i> to
+  <i>ia</i>.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/142.png"><img style="width:100%" src="images/142.png"
+      alt="Using Iacobs staffe in the second kinde." title="Using Iacobs staffe in the second kinde." /></a>
+  </div>
+  <p>This is the first and second kinde of measuring of a Longitude, by one
+  single distance or station: The third which is by a double distance doth
+  now follow. Here the transome, if there be roome enough for the measurer
+  to goe farre enough backe, must be put lower, in the second distance.</p>
+
+  <p><a name="9_e_ix"></a> 9. <i>If the sight be from the beginning of the
+  <!-- Page 122 --><span class="pagenum"><a
+  name="page122"></a>[122]</span>transverie parallell to the length to be
+  measured, as in the index the difference of the greater segment is unto
+  the lesser; so is the difference of the second station unto the
+  length</i>.</p>
+
+  <p>This kinde of Geod�sy is somewhat more subtile than the former were.
+  The figure is thus; in which let the first ayming be from <i>a</i>, the
+  beginning of the transome, and out of <i>ai</i> the length sought by
+  <i>o</i>, the end of the Index, unto <i>e</i>, the toppe of the heighth:
+  And let the segment of the Index be <i>ou</i>: The second ayming let it
+  be from <i>y</i>, the beginning of the transome, out of a greater
+  distance by <i>s</i>, the end of the Index, unto <i>e</i>, the same note
+  of the heighth: And let the segment of the Index be <i>sr</i>.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/143.png"><img style="width:100%" src="images/143.png"
+      alt="Using Iacobs staffe at two distances." title="Using Iacobs staffe at two distances." /></a>
+  </div>
+  <p>Here the measuring performed, is the taking of the difference betweene
+  <i>ou</i> and <i>sr</i>. The rest are faigned onely for demonstrations
+  sake. Therefore in the first station let <i>aml</i>, be from the
+  beginning of the transome, be parallell to <i>ye</i>. Here first
+  <i>mu</i>, is equall to <i>sr</i>. For the triangles <!-- Page 123
+  --><span class="pagenum"><a name="page123"></a>[123]</span><i>mua</i>,
+  and <i>sry</i>, are equall in their shankes <i>ua</i>, and <i>ry</i>, by
+  the grant (Because the transome standeth still in his owne place:) And
+  the angles at <i>mua</i>, <i>uam</i>, are equall to the angles: And all
+  right angles are equall, by the <a href="#14_e_iij">14 e iij</a>. These
+  are the outter and inner opposite one to another: And such are equall by
+  the <span class="correction" title="wrong reference but I cannot deduce the correction."
+  >1 e v</span>. Therefore they are equilaters, by the <a href="#2_e_vij">2
+  e vij</a>; And <i>om</i>, is the difference of the segments of the Index.
+  Then as <i>om</i> is to <i>mu</i>, so is <i>el</i>, to <i>li</i>; as the
+  equation of three degrees doth shew. For, by the <a href="#12_e_vij">12 e
+  vij</a>, as <i>om</i> is to <i>ma</i>: so is <i>el</i> to <i>la</i>: And
+  as <i>ma</i> is to <i>mu</i>; so is <i>la</i>, to <i>li</i>. Therefore by
+  right, as <i>om</i>, is to <i>mu</i>: so is <i>el</i>, to <i>li</i>: And
+  by the <a href="#12_e_vj">12 e vj</a>, so is <i>ya</i>, to <i>ai</i>: As
+  if the difference of the first segment be 36 parts: The second segment be
+  72 parts: The difference of the second station 40 foote. The length
+  sought shall be 80 foote. And here indeed is no heighth definitely given,
+  that may make any bound of the principall proportion. Notwithstanding the
+  Heighth, although it be of an unknowne measure, is the bound of the
+  length sought: And therefore it is an helpe and meanes to argue the
+  question. Because it is conceived to stand plumbe upon the outmost end of
+  the length.</p>
+
+  <p>Therefore that third kinde of measuring of length is oftentimes
+  necessary, when by neither of the former wayes the length may possibly be
+  taken, by reason of some impediment in the way, to wit of a wall, or
+  tree, or house, or mountaine, whereby the end of the length may not be
+  seene, which was the first way: Nor an height next adjoyning to the end
+  of the length is given, which is the second way.</p>
+
+  <p>Hitherto we have spoken of the threefold measure of longitude, the
+  first and second out of an heighth given the third cut of a double
+  distance: The measuring of heighth followeth next, and that is also
+  threefold. Now heighth is a perpendicular line falling from the toppe of
+  the magnitude, unto the ground or plaine whereon the measurer doth stand,
+  after which manner Altitude or <!-- Page 124 --><span class="pagenum"><a
+  name="page124"></a>[124]</span>heighth was defined at the <a
+  href="#9_e_iiij">9 e iiij</a>. The first geodesy or manner of measuring
+  of heighths is thus.</p>
+
+  <p><a name="10_e_ix"></a> 10. <i>If the sight be from the beginning of
+  the transome perpendicular unto the height to be measured, as the segment
+  of the transome, is unto the segment of the Index, so shall the length
+  given be to the height</i>.</p>
+
+  <p>Let the segment of the transome be 60 parts: the segment of the Index
+  36: the Length given 120 foote: the height sought shall be, by the golden
+  rule, 72 foote.</p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/145.png"><img style="width:100%" src="images/145.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>The Figure is thus: And the demonstration is by the <a
+  href="#12_e_vij">12 e vij</a>, as afore: but here is to be added the
+  height of the measurer; which if it be 4 foot, the whole height shall be
+  76 foote.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore in an eversed altitude</p>
+    </div>
+  </div>
+  <p><a name="11_e_ix"></a> 11. <i>If the sight be from the beginning of
+  the Index parallell to the height, as the segment of the transome <!--
+  Page 125 --><span class="pagenum"><a name="page125"></a>[125]</span>is,
+  unto the segment of the index, so shall the length given be, unto the
+  height sought</i>.</p>
+
+  <p><i>Eversa altitudo</i>, An eversed altitude (Reversed, <i>H</i>:) is
+  that which we call depth, which indeed is nothing else, in the Geometers
+  sense, but heighth turned topsie turvie, as we say, or with the heeles
+  upward. For out of the heighth concluded by subducting that which is
+  above ground, the heighth or depth of a Well shall remaine.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/146.png"><img style="width:100%" src="images/146.png"
+      alt="Using Iacobs staffe in an eversed altitude." title="Using Iacobs staffe in an eversed altitude." /></a>
+  </div>
+  <p>Let the segment of the transome <i>ae</i>, be 5 parts: the segment of
+  the Index <i>ei</i>, be 13: the diameter of the Well (which now standeth
+  for the length:) be 10 foote, which at toppe is supposed to be equall to
+  that at bottome: the opposite height, by the <a href="#12_e_vij"><span
+  class="correction" title="text reads `21 e vij'">12 e vij</span></a>, and
+  the golden rule shall <!-- Page 126 --><span class="pagenum"><a
+  name="page126"></a>[126]</span>be 26 foote: From whence you must take the
+  segment of the Index reaching over the mouth of the Well: And the true
+  height (or depth) shall remaine; as if that segment of 13 parts be as
+  much as 2 foote, the height sought shall be 24 foote. The second manner
+  of measuring of heights followeth.</p>
+
+  <p><a name="12_e_ix"></a> 12. <i>If the sight be from the beginning of
+  the Index perpendicular to the heighth to be measured, as the segment of
+  the Index is unto the segment of the Transome, so shall the length given
+  be to the heighth</i>.</p>
+
+  <p>As if the segment of the Index be 60 parts: and the segment also of
+  the transome be 60: And the Length given be 250 foote: By the Rule of
+  three, the height also shall be 250 foote: as thou seest in the example
+  underneath: For as <i>ae</i> is to <i>ei</i>; so is <i>aeo</i> to
+  <i>ou</i>, by the <a href="#12_e_vij">12 e vij</a>. But here unto the
+  height found, you must adde the height of the measurer: Which if it be 4
+  foot, the whole height shall be 254 foote.</p>
+
+  <div class="figcenter" style="width:53%;">
+      <a href="images/147.png"><img style="width:100%" src="images/147.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+<p><!-- Page 127 --><span class="pagenum"><a name="page127"></a>[127]</span></p>
+
+  <p><a name="13_e_ix"></a> 13. <i>If the sight be from the beginning of
+  the Index (perpendicular to the magnitude to be measured) by the names of
+  the transome, unto the ends of some known part of the height, as the
+  distance of the Names is, unto the rest of the transome above them, so
+  shall the known part be unto the part sought</i>.</p>
+
+  <p>Or thus: If the sight passe from the beginning of the Index being
+  right, by the vanes of the transversary, to the tearmes of some parts; as
+  the distance of the vanes is unto the rest of the transversary above the
+  index, so is the part knowne unto the remainder: <i>H</i>.</p>
+
+  <p>This is a consectary of a knowne part of an height, from whence the
+  rest may be knowne, as in the figure.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/148.png"><img style="width:100%" src="images/148.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>As <i>ou</i> is unto <i>uy</i>, so is <i>ei</i> to <i>is</i>. For as
+  <i>ou</i>, is unto <i>ua</i>: so is <i>ei</i> unto <i>ia</i>, by the <a
+  href="#12_e_vij">12 e vij</a>. And as <i>ua</i>, is to <i>uy</i>, so is
+  <i>ia</i> unto <i>is</i>; and by right, as <i>ou</i>, is to <i>uy</i>, so
+  is <i>ei</i>, to <i>is</i>. Here thou hast three bounds of the
+  proportion. Let therefore <i>ou</i>, be 20 parts: <i>uy</i> 30: And
+  <i>ei</i>, the knowne part, let it be <!-- Page 128 --><span
+  class="pagenum"><a name="page128"></a>[128]</span>15 foote: Therefore
+  thou shalt conclude <i>is</i>, the rest to be 22�.</p>
+
+  <p>The first and second kinde of measuring of heights is thus: The third
+  followeth.</p>
+
+  <p><a name="14_e_ix"></a> 14 <i>If the sight be from the beginning of the
+  Index perpendicular to the heighth, as in the Index the difference of the
+  <span class="correction" title="text reads `segmeut'">segment</span>, is
+  unto the difference of the distance or station; so is the segment of the
+  transome unto the heighth</i>.</p>
+
+  <p>Hitherto you must recall that subtilty, which was used in the third
+  manner of measuring of lengths.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/149.png"><img style="width:100%" src="images/149.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>Let the first aime be taken from <i>a</i>, the beginning of the Index
+  perpendicular unto the height to be measured: And from an unknowne length
+  <i>ai</i>, by <i>o</i>, the end of the transome, unto <i>e</i>, the toppe
+  of the height <i>ei</i>: And let the segment of the Index be <i>ua</i>.
+  The second ayme, let it be taken from <i>y</i>, the beginning of the same
+  Index; and out of a <!-- Page 129 --><span class="pagenum"><a
+  name="page129"></a>[129]</span>greater distance, by <i>s</i>, the end of
+  the transome, unto the same toppe <i>e</i>. And the segment of the Index
+  let it be <i><span class="correction" title="text reads `rl'"
+  >ry</span></i>.</p>
+
+  <p>Here, as afore, the measuring is performed and done, by the taking of
+  the difference of the said <i>yr</i>, above <i>au</i>: Now the
+  demonstration is concluded, as in the former was taught. Let the
+  parallell <i>lsm</i>, be erected against <i>aoe</i>.</p>
+
+  <p>Here first the triangles <i>oua</i>, &amp; <i>srl</i>, are equilaters,
+  by the <a href="#2_e_vij">2 e vij</a>.; (seeing that the angles at
+  <i>a</i>, and <i>l</i>, the externall and internall, are equall in bases
+  <i>ou</i>, and <i>sr</i>, for the segment in each distance is the same
+  still:) Therefore <i>ua</i>, is equall to <i>rl</i>. Now the rest is
+  concluded by a sorites of foure degrees: As <i>yr</i>, is unto <i><span
+  class="correction" title="text reads `yu'">yi</span></i>: so by the <a
+  href="#12_e_vij">12. e vij</a>. is <i>sr</i>, that is, <i>ou</i>, unto
+  <i>ei</i>: And as <i>ou</i>, is unto <i>ei</i>, so is <i>au</i>, that is,
+  <i>lr</i>, unto <i>ai</i>. Therefore the remainder <i>yl</i>, unto the
+  remainder <i>ya</i>; shall be as <i>yr</i>, is unto the whole <i>yi</i>,
+  and therefore from the first unto the last, as <i>sr</i>, is to
+  <i>ei</i>.</p>
+
+  <p>Therefore let the difference of the Index be 23. parts: The difference
+  of the distance 30. foote: The segment of the transome <span
+  class="correction" title="text reads `23.'">44.</span> parts: The height
+  shall be 57.9/23. or foote.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="15_e_ix"></a> 15 <i>Out of the Geodesy of heights, the
+  difference of two heights is manifest</i>.</p>
+
+  <p>Or thus: By the measure of one altitude, we may know the difference of
+  two altitudes: <i>H</i>.</p>
+
+  <p>For when thou hast taken or found both of them, by some one of the
+  former wayes, take the lesser out of the greater; and the remaine shall
+  be the heighth desired. From hence therefore by one of the towers of
+  unequall heighth, you may measure the heighth of the other. First out of
+  the lesser, let the length be taken by the first way: Because the height
+  of the lesser, wherein thou art, is easie to be taken, either by a
+  plumbe-line, let fall from the toppe to the bottom, or by some one of the
+  former waies. Then measure <!-- Page 130 --><span class="pagenum"><a
+  name="page130"></a>[130]</span>the heighth, which is above the lesser:
+  And adde that to the lesser, and thou shalt have the whole heighth, by
+  the first or second way. The figure is thus, and the demonstration is out
+  of the <a href="#12_e_vij">12. e vij</a>. For as <i>ae</i>, is to
+  <i>ei</i>, so is <i>ao</i>, to <i>ou</i>. Contrariwise out of an higher
+  Tower, one may measure a lesser.</p>
+
+  <p><a name="16_e_ix"></a> 16 <i>If the sight be first from the toppe,
+  then againe from the base or middle place of the greater, by the vanes of
+  the transome unto the toppe of the lesser heighth; as the said parts of
+  the yards are unto the part of the first yard; so the heighth betweene
+  the stations shall be unto his excesse above the heighth desired</i>.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/151.png"><img style="width:100%" src="images/151.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>Let the unequall heights be these, <i>as</i>, the lesser, and
+  <i>uy</i>, the greater: And out of the assigned greater <i>uy</i>, let
+  the lesser, <i>as</i>, be sought. And let the sight be first from
+  <i>u</i>, the toppe of the greater, unto <i>a</i>, the toppe of the
+  lesser, <!-- Page 131 --><span class="pagenum"><a
+  name="page131"></a>[131]</span>making at the shankes of the staffe the
+  triangle <i>urm</i>. Then againe let the same sight be from the base, or
+  from the lower end of <i>uy</i>, the heighth given, unto <i>a</i>, the
+  same toppe of the lesser, making by the shankes of the staffe the
+  triangle <i>yln</i>, so that the segments of the yard be, the upper one,
+  I meane, <i>ur</i>, the neather one <i>ul</i>: I say the whole of
+  <i>ur</i>, and <i>nl</i>, is unto <i>ur</i>: so is the <i>uy</i>, greater
+  heighth assigned, unto <i>as</i>, the lesser sought.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/152.png"><img style="width:100%" src="images/152.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>The Demonstration, by drawing of <i>ao</i>, a perpendicular unto
+  <i>uy</i>, is a proportion out of two triangles of equall heighth. For
+  the forth of the totall equally heighted triangles <i>uao</i>, and
+  <i>yas</i>, although they be reciprocall in situation, they have their
+  bases <i>uo</i>, and <i>as</i>, as if their were <i>oy</i>. Then they
+  have the same with the whole triangles; as also the subducted triangles
+  <i>urm</i>, and <i>ynl</i>, of equal heighth; to wit whose common heighth
+  is the segment of the transome remained still in the same place, there
+  <i>rm</i>, here <i>yl</i>. And therefore the bases of these, namely, the
+  segments of the yards <i>ur</i>, and <i>nl</i>, have the same rate with
+  <i>uo</i>, unto <i>oy</i>. <!-- Page 132 --><span class="pagenum"><a
+  name="page132"></a>[132]</span>As therefore <i>uo</i>, is unto <i>oy</i>:
+  so is, <i>ur</i>, unto <i>nl</i>. And backward, as <i>nl</i>, is to
+  <i>ur</i>; so is, <i>yo</i>, unto <i>ou</i>, as here thou seest:</p>
+
+<table class="nobctr">
+<tr><td><i>nl</i>,&mdash;&mdash;&mdash;&mdash;<i>ur</i>:</td><td>&nbsp; &nbsp; &nbsp; &nbsp;</td><td><i>yo</i>,&mdash;&mdash;&mdash;&mdash;<i>ou</i>.</td></tr>
+</table>
+
+  <p>Therefore furthermore by composition of the Antecedent with the
+  Consequent unto the Consequent, by the 5 c 9 ij. Arith. As <i>nl</i>, and
+  <i>ur</i>, are unto <i>ur</i>: so are <i>yo</i>, and <i>ou</i>, unto
+  <i>ou</i>, that is <i>yu</i>, unto <i>ou</i>, on this manner.</p>
+
+<table class="nobctr">
+<tr><td><i>nl</i>,&mdash;&mdash;&mdash;&mdash;<i>ur</i>,</td><td>&nbsp; &nbsp; &nbsp; &nbsp;</td><td><i>yo</i>,&mdash;&mdash;&mdash;&mdash;<i>ou</i>,</td></tr>
+<tr><td><i>nr</i>,</td><td>&nbsp; &nbsp; &nbsp; &nbsp;</td><td><i>ou</i>,</td></tr>
+<tr><td colspan="3">&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;</td></tr>
+<tr><td>...&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<i>ur</i>,</td><td>&nbsp; &nbsp; &nbsp; &nbsp;</td><td><i>yu</i>,&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<i>ou</i>,</td></tr>
+</table>
+
+  <p>there is given <i>nl</i>, and <i>ur</i>, for the first proportionall:
+  <i>ur</i>, for the second: and <i>yu</i>, for the third: Therefore there
+  is also given <i>ou</i>, for the fourth: Which <i>ou</i>, subducted out
+  of <i>uy</i>, there remaineth <i>oy</i>, that is, <i>as</i>, the lesser
+  altitude sought.</p>
+
+  <p>For let the parts of the yard be 12. and 6. and the summe of them 18.
+  Now as 18. is <span class="correction" title="omitted in text"
+  >unto</span> 12. so is the whole altitude <i>uy</i>, 190. foote, unto the
+  excesse 126&#x2154; foote. The remainder therefore 63&#x2153; foote,
+  shall be <i>as</i>, the lesser heighth sought.</p>
+
+  <p>But thou maist more fitly dispose and order this proportion thus: As
+  <i>ur</i>, is unto <i>nl</i>: so is <i>uo</i> unto <i>oy</i>. Therefore
+  by Arithmeticall composition, as <i>ur</i>, and <i>nl</i>, are unto
+  <i>nl</i>: so <i>uo</i>, and <i>oy</i>, that is, the whole <i>uy</i>, is
+  unto <i>oy</i>, that is, unto <i>as</i>. For here a subduction of the
+  proportion, after the composition is no way necessary, by the crosse rule
+  of societia, thus:</p>
+
+  <div class="figcenter" style="width:45%;">
+      <a href="images/153.png"><img style="width:100%" src="images/153.png"
+      alt="Crosse rule." title="Crosse rule." /></a>
+  </div>
+  <p>The second station might have beene in <i>o</i>, the end of the
+  perpendicular from <i>a</i>. But by taking the ayme out of the toppe of
+  the lesser altitude, the demonstration shall be yet againe more easie and
+  short, by the two triangles at the yard <i>aei</i>, and <i>aef</i>,
+  resembling the two whole triangles <i>aou</i>, and <i>aoy</i>, in like
+  situation, the parts of the <!-- Page 133 --><span class="pagenum"><a
+  name="page133"></a>[133]</span>shanke cut, are on each side the segments
+  of the transome.</p>
+
+  <p>One may againe also out of the toppe of a Turret measure the distance
+  of two turrets one from another: For it is the first manner of measuring
+  of longitudes, neither doth it here differ any whit from it, more than
+  the yard is hang'd without the heighth given. The figure is thus: And the
+  Demonstration is by the <a href="#12_e_vij">12. e vij</a>. For as
+  <i>ae</i>, the segment of the yard, is unto <i>ei</i> the segment of the
+  transome: so is the assigned altitude <i>ao</i>, unto the length
+  <i>ou</i>.</p>
+
+  <p>The geodesy or measuring of altitude is thus, where either the length,
+  or some part of the length is given, as in the first and second way: Or
+  where the distance is double, as in the third.</p>
+
+  <p><a name="17_e_ix"></a> 17 <i>If the sight be from the beginning of the
+  yard being right or perpendicular, by the vanes of the transome, unto the
+  ends of the breadth; as in the yard the difference of the segment is unto
+  the differ&#x113;ce of the distance, so is the distance of the vanes unto
+  the breadth</i>.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/154.png"><img style="width:100%" src="images/154.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>The measuring of breadth, that is of a thwart or crosse <!-- Page 134
+  --><span class="pagenum"><a name="page134"></a>[134]</span>line,
+  remaineth. The Figure and Demonstration is thus: The first ayming, let it
+  be <i>aei</i>, by <i>o</i>, and <i>u</i>, the vanes of the transome
+  <i>ou</i>. The second, let it be <i>yei</i>, by <i>s</i>, and <i>r</i>,
+  the vanes of the transome <i>sr</i>. Then by the point <i>s</i>, let the
+  parallell <i>lsm</i>, be drawne against <i>aoe</i>. Here first, the
+  triangles <i>oua</i>, and <i>sil</i>, are equilaters, by the <a
+  href="#2_e_vij">2 e vij</a>. Because the angles at <i>n</i> and <i>j</i>,
+  are right angles: And <i>uao</i>, and <i>jls</i>, the outter and inner,
+  are equall in their bases <i>ou</i>, and <i>sj</i>, by the grant: Because
+  here the segment of the transome remaineth the same: Therefore <i>ua</i>,
+  is equall to <i>jl</i>. These grounds thus laid, the demonstration of the
+  third altitude here taken place. For as <i>yl</i>, is unto <i>ya</i>: so
+  is <i>sj</i>, unto <i>er</i>: And, because parts are proportionall unto
+  their multiplicants, so is <i>sr</i>, unto <i>ei</i>: for the rest doe
+  agree.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/155.png"><img style="width:100%" src="images/155.png"
+      alt="Using Iacobs staffe." title="Using Iacobs staffe." /></a>
+  </div>
+  <p>The same shall be the geodesy or manner of measuring, if thou wouldest
+  from some higher place, measure the breadth that is beneath thee, as in
+  the last example. But from the distance of two places, that is, from
+  latitude or breadth, as of Trees, Mountaines, Cities, Geographers and
+  Chorographers do gaine great advantages and helpes. <!-- Page 135
+  --><span class="pagenum"><a name="page135"></a>[135]</span></p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/156.png"><img style="width:100%" src="images/156.png"
+      alt="Geodesy." title="Geodesy." /></a>
+  </div>
+  <p>Wherefore the geodesy or measuring of right lines is thus in length,
+  heighth, and breadth, from whence the Painter, the Architect, and
+  Cosmographer, may view and gather of many famous place the windowes, the
+  statues or imagery, pyramides, signes, and lastly, the length and
+  heighth, either by a single or double: the breadth by a double dimension
+  onely, that is, they may thus behold and take of all places the nature
+  and symmetry; as in the example next following thou mayst make triall
+  when thou pleasest.</p>
+
+<hr class="full" />
+
+<p><!-- Page 136 --><span class="pagenum"><a name="page136"></a>[136]</span></p>
+
+<h2>The tenth Booke of <i>Geometry</i>,
+of a Triangulate and Parallelogramme.</h2>
+
+  <p>And thus much of the geodesy of right lines, by the meanes of
+  rectangled triangles: It followeth now of the triangulate.</p>
+
+  <p><a name="1_e_x"></a> 1. <i>A triangulate is a rectilineall figure
+  compounded of triangles</i>.</p>
+
+  <p>As before (for the dichotomies sake) of a line was made a Lineate, to
+  signifie the <i>genus</i> of surface and a Body: so now is for the same
+  cause of a triangle made a Triangulate, to declare and expresse the
+  <i>genus</i> of a Quadrilater and Multilater, and indeed more justly,
+  then before in a Lineate. For triangles doe compound and make the
+  triangulate, but lines doe not make the lineate.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="2_e_x"></a> 2. <i>The sides of a triangulate are two more
+  than are the triangles of which it is made</i>.</p>
+
+  <div class="figcenter" style="width:60%;">
+      <a href="images/157.png"><img style="width:100%" src="images/157.png"
+      alt="Quadrangle and Quinquangles." title="Quadrangle and Quinquangles." /></a>
+  </div>
+  <p>As the sides of a Quadrangle are 4. Therefore the triangles which doe
+  make the same foure-sided figure are but 2. The sides of a Quinquangle
+  are 5, Therefore the triangles are 3, and so forth of the rest, as here
+  thou seest. And <!-- Page 137 --><span class="pagenum"><a
+  name="page137"></a>[137]</span>that indeed is the least: For even a
+  triangle it selfe, may be cut into as many triangles as one please.</p>
+
+  <p>That both the inner and outter are equall to right angles, in every
+  kinde of right line figure, it was manifest at the <a href="#4_e_vj">4 e
+  vj</a>. The inner <span class="correction" title="text reads `is'"
+  >in</span> a Quadrangle, are equall to 4. In a Quinquangle, to 6: In an
+  Hexangle, to 8; and so forth.</p>
+
+  <div class="figcenter" style="width:40%;">
+      <a href="images/158.png"><img style="width:100%" src="images/158.png"
+      alt="Hexangles." title="Hexangles." /></a>
+  </div>
+  <p>But the outter, in every right-lined figure, are equall to 4 right
+  angles: as here may be demonstrated, by the <a href="#14_e_v">14 e v</a>
+  and <a href="#13_e_vj">13 e vj</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="3_e_x"></a> 3. <i>Homgeneall Triangulates are cut into an
+  equall number of triangles, � 20 p vj</i>.</p>
+
+  <p>For if they be Quadrangles, they be cut into two triangles: If
+  Quinquangles, into 3. If Hexangles, into 4, and so forth.</p>
+
+  <p><a name="4_e_x"></a> 4. <i>Like triangulates are cut into triangles
+  alike one to another and homologall to the whole � 20 p vj</i>.</p>
+
+  <p>Or thus: Like Triangulates are divided into triangles like one unto
+  another, and in porportion correspondent unto the whole: <i>H</i>.</p>
+
+  <p>As in these two <span class="correction" title="text reads `quinqualges'"
+  >quinquangles</span>. First the particular triangles are like betweene
+  themselves. For the shankes of <i>aeu</i> and <i>ysm</i>, equall angles
+  are proportionall, by the grant. Therefore the triangles themselves are
+  equiangles, by <a href="#14_e_vij">14 e vij</a>. And therefore alike, by
+  the <a href="#12_e_vij">12 e vij</a>, and so forth of the rest. <!-- Page
+  138 --><span class="pagenum"><a name="page138"></a>[138]</span></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/159.png"><img style="width:100%" src="images/159.png"
+      alt="Like quinquangles." title="Like quinquangles." /></a>
+  </div>
+  <p>The middle triangles, the equall angles being substracted shall have
+  their other angles equall: And therefore they also shall be equiangles
+  and alike, by the same.</p>
+
+  <p>Secondarily, the triangles <i>aeu</i>. and <i>ysm</i>: <i>eio</i> and
+  <i>srl</i>; <i>eou</i>, and <i>slm</i>, to wit, alike betweene
+  themselves, are by the <a href="#1_e_vj">1 e vj</a>, in a double reason
+  of their homologall sides <i>eu</i>, <i>sm</i>, <i>eo</i>, <i>sl</i>,
+  which reason is the same, by meanes of the common sides. Therefore three
+  triangles are in the same reason: And therefore they are proportionall:
+  And, by the third composition, as one of the antecedents is, unto one of
+  the consequents; so is the whole quinquangle to the whole.</p>
+
+  <p><a name="5_e_x"></a> 5. <i>A triangulate is a Quadrangle or a
+  Multangle</i>.</p>
+
+  <p>The parts of this partition are in Euclide, and yet without any shew
+  of a division. And here also, as before, the species or severall kinds
+  have their denomination their angles, although it had beene better and
+  truer to have beene taken from their sides; as to have beene called a
+  Quadrilater, or a Multilater. But in words use must bee followed as a
+  master.</p>
+
+  <p><a name="6_e_x"></a> 6. <i>A <span class="correction" title="text reads `Quandrangle'"
+  >Quadrangle</span> is that which is comprehended of foure right lines. 22
+  d j</i>.</p>
+
+  <p>As here thou seest. But a Quadrangle may also bee a sphearicall, and a
+  conicall, and a cylindraceall, and that <!-- Page 139 --><span
+  class="pagenum"><a name="page139"></a>[139]</span>those differences are
+  common, we doe foretell at the <a href="#3_e_v">3 e v</a>. And a
+  Quadrangle may be a plaine, which is not a quadrilater, as here.</p>
+
+  <div class="figcenter" style="width:39%;">
+      <a href="images/160a.png"><img style="width:100%" src="images/160a.png"
+      alt="Quadrangles." title="Quadrangles." /></a>
+  </div>
+  <p><a name="7_e_x"></a> 7. <i>A quadrangle is <span class="correction"
+  title="text reads `a a'"></span>a Parallelogramme, or a
+  Trapezium.</i></p>
+
+  <p>This division also in his parts is in the Elements of Euclide, but
+  without any forme or shew of a division. But the difference of the parts
+  shall more fitly be distinguished thus: Because in generall there are
+  many common parallels.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/160b.png"><img style="width:100%" src="images/160b.png"
+      alt="Parallelogramme." title="Parallelogramme." /></a>
+  </div>
+  <p><a name="8_e_x"></a> 8. <i>A Parallelogramme is a quadrangle whose
+  opposite sides are parallell</i>.</p>
+
+  <p>As in the example, the side <i>ae</i>, is parallell to the side
+  <i>io</i>: And the side <i>ei</i>, is parallell to opposite side
+  <i>ao</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="9_e_x"></a> 9. <i>If right lines on one and the same side,
+  doe joyntly bound equall and parallall lines, they shall make a
+  parallelogramme.</i> <!-- Page 140 --><span class="pagenum"><a
+  name="page140"></a>[140]</span></p>
+
+  <p>The reason is, because they shall be equall and parallell betweene
+  themselves, by the <a href="#26_e_v">26. e v</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="10_e_x"></a> 10 <i>A parallelogramme is equall both in his
+  opposite sides, and angles, and segments cut by the diameter</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/161a.png"><img style="width:100%" src="images/161a.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p>Or thus: The opposite, both sides, and angles, and segments cut by the
+  diameter are equall. Three things are here concluded: The first is, that
+  the opposite sides are equall: This manifest by the <a href="#26_e_v">26
+  e v</a>. Because two right lines doe jointly bound equall parallells.</p>
+
+  <p>The second, that the opposite angles are equall, the Diagonall
+  <i>ai</i>, doth shew. For it maketh the triangles <i>aei</i>, and
+  <i>ioa</i>, equilaters: And therefore also equiangles: And seeing that
+  the particular angles at <i>a</i>, and <i>i</i>, are equall, the whole is
+  equall to the whole. This part is the 34. p j;</p>
+
+  <div class="figcenter" style="width:62%;">
+      <a href="images/161b.png"><img style="width:100%" src="images/161b.png"
+      alt="Figure for demonstration 10 the third." title="Figure for demonstration 10 the third." /></a>
+  </div>
+  <p>The third: The segments cut by the diameter are alwayes equall,
+  whether they be triangles, or any manner of quadrangles, as in the
+  figures. For the Diameter doth cut into two equall parts, the
+  parallelogramme by the Angles, or by the opposite sides, or by the <span
+  class="correction" title="text reads `alernall'">alternall</span> equall
+  segments of the sides.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 141 --><span class="pagenum"><a name="page141"></a>[141]</span></p>
+
+  <p><a name="11_e_x"></a> 11. <i>The Diameter of a parallelogramme is cut
+  into two by equall raies</i>.</p>
+
+  <p>As in the three figures <i>aei</i>, next before: This a
+  parallelogramme hath common with a circle, as was manifest at the <a
+  href="#28_e_iiij">28. e iiij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:18%;">
+      <a href="images/162a.png"><img style="width:100%" src="images/162a.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p><a name="12_e_x"></a> 12 <i>A parallelogramme is the double of a
+  triangle of a trinangle of equall base and heighth, 41. p j</i>.</p>
+
+  <p>The comparison first in rate of inequality of a parallelogramme with a
+  triangle, doth follow: As here thou seest in this diagramme. For a
+  parallelogramme is cut into two equall triangles, by the antecedent.
+  Therefore it is the double of the halfe.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="13_e_x"></a> 13 <i>A parallelogramme is equall to a triangle
+  of equall heighth and double base unto it: � 42. p j.</i></p>
+
+  <p>As to <i>aei</i>, the triangle, the parallelogramme <i>aoiu</i>, is
+  equall: because halfe of the parallelogramme is equall to the triangle:
+  Therefore the halfes being equall, whole also shall be equall.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/162b.png"><img style="width:100%" src="images/162b.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>From whence one may</p>
+    </div>
+  </div>
+  <p><a name="14_e_x"></a> 14 <i>To a triangle given, in a rectilineall
+  angle given, make an equall parallelogramme</i>.</p>
+
+  <div class="figleft" style="width:27%;">
+      <a href="images/162c.png"><img style="width:100%" src="images/162c.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>As here to the triangle, <i>aei</i>, given in <i>s</i>, the right
+  lined angle given, you may equall the parallelogramme <i>ouyi</i>. <!--
+  Page 142 --><span class="pagenum"><a name="page142"></a>[142]</span></p>
+
+  <p><a name="15_e_x"></a> 15 <i>A parallelogramme doth consist both of two
+  diagonals, and complements, and gnomons</i>.</p>
+
+  <p>For these three parts of a parallelogramme are much used in
+  Geometricall workes and businesses, and therefore they are to be
+  defined.</p>
+
+  <p><a name="16_e_x"></a> 16 <i>The Diagonall is a particular
+  parallelogramme having both an angle and diagonall diameter common with
+  the whole parallelogramme</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/163.png"><img style="width:100%" src="images/163.png"
+      alt="Diagonall." title="Diagonall." /></a>
+  </div>
+  <p>First the Diagonall is defined: As in the whole parallelogramme
+  <i>aeio</i>, the diagonals are <i>auys</i>, and <i>ylir</i>; Because they
+  are parts of the whole, having both the same common angles at <i>a</i>,
+  and <i>i</i>: and diagonall diameter <i>ai</i>, with the whole
+  parallelogramme: Not that the whole diagonie is common to both: But
+  because the particular diagonies are the parts of the whole diagony.
+  Therefore the diagonalls are two.</p>
+
+  <p><a name="17_e_x"></a> 17 <i>The Diagonall is like, and alike situate
+  to the whole parallelogramme: � 24. p vj</i>.</p>
+
+  <p>There is not any, either rate or proportion of the diagonall
+  propounded, onely similitude is attributed to it, as in the same figure,
+  the Diagonall <i>auys</i>, is like unto the whole parallelogramme
+  <i>aeio</i>. For first it is equianglar to it. For the angle at <i>a</i>,
+  is common to them both: And that is equall to that which is at <i>y</i>,
+  (by the <a href="#10_e_x">10. e x</a>:) And therefore also it is equall
+  to that at <i>i</i> by the <a href="#10_e_x">10. e x</a>. Then the angles
+  <i>auy</i>, and <i>asy</i>, are equall, by the <a href="#21_e_v">21. e
+  v</a>. to the opposite inner angles at <i>e</i>, and <i>o</i>. Therefore
+  it is equiangular unto it.</p>
+
+  <p>Againe, it is proportionall to it in the shankes of the <!-- Page 143
+  --><span class="pagenum"><a name="page143"></a>[143]</span>equall angles.
+  For the triangles <i>auy</i>, and <i>aei</i>, are alike, by the <a
+  href="#12_e_vij">12 e vij</a>, because <i>uy</i> is parallell to the
+  base. Therefore as <i>au</i> is <i>uy</i>; so is <i>ai</i> to <i>ei</i>:
+  Then as <i>uy</i> is to <i>ya</i>; so is <i>ei</i> to <i>ia</i>. Againe
+  by the <a href="#21_e_v">21 e v</a>, because <i>sy</i> is parallell to
+  the base <i>io</i>, as <i>ay</i> is to <i>ys</i>: so is <i>ai</i>, to
+  <i>io</i>: Therefore equiordinately, as <i>uy</i> is to <i>ys</i>: so is
+  <i>ei</i> to <i>io</i>: Item as <i>sy</i> is to <i>ya</i>, so is
+  <i>io</i> to <i>ia</i>: And as <i>ya</i> is to <i>as</i>: so is <i>ia</i>
+  to <i>ao</i>. Therefore equiordinately, as <i>ys</i> is to <i>sa</i>: so
+  is <i>io</i> to <i>oa</i>. Lastly as <i>sa</i> is unto <i>ay</i>; so is
+  <i>oa</i> unto <i>ai</i>: And as <i>ay</i> is to <i>au</i>; so is
+  <i>ai</i> unto <i>ae</i>. Therefore equiordinately, as <i>sa</i> is to
+  <i>au</i>: so is <i>ao</i>, to <i>ae</i>. Wherefore the Diagonall
+  <i>su</i> is proportionall in the shankes of equall angles to the
+  parallelogramme <i>oe</i>.</p>
+
+  <p>The demonstration shall be the same of the Diagonall <i>rl</i>. The
+  like situation is manifest, by the <a href="#21_e_iiij">21 e iiij</a>.
+  And from hence also is manifest, That the diagonall of a Quadrate, is a
+  Quadrate: Of an Oblong, an Oblong: Of a Rhombe, a Rhombe: Of a
+  Rhomboides, a Rhomboides: because it is like unto the whole, and a like
+  situate.</p>
+
+  <p>Now the Diagonalls seeing they are like unto the whole and a like
+  situate, they shall also be like betweene themselves and alike situate
+  one to another, by the <a href="#21_e_iiij">21</a> and <a
+  href="#22_e_iiij">22 e iiij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figleft" style="width:18%;">
+      <a href="images/164.png"><img style="width:100%" src="images/164.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p><a name="18_e_x"></a> 18. <i>If the particular parallelogramme have
+  one and the same angle with the whole, be like and alike situate unto it,
+  it is the Diagonall. 26 p vj</i>.</p>
+
+  <p>This might have beene drawn, as a consectary, out of the former: But
+  it may also as it is by Euclide be forced, by an argument <i>ab
+  impossibili</i>. For otherwise the whole should be equall to the part,
+  which is impossible.</p>
+
+  <p>As for example, Let the particular parallelogramme <i>auys</i>, be
+  <!-- Page 144 --><span class="pagenum"><a
+  name="page144"></a>[144]</span>coangular to the whole parallelogramme
+  <i>aeio</i>; And let it have the same angle with it at <i>a</i>; like
+  unto the whole and alike situate unto it; I say it is the Diagonall.</p>
+
+  <p>Otherwise, let the diverse Diagony be <i>aro</i>: And let <i>lr</i> be
+  parallell against <i>ae</i>: Therefore <i>alrs</i>, shall bee the
+  Diagonall, by the 6 e [<a href="#16_e_x"><span class="correction"
+  title="text reads `15'">16</span></a>.] Now therefore it shall be, by 8 e
+  [<a href="#17_e_x"><span class="correction" title="text reads `16 e'">17
+  e,</span></a>] as <i>ea</i> is to <i>ai</i>: so is <i>sa</i> unto
+  <i>al</i>: Againe,by the grant, as <i>ea</i> is unto <i>ai</i>: so is
+  <i>sa</i> to <i>au</i>: Therefore the same <i>sa</i> is proportionall to
+  <i>al</i>, and to <i>au</i>: And <i>al</i> is equall to <i>au</i>, the
+  part to the whole, which is impossible.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/165a.png"><img style="width:100%" src="images/165a.png"
+      alt="Complements." title="Complements." /></a>
+  </div>
+  <p><a name="19_e_x"></a> 19. <i>The Complement is a particular
+  parallelogramme, comprehended of the conterminall sides of the
+  diagonals</i>.</p>
+
+  <p>Or thus: It is a particular parallelogramme conteined under the next
+  adjoyning sides of the diagonals.</p>
+
+  <p>As in this figure, are <i>ur</i>, and <i>sy</i>: For each of them is
+  comprehended of the continued sides of the two diagonals. And therefore
+  are they called Complements, because they doe with the Diagonals
+  <i>complere</i>, that is, fill or make up the whole parallelogramme.
+  Neither in deed may the two diagonals be described, but withall the
+  complements must needes be described.</p>
+
+  <div class="figleft" style="width:18%;">
+      <a href="images/165b.png"><img style="width:100%" src="images/165b.png"
+      alt="Figure for demonstration 20." title="Figure for demonstration 20." /></a>
+  </div>
+  <p><a name="20_e_x"></a> 20. <i>The complements are equall. 43 p
+  j</i>.</p>
+
+  <p>As in the same figure, are the sayd <i>ur</i>, and <i>sr</i>: For the
+  triangles <i>aei</i>, and <i>aoi</i>, are equall, by the <a
+  href="#12_e_x">12 e</a>. Item, so are <i>asl</i>, and <i>aul</i>: Item,
+  so are <i>lui</i>, and <i>lri</i>. Therefore if you shall on each side
+  take away equall triangles from those which are <!-- Page 145 --><span
+  class="pagenum"><a name="page145"></a>[145]</span>equall, you shall leave
+  the Complements equall betweene themselves.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="21_e_x"></a> 21. <i>If one of the Complements be made equall
+  to a triangle given, in a right-lined angle given, the other made upon a
+  right line given shall be in like manner equall to the same triangle. 44
+  p j</i>.</p>
+
+  <div class="figright" style="width:25%;">
+      <a href="images/166.png"><img style="width:100%" src="images/166.png"
+      alt="Figure for demonstration 21." title="Figure for demonstration 21." /></a>
+  </div>
+  <p>As if thou shouldest desire to have a parallelogramme upon a right
+  line given, and in a right lined angle given, to be made equall to a
+  triangle given, this proposition shall give satisfaction.</p>
+
+  <p>Let <i>aei</i> be the Triangle given: The Angle be <i>o</i>: And the
+  right line given be <i>iu</i>: And the Parallelogramme <i>ay</i> is
+  equall to <i>aei</i>, triangle given in the angle assigned, by the <a
+  href="#13_e_x">13 e</a>. Then let the side <i>ay</i>, bee continued to
+  <i>r</i>, equally to <i>iu</i>, the line given: And let <i>ru</i> be knit
+  by a right line: And from <i>r</i> drawne out a diagony untill it doe
+  meete with <i>as</i>, infinitely continued; which shall meete with it, by
+  the <a href="#19_e_v">19 e v</a>, in <i>l</i>. And the sides <i>yi</i>,
+  and <i>ru</i>, let them be continued equally to <i>sl</i>. in <i>m</i>
+  and <i>n</i>. And knit <i>ln</i> together with a right line. This
+  complement <i>mu</i>, is equall to the complement <i>ys</i>, which is
+  equall to the Triangle assigned, by the former, and that in a right lined
+  angle given.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="22_e_x"></a> 22 <i>If parallelogrammes be continually made
+  equall to all the triangles of an assigned triangulate, in a right lined
+  angle given, the whole parallelogramme shall in like manner be equall to
+  the whole triangulate. 45 p j</i>.</p>
+
+  <p>This is a corollary of the former, of the Reason or rate of a
+  Parallelogramme with a Triangulate; and it needeth no <!-- Page 146
+  --><span class="pagenum"><a name="page146"></a>[146]</span>farther
+  demonstration; but a ready and steddy hand in describing and working of
+  it.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/167a.png"><img style="width:100%" src="images/167a.png"
+      alt="Triangulate." title="Triangulate." /></a>
+  </div>
+  <p>Take therefore an infinite right line; upon the continue the
+  particular parallelogrammes, As if the Triangulate <i>aeiou</i>, were
+  given to be brought into a parallelogramme: Let it be resolved into three
+  triangles, <i>aei</i>, <i>aio</i>, and <i>aou</i>: And let the Angle be
+  <i>y</i>: First in the assigned Angle, upon the Infinite right line, make
+  by the former the Parallelogramme <i>ae</i>, in the angle assigned,
+  equall to <i>aei</i>, the first triangle. Then the second triangle, thou
+  shalt so make upon the said Infinite line, that one of the shankes may
+  fall upon the side of the equall complement; The other be cast on
+  forward, and so forth in more, if neede be.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/167b.png"><img style="width:100%" src="images/167b.png"
+      alt="Complements." title="Complements." /></a>
+  </div>
+  <p>Here thou hast 3 complements continued, and continuing the
+  Parallelogramme: But it is best in making and working of them, to put out
+  the former, and one of the sides of the inferiour or latter Diagonall,
+  least the confusion of lines doe hinder or trouble thee.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="23_e_x"></a> 23. <i>A Parallelogramme is equall to his
+  diagonals and complements</i>.</p>
+
+  <p>For a Parallelogramme doth consist of two diagonals, and as many
+  complements: Wherefore a Parallelogramme is equall to his parts: And
+  againe the parts are equall to their whole.</p>
+
+  <p><a name="24_e_x"></a> 24. <i>The Gnomon is any one of the Diagonall
+  with the two complements</i>.</p>
+
+  <p>There is therefore in every Parallelogramme a double Gnomon; as in
+  these two examples. Of all the space of a <!-- Page 147 --><span
+  class="pagenum"><a name="page147"></a>[147]</span>parallelogramme about
+  his diameter, any parallelogramme with the two complements, let it be
+  called the Gnomon. Therefore the gnomon is compounded, or made of both
+  the kindes of diagonall and complements.</p>
+
+  <div class="figleft" style="width:26%;">
+      <a href="images/168a.png"><img style="width:100%" src="images/168a.png"
+      alt="Gnomons." title="Gnomons." /></a>
+  </div>
+  <p>In the Elements of Geometry there is no other use, as it seemeth of
+  the gnomons than that in one word three parts of a parallelogramme might
+  be signified and called by three letters <i>aei</i>. Otherwise gnomon is
+  a perpendicular.</p>
+
+  <p><a name="25_e_x"></a> 25. <i>Parallelogrames of equall height are one
+  to another as their bases are. 1 p vj</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/168b.png"><img style="width:100%" src="images/168b.png"
+      alt="Figure for demonstration 25." title="Figure for demonstration 25." /></a>
+  </div>
+  <p>As is apparent, by the <a href="#16_e_iiij">16 e iiij</a>. Because
+  they be the double of Triangles, by the <a href="#12_e_x"><span
+  class="correction" title="text reads `10 e'">12 e</span></a>, of first
+  figures: As <i>ae</i>, and <i>ei</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="26_e_x"></a> 26 <i>Parallelogrammes of equall height upon
+  equall bases are equall. 35. 36 pj</i>.</p>
+
+  <p>As is manifest in the same example.</p>
+
+  <p><a name="27_e_x"></a> 27 <i>If equiangle parallelogrammes be
+  reciprocall in the shankes of the equall angle, they are equall: And
+  contrariwise. 15 p vj</i>.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/168c.png"><img style="width:100%" src="images/168c.png"
+      alt="Figure for demonstration 27." title="Figure for demonstration 27." /></a>
+  </div>
+  <p>It is a consectary drawne out of the <a href="#11_e_vij">11 e vij</a>:
+  As here thou seest: And yet indeed both that (as there was sayd) and this
+  is rather a consectary of the <a href="#18_e_iiij">18 e iiij</a>, which
+  here also is more manifest.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+<p><!-- Page 148 --><span class="pagenum"><a name="page148"></a>[148]</span></p>
+
+  <p><a name="28_e_x"></a> 28 <i>If foure right lines be proportionall, the
+  parallelogramme made of the two middle ones, is equall to the equiangled
+  parallelogramme made of the first and last: And contrariwise, e 16 p
+  vj</i>.</p>
+
+  <p>For they shall be equiangled parallelogrammes reciprocall in the
+  shankes of the equall angle.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+  <p><a name="29_e_x"></a> 29 <i>If three right lines be proportionall, the
+  parallelogramme of the middle one is equall to the equiangled
+  parallelogramme of the extremes: And contrariwise</i>.</p>
+
+  <p>It is a consectary drawne out of the former.</p>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i>, the eleventh Booke,
+of a Right angle.</h2>
+
+  <p><a name="1_e_xj"></a> 1. <i>A Parallelogramme is a Right angle or an
+  Obliquangle</i>.</p>
+
+  <p>Hitherto we have spoken of certaine common and generall matters
+  belonging unto parallelogrammes: specials doe follow in Rectangles and
+  Obliquangles, which difference, as is aforesaid, is common to triangles
+  and triangulates. But at this time we finde no fitter words whereby to
+  distinguish the generals.</p>
+
+  <p><a name="2_e_xj"></a> 2. <i>A Right angle is a parallelogramme that
+  hath all his angles right angles</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/169.png"><img style="width:100%" src="images/169.png"
+      alt="A Right angle." title="A Right angle." /></a>
+  </div>
+  <p>As in <i>aeio</i>. And here hence you must understand by one right
+  angle that all are right angles. For the right angle at <i>a</i>, is
+  equall to the opposite angle at <i>i</i>, by the <a href="#10_e_x">10 e
+  x</a>.</p>
+
+  <p>And therefore they are both right angles, by the <a
+  href="#14_e_iij">14 e iij</a>. The other angle at <i>e</i>, and <i>o</i>,
+  by the <a href="#4_e_vj">4 e vj</a>, are equall to two right angles: And
+  they are equall betweene themselves, by the <a href="#10_e_x">10 e x</a>.
+  Therefore all of them are right angles. Neither <!-- Page 149 --><span
+  class="pagenum"><a name="page149"></a>[149]</span>may it indeed possible
+  be, that in a parallelogramme there should be one right angle, but by and
+  by they must be all right angles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="3_e_xj"></a> 3 <i>A rightangle is comprehended of two right
+  lines comprehending the right angle 1. d ij</i>.</p>
+
+  <p><i>Comprehension</i>, in this place doth signifie a certaine kind of
+  Geometricall multiplication. For as of two numbers multiplied betweene
+  themselves there is made a number: so of two sides (<i>ductis</i>) driven
+  together, a right angle is made: And yet every right angle is not
+  rationall, as before was manifest, at the <a href="#12_e_iiij">12. e
+  iiij</a>, and shall after appeare at the <a href="#9_e_xj"><span
+  class="correction" title="text reads `8 e'">9 e</span></a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="4_e_xj"></a> 4 <i>Foure right angles doe fill a
+  place</i>.</p>
+
+  <p>Neither is it any matter at all whether the foure rectangles be
+  equall, or unequall; equilaters, or unequilaters; homogeneals, or
+  heterogenealls. For which way so ever they be turned, the angles shall be
+  right angles: And therefore they shall fill a place.</p>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/170.png"><img style="width:100%" src="images/170.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p><a name="5_e_xj"></a> 5 <i>If the diameter doe cut the side of a right
+  angle into two aquall parts, it doth cut it perpendicularly: And
+  contrariwise</i>.</p>
+
+  <p>As here appeareth by the <a href="#1_e_vij">1 e vij</a>. by drawing of
+  the diagonies of the bisegments. The converse is manifest, by the <a
+  href="#2_e_vij">2 e vij</a>. and <span class="correction" title="wrong reference but I cannot deduce the correction."
+  >17. e vij</span>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="6_e_xj"></a> 6 <i>If an inscribed right line doe
+  perpendicularly cut the side of the right angle into two equall parts, it
+  is the diameter</i>.</p>
+
+  <p>The reason is, because it doth cut the parallelogramme into two equall
+  portions.</p>
+
+  <p><a name="7_e_xj"></a> 7 <i>A right angle is equall to the rightangles
+  <!-- Page 150 --><span class="pagenum"><a
+  name="page150"></a>[150]</span>made of one of his sides and the segments
+  of the other</i>.</p>
+
+  <p>As here the foure particular right angles are equall to the whole,
+  which are made of <i>ae</i>, one of his sides, and of <i>ei</i>,
+  <i>io</i>, <i>ou</i>, <i>uy</i>, the segments of the other.</p>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/171a.png"><img style="width:100%" src="images/171a.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>The Demonstration of this is from the rule of congruency: Because the
+  whole agreeth to all his parts. But the same reason in numbers is more
+  apparent by an induction of the parts: as foure times eight are 32. I
+  breake or divide 8. into 5. and 3. Now foure times 5. are 20. And foure
+  times 3. are 12. And 20. and 12. are 32. And 32. and 32. are equall.
+  Therefore 20. and 12. are also equall to 32.</p>
+
+  <p>Lastly, every arithmeticall multiplication of the whole numbers doth
+  make the same product, that the multiplication of the one of the whole
+  numbers given, by the parts of the other shall make: yea, that the
+  multiplication of the parts by the parts shall make. This proportion is
+  cited by <i>Ptolomey</i> in the 9. Chapter of the 1 booke of his
+  Almagest.</p>
+
+  <p><a name="8_e_xj"></a> 8 <i>If foure right lines be proportionall, the
+  rectangle of the two middle ones, is equall to the rectangle of the two
+  extremes. 16. p vj</i>.</p>
+
+  <div class="figcenter" style="width:28%;">
+      <a href="images/171b.png"><img style="width:100%" src="images/171b.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>It is a speciall consectary out of the <a href="#28_e_x">28 e x</a>.
+  As here are foure right lines proportionall betweene themselves: And the
+  rectangle of the extremes, or first and last let it be <i>ay</i>: Of the
+  middle ones, let it be <i>se</i>.</p>
+
+  <p><a name="9_e_xj"></a> 9 <i>The figurate of a rationall rectangle is
+  called a rectinall plaine. 16. d vij</i>.</p>
+
+  <p>A rationall figure was defined at the <a href="#12_e_iiij">12. e
+  iiij</a>. of which <!-- Page 151 --><span class="pagenum"><a
+  name="page151"></a>[151]</span>sort amongst all the rectilineals hitherto
+  spoken of, we have not had one: The first is a Right angled
+  parallelogramme; And yet not every one indifferently: But that onely
+  whose base is rationall to the highest: And that reason of the base and
+  heighth is expressable by a number, where also the Figurate is defined. A
+  rectangle or irrational sides, such as were mentioned at the <a
+  href="#9_e_j">9 e j</a>. is irrationall. Therefore a rectangled rationall
+  of rationall sides, is here understood: And the figurate thereof, is
+  called, by the generall name, A <i>Plaine:</i> Because of all the kindes
+  of <i>Plaines</i>, this kinde onely is rationall.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/172b.png"><img style="width:100%" src="images/172b.png"
+      alt="Rectangled parallellogramme of 40. square foote." title="Rectangled parallellogramme of 40. square foote." /></a>
+  </div>
+  <div class="figright" style="width:13%;">
+      <a href="images/172a.png"><img style="width:100%" src="images/172a.png"
+      alt="Rectinall plaine." title="Rectinall plaine." /></a>
+  </div>
+  <p>If therefore the Base of a Rectangle be 6. And the height 4. The plot
+  or content shall be 24. And if it be certaine that the rectangles content
+  be 24. And the base be 6. It shall also be certaine that the heighth is
+  4. The example is thus.</p>
+
+  <p>And this multiplication, as appeared at the <a href="#13_e_iiij">13. e
+  iiij</a>. is geometricall: As if thou dost multiply 5. by 8. thou makest
+  40. for the Plaine: And the sides of this Plaine, are 5. and 8. it is all
+  one as if thou hadst made a rectangled parallellogramme of 40. square
+  foote content, whose base should be 5. foote, and the heigth 8. after
+  this manner.</p>
+
+  <p>This manner of multiplication, say I, is Geometricall: Neither are
+  there here, of lines made lines, as there of unities were made unities;
+  but a magnitude one degree higher, to wit, a surface, is here made.</p>
+
+  <p>Here hence is the <i>Geodesy</i> or manner of measuring of a
+  rectangled triangle made knowne unto us. For when thou shalt multiply the
+  shankes of a right angle, the one by the other, thou dost make the whole
+  rectangled parallelogramme, whose halfe is a triangle, by the <a
+  href="#12_e_x">12. e x</a>.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<p><!-- Page 152 --><span class="pagenum"><a name="page152"></a>[152]</span></p>
+
+<h2>Of Geometry the twelfth Booke,
+Of a Quadrate.</h2>
+
+  <p><a name="1_e_xij"></a> 1 <i>A Rectangle is a Quadrate or an
+  Oblong</i>.</p>
+
+  <p>This division is made in the proper termes: but the thing it selfe and
+  the subject difference is common out of the angles and sides.</p>
+
+  <p><a name="2_e_xij"></a> 2 <i>A Quadrate is a rectangle equilater 30. d
+  j</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/173.png"><img style="width:100%" src="images/173.png"
+      alt="Quadrate." title="Quadrate." /></a>
+  </div>
+  <p><i>Quadratum</i>, a Quadrate, or square, is a rectangled
+  parallellogramme of equall sides: as here thou seest <i><span
+  class="correction" title="text reads `aeao'">aeio</span></i>, to be.</p>
+
+  <p>Plaines are with us, according to their diverse natures and qualities,
+  measured with divers and sundry kindes of measures. Boord, Glasse, and
+  Paving-stone are measured by the foote: Cloth, Wainscote, Painting,
+  Paving, and such like, by the yard: Land, and Wood, by the Perch or
+  Rodde.</p>
+
+  <p>Of Measures and sundry sorts thereof commonly used and mentioned in
+  histories we have in the former spoken at large: Yet for the farther
+  confirmation of some thing then spoken, and here againe now upon this
+  particular occasion repeated, it shall not be amisse to heare what our
+  Statutes speake of these three sorts here mentioned.</p>
+
+  <p>It is ordained, saith the Statute, That three Barley-cornes dry and
+  round, doe make an <i>Ynch</i>: twelve ynches doe make a <i>Foote</i>:
+  three foote doe make a <i>Yard</i>: Five yards and an halfe doe make a
+  <i>Perch</i>: Fortie perches in length, and foure in breadth doe make an
+  <i>Aker</i>. <i>33. Edwardi 1. De Terris mensurandis.</i> Item, <i>De
+  compositione Ulnarum &amp; Perticarum</i>.</p>
+
+  <p>Moreover observe, that all those measures there spoken <!-- Page 153
+  --><span class="pagenum"><a name="page153"></a>[153]</span>of were onely
+  lengths: These here now last repeated, are such as the magnitudes by
+  <span class="correction" title="text reads `the'">them</span> measured
+  are, in Planimetry, I meane, they are Plaines: In Stereometry they are
+  solids, as hereafter we shall make manifest. Therefore in that which
+  followeth, An <i>ynch</i> is not onely a length three barley-cornes long:
+  but a plaine three barley-cornes long, and three broad. A <i>Foote</i> is
+  not onely a length of 12. ynches: But a plaine also of 12. ynches square,
+  or containing 144. square ynches: A <i>yard</i> is not onely the length
+  of three foote: But it is also a plaine 3. foote square every way. A
+  <i>Perch</i> is not onely a length of 5�. yards: But it is a plot of
+  ground 5�. yards square every way.</p>
+
+  <p>A Quadrate therefore or square, seeing that it is equilater that is of
+  equall sides: And equiangle by meanes of the equall right angles, of
+  <span class="correction" title="text reads `quandrangles'"
+  >quadrangles</span> that onely is ordinate.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="3_e_xij"></a> 3 <i>The sides of equall quadrates, are
+  equall</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p>The sides of equall quadrates are equally compared: If therefore two
+  or more quadrates be equall, it must needs follow that their sides are
+  equall one to another.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="4_e_xij"></a> 4 <i>The power of a right line is a
+  quadrate</i>.</p>
+
+  <p>Or thus: The possibility of a right line is a square <i>H</i>. A right
+  line is said <i>posse quadratum</i>, to be in power a square; because
+  being multiplied in it selfe, it doth make a square.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/175.png"><img style="width:100%" src="images/175.png"
+      alt="Figure for demonstration 4." title="Figure for demonstration 4." /></a>
+  </div>
+  <p><a name="5_e_xij"></a> 5 <i>If two conterminall perpendicular equall
+  right lines be closed with parallells, they shall make a quadrate. 46. p.
+  j</i>.</p>
+
+  <p>Or thus: If two equall perpendicular lines, ioyning one with another,
+  be inclosed together by parallell lines they will make a square.
+  <i>H</i>. As in <i>aeio</i>, let the perpendiculars <i>ae</i>, and
+  <i>ei</i>, equall betweene themselves, be closed with two parallells,
+  <i>ao</i>, against <i>ei</i>: And <i>oi</i>, against <i>ae</i>; they <!--
+  Page 154 --><span class="pagenum"><a name="page154"></a>[154]</span>shall
+  make the quadrate or square <i>aeio</i>. For it is a parallelogramme, by
+  the grant: Because the opposite sides are parallell: And it is
+  rectangled: because seeing the angle <i>aei</i>, of the perpendicular
+  lines, is a right angle, they shall be all right angles by the <a
+  href="#2_e_xj">2 e xj</a>. Then one side <i>ei</i>, is equall to all the
+  rest. First to <i>ao</i>, that over against it, by the <a href="#8_e_x">8
+  e x</a>. And then to <i>ea</i>, by the grant: And therefore to <i>oi</i>,
+  to that over against it, by the <a href="#8_e_x">8 e x</a>.</p>
+
+  <p><a name="6_e_xij"></a> 6 <i>The plaine of a quadrate is an equilater
+  plaine</i>.</p>
+
+  <p>Or thus: The plaine number of a square, is a plaine number of equall
+  sides, <i>H</i>.</p>
+
+  <p>A quadrate or square number, is that which is equally equall: Or that
+  which is comprehended of two equall numbers, A quadrate of all plaines is
+  especially rationall; and yet not alwayes: But that onely is rationall
+  whose number is a quadrate. Therefore the quadrates of numbers not
+  quadrates, are not rationalls.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="7_e_xij"></a> 7 <i>A quadrate is made of a number multiplied
+  by it self</i>.</p>
+
+  <p>Such quadrates are the first nine. 1, 4, 9, 16, 25, 36, 49, 64, 81,
+  made of once one, twice two, thrise three, foure times foure, five times
+  five, sixe times sixe, seven times seven, eight times eight, and nine
+  times nine. And this is the summe of the making and invention of a
+  quadrate number of multiplication of the side given by it selfe.</p>
+
+<table class="nobctr">
+<tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9.</td><td>The sides given.</td></tr>
+<tr><td>1 &nbsp;</td><td>4 &nbsp;</td><td>9 &nbsp;</td><td>16</td><td>25</td><td>36</td><td>49</td><td>64</td><td>81.</td><td>The quadrates found.</td></tr>
+</table>
+
+  <p>Hereafter diverse comparisons of a quadrate or square, with a
+  rectangle, with a quadrate, <span class="correction" title="text reads `aud'"
+  >and</span> with a rectangle and a quadrate iointly. The comparison or
+  rate of a quadrate with a rectangle is first.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+<p><!-- Page 155 --><span class="pagenum"><a name="page155"></a>[155]</span></p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/176a.png"><img style="width:100%" src="images/176a.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p><a name="8_e_xij"></a> 8 <i>If three right lines be proportionall, the
+  quadrate of the middle one, shall be equall to the rectangle of the
+  extremes: And contrariwise: 17. p vj. and 20. p vij</i>.</p>
+
+  <p>Or thus: If three lines be proportionall, the square made of the
+  middle line is equall to the right angled parallelogramme made of the two
+  outmost lines: <i>H</i>.</p>
+
+  <p>It is a corallary out of the <a href="#28_e_x">28. e x</a>. As in
+  <i>ae</i>, <i>ei</i>, <i>io</i>.</p>
+
+  <p><a name="9_e_xij"></a> 9 <i>If the base of a triangle doe subtend a
+  right angle, the powre of it is as much as of both the shankes: And
+  contrariwise 47, 48. p j</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/176b.png"><img style="width:100%" src="images/176b.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p>It is a consectary out of the <a href="#11_e_viij">11. e viij</a>. But
+  it is sometime rationall, and to be expressed by a number: yet but in a
+  scalene triangle onely. For the sides of an equicrurall right-angled
+  triangle are irrationall; Whereas the sides of a scalene are sometime
+  rationall; and that after two manners, the one of <i>Pythagoras</i>, the
+  other of <i>Plato</i>, as <i>Proclus</i> teacheth, at the 47. p j.
+  <i>Pythagora's</i> way is thus, by an odde number.</p>
+
+  <div class="figright" style="width:13%;">
+      <a href="images/177a.png"><img style="width:100%" src="images/177a.png"
+      alt="Pythagora's way." title="Pythagora's way." /></a>
+  </div>
+  <p><a name="10_e_xij"></a> 10 <i>If the quadrate of an odde number, given
+  for the first shanke, be made lesse by an unity; the halfe of the
+  remainder shall be the other shanke; increased by an unity it shall be
+  the base</i>.</p>
+
+  <p>Or thus: If the square of an odde number given for the first <!-- Page
+  156 --><span class="pagenum"><a name="page156"></a>[156]</span>foote,
+  have an unity taken from it, the halfe of the remainder shall be the
+  other foote, and the same halfe increased by an unitie, shall be the
+  base: <i>H</i>.</p>
+
+  <p>As in the example: The sides are 3, 4, and 5. And 25. the square of 5.
+  the base, is equall to 16. and 9. the squares of the shanks 4. and 3.</p>
+
+  <p>Againe, the quadrate or square of 3. the first shanke is 9. and 9 - 1.
+  is 8, whose halfe 4, is the other shanke. And 9 + 1, is 10. whose halfe
+  5. is the base. <i>Plato's</i> way is thus by an even number.</p>
+
+  <p><a name="11_e_xij"></a> 11 <i>If the halfe of an even number given for
+  the first shanke be squared, the square number diminished by an unity
+  shall be the other shanke, and increased by an unitie it shall be the
+  base</i>.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/177b.png"><img style="width:100%" src="images/177b.png"
+      alt="Plato's way." title="Plato's way." /></a>
+  </div>
+  <p>As in this example where the sides are 6, 8. and 10. For 100. the
+  square of 10. the base is equall to 36. and 64. the squares of the
+  shankes 6. and 8.</p>
+
+  <p>Againe, the quadrate or square of 3. the halfe of 6, the first shanke,
+  is 9. and 9 - 1, is 8, for the second shanke. And out of this rate of
+  rationall powers (as <i>Vitruvius</i>, in the 2. Chapter of his IX.
+  booke) saith <i>Pythagoras</i> taught how to make a most exact and true
+  squire, by joyning of three rulers together in the forme of a triangle,
+  which are one unto another as 3, 4. and 5. are one to another.</p>
+
+  <p>From hence Architecture learned an Arithmeticall proportion in the
+  parts of ladders and stayres. For that rate or proportion, as in many
+  businesses and measures is very commodious; so also in buildings, and
+  making of ladders or staires, that they may have moderate rises of the
+  steps, it is very speedy. For 9 + 1. is, 10, base. <!-- Page 157 --><span
+  class="pagenum"><a name="page157"></a>[157]</span></p>
+
+  <p><a name="12_e_xij"></a> 12. <i>The power of the diagony is twise
+  asmuch, as is the power of the side, and it is unto it also
+  incommensurable</i>.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/178.png"><img style="width:100%" src="images/178.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>Or thus: The diagonall line is in power double to the side, and is
+  incommensurable unto it, <i>H</i>.</p>
+
+  <p>As here thou seest, let the first quadrate bee <i>aeio:</i> Of whose
+  diagony <i>ai</i>, let there be made the quadrate <i>aiuy:</i> This, I
+  say, shall be the double of that: seeing that the diagonies power is
+  equall to the power of both the equall shankes. Therefore it is double to
+  the power of one of them.</p>
+
+  <p>This is the way of doubling of a square taught by <i>Plato</i>, as
+  <i>Vitruvius</i> telleth us: Which notwithstanding may be also doubled,
+  trebled, or according to any reason assigned increased, by the <a
+  href="#25_e_iiij">25 e iiij</a>, as there was foretold.</p>
+
+  <p>But that the Diagony is incommensurable unto the side it is the 116 p
+  x. The reason is, because otherwise there might be given one quadrate
+  number, double to another quadrate number: Which as <i>Theon</i> and
+  <i>Campanus</i> teach us, is impossible to be found. But that reason
+  which <i>Aristotle</i> bringeth is more cleare which is this; Because
+  otherwise an even number should be odde. For if the Diagony be 4, and the
+  side 3: The square of the Diagony 16, shall be double to the square of
+  the side: And so the square of the side shall be 8. and the same square
+  shall be 9, to wit, the square of 3. And so even shall be odde, which is
+  most absurd.</p>
+
+  <p>Hither may be added that at the 42 p x. That the segments of a right
+  line diversly cut; the more unequall they are the greater is their
+  power.</p>
+
+  <p><a name="13_e_xij"></a> 13. <i>If the base of a right angled triangle
+  be cut by a <!-- Page 158 --><span class="pagenum"><a
+  name="page158"></a>[158]</span>perpendicular from the right angle in a
+  doubled reason, the power of it shall be halfe as much more, as is the
+  power of the greater shanke: But thrise so much as is the power of the
+  lesser. If in a quadrupled reason, it shall be foure times and one fourth
+  so much as is the greater: But five times so much as is the lesser, At
+  the 13, 15, 16 p xiij</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/179a.png"><img style="width:100%" src="images/179a.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p>Or thus: If the base of a right angled triangle be cut in double
+  proportion, by a perpendicular comming from the right angle, it is in
+  power sesquialter to the greater foote; and treble to the lesser: But if
+  the base be cut in quadruple proportion, it is sesquiarta to the greater
+  side, and quintuple to the lesser.</p>
+
+  <p>As in <i>aei</i>, let the base <i>ae</i>, be so cut that the segment
+  <i>ao</i>, be double to the segment <i>oe</i>, to wit, as 2 is to 1. The
+  whole <i>ae</i>, shall be unto <i>ao sesquialtera</i>, that is, as 2 is
+  to 3. And therefore by the <a href="#10_e_viij">10 e viij</a>, and <a
+  href="#25_e_iiij">25 e iiij</a>, the square of <i>ae</i>, shall be
+  <i>sesquialterum</i> unto the square of <i>ai</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/179b.png"><img style="width:100%" src="images/179b.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p>And by the same argument it shall be treble unto the quadrate or
+  square of <i>ei</i>.</p>
+
+  <p>The other, of the fourefold or quadruple section, are manifest in the
+  figure following, by the like argument.</p>
+
+  <p><a name="14_e_xij"></a> 14 <i>If a right line be cut into how many
+  parts so ever, the power of it is manifold unto the power of segment,
+  denominated of the square of the number of the section</i>.</p>
+
+  <p>Or thus: if a right <span class="correction" title="omitted in text"
+  >line</span> be cut into how many parts soever it is in power the
+  multiplex of the segment, the square of the number of the section, being
+  denominated thereof: <i>H</i>. <!-- Page 159 --><span class="pagenum"><a
+  name="page159"></a>[159]</span></p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/180a.png"><img style="width:100%" src="images/180a.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>So if it be cut into two equall parts, the power of it shall be foure
+  times so much, as is the power of the halfe, taking demonstration from 4,
+  which is the square of 2, according to which the division was made: If it
+  be cut into three equall parts, the power of it shall be nine fold the
+  power of the third part. If into foure equall parts, it shall be 16 times
+  so much as is the power of the quarter: As here thou seest in these
+  examples.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/180b.png"><img style="width:100%" src="images/180b.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p><a name="15_e_xij"></a> 15. <i>If a right line be cut into two
+  segments, the quadrate of the whole is equall to the quadrats of the
+  segments, and a double rectanguled figure, made of them both. 4 p
+  ij</i>.</p>
+
+  <p>The third rate of a quadrate is hereafter with two rectangles, and two
+  quadrates, and first of equality.</p>
+
+  <p>This is a consectary out of the <a href="#22_e_x">22 e x</a>: Because
+  a parallelogramme is equall to his two diagonals and complements. If the
+  right <i>ae</i>, be cut in <i>i</i>, it maketh the quadrate <i>aeuo</i>,
+  greater than <i>eyi</i>, and <i>yus</i>, the quadrates of the segments,
+  by the two rectangles <i>ay</i> and <i>yo</i>. This is the rate of a
+  quadrate with a rectangle and a quadrate. But the side of a quadrate
+  proposed in a number is oft times sought. Therefore by the <!-- Page 160
+  --><span class="pagenum"><a name="page160"></a>[160]</span>next precedent
+  element and his consectaries, the analysis or finding of the side of a
+  quadrate is made and taught.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="16_e_xij"></a> 16. <i>The side of the first diagonall, is the
+  side of one of the complements; And being doubled, it is the side of them
+  both together: Now the other side of the same complements both together,
+  is the side of the other diagonall</i>.</p>
+
+  <p>The side of a quadrate given is many times in numbers sought.
+  Therefore by the former element and his consectaries the resolution of a
+  quadrates side is framed and performed.</p>
+
+  <p>Let therefore the side now of the quadrate number given be sought: And
+  first let the Genesis or making be considered, such as you see here by
+  the multiplication of numbers in the numbers themselves:</p>
+
+<table class="nobctr">
+<tr><td align="right">10 &nbsp; 2</td><td align="right">10 &nbsp; 2</td><td><i> &nbsp; The number or</i></td></tr>
+<tr><td align="right">10 &nbsp; 2</td><td align="right">10 &nbsp; 2</td><td><i> &nbsp; side divided.</i></td></tr>
+<tr><td align="right">&mdash;&mdash;&mdash;</td><td align="right">&mdash;&mdash;&mdash;</td></tr>
+<tr><td align="right">2 &nbsp; 4</td><td align="right">4</td><td><i> &nbsp; The lesser diagonall.</i></td></tr>
+<tr><td align="right">2 &nbsp; 0</td><td align="right"> &nbsp; &nbsp; &nbsp; Or thus &nbsp; &nbsp; &nbsp; 2 &nbsp; 0</td><td><i> &nbsp; One Complement.</i></td></tr>
+<tr><td align="right">1 &nbsp; 0 &nbsp; 0</td><td align="right">2 &nbsp; 0</td><td><i> &nbsp; Th'other Complement</i></td></tr>
+<tr><td align="right">&mdash;&mdash;&mdash;</td><td align="right">1 &nbsp; 0 &nbsp; 0</td><td><i> &nbsp; The greater diagonall.</i></td></tr>
+<tr><td align="right">1 &nbsp; 4 &nbsp; 4</td><td align="right">&mdash;&mdash;&mdash;</td></tr>
+<tr><td align="right">&nbsp;</td><td align="right">1 &nbsp; 4 &nbsp; 4</td><td><i> &nbsp; The quadrate.</i></td></tr>
+</table>
+
+  <p>This is the rate of a quadrate with a rectangle &amp; a quadrate, from
+  whence is had the analisis or resolution of the side of a quadrate
+  expressable by a number. For it is the same way fro <i>Cambridge</i> to
+  <i>London</i>, that is from <i>London</i> to <i>Cambridge</i>. And this
+  use of geometricall analysis remaineth, as afterward in a Cube, when as
+  otherwise through the whole booke of <i>Euclides</i> Elements there is no
+  other use at all of that.</p>
+
+  <p>Here therefore thou shalt note or marke out the severall quadrates,
+  beginning at the right hand and so proceeding towards the left; after
+  this manner, 144. These notes doe signifie that so many severall sides to
+  be found, to make up <!-- Page 161 --><span class="pagenum"><a
+  name="page161"></a>[161]</span>the whole side of the quadrate given. And
+  here first, it shall not be amisse to warne thee, before thou commest to
+  practice, that for helpe of memory and speed in working, thou know the
+  Quadrats of the nine single numbers of figures; which are these</p>
+
+<table class="nobctr">
+<tr><td align="right">1.</td><td align="right">2.</td><td align="right">3.</td><td align="right">4.</td><td align="right">5.</td><td align="right">6.</td><td align="right">7.</td><td align="right">8.</td><td align="right">9.</td><td><i>&nbsp; Sides.</i></td></tr>
+<tr><td align="right">1.</td><td align="right">&nbsp; 4.</td><td align="right">&nbsp; 9.</td><td>&nbsp; 16.</td><td>&nbsp; 25.</td><td>&nbsp; 36.</td><td>&nbsp; 49.</td><td>&nbsp; 64.</td><td>&nbsp; 81.</td><td><i>&nbsp; Qu.</i></td></tr>
+</table>
+
+<table><tr><td valign="top">
+
+  <p>Then beginning at the left hand, as in Division, that is where we left
+  in multiplication, and I seeke amongst the squares the greatest conteined
+  in the first periode, which here is 1; And the side of it, which is also
+  1, I place with my quotient: Then I square this quotient, that is I
+  multiply it by it selfe, and the product 1, I sect under the same first
+  periode: Lastly, I subtract it from the same periode, and there remaineth
+  not any thing. Then as in division I set up the figures of the next
+  periode one degree higher.</p>
+
+  <p>Secondly double the side now found, and it shall be 2, which I place
+  in manner of a Divisor, on the left hand, within the semicircle: By this
+  I divide the 40, the two complements or Plaines, and I finde the quotient
+  or second side 2; which I place in the quotient by 1, This side I
+  multiply first quadrate like, that is by it selfe; and I make 4, the
+  lesser Diagonall: And therefore I place under the last 4: Then I multiply
+  the said Divisor 2, by the same 2 the quotient, and I make in like
+  manner, 4 which I place under the dividend, or the first 4. Lastly I
+  subtract these products from the numbers above them, and remaineth
+  nothing. Therefore I say first, That 144, the number given is a quadrate:
+  And more-over, That 12 is the true side of it.</p>
+
+</td><td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td><td valign="top" style="width:15%;">
+
+<table>
+<tr><td align="right">&nbsp;</td><td><del>44</del></td></tr>
+<tr><td align="right"><del>1</del>,</td><td><del>44</del></td><td>(12</td></tr>
+<tr><td align="right">2) <del>1</del>&nbsp;</td></tr>
+<tr><td align="right">&nbsp;</td><td><del>44</del></td></tr>
+</table>
+
+  <p> Or thus:</p>
+
+<table>
+<tr><td align="right">&nbsp;</td><td align="right"><del>44</del></td></tr>
+<tr><td align="right"><del>1</del>,</td><td align="right"><del>44</del></td><td>(12</td></tr>
+<tr><td align="right">20) <del>1</del>&nbsp;</td></tr>
+<tr><td align="right">&nbsp;</td><td align="right"><del>40</del></td></tr>
+<tr><td align="right">&nbsp;</td><td align="right"><del>4</del></td></tr>
+</table>
+</td></tr></table>
+
+<table><tr><td valign="top">
+
+  <p>Againe, let the side of 15129 be sought. First divide it into
+  imperfect periods as before was taught; in this manner: 15129. Then I
+  seeke amongst the former quadrates, for the side of 1, the quadrate of
+  the first periode; and I finde it to be 1: This side I place within the
+  quotient or lunular on the right side: Lastly I subtract 1 from 1, and
+  nothing remaineth. Then I double the said side found; and I make 2: This
+  2, I place for my divisor within the lunular or semicircle on the left
+  hand: By which I divide 5; and I finde the <!-- Page 162 --><span
+  class="pagenum"><a name="page162"></a>[162]</span>quotient 2, which I
+  place by the former quotient: Then I multiply the same 2, first
+  quadratelike by it selfe, and I make 4. Then I multiply the sayd divisour
+  by 2, the quotient, and I make likewise 4: which I place underneath 51.
+  Lastly, I subtract the same 44, from 51, and there remaine 7, over the
+  head of 1; By which I place 29, the last periode remaining.</p>
+
+  <p>Againe I double 12, my whole quotient, and I make 24. By this double I
+  divide 72, the double Complement remaining, and I finde 3 for the side or
+  quotient: First this side I multiply quadratelike by it selfe, and I make
+  9, which I place underneath 9, the last figure of my dividende. Then
+  againe, by the same quotient, or side 3, I multiply 24: my divisour, and
+  I make 72; which I place under 72, the two figures of my Dividende:
+  Lastly I subtract the under figures, from the upper, and there is
+  likewise nothing remaining: Wherefore I say, as afore; that the figurate
+  15129 given, is a square: And the side thereof is 123.</p>
+
+</td><td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td><td valign="top" style="width:15%;">
+
+<table>
+<tr><td align="right" colspan="3"><del>7</del>&nbsp;</td><td align="right"><del>29</del></td></tr>
+<tr><td align="right" colspan="3"><del>51</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="2"><del>1</del>,</td><td align="right"><del>51</del>,</td><td align="right"><del>29</del></td><td>(123</td></tr>
+<tr><td align="right"> 2)</td><td align="right"><del>1</del>&nbsp;</td></tr>
+<tr><td align="right">24)</td><td align="right" colspan="2"><del>44</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="3"><del>7</del>&nbsp;</td><td align="right"><del>29</del></td></tr>
+</table>
+
+  <p>  Or thus:</p>
+
+<table>
+<tr><td align="right" colspan="3"><del>7</del>&nbsp;</td><td align="right"><del>29</del></td></tr>
+<tr><td align="right" colspan="3"><del>51</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="2"><del>1</del>,</td><td align="right"><del>51</del>,</td><td align="right"><del>29</del></td><td>(123</td></tr>
+<tr><td align="right"> 20)</td><td align="right"><del>1</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="3"><del>40</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="3"><del>4</del>&nbsp;</td></tr>
+<tr><td align="right" colspan="3">240)&nbsp; &nbsp;<del>7</del>&nbsp;</td><td align="right"><del>20</del></td></tr>
+<tr><td align="right" colspan="4"><del>9</del></td></tr>
+</table>
+</td></tr></table>
+
+  <p>Sometime after the quadrate now found, in the next places, there is
+  neither any plaine nor square to bee found: Therefore the single side
+  thereof shall be <i>O</i>. As in the quadrate 366025, the whole side is
+  605, consisting of three severall sides, of which the middle one is
+  <i>o</i>.</p>
+
+<table><tr><td valign="top">
+Sometime also the middle plaine doth containe a part of
+the quadrate next following: Therfore if the other side remaining
+be greater than the side of the quadrate following,
+it is to be made equall unto it: As for example, Let the side
+of the quadrate 784, be sought; The side of the first quadrate
+shall be 2, and there shall remaine 3, thus: Then the same
+side doubled is 4 for the quotient; Which is found in 38, the
+double plaine remaining 9 times, for the other side: But this
+side is greater than the side of the next following quadrate:
+Take therefore 1 out of it: And for nine take 8, and place
+it in your quotient; Which 8 multiplyed by it selfe maketh
+64, for the Lesser quadrate: And againe the same multiplyed
+by 4 the divisour maketh 32; the summe of which two
+products 384, subtracted from the remaine 384, leave nothing:
+Therefore 784 is a Quadrate: And the side is 28.
+
+</td><td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td><td valign="top" style="width:15%;">
+
+<table>
+<tr><td>&nbsp;</td><td><del>384</del></td></tr>
+<tr><td align="right">4)</td><td><del>784</del></td><td>&nbsp;(28</td></tr>
+<tr><td>&nbsp;</td><td><del>4</del></td></tr>
+</table>
+
+  <p>Or thus:</p>
+
+<table>
+<tr><td>&nbsp;</td><td><del>384</del></td></tr>
+<tr><td align="right">40)</td><td><del>784</del></td><td>&nbsp;(28</td></tr>
+<tr><td>&nbsp;</td><td><del>4</del></td></tr>
+<tr><td>&nbsp;</td><td><del>320</del></td></tr>
+<tr><td>&nbsp;</td><td align="right"><del>64</del></td></tr>
+</table>
+</td></tr></table>
+
+<p><!-- Page 163 --><span class="pagenum"><a name="page163"></a>[163]</span></p>
+
+  <p><i>And from hence the invention of a meane <span class="correction"
+  title="text reads `proportionll'">proportionall</span>, betweene two
+  numbers given, (if there be any such to be found) is manifest.</i> For if
+  the product of two numbers given be a quadrate, the side of the quadrate
+  shall be the meane proportionall, betweene the numbers given; as is
+  apparent by the golden rule: As for example, Betweene 4. and 9. two
+  numbers given, I desire to know what is the meane proportion. I multiply
+  therefore 4 and 9. betweene themselves, and the product is 36: which is a
+  quadrate number; as you see in the former; And the side is 6. Therefore I
+  say, the meane proportionall betweene 4. and 9. is 6, that is, As 4. is
+  to 6. so is 6. to 9.</p>
+
+  <p>If the number given be not a quadrate, there shall no arithmeticall
+  side, and to be expressed by a number be found: And this figurate number
+  is but the shadow of a Geometricall figure, and doth not indeede expresse
+  it fully, neither is such a quadrate rationall: Yet notwithstanding the
+  numerall side of the greatest square in such like number may be found: As
+  in 148. The greatest quadrate continued is 144 and the side is 12. And
+  there doe remaine 4. Therefore of such kinde of number, which is not a
+  quadrate, there is no true or exact side: Neither shall there ever be
+  found any so neare unto the true one; but there may still be one found
+  more neare the truth. Therefore the side is not to be expressed by a
+  number.</p>
+
+  <p>Of the invention of this there are two wayes: The one is by the
+  <i>Addition of the gnomon</i>; The other is by the Reduction of the
+  number assigned unto parts of some greater denomination. The first is
+  thus:</p>
+
+  <p><a name="17_e_xij"></a> 17 <i>If the side found be doubled, and to the
+  double a unity be added, the whole shall be the gnomon of the next
+  greater quadrate</i>.</p>
+
+  <p>For the sides is one of the complements, and being doubled it is the
+  side of both together. And an unity is the latter diagonall. So the side
+  of 148 is 12.4/25.</p>
+
+  <p>The reason of this dependeth on the same proposition, <!-- Page 164
+  --><span class="pagenum"><a name="page164"></a>[164]</span>from whence
+  also the whole side, is found. For seeing that the side of every quadrate
+  lesser than the next follower differeth onely from the side of the
+  quadrate next above greater than it but by an 1. the same unity, both
+  twice multiplied by the side of the former quadrate, and also once by it
+  selfe, doth make the <i>Gnomon</i> of the greater to be added to the
+  quadrate. For it doth make the quadrate 169. Whereby is understood, that
+  looke how much the numerator 4. is short of the denominatour 25. so much
+  is the quadrate 148. short of the next greater quadrate. For <span
+  class="correction" title="text reads `it'">if</span> thou doe adde 21.
+  which is the difference whereby 4 is short of 25. thou shalt make the
+  quadrate 169. whose side is 13. The second is by the reduction, as I
+  said, of the number given unto parts assigned of some great denomination,
+  as 100. or 1000. or some smaller than those, and those quadrates, that
+  their true and certaine may be knowne: Now looke how much the smaller
+  they are, so much nearer to the truth shall the side found be.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/185.png"><img style="width:100%" src="images/185.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+  <p>Let the same example be reduced unto hundreds squared parts, thus:
+  1480000/10000. The side of 10000. by grant is 100. But the side of
+  1480000. the numerator by the former is 1216. and beside there doe
+  remaine 1344: thus, 1216/100. that is, 12.16/100, or 4/25 which was
+  discovered by the former way. But in the side of the numerator there
+  remained 1344. By which little this second way is more accurate and
+  precise than the first. Yet notwithstanding those remaines are not
+  regarded, because they cannot adde so much as 1/100 part unto the side,
+  found: For neither in deed doe 1344/2426. make one hundred part.</p>
+
+  <p>Moreover in lesser parts, the second way beside the other, doth shew
+  the side to be somewhat greater than the side, by the first way found: as
+  in 7. the side by the first way is 3/25. But by the second way the side
+  of 7. reduced unto thousands quadrates, that is unto 7000000/1000000,
+  that is, 2645/1000, and beside there doe remaine 3975. But 645/1000 are
+  greater than 3/5. <!-- Page 165 --><span class="pagenum"><a
+  name="page165"></a>[165]</span>For 3/5. reduced unto 1000. are but
+  600/1000. Therefore the second way, in this example, doth exceed the
+  first by 45/1000. those remaines 3975. being also neglected.</p>
+
+  <p>Therefore this is the Analysis or manner of finding the side of a
+  quadrate, by the first rate of a quadrate, equall to a double rectangle
+  and quadrate.</p>
+
+<p class="cenhead"><i>The Geodesy or measuring of a Triangle</i>.</p>
+
+  <p>There is one generall Geodesy or way of measuring any manner of
+  triangle whatsoever in <i>Hero</i>, by addition of the sides, halving of
+  the summe, subduction, multiplication, and invention of the quadrates
+  side, after this manner.</p>
+
+  <div class="figleft" style="width:18%;">
+      <a href="images/186.png"><img style="width:100%" src="images/186.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p><a name="18_e_xij"></a> 18 <i>If from the halfe of the summe of the
+  sides, the sides be severally subducted, the side of the quadrate
+  continually made of the halfe, and the remaines shall be the content of
+  the triangle</i>.</p>
+
+  <p>As for example, Let the sides of the triangle <i>aei</i>, be 6. 8. 10:
+  The summe 24. the halfe of the summe 12. From which halfe subduct the
+  sides 6. 8. <span class="correction" title="text reads `18'">10</span>.
+  and let the remaines be 6. 4. 2. Now multiply continually these foure
+  numbers 12. 6. 4. 2. and thou shalt make first of 12. and 6. 72. Of 72.
+  and 4. 288. Lastly, of 288. and 2. 576. And the side of 576. by the <a
+  href="#16_e_xij">16. e.</a> shall be found to be 24. for the content of
+  the triangle: which also here will be found to be true, by multiplying
+  the sides <i>ae</i> and <i>ei</i>, containing the right angle, the one by
+  the other; and then taking the halfe of the product.</p>
+
+  <p>This generall way of measuring a triangle is most easie and speedy,
+  where the sides are expressed by whole numbers.</p>
+
+  <p>The speciall geodesy of rectangle triangle was before taught (at the
+  <a href="#9_e_xj">9 e xj</a>.) But of an oblique angle it shall hereafter
+  be spoken. But the generall way is farre more <!-- Page 166 --><span
+  class="pagenum"><a name="page166"></a>[166]</span>excellent than the
+  speciall; For by the reduction of an obliquangle many fraudes and errours
+  doe fall out, which caused the learned <i>Cardine</i> merrily to wish,
+  that hee had but as much land as was lost by that false kinde of
+  measuring.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/187.png"><img style="width:100%" src="images/187.png"
+      alt="Figure for demonstration 19." title="Figure for demonstration 19." /></a>
+  </div>
+  <p><a name="19_e_xij"></a> 19 <i>If the base of a triangle doe subtend an
+  obtuse angle, the power of it is more than the power of the shankes, by a
+  double right angle of the one, and of the continuation from the said
+  obtusangle unto the perpendicular of the toppe. 12. p ij</i>.</p>
+
+  <p>Or thus: If the base of a triangle doe subtend an obtuse angle, it is
+  in power more than the feete, by the right angled figure twise taken,
+  which is contained under one of the feete and the line continued from the
+  said foote unto the perpendicular drawne from the toppe of the triangle.
+  <i>H</i>.</p>
+
+  <p>There is a comparison of a quadrate with two in like manner triangles,
+  and as many quadrates, but of unequality.</p>
+
+  <p>As in the triangle <i>aei</i>, the quadrate of the base <i>ai</i>, is
+  greater in power, than the quadrates of the shankes <i>ae</i>, and
+  <i>ei</i>, by double of the rectangle <i>ar</i>, which is made of
+  <i>ae</i>, one of the shankes, and of <i>eo</i>, the continuation of the
+  same <i>ae</i>, unto <i>o</i>, the perpendicular of the toppe
+  <i>i</i>.</p>
+
+  <p>For by <a href="#9_e_xij">9. e</a>, the quadrate of <i>ai</i>, is
+  equall to the quadrates of <i>ao</i>, and <i>oi</i>, that is, to three
+  quadrates, of <i>io</i>, <i>oe</i>, <i>ea</i>, and the double rectangle
+  aforesaid. But the quadrates of the shankes <i>ae</i>, <i>ei</i>, are
+  equall to those three quadrates, to wit, of <i>ai</i>, his owne quadrate,
+  and of <i>ei</i>, two, the first <i>io</i>, the second <i>oe</i>, by the
+  <a href="#9_e_xij">9. e.</a> Therefore the excesse remaineth of a double
+  rectangle.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<p><!-- Page 167 --><span class="pagenum"><a name="page167"></a>[167]</span></p>
+
+<h2>Of Geometry, the thirteenth Booke,
+Of an Oblong.</h2>
+
+  <div class="figright" style="width:15%;">
+      <a href="images/188a.png"><img style="width:100%" src="images/188a.png"
+      alt="Oblong." title="Oblong." /></a>
+  </div>
+  <p><a name="1_e_xiij"></a> 1 <i>An Oblong is a rectangle of inequall
+  sides, 31. d j</i>.</p>
+
+  <p>Or thus: An Oblong is a rectangled parallelogramme, being not
+  equilater: <i>H</i>. As here is <i>ae</i>, <i>io</i>.</p>
+
+  <p>This second kinde of rectangle is of Euclide in his elements properly
+  named for a definitions sake onely.</p>
+
+  <p>The rate of Oblongs is very copious, out of a threefold section of a
+  right line given, sometime rationall and expresable by a number: The
+  first section is as you please, that is, into two segments, equall or
+  unequall: From whence a five-fold rate ariseth.</p>
+
+  <p><a name="2_e_xiij"></a> 2 <i>An oblong made of an whole line given,
+  and of one segment of the same, is equall to a rectangle made of both the
+  segments, and the square of the said segment. 3. p ij</i>.</p>
+
+  <div class="figleft" style="width:24%;">
+      <a href="images/188b.png"><img style="width:100%" src="images/188b.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p>It is a consectary out of the <a href="#7_e_xj">7 e xj</a>. For the
+  rectangle of the segments, and the quadrate, are made of one side, and of
+  the segments of the other.</p>
+
+  <p>As let the right line <i>ae</i>, be 6. And let it be cut into two
+  parts <i>ai</i>, 2. and <i>ie</i>, 4. The rectangle 12. made of
+  <i>ae</i>, 6. the whole, and of <i>ai</i>, 2. the one segment, shall be
+  equall to <i>iu</i>, 8. the rectangle made <!-- Page 168 --><span
+  class="pagenum"><a name="page168"></a>[168]</span>of the same <i>ai</i>,
+  2. and of <i>ie</i>, 4. And also to <i>ao</i>, 4. the quadrate of the
+  said segment <i>ai</i>, 2.</p>
+
+  <p>Now a rectangle is here therefore proposed, because it may be also a
+  quadrate, to wit, if the line be cut into to equall parts.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Secondarily,</p>
+    </div>
+  </div>
+  <div class="figright" style="width:24%;">
+      <a href="images/189a.png"><img style="width:100%" src="images/189a.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p><a name="3_e_xiij"></a> 3 <i>Oblongs made of the whole line given, and
+  of the segments, are equall to the quadrate of the whole 2 p ij</i>.</p>
+
+  <p>This is also a Consectary out of the <a href="#4_e_xj"><span
+  class="correction" title="text reads `4. e xj'">7. e xj</span></a>.</p>
+
+  <p>As let the line <i>ae</i>, 6. be cut into <i>ai</i>, 2. <span
+  class="correction" title="text reads `10'"><i>io</i></span>. 2. and
+  <i>oe</i>, 2. The Oblongs <i>as</i>, 12. <i>ir</i>, 12. and <i>oy</i>,
+  12. made of the whole <i>ae</i>, and of those segments, are equall to
+  <i>ay</i>, the quadrates of the whole.</p>
+
+  <p>Here the segments are more than two, and yet notwithstanding from the
+  first the rest may be taken for one, seeing that the particular rectangle
+  in like manner is equall to them. This proposition is used in the
+  demonstration of the <a href="#9_e_xviij">9. e xviij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Thirdly,</p>
+    </div>
+  </div>
+  <p><a name="4_e_xiij"></a> 4 <i>Two Oblongs made of the whole line given,
+  and of the one segment, with the third quadrate of the other segment, are
+  equall to the quadrates of the whole, and of the said segment. 7 p
+  ij</i>.</p>
+
+  <div class="figleft" style="width:25%;">
+      <a href="images/189b.png"><img style="width:100%" src="images/189b.png"
+      alt="Figure for demonstration 4." title="Figure for demonstration 4." /></a>
+  </div>
+  <p>As for example, let the right line <i>ae</i>, 8. be cut into
+  <i>ai</i>, 6. and <i>ie</i>, 2. The oblongs <i>ao</i>, and <i>iu</i>, of
+  the whole, and 2. the segments, are 32. The quadrate of 6. the other
+  segment 36. And the whole 68. Now the quadrate, of the whole <i>ae</i>.
+  8. is 64. And the quadrate of the said segment 2, is 4. And the summe of
+  these is 68. <!-- Page 169 --><span class="pagenum"><a
+  name="page169"></a>[169]</span></p>
+
+  <p><a name="5_e_xiij"></a> 5. <i>The base of an acute triangle is of
+  lesse power than the shankes are, by a double oblong made of one of the
+  shankes, and the one segment of the same, from the said angle, unto the
+  perpendicular of the toppe. 13 p. ij</i>.</p>
+
+  <div class="figright" style="width:14%;">
+      <a href="images/190.png"><img style="width:100%" src="images/190.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p>As in the triangle <i>aei</i>, let the angle at <i>i</i>, be taken for
+  an acute angle. Here by the <a href="#4_e_xiij">4. e</a>, two obongs of
+  <i>ei</i>, and <i>oi</i>, with the quadrate of <i>eo</i>, are equall to
+  the quadrates of <i>ei</i>, and <i>oi</i>. Let the quadrate of <i>ao</i>,
+  be added to both in common. Here the quadrate of <i>ei</i>, with the
+  quadrates of <i>io</i>, and <i>oa</i>, that is the <a href="#9_e_xij">9 e
+  xij</a>, with the quadrate of <i>ia</i>, is equall to two oblongs of
+  <i>ei</i>, and <i>oi</i>, with two quadrates of <i>eo</i> &amp;
+  <i>oa</i>, that is by the <a href="#9_e_xij">9 e xij</a>, with the
+  quadrate of <i>ea</i>. Therefore two oblongs with the quadrate of the
+  base, are equall to the quadrates of the shankes: And the base is
+  exceeded of the shankes by two oblongs.</p>
+
+  <p>And from hence is had the segment of the shanke toward the angle, and
+  by that the perpendicular in a triangle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="6_e_xiij"></a> 6. <i>If the square of the base of an acute
+  angle be taken out of the squares of the shankes, the quotient of the
+  halfe of the remaine, divided by the shanke, shall be the segment of the
+  dividing shanke from the said angle unto the perpendicular of the
+  toppe</i>.</p>
+
+  <div class="figleft" style="width:28%;">
+      <a href="images/191.png"><img style="width:100%" src="images/191.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>As in the acute angled triangle <i>aei</i>, let the sides be 13, 20,
+  21. And let <i>ae</i> be the base of the acute angle. Now the quadrate or
+  square of 13 the said base is 169: And the quadrate of 20, or <i>ai</i>,
+  is 400: And of 21, or <i>ei</i>, is 441. The summe of which is 841. And
+  841, 169, are 672: <!-- Page 170 --><span class="pagenum"><a
+  name="page170"></a>[170]</span>Whose halfe is 336. And the quotient of
+  336, divided by 21, is 16, the segment of the dividing shanke <i>ei</i>,
+  from the angle <i>aei</i>, unto <i>ao</i>, the perpendicular of the
+  toppe. Now 21, 16, are 5. Therefore the other segment or portion of the
+  said <i>ei</i>, is 5.</p>
+
+  <p>Now againe from 169, the quadrate of the base 13, take 25, the
+  quadrate of 5, the said segment: And the remaine shall be 144, for the
+  quadrate of the perpendicular <i>ao</i>, by the <a href="#9_e_xij">9 e
+  xij</a>.</p>
+
+  <p>Here the perpendicular now found, and the sides cut, are the sides of
+  the rectangle, whose halfe shall be the content of the Triangle: As here
+  the Rectangle of 21 and 12 is 252; whose halfe 126, is the content of the
+  triangle.</p>
+
+  <p>The second section followeth from whence ariseth the fourth rate or
+  comparison.</p>
+
+  <p><a name="7_e_xiij"></a> 7. <i>If a right line be cut into two equall
+  parts, and otherwise; the oblong of the unequall segments, with the
+  quadrate of the segment betweene them, is equall to the quadrate of the
+  bisegment. 5 p ij</i>.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/192a.png"><img style="width:100%" src="images/192a.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>As for example, Let the right line <i>ae</i> 8, be cut into two equall
+  portions, <i>ai</i> 4, and <i>ie</i> 4. And otherwise that is into two
+  unequall portions, <i>ao</i> 7, and <i>oe</i> 1: The oblong of 7 and 1,
+  with 9, the quadrate of 3, the intersegment, (or portion cut betweene
+  them) that is 16; shall bee equall to the quadrate of <i>ie</i> 4, which
+  is also 16. Which is also manifest by making up the diagramme as here
+  thou seest. For as the parallelogramme <i><span class="correction"
+  title="text reads `missing in text'">as</span></i> is by the <a
+  href="#24_e_x"><span class="correction" title="text reads `24 e x'">26 e
+  x</span></a>, equall to the <!-- Page 171 --><span class="pagenum"><a
+  name="page171"></a>[171]</span>parallelogramme <i>iu</i>; And therefore
+  by the <a href="#19_e_x">19 e x</a>, it is equall to <i>oy</i>. For
+  <i>ou</i>, is common to both the equall complements, Therefore if
+  <i>so</i> be added in common to both; the <i>ar</i>, shall be equall to
+  the gnomon <i>mni</i>: Now the quadrate of the segment betweene them is
+  <i>sl</i>. Wherefore <i>ar</i>, the oblong of the unequall segments, with
+  <i>s</i> the quadrate of the intersegment, is equall to <i>iy</i> the
+  quadrate of the said bisegment.</p>
+
+  <p>The third section doth follow, from whence the fifth reason
+  ariseth.</p>
+
+  <p><a name="8_e_xiij"></a> 8. <i>If a right line be cut into equall
+  parts; and continued; the oblong made of the continued and the
+  continuation, with the quadrate of the bisegment or halfe, is equall to
+  the quadrate of the line compounded of the bisegment and continuation. 6
+  p ij</i>.</p>
+
+  <div class="figleft" style="width:26%;">
+      <a href="images/192b.png"><img style="width:100%" src="images/192b.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>As for example, let the line <i>ae</i> 6, be cut into two equall
+  portions, <i>ai</i> 3, and <i>ie</i> 3: And let it be continued unto
+  <i>eo</i> 2: The oblong 16, made of 8 the continued line, and of 2, the
+  continuation; with 9 the quadrate of 3, the halfe, (that is 25.) shall be
+  equall to 25, the quadrate of 3, the halfe and 2, the continuation, that
+  is 5. This as the former, may geometrically, with the helpe of numbers be
+  expressed. For by the <a href="#26_e_x"><span class="correction"
+  title="text reads `24 e x'">26 e x</span></a>, <i>as</i> is equall to
+  <i>iy</i>: And by the <a href="#19_e_x">19 e x</a>, it is equall to
+  <i>yr</i>, the complement. To these equalls adde <i>so</i>. Now the
+  oblong <i>au</i>, shall be equall to the gnomon <i>nju</i>. Lastly, to
+  the equalls adde the quadrate of the bisegment or halfe. The Oblong of
+  the continued line and of the <!-- Page 172 --><span class="pagenum"><a
+  name="page172"></a>[172]</span>continuation, with the quadrate of the
+  bisegment, shall be equall to the quadrate of the line compounded of the
+  bisegment and continuation. These were the rates of an oblong with a
+  rectangle.</p>
+
+  <p>From hence ariseth the Mesographus or Mesolabus of <i>Heron</i> the
+  mechanicke; so named of the invention of two lines continually
+  proportionall betweene two lines given. Whereupon arose the Deliacke
+  probleme, which troubled <i>Apollo</i> himselfe. Now the Mesographus of
+  <i>Hero</i> is an infinite right line, which is stayed with a
+  scrue-pinne, which is to be moved up and downe in riglet. And it is as
+  <i>Pappus</i> saith, in the beginning of his <span
+  class="scac">III</span> booke, for architects most fit, and more ready
+  than the Plato's mesographus. The mechanicall handling of this
+  mesographus, is described by <i>Eutocius</i> at the 1 Theoreme of the
+  <span class="scac">II</span> booke of the spheare; But it is somewhat
+  more plainely and easily thus layd downe by us.</p>
+
+  <p><a name="9_e_xiij"></a> 9. <i>If the Mesographus, touching the angle
+  opposite to the angle made of the two lines given, doe cut the said two
+  lines given, comprehending a right angled parallelogramme, and infinitely
+  continued, equally distant from the center, the intersegments shall be
+  the meanes continually proportionally, betweene and two lines
+  given</i>.</p>
+
+  <p>Or thus: If a Mesographus, touching the angle opposite to the angle
+  made of the lines given, doe cut the equall distance from the center, the
+  two right lines given, conteining a right angled parallelogramme, and
+  continued out infinitely, the segments shall be meane in continuall
+  proportion with the line given: <i>H</i>.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/194.png"><img style="width:100%" src="images/194.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p>As let the two right-lines given be <i>ae</i>, and <i>ai</i>: And let
+  them comprehend the rectangled parallelogramme <i>ao</i>: And let the
+  said right lines given be continued infinitely, <i>ae</i> toward
+  <i>u</i>; and <i>ai</i> toward <i>y</i>. Now let the Mesographus
+  <i>uy</i>, touch <i>o</i>, the angle opposite to <i>a</i>: And let it cut
+  the sayd continued lines equally distant from the Center. <!-- Page 173
+  --><span class="pagenum"><a name="page173"></a>[173]</span></p>
+
+  <p>(The center is found by the <a href="#8_e_iiij">8 e iiij</a>, to wit,
+  by the meeting of the diagonies: For the equidistance from the center the
+  Mesographus is to be moved up or downe, untill by the Compasses, it be
+  found.)</p>
+
+  <p>Now suppose the points of equidistancy thus found to be <i>u</i>, and
+  <i>y</i>. I say, That the portions of the continued lines thus are the
+  meane proportionalls sought: And as <i>ae</i> is to <i>iy</i>: so is
+  <i>iy</i> to <i>eu</i>, so is <i>eu</i>, to <i>ai</i>.</p>
+
+  <p>First let from <i>s</i>, the center, <i>sr</i> be perpendicular to the
+  side <i>ae</i>: It shall therefore cut the said <i>ae</i>, into two
+  parts, by the <a href="#5_e_xj">5 e xj</a>: And therefore againe, by the
+  <a href="#7_e_xiij">7 e</a>, the oblong made of <i>au</i>, and <i>ue</i>,
+  with the quadrate of <i>re</i>, is equall to the quadrate of <i>ru</i>:
+  And taking to them in common <i>rs</i>, the oblong with two quadrates
+  <i>er</i>, and <i>rs</i>, that is, by the <a href="#9_e_xij">9 e xij</a>,
+  with the quadrate <i>se</i> is equall to the quadrates <i>ru</i> and
+  <i>rs</i>, that is by the <a href="#9_e_xij">9 e xij</a>, to the quadrate
+  <i>su</i>. The like is to be said of the oblong of <i>ay</i>, and
+  <i>yi</i>, by drawing the perpendicular <i>sl</i>, as afore. For this
+  oblong with the quadrates <i>li</i>, and <i>sl</i>, that is, by the <a
+  href="#9_e_xij">9 e xij</a>, with the quadrate <i>is</i>, is equall to
+  the quadrates <i>yl</i>, and <i>ls</i>, that is, by the <a
+  href="#9_e_xij">9 e 12</a>, to <i>ys</i>. Therefore the oblongs equall to
+  equalls, are equall betweene themselves: And taking from each side of
+  equall rayes, by the <a href="#11_e_x">11 e x</a>, equall quadrates
+  <i>se</i> and <i>si</i>, there shall remaine equalls. Wherefore by the <a
+  href="#26_e_x"><span class="correction" title="text reads `26 e x'">27 e
+  x</span></a>, the sides of equall rectangles are reciprocall: And as
+  <i>au</i> is to <i>ay</i>: so by the <a href="#13_e_vij">13 e vij</a>,
+  <i>oi</i>, that is, by the <a href="#8_e_x">8 e x</a>, <i>ea</i>, to
+  <i>iy</i>: And so therefore by the concluded, <i>yi</i> is to <i>ue</i>;
+  And so by the <a href="#13_e_vij">13 e vij</a>, is <i>ue</i> to
+  <i>eo</i>, that is, by the <a href="#8_e_x">8 e x</a>, unto <i>ai</i>.
+  Therefore as <i>ea</i> is to <i>yi</i>: so is <i>yi</i> to <i>ue</i>; and
+  so is <i>ue</i>, to <i>ai</i>. Wherefore <i>eu</i>, <i>iy</i>, the
+  intersegments or portions cut, are the two meane proportionals betweene
+  the two lines given.</p>
+
+<hr class="full" />
+
+<p><!-- Page 174 --><span class="pagenum"><a name="page174"></a>[174]</span></p>
+
+<h2>The fourteenth Booke, of <i>P. Ramus</i> Geometry:
+Of a right line proportionally
+cut: And of other Quadrangles,
+and Multangels.</h2>
+
+  <p>Thus farre of the threefold section, from whence we have the five
+  rationall rates of equality: There followeth of the third section another
+  section, into two segments proportionall to the whole. The section it
+  selfe is first to be defined.</p>
+
+  <p><a name="1_e_xiiij"></a> 1. <i>A right line is cut according to a
+  meane and extreame rate, when as the whole shall be to the greater
+  segment; so the greater shall be unto the lesser. 3 d vj</i>.</p>
+
+  <p>This line is cut so, that the whole line it selfe, with the two
+  segments, doth make the three bounds of the proportion: And the whole it
+  selfe is first bound: The greater segment is the middle bound: The lesser
+  the third bound.</p>
+
+  <p><a name="2_e_xiiij"></a> 2. <i>If a right line cut proportionally be
+  rationall unto the measure given, the segments are unto the same, and
+  betweene themselves irrationall � 6 p xiij</i>.</p>
+
+  <div class="figcenter" style="width:22%;">
+      <a href="images/195.png"><img style="width:100%" src="images/195.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p><i>Euclide</i> calleth each of these segments <span title="Apotom�" class="grk"
+  >&#x1F08;&pi;&omicron;&tau;&omicron;&mu;&#x1F74;</span> that is,
+  <i>Residuum</i>, a Residuall or remaine: And surely these cannot
+  otherwise be expressed, then by the name <i>Residuum</i>; As if a line of
+  7 foote should thus be given or put downe: The greater segment shall be
+  called a line of 7 foote, from whence the lesser is substracted: Neither
+  may the lesser otherwise be expressed, but by saying, It is the part
+  residuall or remnant of the line of 7 foote, from which the greater
+  segment was subtracted or taken.</p>
+
+  <p>A Triangle, and all Triangulates, that is figures made of <!-- Page
+  175 --><span class="pagenum"><a name="page175"></a>[175]</span>triangles,
+  except a Rightangled-parallelogramme, are in Geometry held to be
+  irrationalls. This is therefore the definition of a proportionall
+  section: The section it selfe followeth, which is by the rate of an
+  oblong with a quadrate.</p>
+
+  <p><a name="3_e_xiiij"></a> 3. <i>If a quadrate be made of a right line
+  given, the difference of the right line from the middest of the
+  conterminall side of the said quadrate made, above the same halfe, shall
+  be the greater segment of the line given proportionally cut: 11 p
+  ij</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/196.png"><img style="width:100%" src="images/196.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p>Or thus: If a square be made of a right line given, the difference of
+  a right line drawne from the angle of the square made unto the middest of
+  the next side, above the halfe of the side, shall be the greater segment
+  of the line given, being proportionally cut: <i>H</i>.</p>
+
+  <p>Let the right line gived be <i>ae</i>. The quadrate of the same let it
+  be <i>aeio</i>: And from the angle <i>e</i>, unto <i>u</i>, the middest
+  of the conterminal side, let the right line <i>eu</i>, be drawne; Then
+  compare or lay it to the halfe <i>ua</i>; The difference of it above the
+  said halfe shall be <i>ay</i>, This <i>ay</i>, say 1, is the greater
+  segment of <i>ae</i>, the line given, proportionally cut.</p>
+
+  <p>For of <i>ya</i>, let the quadrate <i>aysr</i>, be made: And let
+  <i>sr</i>, be continued unto <i>l</i>. Now by the <a href="#8_e_xiij">8
+  e, xiij</a>. the <span class="correction" title="text reads `obloug'"
+  >oblong</span> of <i>oy</i>, and <i>ay</i>, with the quadrate of
+  <i>ua</i>, is equall to the quadrate of <i>uy</i>, that is by the
+  construction of <i>ue</i>: And therefore, by the <a href="#9_e_xij">9 e
+  xij</a>. it is equall to the quadrates <i>ea</i>, and <i>au</i>: Take
+  away from each side the common oblong <i>al</i>, and the quadrate
+  <i>yr</i>, shall be equall to the oblong <i>ri</i>. Therefore the three
+  right lines, <i>ea</i>, <i>ar</i>, and <i>re</i>, by the <a
+  href="#8_e_xij">8 e xij</a>. are continuall proportionall. And the right
+  line <i>ae</i>, is cut proportionally.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+<p><!-- Page 176 --><span class="pagenum"><a name="page176"></a>[176]</span></p>
+
+  <p><a name="4_e_xiiij"></a> 4. <i>If a right line cut proportionally, be
+  continued with the greater segment, the whole shall be cut
+  proportionally, and the greater segment shall be the line given. 5 p
+  xiij</i>.</p>
+
+  <p>As in the same example, the right line <i>oy</i>, is continued with
+  the greater segment, and the oblong of the whole and the lesser segment
+  is equall to the quadrate of the greater. And thus one may by infinitely
+  proportionally cutting increase a right line; and againe decrease it. The
+  lesser segment of a right line proportionally cut, is the greater
+  segment, of the greater proportionally cut. And from hence a decreasing
+  may be made infinitely.</p>
+
+  <p><a name="5_e_xiiij"></a> 5. <i>The greater segment continued to the
+  halfe of the whole, is of power quintuple unto the said halfe, that is,
+  five times so great as it is: and if the power of a right line be
+  quintuple to his segment, the remainder made the double of the former is
+  cut proportionally, and the greater segment, is the same remainder. 1.
+  and 2. p xiij</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/197.png"><img style="width:100%" src="images/197.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p>This is the fabricke or manner of making a proportionall section. A
+  threefold rate followeth: The first is of the greater segment.</p>
+
+  <p>Let therefore the right line <i>ae</i>, be cut proportionally in
+  <i>i</i>: And let the greater segment be <i>ia</i>: and let the line cut
+  be continued unto <i>io</i>, so that <i>oa</i>, be the halfe of the line
+  cut. I say, the quadrate of <i>io</i>, is in power five times so great,
+  as <i>ys</i>, the power of the quadrate of <i>ao</i>. Let therefore of
+  <i>ao</i>, be made the quadrate <i>iosr</i>: We doe see the quadrate
+  <i>ua</i>, to be once contained in the quadrate <i>si</i>. Let us now
+  <!-- Page 177 --><span class="pagenum"><a
+  name="page177"></a>[177]</span>teach that it is moreover foure times
+  comprehended in <i>lmn</i>, the gnomon remaining: Let therefore the
+  quadrate <i>aeiu</i>, be made of the line given: And let <i>ri</i>, be
+  continued unto <i>f</i>. Here the quadrate <i>ae</i>, is (<a
+  href="#14_e_xij">14. e xij</a>.) foure times so much as is that
+  <i>au</i>, made of the halfe: and it is also equall to the gnomon
+  <i>lmn</i>: For the part <i>iu</i>, is equall to <i>ry</i>; first by the
+  grant, seeing that <i>ai</i>, is the greater segment, from whence
+  <i>ry</i>, is made the quadrate, because the other Diagonall is also a
+  quadrate: Secondarily the complements <i>sy</i>, and <i>yi</i>, by the <a
+  href="#19_e_x">19. e x</a>, are equall: And to them is equall <i>af</i>.
+  For by the <a href="#23_e_x">23. e x</a>. and by the grant, it is the
+  double of the complement <i>yi</i>. Therefore it is equall to them both.
+  Wherefore the gnomon <i>lmn</i>, is equall to the quadruple quadrate of
+  the said little quadrate: And the greater segment continued to the halfe
+  of the right line given is of power five fold to the power of
+  <i>ao</i>.</p>
+
+  <p>The converse is apparent in the same example: For seeing that
+  <i>io</i>, is of power five times so much as is <i>ao</i>; the gnomon
+  <i>lmn</i>, shall be foure times so much as is <i>ua</i>: Whose quadruple
+  also, by the <a href="#14_e_xij">14. e xij</a>, is <i>av</i>. Therefore
+  it is equall to the gnomon. Now <i>aj</i>, is equall to <i>ae</i>:
+  Therefore it is the double also of <i>ao</i>, that is of <i>ay</i>: And
+  therefore by the <a href="#24_e_x">24. e x</a>. it is the double of
+  <i>at</i>: And therefore it is equall to the complements <i>iy</i>, and
+  <i>ys</i>: Therefore the other diagonall <i>yr</i>, is equall to the
+  other rectangle <i>iv</i>. Wherefore, by the <a href="#8_e_xij">8 e
+  xij</a>. as <i>ev</i>, that is, <i>ae</i>, is to <i>yt</i>, that is
+  <i>ai</i>: so is <i>ai</i>, unto <i>ie</i>; Wherefore by the <a
+  href="#1_e_xiiij">1 e</a>, <i>ae</i>, is proportionall cut: And the
+  greater segment is <i>ai</i>, the same remaine.</p>
+
+  <p>The other propriety of the quintuple doth follow.</p>
+
+  <p><a name="6_e_xiiij"></a> 6 <i>The lesser segment continued to the
+  halfe of the greater, is of power quintuple to the same halfe � 3 p
+  xiij</i>.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/199a.png"><img style="width:100%" src="images/199a.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>As here, the right line <i>ae</i>, let it be cut proportionally in
+  <i>i</i>: And the lesser <i>ie</i>, let it be continued even unto
+  <i>o</i>, the halfe of the greater <i>ai</i>. I say, that the power of
+  <i>oe</i>, shall be five times as much as is the power of <i>io</i>. Let
+  a quadrate <!-- Page 178 --><span class="pagenum"><a
+  name="page178"></a>[178]</span>therefore be made of <i>ae</i>: And let
+  the figure be made up (as you see:) And let the quadrate of the halfe be
+  noted with <i>su</i>: And the gnomon <i>rlm</i>. Here the first quadrate
+  <i>oy</i>, is five times as great, as the second <i>su</i>. For it doe
+  containe it once: And the gnomon <i>rlm</i>, remaining containeth it
+  foure times. For it is equall to the Oblong <i>in</i>; because <i>os</i>,
+  the complement is equall to <i>sy</i>, by the <a href="#19_e_x">19 e
+  x</a>; And therefore also it is equall to <i>in</i>; seeing the whole
+  complement <i>as</i>, is equall to the whole complement <i>sn</i>: And
+  <i>av</i>, is equall to <i>os</i>, by the construction, and <a
+  href="#23_e_x">23. e x</a>: And adding to both the common oblong
+  <i>iy</i>, the whole gnomon is equal to the whole oblong. But the oblong
+  <i>in</i>, is equall to the quadrate <i>ai</i>, by the grant, &amp; <a
+  href="#8_e_xij">8 e xij</a>. which by the <a href="#14_e_xij">14. e
+  xij</a>. is foure times as great, as the quadrate <i>su</i>. Wherefore
+  the lesser segment <i>ie</i>, continued to <i>io</i>, the halfe of the
+  greater segment, is of power five times as much as is the halfe of the
+  same.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p><i>The rate of the triple followeth</i>.</p>
+    </div>
+  </div>
+  <div class="figleft" style="width:18%;">
+      <a href="images/199b.png"><img style="width:100%" src="images/199b.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p><a name="7_e_xiiij"></a> 7 <i>The whole line and the lesser segment
+  are in power treble unto the greater. � 4 p xiij</i>.</p>
+
+  <p>Let the right line <i>ae</i> be proportionally cut in <i>i</i>, and
+  let the figure be made up: The oblong <i>ay</i>, and <i>io</i>, with the
+  quadrate <i>su</i>, by the <a href="#4_e_xiij">4 e xiij</a>, are equall
+  to the quadrates of <i>ae</i>, and <i>ie</i>, whose power is treble to
+  that of <i>ai</i>. For they doe once containe the quadrate <i>su</i>; And
+  each of the oblongs is equall to the same quadrate <i>su</i>, by the
+  grant, and <a href="#8_e_xij">8 e xij</a>. Therefore they doe containe it
+  thrise.</p>
+
+<p><!-- Page 191 --><span class="pagenum"><a name="pageastx191"></a>[191*]</span></p>
+
+  <p><br style="clear : both" /></p>
+  <p><a name="8_e_xiiij"></a> 8 <i>An obliquangled parallelogramme is
+  either a Rhombus, or a Rhomboides</i>.</p>
+
+  <p><a name="9_e_xiiij"></a> 9 <i>A Rhombus is an obliquangled equilater
+  parallelogramme 32 d j</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/200a.png"><img style="width:100%" src="images/200a.png"
+      alt="Rhombi." title="Rhombi." /></a>
+  </div>
+  <p>Whereupon it is apparant that a Rhombus is a square having the angles
+  as it were pressed, or thrust nearer together, by which name, both the
+  Byrt or Turbot, a Fish; and a Wheele or Reele, which Spinners doe use;
+  and the quarrels in glasse windowes, because they are cut commonly of
+  this forme, are by the Greekes and Latines so called.</p>
+
+  <p>It is otherwise of some called a Diamond.</p>
+
+  <p><a name="10_e_xiiij"></a> 10 <i>A Rhomboides is an obliquangled
+  parallelogramme not equilater 33. d j</i>.</p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/200b.png"><img style="width:100%" src="images/200b.png"
+      alt="Rhomboides." title="Rhomboides." /></a>
+  </div>
+  <p>And a Rhomboides is so opposed to an oblong, as a Rhombus is to a
+  quadrate.</p>
+
+  <div class="figright" style="width:28%;">
+      <a href="images/201a.png"><img style="width:100%" src="images/201a.png"
+      alt="Angles of Rhomboides." title="Angles of Rhomboides." /></a>
+  </div>
+  <p>So also looke how much the straightening or pressing <!-- Page 180
+  --><span class="pagenum"><a name="pageastx180"></a>[180*]</span>together
+  is greater, so much is the inequality of the obtuse and acute angles the
+  greater. As here.</p>
+
+  <p>And the Rhomboides is so called as one would say Rhombuslike, although
+  beside the inequality of the angles it hath nothing like to a Rhombus. An
+  example of measuring of a <span class="correction" title="text reads `Rhombus' but diagram shows a Rhomboides"
+  >Rhomboides</span> is thus.</p>
+
+  <p><br style="clear : both" /></p>
+  <p><a name="11_e_xiiij"></a> 11 <i>A Trapezium is a quadrangle not
+  parallelogramme. 34. d j</i>.</p>
+
+  <div class="figcenter" style="width:57%;">
+      <a href="images/201b.png"><img style="width:100%" src="images/201b.png"
+      alt="Trapezia." title="Trapezia." /></a>
+  </div>
+  <p>Of the quadrangles the Trapezium remaineth for the last place:
+  <i>Euclide</i> intreateth this fabricke to be granted him, that a
+  Trapezium may be called as it were a little table: And surely Geometry
+  can yeeld no reason of that name.</p>
+
+  <p>The examples both of the figure and of the measure of the same let
+  these be.</p>
+
+  <p>Therefore triangulate quadrangles are of this sort. <!-- Page 190
+  --><span class="pagenum"><a name="pageastx190"></a>[190*]</span></p>
+
+  <p><a name="12_e_xiiij"></a> 12 <i>A multangle is a figure that is
+  comprehended of more than foure right lines. 23. d j</i>.</p>
+
+  <p>By this generall name, all other sorts of right lined figures
+  hereafter following, are by Euclide comprehended, as are the quinquangle,
+  sexangle, septangle, and such like inumerable taking their names of the
+  number of their angles.</p>
+
+  <p>In every kinde of multangle, there is one ordinate, as we have in the
+  former signified, of which in this place we will say nothing, but this
+  one thing of the quinquangle. The rest shall be reserved untill we come
+  to Adscription.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/202.png"><img style="width:100%" src="images/202.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><a name="13_e_xiiij"></a> 13 <i>Multangled triangulates doe take their
+  measure also from their triangles</i>.</p>
+
+  <p>As here, this quinquangle is measured by his three triangles. The
+  first triangle, whose sides are 9. 10. and 17. by the <a
+  href="#18_e_xij">18. e xij</a>. is 36. The second, whose sides are 6, 17,
+  and 17. by the same e, is 50.20/101. The third, whose sides are 17, 15.
+  and 8. by the same, is 60. And the summe of 36. 50.20/101. and 60. is
+  146.20/101, for the whole content of the Quinquangle given.</p>
+
+  <p><a name="14_e_xiiij"></a> 14 <i>If an equilater quinquangle have three
+  sides equall, it is equiangled. 7 p 13</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/203a.png"><img style="width:100%" src="images/203a.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>This of some, from the Greeke is called Pentagon; of others a
+  Pentangle, by a name partly Greeke partly Latine.</p>
+
+  <p>As in the Quinquangle <i>aeiou</i>, the three angles at <i>a</i>,
+  <i>e</i>, and <i>i</i>, are equall: Therefore the other two are equall:
+  And they are equall unto these. For let <i>eu</i>, <i>ai</i>, <i>ia</i>,
+  be knit together with right lines. Here the triangles <i>aei</i>, and
+  <i>eau</i>. by the grant, and by the <a href="#2_e_vij">2</a> and <a
+  href="#1_e_vij">1 e vij</a>. are equilaters and equiangles: And the Bases
+  <i>ai</i>, and <i>eu</i>, are <!-- Page 200 --><span class="pagenum"><a
+  name="pageastx200"></a>[200*]</span>equall: And the Angles, <i>eai</i>,
+  and <i>aue</i>, are equall: Item <i>aeu</i>, and <i>eia</i>. Therefore
+  <i>ay</i>, and <i>ye</i>, are equall, by the <a href="#17_e_vj">17 e
+  vj</a>. Item the remainder <i>uy</i>, is equall to the remainder
+  <i>yi</i>, when from equals equals be subtracted. Moreover by the grant,
+  and by the <a href="#17_e_vj">17 e vj</a>, <i>oui</i>, and <i>oiu</i>,
+  are equall. Wherefore three are equall; And therefore the whole angle is
+  equall at <i>u</i>, to the whole <span class="correction" title="text reads `anlge'"
+  >angle</span> at <i>i</i>. And therefore it is equall to those which are
+  equall to it.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/203b.png"><img style="width:100%" src="images/203b.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>I say moreover that the angle at <i>o</i>, is likewise equall, if
+  <i>ao</i>, and <i>oe</i>, be knit together with a right line, as here:
+  For three in like manner do come to be equall.</p>
+
+  <div class="figcenter" style="width:53%;">
+      <a href="images/203c.png"><img style="width:100%" src="images/203c.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>But if the three angles <i>non deinceps</i> not successively following
+  be equall, as <i>aio</i>, the businesse will yet be more easie, as here:
+  For the angles <i>eua</i>, and <i>eoi</i>, are equall by the grant: And
+  the inner also <i>eou</i>, and <i>euo</i>. Therefore the wholes of two
+  are equall. Of the other at <i>e</i>, the same will fall out, if
+  <i>iu</i>, be knit together with <span class="correction" title="text reads `e' in italics"
+  >a</span> right line <i>iu</i>, as here: For the wholes of two shall be
+  equall.</p>
+
+<hr class="full" />
+
+<p><!-- Page 201 --><span class="pagenum"><a name="pageastx201"></a>[201*]</span></p>
+
+<h2>The fifteenth Booke of <i>Geometry</i>,
+Of the Lines in a Circle.</h2>
+
+  <p>As yet we have had the Geometry of rectilineals: The Geometry of
+  Curvilineals, of which the Circle is the chiefe, doth follow.</p>
+
+  <p><a name="1_e_xv"></a> 1. <i>A Circle is a round plaine. � 15 d
+  j</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/204.png"><img style="width:100%" src="images/204.png"
+      alt="Circle." title="Circle." /></a>
+  </div>
+  <p>As here thou seest. A Rectilineall plaine was at the <a
+  href="#3_e_vj">3 e vj</a>, defined to be a plaine comprehended of right
+  lines. And so also might a circle have beene defined to be a plaine
+  comprehended of a periphery or bought-line, but this is better.</p>
+
+  <p>The meanes to describe a Circle, is the same, which was to make a
+  Periphery: But with some difference: For there was considered no more but
+  the motion, the point in the end of the ray describing the periphery:
+  Here is considered the motion of the whole ray, making the whole plot
+  conteined within the periphery.</p>
+
+  <p>A Circle of all plaines is the most ordinate figure, as was before
+  taught at the <a href="#10_e_iiij">10 e iiij</a>.</p>
+
+  <p><a name="2_e_xv"></a> 2 <i>Circles are as the quadrates or squares
+  made of their diameters 2 p. xij</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/205a.png"><img style="width:100%" src="images/205a.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p>For Circles are like plaines. And their homologall sides are their
+  diameters, as was foretold at the <a href="#24_e_iiij">24 e iiij</a>. And
+  therefore by the <a href="#1_e_vj">1 e vj</a>, they are one to another,
+  as the quadrates of their diameters are one to another, which indeed is
+  the double reason of their homologall sides. As here the Circle
+  <i>aei</i>, is unto the Circle <i>ouy</i> as 25, is unto 16, which are
+  <!-- Page 202 --><span class="pagenum"><a
+  name="pageastx202"></a>[202*]</span>the quadrates of their Dieameters, 5
+  and 4.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="3_e_xv"></a> 3. <i>The Diameters are, as their peripheries
+  Pappus, 5 l. xj, and 26th. 18</i>.</p>
+
+  <p>As here thou seest in <i>ae</i>, and <i>io</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/205b.png"><img style="width:100%" src="images/205b.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p><a name="4_e_xv"></a> 4. <i>Circular Geometry is either in Lines, or
+  in the segments of a Circle</i>.</p>
+
+  <p>This partition of the subject matters howsoever is taken for the
+  distinguishing and severing with some light a matter somewhat confused;
+  And indeed concerning lines, the consideration of secants is here the
+  foremost, and first of Inscripts.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/206a.png"><img style="width:100%" src="images/206a.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p><a name="5_e_xv"></a> 5. <i>If a right line be bounded by two points
+  in the periphery, it shall fall within the Circle. 2 p iij.</i> <!-- Page
+  203 --><span class="pagenum"><a name="pageastx203"></a>[203*]</span></p>
+
+  <p>As here <i>ae</i>, because the right within the same points is
+  shorter, than the periphery is, by the <a href="#5_e_ij">5 e ij</a>.</p>
+
+  <p>From hence doth follow the Infinite section, of which we spake at the
+  <a href="#6_e_j">6 e j</a>.</p>
+
+  <p>This proposition teacheth how a Rightline is to be inscribed in a
+  circle, to wit, by taking of two points in the periphery.</p>
+
+  <p><a name="6_e_xv"></a> 6. <i>If from the end of the diameter, and with
+  a ray of it equal to the right line given, a periphery be described, a
+  right line drawne from the said end, unto the meeting of the peripheries,
+  shall be inscribed into the circle, equall to the right line given. 1 p
+  iiij</i>.</p>
+
+  <div class="figleft" style="width:25%;">
+      <a href="images/206b.png"><img style="width:100%" src="images/206b.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>As let the right line given be <i>a</i>: And from <i>e</i>, the end of
+  the diameter <i>ei</i>: And with <i>eo</i>, a part of it equall to
+  <i>a</i>, the line given, describe the circle <i>eu</i>: A right line
+  <i>eu</i>, drawne from the end <i>e</i>, unto <i>u</i>, the meeting of
+  the two peripheries, shall be inscribed in the circle given, by the <a
+  href="#5_e_xv">5 e</a>, equall to the line given; because it is equall to
+  <i>eo</i>, by the <a href="#10_e_v">10 e v</a>, seeing it is a ray of the
+  same Circle.</p>
+
+  <p>And this proposition teacheth, How a right line given is to be
+  inscribed into a Circle, equall to a line given.</p>
+
+  <p>Moreover of all inscripts the diameter is the chiefe: For it sheweth
+  the center, and also the reason or proportion of all other inscripts.
+  Therefore the invention and making of the diameter of a Circle is first
+  to be taught.</p>
+
+  <p><a name="7_e_xv"></a> 7. <i>If an inscript do cut into two equall
+  parts, another <!-- Page 174 --><span class="pagenum"><a
+  name="pageastx174"></a>[174*]</span>inscript perpendicularly, it is the
+  diamiter of the Circle, and the middest of it is the center. 1 p
+  iij</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/207.png"><img style="width:100%" src="images/207.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>As let the Inscript <i>ae</i>, cut the inscript <i>iu</i>
+  perpendicularly: dividing it into two equall parts in <i>o</i>. I say
+  that the <span class="correction" title="text reads `once'">one</span>
+  inscript thus halfing the other, is the diameter of the Circle: And that
+  the middest of it is the center thereof: As in the circle, let the
+  Inscript <i>is</i>, cut the inscript <i>ae</i>, and that perpendicularly
+  dividing into two equall parts in <i>o</i>. I say that <i>iu</i>, thus
+  dividing <i>ae</i>, is the Diameter of the Circle: And <i>y</i>, the
+  middest of the said <i>iu</i>, is the Center of the same.</p>
+
+  <p>The cause is the same, which was of the <a href="#5_e_xj">5 e xj</a>.
+  Because the inscript cut into halfes <span class="correction" title="text reads `if'"
+  >is</span> for the side of the inscribed rectangle, and it doth subtend
+  the periphery cut also into two parts; By the which both the Inscript and
+  Periphery also were in like manner cut into two equall parts: Therefore
+  the right line thus halfing in the diameter of the rectangle: But that
+  the middle of the circle is the center, is manifest out of the <a
+  href="#7_e_v">7 e v</a>, and <a href="#29_e_iiij">29 e iiij</a>.</p>
+
+  <p><i>Euclide</i>, thought better of <i>Impossibile</i>, than he did of
+  the cause: And thus he forceth it. For if <i>y</i> be not the Center, but
+  <i>s</i>, the part must be equall to the whole: For the Triangle
+  <i>aos</i>, shall be equilater to the triangle <i>eos</i>. For <i>ao</i>,
+  <i>oe</i>, are equall by the grant: Item <i>sa</i>, and <i>se</i>, are
+  the rayes of the circle: And <i>so</i>, is common to both the triangles.
+  Therefore by the <a href="#1_e_vij">1 e vij</a>, the angles <span
+  class="correction" title="text reads `no'">on</span> each side at
+  <i>o</i> are equall; And by the <a href="#13_e_v">13 e v</a>, they are
+  both right angles. Therefore <i>soe</i> is a right angle; It is therefore
+  equall by the grant, to the right angle <i>yoe</i>, that is, the part is
+  equall to the whole, which is impossible. Wherefore <span
+  class="correction" title="text reads `y'"><i>s</i></span> is not the
+  Center. The same will fall out of any other points whatsoever out of
+  <i>y</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="8_e_xv"></a> 8. <i>If two right lines doe perpendicularly
+  halfe two <!-- Page 175 --><span class="pagenum"><a
+  name="pageastx175"></a>[175*]</span>inscripts, the meeting of these two
+  bisecants shall be the Center of the circle � 25 p iij.</i></p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/208a.png"><img style="width:100%" src="images/208a.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>As here <i>ae</i>, and <i>io</i>, let them cut into halfes the right
+  lines <i>uy</i>, and <i>ys</i>. And let them meete, that they cut one
+  another in <i>r</i>. I say <i>r</i> is the center of the circle
+  <i>ayoseiu</i>. For before, at the <a href="#6_e_xv">6</a>, and <a
+  href="#7_e_xv"><span class="correction" title="text reads `7 e v'">7
+  e</span></a>, it was manifest that the Center was in the Diameter. And in
+  the meeting of the diameters. [Therefore two manner of wayes is the
+  Center found; First by the middle of the diameter: And then againe by the
+  concourse, or meeting of the diameters, in the middest of the lines
+  halfed or cut into two equall portions.] Here is no neede of the meeting
+  of many diameters, one will serve well enough.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And one may</p>
+    </div>
+  </div>
+  <p><a name="9_e_xv"></a> 9. <i>Draw a periphery by three points, which
+  doe not fall in a right line</i>.</p>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/208b.png"><img style="width:100%" src="images/208b.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p>As here, by <i>aei</i>, (First from <i>a</i>, to <i>e</i>, let a right
+  line be drawne; And likewise from <i>e</i> to <i>i</i>. Then, by the <a
+  href="#12_e_v">12 e v</a>, let both these lines be cut into equall parts,
+  by two infinite right lines: These halfing lines also shall meete: And in
+  their meeting shall be the Center, by the <a href="#8_e_xv">8 e</a>. And
+  therefore from that meeting unto any of the sayd points given is the ray
+  of the periphery desired.)</p>
+
+  <p><a name="10_e_xv"></a> 10. <i>If a diameter doe halfe an inscript,
+  that is, not a diameter, it doth cut it perpendicularly: And
+  contrariwise: 3 p iij.</i> <!-- Page 206 --><span class="pagenum"><a
+  name="pageastx206"></a>[206*]</span></p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/209a.png"><img style="width:100%" src="images/209a.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p>As let the diameter <i>ae</i>, halfe the inscript <i>io</i>, which is
+  not a diameter: And let the raies of the circle bee <i>ui</i>, and
+  <i>uo</i>. The cause in all is the same, which was of the <a
+  href="#5_e_xj">5 e xj</a>.</p>
+
+  <p><a name="11_e_xv"></a> 11. <i>If inscripts which are not diameters doe
+  cut one another, the segments shall be unequall. 4 p iij</i>.</p>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/209b.png"><img style="width:100%" src="images/209b.png"
+      alt="Figure for demonstration 11." title="Figure for demonstration 11." /></a>
+  </div>
+  <p>This is a consectary drawne out of the <a href="#28_e_iiij">28 e
+  iiij</a>. For if the inscripts were halfed, they should be diameters,
+  against the grant.</p>
+
+  <p>But rate hath beene hitherto in the parts of inscripts: Proportion in
+  the same parts followeth.</p>
+
+  <p><a name="12_e_xv"></a> 12 <i>If two inscripts doe cut one another, the
+  rectangle of the segments of the one is equall to the rectangle of the
+  segments of the other. 35 p iij</i>.</p>
+
+  <p>If the inscripts thus cut be diameters, the proportion is manifest, as
+  in the first figure. For the Rectangle of the segments, of the one is
+  equall to the rectangle of the segments of the other, seeing they be both
+  quadrates of equall sides. If they be not diameters let them otherwise as
+  <i>ae</i>, and <i>io</i>: I say the Oblong of <i>au</i>, and <i>ue</i>,
+  is equall to the Oblong of <i>ou</i>, and <i>ui</i>. For let the raies
+  from the Center <i>y</i>, be <i>ye</i>, and <i>yi</i>. To the quadrate of
+  each of these both the rectangles of the segments shall be equall. For by
+  the <a href="#7_e_xv">7 e</a>, let the diameter <i>yu</i>, fall upon the
+  point of the common section <i>u</i>; And let <i>ys</i>, and <i>sr</i>,
+  be perpendiculars. Here by the <a href="#5_e_xj">5 e xj</a>. the
+  inscripts are cut equally in the points <i>r</i> and <i>s</i>: And
+  unequally in the point <i>u</i>: Therefore by the <a href="#7_e_xiij">7 e
+  xiij</a>, the <!-- Page 189 --><span class="pagenum"><a
+  name="page189"></a>[189]</span>oblong, of <i>ou</i>, and <i>ui</i>, with
+  the quadrate <i>su</i>, is equall to the quadrates <i>si</i>; And adding
+  <i>ys</i>, the same oblong, with the quadrates <i>us</i> and <i>sy</i>,
+  that is, by the <a href="#9_e_xij">9 e xij</a>, with the quadrate
+  <i>yu</i>, is equall to the quadrates <i>is</i> and <i>sy</i>, that is,
+  by the <a href="#9_e_xij">9 e xij</a>, to the quadrate <i>iy</i>, that
+  is, by the <a href="#5_e_xij">5 e xij</a>, to <i>ye</i>, to the which by
+  the same cause it is manifest the other oblong with the quadrate
+  <i>yu</i> is equall. Let the quadrate <i>yu</i>, bee taken from each of
+  them: And then the oblongs shall be equall to the same: And therefore
+  betweene themselves.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/210a.png"><img style="width:100%" src="images/210a.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>And this is the comparison of the parts inscripts. The rate of whole
+  inscripts doth follow, the which whole one diameter doth make:</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/210b.png"><img style="width:100%" src="images/210b.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><a name="13_e_xv"></a> 13 <i>Inscripts are equall distant from the
+  center, unto which the perpendiculars from the center are equall 4 d
+  iij</i>.</p>
+
+  <p>As it appeareth in the next figure, of the lines <i>ae</i> and
+  <i>io</i>, unto which the perpendiculars <i>uy</i> and <i>us</i>, from
+  the Center <i>u</i>, are equall.</p>
+
+  <p><a name="14_e_xv"></a> 14. <i>If inscripts be equall, they be equally
+  distant from the center: And contrariwise. 13 p iij.</i> <!-- Page 190
+  --><span class="pagenum"><a name="page190"></a>[190]</span></p>
+
+  <p>The diameters in the same circle, by the <a href="#28_e_iiij">28 e
+  iiij</a>, are equall: And they are equally distant from the center,
+  seeing they are by the center, or rather are no whit at all distant from
+  it: Other inscripts are judged to be equall, greater, or lesser one than
+  another, by the diameter, or by the diameters center.</p>
+
+  <p><i>Euclide</i> doth demonstrate this proposition thus: Let first
+  <i>ae</i> and <i>io</i> be equall; I say they are equidistant from the
+  center. For let <i>uy</i>, and <span class="correction" title="text reads `uy'"
+  ><i>us</i></span>, be perpendiculars: They shall cut the assigned
+  <i>ae</i>, &amp; <i>io</i>, into halfes, by the <a href="#5_e_xj">5 e
+  xj</a>: And <i>ya</i> and <i>si</i> are equall, because they are the
+  halfes of equals. Now let the raies of the circle be <i>ua</i>, <span
+  class="correction" title="text reads `aund'">and</span> <i>ui</i>: Their
+  quadrates by the <a href="#9_e_xij">9 e xij</a>, are equall to the paire
+  of quadrates of the shankes, which paires are therefore equall betweene
+  themselves. Take from equalls the quadrates <i>ya</i>, and <i>si</i>,
+  there shall remaine <i>yu</i>, and <i>us</i>, equalls: and therefore the
+  sides are equall, by the <a href="#4_e_xij">4 e 12</a>.</p>
+
+  <p>The converse likewise is manifest: For the perpendiculars given do
+  halfe them: And the halfes as before are equall.</p>
+
+  <p><a name="15_e_xv"></a> 15 <i>Of unequall inscripts the diameter is the
+  greatest: And that which is next to the diameter, is greater than that
+  which is farther off from it: That which is farthest off from it, is the
+  least: And that which is next to the least, is lesser than that which is
+  farther off: And those two onely which are on each side of the diameter
+  are equall � 15 e iij</i>.</p>
+
+  <p>This proposition consisteth of five members: The first is, The
+  diameter is the greatest <span class="correction" title="text reads `iuscript'"
+  >inscript</span>: The second, That which is next to the diameter is
+  greater than that which is farther off: The third, That which is farthest
+  off from the diameter is the least: The fourth, That next to the least is
+  lesser, than that farther off: The fifth, That two onely on each side of
+  the diameter are equall betweene themselves. All which are manifest, out
+  of that same argument of equalitie, that is the center the beginning of
+  decreasing, and the <!-- Page 191 --><span class="pagenum"><a
+  name="page191"></a>[191]</span>end of increasing. For looke how much
+  farther off you goe from the center, or how much nearer you come unto it,
+  so much lesser or greater doe you make the inscript.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/212.png"><img style="width:100%" src="images/212.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>Let there be in a circle; many inscripts, of which one, to wit,
+  <i>ae</i>, let it be the diameter: I say, that it is of them all the
+  greatest or longest. But let <i>io</i>, be nearer to the diameter, (or as
+  in the former Elements was said) nearer to the center, than <i>uy</i>. I
+  say that <i>io</i>, is longer than <i>uy</i>. Moreover, let <i>uy</i>, be
+  the farthest off from the same diameter or center; I say the same
+  <i>uy</i>, is the shortest of them all. Now to this shortest <i>uy</i>,
+  let <i>io</i>, be nearer than <i>ae</i>; I say therefore that <i>io</i>,
+  also is lesser than <i>ae</i>. Let at length <i>io</i>, be not the
+  diameter: I say that beyond the diameter <i>ae</i>, there may onely a
+  line be inscribed equall unto it, such as is <i>sr</i>. And those equal
+  betweene themselves on each side of the diametry may only be given, not
+  three, nor more. And after the same manner also, onely one beyond the
+  diameter, may possibly be equall to <i>uy</i>, to wit, that which is as
+  farre off from the diameter as it is; and so in others.</p>
+
+  <p>But Euclides conclusion is by triangles of two sides greater than the
+  other; and of the greater angle.</p>
+
+  <p>The first part is plaine thus: Because the diameter <i>ae</i>, is
+  equall to <i>il</i>, and <i>lo</i>, <i>viz</i>. to the raies; And to
+  those which are greater than <i>io</i>, the base by the <a
+  href="#9_e_vj">9. e vj</a> &amp;c.</p>
+
+  <p>The second part of the nearer, is manifest by the <a href="#5_e_vij">5
+  e vij</a>. because of the triangle <i>ilo</i>, equicrurall to the
+  triangle <i>uly</i>, is greater in angle: And therefore it is also
+  greater in base.</p>
+
+  <p>The third and fourth are consectaries of the first.</p>
+
+  <p>The fifth part is manifest by the second: For if beside <i>io</i>, and
+  <i>sr</i>, there be supposed a third equall, the same also shall be
+  unequall, because it shall be both nearer and farther off from the
+  diameter. <!-- Page 192 --><span class="pagenum"><a
+  name="page192"></a>[192]</span></p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/213.png"><img style="width:100%" src="images/213.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p><a name="16_e_xv"></a> 16 <i>Of right lines drawne from a point in the
+  diameter which is not the center unto the periphery, that which passeth
+  by the center is the greatest: And that which is nearer to the greatest,
+  is greater than that which is farther off: The other part of the greatest
+  is the left. And that which is nearest to the least, is lesser than that
+  which is farther off: And two on each side of the greater or least are
+  only equall. 7 p iij</i>.</p>
+
+  <p>The first part of <i>ae</i>, and <i>ai</i>, is manifest, as before, by
+  the <a href="#9_e_vj">9 e vj</a>. The second of <i>ai</i>, and <i>ao</i>;
+  Item of <i>ao</i>, and <i>au</i>, is plaine by the <a href="#5_e_vij">5 e
+  vij</a>.</p>
+
+  <p>The third, that <i>ay</i>, is lesser than <i>au</i>, because
+  <i>sy</i>, which is equall to <i>su</i>, is lesser than the right lines
+  <i>sa</i>, and <i>au</i>, by the <a href="#9_e_vj">9 e vj</a>: And the
+  common <i>sa</i>, being taken away, <i>ay</i> shall be left, lesser than
+  <i>au</i>.</p>
+
+  <p>The fourth part followeth of the third.</p>
+
+  <p>The fifth let it be thus: <i>sr</i>, making the angle <i>asr</i>,
+  equall to the angle <i>asu</i>, the bases <i>au</i>, and <i>ar</i>, shall
+  be equall by the <a href="#2_e_vij">2 e vij</a>. To these if the third be
+  supposed to be equall, as <i>al</i>, it would follow by the <a
+  href="#1_e_vij">1 e vij</a>. that the whole angle <i>sa</i>, should be
+  equall to <i>rsa</i>, the particular angle, which is impossible. And out
+  of this fifth part issueth this Consectary.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figleft" style="width:19%;">
+      <a href="images/214a.png"><img style="width:100%" src="images/214a.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+  <p><a name="17_e_xv"></a> 17 <i>If a point in a circle be the bound of
+  three equall right lines determined in the periphery, it is the center of
+  the circle. 9 p iij</i>.</p>
+
+  <p>Let the point <i>a</i>, in a circle be the common bound of three right
+  lines, ending in the periphery and equall betweene themselves, be
+  <i>ae</i>, <i>ai</i>, <i>au</i>. I say this point is the center of the
+  Circle. <!-- Page 193 --><span class="pagenum"><a
+  name="page193"></a>[193]</span></p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/214b.png"><img style="width:100%" src="images/214b.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p>Otherwise from a point of the diameter which is not the center, not
+  onely two right lines on each side should be equall. For by any point
+  whatsoever the diameter may be drawne. Such was before observed in a
+  quinquangle; If three angles be equall, all are equall; so in a Circle:
+  If three right lines falling from the same point unto the perephery be
+  equall, all are equall.</p>
+
+  <p><a name="18_e_xv"></a> 18 <i>Of right lines drawne from a point
+  assigned without the periphery, unto the concavity or hollow of the same,
+  that which is by the center is the greatest; And that next to the
+  greatest, is greater than that which is farther off: But of those which
+  fall upon the convexitie of the circumference, the segment of the
+  greatest is least. And that which is next unto the least is lesser than
+  that is farther off: And two on each side of the greatest or least are
+  onely equall. 8 p iij</i>.</p>
+
+  <p>The demonstration of this is very like unto the above mentioned, of
+  five parts. And thus much of the secants, the Tangents doe follow.</p>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/215a.png"><img style="width:100%" src="images/215a.png"
+      alt="Figure for demonstration 19." title="Figure for demonstration 19." /></a>
+  </div>
+  <p><a name="19_e_xv"></a> 19 <i>If a right line be perpendicular unto the
+  end of the diameter, it doth touch the periphery: And contrariwise � 16 p
+  iij</i>.</p>
+
+<p><!-- Page 194 --><span class="pagenum"><a name="page194"></a>[194]</span></p>
+
+  <p>As for example, Let the circle given <i>ae</i>, be perpendicular to
+  the end of the diameter, or the end of the ray, in the end <i>a</i>, as
+  suppose the ray be <i>ia</i>: I say, that <i>ea</i>, doth touch, not cut
+  the periphery in the common bound <i>a</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/215b.png"><img style="width:100%" src="images/215b.png"
+      alt="Figure for demonstration 19 converse." title="Figure for demonstration 19 converse." /></a>
+  </div>
+  <p>This was to have beene made a <i>postulatum</i> out of the definition
+  of a perpendicle: Because if this should leane never so little, it should
+  cut the periphery, and should not be perpendicular: Notwithstanding
+  <i>Euclide</i> doth force it thus: Otherwise let the right line
+  <i>ae</i>, be perpendicular to the diameter <i>ai</i>. And a right line
+  from <i>o</i>, with the center <i>i</i>, let it fall within the circle at
+  <i>o</i>, and let <i>oi</i>, joyned together. Here in the triangle
+  <i>aoi</i>, two angles, contrary to the <a href="#13_e_vj">13 e vj</a>,
+  should be right angles at <i>a</i>, by the grant: And at <i>o</i>, by the
+  <a href="#17_e_vj">17 e vj</a>.</p>
+
+  <p>The demonstration of the converse is like unto the former. For if the
+  tangent, or touch-line <i>ae</i>, be not perpendicular to the diameter
+  <i>iou</i>, let <i>oe</i>, from the center <i>o</i>, be drawne
+  perpendicular; Then shall the angle <i>oei</i>, be right angle: And
+  <i>oie</i> an acutangle: And therefore by the <a href="#22_e_vj">22 e
+  vj</a>, <i>oi</i>, that is <i>oy</i>, shall be greater then <i>oye</i>,
+  that is the part, then the whole.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="20_e_xv"></a> 20 <i>If a right line doe passe by the center
+  and touch-point, it is perpendicular to the tangent or touch-line. 18 p
+  iij</i>.</p>
+
+  <p>Or thus, as <i>Schoner</i> amendeth it: If a right line be the
+  diameter by the touch point, it is perpendicular to the tangent. <!--
+  Page 195 --><span class="pagenum"><a name="page195"></a>[195]</span></p>
+
+  <p><a name="21_e_xv"></a> 21 <i>If a right line be perpendicular unto the
+  tangent, it doth passe by the center and touch-point. 19. p iij.</i></p>
+
+  <p>Or thus: if it be perpendicular to the tangent, it is a diameter by
+  the touch point: <i>Schoner</i>.</p>
+
+  <p>For a right line either from the center unto the touch-point; or from
+  the touch point unto the center is radius or semidiameter.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="22_e_xv"></a> 22 <i>The touch-point is that, into which the
+  perpendicular from the center doth fall upon the touch line</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/216.png"><img style="width:100%" src="images/216.png"
+      alt="Figure for demonstration 23." title="Figure for demonstration 23." /></a>
+  </div>
+  <p><a name="23_e_xv"></a> 23 <i>A tangent on the same side is onely
+  one</i>.</p>
+
+  <p>Or touch line is but one upon one, and the same side: <i>H</i>. Or. A
+  tangent is but one onely in that point of the periphery
+  <i>Schoner</i>.</p>
+
+  <p>It is a consectary drawne out of the <a href="#13_e_ij">xiij. e
+  ij</a>. Because a tangent is a very perpendicular.</p>
+
+  <p><i>Euclide</i> propoundeth this more specially thus; that no other
+  right line may possibly fall betweene the periphery and the tangent.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="24_e_xv"></a> 24 <i>A touch-angle is lesser than any
+  rectilineall acute angle, � 16 p ij</i>.</p>
+
+  <p><i>Angulus contractus,</i> A touch angle is an angle of a straight
+  touch-line and a periphery. It is commonly called <i>Angulus
+  contingenti�</i>: Of <i>Proclus</i> it is named <i>Cornicularis</i>, an
+  horne-like corner; because it is made of a right line and periphery like
+  unto a horne. It is lesse therefore than any acute or sharpe right-lined
+  angle: Because if it were not lesser, a <!-- Page 196 --><span
+  class="pagenum"><a name="page196"></a>[196]</span>right line might fall
+  between the periphery and the perpendicular.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="25_e_xv"></a> 25 <i>All touch-angles in equall peripheries
+  are equall</i>.</p>
+
+  <p>But in unequall peripheries, the cornicular angle of a lesser
+  periphery, is greater than the Cornicular of a greater.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/217.png"><img style="width:100%" src="images/217.png"
+      alt="Figure for demonstration 26." title="Figure for demonstration 26." /></a>
+  </div>
+  <p><a name="26_e_xv"></a> 26 <i>If from a ray out of the center of a
+  periphery given, a periphery be described unto a point assigned without,
+  and from the meeting of the assigned and the ray, a perpendicular falling
+  upon the said ray unto the now described periphery, be tied by a right
+  line with the said center, a right line drawne from the point given unto
+  the meeting of the periphery given, and the knitting line shall touch the
+  assigned periphery 17 p iij</i>.</p>
+
+  <p>As with the ray <i>ae</i>, from the center <i>a</i>, of the periphery
+  assigned, unto the point assigned <i>e</i>, let the periphery <i>eo</i>,
+  be described: And let <i>io</i>, be perpendicular to the ray unto the
+  described periphery. This knit by a right line unto the center <i>a</i>,
+  let <i>eu</i>, be drawne. I say, that <i>eu</i>, doth touch the periphery
+  <i>iu</i>, assigned: Because it shall be perpendicular unto the end of
+  the diameter. For the triangles <i>eau</i>, and <i>oai</i>, by the <a
+  href="#2_e_vij">2 e vij</a>, seeing they are equicrurall; And equall in
+  shankes of the common angle; they are equall in the angles at the base.
+  But the angle <i>aio</i>, is a right angle: Therefore the angle
+  <i>eua</i>, shall be a right angle. And therefore the right line
+  <i>eu</i>, by the <a href="#13_e_ij">13 e ij</a>, is perpendicular to
+  <i>ao</i>.</p>
+
+  <p>Thus much of the Secants and Tangents severally: It followeth of both
+  kindes joyntly together.</p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/218a.png"><img style="width:100%" src="images/218a.png"
+      alt="Figure for demonstration 27 first case." title="Figure for demonstration 27 first case." /></a>
+  </div>
+  <p><a name="27_e_xv"></a> 27 <i>If of two right lines, from an assigned
+  point without, the first doe cut a periphery unto the concave, <!-- Page
+  197 --><span class="pagenum"><a name="page197"></a>[197]</span>the other
+  do touch the same; the oblong of the secant, and of the outter segment of
+  the secant, is equall to the quadrate of the tangent: and if such a like
+  oblong be equall to the quadrate of the other, that same other doth touch
+  the periphery: 36, and 37. p iij</i>.</p>
+
+  <p>If the secant or cutting line do passe by the center, the matter is
+  more easie and as here, Let <span class="correction" title="text reads `a'"
+  ><i>ae</i></span>, cut; And <i>ai</i>, touch: The outter segment is
+  <i>ao</i>, and the center <i>u</i>, Now <i>ui</i>, shall be perpendicular
+  to the tangent <i>ai</i>, by the 20. e: Then by <a href="#8_e_xiij">8 e
+  xiij</a>, the oblong of <i>ea</i>, and <i>ao</i>, with the quadrate of
+  <i>au</i>, that is, of <i>iu</i>, is equall to the quadrate of <i>au</i>,
+  that is, by the <a href="#9_e_xij">9 e xij</a>. to the quadrates of
+  <i>ai</i>, and <i>iu</i>. Take <i>iu</i>, the common quadrate: The
+  Rectangle shall be equall to the quadrate of the tangent.</p>
+
+  <div class="figleft" style="width:16%;">
+      <a href="images/218b.png"><img style="width:100%" src="images/218b.png"
+      alt="Figure for demonstration 27 second case." title="Figure for demonstration 27 second case." /></a>
+  </div>
+  <p>If the secant doe not passe by the center, as in this figure, the
+  center <i>u</i>, found by the <a href="#7_e_xv">7 e</a>, <i>iu</i>, shall
+  be by the <a href="#20_e_xv">20 e</a> perpendicular unto the tangent
+  <i>ai</i>; then draw <i>ua</i>, and <i>uo</i>, and the perpendicular
+  halving <i>oe</i>, by the <a href="#10_e_xv">10 e</a>. Here by the <a
+  href="#8_e_xiij">8 e xiij</a>, the oblong of <i>ae</i>, and <i>ao</i>,
+  with the quadrate <i>oy</i>, is equal to the quadrate <i>ay</i>:
+  Therefore <i>yu</i>, the common quadrate added, the same oblong, with the
+  quadrates <i>oy</i>, and <i>yu</i>, that is by the <a href="#9_e_xij">9 e
+  xij</a>. with the quadrate <i>ou</i>, is equall to the quadrates
+  <i>ay</i>, and <i>uy</i>, that is, by the <a href="#9_e_xij">9 e xij</a>,
+  to <i>au</i>, that is, againe, to <i>ai</i>, and <i>iu</i>. Lastly, let
+  <i>ur</i>, and <i>iu</i>, two equall quadrates be taken from each, and
+  there wil remaine the oblong equall to the quadrate of the tangent.</p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/219.png"><img style="width:100%" src="images/219.png"
+      alt="Figure for demonstration 27 converse." title="Figure for demonstration 27 converse." /></a>
+  </div>
+  <p>The converse is likewise demonstrated in this figure. Let the
+  Rectangle of <i>ae</i>, and <i>ay</i>, be equall to the quadrate of
+  <i>ai</i>. <!-- Page 198 --><span class="pagenum"><a
+  name="page198"></a>[198]</span>I say, that <i>ai</i> doth touch the
+  circle. For let, by the <a href="#26_e_xv">26 e</a>, <i>ao</i> the
+  tangent be drawne: Item let <i>au</i>, <i>ui</i>, and <i>uo</i> bee
+  drawne. Here the oblong of <i>ea</i>, and <i>ay</i>, is equall to the
+  quadrate of <i>ao</i>, by the <a href="#27_e_xv">27 e</a>: And to the
+  quadrate of <i>ai</i>, by the grant. Therefore <i>ai</i>, and <i>ao</i>,
+  are equall. Then is <i>uo</i>, by the <a href="#20_e_xv">20 e</a>,
+  perpendicular to the tangent. Here the triangles <i>auo</i>, and
+  <i>aui</i>, are equilaters: And by the <a href="#1_e_vij">1 e vij</a>,
+  equiangles. But the angle at <i>o</i> is a right angle: Therefore also a
+  right angle and equall to it is that at <i>i</i>, by the <a
+  href="#13_e_iij">13 e iij</a>, wherefore <i>ai</i> is perpendicular to
+  the end of the diameter: And, by the <a href="#19_e_xv">19 e</a>, it
+  toucheth the periphery.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="28_e_xv"></a> 28. <i>All tangents falling from the same point
+  are equall</i>.</p>
+
+  <p>Or, Touch lines drawne from one and the same point are equall:
+  <i>H</i>.</p>
+
+  <p>Because their quadrates are equall to the same oblong.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="29_e_xv"></a> 29. <i>The oblongs made of any secant from the
+  same point, and of the outter segment of the secant are equall betweene
+  themselves. Camp. 36 p iij</i>.</p>
+
+  <p>The reason is because to the same <i>thing</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="30_e_xv"></a> 30. <i>To two right lines given one may so
+  continue or joyne the third, that the oblong of the continued and the
+  continuation may be equall to the quadrate remaining. Vitellio 127 p
+  j</i>.</p>
+
+  <p>As in the first figure, if the first of the lines given be <i>eo</i>,
+  the second <i>ia</i>, the third <i>oa</i>.</p>
+
+  <p>Now are we come to Circular Geometry, that is to the Geometry of
+  Circles or Peripheries cut and touching one another: And of Right lines
+  and Peripheries. <!-- Page 199 --><span class="pagenum"><a
+  name="page199"></a>[199]</span></p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/220a.png"><img style="width:100%" src="images/220a.png"
+      alt="Figure for demonstration 31 first part." title="Figure for demonstration 31 first part." /></a>
+  </div>
+  <p><a name="31_e_xv"></a> 31. <i>If peripheries doe either cut or touch
+  one another, they are eccentrickes: And they doe cut one another in two
+  points onely, and these by the touch point doe continue their diameters,
+  5. 6. 10, 11, 12 p iij</i>.</p>
+
+  <p>All these might well have beene asked: But they have also their
+  demonstrations, <i>ex impossibili</i>, not very difficult.</p>
+
+  <div class="figleft" style="width:17%;">
+      <a href="images/220b.png"><img style="width:100%" src="images/220b.png"
+      alt="Figure for demonstration 31 second part." title="Figure for demonstration 31 second part." /></a>
+  </div>
+  <p>The first part is manifest, because the part should be equall to the
+  whole, if the Center were the same to both, as <i>a</i>. For two raies
+  are equall to the common raie <i>ao</i>: And therefore <i>ae</i> and
+  <i>ai</i>, that is, the part and the whole, are equall one to
+  another.</p>
+
+  <p>The second part is demonstrated as the first: For otherwise the part
+  must be equall to the whole, as here <i>ae</i> and <i>ai</i>, the raies
+  of the lesser periphery; And <i>ae</i>, and <i>ao</i>, the raies of the
+  greater are equall. Wherefore <i>ai</i>, should be equall to <i>ao</i>
+  the Part to the whole.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/220c.png"><img style="width:100%" src="images/220c.png"
+      alt="Figure for demonstration 31 outwardly contiguall." title="Figure for demonstration 31 outwardly contiguall." /></a>
+  </div>
+  <p>If the Peripheries be outwardly contiguall, the matter is more easie,
+  and by the judgement of <i>Euclide</i> it deserved not a demonstration,
+  as here.</p>
+
+  <p><br style="clear :left" /></p>
+  <div class="figleft" style="width:22%;">
+      <a href="images/220d.png"><img style="width:100%" src="images/220d.png"
+      alt="Figure for demonstration 31 third part." title="Figure for demonstration 31 third part." /></a>
+  </div>
+  <p>The third part is apparent out of the first: Otherwise those which cut
+  one another should be concentrickes. For, by the <a href="#7_e_xv">7
+  e</a>, the center being found: And by the <a href="#9_e_xv">9 e</a>,
+  three right lines being drawne from the center unto three points of <!--
+  Page 200 --><span class="pagenum"><a name="page200"></a>[200]</span>the
+  sections, the three raies must be equall, as here.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/221a.png"><img style="width:100%" src="images/221a.png"
+      alt="Figure for demonstration 31 fourth part." title="Figure for demonstration 31 fourth part." /></a>
+  </div>
+  <p>The fourth part is demonstrated after the same manner: Because
+  otherwise the Part must be greater then the whole. For let the right line
+  <i>aeio</i>, be drawne by the centers <i>a</i> and <i>e</i>: And let the
+  particular raies be <i>eu</i>, and <i>au</i>. Here two sides <i>ue</i>,
+  and <i>ea</i>, of the triangle <i>uea</i>, by the <a href="#9_e_vj">9 e
+  vj</a>, are greater than <i>ua</i>: And therefore also then <i>ao</i>;
+  Take away <i>ae</i>, the remainder <i>ue</i>, shall be greater than
+  <i>eo</i>. But <i>ei</i> is equall to <i>eu</i>. Wherefore <i>ei</i> is
+  greater than <i>eo</i>, the part, than the whole.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/221b.png"><img style="width:100%" src="images/221b.png"
+      alt="Figure for demonstration 31 touching without." title="Figure for demonstration 31 touching without." /></a>
+  </div>
+  <p>The same will fall out, if the touch be without, as here: For, by the
+  <a href="#9_e_vj">9 e vj</a>, <i>ea</i> and <i>ia</i>, are greater than
+  <i>ie</i>. But <i>eo</i> and <i>iu</i>, are equall to <i>ea</i>, and
+  <i>ia</i>. Wherefore <i>eo</i>, and <i>iu</i>, are greater than
+  <i>ie</i>, the parts than the whole.</p>
+
+  <p>Of right lines and Peripheries joyntly the rate is but one.</p>
+
+  <p><a name="32_e_xv"></a> 32. <i>If inscripts be equall, they doe cut
+  equall peripheries: And contrariwise, 28, 29 p iij</i>.</p>
+
+  <p>Or thus: If the inscripts of the same circle or of equall circles be
+  equall, they doe cut equall peripheries: And contrariwise <i>B</i>.</p>
+
+  <p>Or thus: If lines inscribed into equall circles or to the same be
+  equall, they cut equall peripheries: And contrariwise, if they doe cut
+  equall peripheries, they shall themselves be equall: <i>Schoner</i>. <!--
+  Page 201 --><span class="pagenum"><a name="page201"></a>[201]</span></p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/222.png"><img style="width:100%" src="images/222.png"
+      alt="Figure for demonstration 32." title="Figure for demonstration 32." /></a>
+  </div>
+  <p>The matter is apparent by congruency or application: as here in this
+  example. For let the circles agree, and then shall equall inscripts and
+  peripheries agree.</p>
+
+  <p>Except with the learned <i>Rodulphus Snellius</i>, you doe understand
+  aswell two equall peripheries to be given, as two equall right lines, you
+  shall not conclude two equall sections, and therefore we have justly
+  inserted <i>of the same, or of equall Circles</i>; which we doe now see
+  was in like manner by <i>Lazarus Schonerus</i>.</p>
+
+<hr class="full" />
+
+<h2>The sixteenth Booke of <i>Geometry</i>,
+Of the Segments of a Circle.</h2>
+
+  <p><a name="1_e_xvj"></a> 1. <i>A Segment of a Circle is that which is
+  comprehended outterly of a periphery, and innerly of a right
+  line</i>.</p>
+
+  <p>The Geometry of Segments is common also to the spheare: But now this
+  same generall is hard to be declared and taught: And the segment may be
+  comprehended within of an oblique line either single or manifold. But
+  here we follow those things that are usuall and commonly received. First
+  therefore the generall definition is set formost, <!-- Page 202 --><span
+  class="pagenum"><a name="page202"></a>[202]</span>for the more easie
+  distinguishing of the species and severall kindes.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/223a.png"><img style="width:100%" src="images/223a.png"
+      alt="Sector." title="Sector." /></a>
+  </div>
+  <p><a name="2_e_xvj"></a> 2. <i>A segment of a Circle is either a
+  sectour, or a section</i>.</p>
+
+  <p><i>Segmentum</i> a segment, and <i>Sectio</i> a section, and
+  <i>Sector</i> a sectour, are almost the same in common acceptation, but
+  they shall be distinguished by their definitions.</p>
+
+  <p><a name="3_e_xvj"></a> 3. <i>A Sectour is a segment innerly
+  comprehended of two right lines, making an angle in the center; which is
+  called an angle in the center: As the periphery is, the base of the
+  sectour, 9 d iij</i>.</p>
+
+  <p>As <i>aei</i> is a sectour. Here a sectour is defined, and his right
+  lined angle, is absolutely called <i>The greater Sectour</i> which
+  notwithstanding may be cut into two sectours by drawing of a
+  semidiameter, as after shall be seene in the measuring of a section.</p>
+
+  <p><a name="4_e_xvj"></a> 4. <i>An angle in the Periphery is an angle
+  comprehended of two right lines inscribed, and jointly bounded or meeting
+  in the periphery. 8 d iij</i>.</p>
+
+  <div class="figleft" style="width:18%;">
+      <a href="images/223b.png"><img style="width:100%" src="images/223b.png"
+      alt="Angle in the Periphery." title="Angle in the Periphery." /></a>
+  </div>
+  <p>This might have beene called <i>The Sectour in the Periphery</i>, to
+  wit, comprehended innerly of two right lines joyntly bounded in the
+  periphery; as here <i>aei</i>.</p>
+
+  <p><a name="5_e_xvj"></a> 5. <i>The angle in the center, is double to the
+  angle of the periphery standing upon the same base, 20 p iij</i>.</p>
+
+  <p>The variety or the example in <i>Euclide</i> is threefold, and yet
+  <!-- Page 203 --><span class="pagenum"><a
+  name="page203"></a>[203]</span>the demonstration is but one and the same:
+  As here <i>eai</i>, the angle in the center, shall be prooved to be
+  double to <i>eoi</i>, the angle in the periphery, the right line
+  <i>ou</i> cutting it into two triangles on each side equicrurall; And, by
+  the <a href="#17_e_vj">17 e vj</a>, at the base equiangles: Whose doubles
+  severally are the angles, <i>eau</i>, of <i>eoa</i>: And <i>iau</i>, of
+  <i>ioa</i>, For seeing it is equall to the two inner equall betweene
+  themselves by the <a href="#15_e_vj">15 e vj</a>; it shall be the double
+  of one of them. Therefore the whole <i>eai</i>, is the double of the
+  whole <i>eoi</i>.</p>
+
+  <p>The second example is thus of the angle in the center <i>aei</i>: And
+  in the periphery <i>aoi</i>. Here the shankes <i>eo</i>, and <i>ei</i>,
+  by the <a href="#28_e_iiij">28 e iiij</a>, are equall: And by the <a
+  href="#17_e_vj">17 e vj</a>, the angles at <i>o</i> and <i>i</i> are
+  equall: To both which the angle in the center is equall, by the <a
+  href="#15_e_vj">15 e vj</a>. Therefore it is double of the one.</p>
+
+  <div class="figcenter" style="width:73%;">
+      <a href="images/224.png"><img style="width:100%" src="images/224.png"
+      alt="Figures for demonstration 5." title="Figures for demonstration 5." /></a>
+  </div>
+  <p>The third example is of the angle in the center, <i>aei</i>, And in
+  the periphery <i>aoi</i>, Let the diameter be <i>oeu</i>. Here the whole
+  angle <i>ieu</i>, by the <a href="#15_e_vj">15 e vj</a>, is equall to the
+  two inner angles <i>eoi</i>, and <i>eio</i>, which are equall one to
+  another, by the <a href="#17_e_vj">17 e vj</a>: And therefore it is
+  double of the one. Item the particular angle <i>aeu</i>, is equall by the
+  <a href="#15_e_vj">15 e vj</a>, to the angles <i>eoa</i>, and <i>eao</i>,
+  equall also one to another, by the <a href="#17_e_vj">17 e vj</a>.
+  Therefore the remainder <i>aei</i>, is the double of the other
+  <i>aoi</i>, in the periphery.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="6_e_xvj"></a> 6. <i>If the angle in the periphery be equall
+  to the <!-- Page 204 --><span class="pagenum"><a
+  name="page204"></a>[204]</span>angle in the center, it is double to it in
+  base. And contrariwise</i>.</p>
+
+  <p>This followeth out of the former element: For the angle in the center
+  is double to the angle in the periphery standing upon the same base:
+  Wherefore if the angle in the periphery be to be made equall to the angle
+  in the center, his base is to be doubled, and thence shall follow the
+  equality of them both: <i>S</i>.</p>
+
+  <p><a name="7_e_xvj"></a> 7. <i>The angles in the center or periphery of
+  equall circles, are as the Peripheries are upon which they doe insist:
+  And contrariwise. � 33 p vj, and 26, 27 p iij</i>.</p>
+
+  <div class="figcenter" style="width:41%;">
+      <a href="images/225.png"><img style="width:100%" src="images/225.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>Here is a double proportion with the periphery underneath, of the
+  angles in the center: And of angles in the periphery. But it shall
+  suffice to declare it in the angles in the center.</p>
+
+  <p>First therefore let the Angles in the center <i>aei</i>, and
+  <i>ouy</i> be equall: The bases <i>ai</i>, and <i>oy</i>, shall be
+  equall, by the <a href="#11_e_vij">11 e vij</a>: And the peripheries,
+  <i>ai</i>, and <i>oy</i>, by the <a href="#32_e_xv">32 e xv</a>, shall
+  likewise be equall. Therefore if the angles be unequall, the peripheries
+  likewise shall be <span class="correction" title="text reads `equall'"
+  >unequall</span>.</p>
+
+  <p>The same shall also be true of the Angles in the Periphery. The
+  Converse in like manner is true: From whence followeth this
+  consectary:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+<p><!-- Page 205 --><span class="pagenum"><a name="page205"></a>[205]</span></p>
+
+  <p><a name="8_e_xvj"></a> 8. <i>As the sectour is unto the sectour, so is
+  the angle unto the angle: And Contrariwise</i>.</p>
+
+  <p>And thus much of the Sectour.</p>
+
+  <p><a name="9_e_xvj"></a> 9. <i>A section is a segment of a circle within
+  c&#x14D;prehended of one right line, which is termed the base of the
+  section</i>.</p>
+
+  <div class="figcenter" style="width:53%;">
+      <a href="images/226a.png"><img style="width:100%" src="images/226a.png"
+      alt="Sections." title="Sections." /></a>
+  </div>
+  <p>As here, <i>aei</i>, and <i>ouy</i>, and <i>srl</i>, are sections.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/226b.png"><img style="width:100%" src="images/226b.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p><a name="10_e_xvj"></a> 10. <i>A section is made up by finding of the
+  center</i>.</p>
+
+  <p>The Invention of the center was manifest at the <a href="#7_e_xv">7 e
+  xv</a>: And so here thou seest a way to make up a Circle, by the <a
+  href="#8_e_xv">8 e xv</a>.</p>
+
+  <p><a name="11_e_xvj"></a> 11 <i>The periphery of a section is divided
+  into two equall parts by a perpendicular dividing the base into two
+  equall parts. 20. p iij</i>.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/226c.png"><img style="width:100%" src="images/226c.png"
+      alt="Figure for demonstration 11." title="Figure for demonstration 11." /></a>
+  </div>
+  <p>Let the periphery of the section <i>aoe</i>, to be halfed or cut into
+  two equall parts. Let the base <i>ae</i>, be cut into two equall parts by
+  the pendicular <i>io</i>, which shall cut the periphery in <i>o</i>, I
+  say, that <i>ao</i>, and <i>oe</i>, are bisegments. For draw two right
+  lines <i>ao</i>, and <i>oe</i>, and thou shalt have two triangles
+  <i>aio</i>, and <i>eio</i>, equilaters by the <a href="#2_e_vij">2 e
+  vij</a>. Therefore the bases <i>ao</i>, and <i>oe</i>, are <!-- Page 206
+  --><span class="pagenum"><a name="page206"></a>[206]</span>equall: And by
+  the <a href="#32_e_xv">32. e xv</a>. equall peripheries to the
+  subtenses.</p>
+
+  <p>Here <i>Euclide</i> doth by congruency comprehende two peripheries in
+  one, and so doe we comprehend them.</p>
+
+  <p><a name="12_e_xvj"></a> 12 <i>An angle in a section is an angle
+  comprehended of two right lines joyntly bounded in the base and in the
+  periphery joyntly bounded 7 d iij</i>.</p>
+
+  <p>Or thus: An angle in the section, is an angle comprehended under two
+  right lines, having the same tearmes with the bases, and the termes with
+  the circumference: <i>H</i>. As <i>aoe</i>, in the former example.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/227a.png"><img style="width:100%" src="images/227a.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><a name="13_e_xvj"></a> 13 <i>The angles in the same section are
+  equall. 21. p iij</i>.</p>
+
+  <p>Let the section be <i>eauo</i>, And in it the angles at <i>a</i>,
+  &amp; <i>u</i>: These are equall, because, by the <a href="#5_e_xvj">5
+  e</a>, they are the halfes of the angle <i>eyo</i>, in the center: Or
+  else they are equall, by the <a href="#7_e_xvj">7 e</a>, because they
+  insist upon the same periphery.</p>
+
+  <p>Here it is certaine that angles in a section are indeed angles in a
+  periphery, and doe differ onely in base.</p>
+
+  <p><a name="14_e_xvj"></a> 14 <i>The angles in opposite sections are
+  equall to two right angles. 22. p iij</i>.</p>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/227b.png"><img style="width:100%" src="images/227b.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>For here the opposite angles at <i>a</i>, and <i>i</i>, are equall to
+  the three angles of the triangle <i>eoi</i>, which are equall to two
+  right angles, by the <a href="#13_e_vj">13 e vj</a>. For first <i>i</i>,
+  is equall to it selfe: Then <i>a</i>, by parts is equall to the two
+  other. For <i>eai</i>, is equall to <i>eoi</i>, and <i>iao</i>, to
+  <i>oei</i>, by the <a href="#13_e_xvj">13 e</a>. Therefore the opposite
+  angles are equall to two right angles. <!-- Page 207 --><span
+  class="pagenum"><a name="page207"></a>[207]</span></p>
+
+  <p>The reason or rate of a section is thus: The similitude doth
+  follow.</p>
+
+  <p><a name="15_e_xvj"></a> 15 <i>If sections doe receive [or containe]
+  equall angles, they are alike � 10. d iij</i>.</p>
+
+  <div class="figcenter" style="width:41%;">
+      <a href="images/228a.png"><img style="width:100%" src="images/228a.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>As here <i>aei</i>, and <i>ouy</i>. The triangle here inscribed,
+  seeing they are equiangles, by the grant; they shall also be alike, by
+  the <a href="#12_e_vij">12 e vij</a>.</p>
+
+  <p><a name="16_e_xvj"></a> 16 <i>If like sections be upon an equall base,
+  they are equall: and contrariwise. 23, 24. p iij</i>.</p>
+
+  <div class="figcenter" style="width:47%;">
+      <a href="images/228b.png"><img style="width:100%" src="images/228b.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p>In the first figure, let the base be the same. And if they shall be
+  said to unequall sections; and one of them greater than another, the
+  angle in that <i>aoe</i>, shall be lesse than the angle <i>aie</i>, in
+  the lesser section, by the <a href="#16_e_vj">16 e vj</a>. which
+  notwithstanding, by the grant, is equall.</p>
+
+  <p>In the second figure, if one section be put upon another, it will
+  agree with it: Otherwise against the first part, like sections upon the
+  same base, should not be equall. But congruency is here sufficient.</p>
+
+  <p>By the former two propositions, and by the <a href="#9_e_xv">9 e
+  xv</a>. one may finde a section like unto another assigned, or else from
+  a circle given to cut off one like unto it. <!-- Page 208 --><span
+  class="pagenum"><a name="page208"></a>[208]</span></p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/229a.png"><img style="width:100%" src="images/229a.png"
+      alt="Angle of a section." title="Angle of a section." /></a>
+  </div>
+  <p><a name="17_e_xvj"></a> 17 <i>Angle of a section is that which is
+  comprehended of the bounds of a section</i>.</p>
+
+  <p>As here <i>eai</i>: And <i>eia</i>.</p>
+
+  <p><a name="18_e_xvj"></a> 18 <i>A section is either a semicircle: or
+  that which is unequall to a semicircle</i>.</p>
+
+  <p>A section is two fold, a semicircle, to wit, when it is cut by the
+  diameter: or unequall to a semicircle, when it is cut by a line lesser
+  than the diameter.</p>
+
+  <p><a name="19_e_xvj"></a> 19 <i>A semicircle is the half section of a
+  circle</i>.</p>
+
+  <p>Or it is that which is made the diameter.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/229b.png"><img style="width:100%" src="images/229b.png"
+      alt="Semicircle and sections." title="Semicircle and sections." /></a>
+  </div>
+  <p><a name="20_e_xvj"></a> 20 <i>A semicircle is comprehended of a
+  periphery and the diameter 18 d j</i>.</p>
+
+  <p>As <i>aei</i>, is a semicircle: The other sections, as <i>oyu</i>, and
+  <i>oeu</i>, are unequall sections: that greater; this lesser.</p>
+
+  <p><a name="21_e_xvj"></a> 21 <i>The angle in a semicircle is a right
+  angle: The angle of a semicircle is lesser than a rectilineall right
+  angle: But greater than any acute angle: The angle in a greater section
+  is lesser than a right angle: Of a greater, it is a greater. In a lesser
+  it is greater: Of a lesser, it is lesser, � 31. and 16. p iij</i>.</p>
+
+  <p>Or thus: The angle in a semicircle is a right angle, the angle of a
+  semicircle is lesse than a right rightlined angle, but <!-- Page 209
+  --><span class="pagenum"><a name="page209"></a>[209]</span>greater than
+  any acute angle: The angle in the greater section is lesse than a right
+  angle: the angle of the greater section is greater than a right angle:
+  the angle in the lesser section is greater than a right angle, the angle
+  of the lesser section, is lesser than a right angle: <i>H</i>.</p>
+
+  <div class="figleft" style="width:21%;">
+      <a href="images/230a.png"><img style="width:100%" src="images/230a.png"
+      alt="Figure for demonstration 21." title="Figure for demonstration 21." /></a>
+  </div>
+  <p>There are seven parts of this Element: The first is that <i>The angle
+  in a semicircle is a right angle</i>: as in <i>aei</i>: For if the ray
+  <i>oe</i>, be drawne, the angle <i>aei</i>, shall be divided into two
+  angles <i>aeo</i>, and <i>oei</i>, equall to the angles <i>eao</i>, and
+  <i>eio</i>, by the <a href="#17_e_vj">17 e vj</a>. Therefore seeing that
+  one angle is equall to the other two, it is a right angle, by the <a
+  href="#6_e_viij">6 e viij</a>. <i>Aristotle</i> saith that the angle in a
+  semicircle is a right angle, because it is the halfe of two right angles,
+  which is all one in effect.</p>
+
+  <p>The second part, <i>That the angle of a semicircle is lesser than a
+  right angle</i>; is manifest out of that, because it is the part of a
+  right angle. For the angle of <span class="correction" title="text reads `these micircle'"
+  >the semicircle</span> <i>aie</i>, is part of the rectilineall right
+  angle <i>aiu</i>.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/230b.png"><img style="width:100%" src="images/230b.png"
+      alt="Figure for demonstration 21, fourth to seventh." title="Figure for demonstration 21, fourth to seventh." /></a>
+  </div>
+  <p>The third part, That it is greater than any acute angle; is manifest
+  out of the <a href="#23_e_xv">23. e xv</a>. For otherwise a tangent were
+  not on the same part one onely and no more.</p>
+
+  <p>The fourth part is thus made manifest: The angle at <i>i</i>, in the
+  greater section <i>aei</i>, is lesser than a right angle; because it is
+  in the same triangle <i>aei</i>, which at <i>a</i>, is a right angle. And
+  if neither of the shankes be by the center, not withstanding an angle may
+  be made equall to the assigned in the same section.</p>
+
+  <p>The fifth is thus: The angle of the greater section <i>eai</i>, is
+  greater than a right angle: because it containeth a right-angle. <!--
+  Page 210 --><span class="pagenum"><a name="page210"></a>[210]</span></p>
+
+  <p>The sixth is thus, the angle <i>aoe</i>, in a lesser section, is
+  greater than a right angle, by the <a href="#14_e_xvj">14 e xvj</a>.
+  Because that which is in the opposite section, is lesser than a right
+  angle.</p>
+
+  <p>The seventh is thus. The angle <i>eao</i>, is lesser than a
+  right-angle: Because it is part of a right angle, to wit of the outter
+  angle, if <i>ia</i>, be drawne out at length.</p>
+
+  <p>And thus much of the angles of a circle, of all which the most
+  effectuall and of greater power and use is the angle in a semicircle, and
+  therefore it is not without cause so often mentioned of <i>Aristotle</i>.
+  This Geometry therefore of <i>Aristotle</i>, let us somewhat more fully
+  open and declare. For from hence doe arise many things.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="22_e_xvj"></a> 22 <i>If two right lines jointly bounded with
+  the diameter of a circle, be jointly bounded in the periphery, they doe
+  make a right angle</i>.</p>
+
+  <p>Or thus; If two right lines, having the same termes with the diameter,
+  be joyned together in one point, of the circomference, they make a right
+  angle. <i>H</i>.</p>
+
+  <p>This corollary is drawne out of the first part of the former Element,
+  where it was said, that an angle in a semicircle is a right angle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="23_e_xvj"></a> 23 <i>If an infinite right line be cut of a
+  periphery of an externall center, in a point assigned and contingent, and
+  the diameter be drawne from the contingent point, a right line from the
+  point assigned knitting it with the diameter, shall be perpendicular unto
+  the infinite line given</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/232a.png"><img style="width:100%" src="images/232a.png"
+      alt="Figure for demonstration 23." title="Figure for demonstration 23." /></a>
+  </div>
+  <p>Let the infinite right line be <i>ae</i>, from whose point <i>a</i>, a
+  perpendicular is to be raised.</p>
+
+  <p>The right line <i>ae</i>, let it be cut by the periphery <i>aei</i>,
+  (whose center <i>o</i>, is out of the assigned <i>ae</i>,) and that in
+  the point <i>a</i>, and a contingent point, as in <i>e</i>: And from
+  <i>e</i>, let the <!-- Page 211 --><span class="pagenum"><a
+  name="page211"></a>[211]</span>diamiter be <i>eoi</i>: The right line
+  <i>ai</i>, from <i>a</i>, the point given, knitting it with the diameter
+  <i>ioe</i>, shall be perpendicular upon the infinite line <i>ae</i>;
+  Because with the said infinite, it maketh an angle in a semicircle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="24_e_xvj"></a> 24 <i>If a right line from a point given,
+  making an acute angle with an infinite line, be made the diameter of a
+  periphery cutting the infinite, a right line from the point assigned
+  knitting the segment, shall be perpendicular upon the infinite
+  line</i>.</p>
+
+  <p>As in the same example, having an externall point given, let a
+  perpendicular unto the infinite right line <i>ae</i> be sought: Let the
+  right line <i>ioe</i>, be made the diameter of the peripherie; and
+  withall let it make with the infinite right line given an acute angle in
+  <i>e</i>, from whose bisection for the center, let a periphery cut the
+  infinite, &amp;c.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:22%;">
+      <a href="images/232b.png"><img style="width:100%" src="images/232b.png"
+      alt="Figure for demonstration 25." title="Figure for demonstration 25." /></a>
+  </div>
+  <p><a name="25_e_xvj"></a> 25 <i>If of two right lines, the greater be
+  made the diameter of a circle, and the lesser jointly bounded with the
+  greater and inscribed, be knit together, the power of the greater shall
+  be more than the power of the lesser by the quadrate of that which
+  knitteth them both together. ad 13 p. x</i>.</p>
+
+  <p>As in this example; The power of the diameter <i>ae</i>, is greater
+  than the power of <i>ei</i>, by the quadrate of <i>ai</i>. For the
+  triangle <i>aei</i>, shall be a rectangle; And by the <a
+  href="#9_e_xij">9 e xij.</a> <i>ae</i>, the greater shall be of <!-- Page
+  212 --><span class="pagenum"><a name="page212"></a>[212]</span>power
+  equall to the shankes. Out of an angle in a semicircle Euclide raiseth
+  two notable fabrickes; to wit, the invention of a meane proportionall
+  betweene two lines given: And the Reason or rate in opposite sections.
+  The <i>genesis</i> or invention of the meane proportionall, of which we
+  heard at the <a href="#9_e_viij">9 e viij</a>. is thus:</p>
+
+  <p><a name="26_e_xvj"></a> 26 <i>If a right line continued or continually
+  made of two right lines given, be made the diameter of a circle, the
+  perpendicular from the point of their continuation unto the periphery,
+  shall be the meane proportionall betweene the two lines given. 13 p
+  vj</i>.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/233.png"><img style="width:100%" src="images/233.png"
+      alt="Figure for demonstration 26." title="Figure for demonstration 26." /></a>
+  </div>
+  <p>As for example, let the assigned right lines be <i>ae</i>, and
+  <i>ei</i>, of the which <i>aei</i>, is continued. And let <i>eo</i>, be
+  perpendicular from the periphery <i>aoi</i>, unto <i>e</i>, the point of
+  continuation or joyning together of the lines given. This <i>eo</i>, say
+  I, shall be the meane proportionall: Because drawing the right lines
+  <i>ao</i>, and <i>io</i>, you shall make a rectangled triangle, seeing
+  that <i>aoi</i>, is an angle in a semicircle: And, by the <a
+  href="#9_e_viij">9 e viij</a>. <i>oe</i>, shall be proportionall betweene
+  <i>ae</i>, and <i>ei</i>.</p>
+
+  <p>So if the side of a quadrate of 10. foote content, were sought; let
+  the sides 1. foote and 10. foote an oblong equall to that same quadrate,
+  be continued; the meane proportionall shall be the side of the quadrate,
+  that is, the power of it shall be 10. foote. The reason of the angles in
+  opposite sections doth follow.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/234a.png"><img style="width:100%" src="images/234a.png"
+      alt="Figure for demonstration 27." title="Figure for demonstration 27." /></a>
+  </div>
+  <p><a name="27_e_xvj"></a> 27 <i>The angles in opposite sections are
+  equall in the alterne angles made of the secant and touch line. 32. p
+  iij</i>.</p>
+
+  <p>If the sections be equall or alike, then are they the sections of a
+  semicircle, and the matter is plaine by the <a href="#21_e_xvj">21 e</a>.
+  But if they be unequall or unlike the argument of demonstration <!-- Page
+  213 --><span class="pagenum"><a name="page213"></a>[213]</span>is indeed
+  fetch'd from the angle in a semicircle, but by the equall or like angle
+  of the tangent and end of the diameter.</p>
+
+  <p>As let the unequall sections be <i>eio</i>, and <i>eao</i>: the
+  tangent let it be <i>uey</i>: And the angles in the opposite sections,
+  <i>eao</i>, and <i>eio</i>. I say they are equall in the alterne angles
+  of the secant and touch line <i>oey</i>, and <i>oeu</i>. First that which
+  is at <i>a</i>, is equall to the alterne <i>oey</i>: Because also three
+  angles <i>oey</i>, <i>oea</i>, and <i>aeu</i>, are equall to two right
+  angles, by the <a href="#14_e_v">14 e v</a>. Unto which also are equall
+  the three angles in the triangle <i>aeo</i>, by the <a href="#13_e_vj">13
+  e vj</a>. From three equals take away the two right angles <i>aue</i>,
+  and <i>aoe</i>: (For <i>aoe</i>, is a right angle, by the <a
+  href="#21_e_xvj">21 e</a>; because it is in a semicircle:) Take away also
+  the common angle <i>aeo</i>: And the remainders <i>eao</i>, and
+  <i>oey</i>, alterne angles, shall be equall.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/234b.png"><img style="width:100%" src="images/234b.png"
+      alt="Figure for demonstration 27." title="Figure for demonstration 27." /></a>
+  </div>
+  <p>Secondarily, the angles at <i>a</i>, and <i>i</i>, are equall to two
+  right angles, by the <a href="#14_e_xvj">14, e</a>: To these are equall
+  both <i>oey</i>, and <i>oeu</i>. But <i>eao</i>, is equall to the alterne
+  <i>oey</i>. Therefore that which is at <i>i</i>, is equall to, the other
+  alterne <i>oeu</i>. Neither is it any matter, whether the angle at
+  <i>a</i>, be at the diameter or not: For that is onely assumed for
+  demonstrations sake: For wheresoever it is, it is equall, to wit, in the
+  same section. And from hence is the making of a like section, by giving a
+  right line to be subtended.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="28_e_xvj"></a> 28 <i>If at the end of a right line given a
+  right lined angle be made equall to an angle given, and from the <!--
+  Page 214 --><span class="pagenum"><a name="page214"></a>[214]</span>toppe
+  of the angle now made, a perpendicular unto the other side do meete with
+  a perpendicular drawn from the middest of the line given, the meeting
+  shall be the center of the circle described by the equalled angle, in
+  whose opposite section the angle upon the line given shall be made equall
+  to the assigned � 33 p iij</i>.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:64%;">
+      <a href="images/235.png"><img style="width:100%" src="images/235.png"
+      alt="Figure for demonstration 28." title="Figure for demonstration 28." /></a>
+  </div>
+  <p>This you may make triall of in the three kindes of angles, all wayes
+  by the same argument: as here the angle given is <i>a</i>: The right line
+  given <i>ei</i>: at the end <i>e</i>, the equalled angle, <i>ieo</i>: The
+  perpendicular to the side <i>eo</i>, let it be <i>eu</i>: But from the
+  middest of the line given let it be <i>yu</i>. Here <i>u</i>, shall be
+  the center desired. And from hence one may make a section upon a right
+  line given, which shall receive a rectilineall angle equall to an angle
+  assigned.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="29_e_xvj"></a> 29 <i>If the angle of the secant and touch
+  line be equall to an assigned rectilineall angle, the angle in the
+  opposite section shall likewise be equall to the same. 34. p iij</i>.</p>
+
+  <p>As in this figure underneath. And from hence one may from a circle
+  given cut off a section, in which there is an <!-- Page 215 --><span
+  class="pagenum"><a name="page215"></a>[215]</span>angle equall to the
+  assigned. As let the angle given be <i>a</i>: And the circle <i>eio</i>.
+  Thou must make at the point <i>e</i>, of the secant <i>eo</i>, and the
+  tangent <i>yu</i>, an angle equall to the assigned, by the <a
+  href="#11_e_iij">11 e iij</a>. such as here is <i>oeu</i>: Then the
+  section <i>oei</i>, shall containe an angle equall to the assigned.</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/236.png"><img style="width:100%" src="images/236.png"
+      alt="Figure for demonstration 29." title="Figure for demonstration 29." /></a>
+  </div>
+<hr class="full" />
+
+<h2>Of <i>Geometry</i> the seventeenth Booke,
+Of the Adscription of a Circle
+and Triangle.</h2>
+
+  <p>Hitherto we have spoken of the Geometry of Rectilineall plaines, and
+  of a circle: Now followeth the Adscription of both: This was generally
+  defined in the first book <a href="#12_e_j">12 e</a>. Now the periphery
+  of a circle is the bound therof. Therefore a rectilineall is inscribed
+  into a circle, when the periphery doth touch the angles of it 3 d iiij.
+  It is circumscribed when it is touched of every side by the periphery; 4
+  d iij.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/237a.png"><img style="width:100%" src="images/237a.png"
+      alt="Figure for demonstration 1." title="Figure for demonstration 1." /></a>
+  </div>
+  <p><a name="1_e_xvij"></a> 1. <i>If rectilineall ascribed unto a circle
+  be an equilater, it is equiangle</i>.</p>
+
+  <p>Of the inscript it is manifest; And that of a Triangle by it selfe:
+  Because if it be equilater, it is equiangle, by the <a href="#19_e_vj">19
+  e vj</a>. But in a Triangulate the matter is to be prooved by
+  demonstration. As here, if the inscripts <i>ou</i>, and <i>sy</i>, be
+  equall, then doe they subtend equall peripheries, by the 32 <!-- Page 216
+  --><span class="pagenum"><a name="page216"></a>[216]</span>e xv. Then if
+  you doe omit the periphery in the middest betweene them both, as here
+  <i>uy</i>, and shalt adde <i>oies</i> the remainder to each of them, the
+  whole <i>oiesy</i>, subtended to the angle at <i>u</i>: And <i>uoies</i>,
+  subtended to the angle at <i>y</i>, shall be equall. Therefore the angles
+  in the periphery, insisting upon equall peripheries are equall.</p>
+
+  <p>Of the circumscript it is likewise true, if the circumscript be
+  understood to be a circle. For the perpendiculars from the center
+  <i>a</i>, unto the sides of the circumscript, by the <a href="#9_e_xij">9
+  e xij</a>, shal make triangles on each side equilaters, &amp; equiangls,
+  by drawing the semidiameters unto the corners, as in the same
+  ex&#x101;ple.</p>
+
+  <p><a name="2_e_xvij"></a> 2. <i>It is equall to a triangle of equall
+  base to the perimeter, but of heighth to the perpendicular from the
+  center to the side</i>.</p>
+
+<p><!-- Page 217 --><span class="pagenum"><a name="page217"></a>[217]</span></p>
+
+  <p>As here is manifest, by the <a href="#8_e_vij">8 e vij</a>. For there
+  are in one triangle, three triangles of equall heighth.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:59%;">
+      <a href="images/237b.png"><img style="width:100%" src="images/237b.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p>The same will fall out in a Triangulate, as here in a quadrate: For
+  here shal be made foure triangles of equall height.</p>
+
+  <p>Lastly every equilater rectilineall ascribed to a circle, shall be
+  equall to a triangle, of base equall to the perimeter of the adscript.
+  Because the perimeter conteineth the bases of the triangles, into the
+  which the rectilineall is resolved.</p>
+
+  <p><a name="3_e_xvij"></a> 3. <i>Like rectilinealls inscribed into
+  circles, are one to another as the quadrates of their diameters, 1 p.
+  xij</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/238.png"><img style="width:100%" src="images/238.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p>Because by the <a href="#1_e_vj">1 e vj</a>, like plains have a
+  doubled reas� of their homologall sides. But in rectilineals inscribed
+  the diameters are the homologall sides, or they are proportionall to
+  their homologall sides. As let the like rectangled triangles be
+  <i>aei</i>, and <i>ouy</i>; Here because <i>ae</i> and <i>ou</i>, are the
+  diameters, the matter appeareth to be plaine at the first sight. But in
+  the Obliquangled triangles, <i>sei</i>, and <i>ruy</i>, alike also, the
+  diameters are proportionall to their homologall sides, to wit, <i>ei</i>
+  and <i>uy</i>. For by the grant, as <i>se</i> is to <i>ru</i>: so is
+  <i>ei</i> to <i>uy</i>, And therefore, by the former, <span
+  class="correction" title="text is in italics, wrongly">as</span> the
+  diameter <i>ea</i> and <i>uo</i>.</p>
+
+  <p>In like Triangulates, seeing by the <a href="#4_e_x">4 e x</a>, they
+  may be resolved into like triangles, the same will fall out.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+<p><!-- Page 218 --><span class="pagenum"><a name="page218"></a>[218]</span></p>
+
+  <p><a name="4_e_xvij"></a> 4. <i>If it be as the diameter of the circle
+  is unto the side of rectilineall inscribed, so the diameter of the second
+  circle be unto the side of the second rectilineall inscribed, and the
+  severall triangles of the inscripts be alike and likely situate, the
+  rectilinealls inscribed shall be alike and likely situate</i>.</p>
+
+  <p>This <i>Euclide</i> did thus assume at the 2 p xij, and indeed as it
+  seemeth out of the 18 p vj. Both which are conteined in the <a
+  href="#23_e_iiij">23 e iiij</a>. And therefore we also have assumed
+  it.</p>
+
+  <p>Adscription of a Circle is with any triangle: But with a triangulate
+  it is with that onely which is ordinate: And indeed adscription of a
+  Circle is common <i>to all</i>.</p>
+
+  <p><a name="5_e_xvij"></a> 5. <i>If two right lines doe cut into two
+  equall parts two angles of an assigned rectilineall, the circle of the
+  ray from their meeting perpendicular unto the side, shall be inscribed
+  unto the assigned rectilineall. 4 and 8. p. iiij</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/239.png"><img style="width:100%" src="images/239.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p>As in the Triangle <i>aei</i>, let the right lines <i>ao</i>, and
+  <i>eu</i>, halfe the angles <i>a</i> and <i>e</i>: And from <i>y</i>,
+  their meeting, let the perpendiculars unto the sides be <i>yo</i>,
+  <i>yu</i>, <i>ys</i>; I say that the center <i>y</i>, with the ray
+  <i>yo</i>, or <i>ya</i>, or <i>ys</i>, is the circle inscribed, by the <a
+  href="#17_e_xv">17 e xv</a>. Because the halfing lines with the
+  perpendiculars shall make equilater triangles, by the <a
+  href="#2_e_vij">2 e vij</a>. And therefore the three perpendiculars,
+  which are the bases of the equilaters, shall be equall. <!-- Page 219
+  --><span class="pagenum"><a name="page219"></a>[219]</span></p>
+
+  <p>The same argument shall serve in a Triangulate.</p>
+
+  <p><a name="6_e_xvij"></a> 6. <i>If two right lines do right anglewise
+  cut into two equall parts two sides of an assigned rectilineall, the
+  circle of the ray from their meeting unto the angle, shall be
+  circumscribed unto the assigned rectilineall. 5 p iiij</i>.</p>
+
+  <p>As in former figures. The demonstration is the same with the former.
+  For the three rayes, by the <a href="#2_e_vij">2 e vij</a>, are equall:
+  And the meeting of them, by the <a href="#17_e_x">17 e x</a>, is the
+  center.</p>
+
+  <p>And thus is the common adscription of a circle: The adscription of a
+  rectilineall followeth, and first of a Triangle.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/240.png"><img style="width:100%" src="images/240.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p><a name="7_e_xvij"></a> 7. <i>If two inscripts, from the touch point
+  of a right line and a periphery, doe make two angles on each side equall
+  to two angles of the triangle assigned be knit together, they shall
+  inscribe a triangle into the circle given, equiangular to the triangle
+  given � 2 p iiij</i>.</p>
+
+  <p>Let the Triangle <i>aei</i> be given: And the circle, <i>o</i>, into
+  which a Triangle equiangular to the triangle given, is to be inscribed.
+  Therefore let the right line <i>uys</i>, touch the periphery <i>yrl</i>:
+  And from the touch <i>y</i>, let the inscripts <i>yr</i>, and <i>yl</i>,
+  make with the tangent two angles <i>uyr</i>, and <i>syl</i>, equall to
+  the assigned angles <i>aei</i>, and <i>aie</i>: And let them be knit
+  together with the right line <i>rl</i>: They shall by the <a
+  href="#27_e_xvj"><span class="correction" title="text reads `19 e xvj'"
+  >27 e xvj</span></a>, make the angle of the alterne segments equall to
+  the angles <span class="correction" title="text reads `url, and yrl'"
+  ><i>uyr</i>, and <i>syl</i></span>. Therefore by the <a href="#4_e_vij">4
+  e vij</a> seeing that two are equall, the other must needs be equall to
+  the remainder.</p>
+
+  <p>The circumscription here is also speciall. <!-- Page 220 --><span
+  class="pagenum"><a name="page220"></a>[220]</span></p>
+
+  <p><a name="8_e_xvij"></a> 8 <i>If two angles in the center of a circle
+  given, be equall at a common ray to the outter angles of a triangle
+  given, right lines touching a periphery in the shankes of the angles,
+  shall circumscribe a triangle about the circle given like to the triangle
+  given. 3 p iiij</i>.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/241.png"><img style="width:100%" src="images/241.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>Let there be a Triangle, and in it the outter angles <i>aei</i>, and
+  <i>aou</i>: The Circle let it be <i>ysr</i>; And in the center <i>l</i>,
+  let the angles <i>ylr</i>, and <i>slr</i>; at the common side <i>lr</i>,
+  bee made equall to the said outter angles <i>aei</i>, and <i>aou</i>. I
+  say the angles of the circumscribed triangle, are equall to the angles of
+  the triangle given. For the foure inner angles of the quadrangle
+  <i>ylrm</i>, are equall to the foure right angles, by the <a
+  href="#6_e_x">6 e x</a>: And two of them, to wit, at <i>y</i> and
+  <i>r</i>, are right angles, by the construction: For they are made by the
+  secant and touch line, from the touch point by the center, by the <a
+  href="#20_e_xv">20 e xv</a>. Therefore the remainders at <i>l</i> and
+  <i>m</i>, are equall to two right angles: To which two <i>aei</i> and
+  <i>aeo</i> are equall. But the angle at <i>l</i>, is equall to the
+  outter: Therefore the remainder <i>m</i>, is equall to <i>aeo</i>. The
+  same shall be sayd of the angles <i>aoe</i>, and <i>aou</i>. Therefore
+  two being equall, the rest at <i>a</i> and <i>i</i>, shall be equall.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="9_e_xvij"></a> 9. <i>If a triangle be a rectangle, an
+  obtusangle, an acute angle, the center of the circumscribed triangle is
+  in the side, out of the sides, and within the sides: And contrariwise. 5
+  e iiij.</i> <!-- Page 221 --><span class="pagenum"><a
+  name="page221"></a>[221]</span></p>
+
+  <div class="figcenter" style="width:60%;">
+      <a href="images/242.png"><img style="width:100%" src="images/242.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p>As, thou seest in these three figures, underneath, the center
+  <i>a</i>.</p>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i>, the eighteenth Booke,
+Of the adscription of a
+Triangulate.</h2>
+
+  <p>Such is the Adscription of a triangle: The adscription of an ordinate
+  triangulate is now to be taught. And first the common adscription, and
+  yet out of the former adscription, after this manner.</p>
+
+  <p><a name="1_e_xviij"></a> 1. <i>If right lines doe touch a periphery in
+  the angles of the inscript ordinate triangulate, they shall onto a circle
+  cirumscribe a triangulate homogeneall to the inscribed
+  triangulate</i>.</p>
+
+  <p>The examples shall be laid downe according as the species or severall
+  kindes doe come in order. The speciall inscription therefore shall first
+  be taught, and that by one side, which reiterated, as oft as need shall
+  require, may fill up the whole periphery. For that <i>Euclide</i> did in
+  the quindecangle <!-- Page 222 --><span class="pagenum"><a
+  name="page222"></a>[222]</span>one of the kindes, we will doe it in all
+  the rest.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/243.png"><img style="width:100%" src="images/243.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p><a name="2_e_xviij"></a> 2. <i>If the diameters doe cut one another
+  right-angle-wise, a right line subtended or drawne against the right
+  angle, shall be the side of the quadrate. � 6 p iiij</i>.</p>
+
+  <p>As here. For the shankes of the angle are the raies whose diameters
+  knit together shall make foure rectangled triangles, equall in shankes:
+  And by the <a href="#2_e_vij">2 e vij</a>, equall in bases. Therefore
+  they they shall inscribe a quadrate.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="3_e_xviij"></a> 3. <i>A quadrate inscribed is the halfe of
+  that which is circumscribed</i>.</p>
+
+  <p>Because the side of the circumscribed (which here is equall to the
+  diameter of the circle) is of power double, to the side of the inscript,
+  by the <a href="#9_e_xij">9 e xij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="4_e_xviij"></a> 4. <i>It is greater than the halfe of the
+  circumscribed Circle</i>.</p>
+
+  <p>Because the circumscribed quadrate, which is his double, is greater
+  than the whole circle.</p>
+
+  <p>For the inscribing or other multangled odde-sided figures we must
+  needes use the helpe of a triangle, each of whose angles at the base is
+  manifold to the other: In a <span class="correction" title="text reads `Quinguangle'"
+  >Quinquangle</span> first, that which is double unto the remainder, which
+  is thus found.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/244.png"><img style="width:100%" src="images/244.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p><a name="5_e_xviij"></a> 5. <i>If a right line be cut proportionally,
+  the base of that triangle whose shankes shall be equall to the whole line
+  cut, and the base to the greater segment of the same, shall have each of
+  the angles at base double to the <!-- Page 223 --><span
+  class="pagenum"><a name="page223"></a>[223]</span>remainder: And the base
+  shall be the side of the quinquangle inscribed with the triangle into a
+  circle. 10, and 11. p iiij</i>.</p>
+
+  <p>Here first thou shalt take for the fabricke or making of the Triangle,
+  for the ray the right line <i>ae</i> by the <a href="#3_e_xiiij">3 e
+  xiiij</a>, cut proportionally in <i>o</i>: A circle also shalt thou make
+  upon the center <i>a</i>, with the ray <i>ae</i>: And then shalt thou by
+  the <a href="#6_e_xv">6 e xv</a>, inscribe a right line equall to the
+  greater segment: And shalt knit the same inscript with the whole line cut
+  with another right line. This triangle shall be your desire. For by the
+  <a href="#17_e_vj">17 e vj</a>, the angles at the base <i>ei</i> are
+  equall, so that looke whatsoever is prooved of the one, is by and by also
+  prooved of the other. Then let <i>oi</i> be drawne; And a Circle, by the
+  <a href="#8_e_xvij">8 e xvij</a>, circumscribed about the triangle
+  <i>aoi</i>. This circle the right line <i>ei</i>, shall touch, by the <a
+  href="#27_e_xv">27 e xv</a>. Because, by the grant, the right line
+  <i>ae</i>, is cut proportionally, therefore the Oblong of the secant and
+  outter segment, is equall to the quadrate of the greater segment, to
+  which by the grant, the base <i>ei</i>, is equall. Here therefore the
+  angle <i>aie</i> is the double of the angle at <i>a</i>: because it is
+  equall to the angles <i>aio</i>, and <i>oai</i>, which are equall
+  betweene themselves. For by the <a href="#27_e_xvj">27 e xvj</a> it is
+  equall to the angle <i>oai</i> in the alterne segment. And the remainder
+  <i>aio</i>, is equall to it selfe. Therefore also the angle <i>aei</i>,
+  is equall to the same two angles, because it is equall to the angle
+  <i>aie</i>. But the outter angle <i>eoi</i>, is equall to the same two,
+  by the <a href="#15_e_vj">15 e vj</a>. Therefore the angles <i>ioe</i>
+  and <i>oei</i> (because they are equall to the same) they are equall
+  betweene themselves. Wherefore by the <a href="#17_e_vj">17 e vj</a>, the
+  sides <i>oi</i> and <i>ei</i> are equall. And there also <i>ao</i> and
+  <i>oi</i>: And the angles <i>oai</i> &amp; <span class="correction"
+  title="text reads `oai'"><i>oia</i></span> are equall by the <a
+  href="#17_e_vj">17 e vj</a>. Wherefore seeing <!-- Page 224 --><span
+  class="pagenum"><a name="page224"></a>[224]</span>that to both the angle
+  <i>aie</i> is equall, it shall be the double of <span class="correction"
+  title="text reads `ther'">either</span> of the equalls.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/245.png"><img style="width:100%" src="images/245.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p>But the base <i>ei</i>, is the side of the equilater quinquangle. For
+  if two right lines halfing both the angles of a triangle which is the
+  double of the remainder, be knit together with a right line, both one to
+  another, and with the angles, shall inscribe unto a circle an equilater
+  triangle, whose one side shall be the base it selfe: As here seeing the
+  angles <i>eoa</i>, <i>eoi</i>, <i>uio</i>, <i>uia</i>, <i>iao</i>, are
+  equal in the periphery, the peripheries, by the <a href="#7_e_xvj">7 e,
+  xvj</a>. subtending them are equall: And therefore, by the <a
+  href="#32_e_xv">32 e, xv</a>. the subtenses <i>ae</i>, <i>ei</i>,
+  <i>io</i>, <i>ou</i>, <i>ua</i>, are also equall. Now of those five, one
+  is <i>ae</i>. Therefore a right line proportionall cut, doth thus make
+  the adscription of a quinquangle: And from thence againe is afforded a
+  line proportionally cut.</p>
+
+  <p><a name="6_e_xviij"></a> 6 <i>If two right lines doe subtend on each
+  side two angles of an inscript quinquangle, they are cut proportionally,
+  and the greater segments are the sides of the said inscript � 8, p
+  xiij</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/246a.png"><img style="width:100%" src="images/246a.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>As here, Let <i>ai</i>, and <i>eu</i>, subtending the angles on each
+  side <i>aei</i>, and <i>eau</i>: I say, That they are proportionally cut
+  in the point <i>s</i>: And the greater segments <i>si</i>, and <i>su</i>,
+  are equall to <i>ae</i>, the side of the quinquangle. For here two
+  triangles are equiangles: First <i>aei</i>, and <i>uae</i>, are equall by
+  the grant, and by the <a href="#2_e_vij">2 e, vij</a>. Therefore the
+  angles <i>aie</i>, and <i>aes</i>, are equall. Then <i>aei</i>, and
+  <i>ase</i>, are equall: Because the <!-- Page 225 --><span
+  class="pagenum"><a name="page225"></a>[225]</span>angle at <i>a</i>, is
+  common to both: Therefore the other is equall to the remainder, by the <a
+  href="#4_e_vij">4 e, 7</a>. Now, by the <a href="#12_e_vij">12. e,
+  vij.</a> as <i>ia</i>, is unto <i>ae</i>, that is, as by and by shall
+  appeare, unto <i>is</i>: so is <i>ea</i>, unto <i>as</i>: Therefore, by
+  the <a href="#1_e_xiiij">1 e, xiiij</a>. <i>ia</i>, is cut proportionally
+  in <i>s</i>. But the side <i>ea</i>, is equall to <i>is</i>: Because both
+  of them is equall to the side <i>ei</i>, that by the grant, this by the
+  <a href="#17_e_vj">17. e, vj</a>. For the angles at the base, <i>ise</i>,
+  and <i>ies</i>, are equall, as being indeed the doubles of the same. For
+  <i>ise</i>, by the <a href="#16_e_vj">16. e vj</a>. is equall to the two
+  inner, which are equall to the angle at <i>u</i>, by the <a
+  href="#17_e_vj">17 e vj</a>. and by the former conclusion. Therefore it
+  is the double of the angles <i>aes</i>: Whose double also is the angle
+  <i>uei</i>, by the <a href="#7_e_xvj">7 e. xvj</a>. insisting indeede
+  upon a double periphery.</p>
+
+  <p>And from hence the fabricke or construction of an ordinate quinquangle
+  upon a right line given, is manifest.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:26%;">
+      <a href="images/246b.png"><img style="width:100%" src="images/246b.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p><a name="7_e_xviij"></a> 7  <i>If a right line given, cut
+  proportionall, be continued at each end with the greater segment, and
+  sixe peripheries at the distance of the line given shall meete, two on
+  each side from the ends of the line given and the continued, two others
+  from their meetings, right lines drawne from their meetings, &amp; the
+  ends of the assigned shall make an ordinate quinquangle upon the
+  assigned</i>.</p>
+
+  <p>The example is thus.</p>
+
+  <p><a name="8_e_xviij"></a> 8 <i>If the diameter of a circle
+  circumscribed about a quinquangle be rationall, it is irrationall unto
+  the side of the inscribed quinquangle, � 11. p xiij.</i> <!-- Page 226
+  --><span class="pagenum"><a name="page226"></a>[226]</span></p>
+
+  <p>So before the segments of a right line proportionally cut were
+  irrationall.</p>
+
+  <p>The other triangulates hereafter multiplied from the ternary,
+  quaternary, or quinary of the sides, may be inscribed into a circle by an
+  inscript triangle, quadrate, or quinquangle. Therefore by a triangle
+  there may be inscribed a triangulate of 6. 12, 24, <span
+  class="correction" title="text reads `46'">48</span>, angles: By a
+  quadrate, a triangulate of 8. 16, 32, 64, angles. By a quinquangle, a
+  triangulate of 10, 20, 40, 80. angles, &amp;c.</p>
+
+  <p><a name="9_e_xviij"></a> 9 <i>The ray of a circle is the side of the
+  inscript sexangle. � 15 p iiij</i>.</p>
+
+  <div class="figright" style="width:25%;">
+      <a href="images/247.png"><img style="width:100%" src="images/247.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>A sexangle is inscribed by an inscript equilaterall triangle, by
+  halfing of the three angles of the said triangle: But it is done more
+  speedily by the ray or semidiameter of the circle, sixe times continually
+  inscribed. As in the circle given, let the diameter be <i>ae</i>; And
+  upon the center <i>o</i>, with the ray <i>ie</i>, let the periphery
+  <i>uio</i>, be described: And from the points <i>o</i> and <i>u</i>, let
+  the diameters be <i>oy</i>, and <i>us</i>; These knit both one with
+  another, and also with the diameter <i>ae</i> shall inscribe an
+  equilaterall sexangle into the circle given, whose side shal be equal to
+  the ray of the same circle. As <i>eu</i>, is equal to <i>ui</i>, because
+  they both equall to the same <i>ie</i>, by the <a href="#29_e_iiij">29 e,
+  iiij</a>. There fore <i>eiu</i>, is an equilater triangle: And likewise
+  <i>eio</i>, is an equilater. The angles also in the center are &#x2154;
+  of one rightangle: And therefore they are equall. And by the <a
+  href="#14_e_v">14. e v</a>, the angle <i>sio</i>, is &#x2153;. of two
+  rightangles: And by the <a href="#15_e_v">15. e v</a>. the angles at the
+  toppe are also equall. Wherefore sixe are equall: And therefore, by the
+  <a href="#7_e_xvj">7 e xvj</a>. and <a href="#32_e_xv">32. e, xv</a>, all
+  the bases are equall, both betweene themselves, and as was even now made
+  manifest, to the ray of the circle given. Therefore the sexangle inscript
+  by the ray of a circle is an <!-- Page 227 --><span class="pagenum"><a
+  name="page227"></a>[227]</span>equilater; And by the <a
+  href="#1_e_xvij">1 e xvij</a>. equiangled.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="10_e_xviij"></a> 10 <i>Three ordinate sexangles doe fill up a
+  place</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/248a.png"><img style="width:100%" src="images/248a.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p>As here. For they are sixe equilater triangles, if you shal resolve
+  the sexangles into sixe triangls: Or els because the angle of an ordinate
+  sexangle is as much as one right angle and &#x2153;. of a right
+  angle.</p>
+
+  <p>Furthermore also no one figure amongst the plaines doth fill up a
+  place. A Quinquangle doth not: For three angles a quinquangle may make
+  only 3.3/5 angles which is too little. And foure would make 4.4/5 which
+  is as much too great. The angles of a septangle would make onely two
+  rightangles, and 6/7 of one: Three would make 3, and 9/7, that is in the
+  whole 4.2/7, which is too much, &amp;c. to him that by induction shall
+  thus make triall, it will appeare, That a plaine place may be filled up
+  by three sorts of ordinate plaines onely.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="11_e_xviij"></a> 11 <i>If right lines from one angle of an
+  inscript sexangle unto the third angle on each side be knit together,
+  they shall inscribe an equilater triangle into the circle given</i>.</p>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/248b.png"><img style="width:100%" src="images/248b.png"
+      alt="Figure for demonstration 11." title="Figure for demonstration 11." /></a>
+  </div>
+  <p>As here; Because the sides shall be subtended to equall peripheries:
+  Therefore by the <a href="#32_e_xv">32 e xv</a>. they shall be equall
+  betweene themselves: And againe, on the contrary, by such a like
+  triangle, by halfing the angles, a sexangle is inscribed.</p>
+
+  <p><a name="12_e_xviij"></a> 12 <i>The side of an inscribed equilater
+  triangle hath a <!-- Page 228 --><span class="pagenum"><a
+  name="page228"></a>[228]</span>treble power, unto the ray of the circle
+  12. p xiij</i>.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/249a.png"><img style="width:100%" src="images/249a.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>As here, with <i>ae</i>, one side of the triangle <i>aei</i>, two
+  third parts of the halfe periphery are imployed: For with one side one
+  third of the whole <i>eu</i>, is imployed: Therefore <i>eu</i>, is the
+  other third part, that is, the sixth part of the whole periphery.
+  Therefore the inscript <i>eu</i>, is the ray of the circle, by the <a
+  href="#9_e_xviij">9 e</a>. Now the power of the diameter <i>aou</i>, by
+  the <a href="#14_e_xij">14 e xij</a>. is foure times so much as is the
+  power of the ray, that is, of <i>eu</i>: And by <a href="#21_e_xvj">21. e
+  xvj</a>, and <a href="#9_e_xij">9 e xij</a>, <i>ae</i>, and <i>eu</i>,
+  are of the same power; take away <i>eu</i>, and the side <i>ae</i>, shall
+  be of treble power unto the ray.</p>
+
+  <p><a name="13_e_xviij"></a> 13 <i>If the side of a sexangle be cut
+  proportionally, the greater segment shall be the side of the
+  decangle</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/249b.png"><img style="width:100%" src="images/249b.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p><i>Pappus lib. 5. ca. 24.</i> &amp; <i>Campanus ad 3 p xiiij.</i> Let
+  the ray <i>ao</i>, or side of the sexangle be cut proportionally, by the
+  <a href="#3_e_xiiij">3 e xiiij</a>: And let <i>ae</i>, be equall to the
+  greater segment. I say that <i>ae</i>, is the side of the decangle. For
+  if it be moreover continued with the whole ray unto <i>i</i>, the whole
+  <i>aei</i>, shall <span class="correction" title="text reads `be by' (duplicated be)"
+  >by</span> the <a href="#4_e_xiiij">4 e xiiij</a>. be cut proportionally:
+  and the greater segment <i>ei</i>, shal be the same ray. For the if the
+  right line <i>iea</i>, be cut proportionally, it shall be as <i>ia</i>,
+  is unto <i>ie</i>, that is to <i>oa</i>, to wit, unto the ray: so
+  <i>ao</i>, shal be unto <i>ae</i>. Therefore, by the <a
+  href="#15_e_vij">15. e vij</a>. the triangles <i>iao</i>, and <i>oae</i>,
+  are equiangles: And the angle <i>aoe</i>, is equall to the angle
+  <i>oia</i>. But the angle <i>uoe</i>, is foure times as great as the
+  angle <i>aoe</i>: for it is equall to the two inner at <i>a</i>, and
+  <i>e</i>, by the <a href="#15_e_vj">15 e vj</a>: which are equall between
+  themselves, by the <a href="#10_e_v"><span class="correction" title="text reads `9 e v'"
+  >10 e v</span></a>. and by the <a href="#17_e_vj">17 e vj</a>. And
+  therefore it is the double of <!-- Page 249 --><span class="pagenum"><a
+  name="pageastx249"></a>[249*]</span><i>aeo</i>, which is the double, for
+  the same cause, of <i>aio</i>, equall to the same <i>aoe</i>. Therefore
+  <i>uoe</i>, is the quadruple of the said <i>aoe</i>. Therefore <span
+  class="correction" title="text reads `ae'"><i>ue</i></span>, is the
+  quadruple of the periphery <i>ea</i>. Therefore the whole <i>uea</i>, is
+  the quintuple of the same <i>ea</i>: And the whole periphery is decuple
+  unto it. And the subtense <i>ae</i>, is the side of the decangle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="14_e_xviij"></a> 14 <i>If a decangle and a sexangle be
+  inscribed in the same circle, a right line continued and made of both
+  sides, shall be cut proportionally, and the greater segment shall be the
+  side of a sexangle; and if the greater segment of a right line cut
+  proportionally be the side of an hexagon, the rest shall be the side of a
+  decagon. 9. p xiij</i>.</p>
+
+  <p>The comparison of the decangle and the sexangle with the quinangle
+  followeth.</p>
+
+  <p><a name="15_e_xviij"></a> 15 <i>If a decangle, a sexangle, and a
+  pentangle be inscribed into the same circle the side of the pentangle
+  shall in power countervaile the sides of the others. And if a right line
+  inscribed do countervaile the sides of the sexangle and decangle, it is
+  the side of the pentangle. 10. p xiiij</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/250.png"><img style="width:100%" src="images/250.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>Let the side of the inscribed quinquangle be <i>ae</i>: of the
+  sexangle, <i>ei</i>: Of the decangle <i>ao</i>. I say, the side
+  <i>ae</i>, doth in power countervaile the rest. For let there be two
+  perp&#x113;diculars: The first <i>io</i>, the second <i>iu</i>, cutting
+  the sides of the quinquangle and decangle into halves: And the meeting of
+  the second perpendicular with the side of the quinquangle let it be
+  <i>y</i>. The syllogisme of the demonstration is this: The oblongs of the
+  side of the quinquangle, and the segments of the same, are equall to the
+  quadrates of the other sides. But the quadrate of the same whole side, is
+  equall to the oblongs of the whole, and the segments, by the <a
+  href="#3_e_xiij">3 e, xiij</a>. Therefore it is equall to the quadrates
+  of the other sides. <!-- Page 230 --><span class="pagenum"><a
+  name="page230"></a>[230]</span></p>
+
+  <p>Let the proportion of this syllogisme be demonstrated: For this part
+  onely remaineth doubtfull. Therefore two triangles, <i>aei</i>, and
+  <i>yei</i>, are equiangles, having one common angle at <i>e</i>: And also
+  two equall ones <i>aei</i>, and <i>eiy</i>, the halfes, to wit, of the
+  same <i>eis</i>: Because that is, by the <a href="#17_e_vj">17 e, vj</a>:
+  one of the two equalls, unto the which <i>eis</i>, the out angle, is
+  equall, by the <a href="#15_e_vj">15 e. vj</a>. And this doth insist upon
+  a halfe periphery. For the halfe periphery <i>als</i>, is equall to the
+  halfe periphery <i>ars</i>: and also <i>al</i>, is equall to <i>ar</i>.
+  Therefore the remnant <i>ls</i>, is equall to the remnant <i>rs</i>: And
+  the whole <i>rl</i>, is the double of the same <i>rs</i>: And therefore
+  <i>er</i>, is the double of <i>eo</i>: And <i>rs</i>, the double of
+  <i>ou</i>. For the bisegments are manifest by the <a href="#10_e_xv">10
+  e, xv</a>. and the <a href="#11_e_xvj">11 e, xvj</a>. Therefore the
+  periphery <i>ers</i>, is the double of the periphery <i>eou</i>: And
+  therefore the angle <i>eiu</i>, is the halfe of the angle <i>eis</i>, by
+  the <a href="#7_e_xvj">7 e, xvj</a>. Therefore two angles of two
+  triangles are equall: Wherefore the remainder, by the <a
+  href="#4_e_vij">4 e vij</a>, is equall to the remainder. Wherefore by the
+  <a href="#12_e_vij">12 e, vij</a>, as the side <i>ae</i>, is to
+  <i>ei</i>: so is <i>ei</i>, to <i>ey</i>. Therefore by the <a
+  href="#8_e_xij">8 e xij</a>, the oblong of the extreames is equall to the
+  quadrate of the meane.</p>
+
+  <p>Now let <i>oy</i>, be knit together with a straight: Here againe the
+  two triangles <i>aoe</i>, and <i>aoy</i>, are equiangles, having one
+  common angle at <i>a</i>: And <i>aoy</i>, and <i>oea</i>, therefore also
+  equall: Because both are equall to the angle at <i>a</i>: That by the <a
+  href="#17_e_vj">17 e, vj</a>: This by the <a href="#2_e_vij">2 e,
+  vij</a>: Because the perpendicular halfing the side of the decangle, doth
+  make two triangles, equicrurall, and equall by the right angle of their
+  shankes: And therefore they are equiangles. Therefore as <i>ea</i>, is to
+  <i>ao</i>: so is <i>ea</i>, to <i>ay</i>. Wherefore by the <a
+  href="#8_e_xij">8 e, xij</a>. the oblong of the two extremes is equall to
+  the quadrate of the meane: And the proposition of the syllogisme, which
+  was to be demonstrated. The converse from hence as manifest
+  <i>Euclide</i> doth use at the 16 p xiij.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/252a.png"><img style="width:100%" src="images/252a.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p><a name="16_e_xviij"></a> 16. <i>If a triangle and a quinquangle be
+  inscribed into the same Circle at the same point, the right line
+  inscribed betweene the bases of the both opposite to the said <!-- Page
+  231 --><span class="pagenum"><a name="page231"></a>[231]</span>point,
+  shall be the side of the inscribed quindecangle. 16. p. iiij</i>.</p>
+
+  <p>For the side of the equilaterall triangle doth subtend 1/3 of the
+  whole pheriphery. And two sides of the ordinate quinquangle doe subtend
+  2/5 of the same. Now 2/5 - 1/3 is 1/15: Therefore the space betweene the
+  triangle, and the quinquangle shall be the 1/15 of the whole
+  periphery.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="17_e_xviij"></a> 17. <i>If a quinquangle and a sexangle be
+  inscribed into the same circle at the same point, the periphery
+  intercepted beweene both their sides, shall be the thirtieth part of the
+  whole periphery</i>.</p>
+
+  <p>As here. Therefore the inscription of ordinate triangulates, of a
+  Quadrate, Quinquangle, Sexangle, Decangle, Quindecangle is easie to bee
+  performed by one side given or found, which reiterated as oft as need
+  shall require, shal subtend the whole periphery. <i>Jun. 4.</i> A. C. <a
+  href="images/252c.png"><img src="images/252c.png" class="middle"
+  style="height:2ex" alt="MDCXXII in apostrophus form" /></a> <i>Campana
+  pulsante pro</i>. H. W.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/252b.png"><img style="width:100%" src="images/252b.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+<hr class="full" />
+
+<p><!-- Page 252 --><span class="pagenum"><a name="pageastx252"></a>[252*]</span></p>
+
+<h2>Of <i>Geometry</i> the ninteenth Booke;
+Of the Measuring of ordinate
+Multangle and of a
+<i>Circle</i>.</h2>
+
+  <p>Out of the Adscription of a Circle and a Rectilineall is drawne the
+  Geodesy of ordinate Multangles, and first of the Circle it selfe. For the
+  meeting of two right lines equally, dividing two angles is the center of
+  the circumscribed Circle: From the center unto the angle is the ray: And
+  then if the quadrate of halfe the side be taken out of the quadrate of
+  the ray, the side of the remainder shall be the perpendicular, by the <a
+  href="#9_e_xij">9 e xij</a>. Therefore a speciall theoreme is here thus
+  made:</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/253.png"><img style="width:100%" src="images/253.png"
+      alt="Figure for demonstration 1 in a quinquangle." title="Figure for demonstration 1 in a quinquangle." /></a>
+  </div>
+  <p><a name="1_e_xix"></a> 1. <i>A plaine made of the perpendicular from
+  the center unto the side, and of halfe the perimeter, is the content of
+  an ordinate multangle</i>.</p>
+
+  <p>As here; The quadrate of 10, the ray is 100. The quadrate of 6, the
+  halfe of the side 12, is 36: And 100. 36 is 64, the quadrate of the
+  Perpendicular, whose side 8, is the Perpendicular it selfe. Now the whole
+  periphery of the Quinquangle, is 60. The halfe thereof therefore is 30.
+  And the product of 30, by 8, is 240, for the content of the sayd
+  quinquangle.</p>
+
+  <p>The Demonstration here also is of the certaine antecedent cause
+  thereof. For of five triangles in a quinquangle, the plaine of the
+  perpendicular, and of halfe the base is one of them, as in the former
+  hath beene taught: Therefore five <!-- Page 253 --><span
+  class="pagenum"><a name="pageastx253"></a>[253*]</span>such doe make the
+  whole quinquangle. But that multiplication, is a multiplication of the
+  Perpendicular by the Perimeter or bout-line.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/254a.png"><img style="width:100%" src="images/254a.png"
+      alt="Figure for demonstration 1 in a sexangle." title="Figure for demonstration 1 in a sexangle." /></a>
+  </div>
+  <p>In an ordinate Sexangle also the ray, by the <a href="#9_e_xviij">9 e
+  xviij</a>, is knowne by the side of the sexangle. As here, the quadrate
+  of 6, the ray is 36. The quadrate of 3, the halfe of the side, is 9: And
+  36 - 9. are 27, for the quadrate of the Perpendicular, whose side 5.2/11
+  is the perpendicular it selfe. Now the whole perimeter, as you see, is
+  36. Therefore the halfe is 18. And the product of 18 by 5.2/11 is 93.3/11
+  for the content of the sexangle given.</p>
+
+  <p>Lastly in all ordinate Multangles this theoreme shall satisfie
+  thee.</p>
+
+  <p><a name="2_e_xix"></a> 2 <i>The periphery is the triple of the
+  diameter and almost one seaventh part of it</i>.</p>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/254b.png"><img style="width:100%" src="images/254b.png"
+      alt="Figure for demonstration 2." title="Figure for demonstration 2." /></a>
+  </div>
+  <p>Or the Periphery conteineth the diameter three times and almost one
+  seventh of the same diameter. That it is triple of it, sixe raies, (that
+  is three diameters) about which the periphery, the <a href="#9_e_xviij">9
+  e xviij</a>, is circumscribed doth plainely shew: And therefore the
+  continent is the greater: But the excesse is not altogether so much as
+  one seventh part. For there doth want an unity of one seventh: And yet is
+  the same excesse farre greater than one eighth part. Therefore because
+  the difference was neerer to one seventh, than it was to one eighth,
+  therefore one seventh was taken, as neerest unto the truth, for the truth
+  it selfe.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+<p><!-- Page 254 --><span class="pagenum"><a name="pageastx254"></a>[254*]</span></p>
+
+  <p><a name="3_e_xix"></a> 3. <i>The plaine of the ray, and of halfe the
+  periphery is the content of the circle</i>.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/255.png"><img style="width:100%" src="images/255.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p>For here 7, the ray, of halfe the diameter 14, Multiplying 22, the
+  halfe of the periphery 44, maketh the oblong 154, for the content of the
+  circle. In the diameter two opposite sides, and likewise in the perimeter
+  the two other opposite sides of the rectangle are conteined. Therefore
+  the halfes of those two are taken, of the which the rectangle is
+  comprehended.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="4_e_xix"></a> 4. <i>As 14 is unto 11, so is the quadrate of
+  the diameter unto the Circle</i>.</p>
+
+  <p>For here 3 bounds of the proportion are given in <i>potentia</i>: The
+  fourth is found by the multiplication of the third by the second, and by
+  the Division of the product by the first: As here the Quadrate of the
+  diameter 14, is 196. The product of 196 by 11 is 2156. Lastly 2156
+  divided by 14, the first bound, giveth in the Quotient 154, for the
+  content of the circle sought. This ariseth by an analysis out of the
+  quadrate and Circle measured. For the reason of 196, unto a 154; is the
+  reason of 14 unto 11, as will appeare by the reduction of the bounds.</p>
+
+  <p>This is the second manner of squaring of a circle taught by
+  <i>Euclide</i> as <i>Hero</i> telleth us, but otherwise layd downe,
+  namely after this manner. <i>If from the quadrate of the diameter you
+  shall take away 3/14 parts of the same, the remainder shall be the
+  content of the Circle.</i> As if 196, the quadrate be divided by 14, the
+  quotient likewise shall be 14. Now thrise 14, are 42: And 196 - 42, are
+  154, the quadrate equall to the circle.</p>
+
+  <p>Out of that same reason or rate of the pheriphery and <!-- Page 255
+  --><span class="pagenum"><a name="pageastx255"></a>[255*]</span>diameter
+  ariseth the manner of measuring of the Parts of a circle, as of a
+  Semicircle, a Sector, a Section, both greater and lesser.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="5_e_xix"></a> 5. <i>The plaine of the ray and one quarter of
+  the periphery, is the content of the semicircle</i>.</p>
+
+  <div class="figright" style="width:26%;">
+      <a href="images/256a.png"><img style="width:100%" src="images/256a.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p>As here thou seest: For the product of 7, the halfe of the diameter,
+  multiplyed by 11, the quarter of the periphery, doth make 77, for the
+  content of the semicircle.</p>
+
+  <p>This may also be done by taking of the halfe of the circle now
+  measured.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="6_e_xix"></a> 6. <i>The plaine made of the ray and halfe the
+  base, is the content of the Sector</i>.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/256b.png"><img style="width:100%" src="images/256b.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>Here are three sectours, <i>ae</i> the base of 12 foote: And <i>ei</i>
+  in like manner of 12 foote. The other or remainder <i>ia</i> of 7
+  <i>f</i>. and 3/7 of one foote. The diameter is 10 foote. Multiply
+  therefore 5, halfe of the diameter, by 6 halfe of the base, and the
+  product 30, shall be the content of the first sector. The same shall also
+  be for the second sectour. Againe multiply the same ray or semidiameters
+  5, by 3.5/7, the halfe of 7.3/7, the product of 18.4/7 shall be the
+  content of the third sector. Lastly, 30 + 30 + 18.4/7 are 78.4/7, the
+  content of the whole circle.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+<p><!-- Page 256 --><span class="pagenum"><a name="pageastx256"></a>[256*]</span></p>
+
+  <p><a name="7_e_xix"></a> 7. <i>If a triangle, made of two raies and the
+  base of the greater section, be added unto the two sectors in it, the
+  whole shall be the content of the greater section: If the same be taken
+  from his owne sector, the remainder shall be the content of the
+  lesser</i>.</p>
+
+  <p>In the former figure the greater section is <i>aei</i>: The lesser is
+  <i>ai</i>. The base of them both is as you see, 6. The perpendicular from
+  the toppe of the triangle, or his heighth is 4. Therefore the content of
+  the triangle is 12. Wherefore 30 + 30 + 12, that is 72, is the content of
+  the greater section <i>aei</i>. And the lesser sectour, as in the former
+  was taught, is 18.4/7. Therefore 18.4/7 - 12, that is, 6.4/7, is the
+  content of <i>ai</i>, the lesser section.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="8_e_xix"></a> 8. <i>A circle of unequall isoperimetrall
+  plaines is the greatest</i>.</p>
+
+  <div class="figcenter" style="width:70%;">
+      <a href="images/257.png"><img style="width:100%" src="images/257.png"
+      alt="Figures for demonstration 8." title="Figures for demonstration 8." /></a>
+  </div>
+  <p>The reason is because it is the most ordinate, and <!-- Page 257
+  --><span class="pagenum"><a
+  name="pageastx257"></a>[257*]</span>comprehended of most bounds; see the
+  <a href="#7_e_iiij">7</a>, and <a href="#15_e_iiij"><span
+  class="correction" title="text reads `11 e iiij'">15 e iiij</span></a>.
+  As the Circle <i>a</i>, of 24 perimeter, is greater then any rectilineall
+  figure, of equall perimeter to it, as the Quadrate <i>e</i>, or the
+  Triangle <i>i</i>.</p>
+
+  <div class="figcenter" style="width:57%;">
+      <a href="images/258.png"><img style="width:100%" src="images/258.png"
+      alt="Figures for demonstration 8." title="Figures for demonstration 8." /></a>
+  </div>
+<hr class="full" />
+
+<h2>Of <i>Geometry</i> the twentieth Booke,
+Of a Bossed surface.</h2>
+
+  <p><a name="1_e_xx"></a> 1. <i>A bossed surface is a surface which lyeth
+  unequally betweene his bounds</i>.</p>
+
+  <p>It is contrary unto a Plaine surface, as wee heard at the <a
+  href="#4_e_v">4 e v</a>. <!-- Page 258 --><span class="pagenum"><a
+  name="pageastx258"></a>[258*]</span></p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/259a.png"><img style="width:100%" src="images/259a.png"
+      alt="Sphericall surface." title="Sphericall surface." /></a>
+  </div>
+  <p><a name="2_e_xx"></a> 2. <i>A bossed surface is either a sphericall,
+  or varium</i>.</p>
+
+  <p><a name="3_e_xx"></a> 3. <i>A sphericall surface is a bossed surface
+  equally distant from the center of the space inclosed</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="4_e_xx"></a> 4. <i>It is made by the turning about of an
+  halfe circumference the diameter standeth still. � 14 d xj</i>.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/259b.png"><img style="width:100%" src="images/259b.png"
+      alt="Figure for demonstration 4." title="Figure for demonstration 4." /></a>
+  </div>
+  <p>As here if thou shalt conceive the space betweene the periphery and
+  the diameter to be empty.</p>
+
+  <p><a name="5_e_xx"></a> 5. <i>The greatest periphery in a sphericall
+  surface is that which cutteth it into two equall parts</i>.</p>
+
+  <p>Those things which were before spoken of a circle, the same almost are
+  hither to bee referred. The greatest periphery of a sphericall doth
+  answere unto the Diameter of a Circle.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/260a.png"><img style="width:100%" src="images/260a.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p><a name="6_e_xx"></a> 6. <i>That periphery that is neerer to the
+  greatest, is greater than that which is farther off: And on each <!--
+  Page 259 --><span class="pagenum"><a
+  name="pageastx259"></a>[259*]</span>side those two which are equally
+  distant from the greatest, are equall</i>.</p>
+
+  <p>The very like unto those which are taught at the <a
+  href="#15_e_xv">15</a>, <a href="#16_e_xv">16</a>, <a
+  href="#17_e_xv">17</a>, <a href="#18_e_xv">18. e. xv</a>. may here againe
+  be repeated: As here.</p>
+
+  <p><a name="7_e_xx"></a> 7 <i>The plaine made of the greatest periphery
+  and his diameter is the sphericall</i>.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/260b.png"><img style="width:100%" src="images/260b.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>So the plaine made of the diameter 14. and of 44. the greatest
+  periphery, which is 616. is the sphericall surface. So before the content
+  of a circle was measured by a rectangle both of the halfe diameter, and
+  periphery. But here, by the whole periphery and whole diameter, there is
+  made a rectangle for the measure of the sphericall, foure times so great
+  as was that other: Because by the <a href="#1_e_vj">1 e vj</a>. like
+  plaines (such as here are conceived to be made of both halfe the
+  diameter, and halfe the periphery, and both of the whole diameter and
+  whole periphery) are in a doubled reason of their homologall sides.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="8_e_xx"></a> 8 <i>A plaine of the greatest circle and 4, is
+  the sphericall</i>.</p>
+
+  <p>This consectarium is manifest out of the former element.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_xx"></a> 9 <i>As 7 is to 22. so is the quadrate of the
+  diameter unto the sphericall.</i></p>
+
+<p><!-- Page 260 --><span class="pagenum"><a name="pageastx260"></a>[260*]</span></p>
+
+  <p>For 7, and 22, are the two least bounds in the reason of the diameter
+  unto the periphery: But in a circle, as 14, is to 11, so is the quadrate
+  of the diameter unto the circle. The analogie doth answer fitly: Because
+  here thou multipliest by the double, and dividest by the halfe: There
+  contrariwise thou multipliest by the halfe, and dividest by the double.
+  Therefore there one single circle is made, here the quadruple of that.
+  This is, therefore the analogy of a circle and sphericall; from whence
+  ariseth the hemispherical, the greater and the lesser section.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="10_e_xx"></a> 10 <i>The plaine of the greatest periphery and
+  the ray, is the hemisphericall</i>.</p>
+
+  <p>As here, the greatest periphery is 44. the ray 7. The product
+  therefore of 44. by 7. that is, 308. is the hemisphericall.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/261.png"><img style="width:100%" src="images/261.png"
+      alt="Figure for demonstration 11." title="Figure for demonstration 11." /></a>
+  </div>
+  <p><a name="11_e_xx"></a> 11 <i>If looke what the part be of the ray
+  perpendicular from the center unto the base of the greater section, so
+  much the hemisphericall be increased, the whole shall be the greater
+  section of the sphericall: But if it be so much decreased, the remainder
+  shall be the lesser</i>.</p>
+
+  <p>As in the example, the part of the third ray, that is, of 3/7, is from
+  the center: such like part of the hemispherical 308, is 132. (For the 7,
+  part of 308. is 44. And three times 44. is 132.) Therefore 132. added to
+  308. do make 440. for the greater section of the sphericall. And 132.
+  taken from 308. doe leave 176. for the lesser section of the same.</p>
+
+  <p><a name="12_e_xx"></a> 12 <i>The varium is a bossed surface, whose
+  base is a <!-- Page 241 --><span class="pagenum"><a
+  name="page241"></a>[241]</span>periphery, the side a right line from the
+  bound of the toppe, unto the bound of the base</i>.</p>
+
+  <p><a name="13_e_xx"></a> 13 <i>A varium is a conicall or a cylinderlike
+  forme</i>.</p>
+
+  <p><a name="14_e_xx"></a> 14 <i>A conicall surface is that which from the
+  periphery beneath doth equally waxe lesse and lesse unto the very
+  toppe</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figleft" style="width:21%;">
+      <a href="images/262a.png"><img style="width:100%" src="images/262a.png"
+      alt="Conicall surface." title="Conicall surface." /></a>
+  </div>
+  <p><a name="15_e_xx"></a> 15. <i>It is made by turning about of the side
+  about the periphery beneath</i>.</p>
+
+  <p><a name="16_e_xx"></a> 16 <i>The plaine of the side and halfe the base
+  is the conicall surface</i>.</p>
+
+  <p>As in the example next aforegoing, the side is 13. The halfe periphery
+  is 15.5/7: And the product of 15.5/7 by 13. is 204.2/7. for the conicall
+  surface. To which if you shall adde the circle underneath, you shall have
+  the whole surface.</p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/262b.png"><img style="width:100%" src="images/262b.png"
+      alt="Cylinderlike forme." title="Cylinderlike forme." /></a>
+  </div>
+  <p><a name="17_e_xx"></a> 17 <i>A cylinderlike forme is that which from
+  the periphery underneath unto the the upper one, equall and parallell
+  unto it, is equally raised</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="18_e_xx"></a> 18 <i>It is made by the turning of the side
+  about two equall and parallell peripheries</i>.</p>
+
+  <p><a name="19_e_xx"></a> 19 <i>The plaine of his side and heighth is the
+  cylinderlike surface.</i> <!-- Page 242 --><span class="pagenum"><a
+  name="page242"></a>[242]</span></p>
+
+  <p>As here the periphery is 22. as is gathered by the Diameter, which is
+  7. The heighth is 12. The base therefore is 38.1/2. And 38.1/2 by 12. are
+  462. for the cylinderlike surface. To which if you shall adde both the
+  bases on each side, to wit, 38.1/2. twise, or 77. once, the whole surface
+  shall be 539.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<h2><i>Geometry</i>, the one and twentieth Book,
+Of Lines and Surfaces in solids.</h2>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/263.png"><img style="width:100%" src="images/263.png"
+      alt="Body or solid." title="Body or solid." /></a>
+  </div>
+  <p><a name="1_e_xxj"></a> 1 <i>A body or solid is a lineate broad and
+  high 1 d xj</i>.</p>
+
+  <p>For length onely is proper to a line: Length and breadth, to a
+  surface: Length breath, and heighth joyntly, belong unto a body: This
+  threefold perfection of a magnitude, is proper to a body: Whereby wee doe
+  understand that are in a body, not onely lines of length, and surfaces of
+  breadth, (for so a body should consist of lines and surfaces.) But we do
+  conceive a solidity in length, breadth and heighth. For every part of a
+  body is also a body. And therefore a solid we doe understand the body it
+  selfe. As in the body <i>aeio</i>, the length is <i>ae</i>; the breadth,
+  <i>ai</i>, And the heighth, <i>ao</i>.</p>
+
+  <p><a name="2_e_xxj"></a> 2 <i>The bound of a solid is a surface 2 d
+  xj</i>.</p>
+
+  <p>The bound of a line is a point: and yet neither is a point a line, or
+  any part of a line. The bound of a surface is a line: And yet a line is
+  not a surface, or any part of a surface. So now the bound of a body is a
+  surface: And yet a surface is not a body, or any part of a body. A
+  magnitude is one thing; <!-- Page 243 --><span class="pagenum"><a
+  name="page243"></a>[243]</span>a bound of a magnitude is another thing,
+  as appeared at the <a href="#5_e_j">5 e j</a>.</p>
+
+  <p>As they were called plaine lines, which are conceived to be in a
+  plaine, so those are named solid both lines and surfaces which are
+  considered in a solid; And their perpendicle and parallelisme are hither
+  to be recalled from simple lines.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/264.png"><img style="width:100%" src="images/264.png"
+      alt="Figure for demonstration 3." title="Figure for demonstration 3." /></a>
+  </div>
+  <p><a name="3_e_xxj"></a> 3 <i>If a right line be unto right lines cut in
+  a plaine underneath, perpendicular in the common intersection, it is
+  perpendicular to the plaine beneath: And if it be perpendicular, it is
+  unto right lines, cut in the same plaine, perpendicular in the common
+  intersection � 3 d and 4 p xj</i>.</p>
+
+  <p>Perpendicularity was in the former attributed to lines considered in a
+  surface. Therefore from thence is repeated this consectary of the
+  perpendicle of a line with the surface it selfe.</p>
+
+  <p>If thou shalt conceive the right lines, <i>ae</i>, <i>io</i>,
+  <i>uy</i>, to cut one another in the plaine beneath, in the common
+  intersections: And the line <i>rs</i>, falling from above, to be to every
+  one of them perpendicular in the common point <i>s</i>, thou hast an
+  example of this consectary.</p>
+
+  <p><a name="4_e_xxj"></a> 4 <i>If three right lines cutting one another,
+  be unto the same right line perpendicular in the common section, they are
+  in the same plaine 5. p xj</i>.</p>
+
+  <p>For by the perpendicle and common section is understood an equall
+  state on all parts, and therefore the same plaine: as in the former
+  example, <i>as</i>, <i>ys</i>, <i>os</i>, suppose them to be to
+  <i>sr</i>, the same loftie line, perpendicular, they shall be in the same
+  nearer plaine <i>aiueoy</i>.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/265a.png"><img style="width:100%" src="images/265a.png"
+      alt="Figure for demonstration 5." title="Figure for demonstration 5." /></a>
+  </div>
+  <p><a name="5_e_xxj"></a> 5 <i>If two right lines be perpendicular to the
+  under-plaine, they are parallells: And if the one two <!-- Page 244
+  --><span class="pagenum"><a name="page244"></a>[244]</span>parallells be
+  perpendicular to the under plaine, the other is also perpendicular to the
+  same. 6. 8 p xj</i>.</p>
+
+  <p>The cause is out of the first law or rule parallells. For if two right
+  lines be perpendicular to the same under plaine, being joyned together by
+  a right line, they shall make their inner corners equall to two right
+  angles: And therefore they shall be parallells, by the <a
+  href="#21_e_v">21. e v</a>. And if in two parallells knit together with a
+  right line, one of the inner angles, be a right angle: the other also
+  shall be a right angle. Because they are divided by a common
+  perpendicular; As in the example. If the angles at <i>a</i>, and
+  <i>e</i>, be right angles, <i>ai</i>, and <i>eo</i>, are parallells, and
+  contrariwise, if <i>ai</i>, and <i>eo</i> be parallells, and the angle at
+  <i>a</i>, be a right angle, the angle at <i>e</i>, also shall be a right
+  angle.</p>
+
+  <p><a name="6_e_xxj"></a> 6 <i>If right lines in diverse plaines be unto
+  the same right line parallel, they are also parallell betweene
+  themselves. 9 p xj</i>.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/265b.png"><img style="width:100%" src="images/265b.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p>As here <i>ae</i>, and <i>uy</i>, right lines in diverse plaines
+  suppose them to be parallell to <i>io</i>: I say, they are parallell one
+  to another. For from the point <i>i</i>, let <i>ia</i>, and <i>iu</i>, be
+  <span class="correction" title="text reads `these words, or similar, are omitted in the text'"
+  >erected at right angles to <i>io</i> to cut the</span> parallells, by
+  the <a href="#17_e_v">17. e v</a>. Therefore, by the <a href="#3_e_xxj">3
+  e</a>, <i>oi</i>, seeing that it is perpedicular to <i>ia</i>, and
+  <i>iu</i>, two lines cutting one another, it is perpendicular to the
+  plaine beneath. Therefore by the the <a href="#6_e_xxj">6 e</a>,
+  <i>yu</i>, and <i>ea</i>, are perpendicular to the same plaine: And
+  therefore, by the same, they are parallell.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/266a.png"><img style="width:100%" src="images/266a.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p><a name="7_e_xxj"></a> 7 <i>If two right lines be perpendiculars, the
+  first from a point above, unto a right line underneath, the second <!--
+  Page 245 --><span class="pagenum"><a name="page245"></a>[245]</span>from
+  the common section in the plaine underneath, a third, from the sayd point
+  perpendicular to the second, shall be perpendicular to the plaine
+  beneath. � 11 p xj</i>.</p>
+
+  <p>It is a consectary out of the <a href="#3_e_xxj">3 e</a>. As for
+  example, if from a lofty point <i>a</i>, <i>ae</i>, be by the <a
+  href="#18_e_v">18 e v</a>, perpendicular to <i>e</i>, a point of the
+  right line <i>io</i> underneath: And from <i>e</i> the common section, by
+  the <a href="#17_e_v">17 e v</a>, there be <i>eu</i>, another
+  perpendicular: Lastly <i>ay</i>, a lofty right line, be by the <a
+  href="#18_e_v">18 e v</a>, perpendicular unto <i>eu</i>, at the point
+  <i>y</i>, <i>ay</i> shall be perpendicular unto the plaine underneath.
+  For that <i>ae</i> is perpendicular to <i>io</i>, the same <i>ae</i>
+  declineth neither to the right hand, nor to the left, by the <a
+  href="#13_e_ij">13 e ij</a>. And in that againe <i>ay</i> is
+  perpendicular to <i>eu</i>, it leaneth neither forward nor backeward.
+  Therefore it lyeth equally or indifferently, betweene the foure quarters
+  of the world.</p>
+
+  <p>If the right line <i>io</i>, doe with equall angles agree to <i>r</i>,
+  the third element.</p>
+
+  <p><a name="8_e_xxj"></a> 8. <i>If a right line from a point assigned of
+  a plaine underneath, be parallell to a right line perpendicular to the
+  same plaine, it shall also be perpendicular to the plaine underneath. ex
+  12 p xj</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/266b.png"><img style="width:100%" src="images/266b.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>As for example let the plaine be <i>aeio</i>: And the assigned point
+  in it <i>u</i>: From this point a lofty perpendicular is to be erected.
+  Let there be made from the point <i>y</i>, the perpendicular <i>ys</i>,
+  unto the plaine underneath, by the <a href="#7_e_xxj">7 e</a>. And to it
+  let <i>ur</i>, be made parallell by the <a href="#24_e_v">24 e v</a>. Now
+  <i>ur</i>, seeing it is parallell to a perpendicular upon the plaine
+  underneath, it shall be perpendicular to the same, by the <a
+  href="#6_e_xxj"><span class="correction" title="text reads `6 e'">5
+  e</span></a>.</p>
+
+<p><!-- Page 246 --><span class="pagenum"><a name="page246"></a>[246]</span></p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/267.png"><img style="width:100%" src="images/267.png"
+      alt="Plaines perceived in a Booke." title="Plaines perceived in a Booke." /></a>
+  </div>
+  <p><a name="9_e_xxj"></a> 9. <i>If a right line in one of the plaines
+  cut, perpendicular to the common section, be perpendicular to the other,
+  the plaines are perpendicular: And if the plaines be perpendicular, a
+  right line in the one perpendicular to the common section is
+  perpendicular to the other � 4 d, and 38 p xj</i>.</p>
+
+  <p>The perpendicularity of plaines, is drawne out of the former condition
+  of the perpendicle: And the state of plaines on each side equall betweene
+  themselves, is fetch'd from a perpendicularity of a right line falling
+  upon a plaine. Because from hence it is understood that the plaine it
+  selfe doth lye indifferently betweene all parts signified by right lines:
+  Which in a Booke with the pages each way opened, is perceived by the
+  verses or lines of the pages, both to the section and plaine underneath,
+  perpendicular as here thou seest.</p>
+
+  <p><a name="10_e_xxj"></a> 10. <i>If a right line be perpendicular to a
+  plaine, all plaines by it, are perpendicular to the same: And if two
+  plaines be unto any other plaine perpendiculars, the common section is
+  perpendicular to the same. e 15, and 19 p. xj</i>.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/268a.png"><img style="width:100%" src="images/268a.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p>The first is a consectary drawne out of the <a href="#9_e_xxj">9
+  e</a>. And the latter is from hence manifest, because that same common
+  section is a right line, in any manner of lofty plaines intersected,
+  perpendicular both to the common section and plaine underneath. For if
+  the common section, were not perpendicular to the plaine underneath,
+  neither should the plaines <!-- Page 247 --><span class="pagenum"><a
+  name="page247"></a>[247]</span>cutting one another be perpendicular to
+  the plaine underneath, but some one should be oblique, against the grant,
+  as here thou seest.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/268b.png"><img style="width:100%" src="images/268b.png"
+      alt="Parallell plaines." title="Parallell plaines." /></a>
+  </div>
+  <p><a name="11_e_xxj"></a> 11. <i>Plaines are parallell which doe leane
+  no way. 8 d xj</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="12_e_xxj"></a> 12. <i>Those which divided by a common
+  perpendicle. 14 p xj</i>.</p>
+
+  <p>It is a consectary out of the <a href="#3_e_xxj">3</a>, and <a
+  href="#6_e_xxj">6 e</a>. For if the middle right line be perpendicular to
+  both the plaines, it is also to the right lines on either side cut,
+  perpendicular in the common intersection: And the <span
+  class="correction" title="text reads `innner'">inner</span> angles on
+  each side, being right angles, will evince them to be parallels.</p>
+
+  <p>It is also out of the definition of parallels, at the <a
+  href="#15_e_ij"><span class="correction" title="text reads `17 e ij' (no such section)"
+  >15 e ij</span></a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="13_e_xxj"></a> 13. <i>If two paires of right in them be
+  joyntly bounded, they are parallell. 15 p xj</i>.</p>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/269a.png"><img style="width:100%" src="images/269a.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p>Such are the opposite walls in the toppe or ridge of houses. As let
+  <i>aei</i>, and <i>uoy</i>, be plaine which have two payres of <!-- Page
+  248 --><span class="pagenum"><a name="page248"></a>[248]</span>right
+  lines, <i>ea</i>, and <i>ia</i>: Item <i>uo</i>, and <i>yo</i>, joyntly
+  bounded in <i>a</i>, and <i>o</i>: And parallels, to wit <i>ea</i>,
+  against <i>uo</i>: and <i>ia</i>, against <i>yo</i>. I say that the
+  plaines themselves are parallels: For the right lines <i>ue</i>, and
+  <i>oa</i>: item <i>yi</i>, and <i>oa</i>, doe knit together equall
+  parallels, they shal by the <a href="#27_e_v">27 e v</a>, be equall and
+  parallels: And so they shall prove the equidistancie.</p>
+
+  <p>The same will fall out if thou shalt imagine the joyntly bounded to
+  infinitely drawn out; for the plaines also infinitely extended shall be
+  parallell.</p>
+
+  <p><a name="14_e_xxj"></a> 14. <i>If two parallell plaines are cut with
+  another plaine, the common sections are parallels, 16 p xj</i>.</p>
+
+  <div class="figleft" style="width:17%;">
+      <a href="images/269b.png"><img style="width:100%" src="images/269b.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>As here thou seest the parallell plaines <i>aeio</i>, and <i>uysr</i>,
+  cut by the plaine <i>ljvf</i>, the common sections <i>lj</i>, and
+  <i>fv</i>, shall also be parallell: Otherwise they themselves, and
+  therefore also the plaines in which they are, shall meete, as in the
+  point <i>t</i>, which is against the grant.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<p><!-- Page 249 --><span class="pagenum"><a name="page249"></a>[249]</span></p>
+
+<h2>The twenty second Booke, of P.
+<i>Ramus</i> Geometry,
+Of a
+<i>Pyramis</i>.</h2>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/270.png"><img style="width:100%" src="images/270.png"
+      alt="Axis." title="Axis." /></a>
+  </div>
+  <p><a name="1_e_xxij"></a> 1. <i>The axis of a solid is the diameter
+  about which it is turned, e 15, 19, 22 d xj</i>.</p>
+
+  <p>The Axis or Axeltree is commonly thought to be proper to the sphere or
+  globe, as here <i>ae</i>: But it is attributed to other kindes of solids,
+  as well as to that.</p>
+
+  <p><a name="2_e_xxij"></a> 2. <i>A right solid is that whose axis is
+  perpendicular to the center of the base</i>.</p>
+
+  <p>Thus <i>Serenus</i> and <i>Apllonius</i> doe define a Cone and a
+  Cylinder: And these onely <i>Euclide</i> considered: Yea and indeed
+  stereometry entertaineth no other kinde of solid but that which is right
+  or perpendicular.</p>
+
+  <p><a name="3_e_xxij"></a> 3. <i>If solids be comprehended of homogeneall
+  surfaces, equall in multitude and magnitude, they are equall. 10 d
+  xj</i>.</p>
+
+  <p>Equality of lines and surfaces was not informed by any peculiar rule;
+  farther than out of reason and common sense, and in most places
+  congruency and application was enough and did satisfie to the full: But
+  here the congruency of Bodies is judged by their surfaces. Two cubes are
+  equall, whose sixe sides or plaine surfaces, are equall, &amp;c.</p>
+
+  <p><a name="4_e_xxij"></a> 4. <i>If solids be comprehended of surfaces in
+  multitude equall and like, they are equall, 9 d xj.</i> <!-- Page 250
+  --><span class="pagenum"><a name="page250"></a>[250]</span></p>
+
+  <p>This is a consectary drawne out of the general difinition of like
+  figures, at the <a href="#19_e_iiij">19 e. iiij</a>. For there like
+  figures were defined to be equiangled and proportionall in the shankes of
+  the equall angles: But in like plaine solids the angles are esteemed to
+  be equall out of the similitude of their like plaines: And the equall
+  shankes are the same plaine surfaces, and therefore they are
+  proportionall, equall and alike.</p>
+
+  <p><a name="5_e_xxij"></a> 5 <i>Like solids have a treble reason of their
+  homologall sides, and two meane proportionalls. 33. p xj. 8 p
+  xij</i>.</p>
+
+  <div class="figcenter" style="width:62%;">
+      <a href="images/271.png"><img style="width:100%" src="images/271.png"
+      alt="Like solids." title="Like solids." /></a>
+  </div>
+  <p>It is a consectary drawne out of the <a href="#24_e_iiij">24 e.
+  iiij</a>. as the example from thence repeated shall make manifest.</p>
+
+  <p><a name="6_e_xxij"></a> 6 <i>A solid is plaine or embosed</i>.</p>
+
+  <p><a name="7_e_xxij"></a> 7 <i>A plaine solid is that which is
+  comprehended of plaine surfaces</i>.</p>
+
+  <p><a name="8_e_xxij"></a> 8 <i>The plaine angles comprehending a solid
+  angle, are lesse than foure right angles. 21. p xj</i>.</p>
+
+  <p>For if they should be equall to foure right angles, they would fill up
+  a place by the <a href="#27_e_iiij"><span class="correction" title="text reads `22 e, vj'"
+  ></span>27 e, iiij</a>. neither would they at all make an angle, much
+  lesse therefore would they doe it if they were greater.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/272.png"><img style="width:100%" src="images/272.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p><a name="9_e_xxij"></a> 9 <i>If three plaine angles lesse than foure
+  right angles, do comprehend a solid angle, any two of them are greater
+  <!-- Page 251 --><span class="pagenum"><a
+  name="page251"></a>[251]</span>than the other: And if any two of them be
+  greater than the other, then may comprehend a solid angle, 21. and 23. p
+  xj</i>.</p>
+
+  <p>It is an analogy unto the <a href="#10_e_vj">10 e vj</a>. and the
+  cause is in a readinesse. For if two plaine angles be equall to the
+  remainder, they shall with that third include no space betweene them: But
+  if thou shalt conceit to fit the plaine to the shankes, with the
+  congruity they should of two make one: but much lesse if they be
+  lesser.</p>
+
+  <p>The converse from hence also is manifest.</p>
+
+  <p><i>Euclide</i> doth thus demonstrate it: First if three angles are
+  equall, then by and by two are conceived to be greater than the
+  remainder. But if they be unequall, let the angle <i>aei</i>, be greater
+  than the angle <i>aeo</i>: And let <i>aeu</i>, equall to <i>aeo</i>, be
+  cut off from the greater <i>aei</i>: And let <i>eu</i>, be equall to
+  <i>eo</i>. Now by the <a href="#2_e_vij">2 e, vij</a>. two triangles
+  <i>aeu</i>, and <i>aeo</i>, are equall in their bases <i>au</i>, and
+  <i>ao</i>. Item <i>ao</i>, and <i>ei</i>, are greater than <i>ai</i>, and
+  <i>ao</i>: And <i>ao</i>, is equall to <i>au</i>. Therefore <i>oi</i>, is
+  greater than <i>iu</i>. Here two triangles, <i>uei</i>, and <i>ieo</i>,
+  equall in two shankes; and the base <i>oi</i>, greater than the base
+  <i>iu</i>. Therefore, by the <a href="#5_e_vij">5 e vij</a>. the angle
+  <i>oei</i>, is greater than the angle <i>ieu</i>. Therefore two angles
+  <i>aeo</i>, and <i>oei</i>, are greater than <i>aei</i>.</p>
+
+  <div class="figright" style="width:35%;">
+      <a href="images/273a.png"><img style="width:100%" src="images/273a.png"
+      alt="Pyramides." title="Pyramides." /></a>
+  </div>
+  <p><a name="10_e_xxij"></a> 10 <i>A plaine solid is a Pyramis or a
+  Pyramidate</i>.</p>
+
+  <p><a name="11_e_xxij"></a> 11 <i>A Pyramis is a plaine solid from a
+  rectilineall base equally decreasing</i>.</p>
+
+  <p>As here thou conceivest from the triangular base <i>aei</i>, unto the
+  toppe <i>o</i>, the triangles <i>aoe</i>, <i>aoi</i>, and <i>eoi</i>, to
+  be <!-- Page 252 --><span class="pagenum"><a
+  name="page252"></a>[252]</span>reared up.</p>
+
+  <p>In the pyramis <i>aeiou</i>, thou seest from the quadrangular base
+  <i>aeio</i>, unto the toppe <i>u</i>, foure triangles in like manner to
+  be raised.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="12_e_xxij"></a> 12 <i>The sides of a pyramis are one more
+  than are the base</i>.</p>
+
+  <p>The sides are here named <i>Hedr�</i>.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:56%;">
+      <a href="images/273b.png"><img style="width:100%" src="images/273b.png"
+      alt="Sides of Pyramides." title="Sides of Pyramides." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="13_e_xxij"></a> 13 <i>A pyramis is the first figure of
+  solids</i>.</p>
+
+  <p>For a pyramis in solids, is as a triangle is in plaines. For a pyramis
+  may be resolved into other solid figures, but it cannot be resolved into
+  any one more simple than it selfe, and which consists of fewer sides than
+  it doth.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="14_e_xxij"></a> 14 <i>Pyramides of equall heighth, are as
+  their bases are <!-- Page 253 --><span class="pagenum"><a
+  name="page253"></a>[253]</span>5 e, and 6. p xij</i>.</p>
+
+  <div class="figcenter" style="width:32%;">
+      <a href="images/274a.png"><img style="width:100%" src="images/274a.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="15_e_xxij"></a> 15 <i>Those which are reciprocall in base and
+  heighth are equall 9 p xij</i>.</p>
+
+  <div class="figcenter" style="width:35%;">
+      <a href="images/274b.png"><img style="width:100%" src="images/274b.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>These consectaries are drawne out of the <a href="#16_e_iiij">16</a>,
+  <a href="#18_e_iiij">18 e. iiij</a>.</p>
+
+  <div class="figright" style="width:27%;">
+      <a href="images/275.png"><img style="width:100%" src="images/275.png"
+      alt="tetraedrum." title="tetraedrum." /></a>
+  </div>
+  <p><a name="16_e_xxij"></a> 16 <i>A tetraedrum is an ordinate pyramis
+  comprehended of foure triangles 26. d xj</i>.</p>
+
+  <p>As here thou seest. In rectilineall plaines we have in the former
+  signified, in every kinde there is but one ordinate figure: Amongst the
+  triangles the equilater: Amongst the <!-- Page 254 --><span
+  class="pagenum"><a name="page254"></a>[254]</span>quadrangles, the
+  Quadrate: so now of all kinde of Pyramides, there is one kinde ordinate
+  onely, and that is the Tetraedrum. And yet not every Tetraedrum is such,
+  but that only which is comprehended of triangles, not onely severally
+  ordinate, but equall one to another altogether alike.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="17_e_xxij"></a> 17 <i>The edges of a tetraedrum are sixe, the
+  plaine angles twelve, the solide angles foure</i>.</p>
+
+  <p>For a Tetraedrum is comprehended of foure triangles, each of them
+  having three sides, and three corners a peece: And every side is twise
+  taken: Therefore the number of edges is but halfe so many.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="18_e_xxij"></a> 18 <i>Twelve tetraedra's doe fill up a solid
+  place</i>.</p>
+
+  <p>Because 8. solid right angles <span class="correction" title="text reads `fillling'"
+  >filling</span> a place, and 12. angles of the tetraedrum are equall
+  betweene themselves, seeing that both of them are comprehended of 24
+  plaine right-angles. For a solid right angle is comprehended of three
+  plaine right angles: And therefore 8. are comprehended of 24. In like
+  manner the angle of a Tetraedrum is comprehended of three plaine
+  equilaters, that is of sixe third of one right angle: and therefore of
+  two right angles: Therefore 12 are comprehended of 24.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="19_e_xxij"></a> 19. <i>If foure ordinate and equall triangles
+  be joyned together in solid angles, they shall comprehend a
+  tetraedrum.</i> <!-- Page 255 --><span class="pagenum"><a
+  name="page255"></a>[255]</span></p>
+
+  <div class="figcenter" style="width:75%;">
+      <a href="images/276.png"><img style="width:100%" src="images/276.png"
+      alt="Foure triangles comprehend a tetraedrum." title="Foure triangles comprehend a tetraedrum." /></a>
+  </div>
+  <p>This fabricke or construction is very easie, as you may see in these
+  examples: For if thou shalt joyne or fold together these triangles here
+  thus expressed, thou shalt make a tetraedrum.</p>
+
+  <p><a name="20_e_xxij"></a> 20. <i>If a right line whose power is
+  sesquialter unto the side of an equilater triangle, be cut after a double
+  reason, the double segment perpendicular to the center of the triangle,
+  knit together with the angles thereof shall comprehend a tetraedrum. 13 p
+  xiij</i>.</p>
+
+  <p>For a solid to be comprehended of right lines understand plaines
+  comprehended of right lines, as in other places following.</p>
+
+  <div class="figcenter" style="width:75%;">
+      <a href="images/277.png"><img style="width:100%" src="images/277.png"
+      alt="Figures for demonstration 20." title="Figures for demonstration 20." /></a>
+  </div>
+  <p>As here, Let first <i>ae</i> be the right line whose power is
+  sesquialter unto <i>ai</i> the side of the equilater triangle, as in the
+  forme was manifest at the <a href="#13_e_xij">13 e xij</a>. And let it be
+  by the <a href="#29_e_v">29 e v</a>, be cut in a double reason in
+  <i>o</i>: And let the double segment <i>ao</i>, be perpendicular to the
+  equilater triangle <i>uys</i>, unto the center <i>r</i>, by the <a
+  href="#7_e_xxj">7 e xxj</a>. And let <i>lr</i> be knit with the angles,
+  by <i>lu</i>, <i>ls</i>, <i>ly</i>. I say that the triangles <i>uys</i>,
+  <i>usl</i>, <i>uyl</i>, are equilater and equall, because all the sides
+  are equall. First the three lower ones are equall by the grant: And the
+  three higher ones are equall by the <a href="#9_e_xij">9 e xij</a>. And
+  every one of the higher ones are equall to the under one. For if a Circle
+  bee supposed to bee circumscribed about the triangle, the side <!-- Page
+  256 --><span class="pagenum"><a name="page256"></a>[256]</span>shall be
+  of treble power to the ray <i>ur</i>, by the <a href="#12_e_xviij">12 e
+  xviij</a>. But the higher one also is of treble power to the same ray, as
+  is manifest in the first figure of the ray <i>oi</i>, which is for the
+  ray of the second figure <i>ur</i>. For as <i>ao</i>, is to <i>oi</i>, so
+  by the <a href="#9_e_viij">9 e viij</a>, is <i>oi</i>, unto <i>oe</i>:
+  And by the <a href="#25_e_iiij">25 e iiij</a>, as the first rect line
+  <i>ao</i>, is unto the third <i>oe</i>: so is the quadrate <i>ao</i>,
+  unto the quadrate <i>oi</i>. And by compounding <i>ao</i> with <i>oe</i>;
+  As <i>ae</i> is to <i>oe</i>; so are the quadrates <i>ao</i>; and
+  <i>oi</i>, that is, by the <a href="#9_e_xij">9 e xij</a>, the quadrate
+  <i>ai</i>, unto the quadrate <i>oi</i>, But <i>ae</i> is the triple of
+  <i>oe</i>. Therefore the quadrate <i>ai</i>, is the triple of the
+  quadrate <i>oi</i>. Wherefore the higher side equall to <i>ai</i>, is of
+  treble power to the ray: And therefore also all the sides are equall: And
+  therefore againe the triangles themselves are equall.</p>
+
+<hr class="full" />
+
+<h2>The twenty third Booke of <i>Geometry</i>,
+of a <i>Prisma</i>.</h2>
+
+  <div class="figright" style="width:33%;">
+      <a href="images/278a.png"><img style="width:100%" src="images/278a.png"
+      alt="Prisma's." title="Prisma's." /></a>
+  </div>
+  <p><a name="1_e_xxiij"></a> 1 <i>A Pyramidate is a plaine solid
+  comprehended of pyramides</i>.</p>
+
+  <p><a name="2_e_xxiij"></a> 2. <i>A pyramidate is a Prisma, or a mingled
+  polyedrum</i>.</p>
+
+  <p><a name="3_e_xxiij"></a> 3. <i>A prisma is a pyramidate whose opposite
+  plaines are equall, alike, and parallell, the rest parallelogramme. 13 d
+  xj.</i> <!-- Page 257 --><span class="pagenum"><a
+  name="page257"></a>[257]</span></p>
+
+  <p>As here thou seest. The base of a pyramis was but one: Of a Prisma,
+  they are two, and they opposite one against another, First equall; Then
+  like: Next parallell. The other are parallelogramme.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="4_e_xxiij"></a> 4. <i>The flattes of a prisma are two more
+  than are the angles in the base</i>.</p>
+
+  <p>And indeed as the augmentation of a Pyramis from a quaternary is
+  infinite: so is it of a Prisma from a quinary: As if it be from a
+  triangular, quadrangular, or quinquangular base; you shal have a
+  Pentraedrum, Hexaedrum, Heptaedrum, and so in infinite.</p>
+
+  <p><a name="5_e_xxiij"></a> 5. <i>The plaine of the base and heighth is
+  the solidity of a right prisma</i>.</p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/278b.png"><img style="width:100%" src="images/278b.png"
+      alt="Figure for demonstration 6." title="Figure for demonstration 6." /></a>
+  </div>
+  <p><a name="6_e_xxiij"></a> 6. <i>A prisma is the triple of a pyramis of
+  equall base and heighth. � 7 p. xij</i>.</p>
+
+  <p>As in the example a prisma pentaedrum is cut into three equall
+  pyramides. For the first consisting of the plaines <i>aei</i>,
+  <i>aeo</i>, <i>aoi</i>, <i>eio</i>; is equall to the second consisting of
+  the plaines <i>aoi</i>, <i>aou</i>, <i>aiu</i>, <span class="correction"
+  title="text reads `auy'"><i>iou</i></span>, by the <a
+  href="#10_e_vij"><span class="correction" title="wrong reference">10 e
+  vij</span></a>. Because it is equall to it both in common base and
+  heighth. Therefore the first and second are equall. And the same second
+  is equall to it selfe, seeing the base is <i>iou</i>, and the toppe
+  <i>a</i>. Then also it is equall to the third consisting of the plaines
+  <i>aiu</i>, <i>aiy</i>, <i>uiy</i>, <i>auy</i>. Therefore three are
+  equall. <!-- Page 258 --><span class="pagenum"><a
+  name="page258"></a>[258]</span></p>
+
+  <p>If the base be triangular, the Prisma may be resolved into prisma's of
+  triangular bases, and the theoreme shall be concluded as afore.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="7_e_xxiij"></a> 7. <i>The plaine made of the base and the
+  third part of the heighth is the solidity of a pyramis of equall base and
+  heighth</i>.</p>
+
+  <p>The heighth of a pyramis shall be found, if you shall take the square
+  of the ray of the base out of the quadrate of the side: for the side of
+  the remainder, by the <a href="#9_e_xij">9 e xij</a>, shall be the
+  altitude or heighth, as in the example following.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/279a.png"><img style="width:100%" src="images/279a.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>Here the content of the triangle by the <a href="#18_e_xij">18 e
+  xij</a>, is found to be 62.44/125 for the base of the pyramis. The
+  altitude is 9.15/19: Because by the <a href="#12_e_xviij"><span
+  class="correction" title="text reads `6 e xviij'">12 e xviij</span></a>,
+  the side is of treble power to the ray. But if from 144, the quadrate of
+  12 the side, you take the subtriple <i>i</i>. 48, the remainder 96, by
+  the <a href="#9_e_xij">9 e xij</a>, shall be the square of the heighth.
+  And the side of the quadrate shall be 9.15/19. Now the third part of
+  9.15/19 is 3.5/19. And the plaine of 62.44/125 and 3.5/19, shall be
+  203.1103/2375 for the solidity of the pyramis.</p>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/280a.png"><img style="width:100%" src="images/280a.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <div class="figright" style="width:17%;">
+      <a href="images/279b.png"><img style="width:100%" src="images/279b.png"
+      alt="Figure for demonstration 7." title="Figure for demonstration 7." /></a>
+  </div>
+  <p>So in the example following, Let 36, the quadrate of 6 the ray, be
+  taken out of 292.9/1156 the quadrate of the side 17.3/34 the side <span
+  class="correction" title="This value is wrong, it should be approx. 16.1/4110. It follows that the rest of this section is wrong."
+  >16.3/34</span> of 256.9/1156 the remainder shall be the height, whose
+  third part is 5.37/102; the plaine of which by the base 72.1/4 shall be
+  387.11/24 for the solidity of the pyramis given.</p>
+
+  <p>If the pyramis be unperfit, first measure the whole, and then that
+  part which is wanting: Lastly from the whole <!-- Page 259 --><span
+  class="pagenum"><a name="page259"></a>[259]</span>subtract that which was
+  wanting, and the remaine shall be the solidity of the unperfect pyramis
+  given: As here, let <i>ao</i>, the side of the whole be 16.5/12,
+  <i>eo</i> the side of the particular be 8.1/16. Therefore the
+  perpendicular of the whole <i>ou</i>, shall be <span class="correction"
+  title="Wrong again.He is taking sqrt(nn.pp/qq) to be n.p/q!"
+  >15.5/32</span>: Whose third part is 5.5/96: Of which, and the base
+  93.3/11 the plaine shall be 471.134/1056 for the whole pyramis. But in
+  the lesser pyramis, 9 the square of the ray 3, taken out of 65.1/256 the
+  quadrate of the side 8.1/16 the remaine shall be 56.1/256; whose side is
+  almost 7-1/2 for the heighth. The third part of which is 2-1/2. The base
+  likewise is almost 22. The plaine of which two is 55, for the solidity of
+  the lesser pyramis: And 471 - 55 is 416, for the imperfect pyramis.</p>
+
+  <p>After this manner you may measure an imperfect Prisma.</p>
+
+  <p><a name="8_e_xxiij"></a> 8. <i>Homogeneall Prisma's of equall heighth
+  are one to another as their bases are one to another, 29, 30, 31, 32 p
+  xj</i>.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/280b.png"><img style="width:100%" src="images/280b.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>The reason is, because they consist equally of like number <!-- Page
+  260 --><span class="pagenum"><a name="page260"></a>[260]</span>of
+  pyramides. Now it is required that they be homogeneall or of like kindes;
+  Because a Pentaedrum with an Hexaedrum will not so agree.</p>
+
+  <p>This element is a consectary out of the <a href="#16_e_iiij">16 e
+  iiij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_xxiij"></a> 9. <i>If they be reciprocall in base and
+  heighth, they are equall</i>.</p>
+
+  <div class="figcenter" style="width:34%;">
+      <a href="images/281a.png"><img style="width:100%" src="images/281a.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <p>This is a Consectary out the <a href="#18_e_iiij">18 e iiij</a>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:26%;">
+      <a href="images/281b.png"><img style="width:100%" src="images/281b.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p><a name="10_e_xxiij"></a> 10. <i>If a Prisma be cut by a plaine
+  parallell to his opposite flattes, the segments are as the bases are. 25
+  p. xj</i>.</p>
+
+  <p>The segments are homogeneall because the prismas. Therefore seeing
+  they are of equall heighth (by the heighth I meane of plaine dividing
+  them) they shall be as their bases are: And here the bases are to be
+  taken opposite to the heighth.</p>
+
+  <p><a name="11_e_xxiij"></a> 11. <i>A Prisma is either a Pentaedrum, or
+  Compounded of pentaedra's.</i> <!-- Page 261 --><span class="pagenum"><a
+  name="page261"></a>[261]</span></p>
+
+  <p>Here the resolution sheweth the composition.</p>
+
+  <p><a name="12_e_xxiij"></a> 12 <i>If of two pentaedra's, the one of a
+  triangular base, the other of a parallelogramme base, double unto the
+  triangular, be of equall heighth, they are equall 40. p xj</i>.</p>
+
+  <p>The <span class="correction" title="text reads `canse'">cause</span>
+  is manifest and briefe: Because they be the halfes of the same prisma: As
+  here thou maist perceive in a prisma cut into two halfes by the diagoni's
+  of the opposite sides.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/282.png"><img style="width:100%" src="images/282.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p><i>Euclide</i> doth demonstrate it thus: Let the Pentaedra's
+  <i>aeiou</i>, and <i>ysrlm,</i> be of equall heighth: the first of a
+  triangular base <i>eio</i>: The second of a parallelogramme base
+  <i>sl</i>, double unto the triangular. Now let both of them be double and
+  made up, so that first be <i>aeioun.</i> The second <i>ysrlvf</i>. Now
+  againe, by the grant, the base <i>sl</i>, is the double of the base
+  <i>eio</i>,: whose double is the base <i>eo</i>, by the <a
+  href="#12_e_x">12 e x</a>. Therefore the bases <i>sl</i>, and <i>eo</i>,
+  are equall: And therefore seeing the prisma's, by the grant, here are of
+  equall heighth, as the bases by the conclusion are equall, the prisma's
+  are equall; And therefore also their halfes <i>aeiou</i>, and <i><span
+  class="correction" title="text reads `ysnlr'">ysmlr</span></i>, are
+  equall.</p>
+
+  <p>The measuring of a pentaedrall prisma was even now generally taught:
+  The matter in speciall may be conceived in these two examples
+  following.</p>
+
+  <div class="figright" style="width:34%;">
+      <a href="images/283.png"><img style="width:100%" src="images/283.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>The plaine of 18. the perimeter of the triangular base, <!-- Page 262
+  --><span class="pagenum"><a name="page262"></a>[262]</span>and 12, the
+  heighth is 216. This added to the triangular base, 15.18/31. or 15.3/5,
+  almost twise taken, that is, 31.1/5, doth make 247.1/5, for the summe of
+  the whole surface. But the plaine of the same base 15.3/5, and the
+  heighth 12. is 187.1/5, for the whole solidity.</p>
+
+  <p>So in the pentaedrum, the second prisma, which is called
+  <i>Cuneus</i>, (a wedge) of the sharpnesse, and which also more properly
+  of cutting is called a prisma, the whole surface is 150, and the solidity
+  90.</p>
+
+  <p><a name="13_e_xxiij"></a> 13 <i>A prisma compounded of pentaedra's, is
+  either an Hexaedrum or Polyedrum: And the Hexaedrum is either a
+  Parallelepipedum or a Trapezium</i>.</p>
+
+  <p><a name="14_e_xxiij"></a> 14 <i>A parallelepipedum is that whose
+  opposite plaines are parallelogrammes � 24. p xj</i>.</p>
+
+  <p>Therefore a Parallelepipedum in solids, answereth to a Parallelogramme
+  in plaines. For here the opposite <i>Hedr�</i> or flattes are parallell:
+  There the opposite sides are parallell.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:20%;">
+      <a href="images/284a.png"><img style="width:100%" src="images/284a.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p><a name="15_e_xxiij"></a> 15 <i>It is cut into two halfes with a
+  plaine by the diagonies of the opposite sides. 28 p xj. It answereth to
+  the 34. p j.</i> <!-- Page 263 --><span class="pagenum"><a
+  name="page263"></a>[263]</span></p>
+
+  <p>Let the Prisma be of sixe bases <i>ai</i>, <i>yo</i>, <i>ye</i>,
+  <i>ui</i>, <span class="correction" title="text reads `ri'"
+  ><i>si</i></span>, <i>au</i>. The diagonies doe cut into halfes, by the
+  <a href="#10_e_x">10. e x</a>. the opposite bases: And the other opposite
+  bases or the two prisma's cut, are equall by the <a href="#3_e_xxiij">3
+  e</a>. Wherefore two prisma's are comprehended of bases, equall both in
+  multitude and magnitude: therfore they are equall.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="16_e_xxiij"></a> 16 <i>If it be halfed by two plaines halfing
+  the opposite sides, the common bisection and diagony doe halfe one
+  another 39. p xj</i>.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/284b.png"><img style="width:100%" src="images/284b.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p>Because here the diameters (such as is that bisection) are halfed
+  betweene themselves [or doe halfe one another.] Let the parallelepipedum
+  <i>aeiouy</i>, be cut in to <i>y</i> the halfs by two plains, fro
+  <i>srlm</i>, <i><span class="correction" title="The figure is wrongly labelled but the argument is not affected."
+  >uivf</span></i>, halfing the opposite sides: Here the common section
+  <i>ts</i>, and the diagony <i>ao</i>, doe cut one another.</p>
+
+  <p><a name="17_e_xxiij"></a> 17 <i>If three lines be proportionall, the
+  parallelepipedum of meane shall be equall to the equiangled
+  parallelepipedum of all them. � 36. p xj</i>.</p>
+
+  <p>It is a consectary out of the <a href="#8_e_xxiij">8 e</a>.</p>
+
+  <p><a name="18_e_xxiij"></a> 18 <i>Eight rectangled parallelepiped's doe
+  fill a solid place</i>.</p>
+
+  <p><a name="19_e_xxiij"></a> 19 <i>The Figurate of a rectangled
+  parallelepipedum is called a solid, made of three numbers 17. d
+  vij.</i></p>
+
+  <p>As if thou shalt multiply 1, 2, 3. continually, thou shalt make the
+  solid 6. Item if thou shalt in like manner multiply 2, 3, 4. thou shalt
+  make the solid 24. And the sides of that solid <!-- Page 264 --><span
+  class="pagenum"><a name="page264"></a>[264]</span>6 solid shall be 1, 2,
+  3. Of 24, they shall be 2, 3, 4.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="20_e_xxiij"></a> 20 <i>If two solids be alike, they have
+  their sides proportionalls, and two meane proportionalls 21 d vij, 19.
+  21. p viij.</i></p>
+
+  <p>It is a consectary out of the <a href="#5_e_xxij">5 e xxij</a>. But
+  the meane proportionalls are made of the sides of the like solids, to
+  wit, of the second, third, and fourth: Item of the third, fourth, and
+  fifth, as here thou seest.</p>
+
+<table class="nobctr">
+<tr><td>2, &nbsp; &nbsp;</td><td>3, &nbsp; &nbsp;</td><td>5, &nbsp; &nbsp;</td><td>4, &nbsp; &nbsp;</td><td>6, &nbsp; &nbsp;</td><td>10.</td></tr>
+<tr><td colspan="2">30,</td><td>60,</td><td colspan="2" align="center">120,</td><td>240.</td></tr>
+</table>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i> the twentie fourth Book.
+Of a Cube.</h2>
+
+  <p><a name="1_e_xxiiij"></a> 1 <i>A Rightangled parallelepipedum is
+  either a Cube, or an Oblong</i>.</p>
+
+  <p><a name="2_e_xxiiij"></a> 2 <i>A Cube is a right angled
+  parallelepipedum of equall flattes, 25. d. xj</i>.</p>
+
+  <div class="figcenter" style="width:48%;">
+      <a href="images/285.png"><img style="width:100%" src="images/285.png"
+      alt="Cubes." title="Cubes." /></a>
+  </div>
+  <p>As here thou seest in these two figures.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="3_e_xxiiij"></a> 3 <i>The sides of a cube are 12. the plaine
+  angles 24. the solid 8.</i> <!-- Page 265 --><span class="pagenum"><a
+  name="page265"></a>[265]</span></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="4_e_xxiiij"></a> 4 <i>If sixe equall quadrates be joyned with
+  solid angles, they shall comprehend a cube</i>.</p>
+
+  <div class="figcenter" style="width:53%;">
+      <a href="images/286.png"><img style="width:100%" src="images/286.png"
+      alt="Nets of a cube." title="Nets of a cube." /></a>
+  </div>
+  <p>As here in these two examples.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="5_e_xxiiij"></a> 5 <i>If from the angles of a quadrate,
+  perpendiculars equall to the sides be tied together aloft, they shall
+  comprehend a Cube. � 15 p xj</i>.</p>
+
+  <p>It is a consectary following upon the former consectary: For then
+  shall sixe equall quadrates be knit together:</p>
+
+  <p><a name="6_e_xxiiij"></a> 6 <i>The diagony of a Cube is of treble
+  power unto the side</i>.</p>
+
+  <p>For the Diagony of a quadrate is of double power to the side, by the
+  <a href="#12_e_xij">12 e, xij</a>. And the Diagony of a Cube is of as
+  much power as the side the diagony of the quadrate, by the same <i>e</i>.
+  Therefore it is of treble power to the side.</p>
+
+  <p><a name="7_e_xxiiij"></a> 7 <i>If of foure right lines continually,
+  proportionally the first be the halfe of the fourth, the cube of the
+  first shall be the halfe of the Cube of the second � 33 p xj</i>.</p>
+
+  <p>It is a consectary out of the <a href="#25_e_iiij">25 e, iiij</a>.
+  From hence <i>Hippocrates</i> first found how to answer <i>Apollo's</i>
+  Probleme.</p>
+
+  <p><a name="8_e_xxiiij"></a> 8 <i>The solid plaine of a cube is called a
+  Cube, to wit, a solid of equall sides. 19, d vij</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="9_e_xxiiij"></a> 9 <i>It is made of a number multiplied into
+  his owne quadrate.</i> <!-- Page 266 --><span class="pagenum"><a
+  name="page266"></a>[266]</span></p>
+
+  <p>So is a Cube made by multiplying a number by it selfe, and the product
+  againe by the first. Such are these nine first cubes made of the nine
+  first Arithmeticall figures.</p>
+
+<table class="nobctr">
+<tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>Latera.</td></tr>
+<tr><td>1</td><td>4</td><td>9</td><td>16</td><td>25</td><td>36</td><td>49</td><td>64</td><td>81</td><td>Quadrates.</td></tr>
+<tr><td>1 &nbsp;</td><td>8 &nbsp;</td><td>27 &nbsp;</td><td>64 &nbsp;</td><td>125 &nbsp;</td><td>216 &nbsp;</td><td>343 &nbsp;</td><td>512 &nbsp;</td><td>729 &nbsp;</td><td>Cubes.</td></tr>
+</table>
+
+  <p>This is the generall invention of a Cube, both Geometricall and
+  Arithmeticall.</p>
+
+  <p><a name="10_e_xxiiij"></a> 10 <i>If a right line be cut into two
+  segments, the Cube of the whole shall be equall to the Cubes of the
+  segments, and a double solid thrice comprehended of the quadrate of his
+  owne segment and the other segment</i>.</p>
+
+  <p>As for example, the side 12, let it be cut into two segments 10 and 2.
+  The cube of 12. the whole, which is 1728, shall be equall to two cubes
+  1000, and 8 made of the segments 10. and 2. And a double solid; of which
+  the first 600. is thrise comprehended of 100. the quadrate of his segment
+  10. and of 2. the other segment: The second 120. is thrice comprehended
+  of 4, the quadrate of his owne segment, and of 10. the other segment. Now
+  1000 + 600 + 120. + 8, is equall to 1728: And therefore a right.
+  &amp;c.</p>
+
+  <p>But the genesis of the whole cube will make all this whole matter more
+  apparant, to wit, how the extreme and meane solids are made. Let
+  therefore a cube be made of three equall sides, 12, 12, and 12: And first
+  of all let the second side be multiplied by the first, after this manner:
+  And not adding the severall figures of the same degree, as was taught in
+  multiplication, but multiply againe every one of them by the other side;
+  and lastly, add the figures of the same degrees severally, thus: <!--
+  Page 267 --><span class="pagenum"><a name="page267"></a>[267]</span></p>
+
+<table class="nobctr"><tr><td align="right">
+12<br />
+12<br />
+&mdash;&mdash;<br />
+24<br />
+12 &nbsp;<br />
+12<br />
+&mdash;&mdash;<br />
+48<br />
+24 &nbsp;<br />
+24 &nbsp;<br />
+12 &nbsp; &nbsp;<br />
+&mdash;&mdash;<br />
+1,6,12,8
+</td><td align="center">
+&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Or thus, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
+</td><td align="right">
+12<br />
+12<br />
+&mdash;&mdash;<br />
+4<br />
+20<br />
+20<br />
+100<br />
+&mdash;&mdash;<br />
+12<br />
+&mdash;&mdash;<br />
+8<br />
+40<br />
+40<br />
+40<br />
+200<br />
+200<br />
+200<br />
+1000<br />
+&mdash;&mdash;<br />
+1728
+</td></tr></table>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="11_e_xxiiij"></a> 11. <i>The side of the first severall cube
+  is the other side of the second solide: And the quadrate of the same side
+  is the other side of the first solide, whose other side is the side of
+  the second cube; and the quadrate of the same other side is the other
+  side of the second solid</i>.</p>
+
+  <p>In that equation therefore of foure solids with one solid, thou shalt
+  consider a peculiar making and composition: First that the last cube be
+  made of the last segment 2: Then that the second solid of 4, the quadrate
+  of his owne segment, and of 10, the other segment be thrise comprehended:
+  Lastly that the first solid of 100, the square of his owne segment 10 and
+  the other segment 2, be also thrice comprehended: Lastly, that the Cube
+  1000, be made of the greater segment 10. Out of this making &amp;c.</p>
+
+  <p>And thus much of the Cube: Of other sorts of parallelepipedes, as of
+  the Oblong, the Rhombe, the Rhomboides, and of the Trapezium, and many
+  flatted pentaedra's there is no <!-- Page 268 --><span class="pagenum"><a
+  name="page268"></a>[268]</span>peculiar stereometry. The measuring of a
+  Prisma hath in the former beene generally declared, and is now onely
+  farther be made more plaine by speciall examples; as here:</p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/289a.png"><img style="width:100%" src="images/289a.png"
+      alt="Figure for Demonstration 11." title="Figure for Demonstration 11." /></a>
+  </div>
+  <p>The plaine of the perimeter of the base 20, and the altitude 5 is 100.
+  This added to 25 and 25, both the bases that is to 50, maketh 150, for
+  the whole surface. Now the plaine of 25 the base, and the heighth 5 is
+  125, for the whole solidity.</p>
+
+  <p>So in the Oblong, the plaine of the base's perimeter 20, and the
+  heighth 11, is 220, which added to the bases 24 and 24, that is 48,
+  maketh 268, for the whole surface. But the plaine of the base 24, and the
+  height 11, is 264, for the solidity.</p>
+
+  <div class="figcenter" style="width:35%;">
+      <a href="images/289b.png"><img style="width:100%" src="images/289b.png"
+      alt="Figure for Demonstration 11." title="Figure for Demonstration 11." /></a>
+  </div>
+  <p>The same also Geodesie or manner of measuring is used in the measuring
+  of rectangled walls or gates and doores, which have either any window, or
+  any hollow <!-- Page 269 --><span class="pagenum"><a
+  name="page269"></a>[269]</span>or voyde space cut out of them, if those
+  voyde places be taken out of them; as here thou seest in the next
+  following example. The thickenesse is 3 foote; the breadth 12, the
+  heighth 11. Therefore the whole solidity is 396. Now the Gate way is of
+  thickenesse 3 foote, of breadth 4: of heighth 6. And therefore the whole
+  solidity of the Gate is 72 foote. But 396 - 72 are 314. Therefore the
+  solidity of the rest of the wall remaining is 324.</p>
+
+  <p>In the second example, the length is 10. The breadth 8, the heighth 7.
+  Therefore the whole body if it were found, were 560 foote. But there is
+  an hollow in it, whose length is 6, breadth 5, heighth 7. Therefore the
+  cavity or hollow place is 168. Now 560 - 168 is 392, for the solidity of
+  the rest of the sound body.</p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/290.png"><img style="width:100%" src="images/290.png"
+      alt="Figure for Demonstration 11 second example." title="Figure for Demonstration 11 second example." /></a>
+  </div>
+  <p>Thus are such kinde of walls whether of mudde, bricke, or stone, of
+  most large houses to bee measured. The same manner of Geodesy is also to
+  be used in the measuring of a Rhombe, Rhomboides, Trapezium or mensall,
+  and any kinde of multangled body. The base is first to be measured, as in
+  the former: Then out of that and the heighth the solidity shall be
+  manifested: As in the Rhombe the base is 24, the heighth 4. Therefore the
+  solidity is 96.</p>
+
+  <p>In the Rhomboides, the base is 64.35/129: The heigh <span
+  class="correction" title="text reads `11', result requires `16'"
+  >16</span>. Therefore the solidity is 1028.44/129.</p>
+
+  <p>The same is the geodesy of a trapezium, as in these examples: The
+  surface of the first is 198: The solidity 192.1/2.</p>
+
+  <p>The surface of the second is 158.3/49: The solidity is 91.29/49. <!--
+  Page 270 --><span class="pagenum"><a name="page270"></a>[270]</span></p>
+
+  <div class="figcenter" style="width:46%;">
+      <a href="images/291a.png"><img style="width:100%" src="images/291a.png"
+      alt="Figure for Demonstration 11 - a many flatted Prisma." title="Figure for Demonstration 11 - a many flatted Prisma." /></a>
+  </div>
+  <div class="figright" style="width:22%;">
+      <a href="images/291b.png"><img style="width:100%" src="images/291b.png"
+      alt="Figure for Demonstration 11." title="Figure for Demonstration 11." /></a>
+  </div>
+  <p>The same shall be also the geodesy of a many flatted Prisma: As here
+  thou seest in an Octoedrum of a sexangular base: The surface shall bee
+  762.6/11: The solidity 1492.4/11.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/291c.png"><img style="width:100%" src="images/291c.png"
+      alt="Figure for Demonstration 11." title="Figure for Demonstration 11." /></a>
+  </div>
+  <p>And from hence also may the capacity or content of vessels or
+  measures, made after any manner of plaine solid bee esteemed and judged
+  of as here thou seest. For here the plaine of the sexangular base is
+  41.1/7; (For the ray, by the <a href="#9_e_xviij">9 e xviij</a>, is the
+  side:) and the heighth 5, shall be 205.5/7. Therefore if a cubicall foote
+  doe conteine 4 quarters, as we commonly call them, then shall the vessell
+  conteine 822.6/7 quartes, that is almost 823 quartes.</p>
+
+  <p><br style="clear :both" /></p>
+<hr class="full" />
+
+<p><!-- Page 271 --><span class="pagenum"><a name="page271"></a>[271]</span></p>
+
+<h2><i>Of Geometry</i> the twenty fifth Booke;
+Of mingled ordinate <i>Polyedra's</i>.</h2>
+
+  <p><a name="1_e_xxv"></a> 1. <i>A mingled ordinate polyedrum is a
+  pyramidate, compounded of pyramides with their toppes meeting in the
+  center, and their bases onely outwardly appearing</i>.</p>
+
+  <p>Seeing therefore a Mingled ordinate pyramidate is thus made or
+  compounded of pyramides the geodesy of it shall be had from the Geodesy
+  of the pyramides compounding it: And one Base multiplyed by the number of
+  all the bases shall make the surface of the body. And one Pyramis by the
+  number of all the pyramides; shall make the solidity.</p>
+
+  <p><a name="2_e_xxv"></a> 2 <i>The heighth of the compounding pyramis is
+  found by the ray of the circle circumscribed about the base, and by the
+  semidiagony of the polyedrum</i>.</p>
+
+  <p>The base of the pyramis appeareth to the eye: The heighth lieth hidde
+  within, but it is discovered by a right angle triangle, whose base is the
+  semidiagony or halfe diagony, the shankes the ray of the circle, and the
+  perpendicular of the heighth. Therefore subtracting the quadrate of the
+  ray, from the quadrate of the halfe diagony the side of the remainder, by
+  the <a href="#9_e_xij">9 e xij</a>. shall be the heighth. But the ray of
+  the circle shall have a speciall invention, according to the kindes of
+  the base, first of a triangular, and then next of a quinquangular.</p>
+
+  <p><a name="3_e_xxv"></a> 3 <i>A mingled ordinate polyedrum hath either a
+  triangular, or a quinquangular base.</i> <!-- Page 272 --><span
+  class="pagenum"><a name="page272"></a>[272]</span></p>
+
+  <p>The division of a Polyhedron ariseth from the bases upon which it
+  standeth.</p>
+
+  <p><a name="4_e_xxv"></a> 4 <i>If a quadrate of a triangular base be
+  divided into three parts, the side of the third part shall be the ray of
+  the circle circumscribed about the base</i>.</p>
+
+  <p>As is manifest by the <a href="#12_e_xviij">12 e. xviij</a>. And this
+  is the invention or way to finde out the circular ray for an octoedrum,
+  and an icosoedrum.</p>
+
+  <p><a name="5_e_xxv"></a> 5 <i>A mingled ordinate polyedrum of a
+  triangular base, is either an Octoedrum, or an Icosoedrum</i>.</p>
+
+  <p>This division also ariseth from the bases of the figures.</p>
+
+  <p><a name="6_e_xxv"></a> 6 <i>An octoedrum is a mingled ordinate
+  polyedrum, which is comprehended of eight triangles. 27 d xj</i>.</p>
+
+  <div class="figcenter" style="width:52%;">
+      <a href="images/293.png"><img style="width:100%" src="images/293.png"
+      alt="Lines and solid octahedrum." title="Lines and solid octahedrum." /></a>
+  </div>
+  <p>As here thou seest, in this Monogrammum and solidum, that is lines and
+  solid octahedrum.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="7_e_xxv"></a> 7 <i>The sides of an octoedrum are 12. the
+  plaine angles 24, and the solid 6</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="8_e_xxv"></a> 8 <i>Nine octoedra's doe fill a solid
+  place</i>.</p>
+
+  <p>For foure angles of a Tetraedrum are equall to three angles of the
+  Octoedrum: And therefore 12. are equall to <!-- Page 273 --><span
+  class="pagenum"><a name="page273"></a>[273]</span>nine. Therefore nine
+  angles of an octaedrum doe countervaile eight solid right angles.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_xxv"></a> 9 <i>If eight triangles, equilaters and equall
+  be joyned together by their edges; they shall comprehend an
+  octaedreum</i>.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/294a.png"><img style="width:100%" src="images/294a.png"
+      alt="Net of octahedrum." title="Net of octahedrum." /></a>
+  </div>
+  <p>This construction is easie, as it is manifest in the example
+  following: Where thou seest as it were two equilater and equall triangles
+  of a double pentaedrum to cut one another.</p>
+
+  <p><a name="10_e_xxv"></a> 10 <i>If a right line of each side
+  perpendicular to the center of a quadrate and equall to the halfe diagony
+  be tied together with the angles, it shall comprehend an octaedrum, 14. d
+  xiij</i>.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/294b.png"><img style="width:100%" src="images/294b.png"
+      alt="Figure for demonstration 10." title="Figure for demonstration 10." /></a>
+  </div>
+  <p>For the perpendicular <i>yu</i>, and <i>su</i>, with the
+  semidiagoni's, <i>ua</i>, <i>uo</i>, <i>ui</i>, <i>ue</i>, shall be made
+  equall by the <a href="#2_e_vij">2 e vij</a>, the eight sides <i>ya</i>,
+  <i>ye</i>, <i>yo</i>, <i>yi</i>, <i>se</i>, <i>si</i>, <i>sa</i>,
+  <i>so</i>; And also eight triangles.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="11_e_xxv"></a> 11 <i>The Diagony of an octaedrum is of double
+  power to the side</i>.</p>
+
+  <p>As is manifest by the <a href="#9_e_xij">9 e xij</a>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+  <p><a name="12_e_xxv"></a> 12 <i>If the quadrate of the side of an
+  octaedrum, be <!-- Page 274 --><span class="pagenum"><a
+  name="page274"></a>[274]</span>doubled, the side of the double shall be
+  the diagony</i>.</p>
+
+  <p>As in the figure following, the side is 6. The quadrate is 36. the
+  double is 72. whose side 8.8/17, is the diagony.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/295a.png"><img style="width:100%" src="images/295a.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>And from hence doth arise the geodesy of the octaedrum. For the
+  semidiagony is 4.4/17. whose quadrate is 17.171/289. And the quadrate of
+  6, the side of the equilater triangle, being of treble power to the ray,
+  by the <a href="#12_e_xviij"><span class="correction" title="text reads `6 e, xviiij'"
+  >12 e, xviij</span></a>. is 36. And the side of 12. the third part 3.3/7
+  is the ray of the circle. Wherefore 8.8/17. that is 5.21/289. is the
+  quadrate of the perpendicular, whose side 2.1/5 is the height of the same
+  perpendicular: whose third part againe 11/25. <span class="correction"
+  title="text reads `mulliplied'">multiplied</span> by 15.18/31. the
+  triangular base doe make 11.66/155 for one of the eight pyramides:
+  Therefore the same 11.66/155 multiplied by eight, shall make 91.63/155
+  for the whole octoedrum.</p>
+
+  <p><br style="clear : both" /></p>
+  <p><a name="13_e_xxv"></a> 13 <i>An Icosaedrum is an ordinate polyedrum
+  comprehended of 20 triangles 29 d xj</i>.</p>
+
+  <div class="figcenter" style="width:54%;">
+      <a href="images/295b.png"><img style="width:100%" src="images/295b.png"
+      alt="Line and Solid Icosaedrum." title="Line and Solid Icosaedrum." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="14_e_xxv"></a> 14 <i>The sides of an Icosaedrum are 30.
+  plaine angles 60. the solid 12.</i> <!-- Page 275 --><span
+  class="pagenum"><a name="page275"></a>[275]</span></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="15_e_xxv"></a> 15 <i>If twentie ordinate and equall triangles
+  be joyned with solid angles, they shall comprehend an Icosaedrum</i>.</p>
+
+  <div class="figcenter" style="width:28%;">
+      <a href="images/296a.png"><img style="width:100%" src="images/296a.png"
+      alt="Net of Icosaedrum." title="Net of Icosaedrum." /></a>
+  </div>
+  <p>This fabricke is ready end easie, as is to be seene in this example
+  following.</p>
+
+  <p><a name="16_e_xxv"></a> 16. <i>If ordinate figures, to wit, a double
+  quinquangle, and one decangle be so inscribed into the same circle, that
+  the side of both the quinquangle doe subtend two sides of the decangle,
+  sixe right lines perpendicular to the circle and equall to his ray, five
+  from the angles of one of the quinquangles, knit together both betweene
+  themselves, and with the angles of the other quinquangle; the sixth from
+  the center on each side continued with the side of the decangle, and knit
+  therewith the five perpendiculars, here with the angles of the second
+  quinquangle, they shall comprehend an icosaedrum. � 15 p xiij</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/296b.png"><img style="width:100%" src="images/296b.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+  <p>For there shall be made 20 triangles, both equilaters and equall. Let
+  there be therefore two ordinate quinquangles, the first <i>aeiou</i>; The
+  second <i>ysrlm</i>; each of whose sides let them subtend two sides of a
+  decangle; to wit, <i>utym</i>, let it subtend <i>ya</i>, and <i>am</i>.
+  Then let there be five perpendiculars from the angles of the second
+  quinquangle <i>yj</i>, <i>sy</i>, <i>rv</i>, <!-- Page 276 --><span
+  class="pagenum"><a name="page276"></a>[276]</span><i>lf</i>, <i>mt</i>.
+  And let them be knit first one with another, by the lines <i>nj</i>,
+  <i>jv</i>, <i>vf</i>, <i>ft</i>, <i>tn</i>. Secondarily, with the angles
+  of the first quinquangle, by the lines <i>ne</i>, <i>ej</i>, <i>ji</i>,
+  <i>iv</i>, <i>of</i>, <i>fu</i>, <i>ut</i>, <i>ta</i>, <i>an</i>. The
+  sixth perpendicular from the center <i>d</i>, let it be <i>bg</i>, the
+  ray <i>dc</i>, continued at each end with the side of the decangle,
+  <i>cg</i>, and <i>db</i>, tied together about with the perpendiculars, as
+  by the lines <i>ng</i>, <i>tg</i>: Beneath with the angles of the first
+  quinquangle, as by the lines <i>be</i>, <i>bi</i>, and in other places in
+  like manner, and let all the plaines be made up. This say I, is an
+  Icosaedrum; And is comprehended of 20. triangles, both equilaters and
+  equall. First, the tenne middle triangles, leaving out the
+  perpendiculars, that they are equilaters and equall, one shall
+  demonstrate, as <i>nat</i>. For <i>mt</i> and <i>yu</i>, because they are
+  perpendiculars, they are also, by the <a href="#6_e_xxj">6 e xxj</a>.
+  parallells: And by the grant, equall. Therefore by the <a
+  href="#27_e_v">27 e, v</a>, <i>nt</i>, is equall to <i>ym</i>, the side
+  of the quinquangle. Item <i>na</i>, by the <a href="#6_e_xij">6 e
+  xij</a>. is of as great power, as both the shankes <i>ny</i>, and
+  <i>ya</i>, that is, by the construction, as the sides of the sexangle and
+  decangle: And, by the converse of the <a href="#15_e_xviij">15. e
+  xviij</a>. it is the side of the quinquangle. The same shall fall out of
+  <i>ot</i>. Wherefore <i>nat</i>, is an equilater triangle. The same shall
+  fall out of the other nine middle triangles, <i>nae</i>, <i>nej</i>,
+  <i>eji</i>, <i>jiv</i>, <i>ivo</i>, <i>vof</i>, <i>fou</i>, <i>fut</i>,
+  <i>uta</i>, <i>tan</i>.</p>
+
+  <p>In like manner also shall it be proved of the five upper triangles, by
+  drawing the right lines <i>dy</i> and <i>cn</i> which as afore (because
+  they knit together equall parallells, to wit, <i>dc</i>, and <i>yn</i>)
+  they shall be equall. But <i>dy</i>, is the side of a sexangle: Therefore
+  <i>cn</i>, shall be also the side of a sexangle: And <i>cg</i>, is the
+  side of a decangle: Therefore <i>an</i>, whose power is equall to both
+  theirs by the <a href="#9_e_xij">9 e xij</a>. shall by the converse of
+  the <a href="#15_e_xviij">15 e xviij</a>, be the side of a quinquangle:
+  And in like manner <i>gt</i>, shall be concluded to be the side of a
+  quinquangle. Wherefore <i>ngt</i>, is an equilater: And the foure other
+  shall likewise be equilaters.</p>
+
+  <p>The other five triangles beneath shall after the like manner be
+  concluded to be equilaters. Therefore one shall be for all, to wit,
+  <i>ibe</i>, by drawing the raies <i>di</i>, and <i>de</i>. For <i>ib</i>,
+  <!-- Page 277 --><span class="pagenum"><a
+  name="page277"></a>[277]</span>whose power, as afore, is as much as the
+  sides of the sexangle, and decangle, shall be the side of the
+  quinquangle: And in like sort <i>be</i>, being of equall power with
+  <i>de</i>, and <i>do</i>, the sides of the sexangle and decangle, shall
+  be the side of the quinquangle. Wherefore the triangle <i>ebi</i>, is an
+  equilater: And the foure other in like manner may be shewed to be
+  equilaters. Therefore all the side of the twenty triangles, seeing they
+  are equall, they shall be equilater triangles: And by the <a
+  href="#8_e_vij">8 e, vij</a>. equall.</p>
+
+  <p><a name="17_e_xxv"></a> 17 <i>The diagony of an icosaedr&#x16B; is
+  irrational unto the side</i>.</p>
+
+  <p>This is the fourth example of irrationality, or incommensurability.
+  The first was of the Diagony and side of a square or quadrate. The second
+  was of the segments of a line proportionally cut. The third of the
+  Diameter of a circle and the side of a quinquangle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="18_e_xxv"></a> 18 <i>The power of the diagony of an
+  icosaedrum is five times as much as the ray of the circle</i>.</p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/298.png"><img style="width:100%" src="images/298.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p>For by the <a href="#13_e_xviij">13 e, xviij</a>. the line continually
+  made of the side of the sexangle and decangle is cut proportionally, and
+  the greater segment is the side of the sexangle: As here. Let the
+  perpendicular <i>ae</i>, be cut into two equall parts in <i>i</i>. Then
+  <i>eo</i>, that is the lesser segment continued with the halfe of the
+  greater, that is, with <i>ie</i>. it shall by the <a href="#6_e_xiiij">6
+  e xiiij</a>, be of power five times so great as is the power of the same
+  halfe. Therefore seeing that <i>io</i>, the halfe of the diagony is of
+  power fivefold to the halfe: the whole diagony shall be of power fivefold
+  to the whole cut.</p>
+
+  <div class="figleft" style="width:23%;">
+      <a href="images/299.png"><img style="width:100%" src="images/299.png"
+      alt="Figure for demonstration 18." title="Figure for demonstration 18." /></a>
+  </div>
+  <p>And from hence also shall be the geodesy of the Icosaedrum. For the
+  finding out of the heighth of the pyramis, there is the semidiagony of
+  the side of the decangle and the halfe ray of the circle: But the side of
+  the decangle is a right line subtending the halfe periphery of the side
+  of the quinquangle, or else the greater segment of the ray <!-- Page 278
+  --><span class="pagenum"><a name="page278"></a>[278]</span>proportionally
+  cut. For so it may be taken Geometrically, and reckoned for his measure.
+  Therefore if the quadrate of the side of the decangle, be taken out of
+  the quadrate of the side of the quinquangle, there shall by the <a
+  href="#15_e_xviij">15 e xviij</a>, remaine the quadrate of the sexangle,
+  that is of the ray. The side of the decangle (because the side of the
+  quinquangle here is 6) shall be 3.3/35 to wit a right line subtending the
+  halfe periphery. Now the halfe ray shall thus be had. The quadrates of
+  the quinquangle and decangle are 36, and 9.639/1225. And this being
+  subducted fro that, the remaine 26.386/1225 by the <a
+  href="#15_e_xviij">15 e xviij</a>, shall be the quadrate or square of the
+  sexangle: And the side of it, 5, and almost 5/7 shall be the ray: The
+  halfe ray therefore shall be 2.6/7. To the side of the decangle 3.3/35
+  adde 2.6/7: the whole shal be 5.33/35 for the semi-diagony of the
+  Icosaedrum. The ray of the circle circumscribed about the triangle, is by
+  the <a href="#12_e_xviij">12 e xviij</a>, the same which was before 3.3/7
+  to wit of the quadrate 12. Therefore if the quadrate of the circular ray,
+  be taken out of the quadrate of the halfe diagony, there shall remaine
+  the quadrate of the heighth and perpendicular: the quadrate of the
+  halfe-diagony is 35.389/1225: the quadrate of the circular ray is 12.
+  This taken out of that beneath 23.639/1225: whose side is almost 5, for
+  the perpendicular and heighth proposed: From whence now the Pyramis is
+  esteemed. The case of a triangular pyramis is 15.18/31. The Plaine of
+  this base and the third part of the heighth is 25.30/31 for the solidity
+  of one Pyramis. This multiplyed by 20 maketh 519.11/31 for the summe or
+  whole solidity of the Icosaedum. And this is the geodesy or manner of
+  measuring of an Icosaedrum.</p>
+
+  <p><a name="19_e_xxv"></a> 19. <i>A mingled ordinate polyedrum of a <!--
+  Page 279 --><span class="pagenum"><a
+  name="page279"></a>[279]</span>quinquangular base is that which is
+  comprehended of 12 quinquangles, and it is called a Dodecaedrum.</i></p>
+
+  <div class="figcenter" style="width:49%;">
+      <a href="images/300a.png"><img style="width:100%" src="images/300a.png"
+      alt="Line and Solid Dodecaedrum." title="Line and Solid Dodecaedrum." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="20_e_xxv"></a> 20. <i>The sides of a Dodecaedrum are 30, the
+  plaine angles 60. the solid 20</i>.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="21_e_xxv"></a> 21. <i>If 12 ordinate equall quinquangles be
+  joyned with solid angles, they shall comprehend a Dodecaedrum</i>.</p>
+
+  <div class="figcenter" style="width:27%;">
+      <a href="images/300b.png"><img style="width:100%" src="images/300b.png"
+      alt="Net of Dodecaedrum." title="Net of Dodecaedrum." /></a>
+  </div>
+  <p>As here thou seest.</p>
+
+  <p><a name="22_e_xxv"></a> 22. <i>If the sides of a cube be with right
+  lines cut into two equall parts, and three bisegments of the bisecants in
+  the abbuting plaines, neither meeting one the other, nor parallell one
+  unto another, two of one, the third of that next unto the remainder, be
+  so proportionally cut that the lesser segments doe bound the bisecant:
+  three lines without the cube perpendicular unto the sayd <!-- Page 280
+  --><span class="pagenum"><a name="page280"></a>[280]</span>plaines from
+  the points of the proportionall sections, equall to the greater segment
+  knit together, two of the same bisecant, betweene themselves and with the
+  next angles of cube; the third with the same angles, they shall
+  comprehend a dodecaedrum. 17 p xiij</i>.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/301.png"><img style="width:100%" src="images/301.png"
+      alt="Figure for demonstration 22." title="Figure for demonstration 22." /></a>
+  </div>
+  <p>Let there be two plaines for a cube for all, that one quinquangle for
+  twelve may be described, and they abutting one upon another, <i>aeio</i>,
+  and <i>euyi</i>, having their sides halfed by the bisecantes, <i>sr</i>,
+  <i>lm</i>, <i>rn</i>, <i>jv</i>: And the three bisegments or portions of
+  the bisegments <i>lm</i>, and <i>rn</i>, neither concurring or meeting,
+  nor parallell one to another; two of the said <i>lm</i>, to wit,
+  <i>fl</i>, and <i>fm</i>: The third next unto the remainder, that is
+  <i>lr</i>. And let each bisegment be cut proportionally in the points
+  <i>d</i>, <i>c</i>, <i>g</i>; so that the lesser segments doe bound the
+  bisecant, to wit, <i>dl</i>, <i>cm</i>, and <i>gr</i>. Lastly let there
+  be three perpendiculars from the points <i>db</i>, <i>cg</i>, to the said
+  <i>d</i>, <i>cp</i>, <i>gz</i>: And the two first knit one to another, by
+  <i>bp</i>: And againe with the angles of the cube, by <i>be</i>, and
+  <i>pi</i>: The third knit with the same angles, by <i>ze</i>, and
+  <i>zi</i>: And let all the plaines be made up. I say first, that the five
+  sides <i>bp</i>, <i>pi</i>, <i>iz</i>, <i>ze</i>, and <i>eb</i> are
+  equall; Because, every one of them severally are the doubles of the same
+  greater segment. For in drawing the right lines <i>de</i> and <i>eg</i>,
+  <i>ig</i>, it shall be plaine of two of them; And after the same manner
+  of the rest. First therefore <i>cd</i>, and <i>bp</i>, are equall by the
+  <a href="#6_e_xxj">6 e xxj</a>, and by the <a href="#27_e_v">27 e v</a>.
+  Therefore <i>bp</i>, is the double of the greater segment. Then the whole
+  <i>fl</i>, cut proportionally, and the lesser segment <i>dl</i>, they are
+  by the <a href="#7_e_xiiij">7 e xiiij</a>, of treble power to the greater
+  <i>fd</i>, that is, by the fabricke <i>db</i>. Therefore <i>le</i> wich
+  is equall to <i>lf</i>, the line cut, and <i>ld</i>, are of treble power
+  to the same <i>db</i>: But by the <a href="#9_e_xij">9 e xij</a>,
+  <i>de</i> is of as much power as <i>le</i>, and <i>ld</i> too. <!-- Page
+  281 --><span class="pagenum"><a name="page281"></a>[281]</span>Therefore
+  <i>de</i> is of treble power to <i>db</i>. Therefore both <i>ed</i>, and
+  <i>db</i>, are of quadruple power to <i>db</i>. But <i>be</i>, by the <a
+  href="#9_e_xij">9 e xij</a>, is of as much power as <i>ed</i>, and
+  <i>db</i>. And therefore <i>be</i>, is of quadruple value to <span
+  class="correction" title="text reads `eb'"><i>db</i></span>: And by the
+  <a href="#14_e_xij">14 e xij</a>, it is the double of the said <span
+  class="correction" title="text reads `eb'"><i>db</i></span>. Therefore
+  the two sides <i>eb</i>, and <i>bp</i>, are equall: And by the same
+  argument <i>pi</i>, <i>iz</i>, and <i>ze</i>, are equall. Therefore the
+  quinquangle is equilater.</p>
+
+  <p>I say also that it is a Plaine quinquangle: For it may be said to be
+  an oblique quinquangle; and to be seated in two plaines. Let therefore
+  <i>fh</i> be parallell to <i>db</i>, and <i>cp</i>: and be equall unto
+  them. And let <i>hz</i>, be drawne: This <i>hz</i> shall be cut one line,
+  by the <a href="#14_e_vij">14 e vij</a>. For as the whole <i>tr</i>, that
+  is <i>rf</i>, is unto the greater segment that is to <i>fh</i>: so
+  <i>fh</i>, that is <i>zg</i>, is unto <i>gr</i>. And two paire of shankes
+  <i>fh</i>, <i>gr</i>, <i>fc</i>, <i>gz</i>, by the <a href="#6_e_xxj">6 e
+  xxj</a>, are alternely or crosse-wise parallell. Therefore their bases
+  are continuall.</p>
+
+  <p>Hitherto it hath beene prooved that the quinquangle made is an
+  equilater and plaine: It remaineth that it bee prooved to be Equiangled.
+  Let therefore the right lines <i>ep</i>, and <i>ec</i>, be drawne: I say
+  that the angles, <i>pbe</i>, and <i>ezi</i>, are equall: Because they
+  have by the construction, the bases of equall shankes equall, being to
+  wit in value the quadruple of <i>le</i>. For the right line <i>lf</i>,
+  cut proportionally, and increased with the greater segment <i>df</i>,
+  that is <i>fc</i>, is cut also proportionally, by the <a
+  href="#4_e_xiiij">4 e xiiij</a>, and by the <a href="#7_e_xiiij">7 e
+  xiiij</a>, the whole line proportionally cut, and the lesser segment,
+  that is <i>cp</i>, are of treble value to the greater <i>fl</i>, that is
+  of the sayd <i>le</i>. Therefore <i>el</i>, and <i>lc</i>, that is
+  <i>ec</i>, and <i>cp</i>, that is <i>ep</i>, is of quadruple power to
+  <i>el</i>: And therefore by the <a href="#14_e_xij">14 e xij</a>, it is
+  the double of it: And <i>ei</i>, it selfe in like manner, by the fabricke
+  or construction, is the double of the same. Therefore the bases are
+  equall. And after the same manner, by drawing the right lines <i>id</i>,
+  and <i>ib</i>, the third angle <i>bpi</i>, shall be concluded to be equal
+  to the angle <i>ezi</i>. Therefore by the <a href="#13_e_xiiij">13 e
+  xiiij</a>, five angles are equall. <!-- Page 282 --><span
+  class="pagenum"><a name="page282"></a>[282]</span></p>
+
+  <p><a name="23_e_xxv"></a> 23. <i>The Diagony is irrationall unto the
+  side of the dodecahedrum</i>.</p>
+
+  <p>This is the fifth example of irrationality and incommensurability. The
+  first was of the diagony and side of a quadrate or square. The second was
+  of a line proportionally cut and his segments: The third is of the
+  diameter of a Circle and the side of an inscribed quinquangle. The fourth
+  was of the diagony and side of an icosahedrum. The fifth now is of the
+  diagony and side of a dodecahedrum.</p>
+
+  <p><a name="24_e_xxv"></a> 24 <i>If the side of a cube be cut
+  proportionally, the greater segment shall be the side of a
+  dodecahedrum</i>.</p>
+
+  <p>For that hath beene told you even now.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/303.png"><img style="width:100%" src="images/303.png"
+      alt="Figure for demonstration 24." title="Figure for demonstration 24." /></a>
+  </div>
+  <p>But from hence also doth arise the geodesy or m&#x101;ner of measuring
+  of a dodecahedrum. For if the quadrate of the line subtending the angle
+  of a quinquangle be trebled, the half of the treble shall be the side of
+  the semidiagony of the dodecahedrum: Because by the <a
+  href="#6_e_xxiiij">6 e xxiiij</a>, the diagony of the cube, that is of
+  the dodecahedrum is of treble power to the side of the cube. But if the
+  quadrate of the side of the decangle be taken out of the quadrate of the
+  side of the quinquangle; The side of the remainder shall be the ray of
+  the circle circumscribed about a quinquangle. Lastly if the quadrate of
+  the ray, be taken of the quadrate of the half-diagony; the side of the
+  remainder shall be the heighth of perpendicular. As if the side of the
+  decangle be 7.3/5: The quadrate of that shall be 57.19/25: the treble of
+  which is 173.7/25 whose side is about 13.107/131 for the side of the
+  Dodecahedrum, therefore 6.119/131 the halfe shall be the semidiagony of
+  the dodecahedrum. The ray of the <!-- Page 283 --><span
+  class="pagenum"><a name="page283"></a>[283]</span>Circle shall now thus
+  be found. If the quadrate of the side of the decangle be taken out of the
+  quadrate of the side of the sexangle; the side of the remainder, shall be
+  the Ray of the Circle, by the <a href="#15_e_xviij">15</a> and <a
+  href="#9_e_xviij">9 e xviij</a>. As here the side of the Quinquangle is
+  4.2/3. The side of the Decangle 2.2/5: And the quadrates therefore are
+  21.7/9, and 5.19/25. This subducted from that leaveth 16.4/225 whose side
+  is 4.2/15 for the Ray of the Circle.</p>
+
+  <p>The semidiagony and ray of the circle thus found, the altitude
+  remaineth. Take out therefore the quadrate of the ray of the circle,
+  16.4/225 out of the quadrate of the semidiagony 47.12458/17161, the side
+  of the remainder 31.2714406/3861225 is for the altitude or heighth: whose
+  1/3 is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3
+  doth make 63.1/3 for the solidity of one Pyramis; which multiplied by 12,
+  doth make 760. for the soliditie of the whole <span class="correction"
+  title="text reads `dodetacedrum'">dodecaedrum</span>.</p>
+
+  <p><a name="25_e_xxv"></a> 25 <i>There are but five ordinate solid
+  plaines</i>.</p>
+
+  <p>This appeareth plainely out of the nature of a solid angle, by the
+  kindes of plaine figures. Of two plaine angles a solid angle cannot be
+  comprehended. Of three angles of an ordinate triangle is the angle of a
+  Tetrahedrum comprehended: Of foure, an Octahedrum: Of five, an
+  Icosahedrum: Of sixe none can be comprehended: For sixe such like plaine
+  angles, are equall to 12 thirds of one right angle, that is to foure
+  right angles. But plaine angles making a solid angle, are lesser than
+  foure right angles, by the <a href="#8_e_xxij">8 e xxij</a>. Of seven
+  therefore, and of more it is, much lesse possible. Of three quadrate
+  angles the angle of a cube is comprehended: Of 4. such angles none may be
+  comprehended for the same cause. Of three angles of an ordinate
+  quinquangle, is made the angle of a Dodecahedrum. Of 4. none may possibly
+  be made; For every such angle: For every one of them severally doe
+  countervaile one right angle and 1/5 of the same, Therefore they would be
+  foure, and three fifths. Of more therefore much lesse may it be possible.
+  <!-- Page 284 --><span class="pagenum"><a
+  name="page284"></a>[284]</span></p>
+
+  <p>This demonstration doth indeed very accurately and manifestly appeare,
+  Although there may be an innumerable sort of ordinate plaines, yet of the
+  kindes of angles five onely ordinate bodies may be made; From whence the
+  Tetrahedrum, Octahedrum, and Icosahedrum are made upon a triangular base:
+  the Cube upon a quadrangular: And the Dodecahedrum, upon a
+  quinquangular.</p>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i> the twenty sixth Booke;
+Of a <i>Spheare</i>.</h2>
+
+  <p><a name="1_e_xxvj"></a> 1 <i>An imbossed solid is that which is
+  comprehended of an imbossed surface</i>.</p>
+
+  <p><a name="2_e_xxvj"></a> 2. <i>And it is either a spheare or a Mingled
+  forme</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/305.png"><img style="width:100%" src="images/305.png"
+      alt="Sph�ra." title="Sph�ra." /></a>
+  </div>
+  <p><a name="3_e_xxvj"></a> 3. <i>A speare is a round imbossement</i>.</p>
+
+  <p>It may also be defined to be that which is comprehended of a
+  sphearical surface. A sphearicall body in Greeke is called <i>Sph�ra</i>,
+  in Latine <i>Globus</i>, a Globe.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="4_e_xxvj"></a> 4. <i>A Spheare is made by the conversion of a
+  semicircle, the diameter standing still. 14 d xj.</i> <!-- Page 285
+  --><span class="pagenum"><a name="page285"></a>[285]</span></p>
+
+  <div class="figleft" style="width:26%;">
+      <a href="images/306a.png"><img style="width:100%" src="images/306a.png"
+      alt="Figure for demonstration 4." title="Figure for demonstration 4." /></a>
+  </div>
+  <p>As here thou seest.</p>
+
+  <p><a name="5_e_xxvj"></a> 5. <i>The greatest circle of a spheare, is
+  that which cutteth the spheare into two equall parts</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Therefore</p>
+
+  <p><a name="6_e_xxvj"></a> 6. <i>That circle which is neerest to the
+  greatest, is greater than that which is farther off</i>.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+  <p><a name="7_e_xxvj"></a> 7. <i>Those which are equally distant from the
+  greatest are equall</i>.</p>
+
+  <p>As in the example above written.</p>
+
+  <p><a name="8_e_xxvj"></a> 8. <i>The plaine of the diameter and sixth
+  part of the sphearicall is the solidity of the spheare</i>.</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/306b.png"><img style="width:100%" src="images/306b.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <p>As before there was an analogy betweene a Circle and a Sphericall: so
+  now is there betweene a Cube and a spheare. A cubicall surface is
+  comprehended of sixe quadrate or square and equall bases: And a spheare
+  in like manner is comprehended of sixe equall sphearicall bases
+  compassing the <!-- Page 286 --><span class="pagenum"><a
+  name="page286"></a>[286]</span>cubicall bases. A cube is made by the
+  multiplication of the sixth part of the base, by the side: And a spheare
+  likewise is made by multiplying the sixth part of the sphearicall by the
+  diameter, as it were by the side: so the plaine of 616/6 and 14, the
+  diameter is 1437.1/3 for the solidity of the spheare.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="9_e_xxvj"></a> 9. <i>As 21 is unto 11, so is the cube of the
+  diameter unto the spheare</i>.</p>
+
+  <p>As here the Cube of 14 is 2744. For it was an easy matter for him that
+  will compare the cube 2744, with the spheare, to finde that 2744 to be to
+  1437.1/3 in the least boundes of the same reason, as 21 is unto 11.</p>
+
+  <p>Thus much therefore of the Geodesy of the spheare: The geodesy of the
+  <span class="correction" title="text reads `Sctour'">Sectour</span> and
+  section of the spheare shall follow in the next place.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="10_e_xxvj"></a> 10. <i>The plaine of the ray, and of the
+  sixth part of the sphearicall is the hemispheare</i>.</p>
+
+  <p>But it is more accurate and preciser cause to take the halfe of the
+  spheare.</p>
+
+  <p><a name="11_e_xxvj"></a> 11. <i>Spheares have a trebled reason of
+  their diameters</i>.</p>
+
+  <p>So before it was told you; That circles were one to another, as the
+  squares of their diameters were one to another, because they were like
+  plaines: And the diameters in circles were, as now they are in spheares,
+  the homologall sides. Therefore seeing that spheres are figures alike,
+  and of treble dimension, they have a trebled reason of their
+  diameters.</p>
+
+  <p><a name="12_e_xxvj"></a> 12. <i>The five ordinate bodies are inscribed
+  into the same spheare, by the conversion of a semicicle having for the
+  diameter, in a tetrahedrum, a right line of value <!-- Page 287 --><span
+  class="pagenum"><a name="page287"></a>[287]</span>sesquialter unto the
+  side of the said tetrahedrum; in the other foure ordinate bodies, the
+  diagony of the same orginate</i>.</p>
+
+  <p>The Adscription of ordinate plaine bodies is unto a spheare, as before
+  the Adscription plaine surfaces was into a circle; of a triangle, I
+  meane, and ordinate triangulate, as Quadrangle, Quinquangle, Sexangle,
+  Decangle, and Quindecangle. But indeed the Geometer hath both inscribed
+  and circumscribed those plaine figures within a circle. But these five
+  ordinate bodies, and over and above the Polyhedrum the Stereometer hath
+  onely inscribed within the spheare. The Polyhedrum we have passed over,
+  and we purpose onely to touch the other ordinate bodies.</p>
+
+  <p><a name="13_e_xxvj"></a> 13 <i>Out of the reason of the axeltree of
+  the sphearicall the sides of the tetraedrum, cube, octahedrum and
+  dodecahedrum are found out</i>.</p>
+
+  <div class="figright" style="width:24%;">
+      <a href="images/308.png"><img style="width:100%" src="images/308.png"
+      alt="Figure for demonstration 13." title="Figure for demonstration 13." /></a>
+  </div>
+  <p>The axeltree in the three first bodies is rationall unto the side, as
+  was manifested in the former. For it is of the sesquialter valew unto the
+  side of the tetrahedrum; of treble, to the side of the cube: Of double,
+  to the side of the Octahedrum. Therefore if the axis <i>ae</i>, be cut by
+  a double reason in <i>i</i>: And the perpendicular <i>io</i>, be knit to
+  <i>a</i>, and <i>e</i>, shall be the side of the tetrahedrum; and
+  <i>oe</i>, of the cube, as was manifest by the <a href="#10_e_viij">10 e
+  viij</a>, and <a href="#25_e_iiij">25 iiij</a>: And the greater segment
+  of the side of the cube proportionally cut, is by the <a
+  href="#24_e_xxv">24 e, xxv</a>.</p>
+
+  <p>If the same axis be cut into two halfes, as in <i>u</i>: And the
+  perpendicular <i>uy</i>, be erected: And <i>y</i>, and <i>a</i>, be knit
+  together, the same <i>ya</i>, thus knitting them, shall be the side of
+  the Octahedrum, as is manifest in like manner, by the said <a
+  href="#10_e_viij">10 e, viij</a>, and <a href="#25_e_iiij">25 e
+  iiij</a>.</p>
+
+  <p>The side of the Icosahedrum is had by this meanes.</p>
+
+  <p><a name="14_e_xxvj"></a> 14. <i>If a right line equall to the axis of
+  the sphearicall, and to it from the end of the perpendicular be knit unto
+  the center, a right line drawne from the cutting of the <!-- Page 288
+  --><span class="pagenum"><a name="page288"></a>[288]</span>periphery unto
+  the said end shall be the side of the Icosahedrum</i>.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/309.png"><img style="width:100%" src="images/309.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p>As here let the Axis <i>ae</i>; be the diameter of the circle
+  <i>aue</i>, and <i>ai</i>, equall to the same axis, and perpendicular
+  from the end, be knit unto the center, by the right line <i>io</i>: A
+  right drawne from the section <i>u</i>, unto <i>a</i>, shall be the side
+  of the Icosahedrum. From <i>u</i>, let the perpendicular <i>uy</i>, be
+  drawne: Here the two triangles <i>iao</i>, &amp; <i>uyo</i>, are
+  equiangles by the <a href="#13_e_vij">13 e, vij</a>. Therfore by the <a
+  href="#12_e_vij">12 e, vij</a>. as <i>ia</i>, is unto <i>ao</i>: so is
+  <i>uy</i>, unto <i>yo</i>. But <i>ia</i>, is the double of the said
+  <i>ao</i>: Therefore <i>uy</i>, is the double of the same <i>yo</i>:
+  Therefore by the <a href="#14_e_xij">14 e, xij</a>, it is of quadruple
+  power unto it: And therefore also <i>uy</i>, and <i>yo</i>, that is, by
+  the <a href="#9_e_xij">9 e xij</a>, <i>uo</i>, that is againe by the <a
+  href="#28_e_iiij">28 e, iiij</a>, <i>ao</i>, is of quintuple power to
+  <i>yo</i>. But <i>yo</i>, is lesser than <i>ao</i>, that is, than
+  <i>oe</i>: Let therefore <i>os</i>, be cut off equall to it. Now as the
+  halfe of <i>ao</i>, is of quintuple valew to the halfe of <i>yo</i>: so
+  the double <i>ae</i>, is of quintuple power to the double <i>ys</i>.
+  Therefore, by the <a href="#18_e_xxv">18 e xxv</a>. seeing that the
+  diagony <i>ae</i>, is of quintuple power to <i>ys</i>; the said
+  <i>ys</i>, shall be the side of the sexangle inscribed into a circle,
+  circumscribing the quinquangle of the Icosahedrum. But the perpendicular
+  <i>uy</i>, is equall to <i>ys</i>; because each of them is the double of
+  <i>yo</i>. Wherefore <i>uy</i>, is the side of the sexangle. But
+  <i>ay</i>, is the side of the Decangle: For it is equall to <i>se</i>:
+  Because if from equall rayes <i>ao</i>, and <i>oe</i>, you take equall
+  portions <i>oy</i>, and <i>os</i>: There shall remaine equall, <i>ya</i>,
+  and <i>se</i>. And the Diagony of an Icosahedrum by the <a
+  href="#16_e_xxv">16 e xxv</a>, is compounded of the side of the sexangle,
+  continued at each end with the side of the decangle. Wherefore <i>ay</i>,
+  is the side of the decangle. Lastly, <i>ua</i>, whose power is as much as
+  the sides of the <!-- Page 289 --><span class="pagenum"><a
+  name="page289"></a>[289]</span>sexangle and decangle, by the <a
+  href="#15_e_xviij">15. e, xviij</a>, shall be the side of an
+  Icosahedrum.</p>
+
+  <p><a name="15_e_xxvj"></a> 15 <i>Of the five ordinate bodies inscribed
+  into the same spheare, the tetrahedrum in respect of the greatnesse of
+  his side is first, the Octahedrum, the second; the Cube, the third; the
+  Icosahedrum, the fourth; and the Dodecahedrum, the fifth</i>.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/310.png"><img style="width:100%" src="images/310.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <p>As it will plainely appeare, if all of them be gathered into one,
+  thus. For <i>ai</i>, the side of the Tetrahedrum, subtendeth a greater
+  periphery than <i>ao</i>, the side of the Octahedrum; And <i>ao</i>, a
+  greater than <i>ie</i>, the side of the Cube; because it subtendeth but
+  the halfe: And <i>ie</i>, greater than <i>ue</i>, the side of the
+  Icosahedrum: And <i>ue</i>, greater than <i>ye</i>, the side of
+  Dodecahedrum.</p>
+
+  <p>The latter, <i>Euclide</i> doth demonstrate with a greater
+  circumstance. Therefore out of the former figures and demonstrations, let
+  here be repeated, The sections of the axis first into a double reason in
+  <i>s</i>: And the side of the sexangle <i>rl</i>: And the side of the
+  Decangle <i>ar</i>, inscribed into the same circle, circumscribing the
+  quinquangle of an icosahedrum: And the perpendiculars <i>is</i>, and
+  <i>ul</i>.</p>
+
+  <p>Here the two triangles <i>aie</i>, and <i>ies</i>, are by the <a
+  href="#8_e_viij">8 e, viij</a>. alike; And as <i>se</i>, is unto
+  <i>ei</i>: So is <i>ie</i>, unto <i>ea</i>: And by <a
+  href="#25_e_iiij">25 e, iiij</a>, as <i>se</i>, is to <i>ea</i>: so is
+  the quadrate of <i>se</i>, to the quadrate of <i>ei</i>: And inversly or
+  backward, as <i>ae</i>, is to <i>se</i>: so is the quadrate of <i>ie</i>,
+  to the quadrate of <i>se</i>. But <i>ae</i>, is the triple of <i>se</i>.
+  Therefore the quadrate of <i>ie</i>, is the triple of <i>se</i>. But the
+  quadrate of <i>as</i>, by the grant, and <a href="#14_e_xij">14 e
+  xij</a>, the quadruple of the quadrate of <i>se</i>. Therefore also it is
+  greater than the quadrate of <i>ie</i>: And the right line <i>as</i>, is
+  greater than <i>ie</i>, and <i>al</i>, therefore is much greater. But
+  <i>al</i>, is by the grant <!-- Page 290 --><span class="pagenum"><a
+  name="page290"></a>[290]</span>compounded of the sides of the sexangle
+  and decangle <i>rl</i>, and <i>ar</i>. Therefore by the 1 c. <a
+  href="#5_e_xviij">5 e, 18.</a> it is cut proportionally: And the greater
+  segment is the side of the sexangle, to wit, <i>rl</i>: And the greater
+  segment of <i>ie</i>, proportionally also cut, is <i>ye</i>. Therefore
+  the said <i>rl</i>, is greeter than <i>ye</i>: And even now it was shewed
+  <i>ul</i>, was equall to <i>rl</i>. Therefore <i>ul</i>, is greater than
+  <i>ye</i>: But <i>ue</i>, the side of the Icosahedrum, by <a
+  href="#22_e_vj">22. e vj</a>. is greater than <i>ul</i>. Therefore the
+  side of the Icosahedrum is much greater, then the side of the
+  dodecahedrum.</p>
+
+<hr class="full" />
+
+<h2>Of <i>Geometry</i> the twenty seventh Book;
+Of the Cone and Cylinder.</h2>
+
+  <p><a name="1_e_xxvij"></a> 1 <i>A mingled solid is that which is
+  comprehended of a variable surface and of a base</i>.</p>
+
+  <p>For here the base is to be added to the variable surface.</p>
+
+  <p><a name="2_e_xxvij"></a> 2 <i>If variable solids have their axes
+  proportionall to their bases, they are alike. 24. d xj</i>.</p>
+
+  <p>It is a Consectary out of the <a href="#19_e_iiij">19 e, iiij</a>. For
+  here the axes and diameters are, as it were, the shankes of equall
+  angles, to wit, of right angles in the base, and perpendicular axis.</p>
+
+  <p><a name="3_e_xxvij"></a> 3 <i>A mingled body is a Cone or a
+  Cylinder</i>.</p>
+
+  <p>The cause of this division of a varied or mingled body, is to be
+  conceived from the division of surfaces. <!-- Page 291 --><span
+  class="pagenum"><a name="page291"></a>[291]</span></p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/312a.png"><img style="width:100%" src="images/312a.png"
+      alt="Cone." title="Cone." /></a>
+  </div>
+  <p><a name="4_e_xxvij"></a> 4 <i>A Cone is that which is comprehended of
+  a conicall and a base</i>.</p>
+
+  <p>Here the base is a circle.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="5_e_xxvij"></a> 5 <i>It is made by the turning about of a
+  right angled triangle, the one shanke standing still</i>.</p>
+
+  <p>As it appeareth out of the definition of a variable body.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="6_e_xxvij"></a> 6 <i>A Cone is rightangled, if the shanke
+  standing still be equall to that turned about: It is Obtusangeld, if it
+  be lesse: and acutangled, if it be greater. � 18 d xj</i>.</p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/312b.png"><img style="width:100%" src="images/312b.png"
+      alt="Right angled and other cones." title="Right angled and other cones." /></a>
+  </div>
+  <p>Here a threefold difference of the heighth of a Cone is professed, out
+  of the threefold difference of the angles, whereby the toppe of the
+  halfed cone is distinguished: Notwithstanding this consideration
+  belongeth rather to the Optickes, than to Geometry. For a Cone a farre
+  off seeme like triangle. Therefore according to the difference of the
+  heighth, it <!-- Page 292 --><span class="pagenum"><a
+  name="page292"></a>[292]</span>appeareth with a right angled, or
+  obtusangled or acutangled toppe: As here the least Cone is obtusangled:
+  the middle one rightangled: and the highest acutangled. But the cause of
+  this threefold difference in the angles from of the difference of the
+  shankes, is out of the consectaries of the threefold triangle of a right
+  line cutting the base into two equall parts, as appeareth at the end of
+  the <a href="#20_e_viij">viij</a> Booke.</p>
+
+  <p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; And</p>
+
+  <p><a name="7_e_xxvij"></a> 7 <i>A Cone is the first of all
+  variable</i>.</p>
+
+  <p>For a Cone is so the first in variable solids, as a triangle is in
+  rectilineall plaines: As a Pyramis is in solid plaines: For neither may
+  it indeed be divided into any other variable solids more simple.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="8_e_xxvij"></a> 8 <i>Cones of equall heighth are as their
+  bases are 11. p xij</i>.</p>
+
+  <p>As here you see.</p>
+
+  <div class="figcenter" style="width:34%;">
+      <a href="images/313.png"><img style="width:100%" src="images/313.png"
+      alt="Figure for demonstration 8." title="Figure for demonstration 8." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="9_e_xxvij"></a> 9 <i>They which are reciprocall in base and
+  heighth are equall, 15 p xij.</i> <!-- Page 293 --><span
+  class="pagenum"><a name="page293"></a>[293]</span></p>
+
+  <p>These are consectaries drawne out of the <span class="correction"
+  title="text reads `12 and 13 e. iiij' - compare the same discussion for the pyramis at 14, 15 e. xxij."
+  ><a href="#16_e_iiij">16</a> and <a href="#18_e_iiij">18 e.
+  iiij</a></span>. As here you see.</p>
+
+  <div class="figcenter" style="width:35%;">
+      <a href="images/314a.png"><img style="width:100%" src="images/314a.png"
+      alt="Figure for demonstration 9." title="Figure for demonstration 9." /></a>
+  </div>
+  <div class="figright" style="width:19%;">
+      <a href="images/315a.png"><img style="width:100%" src="images/315a.png"
+      alt="Geodesy of Cylinder." title="Geodesy of Cylinder." /></a>
+  </div>
+  <div class="figright" style="width:17%;">
+      <a href="images/314b.png"><img style="width:100%" src="images/314b.png"
+      alt="Cylinder." title="Cylinder." /></a>
+  </div>
+  <p><a name="10_e_xxvij"></a> 10 <i>A Cylinder is that which is
+  comprehended of a <span class="correction" title="text reads `cyliudricall'"
+  >cylindricall</span> surface and the opposite bases</i>.</p>
+
+  <p>For here two circles, parallell one to another are the bases of a
+  Cylinder.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <p><a name="11_e_xxvij"></a> 11 <i>It is made by the turning about of a
+  right angled parallelogramme, the one side standing still. 21. d
+  xj</i>.</p>
+
+  <p>As is apparant out the same definition of a varium. <!-- Page 294
+  --><span class="pagenum"><a name="page294"></a>[294]</span></p>
+
+  <p><a name="12_e_xxvij"></a> 12. <i>A plaine made of the base and heighth
+  is the solidity of a Cylinder</i>.</p>
+
+  <p>The geodesy here is fetch'd from the prisma: As if the base of the
+  cylinder be 38.1/2: Of it and the heighth 12, the solidity of the
+  cylinder is 462.</p>
+
+  <p>This manner of measuring doth answeare, I say, to the manner of
+  measuring of a prisma, and in all respects to the geodesy of a right
+  angled parallelogramme.</p>
+
+  <p>If the cylinder in the opposite bases be oblique, then if what thou
+  cuttest off from one base thou doest adde unto the other, thou shalt have
+  the measure of the whole; as here thou seest in these cylinders, <i>a</i>
+  and <i>b</i>.</p>
+
+  <p><br style="clear : both" /></p>
+  <div class="figcenter" style="width:33%;">
+      <a href="images/315b.png"><img style="width:100%" src="images/315b.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <div class="figright" style="width:17%;">
+      <a href="images/316.png"><img style="width:100%" src="images/316.png"
+      alt="Figure for demonstration 12." title="Figure for demonstration 12." /></a>
+  </div>
+  <p>From hence the capacity or content of cylinder-like <!-- Page 295
+  --><span class="pagenum"><a name="page295"></a>[295]</span>vessell or
+  measure is esteemed and judged of. For the hollow or empty place is to be
+  measured as if it were a solid body.</p>
+
+  <p>As here the diameter of the inner Circle is 6 foote: The periphery is
+  18.6/7: Therefore the plot or content of the circle is 28.2/7. Of which,
+  and the heighth 10, the plaine is 282.6/7 for the capacity of the
+  vessell. Thus therefore shalt thou judge, as afore, how much liquor or
+  any thing else conteined, a cubicall foote may hold.</p>
+
+  <p><a name="13_e_xxvij"></a> 13. <i>A Cylinder is the triple of a cone
+  equall to it in base and heighth. 10 p xij</i>.</p>
+
+  <p>The demonstration of this proposition hath much troubled the
+  interpreters. The reason of a Cylinder unto a Cone, may more easily be
+  assumed from the reason of a Prisme unto a Pyramis: For a Cylinder doth
+  as much resemble a Prisme, as the Cone doth a Pyramis: Yea and within the
+  same sides may a Prisme and a Cylinder, a Pyramis and a Cone be
+  conteined: And if a Prisme and a Pyramis have a very multangled base, the
+  Prisme and <span class="correction" title="text reads `Clinder'"
+  >Cylinder</span>, as also the Pyramis and Cone, do seeme to be the same
+  figure. Lastly within the same sides, as the Cones and Cylinders, so the
+  Prisma and Pyramides, from their axeltrees and diameters may have the
+  similitude of their bases. And with as great reason may the Geometer
+  demand to have it granted him, That the Cylinder is the treble of a Cone:
+  As it was demanded and granted him, That Cylinders and Cones are alike,
+  whose axletees are proportionall to the diameters of their bases.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Therefore</p>
+    </div>
+  </div>
+  <div class="figright" style="width:16%;">
+      <a href="images/317b.png"><img style="width:100%" src="images/317b.png"
+      alt="Archimede's Rhombus." title="Archimede's Rhombus." /></a>
+  </div>
+  <div class="figright" style="width:21%;">
+      <a href="images/317a.png"><img style="width:100%" src="images/317a.png"
+      alt="Figure for demonstration 14." title="Figure for demonstration 14." /></a>
+  </div>
+  <p><a name="14_e_xxvij"></a> 14. <i>A plaine made of the base and thid
+  part of the height, is the solidity of the cone of equall base &amp;
+  height;</i> <!-- Page 296 --><span class="pagenum"><a
+  name="page296"></a>[296]</span></p>
+
+  <p>The heighth is thus had. If the square of the ray of the base, be
+  taken out of the square of the side, the side of the remainder shall bee
+  the heighth, as is manifest by the <a href="#9_e_xij">9 e xij</a>. Here
+  therefore the square of the ray 5, is 25. The square of 13, the side is
+  169. And 169 - 25, are 144; whose side is 12 for the heighth: The third
+  part of which is 4. Now the circular base is 78.4/7: And the plaine of
+  these is 314.2/7 for the solidity of the Cone.</p>
+
+  <p>But the analogie of a conicall unto a Cylinder like surface doth not
+  answeare, that the Conicall should be the subtriple of the Cylindricall,
+  as the Cone is the subtriple of the Cylinder.</p>
+
+  <p>Of two cones of one common base is made <i>Archimede's Rhombus</i>, as
+  here, whose geod�sy shall be cut of two cones.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="15_e_xxvij"></a> 15. <i>Cylinder of equall heighth are as
+  their bases are. 11 p xij</i>.</p>
+
+  <p>Sackes in which they carry corne, are for the most part of <!-- Page
+  297 --><span class="pagenum"><a name="page297"></a>[297]</span>a
+  cylinderlike forme. If an husbandman therefore shall lend unto his
+  neighbour a sacke full or corne, and the base of the sacke be 4 foote
+  over. And the neighbour afterward for that one sacke, shall pay him 4
+  sacke fulls, every sacke being as long as that was, yet but one foote
+  over in the diameter, he may be thought peradventure to have repayed that
+  which he borrowed in equall measure, to wit in heighth and base. But it
+  shall be indeed farre otherwise: For there is a great difference betweene
+  the quadrate of the foure severall diameters, 1. 1. 1. 1. that is 4: and
+  16, the quadrate of 4, the diameter of that sacke by which it was lent.
+  For Circles are one unto another as the quadrates of their diameters are
+  one to another, by the <a href="#2_e_xv">2 e xv</a>. Therefore he payd
+  him but one fourth part of that which he borrowed of him.</p>
+
+  <div class="figcenter" style="width:29%;">
+      <a href="images/318a.png"><img style="width:100%" src="images/318a.png"
+      alt="Figure for demonstration 15." title="Figure for demonstration 15." /></a>
+  </div>
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <p><a name="16_e_xxvij"></a> 16 <i>Cylinders reciprocall in base and
+  heighth are equall. 15 p xij</i>.</p>
+
+  <div class="figcenter" style="width:33%;">
+      <a href="images/318a.png"><img style="width:100%" src="images/318a.png"
+      alt="Figure for demonstration 16." title="Figure for demonstration 16." /></a>
+  </div>
+<p><!-- Page 298 --><span class="pagenum"><a name="page298"></a>[298]</span></p>
+
+  <p>Both these affections are in common attributed to the equally manifold
+  of first figures.</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>And</p>
+    </div>
+  </div>
+  <div class="figright" style="width:15%;">
+      <a href="images/319.png"><img style="width:100%" src="images/319.png"
+      alt="Figure for demonstration 17." title="Figure for demonstration 17." /></a>
+  </div>
+  <p><a name="17_e_xxvij"></a> 17. <i>If a cylinder be cut with a plaine
+  surface parallell to his opposite bases, the segments are, as their axes
+  are 13 p xij</i>.</p>
+
+  <p>As here thou seest. For the axes are the altitudes or heights. It is
+  likwise a consectary following upon that generall theoreme of first
+  figure, but somewhat varyed from it. It doth answere unto the <a
+  href="#10_e_xxiij">10 e 23</a>.</p>
+
+  <p>The unequall sections of a spheare we have reserved for this place:
+  Because they are comprehended of a surface both sphearicall and conicall,
+  as is the sectour. As also of a plaine and sphearicall, as is the
+  section: And in both like as in a Circle, there is but a greater and
+  lesser segment. And the sectour, as before, is considered in the
+  center.</p>
+
+  <p><a name="18_e_xxvij"></a> 18. <i>The sectour of a spheare is a segment
+  of a spheare, which without is comprehended of a sphearicall within of a
+  conicall bounded in the center, the greater of a concave, the lesser of a
+  convex</i>.</p>
+
+  <p><i>Archimedes</i>, maketh mention of such kinde of Sectours, in his 1
+  booke of the Spheare. From hence also is the geodesy following drawne.
+  And here also is there a certaine analogy with a circular sectour.</p>
+
+  <p><a name="19_e_xxvij"></a> 19. <i>A plaine made of the diameter, and
+  sixth part of the greater, or lesser sphearicall, is the greater or
+  lesser sector.</i> <!-- Page 299 --><span class="pagenum"><a
+  name="page299"></a>[299]</span></p>
+
+  <div class="figleft" style="width:22%;">
+      <a href="images/320.png"><img style="width:100%" src="images/320.png"
+      alt="Figure for demonstration 19." title="Figure for demonstration 19." /></a>
+  </div>
+  <p>As here of the Diameter 14, and of 73.1/3 and 4.2/3 (which is the one
+  sixth part of the greater sphearicall) the plaine is 1026.2/3 for the
+  solidity of the greater sectour, so of the same diameter 14, and 29.1/3
+  which is the 1/6 part of 176, the lesser sph�ricall, the plaine is
+  410.2/3 for the solidity of the lesser sectour.</p>
+
+  <p>And from hence lastly doth arise the solidity of the section, by
+  addition and subduction.</p>
+
+  <p><a name="20_e_xxvij"></a> 20. <i>If the greater sectour be increased
+  with the internall cone, the whole shall be the greater section: If the
+  lesser be diminished by it, the remaine shall be the lesser
+  section</i>.</p>
+
+  <p>As here the inner cone measured is 126.4/63. The greater sectour, by
+  the former was 1026.2/3. And <span class="correction" title="text reads `126'"
+  >1026</span>.2/3 + 126.4/63 doe make 1152.46/63.</p>
+
+  <p>Againe the lesser sectour, by the next precedent, was 410.2/3: And
+  here the inner cone is 126.4/63 And therefore 410.2/3 - 126.4/63 that is
+  284.38/63 is the lesser section.</p>
+
+  <p><br style="clear : both" /></p>
+<hr class="full" />
+
+<h3><i>FINIS.</i></h3>
+
+<hr class="full" />
+
+<p><!-- Page 300 --><span class="pagenum"><a name="page300"></a>[300]</span></p>
+
+  <div class="figcenter" style="width:55%;">
+      <a href="images/141.png"><img style="width:100%" src="images/141.png"
+      alt="Use of Jacob's Staffe at Sea (p. 120)." title="Use of Jacob's Staffe at Sea (p. 120)." /></a>
+  </div>
+  <div class="figcenter" style="width:43%;">
+      <a href="images/146.png"><img style="width:100%" src="images/146.png"
+      alt="Use of Jacob's Staffe in a Well (p. 125)." title="Use of Jacob's Staffe in a Well (p. 125)." /></a>
+  </div>
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of The Way To Geometry, by Peter Ramus
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@@ -0,0 +1,434 @@
+The Project Gutenberg EBook of The Way To Geometry, by Peter Ramus
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Way To Geometry
+
+Author: Peter Ramus
+
+Translator: William Bedwell
+
+Release Date: October 2, 2008 [EBook #26752]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE WAY TO GEOMETRY ***
+
+
+
+
+Produced by Jonathan Ingram, Keith Edkins and the Online
+Distributed Proofreading Team at https://www.pgdp.net
+
+
+
+
+
+
+
+
+
+THE WAY
+
+TO
+
+GEOMETRY.
+
+Being necessary and usefull,
+
+For
+
+Astronomers. Engineres. Geographers. Architecks. Land-meaters.
+Carpenters. Sea-men. Paynters. Carvers, &c.
+
+
+
+Written by Peter Ramus
+
+Translated by William Bedwell
+
+
+
+
+Note from submitter:
+
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