--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/texts/XML/echo/en/Harriot_Add_MS_6786_TRMFCPMB.xml	Fri Dec 07 17:05:22 2012 +0100
@@ -0,0 +1,2880 @@
+<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC">
+<metadata>
+<dcterms:identifier>ECHO:TRMFCPMB.xml</dcterms:identifier>
+<dcterms:creator>Harriot, Thomas</dcterms:creator>
+<dcterms:title xml:lang="en">Mss. 6786</dcterms:title>
+<dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date>
+<dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language>
+<dcterms:rights>CC-BY-SA</dcterms:rights>
+<dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license>
+<dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder>
+<echodir>/permanent/library/TRMFCPMB</echodir>
+<log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:54 2011</log>
+</metadata>
+
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+<pb file="add_6786_f174v" o="174v" n="348"/>
+<pb file="add_6786_f175" o="175" n="349"/>
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+<pb file="add_6786_f182" o="182" n="363"/>
+<pb file="add_6786_f182v" o="182v" n="364"/>
+<p>
+<s xml:id="echoid-s1" xml:space="preserve">
+S. JS. Harriman. <lb/>
+Mr Huss. <lb/>
+Mr Alisbury. <lb/>
+Mr Thorperly. <lb/>
+Mr Marlo.
+</s>
+</p>
+<pb file="add_6786_f183" o="183" n="365"/>
+<pb file="add_6786_f183v" o="183v" n="366"/>
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+<pb file="add_6786_f214v" o="214v" n="428"/>
+<pb file="add_6786_f215" o="215" n="429"/>
+<pb file="add_6786_f215v" o="215v" n="430"/>
+<pb file="add_6786_f216" o="216" n="431"/>
+<pb file="add_6786_f216v" o="216v" n="432"/>
+<pb file="add_6786_f217" o="217" n="433"/>
+<pb file="add_6786_f217v" o="217v" n="434"/>
+<pb file="add_6786_f218" o="218" n="435"/>
+<pb file="add_6786_f218v" o="218v" n="436"/>
+<pb file="add_6786_f219" o="219" n="437"/>
+<pb file="add_6786_f219v" o="219v" n="438"/>
+<pb file="add_6786_f220" o="220" n="439"/>
+<pb file="add_6786_f220v" o="220v" n="440"/>
+<pb file="add_6786_f221" o="221" n="441"/>
+<pb file="add_6786_f221v" o="221v" n="442"/>
+<pb file="add_6786_f222" o="222" n="443"/>
+<pb file="add_6786_f222v" o="222v" n="444"/>
+<pb file="add_6786_f223" o="223" n="445"/>
+<head xml:id="echoid-head1" xml:space="preserve">
+Euclides opticks
+</head>
+<pb file="add_6786_f223v" o="223v" n="446"/>
+<pb file="add_6786_f224" o="224" n="447"/>
+<div xml:id="echoid-div1" type="page_commentary" level="2" n="1">
+<p>
+<s xml:id="echoid-s2" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s2" xml:space="preserve">
+This page refers to Propositions I.36 and I.37 of Book I of Apollonius,
+<emph style="it">Conicorum libri quattuor</emph>.
+</s>
+<lb/>
+<quote>
+I.36.
+If some straight line, meeting the transverse side of the figure touches an hyperbola
+or ellipse or circumference of a circle, and if a straight line is dropped from the point of contact
+as an ordinate to the diameter, then as the straight line cut off by the tangent
+from the end of the transverse side is to the straight line cut off by the tangent from the other end of that side,
+so will the straight line cut off by the ordinate from the end of the side be
+to the straight line cut off by the ordinate from the other end of the side in such a way that
+the corresponding straight lines are continuous; and another straight line will not fall into the space
+between the tangent and the section of the cone.
+</quote>
+<lb/>
+<quote>
+I.37
+If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter,
+and from the point of contact to the diameter a straight line is dropped as ordinate,
+then the straight line cut off by the ordinate from the centre of the section
+with the straight line cut off by the tangent from the centre of the section will contain
+an area equal to the square on the radius of the section,
+and with the straight line between the ordinate and the tangent will contain an area
+having the ratio to the square on the ordinate which the transverse has to the upright.
+</quote>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat">
+Appollo: lib. 1.
+<lb/>[<emph style="it">tr: 
+Apollonius, Book I
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s4" xml:space="preserve">
+37.p <lb/>
+36. dia:
+<lb/>[<emph style="it">tr: 
+Propositions 36, 37; diameter:
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f224v" o="224v" n="448"/>
+<pb file="add_6786_f225" o="225" n="449"/>
+<pb file="add_6786_f225v" o="225v" n="450"/>
+<pb file="add_6786_f226" o="226" n="451"/>
+<head xml:id="echoid-head3" xml:space="preserve" xml:lang="lat">
+Atomi
+</head>
+<p>
+ <s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note: 
+A calculation of the number of atoms in 1 foot, 1 pase (= 5 feet), and 1 mile (= 1000 pases).
+</emph>]</s><lb/>
+<s xml:id="echoid-s6" xml:space="preserve">
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> foote.
+</s>
+<lb/>
+<s xml:id="echoid-s7" xml:space="preserve">
+10,000. foote.
+</s>
+<lb/>
+<s xml:id="echoid-s8" xml:space="preserve">
+50,000. pase.
+</s>
+<lb/>
+<s xml:id="echoid-s9" xml:space="preserve">
+50,000,000. mile.
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s10" xml:space="preserve">
+<sc>
+The number of miles in the circumference of the earth.
+</sc>
+60,000. <foreign xml:lang="eng">miles</foreign> <reg norm="circa" type="abbr">circ</reg>. terra
+</s>
+<lb/>
+<s xml:id="echoid-s11" xml:space="preserve">
+<sc>
+The number of atoms in the circumference of the earth.
+</sc>
+3,000,000,000,000. <reg norm="atomi" type="abbr">ato</reg>.
+<reg norm="circa" type="abbr">circ</reg>. <reg norm="terra" type="abbr">ter</reg>.
+</s>
+<lb/>
+<s xml:id="echoid-s12" xml:space="preserve">
+<sc>
+The number of atoms in the diameter of the earth.
+</sc>
+1,000,000,000,000. diameter
+</s>
+<lb/>
+<s xml:id="echoid-s13" xml:space="preserve">
+<sc>
+The number of atoms in 2,000 earth diameters, the supposed distance to the sun.
+</sc>
+2,000. diametri ad solem
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s14" xml:space="preserve">
+<sc>
+The number of atoms from the earth to the sun.
+</sc>
+<reg norm="atomi" type="abbr">ato</reg>. ad solem
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s15" xml:space="preserve">
+<sc>
+The number of atoms in 2000 times the distance to the sun, the supposed distance to the fixed stars.
+</sc>
+<reg norm="atomi" type="abbr">ato</reg>. ad [¿]stellæ[?] [???]
+</s>
+<lb/>
+<s xml:id="echoid-s16" xml:space="preserve">
+<sc>
+Approximately double the previous line, that is, the number of atoms in the diameter of the sphere of the fixed stars.
+</sc>
+maior diameter
+</s>
+<lb/>
+<s xml:id="echoid-s17" xml:space="preserve">
+<sc>
+The cube of the previous line, that is, a rough and ready calculation of the number of atoms in the universe.
+</sc>
+cubus maior sphæræ
+</s>
+</p>
+<pb file="add_6786_f226v" o="226v" n="452"/>
+<pb file="add_6786_f227" o="227" n="453"/>
+<div xml:id="echoid-div2" type="page_commentary" level="2" n="2">
+<p>
+<s xml:id="echoid-s18" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s18" xml:space="preserve">
+Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f227v" o="227v" n="454"/>
+<pb file="add_6786_f228" o="228" n="455"/>
+<pb file="add_6786_f228v" o="228v" n="456"/>
+<pb file="add_6786_f229" o="229" n="457"/>
+<pb file="add_6786_f229v" o="229v" n="458"/>
+<div xml:id="echoid-div3" type="page_commentary" level="2" n="3">
+<p>
+<s xml:id="echoid-s20" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s20" xml:space="preserve">
+This page refers to Euclid X.37. In modern editions the relevant proposition is X.36,
+but Harriot's numbering matches that of both Commandino and Clavius.
+</s>
+<lb/>
+<s xml:id="echoid-s21" xml:space="preserve">
+<emph style="it">
+If two ratioinal straight lines commensurable in square only be added together,
+the whole is irrational; and let it be called binomial.
+</emph>
+</s>
+<lb/>
+<s xml:id="echoid-s22" xml:space="preserve">
+Harriot's example is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><msqrt><mrow><mn>5</mn></mrow></msqrt></mstyle></math>.
+</s>
+<lb/>
+<s xml:id="echoid-s23" xml:space="preserve">
+There is also a reference below the diagram to Euclid VI.1: <lb/>
+Triangles and parallelograms which are under the same height are to one another as their bases.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat">
+Lib, 10: prop. 37.
+<lb/>[<emph style="it">tr: 
+Book X, Proposition 37.
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s25" xml:space="preserve">
+Si duae rationales potentia solum commensurabiles componantur, <lb/>
+tota irrationalis erit, nocetur autem ex binis nominibus.
+<lb/>[<emph style="it">tr: 
+If two quantities commensurable in power only are combined, the whole will be irrational,
+moreover separated into two parts.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f230" o="230" n="459"/>
+<head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat">
+Definitiones decimi Euclidis <lb/>
+explicates per exempla numerorum
+<lb/>[<emph style="it">tr: 
+Definitions from the tenth book of Eculid, shown by numerical examples.
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s26" xml:space="preserve">
+1.) Magnitudeines (sive numeri) commensurabiles
+<lb/>[<emph style="it">tr: 
+Commensurable magnitudes (or numbers)
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s27" xml:space="preserve">
+2.) Incommensurabiles
+<lb/>[<emph style="it">tr: 
+Incommensurables
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s28" xml:space="preserve">
+3.) Commensurabiles potentia
+<lb/>[<emph style="it">tr: 
+Commensurables in power
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s29" xml:space="preserve">
+4.) Incomensurabiles
+<lb/>[<emph style="it">tr: 
+Incommensurables
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f230v" o="230v" n="460"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s30" xml:space="preserve">
+[<emph style="it">Note: 
+See also Add MS 6785, f. 210v.
+ </emph>]<lb/>
+de numeris perfectis
+<lb/>[<emph style="it">tr: 
+on perfect numbers
+</emph>]<lb/>
+</s>
+</p>
+<p>
+<s xml:id="echoid-s31" xml:space="preserve">
+Leida <lb/>
+Haga <lb/>
+Gouda <lb/>
+Leida <lb/>
+Ultrajestorum <lb/>
+Gouda
+</s>
+</p>
+<pb file="add_6786_f231" o="231" n="461"/>
+<pb file="add_6786_f231v" o="231v" n="462"/>
+<pb file="add_6786_f232" o="232" n="463"/>
+<div xml:id="echoid-div4" type="page_commentary" level="2" n="4">
+<p>
+<s xml:id="echoid-s32" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s32" xml:space="preserve">
+This sheet contains an analysis of the first 40 propositions of Book X of
+Euclid's <emph style="it">Elements</emph>, showing how each proposition depends on previous ones.
+The reference to Clavius is to his <emph style="it">Euclidis Elementorum libri XV</emph> (1574, 1589, 1591, 1603, 1607).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat">
+Clavius). 10. lib. Euclid. propositionum
+</head>
+<pb file="add_6786_f232v" o="232v" n="464"/>
+<pb file="add_6786_f233" o="233" n="465"/>
+<pb file="add_6786_f233v" o="233v" n="466"/>
+<p>
+<s xml:id="echoid-s34" xml:space="preserve">
+If you call hym ffather which without respect of parson iudgeth <lb/>
+according to everie mans worke passe the time of your dwelling here in franc. <lb/>
+If you call hym ffather which without respect of parson iudgeth <lb/>
+according to everie mans worke passe the time of your dwelling <lb/>
+here in franc. <lb/>
+Banberynam John
+</s>
+</p>
+<pb file="add_6786_f234" o="234" n="467"/>
+<pb file="add_6786_f234v" o="234v" n="468"/>
+<pb file="add_6786_f235" o="235" n="469"/>
+<pb file="add_6786_f235v" o="235v" n="470"/>
+<pb file="add_6786_f236" o="236" n="471"/>
+<pb file="add_6786_f236v" o="236v" n="472"/>
+<p>
+<s xml:id="echoid-s35" xml:space="preserve">
+ffather <lb/>
+But afarre
+</s>
+</p>
+<pb file="add_6786_f237" o="237" n="473"/>
+<p>
+<s xml:id="echoid-s36" xml:space="preserve">
+A man of wordes and not of deedes is like a garden full of weedes <lb/>
+A man of deedes and not of wordes is like a privifull of tourdes. <lb/>
+A man of wordes
+</s>
+</p>
+<pb file="add_6786_f237v" o="237v" n="474"/>
+<pb file="add_6786_f238" o="238" n="475"/>
+<pb file="add_6786_f238v" o="238v" n="476"/>
+<pb file="add_6786_f239" o="239" n="477"/>
+<pb file="add_6786_f239v" o="239v" n="478"/>
+<pb file="add_6786_f240" o="240" n="479"/>
+<pb file="add_6786_f240v" o="240v" n="480"/>
+<pb file="add_6786_f241" o="241" n="481"/>
+<pb file="add_6786_f241v" o="241v" n="482"/>
+<pb file="add_6786_f242" o="242" n="483"/>
+<pb file="add_6786_f242v" o="242v" n="484"/>
+<pb file="add_6786_f243" o="243" n="485"/>
+<pb file="add_6786_f243v" o="243v" n="486"/>
+<pb file="add_6786_f244" o="244" n="487"/>
+<pb file="add_6786_f244v" o="244v" n="488"/>
+<pb file="add_6786_f245" o="245" n="489"/>
+<pb file="add_6786_f245v" o="245v" n="490"/>
+<pb file="add_6786_f246" o="246" n="491"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s37" xml:space="preserve">
+Vuolffgangius [???], Per aliquot gentium migrationibus &amp; lib. 12. fol. <lb/>
+Decades duæ devoniarum gentium emmigrationibus <lb/>
+in Illyrici occidentalis tractum. <lb/>
+Tonstall. <lb/>
+Lypsius de Machinis.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s38" xml:space="preserve">
+Filtring paper. <lb/>
+4 &amp; 8 &amp; [???] glasses <lb/>
+2 glasses with elliptical <lb/>
+[???] for coyne.
+</s>
+</p>
+<pb file="add_6786_f246v" o="246v" n="492"/>
+<pb file="add_6786_f247" o="247" n="493"/>
+<pb file="add_6786_f247v" o="247v" n="494"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s39" xml:space="preserve">
+Annus lunaris.
+</s>
+<lb/>
+<s xml:id="echoid-s40" xml:space="preserve">
+hoc est,
+</s>
+<lb/>
+<s xml:id="echoid-s41" xml:space="preserve">
+Ergo: Mensis lunaris.
+</s>
+</p>
+<pb file="add_6786_f248" o="248" n="495"/>
+<div xml:id="echoid-div5" type="page_commentary" level="2" n="5">
+<p>
+<s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s42" xml:space="preserve">
+From right to left, the columns list: <lb/>
+units (1, 1, 1, 1,...) <lb/>
+lengths (1, 2, 3, 4,...) <lb/>
+triangular numbers (1, 3, 6, 10,...) <lb/>
+triangular-pyramidal numbers (1, 4, 10, 20,...) <lb/>
+triangular-pyramido-pyramidal numbers (1, 5, 15, 35,...).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f248v" o="248v" n="496"/>
+<pb file="add_6786_f249" o="249" n="497"/>
+<pb file="add_6786_f249v" o="249v" n="498"/>
+<pb file="add_6786_f250" o="250" n="499"/>
+<pb file="add_6786_f250v" o="250v" n="500"/>
+<pb file="add_6786_f251" o="251" n="501"/>
+<pb file="add_6786_f251v" o="251v" n="502"/>
+<pb file="add_6786_f252" o="252" n="503"/>
+<pb file="add_6786_f252v" o="252v" n="504"/>
+<pb file="add_6786_f253" o="253" n="505"/>
+<pb file="add_6786_f253v" o="253v" n="506"/>
+<pb file="add_6786_f254" o="254" n="507"/>
+<pb file="add_6786_f254v" o="254v" n="508"/>
+<pb file="add_6786_f255" o="255" n="509"/>
+<pb file="add_6786_f255v" o="255v" n="510"/>
+<pb file="add_6786_f256" o="256" n="511"/>
+<pb file="add_6786_f256v" o="256v" n="512"/>
+<pb file="add_6786_f257" o="257" n="513"/>
+<pb file="add_6786_f257v" o="257v" n="514"/>
+<pb file="add_6786_f258" o="258" n="515"/>
+<pb file="add_6786_f258v" o="258v" n="516"/>
+<pb file="add_6786_f259" o="259" n="517"/>
+<pb file="add_6786_f259v" o="259v" n="518"/>
+<pb file="add_6786_f260" o="260" n="519"/>
+<div xml:id="echoid-div6" type="page_commentary" level="2" n="6">
+<p>
+<s xml:id="echoid-s44" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s44" xml:space="preserve">
+Harriot 19s letters refer to the diagram on page 3v of Commandino's
+<emph style="it">Ptolemaei planisphaerium</emph> (1558).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat">
+Annotationes in rationes scenographices Federico
+Comandino, post ptolemae planisphaerum
+<lb/>[<emph style="it">tr: 
+Annotations on the scenographic ratios of Federico Comandino, in
+</emph>]<lb/>
+</head>
+<p>
+<s xml:id="echoid-s46" xml:space="preserve">
+pag. 3
+</s>
+</p>
+<pb file="add_6786_f260v" o="260v" n="520"/>
+<pb file="add_6786_f261" o="261" n="521"/>
+<pb file="add_6786_f261v" o="261v" n="522"/>
+<pb file="add_6786_f262" o="262" n="523"/>
+<pb file="add_6786_f262v" o="262v" n="524"/>
+<pb file="add_6786_f263" o="263" n="525"/>
+<pb file="add_6786_f263v" o="263v" n="526"/>
+<pb file="add_6786_f264" o="264" n="527"/>
+<pb file="add_6786_f264v" o="264v" n="528"/>
+<pb file="add_6786_f265" o="265" n="529"/>
+<pb file="add_6786_f265v" o="265v" n="530"/>
+<pb file="add_6786_f266" o="266" n="531"/>
+<pb file="add_6786_f266v" o="266v" n="532"/>
+<pb file="add_6786_f267" o="267" n="533"/>
+<pb file="add_6786_f267v" o="267v" n="534"/>
+<pb file="add_6786_f268" o="268" n="535"/>
+<pb file="add_6786_f268v" o="268v" n="536"/>
+<pb file="add_6786_f269" o="269" n="537"/>
+<pb file="add_6786_f269v" o="269v" n="538"/>
+<pb file="add_6786_f270" o="270" n="539"/>
+<pb file="add_6786_f270v" o="270v" n="540"/>
+<pb file="add_6786_f271" o="271" n="541"/>
+<pb file="add_6786_f271v" o="271v" n="542"/>
+<pb file="add_6786_f272" o="272" n="543"/>
+<pb file="add_6786_f272v" o="272v" n="544"/>
+<pb file="add_6786_f273" o="273" n="545"/>
+<pb file="add_6786_f273v" o="273v" n="546"/>
+<pb file="add_6786_f274" o="274" n="547"/>
+<pb file="add_6786_f274v" o="274v" n="548"/>
+<pb file="add_6786_f275" o="275" n="549"/>
+<pb file="add_6786_f275v" o="275v" n="550"/>
+<pb file="add_6786_f276" o="276" n="551"/>
+<pb file="add_6786_f276v" o="276v" n="552"/>
+<pb file="add_6786_f277" o="277" n="553"/>
+<pb file="add_6786_f277v" o="277v" n="554"/>
+<pb file="add_6786_f278" o="278" n="555"/>
+<pb file="add_6786_f278v" o="278v" n="556"/>
+<pb file="add_6786_f279" o="279" n="557"/>
+<div xml:id="echoid-div7" type="page_commentary" level="2" n="7">
+<p>
+<s xml:id="echoid-s47" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s47" xml:space="preserve">
+This page appears to be related to page 177 of Stevin's <emph style="it">L 19arithmetique pratique</emph> (1585).
+The problem that Stevin addresses there, as stated on page 174, is: <lb/>
+<emph style="it" xml:lang="fr">
+Estant donné ligne droicte, &amp; deux nombres:
+Trouer vne ligne droicte en telle raison à la donnée, comme le nombre au nombre. <lb/>
+Stevin's examples are numerical, but Harriot is working in letters
+(as also in further notes on Stevin in Add MS 6782, f. 231).
+</emph></s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s49" xml:space="preserve">
+Stevin. 177.
+</s>
+</p>
+<pb file="add_6786_f279v" o="279v" n="558"/>
+<pb file="add_6786_f280" o="280" n="559"/>
+<pb file="add_6786_f280v" o="280v" n="560"/>
+<pb file="add_6786_f281" o="281" n="561"/>
+<pb file="add_6786_f281v" o="281v" n="562"/>
+<div xml:id="echoid-div8" type="page_commentary" level="2" n="8">
+<p>
+<s xml:id="echoid-s50" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s50" xml:space="preserve">
+The first half of a diagram continued on Add MS 6786, f. 282.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f282" o="282" n="563"/>
+<div xml:id="echoid-div9" type="page_commentary" level="2" n="9">
+<p>
+<s xml:id="echoid-s52" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s52" xml:space="preserve">
+The second half of the diagram from Add MS 6876, f. 281v.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s54" xml:space="preserve">
+A water worke found below <lb/>
+my Lord Chamberlayne at <lb/>
+his house in blacke friers <lb/>
+the 23 of January 1600/1601.
+</s>
+</p>
+<pb file="add_6786_f282v" o="282v" n="564"/>
+<p>
+<s xml:id="echoid-s55" xml:space="preserve">
+The Lord <lb/>
+Chamberlaynes <lb/>
+Water Worke <lb/>
+Jan. 23. <lb/>
+1600 <lb/>
+1601.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s56" xml:space="preserve">
+The plomber [???]
+</s>
+</p>
+<pb file="add_6786_f283" o="283" n="565"/>
+<pb file="add_6786_f283v" o="283v" n="566"/>
+<div xml:id="echoid-div10" type="page_commentary" level="2" n="10">
+<p>
+<s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s57" xml:space="preserve">
+The numersals 1 to 9 written with dots, and with various other symbols invented by Harriot for the purpose.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f284" o="284" n="567"/>
+<pb file="add_6786_f284v" o="284v" n="568"/>
+<pb file="add_6786_f285" o="285" n="569"/>
+<pb file="add_6786_f285v" o="285v" n="570"/>
+<pb file="add_6786_f286" o="286" n="571"/>
+<pb file="add_6786_f286v" o="286v" n="572"/>
+<pb file="add_6786_f287" o="287" n="573"/>
+<pb file="add_6786_f287v" o="287v" n="574"/>
+<pb file="add_6786_f288" o="288" n="575"/>
+<div xml:id="echoid-div11" type="page_commentary" level="2" n="11">
+<p>
+<s xml:id="echoid-s59" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s59" xml:space="preserve">
+A calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn><mo>.</mo><mn>4</mn><mn>5</mn><mn>8</mn><mn>7</mn><mn>6</mn><mn>4</mn><mn>5</mn><mn>1</mn><mn>3</mn><mo>.</mo><mo>.</mo><mo>.</mo></mrow></msqrt></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f288v" o="288v" n="576"/>
+<pb file="add_6786_f289" o="289" n="577"/>
+<pb file="add_6786_f289v" o="289v" n="578"/>
+<pb file="add_6786_f290" o="290" n="579"/>
+<pb file="add_6786_f290v" o="290v" n="580"/>
+<pb file="add_6786_f291" o="291" n="581"/>
+<div xml:id="echoid-div12" type="page_commentary" level="2" n="12">
+<p>
+<s xml:id="echoid-s61" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s61" xml:space="preserve">
+Combinations of quantities generated by multiplication. <lb/>
+The letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> stand for
+<foreign xml:lang="lat">pondus</foreign> (weight),
+<foreign xml:lang="lat">magnitudo</foreign> (magnitude),
+<foreign xml:lang="lat">figura</foreign> (area),
+<foreign xml:lang="lat">situs</foreign> (place),
+<foreign xml:lang="lat">altitudo</foreign> (altitude)
+(see also Add MS 6782, f. 181 and f. 48v).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f291v" o="291v" n="582"/>
+<pb file="add_6786_f292" o="292" n="583"/>
+<pb file="add_6786_f292v" o="292v" n="584"/>
+<pb file="add_6786_f293" o="293" n="585"/>
+<pb file="add_6786_f293v" o="293v" n="586"/>
+<pb file="add_6786_f294" o="294" n="587"/>
+<pb file="add_6786_f294v" o="294v" n="588"/>
+<pb file="add_6786_f295" o="295" n="589"/>
+<pb file="add_6786_f295v" o="295v" n="590"/>
+<pb file="add_6786_f296" o="296" n="591"/>
+<pb file="add_6786_f296v" o="296v" n="592"/>
+<pb file="add_6786_f297" o="297" n="593"/>
+<pb file="add_6786_f297v" o="297v" n="594"/>
+<pb file="add_6786_f298" o="298" n="595"/>
+<pb file="add_6786_f298v" o="298v" n="596"/>
+<pb file="add_6786_f299" o="299" n="597"/>
+<pb file="add_6786_f299v" o="299v" n="598"/>
+<pb file="add_6786_f300" o="300" n="599"/>
+<pb file="add_6786_f300v" o="300v" n="600"/>
+<pb file="add_6786_f301" o="301" n="601"/>
+<pb file="add_6786_f301v" o="301v" n="602"/>
+<pb file="add_6786_f302" o="302" n="603"/>
+<pb file="add_6786_f302v" o="302v" n="604"/>
+<pb file="add_6786_f303" o="303" n="605"/>
+<div xml:id="echoid-div13" type="page_commentary" level="2" n="13">
+<p>
+<s xml:id="echoid-s63" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s63" xml:space="preserve">
+Here Harriot appears to be devising new words for a counting system,
+with systematic endings for each new power of ten.
+In the first paragraph the basic numerals are: en, ben, tren, quen, quin, sinus, spinus, ...,
+which bear some resemblance to number words in languages derived from Latin,
+but in the second paragraph Harriot has adopted an alphabetical system: hon, an, ban, can, dan, .....
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s65" xml:space="preserve">
+en ogen otem otille. emes emies illies <lb/>
+ben Begen Beten Betillia <lb/>
+tren trigen tritem tritilla <lb/>
+quen quagen quatem quatiliia <lb/>
+quinus quigen quitem quitillia <lb/>
+sinus sigen sitem sigillia <lb/>
+spinus spigen spitem spigillia
+</s>
+</p>
+<p>
+<s xml:id="echoid-s66" xml:space="preserve">
+homilantu hoceman holeman honan holemil hobemil homil hocen hoben hon <lb/>
+amilantu aceman abeman anan acamil abemil amill acen aben an <lb/>
+cabeman banan bucemil cabemil bumilli bacen baben ban <lb/>
+cacen caben can <lb/>
+dacen daben dan <lb/>
+ecen eben en <lb/>
+fecen feben fen <lb/>
+gecen geben gen
+</s>
+</p>
+<pb file="add_6786_f303v" o="303v" n="606"/>
+<pb file="add_6786_f304" o="304" n="607"/>
+<pb file="add_6786_f304v" o="304v" n="608"/>
+<pb file="add_6786_f305" o="305" n="609"/>
+<pb file="add_6786_f305v" o="305v" n="610"/>
+<pb file="add_6786_f306" o="306" n="611"/>
+<pb file="add_6786_f306v" o="306v" n="612"/>
+<pb file="add_6786_f307" o="307" n="613"/>
+<pb file="add_6786_f307v" o="307v" n="614"/>
+<pb file="add_6786_f308" o="308" n="615"/>
+<pb file="add_6786_f308v" o="308v" n="616"/>
+<pb file="add_6786_f309" o="309" n="617"/>
+<pb file="add_6786_f309v" o="309v" n="618"/>
+<pb file="add_6786_f310" o="310" n="619"/>
+<pb file="add_6786_f310v" o="310v" n="620"/>
+<pb file="add_6786_f311" o="311" n="621"/>
+<pb file="add_6786_f311v" o="311v" n="622"/>
+<pb file="add_6786_f312" o="312" n="623"/>
+<pb file="add_6786_f312v" o="312v" n="624"/>
+<pb file="add_6786_f313" o="313" n="625"/>
+<pb file="add_6786_f313v" o="313v" n="626"/>
+<pb file="add_6786_f314" o="314" n="627"/>
+<pb file="add_6786_f314v" o="314v" n="628"/>
+<pb file="add_6786_f315" o="315" n="629"/>
+<pb file="add_6786_f315v" o="315v" n="630"/>
+<pb file="add_6786_f316" o="316" n="631"/>
+<pb file="add_6786_f316v" o="316v" n="632"/>
+<p>
+<s xml:id="echoid-s67" xml:space="preserve">
+STevin problemata <lb/>
+Pappus <lb/>
+Apollonius <lb/>
+Clavius
+</s>
+</p>
+<pb file="add_6786_f317" o="317" n="633"/>
+<div xml:id="echoid-div14" type="page_commentary" level="2" n="14">
+<p>
+<s xml:id="echoid-s68" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s68" xml:space="preserve">
+This page lists and compares the definitions in Book V of Euclid's <emph style="it">Elements</emph>,
+as given by: <lb/>
+Federico Commandino, <emph style="it">Euclidis Elementorum libri XV</emph> (1572), <lb/>
+Christophor Clavius, <emph style="it">Euclidis Elementorum libri XV</emph> (1574, 1589, 1591, 1603, 1607), <lb/>
+and unnamed Greek writers.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat">
+Definitionies Quinti libri
+<lb/>[<emph style="it">tr: 
+Definitions in the fifth book
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s70" xml:space="preserve">
+Comandinus
+</s>
+<lb/>
+<s xml:id="echoid-s71" xml:space="preserve">
+Græces prior
+<lb/>[<emph style="it">tr: 
+Earlier Greeks
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s72" xml:space="preserve">
+Clavius
+</s>
+</p>
+<pb file="add_6786_f317v" o="317v" n="634"/>
+<div xml:id="echoid-div15" type="page_commentary" level="2" n="15">
+<p>
+<s xml:id="echoid-s73" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s73" xml:space="preserve">
+A rough attempt at an analysis of teh contents and structure of Eculid's
+<emph style="it">Elements</emph>, Book I.
+For a more polished version see Add MS 6785, f. 337.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f318" o="318" n="635"/>
+<pb file="add_6786_f318v" o="318v" n="636"/>
+<pb file="add_6786_f319" o="319" n="637"/>
+<pb file="add_6786_f319v" o="319v" n="638"/>
+<pb file="add_6786_f320" o="320" n="639"/>
+<pb file="add_6786_f320v" o="320v" n="640"/>
+<pb file="add_6786_f321" o="321" n="641"/>
+<div xml:id="echoid-div16" type="page_commentary" level="2" n="16">
+<p>
+<s xml:id="echoid-s75" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s75" xml:space="preserve">
+The third line is very nearly an anagram of the first; neither makes much sense in its own right.
+For frequency analyses of the letters see Add MS 6788, f. 251v and Add MS 6789, f, 456.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s77" xml:space="preserve">
+Sum mutatus amati mire præ sole lumen abigui <lb/>
+musa mutata mutis. <lb/>
+Sumamus amemus præbet illum origine Titan.
+</s>
+</p>
+<pb file="add_6786_f321v" o="321v" n="642"/>
+<pb file="add_6786_f322" o="322" n="643"/>
+<pb file="add_6786_f322v" o="322v" n="644"/>
+<pb file="add_6786_f323" o="323" n="645"/>
+<pb file="add_6786_f323v" o="323v" n="646"/>
+<pb file="add_6786_f324" o="324" n="647"/>
+<pb file="add_6786_f324v" o="324v" n="648"/>
+<pb file="add_6786_f325" o="325" n="649"/>
+<pb file="add_6786_f325v" o="325v" n="650"/>
+<pb file="add_6786_f326" o="326" n="651"/>
+<pb file="add_6786_f326v" o="326v" n="652"/>
+<pb file="add_6786_f327" o="327" n="653"/>
+<pb file="add_6786_f327v" o="327v" n="654"/>
+<pb file="add_6786_f328" o="328" n="655"/>
+<pb file="add_6786_f328v" o="328v" n="656"/>
+<pb file="add_6786_f329" o="329" n="657"/>
+<pb file="add_6786_f329v" o="329v" n="658"/>
+<pb file="add_6786_f330" o="330" n="659"/>
+<pb file="add_6786_f330v" o="330v" n="660"/>
+<pb file="add_6786_f331" o="331" n="661"/>
+<pb file="add_6786_f331v" o="331v" n="662"/>
+<pb file="add_6786_f332" o="332" n="663"/>
+<pb file="add_6786_f332v" o="332v" n="664"/>
+<pb file="add_6786_f333" o="333" n="665"/>
+<pb file="add_6786_f333v" o="333v" n="666"/>
+<div xml:id="echoid-div17" type="page_commentary" level="2" n="17">
+<p>
+<s xml:id="echoid-s78" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s78" xml:space="preserve">
+A double-page calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mo maxsize="1">(</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f334" o="334" n="667"/>
+<pb file="add_6786_f334v" o="334v" n="668"/>
+<div xml:id="echoid-div18" type="page_commentary" level="2" n="18">
+<p>
+<s xml:id="echoid-s80" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s80" xml:space="preserve">
+A calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mo maxsize="1">(</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f335" o="335" n="669"/>
+<pb file="add_6786_f335v" o="335v" n="670"/>
+<pb file="add_6786_f336" o="336" n="671"/>
+<pb file="add_6786_f336v" o="336v" n="672"/>
+<pb file="add_6786_f337" o="337" n="673"/>
+<pb file="add_6786_f337v" o="337v" n="674"/>
+<pb file="add_6786_f338" o="338" n="675"/>
+<pb file="add_6786_f338v" o="338v" n="676"/>
+<pb file="add_6786_f339" o="339" n="677"/>
+<pb file="add_6786_f339v" o="339v" n="678"/>
+<pb file="add_6786_f340" o="340" n="679"/>
+<pb file="add_6786_f340v" o="340v" n="680"/>
+<pb file="add_6786_f341" o="341" n="681"/>
+<pb file="add_6786_f341v" o="341v" n="682"/>
+<pb file="add_6786_f342" o="342" n="683"/>
+<pb file="add_6786_f342v" o="342v" n="684"/>
+<pb file="add_6786_f343" o="343" n="685"/>
+<pb file="add_6786_f343v" o="343v" n="686"/>
+<pb file="add_6786_f344" o="344" n="687"/>
+<pb file="add_6786_f344v" o="344v" n="688"/>
+<pb file="add_6786_f345" o="345" n="689"/>
+<pb file="add_6786_f345v" o="345v" n="690"/>
+<pb file="add_6786_f346" o="346" n="691"/>
+<pb file="add_6786_f346v" o="346v" n="692"/>
+<div xml:id="echoid-div19" type="page_commentary" level="2" n="19">
+<p>
+<s xml:id="echoid-s82" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s82" xml:space="preserve">
+The reduction shows that 1101101 represents 64 + 32 + 8 + 4 + 1 = 109. <lb/>
+Conversely, breaking down 109 into 64 + 32 + 8 + 4 + 1 shows that its binary representation is 1101101.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s84" xml:space="preserve">
+Reductio. <lb/>
+<lb/>[<emph style="it">tr: 
+Reduction.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s85" xml:space="preserve">
+Conversio <lb/>
+<lb/>[<emph style="it">tr: 
+Converse
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f347" o="347" n="693"/>
+<div xml:id="echoid-div20" type="page_commentary" level="2" n="20">
+<p>
+<s xml:id="echoid-s86" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s86" xml:space="preserve">
+Examples of binary subtraction, addition, and multiplication,
+the latter done both directly and by successive addition.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s88" xml:space="preserve">
+<emph style="st">Additionis</emph> <emph style="super">Subductionis</emph> exempla.
+<lb/>[<emph style="it">tr: 
+An example of <emph style="st">addition</emph> <emph style="super">subtraction</emph>.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s89" xml:space="preserve">
+Additionis Exempla.
+<lb/>[<emph style="it">tr: 
+An example of addition.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s90" xml:space="preserve">
+Multiplicatio.
+<lb/>[<emph style="it">tr: 
+Multiplication.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s91" xml:space="preserve">
+Aliter, cum additionem <lb/>
+successiva.
+<lb/>[<emph style="it">tr: 
+Another way, as successive addition.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f347v" o="347v" n="694"/>
+<pb file="add_6786_f348" o="348" n="695"/>
+<div xml:id="echoid-div21" type="page_commentary" level="2" n="21">
+<p>
+<s xml:id="echoid-s92" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s92" xml:space="preserve">
+A list of numbers, names, and dates.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s94" xml:space="preserve">
+202. Iantgennius Hessiæ. No. 11 <lb/>
+245. [???] 12. No. <lb/>
+249. <lb/>
+250. 12. oct. <lb/>
+288. Cornelius Gemma. 14. N. <lb/>
+306. Ræslin. <lb/>
+320. Hagesius. 10. N. <lb/>
+369. Scultetus. <lb/>
+425. Nolthinus. <lb/>
+438. Winklerus. <lb/>
+448. alij multi.
+</s>
+</p>
+<pb file="add_6786_f348v" o="348v" n="696"/>
+<pb file="add_6786_f349" o="349" n="697"/>
+<pb file="add_6786_f349v" o="349v" n="698"/>
+<p>
+<s xml:id="echoid-s95" xml:space="preserve">
+Gordianus Nodus. <lb/>
+Labyrinthus Dædali. <lb/>
+Clava Herculis. Democritus his reason pro atomis. <lb/>
+Achilles. [???] reason. <lb/>
+Cerberus. <lb/>
+Tenebræ Farturea. <lb/>
+Cimmeriæ tenebræ. <lb/>
+Tentyritæ crcodilos non timent.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s96" xml:space="preserve">
+The problem of a rectilinear rumbe <lb/>
+infinitely turning about the center <lb/>
+and yet the line finite to be moved <lb/>
+in an hour or any other given time to <lb/>
+be made <lb/>
+double to <lb/>
+passe to two <lb/>
+corners.
+</s>
+</p>
+<pb file="add_6786_f350" o="350" n="699"/>
+<pb file="add_6786_f350v" o="350v" n="700"/>
+<pb file="add_6786_f351" o="351" n="701"/>
+<pb file="add_6786_f351v" o="351v" n="702"/>
+<pb file="add_6786_f352" o="352" n="703"/>
+<pb file="add_6786_f352v" o="352v" n="704"/>
+<pb file="add_6786_f353" o="353" n="705"/>
+<pb file="add_6786_f353v" o="353v" n="706"/>
+<pb file="add_6786_f354" o="354" n="707"/>
+<pb file="add_6786_f354v" o="354v" n="708"/>
+<pb file="add_6786_f355" o="355" n="709"/>
+<pb file="add_6786_f355v" o="355v" n="710"/>
+<pb file="add_6786_f356" o="356" n="711"/>
+<pb file="add_6786_f356v" o="356v" n="712"/>
+<pb file="add_6786_f357" o="357" n="713"/>
+<pb file="add_6786_f357v" o="357v" n="714"/>
+<pb file="add_6786_f358" o="358" n="715"/>
+<pb file="add_6786_f358v" o="358v" n="716"/>
+<pb file="add_6786_f359" o="359" n="717"/>
+<pb file="add_6786_f359v" o="359v" n="718"/>
+<pb file="add_6786_f360" o="360" n="719"/>
+<pb file="add_6786_f360v" o="360v" n="720"/>
+<pb file="add_6786_f361" o="361" n="721"/>
+<pb file="add_6786_f361v" o="361v" n="722"/>
+<pb file="add_6786_f362" o="362" n="723"/>
+<pb file="add_6786_f362v" o="362v" n="724"/>
+<pb file="add_6786_f363" o="363" n="725"/>
+<pb file="add_6786_f363v" o="363v" n="726"/>
+<pb file="add_6786_f364" o="364" n="727"/>
+<pb file="add_6786_f364v" o="364v" n="728"/>
+<pb file="add_6786_f365" o="365" n="729"/>
+<div xml:id="echoid-div22" type="page_commentary" level="2" n="22">
+<p>
+<s xml:id="echoid-s97" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s97" xml:space="preserve">
+Diagrams representing various combinations of points (.), lines (-), and circles (o).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s99" xml:space="preserve">
+melius
+<lb/>[<emph style="it">tr: 
+better
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f365v" o="365v" n="730"/>
+<pb file="add_6786_f366" o="366" n="731"/>
+<pb file="add_6786_f366v" o="366v" n="732"/>
+<pb file="add_6786_f367" o="367" n="733"/>
+<pb file="add_6786_f367v" o="367v" n="734"/>
+<pb file="add_6786_f368" o="368" n="735"/>
+<pb file="add_6786_f368v" o="368v" n="736"/>
+<pb file="add_6786_f369" o="369" n="737"/>
+<div xml:id="echoid-div23" type="page_commentary" level="2" n="23">
+<p>
+<s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s100" xml:space="preserve">
+A plan for supplying water to a number of areas and buildings,
+including one that appears to be for Harriot's personal use.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s102" xml:space="preserve">
+cestern in the bathing room <lb/>
+cellar <lb/>
+cellar <lb/>
+privye <lb/>
+laundry <lb/>
+privy T. H. <lb/>
+The Stables <lb/>
+a cesperall for emptying the pipes <lb/>
+the plane <lb/>
+The farme <lb/>
+S. F. Darcy of Think <lb/>
+The Spring Seat
+</s>
+</p>
+<pb file="add_6786_f369v" o="369v" n="738"/>
+<pb file="add_6786_f370" o="370" n="739"/>
+<pb file="add_6786_f370v" o="370v" n="740"/>
+<pb file="add_6786_f371" o="371" n="741"/>
+<pb file="add_6786_f371v" o="371v" n="742"/>
+<pb file="add_6786_f372" o="372" n="743"/>
+<pb file="add_6786_f372v" o="372v" n="744"/>
+<pb file="add_6786_f373" o="373" n="745"/>
+<pb file="add_6786_f373v" o="373v" n="746"/>
+<pb file="add_6786_f374" o="374" n="747"/>
+<pb file="add_6786_f374v" o="374v" n="748"/>
+<pb file="add_6786_f375" o="375" n="749"/>
+<pb file="add_6786_f375v" o="375v" n="750"/>
+<div xml:id="echoid-div24" type="page_commentary" level="2" n="24">
+<p>
+<s xml:id="echoid-s103" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s103" xml:space="preserve">
+This large folded sheet (f. 375v–376) is one of the few papers that Harriot dated: 12 December 1591.
+It is concerned with the piling of bullets, for which two questions naturally arise:
+1. Given the number of bullets and the required shape of the base, how many bullets must be placed at each level,
+and how many levels?
+2. Given a pile of a certain shape, how many bullets does it contain?
+<lb/>
+(a) Square and oblong numbers: <lb/>
+The numbers 1, 2, 3, 4, ... down the key diagonal indicate the length of the shorter side of the base,
+and therefore also the height of the pyramid. <lb/>
+The numbers 1, 2, 3, 4, ... along the top indicate the length of the longer side of the base. <lb/>
+Thus for an oblong pyramid on a 3 x 7 base,
+move right from 3 in the key diagonal and down from 7 on the top line.
+The entries in the appropriate square show that there are 21 bullets in the base and 38 in the pyramid as a whole. <lb/>
+In particular, the square pyramidal numbers 1, 5, 14, 30, 55, 91 ...
+appear in the squares immediately to the right of the key diagonal;
+thus a square pyramid with base 3 x 3 has 9 bullets in its base and 14 bullets in total.
+<lb/>
+(b) Triangular numbers: <lb/>
+The triangular pyramidal numbers 1, 4, 10, 20, 35, ...
+appear as the lower entries in the squares below and to the left of the key diagonal,
+to which they are joined by short lines.
+Thus a triangular pyramid with three levels contains 6 bullets in its base and 10 bullets in total.
+As Harriot points out in his note on f. 376, the triangular numbers 1, 3, 6, 10, 15, ...
+may be read off as the differences of the oblong numbers in the previous row.
+Conversely, the entire table can be generated by successive addition from the triangular and square numbers.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s105" xml:space="preserve">
+These are those speciall <lb/>
+groundplats upon the which <lb/>
+may be orderly piled bullets: <lb/>
+The triangle: the square: and <lb/>
+the oblonge.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s106" xml:space="preserve">
+Concerning piling there are two <lb/>
+questions: one <emph style="st">which</emph> the nomber <lb/>
+of bullets to be piled being given <lb/>
+with the forme of the groundplat, <lb/>
+to know how many must be placed <lb/>
+in every rounde, with how many <lb/>
+roundes in the sayd groundplat.
+</s>
+</p>
+<pb file="add_6786_f376" o="376" n="751"/>
+<p>
+<s xml:id="echoid-s107" xml:space="preserve">
+The second a pile being made to knowe <lb/>
+the number of bullets therein conteyned. <lb/>
+For the awnswering of which two questions <lb/>
+this table of some <emph style="st">set downe</emph> calculations
+for <emph style="st">for </emph> the purpose. December. 12. 1591.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s108" xml:space="preserve">
+The forme of the progression of the triangle row <lb/>
+is the excesse of the progression <emph style="st">of the</emph> of the <lb/>
+<emph style="st">square</emph> <emph style="super">oblong</emph> plats next adjoyning above.
+</s>
+<lb/>
+<s xml:id="echoid-s109" xml:space="preserve">
+as. 10 of 30, 40, 50 &amp;c. <lb/>
+or. 15 of 55, 70, 85, &amp;c.
+</s>
+<lb/>
+<s xml:id="echoid-s110" xml:space="preserve">
+Thereby the squares &amp; triangles being knowne <lb/>
+<emph style="st">all</emph> <emph style="super">any of</emph> the oblongs may be knowne by rule.
+</s>
+</p>
+<pb file="add_6786_f376v" o="376v" n="752"/>
+<pb file="add_6786_f377" o="377" n="753"/>
+<div xml:id="echoid-div25" type="page_commentary" level="2" n="25">
+<p>
+<s xml:id="echoid-s111" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s111" xml:space="preserve">
+Arrangements of dots as singletons or pairs in rows of four. <lb/>
+Perhaps a counting system in base 16?
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f377v" o="377v" n="754"/>
+<pb file="add_6786_f378" o="378" n="755"/>
+<div xml:id="echoid-div26" type="page_commentary" level="2" n="26">
+<p>
+<s xml:id="echoid-s113" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s113" xml:space="preserve">
+Arrangements of dots in pattersn similar to those on the previous folio (Add MS 6786, f. 377),
+now also in conjunction with signs of the zodiac.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f378v" o="378v" n="756"/>
+<div xml:id="echoid-div27" type="page_commentary" level="2" n="27">
+<p>
+<s xml:id="echoid-s115" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s115" xml:space="preserve">
+Arrangements of dots in patterns similar to those on the previous folio (Add MS 6786, f. 377),
+now arranged in a circle.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f379" o="379" n="757"/>
+<pb file="add_6786_f379v" o="379v" n="758"/>
+<pb file="add_6786_f380" o="380" n="759"/>
+<pb file="add_6786_f380v" o="380v" n="760"/>
+<pb file="add_6786_f381" o="381" n="761"/>
+<pb file="add_6786_f381v" o="381v" n="762"/>
+<pb file="add_6786_f382" o="382" n="763"/>
+<pb file="add_6786_f382v" o="382v" n="764"/>
+<pb file="add_6786_f383" o="383" n="765"/>
+<pb file="add_6786_f383v" o="383v" n="766"/>
+<pb file="add_6786_f384" o="384" n="767"/>
+<pb file="add_6786_f384v" o="384v" n="768"/>
+<pb file="add_6786_f385" o="385" n="769"/>
+<pb file="add_6786_f385v" o="385v" n="770"/>
+<pb file="add_6786_f386" o="386" n="771"/>
+<pb file="add_6786_f386v" o="386v" n="772"/>
+<pb file="add_6786_f387" o="387" n="773"/>
+<pb file="add_6786_f387v" o="387v" n="774"/>
+<pb file="add_6786_f388" o="388" n="775"/>
+<pb file="add_6786_f388v" o="388v" n="776"/>
+<div xml:id="echoid-div28" type="page_commentary" level="2" n="28">
+<p>
+<s xml:id="echoid-s117" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s117" xml:space="preserve">
+Some examples of Stevin's notation.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f389" o="389" n="777"/>
+<pb file="add_6786_f389v" o="389v" n="778"/>
+<pb file="add_6786_f390" o="390" n="779"/>
+<pb file="add_6786_f390v" o="390v" n="780"/>
+<pb file="add_6786_f391" o="391" n="781"/>
+<pb file="add_6786_f391v" o="391v" n="782"/>
+<pb file="add_6786_f392" o="392" n="783"/>
+<pb file="add_6786_f392v" o="392v" n="784"/>
+<pb file="add_6786_f393" o="393" n="785"/>
+<pb file="add_6786_f393v" o="393v" n="786"/>
+<pb file="add_6786_f394" o="394" n="787"/>
+<div xml:id="echoid-div29" type="page_commentary" level="2" n="29">
+<p>
+<s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s119" xml:space="preserve">
+A set of Napier's bones, and some calculations.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f394v" o="394v" n="788"/>
+<div xml:id="echoid-div30" type="page_commentary" level="2" n="30">
+<p>
+<s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s121" xml:space="preserve">
+Calculations using Napier's bones, continued from overleaf Add MS 6786, f. 394.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f395" o="395" n="789"/>
+<pb file="add_6786_f395v" o="395v" n="790"/>
+<pb file="add_6786_f396" o="396" n="791"/>
+<pb file="add_6786_f396v" o="396v" n="792"/>
+<pb file="add_6786_f397" o="397" n="793"/>
+<pb file="add_6786_f397v" o="397v" n="794"/>
+<pb file="add_6786_f398" o="398" n="795"/>
+<p>
+<s xml:id="echoid-s123" xml:space="preserve">
+<sc>
+One chain is 4 poles. One pole is 5.5 yards, or 198 inches.
+</sc>
+792 inches, 4 poles; the length of the chayne.
+</s>
+<lb/>
+<s xml:id="echoid-s124" xml:space="preserve">
+7920 tenths of an inch; 4 poles: or length of the chayne.
+</s>
+<lb/>
+<s xml:id="echoid-s125" xml:space="preserve">
+792 tenths or: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>7</mn><mn>9</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></mstyle></math> inches or 6 foot 7 inches <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> of inches; is <lb/>
+for one tenth parte of the chayne.
+</s>
+<lb/>
+<s xml:id="echoid-s126" xml:space="preserve">
+one of those partes must be divided into 10 <lb/>
+other partes
+</s>
+</p>
+<pb file="add_6786_f398v" o="398v" n="796"/>
+<pb file="add_6786_f399" o="399" n="797"/>
+<pb file="add_6786_f399v" o="399v" n="798"/>
+<pb file="add_6786_f400" o="400" n="799"/>
+<pb file="add_6786_f400v" o="400v" n="800"/>
+<pb file="add_6786_f401" o="401" n="801"/>
+<div xml:id="echoid-div31" type="page_commentary" level="2" n="31">
+<p>
+<s xml:id="echoid-s127" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s127" xml:space="preserve">
+This page referes to Stevin's <emph style="it">L 19arithmétique … aussi l 19algebre</emph>, page 306.
+There Stevin solves the equation 1(3) = 6(1) + 40 (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>4</mn><mn>0</mn></mstyle></math>),
+leading to the cube roots writen down by Harriot at the top of the page.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s129" xml:space="preserve">
+√C.20 + √392 + √C.20 – √392. Stevin pag. <lb/>
+306.
+</s>
+</p>
+<pb file="add_6786_f401v" o="401v" n="802"/>
+<pb file="add_6786_f402" o="402" n="803"/>
+<pb file="add_6786_f402v" o="402v" n="804"/>
+<pb file="add_6786_f403" o="403" n="805"/>
+<pb file="add_6786_f403v" o="403v" n="806"/>
+<pb file="add_6786_f404" o="404" n="807"/>
+<pb file="add_6786_f404v" o="404v" n="808"/>
+<pb file="add_6786_f405" o="405" n="809"/>
+<p>
+<s xml:id="echoid-s130" xml:space="preserve">
+To set the elements of the doctrine of triangles <lb/>
+as deviding of arke by the sines in <lb/>
+datum rationum. &amp; doubling. <lb/>
+The elements <emph style="super">by</emph> versed sines <lb/>
+ut lat. ad lat. ita. signs aut. ad v. ang. <lb/>
+&amp; the prosthephaeresis. <lb/>
+in specie.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s131" xml:space="preserve">
+To set after the doctrine in species. <lb/>
+&amp; in the forme of æquations, whereby <lb/>
+the rules may be [???]: <lb/>
+The compound numbers only <emph style="super">to be</emph> brought <lb/>
+into one æquation.
+</s>
+</p>
+<pb file="add_6786_f405v" o="405v" n="810"/>
+<pb file="add_6786_f406" o="406" n="811"/>
+<pb file="add_6786_f406v" o="406v" n="812"/>
+<pb file="add_6786_f407" o="407" n="813"/>
+<pb file="add_6786_f407v" o="407v" n="814"/>
+<pb file="add_6786_f408" o="408" n="815"/>
+<div xml:id="echoid-div32" type="page_commentary" level="2" n="32">
+<p>
+<s xml:id="echoid-s132" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s132" xml:space="preserve">
+The reference to Pappus is to Commandino's edition of Books III to VIII,
+<emph style="it">Mathematicae collecitones</emph> (1558).
+The proposition on page 47 is Proposition IV.14.
+</s>
+<lb/>
+<quote xml:lang="lat">
+Theorema XIIII. Propositio XIIII. <lb/>
+Sint duo semicirculi BGC BED: &amp; ipsos contingat circulus EFGH: a cuius centro A ad BC basim semicirculorum
+perpendicularis ducatur AM. Dico ut BM as eam, quæ ex centro circuli EFGH,
+ita esse in prima figura vtramque simul CB BD ad earum excessum CD;
+in secunda vero, &amp; tertia figura, ita esse excessum CB BD ad vtramque ipsarum CB BD.
+</quote>
+<lb/>
+<quote>
+Let there be two semicircles BGC and BED, and their touching circle EFGH, from whose centre A to BC,
+the base of the semicircle, there is drawn the perpendicular AM.
+I say that as BM is to that line from the centre of the circle EFGH,
+inthe first figure will be CB and BD togher to their excess, CD;
+but in the second and third figure, it will be as the excess of CB over BD to both of CB and BD together.
+</quote>
+<lb/>
+<s xml:id="echoid-s133" xml:space="preserve">
+For Harriot's diagrams for this proposition, see Add MS 6784, f. 203 and f. 204;
+this page shows only calculations of ratios.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head9" xml:space="preserve">
+1) Pappus pag. 47
+</head>
+<pb file="add_6786_f408v" o="408v" n="816"/>
+<pb file="add_6786_f409" o="409" n="817"/>
+<pb file="add_6786_f409v" o="409v" n="818"/>
+<pb file="add_6786_f410" o="410" n="819"/>
+<div xml:id="echoid-div33" type="page_commentary" level="2" n="33">
+<p>
+<s xml:id="echoid-s135" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s135" xml:space="preserve">
+Rough working, similar to page 14 of the 'Magisteria' (Add MS 6782, f. 121).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f410v" o="410v" n="820"/>
+<pb file="add_6786_f411" o="411" n="821"/>
+<pb file="add_6786_f411v" o="411v" n="822"/>
+<pb file="add_6786_f412" o="412" n="823"/>
+<pb file="add_6786_f412v" o="412v" n="824"/>
+<pb file="add_6786_f413" o="413" n="825"/>
+<pb file="add_6786_f413v" o="413v" n="826"/>
+<pb file="add_6786_f414" o="414" n="827"/>
+<pb file="add_6786_f414v" o="414v" n="828"/>
+<pb file="add_6786_f415" o="415" n="829"/>
+<c>
+Work on eliminating the linear term from an equation of degree 5.
+</c>
+<head xml:id="echoid-head10" xml:space="preserve">
+bb<emph style="super">2</emph>
+</head>
+<pb file="add_6786_f415v" o="415v" n="830"/>
+<pb file="add_6786_f416" o="416" n="831"/>
+<pb file="add_6786_f416v" o="416v" n="832"/>
+<pb file="add_6786_f417" o="417" n="833"/>
+<div xml:id="echoid-div34" type="page_commentary" level="2" n="34">
+<p>
+<s xml:id="echoid-s137" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s137" xml:space="preserve">
+The numbers in the table at the top of the page are, from left to right:
+units, lengths, triangular numbers, triangular-pyramidal numbers, and so on. <lb/>
+The table in the centre of the page repeats the first and third columns from the table at the top,
+but now the triangular numbers are written in multiplicative form, thus,
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>=</mo><mfrac><mrow><mn>3</mn><mo>×</mo><mn>4</mn></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mstyle></math>. <lb/>
+The third table gives all the rows in multiplicative form, thus in the seventh row,
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>4</mn><mo>=</mo><mfrac><mrow><mn>7</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>9</mn></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mrow></mfrac></mstyle></math>. <lb/>
+The final row gives general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th row. <lb/>
+See also page 1 of the 'Magisteria' (Add MS 6782, f. 108), for which this is possibly a draft.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat">
+De numeris triangularibus.
+</head>
+<pb file="add_6786_f417v" o="417v" n="834"/>
+<pb file="add_6786_f418" o="418" n="835"/>
+<pb file="add_6786_f418v" o="418v" n="836"/>
+<pb file="add_6786_f419" o="419" n="837"/>
+<pb file="add_6786_f419v" o="419v" n="838"/>
+<pb file="add_6786_f420" o="420" n="839"/>
+<pb file="add_6786_f420v" o="420v" n="840"/>
+<div xml:id="echoid-div35" type="page_commentary" level="2" n="35">
+<p>
+<s xml:id="echoid-s139" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s139" xml:space="preserve">
+Calulation of the number of days and hours in
+(i) 235 lunar months; (ii) 19 solar years; (iii) Julian years,
+and the differences between them.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s141" xml:space="preserve">
+235. Menses.
+<lb/>[<emph style="it">tr: 
+235 lunar months.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s142" xml:space="preserve">
+19. Solares.
+<lb/>[<emph style="it">tr: 
+19 solar years.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s143" xml:space="preserve">
+19. Juliani.
+<lb/>[<emph style="it">tr: 
+19 Julian years.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f421" o="421" n="841"/>
+<div xml:id="echoid-div36" type="page_commentary" level="2" n="36">
+<p>
+<s xml:id="echoid-s144" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s144" xml:space="preserve">
+Further calculations relating lunar months to solar years.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f421v" o="421v" n="842"/>
+<pb file="add_6786_f422" o="422" n="843"/>
+<head xml:id="echoid-head12" xml:space="preserve">
+Mercators map
+</head>
+<p>
+<s xml:id="echoid-s146" xml:space="preserve">
+Babilon
+</s>
+<lb/>
+<s xml:id="echoid-s147" xml:space="preserve">
+Alexandria
+</s>
+<lb/>
+<s xml:id="echoid-s148" xml:space="preserve">
+Jerusalem
+</s>
+</p>
+<p>
+<s xml:id="echoid-s149" xml:space="preserve">
+London
+</s>
+<lb/>
+<s xml:id="echoid-s150" xml:space="preserve">
+Huena
+</s>
+<lb/>
+<s xml:id="echoid-s151" xml:space="preserve">
+Venetia
+</s>
+<lb/>
+<s xml:id="echoid-s152" xml:space="preserve">
+Praga
+</s>
+<lb/>
+<s xml:id="echoid-s153" xml:space="preserve">
+Roma
+</s>
+</p>
+<p>
+<s xml:id="echoid-s154" xml:space="preserve">
+Ptol: geogr:
+<lb/>[<emph style="it">tr: 
+Prolemy's geography
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s155" xml:space="preserve">
+Hierosolyma. long.
+<lb/>[<emph style="it">tr: 
+Longitude of Jersualem
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s156" xml:space="preserve">
+Alexandria. long.
+</s>
+<lb/>
+<s xml:id="echoid-s157" xml:space="preserve">
+Babilon. long.
+</s>
+<lb/>
+<s xml:id="echoid-s158" xml:space="preserve">
+Roma. long.
+</s>
+<lb/>
+<s xml:id="echoid-s159" xml:space="preserve">
+Londinium. long.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s160" xml:space="preserve">
+Huena <lb/>
+ab Alex:
+</s>
+</p>
+<p>
+<s xml:id="echoid-s161" xml:space="preserve">
+Huena ab <lb/>
+Alexand.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s162" xml:space="preserve">
+Huena <lb/>
+ab Alex.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s163" xml:space="preserve">
+Huena a Londino
+</s>
+</p>
+<p>
+<s xml:id="echoid-s164" xml:space="preserve">
+Huena <lb/>
+ab Alex:
+</s>
+</p>
+<p>
+<s xml:id="echoid-s165" xml:space="preserve">
+Tycho habet 1.35. diff. long <lb/>
+Huenæ ab Alex.
+<lb/>[<emph style="it">tr: 
+Tycho has 1.35. as the difference in longitude betweeh Huena and Alexandria.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f422v" o="422v" n="844"/>
+<pb file="add_6786_f423" o="423" n="845"/>
+<div xml:id="echoid-div37" type="page_commentary" level="2" n="37">
+<p>
+<s xml:id="echoid-s166" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s166" xml:space="preserve">
+Calculations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>7</mn><mo>.</mo><mn>8</mn><mn>6</mn><mn>9</mn><mn>6</mn><mn>0</mn><mn>4</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><msqrt><mrow><mo maxsize="1">(</mo><mi>π</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>-</mo><mn>2</mn></mrow></msqrt></mrow></msqrt></mstyle></math>
+and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mo>.</mo><mn>8</mn><mn>6</mn><mn>9</mn><mn>6</mn><mn>0</mn><mn>4</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><msqrt><mrow><mo maxsize="1">(</mo><mi>π</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>-</mo><mn>4</mn></mrow></msqrt></mrow></msqrt></mstyle></math>
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f423v" o="423v" n="846"/>
+<div xml:id="echoid-div38" type="page_commentary" level="2" n="38">
+<p>
+<s xml:id="echoid-s168" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s168" xml:space="preserve">
+Calculations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mo>.</mo><mn>9</mn><mn>3</mn><mn>4</mn><mn>8</mn><mn>0</mn><mn>2</mn><mn>2</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><msqrt><mrow><mfrac><mrow><mo maxsize="1">(</mo><mi>π</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt></mrow></msqrt></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f424" o="424" n="847"/>
+<div xml:id="echoid-div39" type="page_commentary" level="2" n="39">
+<p>
+<s xml:id="echoid-s170" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s170" xml:space="preserve">
+Note that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mo>.</mo><mn>8</mn><mn>6</mn><mn>9</mn><mn>6</mn><mn>0</mn><mn>4</mn><mn>4</mn><mo>=</mo><mo maxsize="1">(</mo><mi>π</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f424v" o="424v" n="848"/>
+<div xml:id="echoid-div40" type="page_commentary" level="2" n="40">
+<p>
+<s xml:id="echoid-s172" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s172" xml:space="preserve">
+Calculations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mo>.</mo><mn>0</mn><mn>2</mn><mn>1</mn><mn>9</mn><mn>2</mn><mo>.</mo><mo>.</mo><mo>.</mo></mrow></msqrt></mstyle></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn><mn>5</mn><mo>.</mo><mn>9</mn><mn>3</mn><mn>5</mn><mn>2</mn><mn>7</mn><mo>.</mo><mo>.</mo><mo>.</mo></mrow></msqrt></mstyle></math>.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f425" o="425" n="849"/>
+<pb file="add_6786_f425v" o="425v" n="850"/>
+<pb file="add_6786_f426" o="426" n="851"/>
+<pb file="add_6786_f426v" o="426v" n="852"/>
+<pb file="add_6786_f427" o="427" n="853"/>
+<pb file="add_6786_f427v" o="427v" n="854"/>
+<pb file="add_6786_f428" o="428" n="855"/>
+<pb file="add_6786_f428v" o="428v" n="856"/>
+<pb file="add_6786_f429" o="429" n="857"/>
+<p>
+<s xml:id="echoid-s174" xml:space="preserve">
+These wordes are written <lb/>
+with China inke. <lb/>
+so likewise are these: <lb/>
+But these are written with London inke.
+</s>
+</p>
+<p>
+<s xml:id="echoid-s175" xml:space="preserve">
+With China inke these wordes. <lb/>
+With China inke these wordes. <lb/>
+were written with thick <lb/>
+inke &amp;c.
+</s>
+</p>
+<pb file="add_6786_f429v" o="429v" n="858"/>
+<pb file="add_6786_f430" o="430" n="859"/>
+<pb file="add_6786_f430v" o="430v" n="860"/>
+<pb file="add_6786_f431" o="431" n="861"/>
+<pb file="add_6786_f431v" o="431v" n="862"/>
+<pb file="add_6786_f432" o="432" n="863"/>
+<pb file="add_6786_f432v" o="432v" n="864"/>
+<pb file="add_6786_f433" o="433" n="865"/>
+<pb file="add_6786_f433v" o="433v" n="866"/>
+<pb file="add_6786_f434" o="434" n="867"/>
+<pb file="add_6786_f434v" o="434v" n="868"/>
+<pb file="add_6786_f435" o="435" n="869"/>
+<pb file="add_6786_f435v" o="435v" n="870"/>
+<pb file="add_6786_f436" o="436" n="871"/>
+<pb file="add_6786_f436v" o="436v" n="872"/>
+<pb file="add_6786_f437" o="437" n="873"/>
+<pb file="add_6786_f437v" o="437v" n="874"/>
+<pb file="add_6786_f438" o="438" n="875"/>
+<pb file="add_6786_f438v" o="438v" n="876"/>
+<pb file="add_6786_f439" o="439" n="877"/>
+<pb file="add_6786_f439v" o="439v" n="878"/>
+<pb file="add_6786_f440" o="440" n="879"/>
+<div xml:id="echoid-div41" type="page_commentary" level="2" n="41">
+<p>
+<s xml:id="echoid-s176" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s176" xml:space="preserve">This sheet contains Harriot's exploration of Proposition 2, Lemma 4, from the
+<emph style="it">Sphaerica Menelai</emph> of Menelaus, as translated by Maurolico (1558).
+The text includes a supplement by Thabit ibn Qurra (see Add MS 6787, f. 306v to f. 308 and f. 309v). <lb/>
+For Harriot's copy of Propositions 1 and 2 see Add MS 6787, f. 306 to f. 309v.
+The pages relevant to Proposition 2, Lemma 4, are f. 307, f. 307v, and f. 308.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s178" xml:space="preserve">
+Si fuerint sex quantitates, quorum ratio primæ ad secundum: componatur ex rationibus, <lb/>
+tertia ad quartam, et quintæ ad sextam. Tum solidum sub prima quarta et sexta <lb/>
+contentum, æquale erit solido sub secundo tertia et quinta comprehenso.
+<lb/>[<emph style="it">tr: 
+If there are six quantities, where the ratio of the first to the second is composed of
+the ratio of the third to the fourth and of the fifth to the sixth,
+then the solid content of the first, fourth, and sixth will be equal to
+the solid composed of the second, third, and fifth.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s179" xml:space="preserve">
+Sex quantitates.
+<lb/>[<emph style="it">tr: 
+The six quantities
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s180" xml:space="preserve">
+Ergo solidum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>g</mi><mi>e</mi></mstyle></math>.
+<lb/>[<emph style="it">tr: 
+Therefore the solid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>g</mi><mi>e</mi></mstyle></math>.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s181" xml:space="preserve">
+Varia transpositio utrinque <emph style="st">solilidi</emph> solidi terminorum.
+<lb/>[<emph style="it">tr: 
+Various transpositions of the terms of both solids.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s182" xml:space="preserve">
+Habitudines æqualium solidorum. <lb/>[<emph style="it">tr: 
+Forms of equal solids
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s183" xml:space="preserve">
+Thebit ben Corat. <lb/>
+et Ma<emph style="super">u</emph>rolicus.
+<lb/>[<emph style="it">tr: 
+Thabit ibn Qurra and Maurolico.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s184" xml:space="preserve">
+compositiones
+<lb/>[<emph style="it">tr: 
+compositions
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s185" xml:space="preserve">
+componentes rationes transpositæ
+<lb/>[<emph style="it">tr: 
+components of the transposed ratios
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s186" xml:space="preserve">
+Si consequentes fiant antecedentes <lb/>
+et antecedentes consequentes; sunt alia <lb/>
+36 compositiones.
+<lb/>[<emph style="it">tr: 
+If the consequents become antecedents and antecedents consequents, there are another 36 compositions.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f440v" o="440v" n="880"/>
+<pb file="add_6786_f441" o="441" n="881"/>
+<head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat">
+In ptolomaicam propositione
+<lb/>[<emph style="it">tr: 
+On a proposition of Ptolemy
+</emph>]<lb/>
+</head>
+<pb file="add_6786_f441v" o="441v" n="882"/>
+<pb file="add_6786_f442" o="442" n="883"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s187" xml:space="preserve">
+Data magnitudine
+<lb/>[<emph style="it">tr: 
+given magnitude
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s188" xml:space="preserve">
+Data ratione
+<lb/>[<emph style="it">tr: 
+Gien ratio
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s189" xml:space="preserve">
+Quæruntur magnitudines
+<lb/>[<emph style="it">tr: 
+Magnitudes sought
+</emph>]<lb/>
+</s>
+<lb/>
+<s xml:id="echoid-s190" xml:space="preserve">
+vel cuiuslibet alterius
+<lb/>[<emph style="it">tr: 
+or any others one wishes
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f442v" o="442v" n="884"/>
+<pb file="add_6786_f443" o="443" n="885"/>
+<pb file="add_6786_f443v" o="443v" n="886"/>
+<div xml:id="echoid-div42" type="page_commentary" level="2" n="42">
+<p>
+<s xml:id="echoid-s191" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s191" xml:space="preserve">For the proposition referred to here, see Add MS 6785, f. 179.
+See also Add MS 6786, f. 441 to f. 444.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head14" xml:space="preserve" xml:lang="lat">
+In prop. ptolomaica initio Finkij
+<lb/>[<emph style="it">tr: 
+On a proposition of Ptolemy from the beginning of Finck
+</emph>]<lb/>
+</head>
+<pb file="add_6786_f444" o="444" n="887"/>
+<pb file="add_6786_f444v" o="444v" n="888"/>
+<pb file="add_6786_f445" o="445" n="889"/>
+<pb file="add_6786_f445v" o="445v" n="890"/>
+<pb file="add_6786_f446" o="446" n="891"/>
+<pb file="add_6786_f446v" o="446v" n="892"/>
+<pb file="add_6786_f447" o="447" n="893"/>
+<pb file="add_6786_f447v" o="447v" n="894"/>
+<pb file="add_6786_f448" o="448" n="895"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s193" xml:space="preserve">
+Ex Stigelij lib. de circulis <lb/>
+calistibus. et Doctrina <lb/>
+[???]
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s194" xml:space="preserve">
+nostra
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s195" xml:space="preserve">
+invertibus
+</s>
+</p>
+<pb file="add_6786_f448v" o="448v" n="896"/>
+<pb file="add_6786_f449" o="449" n="897"/>
+<pb file="add_6786_f449v" o="449v" n="898"/>
+<pb file="add_6786_f450" o="450" n="899"/>
+<div xml:id="echoid-div43" type="page_commentary" level="2" n="43">
+<p>
+<s xml:id="echoid-s196" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s196" xml:space="preserve">
+One of several pages containing a mnemonic for the digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>π</mi></mstyle></math>,
+by making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mi>a</mi></mstyle></math>, 2 = b <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo></mstyle></math> 3 = c <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>a</mi><mi>n</mi><mi>d</mi><mi>s</mi><mi>o</mi><mi>o</mi><mi>n</mi><mo>.</mo></mstyle></math></s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s198" xml:space="preserve">
+cadaeibf <lb/>
+31415926
+</s>
+</p>
+<pb file="add_6786_f450v" o="450v" n="900"/>
+<pb file="add_6786_f451" o="451" n="901"/>
+<pb file="add_6786_f451v" o="451v" n="902"/>
+<pb file="add_6786_f452" o="452" n="903"/>
+<pb file="add_6786_f452v" o="452v" n="904"/>
+<pb file="add_6786_f453" o="453" n="905"/>
+<pb file="add_6786_f453v" o="453v" n="906"/>
+<div xml:id="echoid-div44" type="page_commentary" level="2" n="44">
+<p>
+<s xml:id="echoid-s199" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s199" xml:space="preserve">
+The reference on this page is to Viète's
+<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 17.
+</s>
+<lb/>
+<quote xml:lang="lat">
+Theorema. <lb/>
+Si fuerint magnitudines continue proportionales: erit ut terminus rationis major ad terminum rationis minorem,
+ita differentia compositæ ex ombinbus &amp; minimæ ad differentiam compositæ ex omnibus &amp; maximæ.
+</quote>
+<lb/>
+<quote>
+If there are magnitudes in continued proportion, then the ratio of the greatest term to the least will be
+as the difference between the sum of all and the minimum to the difference between the sum of all and the maximum.
+</quote>
+<lb/>
+<s xml:id="echoid-s200" xml:space="preserve">
+Harriot's notation here is the same as Viète's.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat">
+In Cap. 17. Resp. pag. 29.
+Vieta
+<lb/>[<emph style="it">tr: 
+From Chapter 17, Responsorum, page 29, Viète
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s202" xml:space="preserve">
+Si fuerint magnitudines continue proportionales: Erit ut terminus <lb/>
+rationis major ad terminum rationis minorem; ita differentia <lb/>
+compositæ ex ombinbus et minimæ ad differentiam compositæ ex <lb/>
+omnibus et maximæ.
+<lb/>[<emph style="it">tr: 
+If there are magnitudes in continued proportion, then the ratio of the greatest term to the least will be
+as the difference between the sum of all and the minimum to the difference between the sum of all and the maximum.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s203" xml:space="preserve">
+Sit maxima. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. <lb/>
+minima. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi></mstyle></math>. <lb/>
+composita ex omnibus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> <lb/>
+ratio Maioris ad minore. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>.
+<lb/>[<emph style="it">tr: 
+Let the greatest be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <lb/>
+the least <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi></mstyle></math>, <lb/>
+the sum of all <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>, <lb/>
+the ratio of a greater to the lesser <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mo>:</mo><mi>B</mi></mstyle></math>.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s204" xml:space="preserve">
+Alius sic: si fuerint magnitudes continue proportionales: <lb/>
+ut prima ad secundam: ita omnes antecedentes ad omnes consequentes.
+<lb/>[<emph style="it">tr: 
+Otherwise thusL if there are magnitudes in continued proportion, as the first is to the second,
+so are all the antecedents to all the consequents.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f454" o="454" n="907"/>
+<pb file="add_6786_f454v" o="454v" n="908"/>
+<pb file="add_6786_f455" o="455" n="909"/>
+<p>
+<s xml:id="echoid-s205" xml:space="preserve">
+12£ sterling. = 1£ gold angel
+</s>
+</p>
+<pb file="add_6786_f455v" o="455v" n="910"/>
+<pb file="add_6786_f456" o="456" n="911"/>
+<pb file="add_6786_f456v" o="456v" n="912"/>
+<pb file="add_6786_f457" o="457" n="913"/>
+<div xml:id="echoid-div45" type="page_commentary" level="2" n="45">
+<p>
+<s xml:id="echoid-s206" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s206" xml:space="preserve">
+Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math>.
+Each power is calculated from the previous one by multiplication. <lb/>
+Note the use of cossist notation:
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for a first power, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> for a square, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> for a cube, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>z</mi></mstyle></math> for a square-suare or fourth power,
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ß</mo></mstyle></math> for a sursolid or fifth power. <lb/>
+Below the main table is a list of the final sums, including the sixth power (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>c</mi></mstyle></math>),
+which has not been calculated on this page but which can be deduced from the pattern for the previous cases. <lb/>
+For a similar table see Add MS 6782, f. 276. <lb/>
+This table underpins the method of root extraction taught by Francois Viete
+in <emph style="it">De numerosa potestatum resolutione</emph> (1600).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s208" xml:space="preserve">
+species figuratorum <lb/>
+ex binomia radice
+<lb/>[<emph style="it">tr: 
+terms of figurate numbers from binomial roots
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s209" xml:space="preserve">
+(sunt canones pro <lb/>
+extractione radicum.)
+<lb/>[<emph style="it">tr: 
+(these are the canons for the extraction of roots.)
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s210" xml:space="preserve">
+(Demptis numeris sunt species <lb/>
+continue proportionalium, <lb/>
+in minimis terminis <lb/>
+ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.)
+<lb/>[<emph style="it">tr: 
+(Without the numbers, the terms are in continual proportion, in the ratio expressed in lowest terms as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>).
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f457v" o="457v" n="914"/>
+<pb file="add_6786_f458" o="458" n="915"/>
+<p xml:lang="lat">
+<s xml:id="echoid-s211" xml:space="preserve">
+vale vale cane ne titules mandataque frangas
+</s>
+</p>
+<pb file="add_6786_f458v" o="458v" n="916"/>
+<pb file="add_6786_f459" o="459" n="917"/>
+<pb file="add_6786_f459v" o="459v" n="918"/>
+<pb file="add_6786_f460" o="460" n="919"/>
+<pb file="add_6786_f460v" o="460v" n="920"/>
+<pb file="add_6786_f461" o="461" n="921"/>
+<pb file="add_6786_f461v" o="461v" n="922"/>
+<pb file="add_6786_f462" o="462" n="923"/>
+<pb file="add_6786_f462v" o="462v" n="924"/>
+<pb file="add_6786_f463" o="463" n="925"/>
+<pb file="add_6786_f463v" o="463v" n="926"/>
+<pb file="add_6786_f464" o="464" n="927"/>
+<pb file="add_6786_f464v" o="464v" n="928"/>
+<pb file="add_6786_f465" o="465" n="929"/>
+<pb file="add_6786_f465v" o="465v" n="930"/>
+<pb file="add_6786_f466" o="466" n="931"/>
+<pb file="add_6786_f466v" o="466v" n="932"/>
+<pb file="add_6786_f467" o="467" n="933"/>
+<pb file="add_6786_f467v" o="467v" n="934"/>
+<pb file="add_6786_f468" o="468" n="935"/>
+<pb file="add_6786_f468v" o="468v" n="936"/>
+<pb file="add_6786_f469" o="469" n="937"/>
+<pb file="add_6786_f469v" o="469v" n="938"/>
+<pb file="add_6786_f470" o="470" n="939"/>
+<pb file="add_6786_f470v" o="470v" n="940"/>
+<pb file="add_6786_f471" o="471" n="941"/>
+<pb file="add_6786_f471v" o="471v" n="942"/>
+<pb file="add_6786_f472" o="472" n="943"/>
+<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat">
+Ratio æqualitatis in infinitum.
+<lb/>[<emph style="it">tr: 
+A ratio of equality at infinity.
+</emph>]<lb/>
+</head>
+<p>
+<s xml:id="echoid-s212" xml:space="preserve">
+A cipher being the last of this lower progression what wilbe <lb/>
+over it, of the hyer, proceeding &amp; continuing, as it is <lb/>
+before?
+</s>
+</p>
+<pb file="add_6786_f472v" o="472v" n="944"/>
+<pb file="add_6786_f473" o="473" n="945"/>
+<pb file="add_6786_f473v" o="473v" n="946"/>
+<pb file="add_6786_f474" o="474" n="947"/>
+<pb file="add_6786_f474v" o="474v" n="948"/>
+<pb file="add_6786_f475" o="475" n="949"/>
+<pb file="add_6786_f475v" o="475v" n="950"/>
+<pb file="add_6786_f476" o="476" n="951"/>
+<pb file="add_6786_f476v" o="476v" n="952"/>
+<pb file="add_6786_f477" o="477" n="953"/>
+<pb file="add_6786_f477v" o="477v" n="954"/>
+<pb file="add_6786_f478" o="478" n="955"/>
+<pb file="add_6786_f478v" o="478v" n="956"/>
+<pb file="add_6786_f479" o="479" n="957"/>
+<p>
+<s xml:id="echoid-s213" xml:space="preserve">
+William Yates William Yates. Christopher Tooke.
+</s>
+</p>
+<pb file="add_6786_f479v" o="479v" n="958"/>
+<pb file="add_6786_f480" o="480" n="959"/>
+<pb file="add_6786_f480v" o="480v" n="960"/>
+<pb file="add_6786_f481" o="481" n="961"/>
+<pb file="add_6786_f481v" o="481v" n="962"/>
+<pb file="add_6786_f482" o="482" n="963"/>
+<pb file="add_6786_f482v" o="482v" n="964"/>
+<pb file="add_6786_f483" o="483" n="965"/>
+<pb file="add_6786_f483v" o="483v" n="966"/>
+<pb file="add_6786_f484" o="484" n="967"/>
+<pb file="add_6786_f484v" o="484v" n="968"/>
+<pb file="add_6786_f485" o="485" n="969"/>
+<head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat">
+Ex bin: med: 4.
+<lb/>[<emph style="it">tr: 
+From a fourth bimedial.
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s214" xml:space="preserve">
+bin. 6
+<lb/>[<emph style="it">tr: 
+A sixth binome.
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s215" xml:space="preserve">
+bin. 5 ex bin: med. 1.
+<lb/>[<emph style="it">tr: 
+A fifth binome from a first bimedial.
+</emph>]<lb/>
+</s>
+</p>
+<head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat">
+Ex bin: med: 5.
+<lb/>[<emph style="it">tr: 
+From fifth bimedials.
+</emph>]<lb/>
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s216" xml:space="preserve">
+bin. 5 ex bin: med. 1.
+<lb/>[<emph style="it">tr: 
+A fifth binome from a first bimedial.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f485v" o="485v" n="970"/>
+<pb file="add_6786_f486" o="486" n="971"/>
+<pb file="add_6786_f486v" o="486v" n="972"/>
+<pb file="add_6786_f487" o="487" n="973"/>
+<pb file="add_6786_f487v" o="487v" n="974"/>
+<pb file="add_6786_f488" o="488" n="975"/>
+<pb file="add_6786_f488v" o="488v" n="976"/>
+<pb file="add_6786_f489" o="489" n="977"/>
+<pb file="add_6786_f489v" o="489v" n="978"/>
+<pb file="add_6786_f490" o="490" n="979"/>
+<pb file="add_6786_f490v" o="490v" n="980"/>
+<div xml:id="echoid-div46" type="page_commentary" level="2" n="46">
+<p>
+<s xml:id="echoid-s217" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s217" xml:space="preserve">
+The author of this fragment is unkown.
+However, the same form of address was used by Harriot's friend Nathaniel Trporley, see Add MS 67888, f. 117.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p>
+<s xml:id="echoid-s219" xml:space="preserve">
+TO HIS VERY LOVING FRENDE <lb/>
+MASTER THOMAS HARRIOT <lb/>
+GEVE THESE AT SION.
+</s>
+</p>
+<pb file="add_6786_f491" o="491" n="981"/>
+<pb file="add_6786_f491v" o="491v" n="982"/>
+<pb file="add_6786_f492" o="492" n="983"/>
+<pb file="add_6786_f492v" o="492v" n="984"/>
+<pb file="add_6786_f493" o="493" n="985"/>
+<pb file="add_6786_f493v" o="493v" n="986"/>
+<pb file="add_6786_f494" o="494" n="987"/>
+<pb file="add_6786_f494v" o="494v" n="988"/>
+<pb file="add_6786_f495" o="495" n="989"/>
+<pb file="add_6786_f495v" o="495v" n="990"/>
+<pb file="add_6786_f496" o="496" n="991"/>
+<pb file="add_6786_f496v" o="496v" n="992"/>
+<pb file="add_6786_f497" o="497" n="993"/>
+<pb file="add_6786_f497v" o="497v" n="994"/>
+<pb file="add_6786_f498" o="498" n="995"/>
+<pb file="add_6786_f498v" o="498v" n="996"/>
+<pb file="add_6786_f499" o="499" n="997"/>
+<pb file="add_6786_f499v" o="499v" n="998"/>
+<pb file="add_6786_f500" o="500" n="999"/>
+<pb file="add_6786_f500v" o="500v" n="1000"/>
+<pb file="add_6786_f501" o="501" n="1001"/>
+<pb file="add_6786_f501v" o="501v" n="1002"/>
+<pb file="add_6786_f502" o="502" n="1003"/>
+<pb file="add_6786_f502v" o="502v" n="1004"/>
+<pb file="add_6786_f503" o="503" n="1005"/>
+<pb file="add_6786_f503v" o="503v" n="1006"/>
+<pb file="add_6786_f504" o="504" n="1007"/>
+<pb file="add_6786_f504v" o="504v" n="1008"/>
+<pb file="add_6786_f505" o="505" n="1009"/>
+<pb file="add_6786_f505v" o="505v" n="1010"/>
+<pb file="add_6786_f506" o="506" n="1011"/>
+<pb file="add_6786_f506v" o="506v" n="1012"/>
+<pb file="add_6786_f507" o="507" n="1013"/>
+<pb file="add_6786_f507v" o="507v" n="1014"/>
+<pb file="add_6786_f508" o="508" n="1015"/>
+<pb file="add_6786_f508v" o="508v" n="1016"/>
+<c>
+See also Add MS 6783, f. 174 (d.10) where the same quadratic equation is solved for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
+</c>
+<head xml:id="echoid-head19" xml:space="preserve">
+as<emph style="super">2</emph>)
+</head>
+<p xml:lang="lat">
+<s xml:id="echoid-s220" xml:space="preserve">
+ad expurgandam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>
+<lb/>[<emph style="it">tr: 
+for the removal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f509" o="509" n="1017"/>
+<pb file="add_6786_f509v" o="509v" n="1018"/>
+<pb file="add_6786_f510" o="510" n="1019"/>
+<div xml:id="echoid-div47" type="page_commentary" level="2" n="47">
+<p>
+<s xml:id="echoid-s221" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s221" xml:space="preserve">
+Examples of arithmetic, geometric, and harmonic porgressions, and rules for their general terms.
+(The terms of a harmonic progression are the reciprocals of an arithmetic porgression.)
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s223" xml:space="preserve">
+<reg norm="Arithmetica" type="abbr">Arith</reg>. 3. 6. 9.
+</s>
+<lb/>
+<s xml:id="echoid-s224" xml:space="preserve">
+<reg norm="Geometrica" type="abbr">Geo</reg>. 3. 6. 12.
+</s>
+<lb/>
+<s xml:id="echoid-s225" xml:space="preserve">
+<reg norm="Harmonica" type="abbr">Har</reg>. 6. 3. 2.
+</s>
+</p>
+<pb file="add_6786_f510v" o="510v" n="1020"/>
+<pb file="add_6786_f511" o="511" n="1021"/>
+<pb file="add_6786_f511v" o="511v" n="1022"/>
+<pb file="add_6786_f512" o="512" n="1023"/>
+<pb file="add_6786_f512v" o="512v" n="1024"/>
+<pb file="add_6786_f513" o="513" n="1025"/>
+<pb file="add_6786_f513v" o="513v" n="1026"/>
+<pb file="add_6786_f514" o="514" n="1027"/>
+<pb file="add_6786_f514v" o="514v" n="1028"/>
+<pb file="add_6786_f515" o="515" n="1029"/>
+<pb file="add_6786_f515v" o="515v" n="1030"/>
+<pb file="add_6786_f516" o="516" n="1031"/>
+<pb file="add_6786_f516v" o="516v" n="1032"/>
+<div xml:id="echoid-div48" type="page_commentary" level="2" n="48">
+<p>
+<s xml:id="echoid-s226" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s226" xml:space="preserve">
+This page contains jottings on both binary and ternary arithmetic. <lb/>
+The calculation on the left shows 101101 (= 45) multiplied by 1011 (= 11);
+the answer, 111,101,111 (= 495), is then divided by 1011 (= 11) to recover 101101 (= 45).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<p xml:lang="lat">
+<s xml:id="echoid-s228" xml:space="preserve">
+ternaria <lb/>
+<lb/>[<emph style="it">tr: 
+ternary
+</emph>]<lb/>
+</s>
+</p>
+<p xml:lang="lat">
+<s xml:id="echoid-s229" xml:space="preserve">
+binaria <lb/>
+<lb/>[<emph style="it">tr: 
+binary
+</emph>]<lb/>
+</s>
+</p>
+<pb file="add_6786_f517" o="517" n="1033"/>
+<pb file="add_6786_f517v" o="517v" n="1034"/>
+<pb file="add_6786_f518" o="518" n="1035"/>
+<div xml:id="echoid-div49" type="page_commentary" level="2" n="49">
+<p>
+<s xml:id="echoid-s230" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s230" xml:space="preserve">
+The beginning of a page 34.3 for the 'Magisteria',
+not used in the final version which has only pages 34.1 and 34.2 (see Add MS 6782, f. 141 and f. 142).
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f518v" o="518v" n="1036"/>
+<pb file="add_6786_f519" o="519" n="1037"/>
+<pb file="add_6786_f519v" o="519v" n="1038"/>
+<pb file="add_6786_f520" o="520" n="1039"/>
+<pb file="add_6786_f520v" o="520v" n="1040"/>
+<pb file="add_6786_f521" o="521" n="1041"/>
+<pb file="add_6786_f521v" o="521v" n="1042"/>
+<pb file="add_6786_f522" o="522" n="1043"/>
+<pb file="add_6786_f522v" o="522v" n="1044"/>
+<pb file="add_6786_f523" o="523" n="1045"/>
+<pb file="add_6786_f523v" o="523v" n="1046"/>
+<pb file="add_6786_f524" o="524" n="1047"/>
+<pb file="add_6786_f524v" o="524v" n="1048"/>
+<pb file="add_6786_f525" o="525" n="1049"/>
+<pb file="add_6786_f525v" o="525v" n="1050"/>
+<pb file="add_6786_f526" o="526" n="1051"/>
+<pb file="add_6786_f526v" o="526v" n="1052"/>
+<pb file="add_6786_f527" o="527" n="1053"/>
+<pb file="add_6786_f527v" o="527v" n="1054"/>
+<pb file="add_6786_f528" o="528" n="1055"/>
+<pb file="add_6786_f528v" o="528v" n="1056"/>
+<pb file="add_6786_f529" o="529" n="1057"/>
+<pb file="add_6786_f529v" o="529v" n="1058"/>
+<pb file="add_6786_f530" o="530" n="1059"/>
+<pb file="add_6786_f530v" o="530v" n="1060"/>
+<pb file="add_6786_f531" o="531" n="1061"/>
+<pb file="add_6786_f531v" o="531v" n="1062"/>
+<pb file="add_6786_f532" o="532" n="1063"/>
+<pb file="add_6786_f532v" o="532v" n="1064"/>
+<pb file="add_6786_f533" o="533" n="1065"/>
+<pb file="add_6786_f533v" o="533v" n="1066"/>
+<pb file="add_6786_f534" o="534" n="1067"/>
+<pb file="add_6786_f534v" o="534v" n="1068"/>
+<pb file="add_6786_f535" o="535" n="1069"/>
+<pb file="add_6786_f535v" o="535v" n="1070"/>
+<pb file="add_6786_f536" o="536" n="1071"/>
+<pb file="add_6786_f536v" o="536v" n="1072"/>
+<pb file="add_6786_f537" o="537" n="1073"/>
+<pb file="add_6786_f537v" o="537v" n="1074"/>
+<pb file="add_6786_f538" o="538" n="1075"/>
+<pb file="add_6786_f538v" o="538v" n="1076"/>
+<pb file="add_6786_f539" o="539" n="1077"/>
+<pb file="add_6786_f539v" o="539v" n="1078"/>
+<pb file="add_6786_f540" o="540" n="1079"/>
+<pb file="add_6786_f540v" o="540v" n="1080"/>
+<pb file="add_6786_f541" o="541" n="1081"/>
+<pb file="add_6786_f541v" o="541v" n="1082"/>
+<pb file="add_6786_f542" o="542" n="1083"/>
+<pb file="add_6786_f542v" o="542v" n="1084"/>
+<pb file="add_6786_f543" o="543" n="1085"/>
+<pb file="add_6786_f543v" o="543v" n="1086"/>
+<pb file="add_6786_f544" o="544" n="1087"/>
+<pb file="add_6786_f544v" o="544v" n="1088"/>
+<pb file="add_6786_f545" o="545" n="1089"/>
+<pb file="add_6786_f545v" o="545v" n="1090"/>
+<pb file="add_6786_f546" o="546" n="1091"/>
+<pb file="add_6786_f546v" o="546v" n="1092"/>
+<pb file="add_6786_f547" o="547" n="1093"/>
+<pb file="add_6786_f547v" o="547v" n="1094"/>
+<pb file="add_6786_f548" o="548" n="1095"/>
+<pb file="add_6786_f548v" o="548v" n="1096"/>
+<pb file="add_6786_f549" o="549" n="1097"/>
+<pb file="add_6786_f549v" o="549v" n="1098"/>
+<pb file="add_6786_f550" o="550" n="1099"/>
+<pb file="add_6786_f550v" o="550v" n="1100"/>
+<pb file="add_6786_f551" o="551" n="1101"/>
+<pb file="add_6786_f551v" o="551v" n="1102"/>
+<p>
+<s xml:id="echoid-s232" xml:space="preserve">
+red sage. 624. <lb/>
+une. 1073. <lb/>
+cummin. 908. <lb/>
+cammomill. 616. <lb/>
+otes. 69. <lb/>
+wall une. 984.i.
+</s>
+</p>
+<pb file="add_6786_f552" o="552" n="1103"/>
+<pb file="add_6786_f552v" o="552v" n="1104"/>
+<pb file="add_6786_f553" o="553" n="1105"/>
+<pb file="add_6786_f553v" o="553v" n="1106"/>
+<pb file="add_6786_f554" o="554" n="1107"/>
+<pb file="add_6786_f554v" o="554v" n="1108"/>
+<p>
+<s xml:id="echoid-s233" xml:space="preserve">
+poles <lb/>
+London <lb/>
+Syon <lb/>
+London Durham house <lb/>
+The East <lb/>
+Syon
+</s>
+</p>
+<pb file="add_6786_f555" o="555" n="1109"/>
+<p>
+<s xml:id="echoid-s234" xml:space="preserve">
+Brought by Mr Alisbury from Bristow <lb/>
+Avacardus <lb/>
+The kernel <lb/>
+good to eat. <lb/>
+The ioyce of the vine <lb/>
+good for tetters. <lb/>
+quod de Loan. <lb/>
+The Apot: <lb/>
+in Blackfriars.
+</s>
+</p>
+<pb file="add_6786_f555v" o="555v" n="1110"/>
+<pb file="add_6786_f556" o="556" n="1111"/>
+<pb file="add_6786_f556v" o="556v" n="1112"/>
+<div xml:id="echoid-div50" type="page_commentary" level="2" n="50">
+<p>
+<s xml:id="echoid-s235" xml:space="preserve">[<emph style="it">Note: 
+<p>
+<s xml:id="echoid-s235" xml:space="preserve">
+Pythagoreadn triples with hypotenuse 65, 85, 221, and 1105.
+</s>
+</p>
+</emph>]
+<lb/><lb/></s></p></div>
+<pb file="add_6786_f557" o="557" n="1113"/>
+<pb file="add_6786_f557v" o="557v" n="1114"/>
+<pb file="add_6786_f558" o="558" n="1115"/>
+<pb file="add_6786_f558v" o="558v" n="1116"/>
+<pb file="add_6786_f559" o="559" n="1117"/>
+<pb file="add_6786_f559v" o="559v" n="1118"/>
+<pb file="add_6786_f560" o="560" n="1119"/>
+<pb file="add_6786_f560v" o="560v" n="1120"/>
+<pb file="add_6786_f561" o="561" n="1121"/>
+<pb file="add_6786_f561v" o="561v" n="1122"/>
+</div>
+</text>
+</echo>
\ No newline at end of file