<?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>
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<pb file="add_6786_f167" o="167" n="333"/>
<pb file="add_6786_f167v" o="167v" n="334"/>
<pb file="add_6786_f168" o="168" n="335"/>
<pb file="add_6786_f168v" o="168v" n="336"/>
<pb file="add_6786_f169" o="169" n="337"/>
<pb file="add_6786_f169v" o="169v" n="338"/>
<pb file="add_6786_f170" o="170" n="339"/>
<pb file="add_6786_f170v" o="170v" n="340"/>
<pb file="add_6786_f171" o="171" n="341"/>
<pb file="add_6786_f171v" o="171v" n="342"/>
<pb file="add_6786_f172" o="172" n="343"/>
<pb file="add_6786_f172v" o="172v" n="344"/>
<pb file="add_6786_f173" o="173" n="345"/>
<pb file="add_6786_f173v" o="173v" n="346"/>
<pb file="add_6786_f174" o="174" n="347"/>
<pb file="add_6786_f174v" o="174v" n="348"/>
<pb file="add_6786_f175" o="175" n="349"/>
<pb file="add_6786_f175v" o="175v" n="350"/>
<pb file="add_6786_f176" o="176" n="351"/>
<pb file="add_6786_f176v" o="176v" n="352"/>
<pb file="add_6786_f177" o="177" n="353"/>
<pb file="add_6786_f177v" o="177v" n="354"/>
<pb file="add_6786_f178" o="178" n="355"/>
<pb file="add_6786_f178v" o="178v" n="356"/>
<pb file="add_6786_f179" o="179" n="357"/>
<pb file="add_6786_f179v" o="179v" n="358"/>
<pb file="add_6786_f180" o="180" n="359"/>
<pb file="add_6786_f180v" o="180v" n="360"/>
<pb file="add_6786_f181" o="181" n="361"/>
<pb file="add_6786_f181v" o="181v" n="362"/>
<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"/>
<pb file="add_6786_f184" o="184" n="367"/>
<pb file="add_6786_f184v" o="184v" n="368"/>
<pb file="add_6786_f185" o="185" n="369"/>
<pb file="add_6786_f185v" o="185v" n="370"/>
<pb file="add_6786_f186" o="186" n="371"/>
<pb file="add_6786_f186v" o="186v" n="372"/>
<pb file="add_6786_f187" o="187" n="373"/>
<pb file="add_6786_f187v" o="187v" n="374"/>
<pb file="add_6786_f188" o="188" n="375"/>
<pb file="add_6786_f188v" o="188v" n="376"/>
<pb file="add_6786_f189" o="189" n="377"/>
<pb file="add_6786_f189v" o="189v" n="378"/>
<pb file="add_6786_f190" o="190" n="379"/>
<pb file="add_6786_f190v" o="190v" n="380"/>
<pb file="add_6786_f191" o="191" n="381"/>
<pb file="add_6786_f191v" o="191v" n="382"/>
<pb file="add_6786_f192" o="192" n="383"/>
<pb file="add_6786_f192v" o="192v" n="384"/>
<pb file="add_6786_f193" o="193" n="385"/>
<pb file="add_6786_f193v" o="193v" n="386"/>
<pb file="add_6786_f194" o="194" n="387"/>
<pb file="add_6786_f194v" o="194v" n="388"/>
<pb file="add_6786_f195" o="195" n="389"/>
<pb file="add_6786_f195v" o="195v" n="390"/>
<pb file="add_6786_f196" o="196" n="391"/>
<pb file="add_6786_f196v" o="196v" n="392"/>
<pb file="add_6786_f197" o="197" n="393"/>
<pb file="add_6786_f197v" o="197v" n="394"/>
<pb file="add_6786_f198" o="198" n="395"/>
<pb file="add_6786_f198v" o="198v" n="396"/>
<pb file="add_6786_f199" o="199" n="397"/>
<pb file="add_6786_f199v" o="199v" n="398"/>
<pb file="add_6786_f200" o="200" n="399"/>
<pb file="add_6786_f200v" o="200v" n="400"/>
<pb file="add_6786_f201" o="201" n="401"/>
<pb file="add_6786_f201v" o="201v" n="402"/>
<pb file="add_6786_f202" o="202" n="403"/>
<pb file="add_6786_f202v" o="202v" n="404"/>
<pb file="add_6786_f203" o="203" n="405"/>
<pb file="add_6786_f203v" o="203v" n="406"/>
<pb file="add_6786_f204" o="204" n="407"/>
<pb file="add_6786_f204v" o="204v" n="408"/>
<pb file="add_6786_f205" o="205" n="409"/>
<pb file="add_6786_f205v" o="205v" n="410"/>
<pb file="add_6786_f206" o="206" n="411"/>
<pb file="add_6786_f206v" o="206v" n="412"/>
<pb file="add_6786_f207" o="207" n="413"/>
<pb file="add_6786_f207v" o="207v" n="414"/>
<pb file="add_6786_f208" o="208" n="415"/>
<pb file="add_6786_f208v" o="208v" n="416"/>
<pb file="add_6786_f209" o="209" n="417"/>
<pb file="add_6786_f209v" o="209v" n="418"/>
<pb file="add_6786_f210" o="210" n="419"/>
<pb file="add_6786_f210v" o="210v" n="420"/>
<pb file="add_6786_f211" o="211" n="421"/>
<pb file="add_6786_f211v" o="211v" n="422"/>
<pb file="add_6786_f212" o="212" n="423"/>
<pb file="add_6786_f212v" o="212v" n="424"/>
<pb file="add_6786_f213" o="213" n="425"/>
<pb file="add_6786_f213v" o="213v" n="426"/>
<pb file="add_6786_f214" o="214" n="427"/>
<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>
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<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>