Mercurial > hg > mpdl-xml-content

<?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:MAH52R5E.xml</dcterms:identifier>
<dcterms:creator>Harriot, Thomas</dcterms:creator>
<dcterms:title xml:lang="en">Mss. 6787</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/MAH52R5E</echodir>
<log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log>
</metadata>

<text xml:lang="eng" type="free">
<div xml:id="echoid-div1" type="section" level="1" n="1">
<pb file="add_6787_f001" o="1" n="1"/>
<pb file="add_6787_f001v" o="1v" n="2"/>
<pb file="add_6787_f002" o="2" n="3"/>
<pb file="add_6787_f002v" o="2v" n="4"/>
<pb file="add_6787_f003" o="3" n="5"/>
<pb file="add_6787_f003v" o="3v" n="6"/>
<pb file="add_6787_f004" o="4" n="7"/>
<pb file="add_6787_f004v" o="4v" n="8"/>
<pb file="add_6787_f005" o="5" n="9"/>
<pb file="add_6787_f005v" o="5v" n="10"/>
<pb file="add_6787_f006" o="6" n="11"/>
<pb file="add_6787_f006v" o="6v" n="12"/>
<pb file="add_6787_f007" o="7" n="13"/>
<pb file="add_6787_f007v" o="7v" n="14"/>
<pb file="add_6787_f008" o="8" n="15"/>
<pb file="add_6787_f008v" o="8v" n="16"/>
<pb file="add_6787_f009" o="9" n="17"/>
<pb file="add_6787_f009v" o="9v" n="18"/>
<pb file="add_6787_f010" o="10" n="19"/>
<pb file="add_6787_f010v" o="10v" n="20"/>
<pb file="add_6787_f011" o="11" n="21"/>
<pb file="add_6787_f011v" o="11v" n="22"/>
<pb file="add_6787_f012" o="12" n="23"/>
<pb file="add_6787_f012v" o="12v" n="24"/>
<pb file="add_6787_f013" o="13" n="25"/>
<pb file="add_6787_f013v" o="13v" n="26"/>
<pb file="add_6787_f014" o="14" n="27"/>
<pb file="add_6787_f014v" o="14v" n="28"/>
<pb file="add_6787_f015" o="15" n="29"/>
<pb file="add_6787_f015v" o="15v" n="30"/>
<pb file="add_6787_f016" o="16" n="31"/>
<pb file="add_6787_f016v" o="16v" n="32"/>
<pb file="add_6787_f017" o="17" n="33"/>
<div xml:id="echoid-div1" type="page_commentary" level="2" n="1">
<p>
<s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1" xml:space="preserve">
The columns show the first, second, third, and fourth differences of an interpolated table
with constant fourth difference.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat">
3) Ad calculum sinuum 4<emph style="super">or</emph> differentiarum succedentium
<lb/>[<emph style="it">tr:
For the calculation of four successive differences of sines.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s3" xml:space="preserve">
Æquatio ista <lb/>
examinata <lb/>
fuit per induct<lb/>
ione ad aliam <lb/>
probatam.
<lb/>[<emph style="it">tr:
This equation has been examined by the induction used to demonstrate the other.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s4" xml:space="preserve">
Reductio <lb/>
superioris <lb/>
æquationis <lb/>
in [¿?].
<lb/>[<emph style="it">tr:
The separation of the above equation into <emph style="super">parts</emph>
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f017v" o="17v" n="34"/>
<pb file="add_6787_f018" o="18" n="35"/>
<pb file="add_6787_f018v" o="18v" n="36"/>
<pb file="add_6787_f019" o="19" n="37"/>
<div xml:id="echoid-div2" type="page_commentary" level="2" n="2">
<p>
<s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s5" xml:space="preserve">
At the top of the page the entries from Add MS 6787, f. 20, now written with common denominator 24. <lb/>
The lower half of the page shows the first and second differences of the above.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head2" xml:space="preserve">
2)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s7" xml:space="preserve">
Superiorum primariæ differentiæ
<lb/>[<emph style="it">tr:
First differences of the above.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s8" xml:space="preserve">
Secundariæ differentiæ
<lb/>[<emph style="it">tr:
Second differences.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f019v" o="19v" n="38"/>
<pb file="add_6787_f020" o="20" n="39"/>
<pb file="add_6787_f020v" o="20v" n="40"/>
<pb file="add_6787_f021" o="21" n="41"/>
<pb file="add_6787_f021v" o="21v" n="42"/>
<pb file="add_6787_f022" o="22" n="43"/>
<pb file="add_6787_f022v" o="22v" n="44"/>
<pb file="add_6787_f023" o="23" n="45"/>
<div xml:id="echoid-div3" type="page_commentary" level="2" n="3">
<p>
<s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s9" xml:space="preserve">
This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of
<emph style="it">Variorum responsorum liber VIII</emph>.
</s>
<lb/>
<quote xml:lang="lat">
Quæ per factionem sub sinibus peripheriarum &amp; adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt. <lb/>
Cum duæ peripheriæ angulum acutum componunt, est <lb/>
Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia,
minus sinu complementi composita.
</quote>
<lb/>
<quote>
What appears from a combination of the sine of the arcs, dividing the sine of the total,
is also shown by the operations of addition and subtraction. <lb/>
When two arcs contain acute angles, then as the whole sine is to twice the sine of the first,
so is the sine of the second to the sum of
the sine of the complement of the difference minus the sine of the complement of the sum.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head3" xml:space="preserve" xml:lang="lat">
De 3<emph style="super">o</emph> et 4<emph style="super">o</emph> casu
<foreign xml:lang="gre">ton syntomon</foreign> (1.
<lb/>[<emph style="it">tr:
On cases 3 and 4 of Syntomon
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s11" xml:space="preserve">
Etsi duo priores casus <lb/>
sufficiunt ad operationes: <lb/>
duo tamen sequentes ad <lb/>
argumentationes sunt ali-<lb/>
quando necessarij.
<lb/>[<emph style="it">tr:
Although the two first cases suffice for working,
nevertheless the two following arguments are sometimes necessary.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s12" xml:space="preserve">
3<emph style="super">us</emph> casus est, quando unus <lb/>
datorum arcuum sit maior <lb/>
quadrante; et differentia sit <lb/>
etiam quadrante maior.
<lb/>[<emph style="it">tr:
The 3rd case is when one of the given arcs is greater than a quadrant, and the difference is also greater than a quadrant.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s13" xml:space="preserve">
4<emph style="super">us</emph> casus est, quando unus datorum <lb/>
arcuum sit maior quadrante; <lb/>
sed differentia sit quadrante minore.
<lb/>[<emph style="it">tr:
The 4th case is when one of the given arcs is greater than a quadrant, but the difference is less.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f023v" o="23v" n="46"/>
<pb file="add_6787_f024" o="24" n="47"/>
<div xml:id="echoid-div4" type="page_commentary" level="2" n="4">
<p>
<s xml:id="echoid-s14" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s14" xml:space="preserve">
This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of
<emph style="it">Variorum responsorum liber VIII</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat">
Ad triangula sphærica obliquangula Vietæ (2.
<lb/>[<emph style="it">tr:
On Viète's obtuse-angle spherical triangles.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s16" xml:space="preserve">
Sint arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, et, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>, sigillatim minores quadrante: <lb/>
Dico quod: <lb/>[...]<lb/> hoc est sinuui comple-<lb/>
menti differentiæ <lb/>
arcuum datorum.
<lb/>[<emph style="it">tr:
Let the arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> each be less than a quadrant. <lb/>
I say that <lb/>[...]<lb/> this is the sine of the complement of the difference of the given arcs.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s17" xml:space="preserve">
Ad sinus magisterij demonstrationem, dico quod:
<lb/>[<emph style="it">tr:
For demonstrating the doctrine of sines, I say that:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s18" xml:space="preserve">
Quæ demonstrari oportuit.
<lb/>[<emph style="it">tr:
Which was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f024v" o="24v" n="48"/>
<pb file="add_6787_f025" o="25" n="49"/>
<div xml:id="echoid-div5" type="page_commentary" level="2" n="5">
<p>
<s xml:id="echoid-s19" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s19" xml:space="preserve">
This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of
<emph style="it">Variorum responsorum liber VIII</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat">
(3.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s21" xml:space="preserve">
Aliter. <lb/>
In diagrammatis syntomi
(<emph style="super">In</emph> utriusque casibus, 1<emph style="super">o</emph> et 2<emph style="super">o</emph>) <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, secat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>. <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Another way, <lb/>
In the diagram for syntomon, (in either case 1 or 2) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> cuts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>.
Make <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s22" xml:space="preserve">
1<emph style="super">m</emph> Dico quod: <lb/>[...]<lb/> hoc est sinuui <lb/>
complementi <reg norm="differentiaæ" type="abbr">dræ</reg> <lb/>
arcuum datorum.
<lb/>[<emph style="it">tr:
1st, I say that this is the sine of the complement of the difference of the given arcs.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s23" xml:space="preserve">
Quod fiat demonstrandum.
<lb/>[<emph style="it">tr:
Which was to be demonstrated. </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s24" xml:space="preserve">
Sequitur etiam per 1<emph style="super">m</emph>: quod: <lb/>
in utroque <lb/>
casu.
<lb/>[<emph style="it">tr:
There follows from the first, because in either case:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f025v" o="25v" n="50"/>
<pb file="add_6787_f026" o="26" n="51"/>
<div xml:id="echoid-div6" type="page_commentary" level="2" n="6">
<p>
<s xml:id="echoid-s25" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s25" xml:space="preserve">
This page contains further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat">
(4.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s27" xml:space="preserve">
In utriusque casibus, 3<emph style="super">o</emph> et 4<emph style="super">o</emph>
<foreign xml:lang="gre">ton syntomon</foreign> <lb/>
1<emph style="super">m</emph>) Dico quod:
<lb/>[<emph style="it">tr:
In either the 3rd or 4th case of Syntomon <lb/>
1) I say that:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s28" xml:space="preserve">
2<emph style="super">m</emph>) Dico quod in 3<emph style="super">m</emph> casu:
<lb/>[<emph style="it">tr:
2) I say that in the 3rd case:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s29" xml:space="preserve">
3<emph style="super">m</emph>) Dico quod in 4<emph style="super">m</emph> casu:
<lb/>[<emph style="it">tr:
3) I say that in the 4th case:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s30" xml:space="preserve">
Sequitur etiam per 1<emph style="super">m</emph>: quod: <lb/>
in utroque casu, 3<emph style="super">o</emph> et 4<emph style="super">o</emph>.
<lb/>[<emph style="it">tr:
There follows also from the first, because in either the 3rd or 4th case:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f026v" o="26v" n="52"/>
<pb file="add_6787_f027" o="27" n="53"/>
<div xml:id="echoid-div7" type="page_commentary" level="2" n="7">
<p>
<s xml:id="echoid-s31" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s31" xml:space="preserve">
This page contains further work on spherical triangles. <lb/>
There are references to both Regiomontanus and Clavius,
who both gave a version of the theorem given here. <lb/>
The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis libri quinque</emph> (1533),
Book V, Proposition II. (For another reference to the same proposition, see also Add MS 6782, f. 490.)
</s>
<lb/>
<quote xml:lang="lat">
V.II In omni triangulo sphaerali ex arcubus circulorum magnorum constante,
proportio sinus uersi anguli cuislibet ad differentiam duorum sinuum uersorum,
quorum unus est lateris eum angulum subtendentis:
alius uerÃ² differentiae duorum arcuum ipsi angulo circumiacentium
est tanquam proportio quadrati sinus recti totius ad id,
quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum
</quote>
<lb/>
<quote>
In all spherical triangles composed from great arcs of circles, the ratio of the versed sine of any angle
to the difference of two versed sines, one of which is the side subtending the angle,
the other the difference of the two arcs adjacent to the angle,
is the proportion of the the square of the whole sine
to the product of the sines of the surrounding arcs by which the said angle is contained.
</quote>
<lb/>
<s xml:id="echoid-s32" xml:space="preserve">
The reference to Clavius is to his <emph style="it">Triangula sphærica</emph> in
<emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586).
Proposition 58, on page 445.
</s>
<lb/>
<quote xml:lang="lat">
Theorema 56, Propositio 58. <lb/>
In omni triangulo sphærico, cuius duo arcus sint inæquales;
quadratum sinus totius ad rectangulum sub sinubus rectis duorum arcuum inæqulium contentum,
eandem proportionem habet, quam sinus versus anguli a dictis arcubus comprehensi
ad differentiam duorum sinuum versorum, quorum vnus differentiæ eorundem arcuum debetur,
alter vero tertio arcui, qui prædicto angulo oppostitus est, respondet.
</quote>
<lb/>
<quote>
In all spherical triangles, whose two arcs are unequal,
the square of the whole sine to the product of the sines of the two unequal arcs
is in the same ratio as the versed sine of the angle between the said arcs
to the difference of two versed sines, one of which is of the difference of the arcs,
the other corresponding to the third arc, which is opposite the aforesaid angle.
</quote>
<lb/>
<s xml:id="echoid-s33" xml:space="preserve">
Harrot translates Clavius's statement into symbols for the particular triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mi>b</mi></mstyle></math> shown in his diagram.
He then goes on to prove that the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>b</mi></mstyle></math> is greater than the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
For another version of this page, see Add MS 6787, f. 51.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat">
(5. <lb/>
Analogia per sinus versos, et universalis ad triangula <lb/>
sphærica cuiuscunque conditionis.
<lb/>[<emph style="it">tr:
Ratio by versed sines, and generally for a spherical trianlge under any conditions.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s35" xml:space="preserve">
Demonstratur a Regiomontano <lb/>
lib. 5<emph style="super">o</emph>. pr. 2, de triang. <lb/>
A clavio pr. 58. de sphæricis <lb/>
Ab alijs Trigonistis. <lb/>
Et a nobis alibi in notis.
<lb/>[<emph style="it">tr:
Demonstrated by Regiomontanus in <emph style="it">De triangulis</emph>, Book 5, Proposition 2. <lb/>
by Clavius in Proposition 58 of <emph style="it">De sphæricis triangulis</emph> <lb/>
By other triangulists. <lb/>
And by me elsewhere in notation.
</emph>]<lb/>
[<emph style="it">Note:
The page 'elsewhere' referred to here is probably Add MS 6787, f. 27.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s36" xml:space="preserve">
Dico quod: <lb/>[...]<lb/> (<foreign xml:lang="gre">catolicos</foreign>)
<lb/>[<emph style="it">tr:
I say that <lb/>[...]<lb/> (generally)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s37" xml:space="preserve">
Consectarium <lb/>
ponatur quod: <lb/>
In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, datis duobus lateribus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>; cum angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/>
queratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
per superiorem analogiam, sit <emph style="super">data</emph> quarta proportionalis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Consequence <lb/>
It is supposed that, in triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, from given two sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> with the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
by the above ratio, let the given fourth proportional be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> <lb/>
<omissions/> <lb/>
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is given
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f027v" o="27v" n="54"/>
<pb file="add_6787_f028" o="28" n="55"/>
<div xml:id="echoid-div8" type="page_commentary" level="2" n="8">
<p>
<s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s38" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat">
(6. <lb/>
Investigatio analogiæ <lb/>
Vietanæ
<lb/>[<emph style="it">tr:
Investigation of Viète's ratios.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s40" xml:space="preserve">
Hoc est magisterium.
<lb/>[<emph style="it">tr:
This is the rule.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s41" xml:space="preserve">
Ut Vieta <lb/>
pag. 47b.
<lb/>[<emph style="it">tr:
As Viète page 47v.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s42" xml:space="preserve">
Ut Vieta <lb/>
pag. 35b.
<lb/>[<emph style="it">tr:
As Viete, page 35v.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s43" xml:space="preserve">
Conditiones <lb/>
alterius trianguli <lb/>
contrariæ, sub eadem <lb/>
analogia.
<lb/>[<emph style="it">tr:
Conditions for another triangle, conversely, by the same ratio.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s44" xml:space="preserve">
Nota. Etsi signum (&lt;) ponatur sub <lb/>[...]<lb/>, intelligitur quod, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
Ita signum (&gt;) sub (<lb/>[...]<lb/>) denotat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
Istud signum (<lb/>[...]<lb/>), denotat unum latus maius altera minus 90 &amp;c. <lb/>
Et in alijs locis, (<lb/>[...]<lb/>), utrinque minus, utrinque maius 90.
<lb/>[<emph style="it">tr:
Note. If this sign (&lt;) is put below <lb/>[...]<lb/>, it is to be understood that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
This sign (&gt;) below (<lb/>[...]<lb/>) denotes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
This sign (<lb/>[...]<lb/>) denotes one side is greater than the other minus 90. <lb/>
An in other places, (<omision/>) eother less than or greater than 90.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f028v" o="28v" n="56"/>
<pb file="add_6787_f029" o="29" n="57"/>
<div xml:id="echoid-div9" type="page_commentary" level="2" n="9">
<p>
<s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s45" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head9" xml:space="preserve" xml:lang="lat">
(7.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s47" xml:space="preserve">
Investigatio trium analogiarum <lb/>
sive casuum. <lb/>
ex positis sequentibus
<lb/>[<emph style="it">tr:
Investigation of three ratios or cases, supposing the following.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f029v" o="29v" n="58"/>
<pb file="add_6787_f030" o="30" n="59"/>
<pb file="add_6787_f030v" o="30v" n="60"/>
<pb file="add_6787_f031" o="31" n="61"/>
<div xml:id="echoid-div10" type="page_commentary" level="2" n="10">
<p>
<s xml:id="echoid-s48" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s48" xml:space="preserve">
The reference on this page is to Proposition 21 from Chapter 19 of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
XXI. <lb/>
Trianguli cujuslibet sphærici. <lb/>
Datis duobus lateribus, &amp; angulo cui unum ex illis lateribus opponitur,
datur angulus cui alterum datorum laterum opponitur. <lb/>
Vel, <lb/>
Datis duobs angulis, &amp; latere quod alteri datorum angulorum opponitur,
datur latus reliquo oppositum.
</quote>
<lb/>
<quote>
Given two sides and the angle opposite one of those sides, the angle opposite the other is known. <lb/>
Or, <lb/>
Given two angles, and the side opposite one of the given angles, the side opposite the other is known.
</quote>
<lb/>
<s xml:id="echoid-s49" xml:space="preserve">
Immediately after the statement of the proposition, Viète gave the following statement,
under the heading Syntomon.
</s>
<lb/>
<quote xml:lang="lat">
Quæ per factionem sub sinibus peripheriarum &amp; adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt. <lb/>
Cum duæ peripheriæ angulum acutum componunt, est <lb/>
Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia,
minus sinu complementi composita.
</quote>
<lb/>
<quote>
What appears from a combination of the sine of the arcs, dividing the sine of the total,
is also shown by the operations of addition and subtraction. <lb/>
When two arcs contain acute angles, then as the whole sine is to twice the sine of the first,
so is the sine of the second to the sum of
the sine of the complement of the difference minus the sine of the complement of the sum.
</quote>
<lb/>
<s xml:id="echoid-s50" xml:space="preserve">
In modern notation this statement may be written as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>:</mo><mrow><mn>2</mn><mspace width="0.167em"/></mrow><mrow><mi>sin</mi><mrow><mi>a</mi></mrow></mrow><mo>=</mo><mrow><mi>sin</mi><mrow><mi>b</mi></mrow></mrow><mo>:</mo><mrow><mi>cos</mi><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mrow><mo>+</mo><mrow><mi>cos</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mrow></mstyle></math>.
This is the ratio Harriot has written nex to diagram 1, where both angles are acute.
The other diagrams are for cases where one or both the angles are obtuse.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat">
Vieta lib. 8. resp.
pag. 39. <lb/>
<foreign xml:lang="gre">Syntomon</foreign>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII. <lb/>
Syntomon
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s52" xml:space="preserve">
[???] in alia charta
<lb/>[<emph style="it">tr:
[???] in another sheet
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s53" xml:space="preserve">
Quæ per factionem sub sinibus peripherieriarum et adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt.
<lb/>[<emph style="it">tr:
What appears from a combination of the sine of the arcs, dividing the sine of the total,
is also shown by the operations of addition and subtraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s54" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>, differentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, aggregatum &amp;c.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is one arc, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> the sum.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s55" xml:space="preserve">
Hæc quarta analogia est re eadem <lb/>
cum secunda. <lb/>
porro, prima et tertia analogiæ <lb/>
reducuntur ad unam si quartus <lb/>
terminus ita notetur.
<lb/>[<emph style="it">tr:
This fourth ratio is the same thing as the second. <lb/>
Further, the first and third ratios are reduced to one if the fourth term is written thus.
</emph>]<lb/>
[<emph style="it">Note:
Here the symbols that looks like an equals sign is to be read as a minus sign,
where the smaller quantity is always understood to be subtracted from the larger.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s56" xml:space="preserve">
Nota <lb/>
Quando una peripheria est maior quadranti <lb/>
ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> est <emph style="super">in</emph>
3,<emph style="super">a</emph> et 4,<emph style="super">a</emph> diagrammati; summatur <lb/>
eius residuum ad semicirculum. Et tum <lb/>
operatio erit secundum primum vel secundum casum. <lb/>
Quare hoc modo sunt duo tantummodo <lb/>
casus.
<lb/>[<emph style="it">tr:
Note. <lb/>
When one arc is greater than a quadrant, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is in the 3rd and 4th diagrams,
there are taken their residuals from a semicircle.
And then the operation witll be as the first or second case. <lb/>
Therefore by this method there are only two cases.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f031v" o="31v" n="62"/>
<pb file="add_6787_f032" o="32" n="63"/>
<div xml:id="echoid-div11" type="page_commentary" level="2" n="11">
<p>
<s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s57" xml:space="preserve">
This page continues Harriot's work from Add MS 6787, f. 61,
on Viète's statement of 'Syntomon'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat">
2) 	syntomon
<lb/>[<emph style="it">tr:
2) Syntomon
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s59" xml:space="preserve">
primus casus. quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
Ponatur analogia
<lb/>[<emph style="it">tr:
First case, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
There is put the ratio:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s60" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> maior arcus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> minor <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
ideo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
sit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math>: pro 1<emph style="super">o</emph> casu. <lb/>
Dico quod <lb/>
vel sub hæc speciæ
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be the greater arc, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> the lesser, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
Let also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi><mo>&lt;</mo><mn>9</mn><mn>0</mn></mstyle></math> for the first case. <lb/>
I say that: <lb/>
Or in this general form:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s61" xml:space="preserve">
nota <lb/>
complementum differentiæ <lb/>
complementum aggregati
<lb/>[<emph style="it">tr:
Note. <lb/>
Complement of the difference. <lb/>
Complement of the sum.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s62" xml:space="preserve">
Secundus casus est quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>&gt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Quod etiam demonstrandum fuit.
<lb/>[<emph style="it">tr:
The second case is wehn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>&gt;</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Which was also to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f032v" o="32v" n="64"/>
<pb file="add_6787_f033" o="33" n="65"/>
<div xml:id="echoid-div12" type="page_commentary" level="2" n="12">
<p>
<s xml:id="echoid-s63" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s63" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head12" xml:space="preserve" xml:lang="lat">
(8.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s65" xml:space="preserve">
Investigatio trium aliarum <lb/>
analogiarum sive casuum. <lb/>
ex positis sequentibus
<lb/>[<emph style="it">tr:
Investigation of three other ratios or cases, supposing the following.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s66" xml:space="preserve">
per antithesin <lb/>
vel conversione <lb/>
signorum: <lb/>
istæ æquationes <lb/>
sunt eadem <lb/>
cum illis in <lb/>
antecedente <lb/>
charta.
<lb/>[<emph style="it">tr:
By antithesis or change of sign, these equations are the same as those in the preceding sheet.
</emph>]<lb/>
[<emph style="it">Note:
The preceding sheet was Add MS 6787, f. 29.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s67" xml:space="preserve">
Ergo analogiæ sunt etiam eædem ut antea, nisi quod <lb/>
conditiones sunt contrariæ.
<lb/>[<emph style="it">tr:
Therefore the ratios are also the samse as before, unless the conditions are opposite.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f033v" o="33v" n="66"/>
<pb file="add_6787_f034" o="34" n="67"/>
<div xml:id="echoid-div13" type="page_commentary" level="2" n="13">
<p>
<s xml:id="echoid-s68" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s68" xml:space="preserve">
Further work on spherical triangles. <lb/>
As on Add MS 6787, f. 27, there is reference to Clavius,
<emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586),
this time to Proposition 27.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat">
(9. <lb/>
Casuum Limitatio cum designatione adcomodum
<lb/>[<emph style="it">tr:
Determination of cases with a useful specification
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s70" xml:space="preserve">
Animadvertendum quod in superioribus investigationibus analogiarum <lb/>
sive casuum, ubi latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> signantur &lt; &lt;; debeat etiam <lb/>
intelligi signata &gt; &gt;. Quoniam si sint eiusdem affectionis <lb/>
sive utrique minora quadrantibus, sive maiora; non variatur <lb/>
inde illatio neque casus. <lb/>
Hinc in 6 notatis casibus positi trianguli, sunt 9 variationes <lb/>
signorum ut sequitur.
<lb/>[<emph style="it">tr:
It is to be noted that in the above investigations of ratios, or cases,
where the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> are marked with &lt;, &lt;, respectively,
it is also to be understood that they could be marked with &gt;, &gt;, respectively.
Because they have the same relationship, whether both less than a quadrant, or greater;
therefore the result does not vary, nor the cases. <lb/>
Here in the 6 denoted cases of the supposed triangle, there are 9 variations.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s71" xml:space="preserve">
Sunt præter illas novem, tres aliæ variationes; et non dantur <lb/>
plures: sed istæ sunt impossibiles (hypostatice).
<lb/>[<emph style="it">tr:
Besides those nine, there are three other variations; more are not given, but these are impossible.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s72" xml:space="preserve">
Una <emph style="super">et ultima</emph> impossibilium probatur per Clavium pro: 27 de Sphæricis. <lb/>
et a me magis perspicus in notis de conversione triangulorum: <lb/>
reliquæ duæ conseqununtur per paraplerosin.
<lb/>[<emph style="it">tr:
One, the final impossibility, is proved by Clavus in Proposition 27 of De sphærica;
and by me more clearly in notation in the conversion of triangles; the remaining two follow by paraplerosis.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s73" xml:space="preserve">
Alia designatio analogiarum et casuum, <lb/>
usui magi adcommoda.
<lb/>[<emph style="it">tr:
Another specification of ratios and cases, more convenient in practice.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f034v" o="34v" n="68"/>
<pb file="add_6787_f035" o="35" n="69"/>
<div xml:id="echoid-div14" type="page_commentary" level="2" n="14">
<p>
<s xml:id="echoid-s74" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s74" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head14" xml:space="preserve" xml:lang="lat">
(10.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s76" xml:space="preserve">
Duo triangula <lb/>
sub canone 1<emph style="super">æ</emph> analogiæ <lb/>
et duobus primis <lb/>
casibus.
<lb/>[<emph style="it">tr:
Two triangles under the canon of the first ratio and the two first cases.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s77" xml:space="preserve">
(operationum designationes <lb/>
sunt in Chartis nostris de <lb/>
triangulis sphæricis.)
<lb/>[<emph style="it">tr:
(the specifications of the operations are in my sheets on spherical triangles)
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f035v" o="35v" n="70"/>
<pb file="add_6787_f036" o="36" n="71"/>
<div xml:id="echoid-div15" type="page_commentary" level="2" n="15">
<p>
<s xml:id="echoid-s78" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s78" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat">
(11.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s80" xml:space="preserve">
Analogia per sinus versos
<lb/>[<emph style="it">tr:
Ratio in terms of versed sines.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f036v" o="36v" n="72"/>
<pb file="add_6787_f037" o="37" n="73"/>
<div xml:id="echoid-div16" type="page_commentary" level="2" n="16">
<p>
<s xml:id="echoid-s81" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s81" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat">
(12.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s83" xml:space="preserve">
Analogia per sinus versos
<lb/>[<emph style="it">tr:
Ratio in terms of versed sines.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f037v" o="37v" n="74"/>
<pb file="add_6787_f038" o="38" n="75"/>
<div xml:id="echoid-div17" type="page_commentary" level="2" n="17">
<p>
<s xml:id="echoid-s84" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s84" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat">
(13.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s86" xml:space="preserve">
Analogia per sinus versos
<lb/>[<emph style="it">tr:
Ratio in terms of versed sines.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f038v" o="38v" n="76"/>
<pb file="add_6787_f039" o="39" n="77"/>
<div xml:id="echoid-div18" type="page_commentary" level="2" n="18">
<p>
<s xml:id="echoid-s87" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s87" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat">
(14.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s89" xml:space="preserve">
Analogia per sinus versos
<lb/>[<emph style="it">tr:
Ratio in terms of versed sines.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f039v" o="39v" n="78"/>
<pb file="add_6787_f040" o="40" n="79"/>
<div xml:id="echoid-div19" type="page_commentary" level="2" n="19">
<p>
<s xml:id="echoid-s90" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s90" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat">
(15. <lb/>
De 5<emph style="super">o</emph>, et 6<emph style="super">o</emph>, casu
<foreign xml:lang="gre">ton syntomon</foreign>
<lb/>[<emph style="it">tr:
On the 5th and 6th cases of syntomon.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s92" xml:space="preserve">
5<emph style="super">us</emph> casus est, quando uterque arcus sit maior quadrante <lb/>
et aggregatum maius 3<emph style="super">bus</emph> quadrantibus.
<lb/>[<emph style="it">tr:
The 5th case is when either arc is greater than a quadrant, and the sum is greater than three quadrants.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s93" xml:space="preserve">
Dico etiam quod: in utroque casu: <lb/>
<lb/>[...]<lb/> <lb/>
Quod fuit demosntradum.
<lb/>[<emph style="it">tr:
I also say that in eithe case <lb/>
<lb/>[...]<lb/> <lb/>
Which was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s94" xml:space="preserve">
6<emph style="super">us</emph> casus est, quando uterque arcus sit maior quadrante <lb/>
et aggregatum minus 3<emph style="super">bus</emph> quadrantibus.
<lb/>[<emph style="it">tr:
The 6th case is when either arc is greater than a quadrant, and the sum is less than three quadrants.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f040v" o="40v" n="80"/>
<pb file="add_6787_f041" o="41" n="81"/>
<div xml:id="echoid-div20" type="page_commentary" level="2" n="20">
<p>
<s xml:id="echoid-s95" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s95" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat">
(16. <lb/>
<foreign xml:lang="gre">ton syntomon</foreign> ruminatio. 6 casus exhibantur et omnes <lb/>
varia enunicantur et designantur; ut magis canonicæ <lb/>
formæ ad usus seligantur.
<lb/>[<emph style="it">tr:
Further thoughts on syntomon. <lb/>
6 cases are exhibeted and all varations enunicated and specified;
so that canonical forms can further be picked out in practice.
</emph>]<lb/>
</head>
<pb file="add_6787_f041v" o="41v" n="82"/>
<pb file="add_6787_f042" o="42" n="83"/>
<div xml:id="echoid-div21" type="page_commentary" level="2" n="21">
<p>
<s xml:id="echoid-s97" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s97" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head21" xml:space="preserve" xml:lang="lat">
(17. <lb/>
sex casuum <foreign xml:lang="gre">ton syntomon</foreign> <lb/>
duæ selectæ formæ.
<lb/>[<emph style="it">tr:
Six cases of Syntomon; <lb/>
two chosen forms.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s99" xml:space="preserve">
Exempla analogiarum, accommodata casibus.
<lb/>[<emph style="it">tr:
Examples of ratios, accommodated to cases.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s100" xml:space="preserve">
sex sunt diversæ analogiæ sub speciebus arcuum <lb/>
cum tamen duæ solummodo sub numerus sinuum. <lb/>
Unde manifestum quod 1<emph style="super">us</emph> et 2<emph style="super">us</emph> casus
<foreign xml:lang="gre">ton syntomon</foreign> possunt <lb/>
satis ad operationes. reliquæ tamen ad argumentationes sunt <lb/>
utilis.
<lb/>[<emph style="it">tr:
There are six different ratios for types of arc, but with only two for sines in numbers. <lb/>
From which is is clear that the 1st and 2nd cases may be enough for working;
nevertheless, the others are useful for arguments.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f042v" o="42v" n="84"/>
<pb file="add_6787_f043" o="43" n="85"/>
<div xml:id="echoid-div22" type="page_commentary" level="2" n="22">
<p>
<s xml:id="echoid-s101" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s101" xml:space="preserve">
Further work on spherical triangles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat">
(18. <lb/>
Exempla operandi per <foreign xml:lang="gre">syntomon</foreign>, <lb/>
secundum duas formas selectas.
<lb/>[<emph style="it">tr:
Examples of working with Syntomon according to two chosen forms.
</emph>]<lb/>
</head>
<pb file="add_6787_f043v" o="43v" n="86"/>
<pb file="add_6787_f044" o="44" n="87"/>
<pb file="add_6787_f044v" o="44v" n="88"/>
<pb file="add_6787_f045" o="45" n="89"/>
<pb file="add_6787_f045v" o="45v" n="90"/>
<pb file="add_6787_f046" o="46" n="91"/>
<pb file="add_6787_f046v" o="46v" n="92"/>
<pb file="add_6787_f047" o="47" n="93"/>
<pb file="add_6787_f047v" o="47v" n="94"/>
<pb file="add_6787_f048" o="48" n="95"/>
<pb file="add_6787_f048v" o="48v" n="96"/>
<pb file="add_6787_f049" o="49" n="97"/>
<pb file="add_6787_f049v" o="49v" n="98"/>
<pb file="add_6787_f050" o="50" n="99"/>
<pb file="add_6787_f050v" o="50v" n="100"/>
<pb file="add_6787_f051" o="51" n="101"/>
<div xml:id="echoid-div23" type="page_commentary" level="2" n="23">
<p>
<s xml:id="echoid-s103" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s103" xml:space="preserve">
This page contains references to both Regiomontanus and Clavius,
who both gave a version of this theorem. <lb/>
The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis libri quinque</emph> (1533),
Book V, Proposition II. (For another reference to the same proposition, see also Add MS 6782, f. 490.)
</s>
<lb/>
<quote xml:lang="lat">
V.II In omni triangulo sphaerali ex arcubus circulorum magnorum constante,
proportio sinus uersi anguli cuislibet ad differentiam duorum sinuum uersorum,
quorum unus est lateris eum angulum subtendentis:
alius uerÃ² differentiae duorum arcuum ipsi angulo circumiacentium
est tanquam proportio quadrati sinus recti totius ad id,
quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum
</quote>
<lb/>
<quote>
In all spherical triangles composed from great arcs of circles, the ratio of the versed sine of any angle
to the difference of two versed sines, one of which is the side subtending the angle,
the other the difference of the two arcs adjacent to the angle,
is the proportion of the the square of the whole sine
to the product of the sines of the surrounding arcs by which the said angle is contained.
</quote>
<lb/>
<s xml:id="echoid-s104" xml:space="preserve">
The reference to Clavius is to his <emph style="it">Triangula sphærica</emph> in
<emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586).
Proposition 58, on page 445.
</s>
<lb/>
<quote xml:lang="lat">
Theorema 56, Propositio 58. <lb/>
In omni triangulo sphærico, cuius duo arcus sint inæquales;
quadratum sinus totius ad rectangulum sub sinubus rectis duorum arcuum inæqulium contentum,
eandem proportionem habet, quam sinus versus anguli a dictis arcubus comprehensi
ad differentiam duorum sinuum versorum, quorum vnus differentiæ eorundem arcuum debetur,
alter vero tertio arcui, qui prædicto angulo oppostitus est, respondet.
</quote>
<lb/>
<quote>
In all spherical triangles, whose two arcs are unequal,
the square of the whole sine to the product of the sines of the two unequal arcs
is in the same ratio as the versed sine of the angle between the said arcs
to the difference of two versed sines, one of which is of the difference of the arcs,
the other corresponding to the third arc, which is opposite the aforesaid angle.
</quote>
<lb/>
<s xml:id="echoid-s105" xml:space="preserve">
Harrot translates Clavius's statement into symbols for the particular triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mi>b</mi></mstyle></math> shown in his diagram.
He then goes on to prove that the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>b</mi></mstyle></math> is greater than the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
For another version of this page, see Add MS 6787, f. 27.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat">
Analogia per sinus versos, et universalis ad triangula <lb/>
sphærica cuiuscunque <emph style="st">affectionis</emph> conditionis.
<lb/>[<emph style="it">tr:
Ratio by versed sines, and generally for a spherical trianlge under any conditions.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s107" xml:space="preserve">
Demonstratur a Regiomontano <lb/>
lib. 5<emph style="super">to</emph>. pr. 2, de triang. <lb/>
a clavio pr. 58. de sphær. Tri. <lb/>
et ab alijs Trigonistis.
<lb/>[<emph style="it">tr:
Demonstrated by Regiomontanus in <emph style="it">De triangulis</emph>, Book 5, Proposition 2. <lb/>
by Clavius in Proposition 58 of <emph style="it">De sphæricis triangulis</emph> <lb/>
and by other triangulists.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s108" xml:space="preserve">
Dico quod: <lb/>[...]<lb/> (universitaliter)
<lb/>[<emph style="it">tr:
I say that <lb/>[...]<lb/> (generally)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s109" xml:space="preserve">
Consectarium
<lb/>[<emph style="it">tr:
Consequence
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s110" xml:space="preserve">
ponatur quod: <lb/>
In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, Datis duobus lateribus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>; cum ang: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/>
queratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
per superiorem analogiam, sit quarta proportionalis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
It is supposed that, in triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, from given two sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> with the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
by the above ratio, let the given fourth proportional be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> <lb/>
<omissions/> <lb/>
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is given
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s111" xml:space="preserve">
<emph style="st">omittuntur sequentia</emph>
<lb/>[<emph style="it">tr:
The following are omitted.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f051v" o="51v" n="102"/>
<pb file="add_6787_f052" o="52" n="103"/>
<pb file="add_6787_f052v" o="52v" n="104"/>
<pb file="add_6787_f053" o="53" n="105"/>
<div xml:id="echoid-div24" type="page_commentary" level="2" n="24">
<p>
<s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s112" xml:space="preserve">
Another version of the information in Add MS 6787, f. 54.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat">
5) Ad calculum sinuum trium differentiarum succedentium
<lb/>[<emph style="it">tr:
For the calculation of three successive differences of sines.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s114" xml:space="preserve">
Universalis aequatio. examinata <lb/>
per rationales progressiones, charta 6a <lb/>
et per sinuus.
<lb/>[<emph style="it">tr:
General equation, examined by rational progressions, sheet 6a, and by sines.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s115" xml:space="preserve">
Ista aequatione fit ex reductione <lb/>
superioris.
<lb/>[<emph style="it">tr:
This equation arises by reduction of the above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f053v" o="53v" n="106"/>
<pb file="add_6787_f054" o="54" n="107"/>
<div xml:id="echoid-div25" type="page_commentary" level="2" n="25">
<p>
<s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s116" xml:space="preserve">
The columns show the first, second and third differences of an interpolated table
with constant third difference.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head25" xml:space="preserve">
4)
</head>
<pb file="add_6787_f054v" o="54v" n="108"/>
<pb file="add_6787_f055" o="55" n="109"/>
<div xml:id="echoid-div26" type="page_commentary" level="2" n="26">
<p>
<s xml:id="echoid-s118" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s118" xml:space="preserve">
Formulae for the entries in a table that has been interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length;
for full details see the 'Magisteria magna', Add MS 6782, f. 107 to f. 146v. <lb/>
The third difference is constant, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi><mi>n</mi><mi>n</mi></mstyle></math>. <lb/>
The second differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, the first differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>;
these are superscripts, not powers. <lb/>
The differences alternately increase and decrease, as in a table of sines. <lb/>
Values are calculated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>n</mi></mstyle></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>n</mi></mstyle></math>th entries
using the result obtained on Add MS 6787. f. 56.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head26" xml:space="preserve">
3)
</head>
<pb file="add_6787_f055v" o="55v" n="110"/>
<pb file="add_6787_f056" o="56" n="111"/>
<div xml:id="echoid-div27" type="page_commentary" level="2" n="27">
<p>
<s xml:id="echoid-s120" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s120" xml:space="preserve">
Calculations of formulae for the rows of Pascal's triangle, beginning from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>n</mi></mstyle></math>. <lb/>
In each case the numerators are denoted <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>3</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>4</mn></msup></mrow></mstyle></math>;
these are superscripts, not powers. <lb/>
For calculations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>5</mn></msup></mrow></mstyle></math> see Add MS 6787, f. 252v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head27" xml:space="preserve">
2)
</head>
<pb file="add_6787_f056v" o="56v" n="112"/>
<pb file="add_6787_f057v" o="57v" n="113"/>
<pb file="add_6787_f058" o="58" n="114"/>
<div xml:id="echoid-div28" type="page_commentary" level="2" n="28">
<p>
<s xml:id="echoid-s122" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s122" xml:space="preserve">
On the left is a table of 'pretend' sines with constant third difference 81. <lb/>
The final row of the tables is 7240, 600, 405, 81. <lb/>
The task is to interpolate the table with two new entries between each existing entry,
using the formulae calculated in the previous sheets. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> by hypothesis. <lb/>
Thus the constant difference in the interpolated table will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>8</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn></mstyle></math>. <lb/>
Further calculations give <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math> for the first new entries
after the row beginning 6640.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head28" xml:space="preserve">
6)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s124" xml:space="preserve">
data
<lb/>[<emph style="it">tr:
given
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s125" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> per hypothesi numero dividendum <lb/>
partium.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> by hypothesis, the number of parts to be divided.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f058v" o="58v" n="115"/>
<pb file="add_6787_f059" o="59" n="116"/>
<div xml:id="echoid-div29" type="page_commentary" level="2" n="29">
<p>
<s xml:id="echoid-s126" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s126" xml:space="preserve">
At the top of the page is a table of sines for every 45 minutes, with fourth differences.
The sine of 48.45' is highlighted. <lb/>
In the centre is a table of sines for every 15 minutes, with third differences.
The sine of 48.45' is in the final row. <lb/>
At the bottom is a table of sines for every 15 minutes, calculated from the row for 48.45
on the assumption that the fourth difference is constant at
56 for the upper half of the table, 55 for the lower half.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat">
Ad calculum sinuum, per progressiones.
<lb/>[<emph style="it">tr:
For the calculation of sines, by progressisons
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s128" xml:space="preserve">
data
<lb/>[<emph style="it">tr:
given
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s129" xml:space="preserve">
E canone sinuum
<lb/>[<emph style="it">tr:
From the table of sines
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s130" xml:space="preserve">
Per caclulum
<lb/>[<emph style="it">tr:
By calculation
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f059v" o="59v" n="117"/>
<pb file="add_6787_f060" o="60" n="118"/>
<pb file="add_6787_f060v" o="60v" n="119"/>
<pb file="add_6787_f061" o="61" n="120"/>
<pb file="add_6787_f061v" o="61v" n="121"/>
<pb file="add_6787_f062" o="62" n="122"/>
<pb file="add_6787_f062v" o="62v" n="123"/>
<pb file="add_6787_f063" o="63" n="124"/>
<div xml:id="echoid-div30" type="page_commentary" level="2" n="30">
<p>
<s xml:id="echoid-s131" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s131" xml:space="preserve">
Selections of sines from the beginning, end, and middle of the table,
each showing a pattern of alternately decreasing and increasing columns of differences. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s133" xml:space="preserve">
<reg norm="Differentiae" type="abbr">Dræ</reg>
<lb/>[<emph style="it">tr:
Differences
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s134" xml:space="preserve">
In principio tablum sinuum
<lb/>[<emph style="it">tr:
From the beginning of the table of sines
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s135" xml:space="preserve">
In finis
<lb/>[<emph style="it">tr:
From the end
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s136" xml:space="preserve">
In medio
<lb/>[<emph style="it">tr:
From the middle
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s137" xml:space="preserve">
In principio iterum
<lb/>[<emph style="it">tr:
From the beginning again
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f063v" o="63v" n="125"/>
<pb file="add_6787_f064" o="64" n="126"/>
<pb file="add_6787_f064v" o="64v" n="127"/>
<pb file="add_6787_f065" o="65" n="128"/>
<pb file="add_6787_f065v" o="65v" n="129"/>
<pb file="add_6787_f066" o="66" n="130"/>
<div xml:id="echoid-div31" type="page_commentary" level="2" n="31">
<p>
<s xml:id="echoid-s138" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s138" xml:space="preserve">
The columns show the first, second, third, fourth, and fifth differences of an interpolated table
with constant fifth difference.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head30" xml:space="preserve">
6) Ad calculum sinuum 5 differentiarum succedentium
<lb/>[<emph style="it">tr:
For the calculation of 5 successive differences of sines
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s140" xml:space="preserve">
Superioris 5, aequationes probatæ sunt <lb/>
per rationales progressiones.
<lb/>[<emph style="it">tr:
The above five equations may be proved from known progressions.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f066v" o="66v" n="131"/>
<pb file="add_6787_f067" o="67" n="132"/>
<div xml:id="echoid-div32" type="page_commentary" level="2" n="32">
<p>
<s xml:id="echoid-s141" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s141" xml:space="preserve">
Third differences of the entries from Add MS 6787, f. 71.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat">
3) Tertariæ differentiæ
<lb/>[<emph style="it">tr:
Of third differences
</emph>]<lb/>
</head>
<pb file="add_6787_f067v" o="67v" n="133"/>
<pb file="add_6787_f068" o="68" n="134"/>
<div xml:id="echoid-div33" type="page_commentary" level="2" n="33">
<p>
<s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s143" xml:space="preserve">
Second differences of the entries from Add MS 6787, f. 71.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head32" xml:space="preserve" xml:lang="lat">
3) Secundariæ differentiæ
<lb/>[<emph style="it">tr:
Of second differences
</emph>]<lb/>
</head>
<pb file="add_6787_f068v" o="68v" n="135"/>
<pb file="add_6787_f069" o="69" n="136"/>
<div xml:id="echoid-div34" type="page_commentary" level="2" n="34">
<p>
<s xml:id="echoid-s145" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s145" xml:space="preserve">
First differences of the entries from Add MS 6787, f. 71.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat">
3) Primariæ differentiæ
<lb/>[<emph style="it">tr:
Of first differences
</emph>]<lb/>
</head>
<pb file="add_6787_f069v" o="69v" n="137"/>
<pb file="add_6787_f070" o="70" n="138"/>
<div xml:id="echoid-div35" type="page_commentary" level="2" n="35">
<p>
<s xml:id="echoid-s147" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s147" xml:space="preserve">
The entries from Add MS 6787, f. 71, now written over the common denominator 120.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head34" xml:space="preserve">
2)
</head>
<pb file="add_6787_f070v" o="70v" n="139"/>
<pb file="add_6787_f071" o="71" n="140"/>
<div xml:id="echoid-div36" type="page_commentary" level="2" n="36">
<p>
<s xml:id="echoid-s149" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s149" xml:space="preserve">
Formulae for the first differences in a table that has been interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length;
for full details see the 'Magisteria magna', Add MS 6782, f. 107 to f. 146v. <lb/>
The fourth differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, the third differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>,
the second differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>, the first differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>4</mn></msup></mrow></mstyle></math>;
these are superscripts, not powers. <lb/>
The difference columns increase and decrease alternately, as in a table of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head35" xml:space="preserve">
1)
</head>
<pb file="add_6787_f071v" o="71v" n="141"/>
<pb file="add_6787_f072" o="72" n="142"/>
<div xml:id="echoid-div37" type="page_commentary" level="2" n="37">
<p>
<s xml:id="echoid-s151" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s151" xml:space="preserve">
A difference table with constant fifth difference 1,
and with alternating increasing and decreasing columns as for sines.
Negative entries appear in every column except the last.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head36" xml:space="preserve">
7)
</head>
<pb file="add_6787_f072v" o="72v" n="143"/>
<pb file="add_6787_f073" o="73" n="144"/>
<pb file="add_6787_f073v" o="73v" n="145"/>
<pb file="add_6787_f074" o="74" n="146"/>
<pb file="add_6787_f074v" o="74v" n="147"/>
<pb file="add_6787_f075" o="75" n="148"/>
<pb file="add_6787_f075v" o="75v" n="149"/>
<pb file="add_6787_f076" o="76" n="150"/>
<pb file="add_6787_f076v" o="76v" n="151"/>
<pb file="add_6787_f077" o="77" n="152"/>
<pb file="add_6787_f077v" o="77v" n="153"/>
<pb file="add_6787_f078" o="78" n="154"/>
<pb file="add_6787_f078v" o="78v" n="155"/>
<pb file="add_6787_f079" o="79" n="156"/>
<pb file="add_6787_f079v" o="79v" n="157"/>
<pb file="add_6787_f080" o="80" n="158"/>
<pb file="add_6787_f080v" o="80v" n="159"/>
<pb file="add_6787_f081" o="81" n="160"/>
<pb file="add_6787_f081v" o="81v" n="161"/>
<pb file="add_6787_f082" o="82" n="162"/>
<pb file="add_6787_f082v" o="82v" n="163"/>
<pb file="add_6787_f083" o="83" n="164"/>
<pb file="add_6787_f083v" o="83v" n="165"/>
<p>
<s xml:id="echoid-s153" xml:space="preserve">
Mr Vincent I received your last letter being wihout water by Mr <lb/>
Fowler the 28th of March last. Other letter I received about 6 weekes <lb/>
past of Mr Cook. I received other some whether once or twice I can <lb/>
not tell about two years since as I gesse or I thinke some what later. <lb/>
The first letters by reason of some way into the country I lost <lb/>
the means that brought them convey my answer.
</s>
</p>
<pb file="add_6787_f084" o="84" n="166"/>
<pb file="add_6787_f084v" o="84v" n="167"/>
<pb file="add_6787_f085" o="85" n="168"/>
<pb file="add_6787_f085v" o="85v" n="169"/>
<pb file="add_6787_f086" o="86" n="170"/>
<pb file="add_6787_f086v" o="86v" n="171"/>
<pb file="add_6787_f087" o="87" n="172"/>
<pb file="add_6787_f087v" o="87v" n="173"/>
<pb file="add_6787_f088" o="88" n="174"/>
<pb file="add_6787_f088v" o="88v" n="175"/>
<pb file="add_6787_f089" o="89" n="176"/>
<pb file="add_6787_f089v" o="89v" n="177"/>
<pb file="add_6787_f090" o="90" n="178"/>
<pb file="add_6787_f090v" o="90v" n="179"/>
<pb file="add_6787_f091" o="91" n="180"/>
<pb file="add_6787_f091v" o="91v" n="181"/>
<pb file="add_6787_f092" o="92" n="182"/>
<pb file="add_6787_f092v" o="92v" n="183"/>
<pb file="add_6787_f093" o="93" n="184"/>
<pb file="add_6787_f093v" o="93v" n="185"/>
<pb file="add_6787_f094" o="94" n="186"/>
<pb file="add_6787_f094v" o="94v" n="187"/>
<pb file="add_6787_f095" o="95" n="188"/>
<pb file="add_6787_f095v" o="95v" n="189"/>
<pb file="add_6787_f096" o="96" n="190"/>
<pb file="add_6787_f096v" o="96v" n="191"/>
<pb file="add_6787_f097" o="97" n="192"/>
<pb file="add_6787_f097v" o="97v" n="193"/>
<pb file="add_6787_f098" o="98" n="194"/>
<pb file="add_6787_f098v" o="98v" n="195"/>
<pb file="add_6787_f099" o="99" n="196"/>
<pb file="add_6787_f099v" o="99v" n="197"/>
<pb file="add_6787_f100" o="100" n="198"/>
<pb file="add_6787_f100v" o="100v" n="199"/>
<pb file="add_6787_f101" o="101" n="200"/>
<pb file="add_6787_f101v" o="101v" n="201"/>
<pb file="add_6787_f102" o="102" n="202"/>
<pb file="add_6787_f102v" o="102v" n="203"/>
<pb file="add_6787_f103" o="103" n="204"/>
<pb file="add_6787_f103v" o="103v" n="205"/>
<pb file="add_6787_f104" o="104" n="206"/>
<pb file="add_6787_f104v" o="104v" n="207"/>
<pb file="add_6787_f105" o="105" n="208"/>
<pb file="add_6787_f105v" o="105v" n="209"/>
<pb file="add_6787_f106" o="106" n="210"/>
<pb file="add_6787_f106v" o="106v" n="211"/>
<pb file="add_6787_f107" o="107" n="212"/>
<pb file="add_6787_f107v" o="107v" n="213"/>
<pb file="add_6787_f108" o="108" n="214"/>
<pb file="add_6787_f108v" o="108v" n="215"/>
<pb file="add_6787_f109" o="109" n="216"/>
<pb file="add_6787_f109v" o="109v" n="217"/>
<pb file="add_6787_f110" o="110" n="218"/>
<pb file="add_6787_f110v" o="110v" n="219"/>
<pb file="add_6787_f111" o="111" n="220"/>
<pb file="add_6787_f111v" o="111v" n="221"/>
<div xml:id="echoid-div38" type="page_commentary" level="2" n="38">
<p>
<s xml:id="echoid-s154" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s154" xml:space="preserve">
S. W. R. was presumably Sir Walter Ralegh.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s156" xml:space="preserve">
S.W.R. <lb/>
L. Guy. <lb/>
L. Cobsam. <lb/>
M. G. Brooke. <lb/>
S. Ar. Gorge. <lb/>
S. Ar. Sanaher. <lb/>
S. Griffin Marcum
</s>
</p>
<pb file="add_6787_f112" o="112" n="222"/>
<p>
<s xml:id="echoid-s157" xml:space="preserve">
At Rouhampton: : at Chalkhill <lb/>
house <lb/>
half of <lb/>
party
</s>
</p>
<pb file="add_6787_f112v" o="112v" n="223"/>
<pb file="add_6787_f113" o="113" n="224"/>
<p>
<s xml:id="echoid-s158" xml:space="preserve">
Martins booke <lb/>
Whitewell of the [???] <lb/>
[???] of the vine <lb/>
Tobacco
</s>
</p>
<pb file="add_6787_f113v" o="113v" n="225"/>
<pb file="add_6787_f114" o="114" n="226"/>
<head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat">
Archimedes. de cylindro <lb/>
pa. 26.
<lb/>[<emph style="it">tr:
Archimedes, <emph style="it">De cylindro</emph>, page 26.
</emph>]<lb/>
</head>
<pb file="add_6787_f114v" o="114v" n="227"/>
<pb file="add_6787_f115" o="115" n="228"/>
<head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat">
Archimedes. de cylindro <lb/>
pa. 19.
<lb/>[<emph style="it">tr:
Archimedes, <emph style="it">De cylindro</emph>, page 19.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s159" xml:space="preserve">
pa. 19
<lb/>[<emph style="it">tr:
page 19
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s160" xml:space="preserve">
coni igitur æquales
<lb/>[<emph style="it">tr:
therefore the cones are equal
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s161" xml:space="preserve">
pa. 26. pro. 4
<lb/>[<emph style="it">tr:
page 26, Proposition 4
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f115v" o="115v" n="229"/>
<pb file="add_6787_f116" o="116" n="230"/>
<head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat">
Archimedes. De cylindro. pag. 14.
<lb/>[<emph style="it">tr:
Archimedes, <emph style="it">De cylindro</emph>, page 14.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s162" xml:space="preserve">
pa. 22
<lb/>[<emph style="it">tr:
page 22
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s163" xml:space="preserve">
pa. 23
<lb/>[<emph style="it">tr:
page 23
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s164" xml:space="preserve">
pa. 16
<lb/>[<emph style="it">tr:
page 16
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s165" xml:space="preserve">
superficies <lb/>
coni
<lb/>[<emph style="it">tr:
surface of a cone
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s166" xml:space="preserve">
periferia circuli (a), et basis <lb/>
coni eadem est
<lb/>[<emph style="it">tr:
the periphery of the circle (a) and the base of the cone are the same
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s167" xml:space="preserve">
pa. 18
<lb/>[<emph style="it">tr:
page 18
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f116v" o="116v" n="231"/>
<pb file="add_6787_f117" o="117" n="232"/>
<pb file="add_6787_f117v" o="117v" n="233"/>
<pb file="add_6787_f118" o="118" n="234"/>
<pb file="add_6787_f118v" o="118v" n="235"/>
<pb file="add_6787_f119" o="119" n="236"/>
<pb file="add_6787_f119v" o="119v" n="237"/>
<div xml:id="echoid-div39" type="page_commentary" level="2" n="39">
<p>
<s xml:id="echoid-s168" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s168" xml:space="preserve">
The verses referred to on this page are the following.
</s>
<lb/>
<quote>
Numbers 35.30: I any one kill a person, the murderer shall be put to death on the evidence of witnesses;
but no person shall be put to death on the tetimony of one witness.
</quote>
<lb/>
<quote>
Deuteronomy 17.6: On the evidence of two witnesses or of three witnesses he that is to die shall be put to death;
a person shall not be put to death on the evidence of one witness.
</quote>
<lb/>
<quote>
Deuteronomy 19.5: as when a man goes into the forest with his neighbour to cut wood,
and his hand swings the axe to cut down the tree, and the head slips from the handle
and strikes his neighbour so that he dies–he may flee to one of these cities and save his life;
</quote>
<lb/>
<quote>
Matthew 18.15–17: If your brother sins against you, go and tell him his fault, between you and him alone.
If he listens to you, you have gained yoru brother. But if he does not listen,
take one or two others along with you, that every word may be confirmed by the evidence of two or three witnesses.
If he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.
</quote>
<lb/>
<quote>
John 8.16–18: Yet even if I do judge, my judgement is true, for it is not I alone that judge,
but I and he who sent me. In your law it is written that the testimony of two men is true;
</quote>
<lb/>
<quote>
2 Corinthians 13.1: This is the third time I am comoing to you.
Any charge must be sustained by the evidence of two or three witnesses.
</quote>
<lb/>
<quote>
Matthew 7: 12: So whatever you wish that men would do to you, do so to them; for this is the law and the prophets.
</quote>
<lb/>
<quote>
Luke 6.13: And when it was day, he called his disciples, and chose from them twelve, whom he named apostles.
</quote>
<lb/>
<quote>
Tobit 4.15: And what you hate, do not do to anyone. Do not drink wine to excess or let drunkenness go with you on your way.
</quote>
<lb/>
<quote>
Matthew 7.2: For with the judgement you pronounce you will be judged,
and the measure you give will be the the measure you get.
</quote>
<lb/>
<quote>
Proverbs 28.27: He who gives to the poor will not want, but he who hides his eyes will get many a curse.
</quote>
<lb/>
<quote>
Ecclesiasticus 27.26: Whoever digs a pit will fall into it, and whoever sets a snare will be caught in it.
</quote>
<lb/>
<quote>
Proverbs 19.5: A false witness will not go unpunished, and he who utters lies will not escape.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s170" xml:space="preserve">
Numbers. 35, 30. <lb/>
Deut. 17, 6. <lb/>
* Deut. 19, 5. <lb/>
Math. 18, 15. 16, 17, <lb/>
John. 8, 16. 17. 18. <lb/>
2. Cor, 13, 1, <lb/>
Math. 7, 12. <lb/>
Luke. 6, 13. <lb/>
Tobit. 4, 15, <lb/>
Math. 7, 2. <lb/>
prob. 28, 27. <lb/>
Eccus. 27, 26. <lb/>
prob. 19. 5
</s>
</p>
<pb file="add_6787_f120" o="120" n="238"/>
<pb file="add_6787_f120v" o="120v" n="239"/>
<pb file="add_6787_f121" o="121" n="240"/>
<pb file="add_6787_f121v" o="121v" n="241"/>
<pb file="add_6787_f122" o="122" n="242"/>
<pb file="add_6787_f122v" o="122v" n="243"/>
<pb file="add_6787_f123" o="123" n="244"/>
<pb file="add_6787_f123v" o="123v" n="245"/>
<pb file="add_6787_f124" o="124" n="246"/>
<pb file="add_6787_f124v" o="124v" n="247"/>
<pb file="add_6787_f125" o="125" n="248"/>
<pb file="add_6787_f125v" o="125v" n="249"/>
<pb file="add_6787_f126" o="126" n="250"/>
<pb file="add_6787_f126v" o="126v" n="251"/>
<pb file="add_6787_f127" o="127" n="252"/>
<pb file="add_6787_f127v" o="127v" n="253"/>
<pb file="add_6787_f128" o="128" n="254"/>
<pb file="add_6787_f128v" o="128v" n="255"/>
<pb file="add_6787_f129" o="129" n="256"/>
<pb file="add_6787_f129v" o="129v" n="257"/>
<pb file="add_6787_f130" o="130" n="258"/>
<pb file="add_6787_f130v" o="130v" n="259"/>
<pb file="add_6787_f131" o="131" n="260"/>
<div xml:id="echoid-div40" type="page_commentary" level="2" n="40">
<p>
<s xml:id="echoid-s171" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s171" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>
this triangle is to be found on page 421.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head40" xml:space="preserve" xml:lang="lat">
Vieta. resp. pag. 42, b. A. De Triangulis rectangulis sphaericis
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 42v. On right-angled spherical triangles.
</emph>]<lb/>
</head>
<pb file="add_6787_f131v" o="131v" n="261"/>
<pb file="add_6787_f132" o="132" n="262"/>
<pb file="add_6787_f132v" o="132v" n="263"/>
<pb file="add_6787_f133" o="133" n="264"/>
<pb file="add_6787_f133v" o="133v" n="265"/>
<pb file="add_6787_f134" o="134" n="266"/>
<pb file="add_6787_f134v" o="134v" n="267"/>
<pb file="add_6787_f135" o="135" n="268"/>
<pb file="add_6787_f135v" o="135v" n="269"/>
<pb file="add_6787_f136" o="136" n="270"/>
<pb file="add_6787_f136v" o="136v" n="271"/>
<pb file="add_6787_f137" o="137" n="272"/>
<pb file="add_6787_f137v" o="137v" n="273"/>
<pb file="add_6787_f138" o="138" n="274"/>
<pb file="add_6787_f138v" o="138v" n="275"/>
<pb file="add_6787_f139" o="139" n="276"/>
<pb file="add_6787_f139v" o="139v" n="277"/>
<pb file="add_6787_f140" o="140" n="278"/>
<pb file="add_6787_f140v" o="140v" n="279"/>
<pb file="add_6787_f141" o="141" n="280"/>
<pb file="add_6787_f141v" o="141v" n="281"/>
<pb file="add_6787_f142" o="142" n="282"/>
<pb file="add_6787_f142v" o="142v" n="283"/>
<pb file="add_6787_f143" o="143" n="284"/>
<pb file="add_6787_f143v" o="143v" n="285"/>
<pb file="add_6787_f144" o="144" n="286"/>
<pb file="add_6787_f144v" o="144v" n="287"/>
<pb file="add_6787_f145" o="145" n="288"/>
<pb file="add_6787_f145v" o="145v" n="289"/>
<pb file="add_6787_f146" o="146" n="290"/>
<pb file="add_6787_f146v" o="146v" n="291"/>
<pb file="add_6787_f147" o="147" n="292"/>
<pb file="add_6787_f147v" o="147v" n="293"/>
<pb file="add_6787_f148" o="148" n="294"/>
<pb file="add_6787_f148v" o="148v" n="295"/>
<pb file="add_6787_f149" o="149" n="296"/>
<pb file="add_6787_f149v" o="149v" n="297"/>
<pb file="add_6787_f150" o="150" n="298"/>
<pb file="add_6787_f150v" o="150v" n="299"/>
<pb file="add_6787_f151" o="151" n="300"/>
<pb file="add_6787_f151v" o="151v" n="301"/>
<pb file="add_6787_f152" o="152" n="302"/>
<pb file="add_6787_f152v" o="152v" n="303"/>
<pb file="add_6787_f153" o="153" n="304"/>
<pb file="add_6787_f153v" o="153v" n="305"/>
<pb file="add_6787_f154" o="154" n="306"/>
<pb file="add_6787_f154v" o="154v" n="307"/>
<pb file="add_6787_f155" o="155" n="308"/>
<pb file="add_6787_f155v" o="155v" n="309"/>
<pb file="add_6787_f156" o="156" n="310"/>
<pb file="add_6787_f156v" o="156v" n="311"/>
<pb file="add_6787_f157" o="157" n="312"/>
<pb file="add_6787_f157v" o="157v" n="313"/>
<pb file="add_6787_f158" o="158" n="314"/>
<pb file="add_6787_f158v" o="158v" n="315"/>
<pb file="add_6787_f159" o="159" n="316"/>
<pb file="add_6787_f159v" o="159v" n="317"/>
<pb file="add_6787_f160" o="160" n="318"/>
<div xml:id="echoid-div41" type="page_commentary" level="2" n="41">
<p>
<s xml:id="echoid-s173" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s173" xml:space="preserve">
At the end of Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593)
there is a lengthy section entitled '<foreign xml:lang="gre">Eis procheiron scholia</foreign>.
Section IV is on spherical geometry. The 18th and final proposition contains four statements,
which Harriot here translated into symbolic notation.
</s>
<lb/>
<quote xml:lang="lat">
18 Sit triangulum sphaericum ABD, &amp; in peripheria BD cadat segmentum orthogonii AC. <lb/>
Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC,
sicut prosinum peripheria AB ad prosinum peripheri AD. <lb/>
Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD,
sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD. <lb/>
Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D. <lb/>
Denique &amp; quarto dico esse sinum anguli BAC ad sinum anguli DAC,
sicut transsinuosam anguli D ad transsinuosam anguli B.
</quote>
<lb/>
<quote>
18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC. <lb/>
First I say that the secant of angle BAC to the secant of angle DAC is
as the tangent of the arc AB to the tangent of the arc AD. <lb/>
Second I say that the secant of the arc CB to the secant of the arc CD is
as the secant of the arc AB to the secant of the arc AD. <lb/>
Third I say that the sine of CD to the sine of DB is
as the tangent of angle B to the tangent of angle D. <lb/>
Fourth and last I say that the sine of angle BAC to the sine of angle DAC is
as the secant of angle D to the secant of angle B.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat">
Demonstratio eorum quæ desiderant <lb/>
in Vieta. lib. 8. respons. pag. 41.b. <lb/>
sectione 18.
<lb/>[<emph style="it">tr:
Demonstration of what is missing in Viète, Responsorum liber VIII, page 41v, section 18.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s175" xml:space="preserve">
Hinc apparet mendam <lb/>
esse apud Vietam. nam <lb/>
ille:
<lb/>[<emph style="it">tr:
Here is is clear that it is wrong in Viète, for this:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s176" xml:space="preserve">
Istæ quatuor conclusiones <lb/>
sexeis variari possunt ut <lb/>
ex analogijs rectangulorm est <lb/>
manifestum.
<lb/>[<emph style="it">tr:
These four conclusions have six variations as is clear from the ratios for products.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f160v" o="160v" n="319"/>
<pb file="add_6787_f161" o="161" n="320"/>
<div xml:id="echoid-div42" type="page_commentary" level="2" n="42">
<p>
<s xml:id="echoid-s177" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s177" xml:space="preserve">
Here Harriot examines a statement that appears as Proposition VI of the 'Dati sexti', Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
See also Add MS 6782, f. 439.
</s>
<lb/>
<quote xml:lang="lat">
VI. <lb/>
Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem,
dantur singulares peripheriæ.
</quote>
<lb/>
<quote>
VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually.
</quote>
<lb/>
<s xml:id="echoid-s178" xml:space="preserve">
The reference to Pitiscus is to
<emph style="it">Trigonometria: sive de solutione triangulorum tractarus brevis et perpsicuus</emph> (1595).
</s>
<lb/>
<s xml:id="echoid-s179" xml:space="preserve">
The reference to Lansberg is to <emph style="it">Triangulorum geometriae libri quatuor</emph> (1591).
</s>
<lb/>
<s xml:id="echoid-s180" xml:space="preserve">
The reference to Regiomontanus to <emph style="it">De triangulis omnimodis libri quinque</emph> ([1464], 1533, 1561),
Book IV, Proposition 31.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
Data differentia.) <lb/>
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> differentia <emph style="super">duarum</emph> peripheriam. <lb/>
ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/>
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/>
Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
Cætera ut supra. et habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> arcus. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus minor <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis <lb/>
Tum etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior quæsitis.
<head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat">
12.) Data summa vel differentia duarum periferiarum, <lb/>
quarum sinus datam habeant rationem: dantur singulares <lb/>
peripheriæ.
<lb/>[<emph style="it">tr:
Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are given.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s182" xml:space="preserve">
per Tangentes exhibit <lb/>
Vieta in responsis pag. 37. <lb/>
Pitiscus pag. 92. <lb/>
Lansbergis. pag. 162
<lb/>[<emph style="it">tr:
Shown by tangents <lb/>
by Viète in Responsorum, page 37, <lb/>
Pitiscus, page 92, <lb/>
Lansberg, page 162.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s183" xml:space="preserve">
Quæ modo usui accomodatior est, quam per <lb/>
sinus solos, quando tangentibus ut liceat.
<lb/>[<emph style="it">tr:
Which method of use is more convenient than by sines alone, when by tangents, as one pleases.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s184" xml:space="preserve">
Sed quando non licet Tangentibus uti, modus per solos sinus (etsi laboriosior) <lb/>
adhibendus est. Exhibatur a Regiomontano lib. 4. prop. 31. de triangulis <lb/>
Modus ille hic apponitur paucis explicatur.
<lb/>[<emph style="it">tr:
But when one does not want to use tangents, the method by sines alone (though more laborious) is shown.
It is given by Regiomontanus, Book IV, Proposition 31 of De triangulis.
That method set out here is explained a little.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s185" xml:space="preserve">
Data summa.) <lb/>
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> summa duarum peripheriam. <lb/>
ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/>
fiat: <lb/>
Datur ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>, nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>-</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> est sinus dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>+</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> arcus minor quæsita <lb/>
Tum etiam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis.
<lb/>[<emph style="it">tr:
Given the sum.) <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be the sum of the two arcs,
and the ratio of the two sines of the sought arcs as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/>
construct: <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> is given, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>-</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the sine of half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math> is gien, for the sine of the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>, or half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>+</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> the lesser arc sought. <lb/>
Then also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s186" xml:space="preserve">
Data differentia.) <lb/>
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> differentia <emph style="super">duarum</emph> peripheriam. <lb/>
ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/>
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/>
Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
Cætera ut supra. et habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> arcus. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus minor <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis <lb/>
Tum etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior quæsitis.
<lb/>[<emph style="it">tr:
Given the difference.) <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be the difference of the two sought arcs,
and the ratio of the sines of the sought arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
sine of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> is given, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/>
Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math> is given, for the sine of the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>, or half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/>
The rest as above. And we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math>. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the lesser arc. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought. <lb/>
Then also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f161v" o="161v" n="321"/>
<pb file="add_6787_f162" o="162" n="322"/>
<head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat">
13.) Data summa vel differentia <lb/>
duarum peripheriarum, <lb/>
quarum sinus datam habeant <lb/>
rationem: dantur singulares <lb/>
peripheriæ.
<lb/>[<emph style="it">tr:
Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are given.
</emph>]<lb/>
</head>
<pb file="add_6787_f162v" o="162v" n="323"/>
<pb file="add_6787_f163" o="163" n="324"/>
<pb file="add_6787_f163v" o="163v" n="325"/>
<pb file="add_6787_f164" o="164" n="326"/>
<pb file="add_6787_f164v" o="164v" n="327"/>
<pb file="add_6787_f165" o="165" n="328"/>
<pb file="add_6787_f165v" o="165v" n="329"/>
<pb file="add_6787_f166" o="166" n="330"/>
<pb file="add_6787_f166v" o="166v" n="331"/>
<pb file="add_6787_f167" o="167" n="332"/>
<div xml:id="echoid-div43" type="page_commentary" level="2" n="43">
<p>
<s xml:id="echoid-s187" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s187" xml:space="preserve">
This page refers to Propositions 48 and 49 of Book III of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
III.48
With the same things being so, it must be shown that the straight lines drawn from the point of contact
to the points produced by the application make equal angles with the tangent.
</quote>
<lb/>
<quote>
III.49
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s189" xml:space="preserve">
Sit ellipsis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mi>k</mi></mstyle></math>: <lb/>
cuius axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math> <lb/>
centroides puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. <lb/>
diametroides, recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math> <lb/>
centrum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
circulus circa axim, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>d</mi><mi>k</mi></mstyle></math>. <lb/>
circulus circa diametroides, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>w</mi><mi>a</mi></mstyle></math>. <lb/>
recta contingens ellipsin in <lb/>
puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>d</mi></mstyle></math>. <lb/>
perpendicularis a centroide <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math> <lb/>
ad illam fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>e</mi></mstyle></math> <lb/>
per 49.3 conicorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>g</mi></mstyle></math> est angulus <lb/>
rectus <lb/>
ergo punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in periferia.
<lb/>[<emph style="it">tr:
Let there be an ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mi>k</mi></mstyle></math> with axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, centroids at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>, diametroid the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>,
centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. The circle about the axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>d</mi><mi>k</mi></mstyle></math>; the circle about the diametroid is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>w</mi><mi>a</mi></mstyle></math>;
the line touching the ellipse at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>d</mi></mstyle></math>. Perpendicular to it from the centroid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>,
construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>e</mi></mstyle></math>. By Proposition III.49 of the Conics, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>g</mi></mstyle></math> is a right angle. Therefore, the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
is on the periphery.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s190" xml:space="preserve">
hinc sequitur <lb/>
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>w</mi></mstyle></math> producatur ad periferium in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> <lb/>
et ducatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>x</mi></mstyle></math> parallela ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/>
continget etiam ellipsin
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>w</mi></mstyle></math> is produced to the perpiphery at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>x</mi></mstyle></math> is taken paralle to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, it will also touch the ellipse.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s191" xml:space="preserve">
Si puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> in periferia <lb/>
connectantur <lb/>
linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>x</mi></mstyle></math> transibit per alterum <lb/>
centroides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
If the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> in the periphery are joined, the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>x</mi></mstyle></math> will pass through the other centroid, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s192" xml:space="preserve">
Hinc. conclusio <lb/>
Si circa <emph style="super">axim</emph> ellipseos describatur circulus <lb/>
et in circulo inscribatur parallelogrammum <lb/>
ita ut duo latera transeant per centroides: <lb/>
reliqua duo contingent ellipsin. <lb/>
et si duo latera contingent ellipsin; reliqua <lb/>
duo transibunt per centroides. <lb/>
Ita etam: <lb/>
Si circa axim Hyperboles &amp;c.
<lb/>[<emph style="it">tr:
Hence, the conclusion. <lb/>
If around the axis of an ellipse there is described a circle, and in the circle there is inscribed a parallelogram
so that two sides pass thorugh the centroids, the other two are tangents to the ellipse.
And if two sides are tangents to the ellipse, the other two will pass throug the centroids. <lb/>
Thus also: if around the axis of a hyperbola, etc.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s193" xml:space="preserve">
Alia conclusiones <lb/>
iisdem positis. <lb/>
per 48.3. conicorum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>f</mi></mstyle></math> <lb/>
faciunt æquale angulos ad <lb/>
contingentem. <lb/>
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> agantur <lb/>
parallelæ ad contingentes: <lb/>
puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sunt in peri- <lb/>
feria cuius diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Other conclusions form the same assumptions. <lb/>
By Proposition III.48 of the Cinics, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>f</mi></mstyle></math> make equal angles to the tangent. <lb/>
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> are taken parallel to the tangents, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> are on
the circumference whose diamter is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s194" xml:space="preserve">
Conveniat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>z</mi></mstyle></math> cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>w</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>. <lb/>
Dico quod: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>t</mi><mo>=</mo><mi>v</mi><mi>a</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>-</mo><mi>f</mi><mi>w</mi></mstyle></math> <lb/>
nam: <lb/>
anguli:
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>z</mi></mstyle></math> meet with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>w</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>. I say that: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>t</mi><mo>=</mo><mi>v</mi><mi>a</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>-</mo><mi>f</mi><mi>w</mi></mstyle></math> <lb/>
for the angles:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s195" xml:space="preserve">
Dico etiam:
<lb/>[<emph style="it">tr:
I also say:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s196" xml:space="preserve">
Dico etiam:
<lb/>[<emph style="it">tr:
I also say:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f167v" o="167v" n="333"/>
<pb file="add_6787_f168" o="168" n="334"/>
<pb file="add_6787_f168v" o="168v" n="335"/>
<pb file="add_6787_f169" o="169" n="336"/>
<pb file="add_6787_f169v" o="169v" n="337"/>
<pb file="add_6787_f170" o="170" n="338"/>
<pb file="add_6787_f170v" o="170v" n="339"/>
<pb file="add_6787_f171" o="171" n="340"/>
<pb file="add_6787_f171v" o="171v" n="341"/>
<pb file="add_6787_f172" o="172" n="342"/>
<pb file="add_6787_f172v" o="172v" n="343"/>
<pb file="add_6787_f173" o="173" n="344"/>
<pb file="add_6787_f173v" o="173v" n="345"/>
<pb file="add_6787_f174" o="174" n="346"/>
<pb file="add_6787_f174v" o="174v" n="347"/>
<pb file="add_6787_f175" o="175" n="348"/>
<head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat">
a.) 1.) De anomalijs Kepler. 284. 290.
<lb/>[<emph style="it">tr:
Kepler, De anomalijs, pages 284, 290.
</emph>]<lb/>
</head>
<pb file="add_6787_f175v" o="175v" n="349"/>
<pb file="add_6787_f176" o="176" n="350"/>
<head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat">
De anomalijs (a.1.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s197" xml:space="preserve">
Quod demonstrandum fuerit
<lb/>[<emph style="it">tr:
Which was to be proved
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s198" xml:space="preserve">
4<emph style="super">o</emph>.) ut supra 5<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
4 and the rest as above, and 5.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f176v" o="176v" n="351"/>
<pb file="add_6787_f177" o="177" n="352"/>
<head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat">
a.1 De anomalijs
</head>
<pb file="add_6787_f177v" o="177v" n="353"/>
<pb file="add_6787_f178" o="178" n="354"/>
<head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat">
a.1) De anomalijs 284.290.
</head>
<pb file="add_6787_f178v" o="178v" n="355"/>
<pb file="add_6787_f179" o="179" n="356"/>
<head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat">
a.1. Aliter et optime
</head>
<pb file="add_6787_f179v" o="179v" n="357"/>
<pb file="add_6787_f180" o="180" n="358"/>
<head xml:id="echoid-head49" xml:space="preserve" xml:lang="lat">
a.1. De anomalijs 299.
</head>
<pb file="add_6787_f180v" o="180v" n="359"/>
<pb file="add_6787_f181" o="181" n="360"/>
<head xml:id="echoid-head50" xml:space="preserve" xml:lang="lat">
a.1	De anomalijs Emendatur. <lb/>
Keplerus. 299.
<lb/>[<emph style="it">tr:
De anomalijs. Amended. Kepler, page 299.
</emph>]<lb/>
</head>
<pb file="add_6787_f181v" o="181v" n="361"/>
<pb file="add_6787_f182" o="182" n="362"/>
<head xml:id="echoid-head51" xml:space="preserve" xml:lang="lat">
a.2	De anomalijs
</head>
<pb file="add_6787_f182v" o="182v" n="363"/>
<pb file="add_6787_f183" o="183" n="364"/>
<head xml:id="echoid-head52" xml:space="preserve" xml:lang="lat">
a.3.) De anomalijs
</head>
<pb file="add_6787_f183v" o="183v" n="365"/>
<pb file="add_6787_f184" o="184" n="366"/>
<head xml:id="echoid-head53" xml:space="preserve" xml:lang="lat">
Kepler. pa. 290.
</head>
<pb file="add_6787_f184v" o="184v" n="367"/>
<pb file="add_6787_f185" o="185" n="368"/>
<pb file="add_6787_f185v" o="185v" n="369"/>
<pb file="add_6787_f186" o="186" n="370"/>
<pb file="add_6787_f186v" o="186v" n="371"/>
<head xml:id="echoid-head54" xml:space="preserve" xml:lang="lat">
299. Kepr.
<lb/>[<emph style="it">tr:
Kepler, page 299.
</emph>]<lb/>
</head>
<pb file="add_6787_f187" o="187" n="372"/>
<head xml:id="echoid-head55" xml:space="preserve" xml:lang="lat">
299. <lb/>
Charta. <lb/>
(a.1) de <lb/>
Anomalijs
<lb/>[<emph style="it">tr:
Page 299, Sheet (a.1), De anomalijs
</emph>]<lb/>
</head>
<pb file="add_6787_f187v" o="187v" n="373"/>
<pb file="add_6787_f188" o="188" n="374"/>
<pb file="add_6787_f188v" o="188v" n="375"/>
<pb file="add_6787_f189" o="189" n="376"/>
<pb file="add_6787_f189v" o="189v" n="377"/>
<pb file="add_6787_f190" o="190" n="378"/>
<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 text referred to here is Johan Philip Lansberg,
<emph style="it">Triangulorum geometriae libri quatuor</emph> (1591).
Lansberg 19s rules for spherical triangles are in Book 4, pages 167–207.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head56" xml:space="preserve" xml:lang="lat">
Anapherosis trianguli obliquanguli <lb/>
Lansbergi
<lb/>[<emph style="it">tr:
Anapherosis for oblique-angled triangles, from Lansberg
</emph>]<lb/>
</head>
<p xml:lang="lat">
conversio
<lb/>[<emph style="it">tr:
conversion
</emph>]<lb/>
</p>
<pb file="add_6787_f190v" o="190v" n="379"/>
<pb file="add_6787_f191" o="191" n="380"/>
<div xml:id="echoid-div45" type="page_commentary" level="2" n="45">
<p>
<s xml:id="echoid-s201" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s201" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>,
diagrams related to the one drawn here are to be found on pages 423 and 424.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head57" xml:space="preserve" xml:lang="lat">
Anapherosis Trianguli Vietani
<lb/>[<emph style="it">tr:
Anapherosis in a triangle of Viète
</emph>]<lb/>
</head>
<pb file="add_6787_f191v" o="191v" n="381"/>
<pb file="add_6787_f192" o="192" n="382"/>
<pb file="add_6787_f192v" o="192v" n="383"/>
<pb file="add_6787_f193" o="193" n="384"/>
<div xml:id="echoid-div46" type="page_commentary" level="2" n="46">
<p>
<s xml:id="echoid-s203" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s203" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>,
diagrams related to the one drawn here are to be found on pages 423 and 424.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f193v" o="193v" n="385"/>
<pb file="add_6787_f194" o="194" n="386"/>
<div xml:id="echoid-div47" type="page_commentary" level="2" n="47">
<p>
<s xml:id="echoid-s205" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s205" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>,
diagrams related to the one drawn here are to be found on pages 423 and 424.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat">
Conversio triangul Vietani
<lb/>[<emph style="it">tr:
Conversion of Viète's triangle
</emph>]<lb/>
</head>
<pb file="add_6787_f194v" o="194v" n="387"/>
<pb file="add_6787_f195" o="195" n="388"/>
<div xml:id="echoid-div48" type="page_commentary" level="2" n="48">
<p>
<s xml:id="echoid-s207" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s207" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>,
the diagram drawn here is to be found on pages 423 and 424.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f195v" o="195v" n="389"/>
<pb file="add_6787_f196" o="196" n="390"/>
<div xml:id="echoid-div49" type="page_commentary" level="2" n="49">
<p>
<s xml:id="echoid-s209" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s209" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>,
diagrams related to the one drawn here are to be found on pages 421 to 426.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f196v" o="196v" n="391"/>
<pb file="add_6787_f197" o="197" n="392"/>
<div xml:id="echoid-div50" type="page_commentary" level="2" n="50">
<p>
<s xml:id="echoid-s211" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s211" xml:space="preserve">
The text referred to here is Johan Philip Lansberg,
<emph style="it">Triangulorum geometriae libri quatuor</emph> (1591).
Lansberg 19s rule for finding a side of a spherical triangles, given its angles,
is the final rule in Book 4, on page 201.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat">
Datis tribus angulis quaesitur latus. Lansbergi trianguli<lb/>
solutio.
<lb/>[<emph style="it">tr:
Given three angles, a side is sought. Solution from Lansberg, Triangulorum geometriae.
</emph>]<lb/>
</head>
<pb file="add_6787_f197v" o="197v" n="393"/>
<pb file="add_6787_f198" o="198" n="394"/>
<div xml:id="echoid-div51" type="page_commentary" level="2" n="51">
<p>
<s xml:id="echoid-s213" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s213" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
The table relates to the various diagrams in Add MS 6787, f. 191 to f. 195.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat">
De Anapherosi et conversione traingulorum
<lb/>[<emph style="it">tr:
On Anapherosis and the conversion of triangles
</emph>]<lb/>
</head>
<pb file="add_6787_f198v" o="198v" n="395"/>
<pb file="add_6787_f199" o="199" n="396"/>
<div xml:id="echoid-div52" type="page_commentary" level="2" n="52">
<p>
<s xml:id="echoid-s215" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s215" xml:space="preserve">
Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>
the triangle referred to here is to be found on page 424.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat">
Datis tribus lateri, quaeruntur anguli
<lb/>[<emph style="it">tr:
Given three sides, the angles the angles are sought.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s217" xml:space="preserve">
Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is sought
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f199v" o="199v" n="397"/>
<pb file="add_6787_f200" o="200" n="398"/>
<p>
<s xml:id="echoid-s218" xml:space="preserve">
Wernerus <emph style="st">discipulus Regiomontani</emph><lb/>
Duiditius <lb/>
Mr Savill <lb/>
Mr Warner
</s>
</p>
<pb file="add_6787_f200v" o="200v" n="399"/>
<pb file="add_6787_f201" o="201" n="400"/>
<pb file="add_6787_f201v" o="201v" n="401"/>
<pb file="add_6787_f202" o="202" n="402"/>
<pb file="add_6787_f202v" o="202v" n="403"/>
<pb file="add_6787_f203" o="203" n="404"/>
<pb file="add_6787_f203v" o="203v" n="405"/>
<pb file="add_6787_f204" o="204" n="406"/>
<pb file="add_6787_f204v" o="204v" n="407"/>
<pb file="add_6787_f205" o="205" n="408"/>
<pb file="add_6787_f205v" o="205v" n="409"/>
<pb file="add_6787_f206" o="206" n="410"/>
<pb file="add_6787_f206v" o="206v" n="411"/>
<pb file="add_6787_f207" o="207" n="412"/>
<pb file="add_6787_f207v" o="207v" n="413"/>
<pb file="add_6787_f208" o="208" n="414"/>
<div xml:id="echoid-div53" type="page_commentary" level="2" n="53">
<p>
<s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s219" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13,
entitled 'Angulus cornicularis'.
There Viète gave 6 arguments about the angle between the tangent and circumference of a circle.
Harriot has converted each of them into ratios, written in the lower half of the page in symbolic notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head62" xml:space="preserve">
Cornus. Vieta habet 6 sequentes analogias
<lb/>[<emph style="it">tr:
Horn angle. Viète has the 6 following ratios.
</emph>]<lb/>
</head>
<pb file="add_6787_f208v" o="208v" n="415"/>
<pb file="add_6787_f209" o="209" n="416"/>
<div xml:id="echoid-div54" type="page_commentary" level="2" n="54">
<p>
<s xml:id="echoid-s221" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s221" xml:space="preserve">
This page appears to be a continuation of Add MS 6787, f. 208,
concerned with Viète's arguments about the horn angle in
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head63" xml:space="preserve">
De Triangulis Sphæricis Rectangulis <lb/>
Cornus
<lb/>[<emph style="it">tr:
On right-angles spherical triangles. Horn angle.
</emph>]<lb/>
</head>
<pb file="add_6787_f209v" o="209v" n="417"/>
<pb file="add_6787_f210" o="210" n="418"/>
<div xml:id="echoid-div55" type="page_commentary" level="2" n="55">
<p>
<s xml:id="echoid-s223" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s223" xml:space="preserve">
This page appears to be a continuation of Add MS 6787, f. 208 and f. 209,
concerned with Viète's arguments about the horn angle in
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head64" xml:space="preserve">
Siphon
</head>
<p xml:lang="lat">
<s xml:id="echoid-s225" xml:space="preserve">
Ista 6 analogiæ sunt <lb/>
apud vieta.
<lb/>[<emph style="it">tr:
These 6 ratios are in Viète.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f210v" o="210v" n="419"/>
<pb file="add_6787_f211" o="211" n="420"/>
<pb file="add_6787_f211v" o="211v" n="421"/>
<pb file="add_6787_f212" o="212" n="422"/>
<pb file="add_6787_f212v" o="212v" n="423"/>
<pb file="add_6787_f213" o="213" n="424"/>
<pb file="add_6787_f213v" o="213v" n="425"/>
<pb file="add_6787_f214" o="214" n="426"/>
<pb file="add_6787_f214v" o="214v" n="427"/>
<pb file="add_6787_f215" o="215" n="428"/>
<pb file="add_6787_f215v" o="215v" n="429"/>
<pb file="add_6787_f216" o="216" n="430"/>
<pb file="add_6787_f216v" o="216v" n="431"/>
<pb file="add_6787_f217" o="217" n="432"/>
<pb file="add_6787_f217v" o="217v" n="433"/>
<pb file="add_6787_f218" o="218" n="434"/>
<pb file="add_6787_f218v" o="218v" n="435"/>
<pb file="add_6787_f219" o="219" n="436"/>
<pb file="add_6787_f219v" o="219v" n="437"/>
<pb file="add_6787_f220" o="220" n="438"/>
<pb file="add_6787_f220v" o="220v" n="439"/>
<pb file="add_6787_f221" o="221" n="440"/>
<div xml:id="echoid-div56" type="page_commentary" level="2" n="56">
<p>
<s xml:id="echoid-s226" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s226" xml:space="preserve">
At the end of Chapter XIX of <emph style="it">Variorum responsorum liber VIII</emph> (1593),
under the heading 'ALIUD', Viète listed sixteen propositions connecting sines, tangents, and secants.
In the first edition of the <emph style="it">Responsorum</emph>, the pages are numbered only on the recto side.
However, the pagination goes badly wrong, so that in Chapter XIX we have the sequence:
37, 38, 39, 38. This sometimes makes it difficult to follow Harriot's references correctly.
Here it seems that he has seen the number '38' on the right-hand (recto) page,
and thus inferred that the left-hand (verso) page must be 37v, whereas it is in face 39v.
In the 1646 edition of Viète's <emph style="it">Opera mathematica</emph>
the sixteen propositions are to be found on pages 412–413.
</s>
<lb/>
<s xml:id="echoid-s227" xml:space="preserve">
On this and the following pages, Harriot worked through the sixteen propositions systematically.
On this page he lists the first six.
Note that for Harriot, as for Viète, trigonometrical relationships arose from astronomy.
Thus the concepts of sine, tangent, and secant related not to angles defined by a pair of lines meeting at a point,
but to arcs of a circle with a given radius, and therefore only by implication to the angles subtended by them.
</s>
<lb/>
<s xml:id="echoid-s228" xml:space="preserve">
At the top of the page Harriot listed the relevant quantities:
sine, tangent, secant, radius (or whole sine), and the symbols he regularly used for them.
The equivalent names used by Viète were sinus, prosinus, transinuosa, totus.
Harriot had no words for cosine, cotangent, or coseceant.
Where we would use 'cosine', for example, he spoke of the sine of the complement.
Thus he wrote <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> BC for sine(arc BC)
but <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> <emph style="st">BC</emph> for the sine of the complement of BC, that is, cosine(BC).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head65" xml:space="preserve" xml:lang="lat">
vieta in lib. 8. respons. <lb/>
pag. 37. <lb/>
proportionalia
<lb/>[<emph style="it">tr:
Viète, in Responsorum liber VIII, page 37, proportionals
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s230" xml:space="preserve">
sinus <lb/>
tangens <lb/>
secans <lb/>
radius
<lb/>[<emph style="it">tr:
sine <lb/>
tangent <lb/>
secans <lb/>
radius
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s231" xml:space="preserve">
1. Sinus peripheriæ, Radius: Radius, Secans complementi. <lb/>
2. Sinus comp. peripheriæ. Radius. Radius. Secans peripheriæ. <lb/>
3. Tangens peripheriæ. Radius. Radius. Tangens complementi.
<lb/>[<emph style="it">tr:
1. Sine of the arc : Radius = Radius : Secant of the complement. <lb/>
2. Sine of the compplement of the arc : Radius = Radius : Secant of the arc. <lb/>
3. Tangent of the arc : Radius = Radius : Tangent of the complement.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s232" xml:space="preserve">
Ergo: cum proponuntur duæ peripheriæ <lb/>
4. Sinus peripheriæ primæ. Sinus secundæ. Secans. compl. secundæ. Secans compl. primæ. <lb/>
5. Sinus compl. primæ. Sinus com. secundæ. Secans 2<emph style="super">æ</emph>. Secans primæ. <lb/>
6. Tangens primæ. Tangens 2<emph style="super">æ</emph>.
Tangens comp. 2<emph style="super">æ</emph>. Tangens compl. primæ
<lb/>[<emph style="it">tr:
Therefore, when there are given two arcs: <lb/>
4. Sine of the first arc : Sine of the second =
Secant of the complement of the second : Secant of the complement of the first. <lb/>
5. Sine of the complement of the first : SIne of the complement of the second =
Secant of the second : Secant of the first. <lb/>
6. Tangent of the first : Tangent of the second =
Tangent of the complemnt of the second : Tangent of the complement of the first.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s233" xml:space="preserve">
Menda in Vieta. <lb/>
correctio.
<lb/>[<emph style="it">tr:
Wrong in Viète. <lb/>
Correction. </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s234" xml:space="preserve">
sinus <lb/>
tangens <lb/>
secans <lb/>
radius
</s>
</p>
<pb file="add_6787_f221v" o="221v" n="441"/>
<pb file="add_6787_f222" o="222" n="442"/>
<div xml:id="echoid-div57" type="page_commentary" level="2" n="57">
<p>
<s xml:id="echoid-s235" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s235" xml:space="preserve">
On this page, a continuation from Add MS 6787, f. 221, Harriot lists and proves Propositions 7 to 11
from the 'ALIUD' in Chapter XIX of Viète's <emph style="it">Variorum resposorum liber VIII</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat">
Vieta lib. 8. resp. <lb/>
pag. 37
<lb/>[<emph style="it">tr:
Viète, in Responsorum liber VIII, page 37.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s237" xml:space="preserve">
Lemma, satis evidens ex elementis: <lb/>
ad demonstrandam 7<emph style="super">am</emph> prop. <lb/>
1.
<lb/>[<emph style="it">tr:
Lemma, sufficiently evident from the fundamentals;
used for the demonstration of the 7th proposition. <lb/>
1.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s238" xml:space="preserve">
Ergo pro 7<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
Therefore for the 7th proposition
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s239" xml:space="preserve">
2. Alia dispositio terminorum lemmatis <lb/>
ad demonstrandam 7<emph style="super">am</emph> prop.
<lb/>[<emph style="it">tr:
2. Another arrangement of the terms in the lemma.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s240" xml:space="preserve">
Ergo pro 8<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
Therefore for the 8th proposition.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s241" xml:space="preserve">
per 5tam <lb/>
Et alterne <lb/>
Ergo <lb/>
Et alterne
<lb/>[<emph style="it">tr:
By the 5th <lb/>
And alternately <lb/>
Therefore <lb/>
And alternately
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s242" xml:space="preserve">
Ergo pro 9<emph style="super">a</emph>: per 7<emph style="super">am</emph>.
<lb/>[<emph style="it">tr:
Therefore for the 9th; by the 7th.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s243" xml:space="preserve">
3. Lemmatis variato
<lb/>[<emph style="it">tr:
3. A variation of the lemmma.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s244" xml:space="preserve">
Ergo pro 10<emph style="super">a</emph>: per 7<emph style="super">am</emph>. <lb/>
per 6<emph style="super">a</emph> alterne.
<lb/>[<emph style="it">tr:
Therefore for the 10th; by the 7th.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s245" xml:space="preserve">
4. Lemmatis variato
<lb/>[<emph style="it">tr:
4. A variation of the lemma.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s246" xml:space="preserve">
Ergo pro 11<emph style="super">a</emph>: per 7<emph style="super">am</emph>. <lb/>
per 5<emph style="super">a</emph> alterne.
<lb/>[<emph style="it">tr:
Therefore for the 11th; by the 7th.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f222v" o="222v" n="443"/>
<pb file="add_6787_f223" o="223" n="444"/>
<div xml:id="echoid-div58" type="page_commentary" level="2" n="58">
<p>
<s xml:id="echoid-s247" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s247" xml:space="preserve">
On this page, a continuation from Add MS 6787, f. 221, f. 222, and f. 225,
Harriot lists and proves Propositions 13 to 16 from the 'ALIUD' in Chapter XIX of Viète's
<emph style="it">Variorum resposorum liber VIII</emph>.
</s>
<lb/>
<s xml:id="echoid-s248" xml:space="preserve">
The reference to Regiomontanus to <emph style="it">De triangulis omnimodis libri quinque</emph> ([1464], 1533, 1561),
Book V, Proposition 1.
</s>
<lb/>
<s xml:id="echoid-s249" xml:space="preserve">
The reference to Fink is to <emph style="it">Geometriae rotundi libri XIIII</emph> (1583), page 364.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. responsorum. <lb/>
pag. 37b.
<lb/>[<emph style="it">tr:
Viète, in Responsorum liber VIII, page 37.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s251" xml:space="preserve">
Et cum proponuntur tres peripheriæ. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
And when there are given three arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s252" xml:space="preserve">
Superiores Analogiæ demonstrantur <lb/>
per istas quæ sequntur.
<lb/>[<emph style="it">tr:
The above ratios may be demonstrated by these which follow.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s253" xml:space="preserve">
[???] 13 habetur in Reg. lib. 5. 1. <lb/>
et Finkio pag. 364. <lb/>
Notavi in alia charta <lb/>
[???]
<lb/>[<emph style="it">tr:
[???] 13 is to be had in Regiomontanus. Book V, Proposition 1, and Fink, page 364. <lb/>
I have noted in another sheet [???].
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f223v" o="223v" n="445"/>
<pb file="add_6787_f224" o="224" n="446"/>
<div xml:id="echoid-div59" type="page_commentary" level="2" n="59">
<p>
<s xml:id="echoid-s254" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s254" xml:space="preserve">
At the end of Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593)
there is a lengthy section entitled '<foreign xml:lang="gre">Eis procheiron scholia</foreign>.
Section IV is on spherical geometry. The 18th and final proposition contains four statements,
which Harriot here translated into symbolic notation.
</s>
<lb/>
<quote xml:lang="lat">
18 Sit triangulum sphaericum ABD, &amp; in peripheria BD cadat segmentum orthogonii AC. <lb/>
Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC,
sicut prosinum peripheria AB ad prosinum peripheri AD. <lb/>
Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD,
sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD. <lb/>
Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D. <lb/>
Denique &amp; quarto dico esse sinum anguli BAC ad sinum anguli DAC,
sicut transsinuosam anguli D ad transsinuosam anguli B.
</quote>
<lb/>
<quote>
18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC. <lb/>
First I say that the secant of angle BAC to the secant of angle DAC is
as the tangent of the arc AB to the tangent of the arc AD. <lb/>
Second I say that the secant of the arc CB to the secant of the arc CD is
as the secant of the arc AB to the secant of the arc AD. <lb/>
Third I say that the sine of CD to the sine of DB is
as the tangent of angle B to the tangent of angle D. <lb/>
Fourth and last I say that the sine of angle BAC to the sine of angle DAC is
as the secant of angle D to the secant of angle B.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat">
Vieta lib. 8. resp. <lb/>
pag. 41. b. <lb/>
prop. 18
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 41v, Proposition 18.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s256" xml:space="preserve">
Hinc apparet mendam <lb/>
esse apud Vietam.
<lb/>[<emph style="it">tr:
Here is is clear that it is wrong in Viète.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f224v" o="224v" n="447"/>
<pb file="add_6787_f225" o="225" n="448"/>
<div xml:id="echoid-div60" type="page_commentary" level="2" n="60">
<p>
<s xml:id="echoid-s257" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s257" xml:space="preserve">
On this page, a continuation from Add MS 6787, f. 221 and f. 222,
Harriot gives a diagram for the proof of Proposition 12
from the 'ALIUD' in Chapter XIX of Viète's <emph style="it">Variorum resposorum liber VIII</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head69" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. responsorum. <lb/>
pag. 37b. <lb/>
Diagramma ad demonstrationem prop. 12.
<lb/>[<emph style="it">tr:
Viète, in Responsorum liber VIII, page 37. <lb/>
Diagram for the demonstration of Proposition 12.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s259" xml:space="preserve">
Manifesta ex diagrammate
<lb/>[<emph style="it">tr:
Obvious from the diagram.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s260" xml:space="preserve">
Anaologiæ eædem in notis universalibus.
<lb/>[<emph style="it">tr:
The same ratios in general notation.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s261" xml:space="preserve">
Ergo pro 12<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
Therefore for the 12th.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s262" xml:space="preserve">
Hinc notatur Menda in Vieta <lb/>
et emendatur.
<lb/>[<emph style="it">tr:
Here it is noted that it is wrong in Viète, and amended.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f225v" o="225v" n="449"/>
<pb file="add_6787_f226" o="226" n="450"/>
<div xml:id="echoid-div61" type="page_commentary" level="2" n="61">
<p>
<s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s263" xml:space="preserve">
The first three propositions from the 'ALIUD' in Chapter XIX of Viète's
<emph style="it">Variorum resposorum liber VIII</emph>,
written in symbolic notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f226v" o="226v" n="451"/>
<pb file="add_6787_f227" o="227" n="452"/>
<div xml:id="echoid-div62" type="page_commentary" level="2" n="62">
<p>
<s xml:id="echoid-s265" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s265" xml:space="preserve">
This is the first of four pages devoted to Proposition 14 from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
a lengthy chapter on plane and spherical triangles.
</s>
<lb/>
<quote xml:lang="lat">
XIV. <lb/>
Prothechidion. <lb/>
Data duorum maximorum in sphæra circulorum inclinatione, quorum unus secatur a tertio per alterius polos,
arguitur quanta fit maxima differentia suarum a nodo longitudinum. <lb/>
Et contra. Ex maxima differentia longitudinum a nodo, arguitur quanta fit circulum inclinatio.
</quote>
<lb/>
<quote>
Given the inclination of two great circles on a sphere,
one of which is cut by a third through the pole of the other,
there is to be found the greatest difference in their longitudes from the node.
Conversely, from the greatest difference of longitudes from the node,
there may be found the inclination of the circles.
</quote>
<s xml:id="echoid-s266" xml:space="preserve">
In the crossed out sentence halfway down the page there are references to Fink and Clavius. <lb/>
The reference to Fink is to his <emph style="it">Geometriae rotundi libri XIIII</emph> (1583). <lb/>
The reference to Clavius is to his <emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head70" xml:space="preserve">
Vieta. lib. 8. resp.
pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 35, Proposition 14,
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s268" xml:space="preserve">
Ista propositio est utilis in calcu-<lb/>
lationibus astronimicis. per eam cog-<lb/>
noscitur maxima differentia inter <lb/>
numerationes per Eclipticam et proprios <lb/>
circulos planetorum, et ubi est &amp;c. <lb/>
Etiam: <lb/>
æquationibus <lb/>
dierum.
<lb/>[<emph style="it">tr:
This proposition is useful in astronomical calculations.
By it may be known the maximum difference between observations by the ecliptic and the nearest orbits of planets,
and where it is. <lb/>
Also, the equations of the days.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s269" xml:space="preserve">
Sit triangulum rectangulum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, angulus rectus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. Quæritur <lb/>
Maxima differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. Nam arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>C</mi></mstyle></math> in diversis <lb/>
positionibus inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>, facit diversis <lb/>
differentias longitudinum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> be a right-angled triangle with right angle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
There is sought the maximum differene between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
For the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>C</mi></mstyle></math> in various positions between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>
makes various differences of longitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s270" xml:space="preserve">
polo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. <emph style="super">et</emph> per punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> describatur <lb/>
parallelus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>B</mi><mi>D</mi></mstyle></math>. Ergo: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> est differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> <lb/>
sit diameter paralleli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>O</mi></mstyle></math> et <lb/>
sit perpendcularis illi, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>M</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Taking the pole <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, there is drawn through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> parallel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>B</mi><mi>D</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> is the difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
Let the diameter parallel to it be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>O</mi></mstyle></math> and the perpendicuar to it <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>M</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s271" xml:space="preserve">
<emph style="st">
Dico quando <emph style="super">*</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> linea est æqualis sinui <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>, hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>O</mi></mstyle></math> erit æqualis <lb/>
semidiametro sphæræ, scilicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>F</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> erit differentia quæsita maxima. <lb/>
Pro demonstratione nota diagramma in Finkio pag. 393. et Clavium 445.
</emph>
<lb/>[<emph style="it">tr:
I say that when the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> is equal to the sine <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>,
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>O</mi></mstyle></math> will be equal to the semidiameter of a spehre, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>F</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> will be the sought maximum difference.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s272" xml:space="preserve">
Hic notabo solummodo proportiones in Vieta. <lb/>
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>, sinus arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>, hoc est anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math> erit sinus versus [???]. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> est arcus similis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>. Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi><mi>L</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>O</mi><mo>=</mo><mi>D</mi><mi>M</mi></mstyle></math>. cætera
<emph style="st">ex</emph> <emph style="super">in</emph> diagramata.
<lb/>[<emph style="it">tr:
Here I have noted only the proportions in Viète.
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>, be the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>, that is of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. The versed sine will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math>.
The arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> is similar to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi><mi>L</mi></mstyle></math> be equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>O</mi><mo>=</mo><mi>D</mi><mi>M</mi></mstyle></math>.
The rest from the diagram.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s273" xml:space="preserve">
* <lb/>
Dico quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> habet <lb/>
<emph style="st">ratio</emph> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math> sinum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>: <lb/>
eandem rationem quam <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>M</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>D</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>L</mi><mi>I</mi></mstyle></math> ad <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>F</mi></mstyle></math>. Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> erit <lb/>
differentia maxima. <lb/>
Vel melius ita: <lb/>
Quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> sit <lb/>
parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. hoc <lb/>
est quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>H</mi><mi>O</mi></mstyle></math> est <lb/>
rectus angulus. <lb/>
Demonstratio <lb/>
Habetur in alia <lb/>
charta annexa.
<lb/>[<emph style="it">tr:
I say that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> has the same ratio to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>, the sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>M</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>D</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>L</mi><mi>I</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>F</mi></mstyle></math>,
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> will be the maximum difference. <lb/>
Or better thus: when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> is a right angle. <lb/>
The demonstration is to be found in another sheet, adjoined.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f227v" o="227v" n="453"/>
<pb file="add_6787_f228" o="228" n="454"/>
<div xml:id="echoid-div63" type="page_commentary" level="2" n="63">
<p>
<s xml:id="echoid-s274" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s274" xml:space="preserve">
Further work on Proposition 14 from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
continued from Add MS 6787, f. 227.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head71" xml:space="preserve">
Vieta. lib. 8. resp.
pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 35, Proposition 14,
</emph>]<lb/>
</head>
<pb file="add_6787_f228v" o="228v" n="455"/>
<pb file="add_6787_f229" o="229" n="456"/>
<div xml:id="echoid-div64" type="page_commentary" level="2" n="64">
<p>
<s xml:id="echoid-s276" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s276" xml:space="preserve">
Further work on Proposition 14 from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
continued from Add MS 6787, f. 227 and f. 228.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head72" xml:space="preserve">
Vieta. lib. 8. resp.
pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 35, Proposition 14,
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s278" xml:space="preserve">
Habita maxima differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> et eius sinu <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>: sinus arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, <lb/>
hoc est linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> ita invenietur. <lb/>
<omision/> <lb/>
Ergo nota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> quæsita
<lb/>[<emph style="it">tr:
Having the maximum difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> and its sine <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>,
then the sine of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, that is, the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> is found thus. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math>, the line sought
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s279" xml:space="preserve">
vel ita Brevius <lb/>
In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, cum etiam datur angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <lb/>
per doctrinam triangulorum dabitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
In the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, since the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is also given,
by the teaching on triangles there will be given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f229v" o="229v" n="457"/>
<pb file="add_6787_f230" o="230" n="458"/>
<div xml:id="echoid-div65" type="page_commentary" level="2" n="65">
<p>
<s xml:id="echoid-s280" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s280" xml:space="preserve">
Further work on Proposition 14 from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
continued from Add MS 6787, f. 227, f. 228, and f. 229.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head73" xml:space="preserve">
Vieta. lib. 8. resp.
pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 35, Proposition 14,
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s282" xml:space="preserve">
Synopsis Demonstrationis illius proportionis.
<lb/>[<emph style="it">tr:
Sysnopsis of the demonstration of this proportion.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s283" xml:space="preserve">
sunt in eadem ratione
<lb/>[<emph style="it">tr:
are in the same ratio
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s284" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>, angulus rectus <lb/>
Maior est ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>α</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> est in eadem ratione <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi></mstyle></math> maior est quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math>. <lb/>
Maior est etiam ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>ɛ</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> est eadem,
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mo>\</mo><mi>v</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>p</mi><mi>s</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>n</mi></mstyle></math>. <lb/>
Hæc ita se habent, quia <emph style="super">angulus</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi><mi>V</mi></mstyle></math> est obtusus maior scilicet recto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>.
et alter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi><mi>X</mi></mstyle></math> est <lb/>
minor eadem ratio. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. <lb/>
Ac etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Y</mi><mi>X</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math> be a right angle <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>α</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is greater, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is in the same ratio
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math>. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> is greater, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>ɛ</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> is the same,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mo>\</mo><mi>v</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>p</mi><mi>s</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>n</mi></mstyle></math>. <lb/>
These are had thus, because angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi><mi>V</mi></mstyle></math> is obtuse, clearly greater than the right angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>,
and the other, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi><mi>X</mi></mstyle></math>, is less than the same ratio. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. <lb/>
And also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Y</mi><mi>X</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s285" xml:space="preserve">
Hæc facile applicantur <lb/>
ad diagramma <lb/>
precedens
<lb/>[<emph style="it">tr:
These are easily applied to the preceding diagram.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f230v" o="230v" n="459"/>
<pb file="add_6787_f231" o="231" n="460"/>
<div xml:id="echoid-div66" type="page_commentary" level="2" n="66">
<p>
<s xml:id="echoid-s286" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s286" xml:space="preserve">
This page gives numerical examples for Propositions III and VI from the 'Sexti dati' in Chapter 19
of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
III. <lb/>
Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ.
</quote>
<lb/>
<quote>
III.Given the sum or difference of two arcs, whose secants are in a given ratio, the arcs are given individually.
</quote>
<lb/>
<quote xml:lang="lat">
VII. <lb/>
Data summa vel differentia duarum peripheriarum, quarum prosinus datam habeant rationem, dantur singulares peripheriæ.
</quote>
<lb/>
<quote>
VII.Given the sum or difference of two arcs, whose tangents are in a given ratio, the arcs are given individually.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head74" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 38. <lb/>
<foreign xml:lang="gre">parapompe</foreign>. 3. Data summa vel differentia. <lb/>
pag. 37.b. <lb/>
<foreign xml:lang="gre">parapompe</foreign>. data septimi <lb/>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38v. On sines.
</emph>]<lb/>
</head>
<pb file="add_6787_f231v" o="231v" n="461"/>
<pb file="add_6787_f232" o="232" n="462"/>
<div xml:id="echoid-div67" type="page_commentary" level="2" n="67">
<p>
<s xml:id="echoid-s288" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s288" xml:space="preserve">
This page gives numerical examples for Propositions I and II from the 'Sexti dati' in Chapter 19
of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
I. <lb/>
Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ.
</quote>
<lb/>
<quote>
I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually.
</quote>
<lb/><quote xml:lang="lat">
II. <lb/>
Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ.
</quote>
<lb/>
<quote>
II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 38. <lb/>
Data peripheria <lb/>
compositam <lb/>
Data <emph style="st">peripheria</emph> differentia <lb/>
duarum perpheriarum
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38. <lb/>
Given the sum of the arcs <lb/>
Given the difference of two arcs
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s290" xml:space="preserve">
Menda in Vieta
<lb/>[<emph style="it">tr:
Wrong in Viète
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f232v" o="232v" n="463"/>
<pb file="add_6787_f233" o="233" n="464"/>
<div xml:id="echoid-div68" type="page_commentary" level="2" n="68">
<p>
<s xml:id="echoid-s291" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s291" xml:space="preserve">
This page gives numerical examples for Proposition VI from the 'Sexti dati' in Chapter 19
of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<quote xml:lang="lat">
VI. <lb/>
Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem,
dantur singulares peripheriæ.
</quote>
<lb/>
<quote>
VI.Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. pag. 37. b.		De sinubus
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38v. On sines.
</emph>]<lb/>
</head>
<pb file="add_6787_f233v" o="233v" n="465"/>
<pb file="add_6787_f234" o="234" n="466"/>
<pb file="add_6787_f234v" o="234v" n="467"/>
<pb file="add_6787_f235" o="235" n="468"/>
<pb file="add_6787_f235v" o="235v" n="469"/>
<div xml:id="echoid-div69" type="page_commentary" level="2" n="69">
<p>
<s xml:id="echoid-s293" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s293" 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><mi>I</mi><mi>n</mi><mi>t</mi><mi>h</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>H</mi><mi>a</mi><mi>r</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>t</mi><mi>g</mi><mi>o</mi><mi>e</mi><mi>s</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>a</mi><mi>k</mi><mi>e</mi><mi>w</mi><mi>o</mi><mi>r</mi><mi>d</mi><mi>s</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>p</mi><mi>h</mi><mi>r</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>h</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>n</mi><mi>c</mi><mi>l</mi><mi>u</mi><mi>d</mi><mi>e</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>t</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>e</mi><mi>c</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>l</mi><mi>y</mi><mo>.</mo></mstyle></math></s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s295" xml:space="preserve">
31415926535897 ...<lb/>
ecehigcadaeibf ...
</s>
</p>
<pb file="add_6787_f236" o="236" n="470"/>
<pb file="add_6787_f236v" o="236v" n="471"/>
<pb file="add_6787_f237" o="237" n="472"/>
<pb file="add_6787_f237v" o="237v" n="473"/>
<pb file="add_6787_f238" o="238" n="474"/>
<pb file="add_6787_f238v" o="238v" n="475"/>
<pb file="add_6787_f239" o="239" n="476"/>
<pb file="add_6787_f239v" o="239v" n="477"/>
<pb file="add_6787_f240" o="240" n="478"/>
<pb file="add_6787_f240v" o="240v" n="479"/>
<pb file="add_6787_f241" o="241" n="480"/>
<pb file="add_6787_f241v" o="241v" n="481"/>
<pb file="add_6787_f242" o="242" n="482"/>
<pb file="add_6787_f242v" o="242v" n="483"/>
<pb file="add_6787_f243" o="243" n="484"/>
<pb file="add_6787_f243v" o="243v" n="485"/>
<pb file="add_6787_f244" o="244" n="486"/>
<pb file="add_6787_f244v" o="244v" n="487"/>
<pb file="add_6787_f245" o="245" n="488"/>
<div xml:id="echoid-div70" type="page_commentary" level="2" n="70">
<p>
<s xml:id="echoid-s296" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s296" xml:space="preserve">
For the historical and mathematical context of Add MS 6787, f. 245 to f. 248 see Beery and Stedall, 2009. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head77" xml:space="preserve">
Of unæquall progression <lb/>
of sines. For S.W.L.
</head>
<pb file="add_6787_f245v" o="245v" n="489"/>
<pb file="add_6787_f246" o="246" n="490"/>
<div xml:id="echoid-div71" type="page_commentary" level="2" n="71">
<p>
<s xml:id="echoid-s298" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s298" xml:space="preserve">
Propositions on triangular numbers;
this sheet was probably placed here deliberately because of the importance of triangular numbers
in Harriot's method of interpolation (se Beery and Stedall, 2009.)
</s>
<lb/>
<s xml:id="echoid-s299" xml:space="preserve">
The refrence to Viète is to his <emph style="it">Variotum responsorum liber VII</emph>,
Chapter IX, Proposition 14.
</s>
<lb/>
<quote xml:lang="lat">
Propositio XIV. <lb/>
Si fuerint lineae quotcunque æqualiter sese excedentes, fit autem prima excessui æqualis:
octuplum ejus quod fit sub minima &amp; composita ex omibus, adjunctum minimae quadrato,
æquatur quadrato compositæ ex minima &amp; extrema dupla.
</quote>
<lb/>
<quote>
If there are any number of lines exceeding each other, and moreover the first differences are equal,
then eight times the product of the least and the sum of all, added to the square of the least,
is equal to the square of the sum of the least and twice the greatest.
</quote>
<lb/>
<s xml:id="echoid-s300" xml:space="preserve">
The reference to Stevin is to his <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
Pages 558–642 contain Stevin's treatment of the 'Quatriesme livre d'algebre de Diophante d'Alexandrie'.
On page 634, Stevin has the following Theorem.
</s>
<lb/>
<quote xml:lang="fre">
Nombre triangulaire multiplié par 8, &amp; plus 1 faict quarré a sa racine commensurable.
</quote>
<lb/>
<quote>
A triangular number multiplied by 8, plus 1, makes a square commensurable with its root.
</quote>
<lb/>
<s xml:id="echoid-s301" xml:space="preserve">
The reference to Maurolico is to his <emph style="it">Arithmeticorum libri duo</emph> (1575),
Propostion 54 (page 24).
</s>
<lb/>
<quote xml:lang="lat">
Omnis triangulus octuplatus cum unitate, conficit sequentis imparis quadratum.
</quote>
<lb/>
<quote>
Eight times any triangular number, plus one, makes the square of the next odd number.
</quote>
<lb/>
<s xml:id="echoid-s302" xml:space="preserve">
As an example, Maurolico gave <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>1</mn><mn>2</mn><mn>1</mn></mstyle></math>;
note that 15 is the fifth triangular number, 11 is the sixth odd number.
</s>
<lb/>
<s xml:id="echoid-s303" xml:space="preserve">
There is also a reference lower down to Maurolico's Proposition 11 (page 6):
</s>
<lb/>
<quote xml:lang="lat">
Omnis numerus triangulo deinceps: aequatur quadratis lateris trianguli maioris.
</quote>
<lb/>
<quote>
Every triangular number joined with the preceding triangular number makes the square of its side.
</quote>
<lb/>
<s xml:id="echoid-s304" xml:space="preserve">
Maurolico's example is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>5</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat">
Propositio
<lb/>[<emph style="it">tr:
Proposition
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s306" xml:space="preserve">
plutarchus platonica <lb/>
quæstione 4<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
Plutarch on Plato, question 4.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s307" xml:space="preserve">
Vieta resp. pa. 15.
<lb/>[<emph style="it">tr:
Viete, <emph style="it">Liber variorum responsorum</emph>, page 15.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s308" xml:space="preserve">
Maurolicus pr. 54. arith.
<lb/>[<emph style="it">tr:
Maurolico, <emph style="it">Arithmetica</emph>, Proposition 54.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s309" xml:space="preserve">
Stevin . pag. 634. arith. <lb/>
in diophanto.
<lb/>[<emph style="it">tr:
Stevin, page 634, <emph style="it">Arithmetic</emph>, on Diophantus.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s310" xml:space="preserve">
Trianguli numeri octuplum, plus quadrato unitatis:
æqualie est quadrato <lb/>
facto a duplo latere trianguli plus unitate.
<lb/>[<emph style="it">tr:
Eight times a trinagular number plus the square of one
is equal to the square of double the side of the triangular number, plus one.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s311" xml:space="preserve">
Sit latus trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the side of the triangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s312" xml:space="preserve">
Tum triangulus erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
Then the triangular number will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s313" xml:space="preserve">
Unde propositio cum demonstratione in notis logisticis ita se habet:
<lb/>[<emph style="it">tr:
Whence we have the proposition with its demonstration in arithmetic notation thus:
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat">
Propositio
<lb/>[<emph style="it">tr:
Proposition
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s314" xml:space="preserve">
Marol. pr. 11
<lb/>[<emph style="it">tr:
Maurolico, Proposition 11
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s315" xml:space="preserve">
Omnis numerus triangulus, plus triangulo deinceps <emph style="st">priori</emph>: æquatur <lb/>
quadrato lateris trianguli maioris.
<lb/>[<emph style="it">tr:
Every triangular number, plus the triangular numebr following,
is equal to the square of the side of the greater triangular number.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s316" xml:space="preserve">
Sit latus maioris trianguli. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the side of the greater triangular number be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s317" xml:space="preserve">
latus minoris triang: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>-</mo><mn>1</mn></mstyle></math>.
<lb/>[<emph style="it">tr:
the side of the smaller triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>-</mo><mn>1</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s318" xml:space="preserve">
Ergo: maior triangulus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore the greater triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s319" xml:space="preserve">
Minor triangulus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
 the smaller triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s320" xml:space="preserve">
Unde propositio cum demonstration in notis logisticis ita se habet:
<lb/>[<emph style="it">tr:
Whence we have the proposition with its demonstration in arithmetic notation thus:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f246v" o="246v" n="491"/>
<pb file="add_6787_f247" o="247" n="492"/>
<head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat">
Problema
<lb/>[<emph style="it">tr:
Problem
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s321" xml:space="preserve">
A progression increasing being given, whose
<emph style="super"><emph style="st">next</emph> first progresionall </emph>
<emph style="st">first</emph> differences <lb/>
are unæquall, &amp; the <emph style="st">second</emph>
<emph style="super">next <emph style="st">after</emph> differences</emph> æquall:
to devide the sayde <lb/>
progression into a fewer nomber of progresionall partes.
</s>
<lb/>
<s xml:id="echoid-s322" xml:space="preserve">
The first case. as of the <emph style="super">progresionall</emph> differences decreasing, <lb/>
as it is in the progression of sines.
</s>
</p>
<p>
<s xml:id="echoid-s323" xml:space="preserve">
Example of the progresion <emph style="super">given</emph>.
</s>
</p>
<p>
<s xml:id="echoid-s324" xml:space="preserve">
<sc>
'Arkes' are arcs of a fixed circle, that is, measures of angle.
Each sine (or number) corresponds to, or is 'answerable to', an arc (or angle).
(Hence the modern terminology 'arcsin'.)
</sc>
This example I have <lb/>
so set downe as though the <lb/>
nombers were answerable <lb/>
to arkes.
</s>
<s xml:id="echoid-s325" xml:space="preserve">
That by it you may <lb/>
se the use of the problem <lb/>
for sines.
</s>
</p>
<p>
<s xml:id="echoid-s326" xml:space="preserve">
Suppose that it be required to find the nomber answerable to 37''.
</s>
<lb/>
<s xml:id="echoid-s327" xml:space="preserve">
The number answerable <emph style="super">to</emph> 30'' is 2280. that which is answerable <lb/>
to 40'' is 2925. there difference is 645.
</s>
<s xml:id="echoid-s328" xml:space="preserve">
And because the other <lb/>
differences of the same <emph style="super">order</emph> ranke are unæquall, the rule of pro-<lb/>
portion will not find the number desired. But it must be <lb/>
found by a speciall Canon. <emph style="st">which followeth</emph>
</s>
<s xml:id="echoid-s329" xml:space="preserve">
The purpose <lb/>
of <emph style="st">which canon</emph> <emph style="super">thereof</emph> must be
<emph style="st">for to find</emph> (because 30''&amp; 40'' do differ <lb/>
[by] 10''.) to find the <emph style="st"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math></emph>
<emph style="super">first tenth</emph> parte progresionall, that is to say that number.
</s>
</p>
<pb file="add_6787_f247v" o="247v" n="493"/>
<pb file="add_6787_f248" o="248" n="494"/>
<div xml:id="echoid-div72" type="page_commentary" level="2" n="72">
<p>
<s xml:id="echoid-s330" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s330" xml:space="preserve">
The table of 'pretend' sines from Add MS 6787, f. 247, now interpolated to give sines for every minute
from 30minutes to 50 minutes.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f248v" o="248v" n="495"/>
<pb file="add_6787_f249" o="249" n="496"/>
<pb file="add_6787_f249v" o="249v" n="497"/>
<pb file="add_6787_f250" o="250" n="498"/>
<pb file="add_6787_f250v" o="250v" n="499"/>
<p>
<s xml:id="echoid-s332" xml:space="preserve">
Serjeant Harris. <lb/>
Mr Martin. <lb/>
Mr Waldon. <lb/>
Mr More. <lb/>
Mr Karbile <lb/>
Mr Walters. <lb/>
Mr Dorrel
</s>
</p>
<p>
<s xml:id="echoid-s333" xml:space="preserve">
April 26. ho. 7. <lb/>
first subpena <lb/>
George Sanderson's man
</s>
</p>
<p>
<s xml:id="echoid-s334" xml:space="preserve">
Mr Nicholas Lower <lb/>
at a dues mans home <emph style="super">[???]</emph> <lb/>
within the meremayde <lb/>
in cheapside.
</s>
</p>
<p>
<s xml:id="echoid-s335" xml:space="preserve">
Bish. of Elyes booke. <lb/>
Wright. <lb/>
Thesaro politico. 2. <lb/>
Onuphrius in platinam.
</s>
</p>
<p>
<s xml:id="echoid-s336" xml:space="preserve">
[???] <lb/>
Mappes <lb/>
globes
</s>
</p>
<pb file="add_6787_f251" o="251" n="500"/>
<pb file="add_6787_f251v" o="251v" n="501"/>
<pb file="add_6787_f252" o="252" n="502"/>
<pb file="add_6787_f252v" o="252v" n="503"/>
<div xml:id="echoid-div73" type="page_commentary" level="2" n="73">
<p>
<s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s337" xml:space="preserve">
A continuation of the calculations on Add MS 6787, f. 56, now extended to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>5</mn></msup></mrow></mstyle></math>. <lb/>
On this page Harriot also changes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>b</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f253" o="253" n="504"/>
<pb file="add_6787_f253v" o="253v" n="505"/>
<pb file="add_6787_f254" o="254" n="506"/>
<pb file="add_6787_f254v" o="254v" n="507"/>
<pb file="add_6787_f255" o="255" n="508"/>
<pb file="add_6787_f255v" o="255v" n="509"/>
<pb file="add_6787_f256" o="256" n="510"/>
<pb file="add_6787_f256v" o="256v" n="511"/>
<pb file="add_6787_f257" o="257" n="512"/>
<pb file="add_6787_f257v" o="257v" n="513"/>
<pb file="add_6787_f258" o="258" n="514"/>
<pb file="add_6787_f258v" o="258v" n="515"/>
<pb file="add_6787_f259" o="259" n="516"/>
<pb file="add_6787_f259v" o="259v" n="517"/>
<pb file="add_6787_f260" o="260" n="518"/>
<pb file="add_6787_f260v" o="260v" n="519"/>
<pb file="add_6787_f261" o="261" n="520"/>
<pb file="add_6787_f261v" o="261v" n="521"/>
<pb file="add_6787_f262" o="262" n="522"/>
<pb file="add_6787_f262v" o="262v" n="523"/>
<pb file="add_6787_f263" o="263" n="524"/>
<pb file="add_6787_f263v" o="263v" n="525"/>
<pb file="add_6787_f264" o="264" n="526"/>
<pb file="add_6787_f264v" o="264v" n="527"/>
<pb file="add_6787_f265" o="265" n="528"/>
<pb file="add_6787_f265v" o="265v" n="529"/>
<pb file="add_6787_f266" o="266" n="530"/>
<pb file="add_6787_f266v" o="266v" n="531"/>
<pb file="add_6787_f267" o="267" n="532"/>
<pb file="add_6787_f267v" o="267v" n="533"/>
<pb file="add_6787_f268" o="268" n="534"/>
<pb file="add_6787_f268v" o="268v" n="535"/>
<pb file="add_6787_f269" o="269" n="536"/>
<pb file="add_6787_f269v" o="269v" n="537"/>
<pb file="add_6787_f270" o="270" n="538"/>
<pb file="add_6787_f270v" o="270v" n="539"/>
<pb file="add_6787_f271" o="271" n="540"/>
<pb file="add_6787_f271v" o="271v" n="541"/>
<pb file="add_6787_f272" o="272" n="542"/>
<pb file="add_6787_f272v" o="272v" n="543"/>
<pb file="add_6787_f273" o="273" n="544"/>
<pb file="add_6787_f273v" o="273v" n="545"/>
<pb file="add_6787_f274" o="274" n="546"/>
<p>
<s xml:id="echoid-s339" xml:space="preserve">
Cloth for a [???] <lb/>
4 yardes of [???] <lb/>
Velvet <lb/>
[???] for doublet and hose <lb/>
&amp; taffeta <lb/>
4 yards
</s>
</p>
<pb file="add_6787_f274v" o="274v" n="547"/>
<pb file="add_6787_f275" o="275" n="548"/>
<pb file="add_6787_f275v" o="275v" n="549"/>
<pb file="add_6787_f276" o="276" n="550"/>
<pb file="add_6787_f276v" o="276v" n="551"/>
<pb file="add_6787_f277" o="277" n="552"/>
<pb file="add_6787_f277v" o="277v" n="553"/>
<pb file="add_6787_f278" o="278" n="554"/>
<pb file="add_6787_f278v" o="278v" n="555"/>
<pb file="add_6787_f279" o="279" n="556"/>
<pb file="add_6787_f279v" o="279v" n="557"/>
<pb file="add_6787_f280" o="280" n="558"/>
<pb file="add_6787_f280v" o="280v" n="559"/>
<pb file="add_6787_f281" o="281" n="560"/>
<pb file="add_6787_f281v" o="281v" n="561"/>
<pb file="add_6787_f282" o="282" n="562"/>
<pb file="add_6787_f282v" o="282v" n="563"/>
<pb file="add_6787_f283" o="283" n="564"/>
<pb file="add_6787_f283v" o="283v" n="565"/>
<pb file="add_6787_f284" o="284" n="566"/>
<pb file="add_6787_f284v" o="284v" n="567"/>
<pb file="add_6787_f285" o="285" n="568"/>
<pb file="add_6787_f285v" o="285v" n="569"/>
<pb file="add_6787_f286" o="286" n="570"/>
<pb file="add_6787_f286v" o="286v" n="571"/>
<pb file="add_6787_f287" o="287" n="572"/>
<pb file="add_6787_f287v" o="287v" n="573"/>
<pb file="add_6787_f288" o="288" n="574"/>
<pb file="add_6787_f288v" o="288v" n="575"/>
<pb file="add_6787_f289" o="289" n="576"/>
<pb file="add_6787_f289v" o="289v" n="577"/>
<pb file="add_6787_f290" o="290" n="578"/>
<pb file="add_6787_f290v" o="290v" n="579"/>
<pb file="add_6787_f291" o="291" n="580"/>
<pb file="add_6787_f291v" o="291v" n="581"/>
<pb file="add_6787_f292" o="292" n="582"/>
<pb file="add_6787_f292v" o="292v" n="583"/>
<pb file="add_6787_f293" o="293" n="584"/>
<pb file="add_6787_f293v" o="293v" n="585"/>
<pb file="add_6787_f294" o="294" n="586"/>
<pb file="add_6787_f294v" o="294v" n="587"/>
<pb file="add_6787_f295" o="295" n="588"/>
<pb file="add_6787_f295v" o="295v" n="589"/>
<pb file="add_6787_f296" o="296" n="590"/>
<pb file="add_6787_f296v" o="296v" n="591"/>
<pb file="add_6787_f297" o="297" n="592"/>
<pb file="add_6787_f297v" o="297v" n="593"/>
<pb file="add_6787_f298" o="298" n="594"/>
<pb file="add_6787_f298v" o="298v" n="595"/>
<div xml:id="echoid-div74" type="page_commentary" level="2" n="74">
<p>
<s xml:id="echoid-s340" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s340" xml:space="preserve">
The reference on this page is to Viète,
<emph style="it">Adrianus Romanus responsum</emph> (1595).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat">
Ad Auctorium Vietæ in responso ad Romanum
<lb/>[<emph style="it">tr:
On teh authority of Viète in his response to Romanus
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s342" xml:space="preserve">
occasio doctrinæ <lb/>
in his chartis
<lb/>[<emph style="it">tr:
the occasion for the teaching in these sheets
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f299" o="299" n="596"/>
<pb file="add_6787_f299v" o="299v" n="597"/>
<pb file="add_6787_f300" o="300" n="598"/>
<pb file="add_6787_f300v" o="300v" n="599"/>
<pb file="add_6787_f301" o="301" n="600"/>
<pb file="add_6787_f301v" o="301v" n="601"/>
<pb file="add_6787_f302" o="302" n="602"/>
<pb file="add_6787_f302v" o="302v" n="603"/>
<pb file="add_6787_f303" o="303" n="604"/>
<pb file="add_6787_f303v" o="303v" n="605"/>
<pb file="add_6787_f304" o="304" n="606"/>
<div xml:id="echoid-div75" type="page_commentary" level="2" n="75">
<p>
<s xml:id="echoid-s343" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s343" xml:space="preserve">
Here Harriot creates further Pythagorean triples, not necessarily all integers. <lb/>
In the first few lines, for example, he multiplies 21 by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>
to obtain (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>0</mn><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>8</mn><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>) or
(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>2</mn><mn>6</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>1</mn><mn>5</mn><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math>).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f304v" o="304v" n="607"/>
<pb file="add_6787_f305" o="305" n="608"/>
<div xml:id="echoid-div76" type="page_commentary" level="2" n="76">
<p>
<s xml:id="echoid-s345" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s345" xml:space="preserve">
This is the cover sheet of eight pages copied from Book 3, Propositions 1 and 2,
of the <emph style="it">Sphaerica Menelai</emph> of Menelaus of Alexandria,
as translated by Francesco Maurolico (1588). <lb/>
The text includes a supplement by Thabit ibn Qurra (826–901).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat">
Propositiones quædam <lb/>
ex Menelauo <lb/>
et <lb/>
Thebit
<lb/>[<emph style="it">tr:
Certain propositions from Menelaus and Thebit
</emph>]<lb/>
</head>
<pb file="add_6787_f305v" o="305v" n="609"/>
<pb file="add_6787_f306" o="306" n="610"/>
<div xml:id="echoid-div77" type="page_commentary" level="2" n="77">
<p>
<s xml:id="echoid-s347" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s347" xml:space="preserve">
Text from pages 36v–37v of <emph style="it">Sphaerica Menelai</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat">
Ex libro 3. Spæricorum Menelai <lb/>
secundum traditionem Maurolyci <lb/>
1
<lb/>[<emph style="it">tr:
From Book 3 of Sphærica Menelai, as translated by Maurolicus
</emph>]<lb/>
</head>
<pb file="add_6787_f306v" o="306v" n="611"/>
<div xml:id="echoid-div78" type="page_commentary" level="2" n="78">
<p>
<s xml:id="echoid-s349" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s349" xml:space="preserve">
Text from pages 37v–38 of <emph style="it">Sphaerica Menelai</emph>. <lb/>
The supplement by Thabit ibn Qurra begins here and continues to f. 308.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s351" xml:space="preserve">
<lb/>[...]<lb/>
</s>
<lb/>
<s xml:id="echoid-s352" xml:space="preserve">
Suppementum Thebitij
<lb/>[<emph style="it">tr:
Supplement by Thebit
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s353" xml:space="preserve">
<lb/>[...]<lb/>
</s>
</p>
<pb file="add_6787_f307" o="307" n="612"/>
<div xml:id="echoid-div79" type="page_commentary" level="2" n="79">
<p>
<s xml:id="echoid-s354" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s354" xml:space="preserve">
Text from pages 38 from <emph style="it">Sphaerica Menelai</emph>. <lb/>
For further work by Harriot on the 18 cases of Lemma 4, see Ad MS 6786, f. 440.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head84" xml:space="preserve">
2
</head>
<pb file="add_6787_f307v" o="307v" n="613"/>
<div xml:id="echoid-div80" type="page_commentary" level="2" n="80">
<p>
<s xml:id="echoid-s356" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s356" xml:space="preserve">
Text from page 38v of <emph style="it">Sphaerica Menelai</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f308" o="308" n="614"/>
<div xml:id="echoid-div81" type="page_commentary" level="2" n="81">
<p>
<s xml:id="echoid-s358" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s358" xml:space="preserve">
Text from pages 38v–39 of <emph style="it">Sphaerica Menelai</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head85" xml:space="preserve">
3
</head>
<pb file="add_6787_f308v" o="308v" n="615"/>
<div xml:id="echoid-div82" type="page_commentary" level="2" n="82">
<p>
<s xml:id="echoid-s360" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s360" xml:space="preserve">
Text from pages 39 of <emph style="it">Sphaerica Menelai</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6787_f309" o="309" n="616"/>
<div xml:id="echoid-div83" type="page_commentary" level="2" n="83">
<p>
<s xml:id="echoid-s362" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s362" xml:space="preserve">
Text from pages 39–39v of <emph style="it">Sphaerica Menelai</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head86" xml:space="preserve">
4
</head>
<pb file="add_6787_f309v" o="309v" n="617"/>
<div xml:id="echoid-div84" type="page_commentary" level="2" n="84">
<p>
<s xml:id="echoid-s364" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s364" xml:space="preserve">
Text from pages 39v–40 of <emph style="it">Sphaerica Menelai</emph>.<lb/>
Thabit ibn Qurra is mentioned in the final paragraph; see also Add MS 6787, f. 306v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s366" xml:space="preserve">
Liber impressus fuit <lb/>
Messanæ in freto Siculo <lb/>
1558
<lb/>[<emph style="it">tr:
The book was printed in Messina, in the Straits of Sicily, 1558.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f310" o="310" n="618"/>
<pb file="add_6787_f310v" o="310v" n="619"/>
<pb file="add_6787_f311" o="311" n="620"/>
<pb file="add_6787_f311v" o="311v" n="621"/>
<pb file="add_6787_f312" o="312" n="622"/>
<pb file="add_6787_f312v" o="312v" n="623"/>
<pb file="add_6787_f313" o="313" n="624"/>
<pb file="add_6787_f313v" o="313v" n="625"/>
<pb file="add_6787_f314" o="314" n="626"/>
<pb file="add_6787_f314v" o="314v" n="627"/>
<pb file="add_6787_f315" o="315" n="628"/>
<pb file="add_6787_f315v" o="315v" n="629"/>
<pb file="add_6787_f316" o="316" n="630"/>
<pb file="add_6787_f316v" o="316v" n="631"/>
<pb file="add_6787_f317" o="317" n="632"/>
<div xml:id="echoid-div85" type="page_commentary" level="2" n="85">
<p>
<s xml:id="echoid-s367" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s367" 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-s369" xml:space="preserve">
cadaeibf <lb/>
31415926
</s>
</p>
<pb file="add_6787_f317v" o="317v" n="633"/>
<pb file="add_6787_f318" o="318" n="634"/>
<pb file="add_6787_f318v" o="318v" n="635"/>
<div xml:id="echoid-div86" type="page_commentary" level="2" n="86">
<p>
<s xml:id="echoid-s370" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s370" 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 xml:lang="lat">
<s xml:id="echoid-s372" xml:space="preserve">
Inquisitio indentata hasta apud Stratford Langthornes
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s373" xml:space="preserve">
Bis non spondebis <lb/>
quod præsto solvere valebis
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s374" xml:space="preserve">
In Principio
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s375" xml:space="preserve">
Bis duo notam
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s376" xml:space="preserve">
Arbor ut ex fractu <lb/>
sit nequam noscitur acta
</s>
</p>
<p>
<s xml:id="echoid-s377" xml:space="preserve">
James
</s>
</p>
<p>
<s xml:id="echoid-s378" xml:space="preserve">
Alexand
</s>
</p>
<p>
<s xml:id="echoid-s379" xml:space="preserve">
cadaeibf <lb/>
31415926
</s>
</p>
<pb file="add_6787_f319" o="319" n="636"/>
<p><s xml:id="echoid-s380"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math></s></p>
<p><s xml:id="echoid-s381"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>,</mo><mi>a</mi><mi>c</mi><mo>,</mo><mi>z</mi><mi>x</mi></mstyle></math></s><lb/>
 <s xml:id="echoid-s382"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>,</mo><mi>z</mi><mi>x</mi><mo>:</mo><mi>a</mi><mi>b</mi><mi>e</mi><mo>,</mo><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s383"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>,</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>z</mi><mi>x</mi></mstyle></math></s><lb/>
 <s xml:id="echoid-s384"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>,</mo><mi>z</mi><mi>x</mi><mo>:</mo><mi>a</mi><mi>c</mi><mi>d</mi><mo>,</mo><mi>a</mi><mi>b</mi><mi>e</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s385"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>-</mo><mo>√</mo><mn>2</mn><mo>,</mo><mo>√</mo><mn>2</mn><mo>,</mo><mn>1</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s386"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mo>=</mo><mn>5</mn><mn>8</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s387"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mo maxsize="1.01">/</mo><mn>5</mn><mn>8</mn><mn>6</mn><mo>=</mo><mn>2</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math></s></p>
<pb file="add_6787_f319v" o="319v" n="637"/>
<p><s xml:id="echoid-s388"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>3</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s389"><reg norm="Tempus" type="abbr">Temp.</reg> <reg norm="Spatium" type="abbr">Spac.</reg></s></p>
<p><s xml:id="echoid-s390"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>1</mn><mn>5</mn><mn>0</mn><mn>0</mn></mstyle></math></s></p>
<pb file="add_6787_f320" o="320" n="638"/>
<pb file="add_6787_f320v" o="320v" n="639"/>
<pb file="add_6787_f321" o="321" n="640"/>
<pb file="add_6787_f321v" o="321v" n="641"/>
<pb file="add_6787_f322" o="322" n="642"/>
<pb file="add_6787_f322v" o="322v" n="643"/>
<pb file="add_6787_f323" o="323" n="644"/>
<pb file="add_6787_f323v" o="323v" n="645"/>
<pb file="add_6787_f324" o="324" n="646"/>
<p><s xml:id="echoid-s391"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>1</mn><mn>0</mn><mn>8</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>8</mn><mn>6</mn><mn>0</mn><mn>5</mn><mn>1</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>4</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>5</mn><mn>4</mn><mn>2</mn><mn>9</mn><mn>9</mn><mn>0</mn><mn>1</mn><mn>1</mn><mn>1</mn><mn>2</mn><mo>.</mo><mn>8</mn><mn>0</mn><mn>2</mn><mn>7</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s392"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>5</mn><mn>4</mn><mn>2</mn><mn>9</mn><mn>9</mn><mn>0</mn><mn>1</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>8</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>7</mn><mn>5</mn><mn>0</mn><mn>5</mn><mn>0</mn><mo>.</mo><mn>2</mn><mn>0</mn><mn>2</mn><mn>4</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s393"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>7</mn><mn>5</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>2</mn><mn>0</mn><mn>2</mn><mn>4</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>4</mn><mn>7</mn><mn>1</mn><mn>9</mn><mn>8</mn><mn>9</mn><mn>2</mn><mo>.</mo><mn>1</mn><mn>0</mn><mn>8</mn><mn>1</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s394"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>4</mn><mn>7</mn><mn>1</mn><mn>9</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>1</mn><mn>0</mn><mn>8</mn><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>9</mn><mn>3</mn><mo>.</mo><mn>0</mn><mn>6</mn><mn>6</mn><mn>3</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s395"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>9</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>6</mn><mn>3</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>3</mn><mn>3</mn><mn>8</mn><mn>5</mn><mn>0</mn><mn>8</mn><mn>0</mn><mn>5</mn><mn>2</mn><mo>.</mo><mn>6</mn><mn>8</mn><mn>2</mn><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s396"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>3</mn><mn>3</mn><mn>8</mn><mn>5</mn><mn>0</mn><mn>8</mn><mn>0</mn><mn>5</mn><mn>2</mn><mn>6</mn><mn>8</mn><mn>2</mn><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>6</mn><mn>9</mn><mn>2</mn><mn>3</mn><mn>9</mn><mn>7</mn><mn>0</mn><mn>5</mn><mo>.</mo><mn>3</mn><mn>0</mn><mn>2</mn><mn>1</mn></mstyle></math></s></p>
<pb file="add_6787_f324v" o="324v" n="647"/>
<pb file="add_6787_f325" o="325" n="648"/>
<p><s xml:id="echoid-s397"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>8</mn><mn>5</mn><mn>7</mn><mn>8</mn><mn>7</mn><mn>9</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo maxsize="1.01">/</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>3</mn><mo>=</mo><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s398"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>6</mn><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>1</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>0</mn><mn>7</mn><mo>.</mo><mn>1</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s399"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>D</mi><mi>c</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn><mi>e</mi><mi>t</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>2</mn></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s><lb/>
 <s xml:id="echoid-s400"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mi>g</mi><mi>y</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>4</mn></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>5</mn><mn>1</mn><mn>2</mn><mo>-</mo><mn>4</mn></mstyle></math></s><lb/>
 <s xml:id="echoid-s401"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mi>r</mi><mi>H</mi></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>8</mn></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn><mo>-</mo><mn>8</mn></mstyle></math></s><lb/></p>
<p><s xml:id="echoid-s402"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>1</mn><mn>4</mn><mo>×</mo><mn>4</mn><mn>1</mn><mn>4</mn><mo>=</mo><mn>1</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>9</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s403"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mn>4</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>5</mn><mn>1</mn><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s404"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s405"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mo>×</mo><mn>1</mn><mn>6</mn><mo>=</mo><mn>2</mn><mn>5</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s406"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mn>6</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>5</mn><mn>1</mn><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s407"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>6</mn><mn>4</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s408"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mn>4</mn><mo>×</mo><mn>6</mn><mn>4</mn><mo>=</mo><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s409"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn><mo>×</mo><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn><mo>=</mo><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s410"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn><mo>+</mo><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn><mo>=</mo><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s411"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>1</mn><mn>2</mn><mo>×</mo><mn>5</mn><mn>1</mn><mn>2</mn><mo>=</mo><mn>2</mn><mn>6</mn><mn>2</mn><mn>1</mn><mn>4</mn><mn>4</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s412"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>6</mn><mn>2</mn><mn>1</mn><mn>4</mn><mn>4</mn><mo>×</mo><mn>5</mn><mn>1</mn><mn>2</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>8</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s413"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>1</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>9</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>7</mn><mo>.</mo><mn>7</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s414"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>7</mn><mn>7</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>7</mn><mn>2</mn><mn>4</mn><mn>0</mn><mn>6</mn><mn>1</mn><mn>6</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s415"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s416"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s417"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mrow><mn>4</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>4</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s418"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mrow><mn>8</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>8</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s419"><emph style="st"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn></mstyle></math><lb/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></emph></s></p>
<p><s xml:id="echoid-s420"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s421"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>5</mn><mn>1</mn><mn>2</mn><mo>-</mo><mn>4</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s422"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn><mo>-</mo><mn>8</mn></mstyle></math></s></p>
<p><s xml:id="echoid-s423">Quæritur Summa</s></p>
<pb file="add_6787_f325v" o="325v" n="649"/>
<pb file="add_6787_f326" o="326" n="650"/>
<p><s xml:id="echoid-s424"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>,</mo><mn>2</mn><mn>2</mn><mo>,</mo><mn>1</mn><mn>6</mn></mstyle></math></s></p>
<pb file="add_6787_f326v" o="326v" n="651"/>
<pb file="add_6787_f327" o="327" n="652"/>
<pb file="add_6787_f327v" o="327v" n="653"/>
<pb file="add_6787_f328" o="328" n="654"/>
<pb file="add_6787_f328v" o="328v" n="655"/>
<pb file="add_6787_f329" o="329" n="656"/>
<pb file="add_6787_f329v" o="329v" n="657"/>
<pb file="add_6787_f330" o="330" n="658"/>
<pb file="add_6787_f330v" o="330v" n="659"/>
<pb file="add_6787_f331" o="331" n="660"/>
<pb file="add_6787_f331v" o="331v" n="661"/>
<pb file="add_6787_f332" o="332" n="662"/>
<pb file="add_6787_f332v" o="332v" n="663"/>
<pb file="add_6787_f333" o="333" n="664"/>
<pb file="add_6787_f333v" o="333v" n="665"/>
<pb file="add_6787_f334" o="334" n="666"/>
<pb file="add_6787_f334v" o="334v" n="667"/>
<pb file="add_6787_f335" o="335" n="668"/>
<pb file="add_6787_f335v" o="335v" n="669"/>
<pb file="add_6787_f336" o="336" n="670"/>
<pb file="add_6787_f336v" o="336v" n="671"/>
<pb file="add_6787_f337" o="337" n="672"/>
<pb file="add_6787_f337v" o="337v" n="673"/>
<pb file="add_6787_f338" o="338" n="674"/>
<pb file="add_6787_f338v" o="338v" n="675"/>
<div xml:id="echoid-div87" type="page_commentary" level="2" n="87">
<p>
<s xml:id="echoid-s425" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s425" xml:space="preserve">
Calulations based on on the series 3, 5, 7, 9, ..., with constant difference 2. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> denote sums of terms taken two, three, four, or five at a time.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat">
4. Differentiæ differentiarum
<lb/>[<emph style="it">tr:
Differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f339" o="339" n="676"/>
<div xml:id="echoid-div88" type="page_commentary" level="2" n="88">
<p>
<s xml:id="echoid-s427" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s427" xml:space="preserve">
Calculations based on a general series with constant second difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
The second and first rows begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> respectively. <lb/>
The series 5, 8, 13, 20, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>.
(A second version takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>.)
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken two at a time.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat">
5. Differentiæ differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s429" xml:space="preserve">
differentiæ
<lb/>[<emph style="it">tr:
differences
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s430" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr:
Another way
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f339v" o="339v" n="677"/>
<pb file="add_6787_f340" o="340" n="678"/>
<pb file="add_6787_f340v" o="340v" n="679"/>
<div xml:id="echoid-div89" type="page_commentary" level="2" n="89">
<p>
<s xml:id="echoid-s431" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s431" xml:space="preserve">
Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken two at a time. <lb/>
Fingers in the margin point to formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head89" xml:space="preserve">
5).2.
</head>
<pb file="add_6787_f341" o="341" n="680"/>
<div xml:id="echoid-div90" type="page_commentary" level="2" n="90">
<p>
<s xml:id="echoid-s433" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s433" xml:space="preserve">
Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken two at a time. <lb/>
A finger in the margin points to a formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head90" xml:space="preserve">
5).3.
</head>
<pb file="add_6787_f341v" o="341v" n="681"/>
<div xml:id="echoid-div91" type="page_commentary" level="2" n="91">
<p>
<s xml:id="echoid-s435" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s435" xml:space="preserve">
Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken two at a time. <lb/>
Fingers in the margin point to formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head91" xml:space="preserve">
5).4.
</head>
<pb file="add_6787_f342" o="342" n="682"/>
<div xml:id="echoid-div92" type="page_commentary" level="2" n="92">
<p>
<s xml:id="echoid-s437" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s437" xml:space="preserve">
Calculations similar to those in Add MS 6787, f. 339. <lb/>
Here the series 7, 9, 14, 32, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>. <lb/>
The second example is 5, 8, 13, ..., as on Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken three at a time.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head92" xml:space="preserve" xml:lang="lat">
6. Differentiæ differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f342v" o="342v" n="683"/>
<pb file="add_6787_f343" o="343" n="684"/>
<pb file="add_6787_f343v" o="343v" n="685"/>
<div xml:id="echoid-div93" type="page_commentary" level="2" n="93">
<p>
<s xml:id="echoid-s439" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s439" xml:space="preserve">
Calculations similar to those on Add MS 6787, f. 341v, now for the series 7, 9, 14, ...from Add MS 6787, f. 342. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken three at a time. <lb/>
Fingers in the margin point to formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head93" xml:space="preserve">
6).
</head>
<pb file="add_6787_f344" o="344" n="686"/>
<pb file="add_6787_f344v" o="344v" n="687"/>
<div xml:id="echoid-div94" type="page_commentary" level="2" n="94">
<p>
<s xml:id="echoid-s441" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s441" xml:space="preserve">
Further calculations for the series 5, 8, 13, ... from Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken four at a time. <lb/>
Fingers in the margin point to formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat">
7.) Differentiæ differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f345" o="345" n="688"/>
<div xml:id="echoid-div95" type="page_commentary" level="2" n="95">
<p>
<s xml:id="echoid-s443" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s443" xml:space="preserve">
Further calculations for the series 5, 8, 13, ... from Add MS 6787, f. 339. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> denote successive sums of terms, taken five at a time. <lb/>
Fingers in the margin point to formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat">
8.) Differentiæ differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f345v" o="345v" n="689"/>
<pb file="add_6787_f346" o="346" n="690"/>
<div xml:id="echoid-div96" type="page_commentary" level="2" n="96">
<p>
<s xml:id="echoid-s445" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s445" xml:space="preserve">
Calculations on a general series with constant third difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
The third, second, and first rows begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> respectively. <lb/>
The series 10, 15, 23, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <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>f</mi></mstyle></math> denote successive sums of terms, taken two at a time. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head96" xml:space="preserve" xml:lang="lat">
9) Differentiæ differentiarum differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f346v" o="346v" n="691"/>
<pb file="add_6787_f347" o="347" n="692"/>
<pb file="add_6787_f347v" o="347v" n="693"/>
<div xml:id="echoid-div97" type="page_commentary" level="2" n="97">
<p>
<s xml:id="echoid-s447" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s447" xml:space="preserve">
Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <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>f</mi></mstyle></math> denote successive sums of terms, taken two at a time. <lb/>
The page shows formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head97" xml:space="preserve">
9)
</head>
<pb file="add_6787_f348" o="348" n="694"/>
<pb file="add_6787_f348v" o="348v" n="695"/>
<div xml:id="echoid-div98" type="page_commentary" level="2" n="98">
<p>
<s xml:id="echoid-s449" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s449" xml:space="preserve">
Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <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>f</mi></mstyle></math> denote successive sums of terms, taken three at a time. <lb/>
The page shows formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat">
10) Differentiæ differentiarum differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f349" o="349" n="696"/>
<pb file="add_6787_f349v" o="349v" n="697"/>
<div xml:id="echoid-div99" type="page_commentary" level="2" n="99">
<p>
<s xml:id="echoid-s451" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s451" xml:space="preserve">
Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <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>f</mi></mstyle></math> denote successive sums of terms, taken four at a time. <lb/>
The page shows formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat">
11) Differentiæ differentiarum differentiarum differentiarum
<lb/>[<emph style="it">tr:
Differences of differences of differences of differences
</emph>]<lb/>
</head>
<pb file="add_6787_f350" o="350" n="698"/>
<pb file="add_6787_f350v" o="350v" n="699"/>
<pb file="add_6787_f351" o="351" n="700"/>
<pb file="add_6787_f351v" o="351v" n="701"/>
<pb file="add_6787_f352" o="352" n="702"/>
<pb file="add_6787_f352v" o="352v" n="703"/>
<pb file="add_6787_f353" o="353" n="704"/>
<pb file="add_6787_f353v" o="353v" n="705"/>
<pb file="add_6787_f354" o="354" n="706"/>
<pb file="add_6787_f354v" o="354v" n="707"/>
<pb file="add_6787_f355" o="355" n="708"/>
<pb file="add_6787_f355v" o="355v" n="709"/>
<pb file="add_6787_f356" o="356" n="710"/>
<pb file="add_6787_f356v" o="356v" n="711"/>
<pb file="add_6787_f357" o="357" n="712"/>
<div xml:id="echoid-div100" type="page_commentary" level="2" n="100">
<p>
<s xml:id="echoid-s453" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s453" xml:space="preserve">
This is the first of a set of 11 pages containing propositions from Book I of the
<emph style="it">Conics</emph> of Apollonius. The edition used by Harriot was Commandino's
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566),
but Harriot translated the verbal propositions and proofs into his own symbolic notation. <lb/>
English translations of the propositions are taken from Dana Densmore (ed),
<emph style="it">Apollonius of Perga: Conics Books I–III</emph>, Green Lion Press, 1998. <lb/>
</s>
<lb/>
<s xml:id="echoid-s454" xml:space="preserve">
Proposition 11 of Book I is Apollonius's definition of a parabola.
</s>
<lb/>
<quote>
I.11
If a cone is cut by a plane through its axis, and also cut by another plane
cutting the base of the cone in a straight line perpendicular to the base of the axial triangle,
and if, further, the diameter of the section is parallel to one side of the axial triangle,
and if any straight line is drawn from the section of the cone to its diameter
such that this straight line is parallel to the common section of the cutting plane and of the cone 19s base,
then this straight line to the diameter will equal in square the rectangle contained
by the straight line from the section 19s vertex to where the straight line to the diameter cuts it off,
and another straight line which has the same ratio to the straight line between the angle of the cone
and the vertex of the section as the square on the base of the axial triangle
has to the rectangle contained by the remaining two sides of the triangle.
And let such a section be called a parabola.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat">
Appol. lib. 1. prop. 11. De conicis
<lb/>[<emph style="it">tr:
Apollonius, Book I, Proposition 11. On conics
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s456" xml:space="preserve">
parabola
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s457" xml:space="preserve">
Aliter demonstrationem <lb/>
ordinavimus ut <lb/>
sequitur.
<lb/>[<emph style="it">tr:
Another demonstration, which we have ordered as follows.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s458" xml:space="preserve">
Ratio componitur ex
<lb/>[<emph style="it">tr:
Ratio composed from
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s459" xml:space="preserve">
Quæsitum
<lb/>[<emph style="it">tr:
Sought
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f357v" o="357v" n="713"/>
<pb file="add_6787_f358" o="358" n="714"/>
<div xml:id="echoid-div101" type="page_commentary" level="2" n="101">
<p>
<s xml:id="echoid-s460" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s460" xml:space="preserve">
Proposition 20 of Book I of Apollonius, edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566),
is the defining property of a parabola in terms of its ordinates to the diameter.
</s>
<lb/>
<quote>
I.20
If in a parabola two straight lines are dropped as ordinates to the diameter,
the squares on them will be to each other as the straight lines cut off by them on the diameter
beginning from the vertex are to each other.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat">
Appol. lib. 1. prop. 20. De conicis
<lb/>[<emph style="it">tr:
Apollonius, Book I, Proposition 20. On conics
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s462" xml:space="preserve">
parabola
</s>
</p>
<pb file="add_6787_f358v" o="358v" n="715"/>
<div xml:id="echoid-div102" type="page_commentary" level="2" n="102">
<p>
<s xml:id="echoid-s463" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s463" xml:space="preserve">
This page refers to Propositions II.1, II.2, and I.21 from Apollonius,
<emph style="it">Conicorum libri quattuor</emph>.
</s>
<lb/>
<quote>
II.1
If a straight line touch an hyperbola at its vertex, and from it on both sides of the diamter
a straight line is cut off equal in square to the fourth of the figure,
then the straight lines drawn from the centre of the section to the ends thus taken on the tangent
will not meet the section.
</quote>
<lb/>
<quote>
II.2
With the same things it is to be shown that a straight line cutting the angle contained by
the straight lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> is not another asymptote.
</quote>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s465" xml:space="preserve">
prop. 2
<lb/>[<emph style="it">tr:
Proposition 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s466" xml:space="preserve">
lib. 2. prop: 1.
<lb/>[<emph style="it">tr:
Book 2, Proposition 1.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s467" xml:space="preserve">
per. 21
<lb/>[<emph style="it">tr:
by Proposition 21.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s468" xml:space="preserve">
absurdum
<lb/>[<emph style="it">tr:
absurd
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s469" xml:space="preserve">
absurdum
<lb/>[<emph style="it">tr:
absurd
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s470" xml:space="preserve">
prop. 2
<lb/>[<emph style="it">tr:
Proposition 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s471" xml:space="preserve">
Ergo absurdum
<lb/>[<emph style="it">tr:
Therefore absurd
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f359" o="359" n="716"/>
<div xml:id="echoid-div103" type="page_commentary" level="2" n="103">
<p>
<s xml:id="echoid-s472" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s472" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
Proposition 30 is a property of the ellipse.
</s>
<lb/>
<quote>
I.30
If in an ellipse or in opposite sections a straight line is drawn in both directions from the centre,
meeting the section, it will be bisected at the centre.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat">
Appol. 22.b. <lb/>
lib. 1. pro. 30.
<lb/>[<emph style="it">tr:
Apollonius, page 22v, Book I, Proposition 30.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s474" xml:space="preserve">
In elipsij
<lb/>[<emph style="it">tr:
In an ellipse
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f359v" o="359v" n="717"/>
<pb file="add_6787_f360" o="360" n="718"/>
<div xml:id="echoid-div104" type="page_commentary" level="2" n="104">
<p>
<s xml:id="echoid-s475" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s475" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
Proposition 33 explains how to find the tangent to any point of a parabola.
</s>
<lb/>
<quote>
I.33
If on a parabola some point is taken, and from it an ordinate is dropped to the diameter, and,
to the straight line cut off by it on the diameter from the vertex,
a straight line in the same straight line from its extremity is made equal,
then the straight line joined from the point thus resulting to the point taken will touch the section.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head103" xml:space="preserve" xml:lang="lat">
Appoll. pa: 24.b <lb/>
lib. 1. pr. 33
<lb/>[<emph style="it">tr:
Apollonius, page 24v, Book I, Proposition 33.
</emph>]<lb/>
</head>
<pb file="add_6787_f360v" o="360v" n="719"/>
<pb file="add_6787_f361" o="361" n="720"/>
<div xml:id="echoid-div105" type="page_commentary" level="2" n="105">
<p>
<s xml:id="echoid-s477" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s477" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
Proposition 37 gives a property of a tangent to a hyperbola or ellipse or circle.
</s>
<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-head104" xml:space="preserve" xml:lang="lat">
lib. 1. <lb/>
Appol. pag. 27. <lb/>
prop. 37.
<lb/>[<emph style="it">tr:
Book I, Apollonius, page 27, Proposition 37.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s479" xml:space="preserve">
Dico quod
<lb/>[<emph style="it">tr:
I say that
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s480" xml:space="preserve">
In hyperbola
<lb/>[<emph style="it">tr:
In a hyerbola
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s481" xml:space="preserve">
In elipsi &amp; circulo
<lb/>[<emph style="it">tr:
In an ellipse and circle
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f361v" o="361v" n="721"/>
<pb file="add_6787_f362" o="362" n="722"/>
<div xml:id="echoid-div106" type="page_commentary" level="2" n="106">
<p>
<s xml:id="echoid-s482" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s482" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
Proposition 38 gives a further property of a tangent to a hyperbola or ellipse or circle
(see also Add MS 6787, f. 361).
</s>
<lb/>
<quote>
I.38
If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter,
and if from the point of contact a straight line is dropped to that same [second] diameter parallel
to the other diameter, then the straight line cut off from the centre of the section
by the dropped straight line, together with the straight line cut off [on the second diameter]
by the tangent from the centre of the section will contain an area
equal to the square on the half of the second diameter, and,
together with the straight line [on the second diameter] between the dropped straight line and the tangent,
will contain an area having a ratio to the square on the dropped straight line
which the upright side of the figure has to the transverse.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat">
Appol. pag. 28 <lb/>
pro: 38.
<lb/>[<emph style="it">tr:
Apollonius, page 28, Proposition 38.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s484" xml:space="preserve">
Iisdem positis.
<lb/>[<emph style="it">tr:
Under the same suppositions.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s485" xml:space="preserve">
1. casus: hyperboles
<lb/>[<emph style="it">tr:
Case 1. Hyperbolas
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s486" xml:space="preserve">
Sed: <lb/>
in elipsi <lb/>
per 36
<lb/>[<emph style="it">tr:
But in an ellipse, by proposition 36
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s487" xml:space="preserve">
In Ellipsi
<lb/>[<emph style="it">tr:
In an ellipse
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f362v" o="362v" n="723"/>
<div xml:id="echoid-div107" type="page_commentary" level="2" n="107">
<p>
<s xml:id="echoid-s488" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s488" xml:space="preserve">
This page refers to Propositions 21, 36, and 37 from Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21.
If in a hyperbola or ellipse or circumference of a circle straight lines
are dropped as ordinates to the diameter, the square on them will be to the areas contained
by the straight lines cut off by them beginning from the ends of the transverse side of the figure,
as the upright side of the figure is to the transverse, and to each other as the areas contained
by the straight lines cut off, as we have said.
</quote>
<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>
<p xml:lang="lat">
<s xml:id="echoid-s490" xml:space="preserve">
Nota
<lb/>[<emph style="it">tr:
Note
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s491" xml:space="preserve">
per 37:
<lb/>[<emph style="it">tr:
by proposition 37
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s492" xml:space="preserve">
per 21:
<lb/>[<emph style="it">tr:
by proposition 21
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s493" xml:space="preserve">
per, 36: ita:
<lb/>[<emph style="it">tr:
by proposition 36, thus:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s494" xml:space="preserve">
igitur: ita:
<lb/>[<emph style="it">tr:
therefore, thus:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f363" o="363" n="724"/>
<div xml:id="echoid-div108" type="page_commentary" level="2" n="108">
<p>
<s xml:id="echoid-s495" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s495" xml:space="preserve">
For proposition 38, see Add MS 6787, f. 362.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat">
Appol: lib. 1. pag. 28. <lb/>
prop: 38
<lb/>[<emph style="it">tr:
Apollonius, page 28, Proposition 38.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s497" xml:space="preserve">
Dico quod:
<lb/>[<emph style="it">tr:
I say that
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s498" xml:space="preserve">
2. Casus
<lb/>[<emph style="it">tr:
Case 2.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f363v" o="363v" n="725"/>
<pb file="add_6787_f364" o="364" n="726"/>
<div xml:id="echoid-div109" type="page_commentary" level="2" n="109">
<p>
<s xml:id="echoid-s499" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s499" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
Proposition 39 gives a further property of a tangent to a hyperbola or ellipse or circle
(see also Add MS 6787, f. 361 and f. 362).
</s>
<lb/>
<quote>
I.39
If a straight line touching a hyperbola or ellipse or circumference of a circle meets the diameter,
and if from the point of contact a straight line is dropped as ordinate to the diameter,
then whichever of the two straight lines is taken, of which one is the straight line between the
[intersection of the] ordinate [with the diameter] and the centre of the section,
and the other is between [the intersection of] the ordinate and the tangent [with the diameter],
the ordinate will have to it the ratio compounded of the ratio of the other of the two straight lines
to the ordinate and of the ratio of the upright side of the figure to the transverse.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat">
lib. 1. <lb/>
App. pag. 29. <lb/>
pro: 39.
<lb/>[<emph style="it">tr:
Book I, Apollonius, page 29, Proposition 39.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s501" xml:space="preserve">
Nostro modo <lb/>
brevissime.
<lb/>[<emph style="it">tr:
My method, very brief.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s502" xml:space="preserve">
Sed: <lb/>
in elipsi <lb/>
per 36
<lb/>[<emph style="it">tr:
But in an ellipse, by proposition 36
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s503" xml:space="preserve">
In Ellipsi
<lb/>[<emph style="it">tr:
In an ellipse
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f364v" o="364v" n="727"/>
<div xml:id="echoid-div110" type="page_commentary" level="2" n="110">
<p>
<s xml:id="echoid-s504" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s504" xml:space="preserve">
This page refers to Propositions I.38 and I.40 from Apollonius,
<emph style="it">Conicorum libri quattuor</emph>.
</s>
<lb/>
<quote>
I.38
If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter,
and if from the point of contact a straight line is dropped to that same [second] diameter parallel
to the other diameter, then the straight line cut off from the centre of the section
by the dropped straight line, together with the straight line cut off [on the second diameter]
by the tangent from the centre of the section will contain an area
equal to the square on the half of the second diameter, and,
together with the straight line [on the second diameter] between the dropped straight line and the tangent,
will contain an area having a ratio to the square on the dropped straight line
which the upright side of the figure has to the transverse.
</quote>
<lb/>
<quote>
I.40.
If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter,
and if from the point of contact a straight line is dropped to the same diameter parallel to the other diameter,
then whichever of the two straight lines is taken [along the second diameter], of which one is
the straight line between the dropped straight line and the centre of the section, and the other is
between the dropped straight line and the tangent, the dropped straight line will have to it
the ratio compounded of the ratio of the transverse side to the upright and of
the ratio of the other of the two straight lines to the dropped straight line.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s506" xml:space="preserve">
prop: 40
<lb/>[<emph style="it">tr:
proposition 40
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s507" xml:space="preserve">
per 38:
<lb/>[<emph style="it">tr:
by proposition 38
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f365" o="365" n="728"/>
<div xml:id="echoid-div111" type="page_commentary" level="2" n="111">
<p>
<s xml:id="echoid-s508" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s508" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.41
If in a hyperbola or ellipse or circumference of a circle a straight line is dropped as ordinate to the diameter,
and if equiangular parallelograms are described both on the ordinate and on the radius,
and if the ordinate side has to the remaining side of the figure the ratio compounded of the ratio of the radius
to the remaining side of its figure, and the ratio of the upright side of the section 19s figure to the transverse,
then the figure on the straight line between the centre and the ordinate, similar to the figure on the radius,
is in the case of the hyperbola greater than the figure on the ordinate by the figure on the radius, and,
in the case of the ellipse and circumference of a circle, together with the figure on the ordinate
is equal to the figure on the radius.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat">
pag: 29. b. <lb/>
Appol. pro: 41
<lb/>[<emph style="it">tr:
page 29v, Apollonius, Proposition 41.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s510" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> ordinata <lb/>
et parallelogramma <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> æquiangula
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> be an ordinate, and parallelograms <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> equiangular.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s511" xml:space="preserve">
composita ex
<lb/>[<emph style="it">tr:
composed from
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s512" xml:space="preserve">
Dico quod: in Hyperbole
<lb/>[<emph style="it">tr:
I say that, in a hyperbola
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s513" xml:space="preserve">
In ellipsi et circulo
<lb/>[<emph style="it">tr:
In an ellipse and circle
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f365v" o="365v" n="729"/>
<pb file="add_6787_f366" o="366" n="730"/>
<div xml:id="echoid-div112" type="page_commentary" level="2" n="112">
<p>
<s xml:id="echoid-s514" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s514" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/>
For proposition 41, see Add MS 6787, f. 365.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat">
pag: 29. <lb/>
Ap. pro: 41
<lb/>[<emph style="it">tr:
page 29, Apollonius, Proposition 41.
</emph>]<lb/>
</head>
<pb file="add_6787_f366v" o="366v" n="731"/>
<pb file="add_6787_f367" o="367" n="732"/>
<div xml:id="echoid-div113" type="page_commentary" level="2" n="113">
<p>
<s xml:id="echoid-s516" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s516" xml:space="preserve">
The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/>
For Proposition 11, the original definition of a parabola, see Add MS 6787, f. 357.
</s>
<lb/>
<quote>
I. 52
Given a straight line in a plane bounded at one point, to find in the plane the section of a cone called parabola,
whose diameter is the given straight line, and whose vertex is the end of the straight line,
and where whatever straight line is dropped from the section to the diameter at a given angle,
will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section
and by some other given straight line.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat">
pag. 37. <lb/>
Appol. pro: 52
<lb/>[<emph style="it">tr:
page 37, Apollonius, Proposition 52.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s518" xml:space="preserve">
ad latus rectum
<lb/>[<emph style="it">tr:
for the latus rectum</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s519" xml:space="preserve">
per. 11. ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>a</mi><mi>l</mi></mstyle></math> est sectio <lb/>
cuius axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
by proposition 11, therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>a</mi><mi>l</mi></mstyle></math> is the section whose axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> with line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s520" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> recta <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi><mi>e</mi></mstyle></math> angulus appl. <lb/>
non recta
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> be the diameter, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> the line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi><mi>e</mi></mstyle></math> the angle of application, not a right angle.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s521" xml:space="preserve">
unde fit sectio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math> ex cono recto <lb/>
ut supra. et transit per <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> est contingens, quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>l</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
whence arises the section <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math> from the right cone as above, and the crossing line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> is a tangent,
because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>l</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s522" xml:space="preserve">
Ergo per 49 <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> est latus <lb/>
rectum. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter &amp;c.
<lb/>[<emph style="it">tr:
Therefore by pproposition 49, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> is the latus rectum and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> the diameter.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f367v" o="367v" n="733"/>
<pb file="add_6787_f368" o="368" n="734"/>
<pb file="add_6787_f368v" o="368v" n="735"/>
<pb file="add_6787_f369" o="369" n="736"/>
<pb file="add_6787_f369v" o="369v" n="737"/>
<pb file="add_6787_f370" o="370" n="738"/>
<pb file="add_6787_f370v" o="370v" n="739"/>
<pb file="add_6787_f371" o="371" n="740"/>
<pb file="add_6787_f371v" o="371v" n="741"/>
<pb file="add_6787_f372" o="372" n="742"/>
<div xml:id="echoid-div114" type="page_commentary" level="2" n="114">
<p>
<s xml:id="echoid-s523" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s523" xml:space="preserve">
This page refers to Proposition 21 of Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head111" xml:space="preserve" xml:lang="lat">
De ellipsi.
<lb/>[<emph style="it">tr:
On the ellipse
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s525" xml:space="preserve">
Theorema
</s>
<lb/>
<s xml:id="echoid-s526" xml:space="preserve">
Si quotlibet lineis ordinatim applicatis ad diametrum circuli, aliæ numero <lb/>
et quantitate æquales similiter applicentur ad similes partes lineæ maioris <emph style="st">vel</emph> <lb/>
minoris <emph style="super">de</emph> data diametro circuli: termini illarum linearum sunt in ellipsi.
<lb/>[<emph style="it">tr:
If any number of ordinate lines are dropped to the diameter of a circle, that number of other lines,
equal in quantity and similarly applied to similar parts of lines greater or less than the given diameter of a circe,
then the ends of those lines are on an ellipse.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s527" xml:space="preserve">
Sit diameter circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>. lineæ ordinatim applicatæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. sit etiam <lb/>
linea maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> cui applicentur ad angulos rectos <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>β</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math> æquales <lb/>
<emph style="super">lineis</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> circuli.
<emph style="st">Et sint partes lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> videlicet [???] quotlibet </emph> <lb/>
<emph style="st">[???]</emph> <emph style="super">sed ita</emph>
fiat ud <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>β</mo></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>θ</mo></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math>. Quod etiam fit <lb/>
si utraque lineæ <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math></emph> similter dividantur et ad utraque partes
<emph style="st">æquales</emph> similes et <lb/>
similiter sitas æquales lineæ ordinatim applicentur. <lb/>
Dico quod puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> termini linearum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>θ</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math> sunt in ellipsi. <lb/>
Quoniam ex hypothesi <lb/>
<lb/>[...]<lb/>
Ergo: per 21, prop: primi Apollonij, puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> <lb/>
sunt in ellipsi. quod demonstrare oportuit.
<lb/>[<emph style="it">tr:
Let the diameter of the circle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>, and the ordinate lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>;
also let the line greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math>, to which are applied at right angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>β</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math>
euqal to the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> in the circle.
But thus, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> so is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>β</mo></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>θ</mo></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math>.
Whcih also happens if the two lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> are similarly divided and to both parts,
similar and similarly situated, equal ordinate lines are applied. <lb/>
I say that the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math>, the ends of the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>θ</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math>, are on an ellipse. <lb/>
Because from the hypothesis: <lb/>
<lb/>[...]<lb/>
Therefore, by Proposition 21 of the first Book of Apollonius, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> are on an ellipse;
which was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f372v" o="372v" n="743"/>
<pb file="add_6787_f373" o="373" n="744"/>
<pb file="add_6787_f373v" o="373v" n="745"/>
<pb file="add_6787_f374" o="374" n="746"/>
<pb file="add_6787_f374v" o="374v" n="747"/>
<pb file="add_6787_f375" o="375" n="748"/>
<pb file="add_6787_f375v" o="375v" n="749"/>
<pb file="add_6787_f376" o="376" n="750"/>
<pb file="add_6787_f376v" o="376v" n="751"/>
<pb file="add_6787_f377" o="377" n="752"/>
<pb file="add_6787_f377v" o="377v" n="753"/>
<p>
<s xml:id="echoid-s528" xml:space="preserve">
Amen <lb/>
Alexander King of M. <lb/>
</s>
</p>
<p>
<s xml:id="echoid-s529" xml:space="preserve">
[tomas haryots] <lb/>
[Tomas Haryots] <lb/>
[yn the begeneng]
</s>
</p>
<pb file="add_6787_f378" o="378" n="754"/>
<pb file="add_6787_f378v" o="378v" n="755"/>
<pb file="add_6787_f379" o="379" n="756"/>
<pb file="add_6787_f379v" o="379v" n="757"/>
<pb file="add_6787_f380" o="380" n="758"/>
<pb file="add_6787_f380v" o="380v" n="759"/>
<pb file="add_6787_f381" o="381" n="760"/>
<pb file="add_6787_f381v" o="381v" n="761"/>
<pb file="add_6787_f382" o="382" n="762"/>
<pb file="add_6787_f382v" o="382v" n="763"/>
<pb file="add_6787_f383" o="383" n="764"/>
<pb file="add_6787_f383v" o="383v" n="765"/>
<pb file="add_6787_f384" o="384" n="766"/>
<pb file="add_6787_f384v" o="384v" n="767"/>
<pb file="add_6787_f385" o="385" n="768"/>
<pb file="add_6787_f385v" o="385v" n="769"/>
<pb file="add_6787_f386" o="386" n="770"/>
<pb file="add_6787_f386v" o="386v" n="771"/>
<pb file="add_6787_f387" o="387" n="772"/>
<pb file="add_6787_f387v" o="387v" n="773"/>
<pb file="add_6787_f388" o="388" n="774"/>
<pb file="add_6787_f388v" o="388v" n="775"/>
<pb file="add_6787_f389" o="389" n="776"/>
<div xml:id="echoid-div115" type="page_commentary" level="2" n="115">
<p>
<s xml:id="echoid-s530" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s530" 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 pages 320v–321 is Proposition 14.
</s>
<lb/>
<quote xml:lang="lat">
Problema X. Propositio XIV. <lb/>
Facile autem est inuentis quibuscumque coniugationibus diametrorum ellipsis,
axes cuius organice inuenire. quod quidem hac ratione fiet.
</quote>
<lb/>
<quote>
Moreover, having found any conjugates of the diamters of the ellipse,
it is easy to find the axes mechancially, which indeed are in this ratio.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat">
pappus. 321 <lb/>
Datæ, coniugatæ elypseos <lb/>
diametri <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>. <lb/>
Quæruntur axes.
<lb/>[<emph style="it">tr:
Pappus, page 321. <lb/>
Given conjugate ellipses with diameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>, there are sought the axes.
</emph>]<lb/>
</head>
<pb file="add_6787_f389v" o="389v" n="777"/>
<pb file="add_6787_f390" o="390" n="778"/>
<pb file="add_6787_f390v" o="390v" n="779"/>
<pb file="add_6787_f391" o="391" n="780"/>
<pb file="add_6787_f391v" o="391v" n="781"/>
<pb file="add_6787_f392" o="392" n="782"/>
<pb file="add_6787_f392v" o="392v" n="783"/>
<pb file="add_6787_f393" o="393" n="784"/>
<pb file="add_6787_f393v" o="393v" n="785"/>
<pb file="add_6787_f394" o="394" n="786"/>
<div xml:id="echoid-div116" type="page_commentary" level="2" n="116">
<p>
<s xml:id="echoid-s532" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s532" xml:space="preserve">
The reference on this page is to Giambattista Benedetti,
<emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat">
Locus in sphaeram concavu et convexa, videtur in catheto. <lb/>
fallitur Baptista de Ben: <lb/>
pag: 339 <lb/>
343 <lb/>
344
</head>
<pb file="add_6787_f394v" o="394v" n="787"/>
<pb file="add_6787_f395" o="395" n="788"/>
<pb file="add_6787_f395v" o="395v" n="789"/>
<div xml:id="echoid-div117" type="page_commentary" level="2" n="117">
<p>
<s xml:id="echoid-s534" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s534" xml:space="preserve">
Some rough work on Proposition 14 from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
continued from Add MS 6787, f. 227 to f. 230.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head114" xml:space="preserve">
Viet. lib. 8. res. prop. 14. <foreign xml:lang="gre">proch?on</foreign> <lb/>
pag. 35.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, Proposition 14, page 35.
</emph>]<lb/>
</head>
<pb file="add_6787_f396" o="396" n="790"/>
<pb file="add_6787_f396v" o="396v" n="791"/>
<pb file="add_6787_f397" o="397" n="792"/>
<pb file="add_6787_f397v" o="397v" n="793"/>
<pb file="add_6787_f398" o="398" n="794"/>
<pb file="add_6787_f398v" o="398v" n="795"/>
<div xml:id="echoid-div118" type="page_commentary" level="2" n="118">
<p>
<s xml:id="echoid-s536" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s536" xml:space="preserve">The reference on this page is to Viète's
<emph style="it">Variorum de rebus mathematicis responsorum lilber VIII</emph> (1593).
Viète investigated the properties of the quadratrix in Chapter VII.
Harriot's diagram is the same as Viète's.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat">
Vieta resp. pag. 12. lin: 9, De quadrataria
<lb/>[<emph style="it">tr:
Viète, Responsorum, page 12, line 9, On the quadratirix
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s538" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>E</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>E</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s539" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> maior, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>H</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math> absurdum
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>H</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math>, absurd
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s540" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math> absurdum
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math>, absurd
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s541" xml:space="preserve">
Et inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>,</mo><mi>B</mi><mi>D</mi></mstyle></math> quæsitum
<lb/>[<emph style="it">tr:
And hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>B</mi><mi>D</mi></mstyle></math>, which was sought.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f399" o="399" n="796"/>
<pb file="add_6787_f399v" o="399v" n="797"/>
<pb file="add_6787_f400" o="400" n="798"/>
<pb file="add_6787_f400v" o="400v" n="799"/>
<div xml:id="echoid-div119" type="page_commentary" level="2" n="119">
<p>
<s xml:id="echoid-s542" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s542" xml:space="preserve">
This page refers to Proposition 51 of Book III of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
III.51
If a rectangle equal to a fourth part of the figure is applied from both sides to the axis of an hyperbola
or opposite sections and exceeding by a square figure, and straight lines are deflected from
the resulting points of application to either one of the sections,
then the greater of the two straight lines exceeds the less by exactly as much as the axis.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat">
Hyperboles descriptio. per 51. pr. 3: lib. Appollonij
<lb/>[<emph style="it">tr:
A description of a hyperbola, by Propostion 53, Book 3, of Apollonius.
</emph>]<lb/>
</head>
<pb file="add_6787_f401" o="401" n="800"/>
<pb file="add_6787_f401v" o="401v" n="801"/>
<pb file="add_6787_f402" o="402" n="802"/>
<pb file="add_6787_f402v" o="402v" n="803"/>
<div xml:id="echoid-div120" type="page_commentary" level="2" n="120">
<p>
<s xml:id="echoid-s544" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s544" xml:space="preserve">
The reference on this page is to Giambattista Benedetti,
<emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head117" xml:space="preserve" xml:lang="lat">
Bap: de Ben: <lb/>
pag: 349 <lb/>
334
<lb/>[<emph style="it">tr:
Baptista de Benedictis, page 349, 334
</emph>]<lb/>
</head>
<pb file="add_6787_f403" o="403" n="804"/>
<pb file="add_6787_f403v" o="403v" n="805"/>
<pb file="add_6787_f404" o="404" n="806"/>
<div xml:id="echoid-div121" type="page_commentary" level="2" n="121">
<p>
<s xml:id="echoid-s546" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s546" xml:space="preserve">
The references on this page are to Francesco Barozzi,
<emph style="it">Admirandum illud geometricum problema tredecim modis demonstratum </emph> (1586).
Page 104 contains a diagram.
Harriot's (lower case) letters correspond to the (upper case) letters in Barozzi's diagram.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head118" xml:space="preserve" xml:lang="lat">
Barocius. pag. 104.
</head>
<pb file="add_6787_f404v" o="404v" n="807"/>
<pb file="add_6787_f405" o="405" n="808"/>
<pb file="add_6787_f405v" o="405v" n="809"/>
<pb file="add_6787_f406" o="406" n="810"/>
<pb file="add_6787_f406v" o="406v" n="811"/>
<pb file="add_6787_f407" o="407" n="812"/>
<pb file="add_6787_f407v" o="407v" n="813"/>
<pb file="add_6787_f408" o="408" n="814"/>
<pb file="add_6787_f408v" o="408v" n="815"/>
<pb file="add_6787_f409" o="409" n="816"/>
<pb file="add_6787_f409v" o="409v" n="817"/>
<pb file="add_6787_f410" o="410" n="818"/>
<pb file="add_6787_f410v" o="410v" n="819"/>
<pb file="add_6787_f411" o="411" n="820"/>
<pb file="add_6787_f411v" o="411v" n="821"/>
<pb file="add_6787_f412" o="412" n="822"/>
<pb file="add_6787_f412v" o="412v" n="823"/>
<pb file="add_6787_f413" o="413" n="824"/>
<pb file="add_6787_f413v" o="413v" n="825"/>
<pb file="add_6787_f414" o="414" n="826"/>
<pb file="add_6787_f414v" o="414v" n="827"/>
<pb file="add_6787_f415" o="415" n="828"/>
<pb file="add_6787_f415v" o="415v" n="829"/>
<pb file="add_6787_f416" o="416" n="830"/>
<pb file="add_6787_f416v" o="416v" n="831"/>
<pb file="add_6787_f417" o="417" n="832"/>
<pb file="add_6787_f417v" o="417v" n="833"/>
<pb file="add_6787_f418" o="418" n="834"/>
<pb file="add_6787_f418v" o="418v" n="835"/>
<pb file="add_6787_f419" o="419" n="836"/>
<pb file="add_6787_f419v" o="419v" n="837"/>
<pb file="add_6787_f420" o="420" n="838"/>
<pb file="add_6787_f420v" o="420v" n="839"/>
<pb file="add_6787_f421" o="421" n="840"/>
<pb file="add_6787_f421v" o="421v" n="841"/>
<pb file="add_6787_f422" o="422" n="842"/>
<pb file="add_6787_f422v" o="422v" n="843"/>
<pb file="add_6787_f423" o="423" n="844"/>
<pb file="add_6787_f423v" o="423v" n="845"/>
<pb file="add_6787_f424" o="424" n="846"/>
<pb file="add_6787_f424v" o="424v" n="847"/>
<pb file="add_6787_f425" o="425" n="848"/>
<div xml:id="echoid-div122" type="page_commentary" level="2" n="122">
<p>
<s xml:id="echoid-s548" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s548" xml:space="preserve">
The reference towards the end of this page is to Giambattista Benedetti,
<emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585),
thought the theorem on page 26 is actually Theorem 41, not Theorem 45. <lb/>
There is also a reference to Proposition 13 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593),
which explains how to find two quantities from their geometric mean and their sum.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s550" xml:space="preserve">
Data
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s551" xml:space="preserve">
Data in partibus canonis
<lb/>[<emph style="it">tr:
Given in parts of the canon
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s552" xml:space="preserve">
Ex Methodo <lb/>
Adde [???] <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>d</mi></mstyle></math> <lb/>
vel illa æqualium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi><mi>o</mi></mstyle></math> <lb/>
qui in centro <lb/>
sed in principia
<lb/>[<emph style="it">tr:
By the method: add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>d</mi></mstyle></math> or its equal <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi><mi>o</mi></mstyle></math> which in the centre is <lb/>[...]<lb/> but in the beginning <lb/>[...]<lb/>.
</emph>]<lb/>
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s553" xml:space="preserve">
tum quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>l</mi></mstyle></math>. per Theor. 45. pag. 26. Joh. Baptistæ <lb/>
de Benedictis
<lb/>[<emph style="it">tr:
then there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>l</mi></mstyle></math>, by Theorem 45, page 26, Johan Baptista de Benedictis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s554" xml:space="preserve">
Vel per Algebræ
<lb/>[<emph style="it">tr:
Or by algebra
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s555" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math>. tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>b</mi></mstyle></math> erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mi>r</mi></mstyle></math>. hoc multiplicatum per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math> faciet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi></mstyle></math>. <lb/>
quod æquale erit rectangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>m</mi></mstyle></math>, hoc est 78,545,532. <lb/>
Forma æquationis ita erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi><mo>=</mo><mn>7</mn><mn>8</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>5</mn><mn>3</mn><mn>2</mn></mstyle></math>. <lb/>
Et duplis erit responsum, videlicet
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi><mo>=</mo><mn>1</mn><mi>r</mi></mstyle></math>. then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>b</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mi>r</mi></mstyle></math>. This multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math> makes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi></mstyle></math>. <lb/>
which is equal to the rectangle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>m</mi></mstyle></math>, that is 78,545,532. <lb/>
Thus the form of the equation will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi><mo>=</mo><mn>7</mn><mn>8</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>5</mn><mn>3</mn><mn>2</mn></mstyle></math>. <lb/>
And it will be twice the answer, namely
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s556" xml:space="preserve">
Habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> alias per 13 prop. Geom. Effect. <lb/>
Vietæ
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> can be had otherwise by Proposition 13 of ViÂ´te, <emph style="it">Effectionum geometricarum</emph>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f425v" o="425v" n="849"/>
<pb file="add_6787_f426" o="426" n="850"/>
<pb file="add_6787_f426v" o="426v" n="851"/>
<pb file="add_6787_f427" o="427" n="852"/>
<pb file="add_6787_f427v" o="427v" n="853"/>
<pb file="add_6787_f428" o="428" n="854"/>
<pb file="add_6787_f428v" o="428v" n="855"/>
<pb file="add_6787_f429" o="429" n="856"/>
<pb file="add_6787_f429v" o="429v" n="857"/>
<pb file="add_6787_f430" o="430" n="858"/>
<pb file="add_6787_f430v" o="430v" n="859"/>
<pb file="add_6787_f431" o="431" n="860"/>
<pb file="add_6787_f431v" o="431v" n="861"/>
<pb file="add_6787_f432" o="432" n="862"/>
<pb file="add_6787_f432v" o="432v" n="863"/>
<pb file="add_6787_f433" o="433" n="864"/>
<pb file="add_6787_f433v" o="433v" n="865"/>
<pb file="add_6787_f434" o="434" n="866"/>
<pb file="add_6787_f434v" o="434v" n="867"/>
<pb file="add_6787_f435" o="435" n="868"/>
<pb file="add_6787_f435v" o="435v" n="869"/>
<pb file="add_6787_f436" o="436" n="870"/>
<pb file="add_6787_f436v" o="436v" n="871"/>
<pb file="add_6787_f437" o="437" n="872"/>
<pb file="add_6787_f437v" o="437v" n="873"/>
<pb file="add_6787_f438" o="438" n="874"/>
<pb file="add_6787_f438v" o="438v" n="875"/>
<pb file="add_6787_f439" o="439" n="876"/>
<pb file="add_6787_f439v" o="439v" n="877"/>
<pb file="add_6787_f440" o="440" n="878"/>
<pb file="add_6787_f440v" o="440v" n="879"/>
<pb file="add_6787_f441" o="441" n="880"/>
<pb file="add_6787_f441v" o="441v" n="881"/>
<pb file="add_6787_f442" o="442" n="882"/>
<pb file="add_6787_f442v" o="442v" n="883"/>
<pb file="add_6787_f443" o="443" n="884"/>
<pb file="add_6787_f443v" o="443v" n="885"/>
<pb file="add_6787_f444" o="444" n="886"/>
<pb file="add_6787_f444v" o="444v" n="887"/>
<pb file="add_6787_f445" o="445" n="888"/>
<pb file="add_6787_f445v" o="445v" n="889"/>
<pb file="add_6787_f446" o="446" n="890"/>
<pb file="add_6787_f446v" o="446v" n="891"/>
<pb file="add_6787_f447" o="447" n="892"/>
<pb file="add_6787_f447v" o="447v" n="893"/>
<pb file="add_6787_f448" o="448" n="894"/>
<pb file="add_6787_f448v" o="448v" n="895"/>
<pb file="add_6787_f449" o="449" n="896"/>
<pb file="add_6787_f449v" o="449v" n="897"/>
<div xml:id="echoid-div123" type="page_commentary" level="2" n="123">
<p>
<s xml:id="echoid-s557" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s557" xml:space="preserve">
This page refers to Proposition 37 of Book III of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
If two straight lines touching a section of a cone or circumference of a circle or opposite sections meet,
and a straight line is joined to their points of contact, and from the point of meeting of the tangents
some straight lin is drawn across cutting the line (of the section) at two points,
then as the whole straight line is to the straight line cut off from outside,
so will the segments proeuced by the straight line joining the points be to each other.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s559" xml:space="preserve">
37.3<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
Proposition III.37
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f450" o="450" n="898"/>
<pb file="add_6787_f450v" o="450v" n="899"/>
<pb file="add_6787_f451" o="451" n="900"/>
<div xml:id="echoid-div124" type="page_commentary" level="2" n="124">
<p>
<s xml:id="echoid-s560" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s560" xml:space="preserve">
This page refers to Propositions I.21 and I.34 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
<lb/>
<quote>
I.34 If on a hyperbola or ellipse or circumference of a circle some point is taken,
and from it a straight line is dropped as ordinate to the diameter,
and if the straight lines which the ordinate cuts off from the ends of the figure 19s transverse side
have to each other a ratio which other segments of the transverse side have to each other,
so that the segments from the vertex are corresponding, then the straight line joining the point
taken on the transverse side and that taken on the section will touch the section.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat">
Appol. lib: 1. pag. 25 <lb/>
prop: 34. Elementa tactus
<lb/>[<emph style="it">tr:
Apollonius, Book 1, page 25, Proposition 34. Elements of tangents.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s562" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi></mstyle></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> <lb/>
iugatur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> <lb/>
Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> tangit <lb/>
si non secet ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi><mi>f</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> be joined; I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> is a tangent. <lb/>
If not, it cuts, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s563" xml:space="preserve">
per 21, 1 <lb/>[...]<lb/> ex hypoth
<lb/>[<emph style="it">tr:
by Propositon I.21 <lb/>[...]<lb/> by hypothesis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s564" xml:space="preserve">
absurdum
<lb/>[<emph style="it">tr:
absurd
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f451v" o="451v" n="901"/>
<pb file="add_6787_f452" o="452" n="902"/>
<pb file="add_6787_f452v" o="452v" n="903"/>
<pb file="add_6787_f453" o="453" n="904"/>
<pb file="add_6787_f453v" o="453v" n="905"/>
<pb file="add_6787_f454" o="454" n="906"/>
<pb file="add_6787_f454v" o="454v" n="907"/>
<pb file="add_6787_f455" o="455" n="908"/>
<div xml:id="echoid-div125" type="page_commentary" level="2" n="125">
<p>
<s xml:id="echoid-s565" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s565" xml:space="preserve">
This page refers to Proposition 17 of Book III of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
III. 17
If two straight lines touching a section of a cone or circumference of a circle meet,
and two points are taken at random on the section, and from them in the section are drawn parallel
to the tangents straight lines cutting each other and the line of the section,
then as the squares on the tangents are to each other, so will the rectangles contained by the straight lines
taken similarly.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head120" xml:space="preserve" xml:lang="lat">
1) circa 5 data puncta <emph style="super">quæ sunt in ellipsi</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>l</mi><mi>m</mi><mi>n</mi></mstyle></math> <lb/>
ellipsin describere.
<lb/>[<emph style="it">tr:
Around 5 given points which are on an ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>l</mi><mi>m</mi><mi>n</mi></mstyle></math>, describe the ellipse.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s567" xml:space="preserve">
sit descripta.
<lb/>[<emph style="it">tr:
suppose it described
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s568" xml:space="preserve">
1. casus. sint <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math>, parallelæ. <lb/>
dividantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> bisarium <lb/>
in punctis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. et per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
ducatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi></mstyle></math>. quæ diameter <lb/>
est ellipseos <lb/>
data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Case 1. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> are parallel. <lb/>
Bisect <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, and through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi></mstyle></math>,
which is the diameter of the ellipse. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> are given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s569" xml:space="preserve">
17.3.coni
<lb/>[<emph style="it">tr:
Proposition III.17 of the conics.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s570" xml:space="preserve">
datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi><mo>.</mo><mi>c</mi><mi>g</mi></mstyle></math> quoniam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math>, secat <lb/>
datam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> ergo datam et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>h</mi></mstyle></math>. <lb/>
ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> datam quoniam
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi><mo>.</mo><mi>c</mi><mi>g</mi></mstyle></math> is given because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math>, and cuts the given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is therefore given, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>h</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> is geven because
ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> datam quoniam </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s571" xml:space="preserve">
sed et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> datur. Igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>p</mi><mi>x</mi></mstyle></math> postione <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> positione. Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>, dabitur, quod est in ellipsi.
<lb/>[<emph style="it">tr:
but also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> is given. Therefore the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>p</mi><mi>x</mi></mstyle></math> and the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math>.
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> will be given, which is on the ellipse.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s572" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>.</mo><mi>a</mi><mi>s</mi></mstyle></math> datum. quia, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi></mstyle></math>, data. <lb/>
Igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mo>.</mo><mi>e</mi><mi>f</mi></mstyle></math> datur.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>.</mo><mi>a</mi><mi>s</mi></mstyle></math> is fiven, because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi></mstyle></math> are given. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mo>.</mo><mi>e</mi><mi>f</mi></mstyle></math> is given. </emph>]<lb/>
</s>
</p>
<pb file="add_6787_f455v" o="455v" n="909"/>
<pb file="add_6787_f456" o="456" n="910"/>
<pb file="add_6787_f456v" o="456v" n="911"/>
<pb file="add_6787_f457" o="457" n="912"/>
<pb file="add_6787_f457v" o="457v" n="913"/>
<pb file="add_6787_f458" o="458" n="914"/>
<div xml:id="echoid-div126" type="page_commentary" level="2" n="126">
<p>
<s xml:id="echoid-s573" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s573" xml:space="preserve">
This page refers to Propositions I.21 and I.54 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
<lb/>
<quote>
I.54 Given two bounded straight lines perpendicular to each other,
one of them being produced on the side of the right angle,
to find on the straight line produced the section of a cone called hyperbola in the same plane with the straight line,
so that the straight line produced is a diameter of the section and the point at the angle is the vertex,
and where whatever straight line is dropped from the section to the diameter, making an angle equal to a given angle,
will equal in square the rectangle applied to the other straight line having as breadth
the straight line cut off the dropped straight line beginning with the vertex and projecting beyond
by a figure similar and similarly situated to that contained by the original straight lines.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head121" xml:space="preserve" xml:lang="lat">
2) Circa 5 data puncta ellispin describere
<lb/>[<emph style="it">tr:
To describe an ellipse around five given points.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s575" xml:space="preserve">
simili ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> dabitur
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> are given in similar ratio
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s576" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>z</mi></mstyle></math> datur. ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>f</mi></mstyle></math> datur.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>z</mi></mstyle></math> is given, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>f</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s577" xml:space="preserve">
dantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, puncta; ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> ut deinceps. (alia charta) <lb/>
quære <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> diameter maagnitudine data est, et diameter <lb/>
ipsi coniugata, cum datur proportio transversis lateris et ad <lb/>
rectum. eadem enim est ut
<lb/>[<emph style="it">tr:
there are given the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>; and thereafter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> (another sheet);
whereby the magnitude of the diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> is given, and the diameter of the conjugate,
since the ratio to the transverse side is given and to the latus rectum. For it is the same as:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s578" xml:space="preserve">
nam <lb/>
21.1.con.)
<lb/>[<emph style="it">tr:
for by proposition I.21 of the conics
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s579" xml:space="preserve">
ergo data coniugata diameter, hoc est 2<emph style="super">a</emph> diameter nam <lb/>
<lb/>[...]<lb/> <lb/>
ergo per <lb/>
54.1.con <lb/>
ellipsis describatur, vel datur
<lb/>[<emph style="it">tr:
therefore the conjugate diameter is given, that is, the 2nd diameter, for <lb/>
<lb/>[...]<lb/> <lb/>
therefore by proposition I.54 of the conics, an ellipse is described, or given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s580" xml:space="preserve">
quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> are sought, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s581" xml:space="preserve">
[<emph style="it">Note:
Sheet 5 is Add MS 6787, f. 461.
 </emph>]<lb/>
in charta <lb/>
5 <lb/>
ita est
<lb/>[<emph style="it">tr:
in sheet 5, it is thus
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f458v" o="458v" n="915"/>
<pb file="add_6787_f459" o="459" n="916"/>
<pb file="add_6787_f459v" o="459v" n="917"/>
<pb file="add_6787_f460" o="460" n="918"/>
<div xml:id="echoid-div127" type="page_commentary" level="2" n="127">
<p>
<s xml:id="echoid-s582" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s582" 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 diagram given by Harriot at the top of this folio appears on page 320,
as part of Commandino's long commentary to Proposition 13.
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. Propositio XIII. <lb/>
Cum autem quæsitum fit circa quinque data puncta HKLMN ellispin describere.
Sit iam descripta: &amp; iunctæ MK NH primum sint parallelæ diuidanturque bisariam in punctis AB.
&amp; ducta AB ad EF puncta ellipsis producatur. est igitur EF ipsius diameter
per diffinitionem conicorum positione data. etenim vnumquodque punctorum AB datum est positione.
</quote>
<lb/>
<quote>
Moreover, when it is sought to draw an ellipse about the five given points H, K, L, M, N, suppose it is already done.
And first join the lines MK and NH, letting them be parallel, and bisected at the points A and B.
And AB is drawn and extended to E and F, points on the ellipse. Therefore, EF is the diameter of that ellipse,
by the rules of conics in a given position, and indeed, each of the points A and B is given in position.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head122" xml:space="preserve">
4.) Circa 5 data puncta <lb/>
ellipsin describere. <lb/>
pappus. 320
<lb/>[<emph style="it">tr:
To draw an ellipse through 5 given points. Pappus, page 320.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s584" xml:space="preserve">
melius in <lb/>
5 charta
<lb/>[<emph style="it">tr:
better in sheet 5
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f460v" o="460v" n="919"/>
<pb file="add_6787_f461" o="461" n="920"/>
<pb file="add_6787_f461v" o="461v" n="921"/>
<pb file="add_6787_f462" o="462" n="922"/>
<pb file="add_6787_f462v" o="462v" n="923"/>
<pb file="add_6787_f463" o="463" n="924"/>
<pb file="add_6787_f463v" o="463v" n="925"/>
<pb file="add_6787_f464" o="464" n="926"/>
<pb file="add_6787_f464v" o="464v" n="927"/>
<pb file="add_6787_f465" o="465" n="928"/>
<pb file="add_6787_f465v" o="465v" n="929"/>
<pb file="add_6787_f466" o="466" n="930"/>
<pb file="add_6787_f466v" o="466v" n="931"/>
<pb file="add_6787_f467" o="467" n="932"/>
<div xml:id="echoid-div128" type="page_commentary" level="2" n="128">
<p>
<s xml:id="echoid-s585" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s585" xml:space="preserve">
The reference on this page is to Guidobaldi del Monte (Guido Ubaldi),
<emph style="it">In duos Archimedies aequiponderantium libros paraphrasis</emph> (1588), page 172.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head123" xml:space="preserve" xml:lang="lat">
Archimedes <emph style="st">de quad</emph>: <emph style="super">centro gravitatis</emph>
parabolæ Ubaldus. pag: 172. <lb/>
de centro gravitatis
<lb/>[<emph style="it">tr:
Archimedes, on the centre of gravity of a parabola, Ubaldus page 172
</emph>]<lb/>
</head>
<pb file="add_6787_f467v" o="467v" n="933"/>
<pb file="add_6787_f468" o="468" n="934"/>
<pb file="add_6787_f468v" o="468v" n="935"/>
<pb file="add_6787_f469" o="469" n="936"/>
<pb file="add_6787_f469v" o="469v" n="937"/>
<pb file="add_6787_f470" o="470" n="938"/>
<pb file="add_6787_f470v" o="470v" n="939"/>
<pb file="add_6787_f471" o="471" n="940"/>
<pb file="add_6787_f471v" o="471v" n="941"/>
<pb file="add_6787_f472" o="472" n="942"/>
<pb file="add_6787_f472v" o="472v" n="943"/>
<pb file="add_6787_f473" o="473" n="944"/>
<pb file="add_6787_f473v" o="473v" n="945"/>
<pb file="add_6787_f474" o="474" n="946"/>
<pb file="add_6787_f474v" o="474v" n="947"/>
<pb file="add_6787_f475" o="475" n="948"/>
<div xml:id="echoid-div129" type="page_commentary" level="2" n="129">
<p>
<s xml:id="echoid-s587" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s587" xml:space="preserve">
The references on this page are to Francesco Barozzi,
<emph style="it">Admirandum illud geometricum problema tredecim modis demonstratum </emph> (1586)
and to Federico Commandino,
<emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/>
Pages 98 and 52 of the <emph style="it">Amirandum</emph> both contain diagrams.
Harriot's (lower case) letters correspond to the (upper case) letters in Barozzi's diagrams. <lb/>
The proposition of page 14 of <emph style="it">Conicorum libri quattuor</emph> is Proposition 11,
Apollonius's definition of a parabola. For the full statement of this proposition see Add MS 6787, f. 357.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head124" xml:space="preserve" xml:lang="lat">
Barocius. pag. 98.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s589" xml:space="preserve">
Hyperbola
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s590" xml:space="preserve">
Barocius <lb/>
pag. 52.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s591" xml:space="preserve">
Appol. pag. 14. parabola
<lb/>[<emph style="it">tr:
Apollonius, page 14, parabola
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s592" xml:space="preserve">
Brevissime
<lb/>[<emph style="it">tr:
very briefly
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f475v" o="475v" n="949"/>
<pb file="add_6787_f476" o="476" n="950"/>
<p>
<s xml:id="echoid-s593" xml:space="preserve">
<sc>
A span is approximately 8 inches, or 20 cm.
</sc>
25. spannes is a pole in length.
</s>
<lb/>
<s xml:id="echoid-s594" xml:space="preserve">
625. square spannes is a pole square.
</s>
<lb/>
<s xml:id="echoid-s595" xml:space="preserve">
1250. spannes is 2 square poles <emph style="st">square</emph>
</s>
</p>
<p>
<s xml:id="echoid-s596" xml:space="preserve">
<sc>
There are 4 poles to one chain, so 16 square poles to one square chain.
</sc>
or a square of the <lb/>
whole [¿]chayne[?]
</s>
<lb/>
<s xml:id="echoid-s597" xml:space="preserve">
<sc>
There are 10 square chains to an acre.
</sc>
or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> of an acre
</s>
<lb/>
<s xml:id="echoid-s598" xml:space="preserve">
or: a day worke.
</s>
</p>
<pb file="add_6787_f476v" o="476v" n="951"/>
<pb file="add_6787_f477" o="477" n="952"/>
<pb file="add_6787_f477v" o="477v" n="953"/>
<pb file="add_6787_f478" o="478" n="954"/>
<pb file="add_6787_f478v" o="478v" n="955"/>
<pb file="add_6787_f479" o="479" n="956"/>
<pb file="add_6787_f479v" o="479v" n="957"/>
<pb file="add_6787_f480" o="480" n="958"/>
<pb file="add_6787_f480v" o="480v" n="959"/>
<pb file="add_6787_f481" o="481" n="960"/>
<pb file="add_6787_f481v" o="481v" n="961"/>
<pb file="add_6787_f482" o="482" n="962"/>
<pb file="add_6787_f482v" o="482v" n="963"/>
<pb file="add_6787_f483" o="483" n="964"/>
<pb file="add_6787_f483v" o="483v" n="965"/>
<pb file="add_6787_f484" o="484" n="966"/>
<pb file="add_6787_f484v" o="484v" n="967"/>
<pb file="add_6787_f485" o="485" n="968"/>
<pb file="add_6787_f485v" o="485v" n="969"/>
<pb file="add_6787_f486" o="486" n="970"/>
<pb file="add_6787_f486v" o="486v" n="971"/>
<pb file="add_6787_f487" o="487" n="972"/>
<pb file="add_6787_f487v" o="487v" n="973"/>
<pb file="add_6787_f488" o="488" n="974"/>
<pb file="add_6787_f488v" o="488v" n="975"/>
<pb file="add_6787_f489" o="489" n="976"/>
<pb file="add_6787_f489v" o="489v" n="977"/>
<pb file="add_6787_f490" o="490" n="978"/>
<pb file="add_6787_f490v" o="490v" n="979"/>
<pb file="add_6787_f491" o="491" n="980"/>
<pb file="add_6787_f491v" o="491v" n="981"/>
<pb file="add_6787_f492" o="492" n="982"/>
<pb file="add_6787_f492v" o="492v" n="983"/>
<pb file="add_6787_f493" o="493" n="984"/>
<pb file="add_6787_f493v" o="493v" n="985"/>
<pb file="add_6787_f494" o="494" n="986"/>
<pb file="add_6787_f494v" o="494v" n="987"/>
<pb file="add_6787_f495" o="495" n="988"/>
<pb file="add_6787_f495v" o="495v" n="989"/>
<p>
<s xml:id="echoid-s599" xml:space="preserve">
Hilles. a joyner. to cutte the parabola <lb/>
at Ponden in Essex by Walton abbey <lb/>
Well. Borne.
</s>
</p>
<p>
<s xml:id="echoid-s600" xml:space="preserve">
Nic. Burcot <emph style="super">Burked</emph> a turner <lb/>
of town or shire <lb/>
of Chisleworth <lb/>
mint miller <lb/>
In St Annes Lane <lb/>
within Aldersgate
</s>
</p>
<pb file="add_6787_f496" o="496" n="990"/>
<pb file="add_6787_f496v" o="496v" n="991"/>
<pb file="add_6787_f497" o="497" n="992"/>
<pb file="add_6787_f497v" o="497v" n="993"/>
<pb file="add_6787_f498" o="498" n="994"/>
<pb file="add_6787_f498v" o="498v" n="995"/>
<pb file="add_6787_f499" o="499" n="996"/>
<pb file="add_6787_f499v" o="499v" n="997"/>
<pb file="add_6787_f500" o="500" n="998"/>
<pb file="add_6787_f500v" o="500v" n="999"/>
<pb file="add_6787_f501" o="501" n="1000"/>
<pb file="add_6787_f501v" o="501v" n="1001"/>
<pb file="add_6787_f502" o="502" n="1002"/>
<pb file="add_6787_f502v" o="502v" n="1003"/>
<pb file="add_6787_f503" o="503" n="1004"/>
<pb file="add_6787_f503v" o="503v" n="1005"/>
<pb file="add_6787_f504" o="504" n="1006"/>
<pb file="add_6787_f504v" o="504v" n="1007"/>
<pb file="add_6787_f505" o="505" n="1008"/>
<pb file="add_6787_f505v" o="505v" n="1009"/>
<pb file="add_6787_f506" o="506" n="1010"/>
<pb file="add_6787_f506v" o="506v" n="1011"/>
<pb file="add_6787_f507" o="507" n="1012"/>
<pb file="add_6787_f507v" o="507v" n="1013"/>
<pb file="add_6787_f508" o="508" n="1014"/>
<pb file="add_6787_f508v" o="508v" n="1015"/>
<pb file="add_6787_f509" o="509" n="1016"/>
<pb file="add_6787_f509v" o="509v" n="1017"/>
<pb file="add_6787_f510" o="510" n="1018"/>
<pb file="add_6787_f510v" o="510v" n="1019"/>
<pb file="add_6787_f511" o="511" n="1020"/>
<pb file="add_6787_f511v" o="511v" n="1021"/>
<pb file="add_6787_f512" o="512" n="1022"/>
<pb file="add_6787_f512v" o="512v" n="1023"/>
<pb file="add_6787_f513" o="513" n="1024"/>
<pb file="add_6787_f513v" o="513v" n="1025"/>
<pb file="add_6787_f514" o="514" n="1026"/>
<pb file="add_6787_f514v" o="514v" n="1027"/>
<pb file="add_6787_f515" o="515" n="1028"/>
<pb file="add_6787_f515v" o="515v" n="1029"/>
<pb file="add_6787_f516" o="516" n="1030"/>
<pb file="add_6787_f516v" o="516v" n="1031"/>
<pb file="add_6787_f517" o="517" n="1032"/>
<pb file="add_6787_f517v" o="517v" n="1033"/>
<pb file="add_6787_f518" o="518" n="1034"/>
<pb file="add_6787_f518v" o="518v" n="1035"/>
<pb file="add_6787_f519" o="519" n="1036"/>
<pb file="add_6787_f519v" o="519v" n="1037"/>
<pb file="add_6787_f520" o="520" n="1038"/>
<pb file="add_6787_f520v" o="520v" n="1039"/>
<pb file="add_6787_f521" o="521" n="1040"/>
<pb file="add_6787_f521v" o="521v" n="1041"/>
<pb file="add_6787_f522" o="522" n="1042"/>
<pb file="add_6787_f522v" o="522v" n="1043"/>
<pb file="add_6787_f523" o="523" n="1044"/>
<pb file="add_6787_f523v" o="523v" n="1045"/>
<pb file="add_6787_f524" o="524" n="1046"/>
<pb file="add_6787_f524v" o="524v" n="1047"/>
<pb file="add_6787_f525" o="525" n="1048"/>
<div xml:id="echoid-div130" type="page_commentary" level="2" n="130">
<p>
<s xml:id="echoid-s601" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s601" xml:space="preserve">
This page refers to Proposition I.34 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.34 If on a hyperbola or ellipse or circumference of a circle some point is taken,
and from it a straight line is dropped as ordinate to the diameter,
and if the straight lines which the ordinate cuts off from the ends of the figure 19s transverse side
have to each other a ratio which other segments of the transverse side have to each other,
so that the segments from the vertex are corresponding, then the straight line joining the point
taken on the transverse side and that taken on the section will touch the section.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head125" xml:space="preserve" xml:lang="lat">
ut in elementa <lb/>
conico ad tactus <lb/>
prop. 34. lib. 1. <lb/>
Appol.
<lb/>[<emph style="it">tr:
as in the elements of tangents to a cone, Proposition 34, Book 1, of Apollonius.
</emph>]<lb/>
</head>
<pb file="add_6787_f525v" o="525v" n="1049"/>
<pb file="add_6787_f526" o="526" n="1050"/>
<pb file="add_6787_f526v" o="526v" n="1051"/>
<pb file="add_6787_f527" o="527" n="1052"/>
<pb file="add_6787_f527v" o="527v" n="1053"/>
<pb file="add_6787_f528" o="528" n="1054"/>
<pb file="add_6787_f528v" o="528v" n="1055"/>
<pb file="add_6787_f529" o="529" n="1056"/>
<pb file="add_6787_f529v" o="529v" n="1057"/>
<pb file="add_6787_f530" o="530" n="1058"/>
<pb file="add_6787_f530v" o="530v" n="1059"/>
<pb file="add_6787_f531" o="531" n="1060"/>
<pb file="add_6787_f531v" o="531v" n="1061"/>
<pb file="add_6787_f532" o="532" n="1062"/>
<pb file="add_6787_f532v" o="532v" n="1063"/>
<pb file="add_6787_f533" o="533" n="1064"/>
<pb file="add_6787_f533v" o="533v" n="1065"/>
<pb file="add_6787_f534" o="534" n="1066"/>
<pb file="add_6787_f534v" o="534v" n="1067"/>
<pb file="add_6787_f535" o="535" n="1068"/>
<pb file="add_6787_f535v" o="535v" n="1069"/>
<pb file="add_6787_f536" o="536" n="1070"/>
<pb file="add_6787_f536v" o="536v" n="1071"/>
<pb file="add_6787_f537" o="537" n="1072"/>
<div xml:id="echoid-div131" type="page_commentary" level="2" n="131">
<p>
<s xml:id="echoid-s603" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s603" xml:space="preserve">
This page refers to Propositions I.21 and II.1 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
<lb/>
<quote>
II.1
If a straight line touch an hyperbola at its vertex, and from it on both sides of the diamter
a straight line is cut off equal in square to the fourth of the figure,
then the straight lines drawn from the centre of the section to the ends thus taken on the tangent
will not meet the section.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head126" xml:space="preserve" xml:lang="lat">
B.2. Appol. lib. 1. pr. 21 De hyperbola
<lb/>[<emph style="it">tr:
Apollonius, Book I, Proposition 21. On the hyperbola.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s605" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>x</mi></mstyle></math> latus <lb/>
rectum.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>x</mi></mstyle></math> is the latus rectum
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s606" xml:space="preserve">
2<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>u</mi></mstyle></math> = diametro 2<emph style="super">a</emph>
<lb/>[<emph style="it">tr:
2<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>u</mi></mstyle></math> is the second diameter
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s607" xml:space="preserve">
[<emph style="it">Note:
For two nearby sheets labelled B.4 see Add MS 6787, f. 545, f. 546.
 </emph>]<lb/>
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>u</mi></mstyle></math> asymptotos. Ut est per B.4 <lb/>
per 1p. 2, lib: Apol.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>u</mi></mstyle></math> is an asymptote, as it is by B.4., then by Proposition 1, Book 2 of Apollonius
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f537v" o="537v" n="1073"/>
<pb file="add_6787_f538" o="538" n="1074"/>
<pb file="add_6787_f538v" o="538v" n="1075"/>
<pb file="add_6787_f539" o="539" n="1076"/>
<pb file="add_6787_f539v" o="539v" n="1077"/>
<pb file="add_6787_f540" o="540" n="1078"/>
<pb file="add_6787_f540v" o="540v" n="1079"/>
<pb file="add_6787_f541" o="541" n="1080"/>
<pb file="add_6787_f541v" o="541v" n="1081"/>
<pb file="add_6787_f542" o="542" n="1082"/>
<pb file="add_6787_f542v" o="542v" n="1083"/>
<pb file="add_6787_f543" o="543" n="1084"/>
<pb file="add_6787_f543v" o="543v" n="1085"/>
<pb file="add_6787_f544" o="544" n="1086"/>
<pb file="add_6787_f544v" o="544v" n="1087"/>
<pb file="add_6787_f545" o="545" n="1088"/>
<pb file="add_6787_f545v" o="545v" n="1089"/>
<pb file="add_6787_f546" o="546" n="1090"/>
<pb file="add_6787_f546v" o="546v" n="1091"/>
<pb file="add_6787_f547" o="547" n="1092"/>
<pb file="add_6787_f547v" o="547v" n="1093"/>
<pb file="add_6787_f548" o="548" n="1094"/>
<pb file="add_6787_f548v" o="548v" n="1095"/>
<pb file="add_6787_f549" o="549" n="1096"/>
<div xml:id="echoid-div132" type="page_commentary" level="2" n="132">
<p>
<s xml:id="echoid-s608" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s608" xml:space="preserve">
The references on this page are to the verses of Leviticus, Chapter 18. <lb/>
The rules in the uper left column are taken directly from the verses of Leviticus,
and refer to sexual relations prohibited to a man. <lb/>
In the upper right column, Harriot has given the equivalent relations for a woman. <lb/>
In the lower columns, Harriot has reversed the generations,
so that 'father' is replaced by 'son', 'mother' by 'daughter', and so on.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head127" xml:space="preserve">
Leviticus. cap. 18
</head>
<p>
<s xml:id="echoid-s610" xml:space="preserve">
A man shall not <lb/>
marry. <lb/>
7) <emph style="super">verse</emph> his mother <lb/>
8. his fathers wife <lb/>
or stepmother <lb/>
9. <emph style="super">his sister being:</emph> the daughter of his father <lb/>
or the daughter of his mother. <lb/>
10. his sonnes daughter. <lb/>
or his daughters daughter. <lb/>
11. his fathers wifes daughter, <lb/>
begotten by the father. <lb/>
12. his fathers sister. <lb/>
13. his mothers sister. <lb/>
14. his fathers brothers wife. <lb/>
15. his sonnes wife. <lb/>
16. his brothers wife. <lb/>
17. his wives daughter. <lb/>
(his wives daughters daughter) <lb/>
(his wives sonnes daughter).
</s>
</p>
<p>
<s xml:id="echoid-s611" xml:space="preserve">
A woman shall <lb/>
not marry. <lb/>
7) <emph style="super">verse</emph> her father <lb/>
8. her mothers hsuband <lb/>
or stepmother <lb/>
9. her brother being: <lb/>
the sonne of her father <lb/>
or the sonne of her mother. <lb/>
10. her daughters sonne. <lb/>
or her sonnes sonne. <lb/>
11. her mothers husbands sonne. <lb/>
12. her mothers brother. <lb/>
13. her fathers brother. <lb/>
14. her mothers sisters husband. <lb/>
15. her daughters hsuband. <lb/>
16. her sisters husband. <lb/>
17. her husbands sonne. <lb/>
(her husbands sonnes sonne) <lb/>
(her husbands daughters sonne).
</s>
</p>
<p>
<s xml:id="echoid-s612" xml:space="preserve">
A woman shall <lb/>
not marry. <lb/>
7) <emph style="super">verse</emph> her sonne <lb/>
8. her husbandes sonne <lb/>
or stepmother <lb/>
9. her fathers sonne <lb/>
her mothers sonne <lb/>
10. her fathers father. <lb/>
her mothers mother. <lb/>
11. her mothers husbands sonne. <lb/>
12. her brothers sonne. <lb/>
13. her sisters sonne. <lb/>
14. her husbands brothers sonne. <lb/>
15. her hsubands father. <lb/>
16. her husbands brother. <lb/>
17. her mothers husband. <lb/>
(her mothers mothers hsuband) <lb/>
(her fathers mothesr husband).
</s>
</p>
<p>
<s xml:id="echoid-s613" xml:space="preserve">
A man shall <lb/>
not marry. <lb/>
7) <emph style="super">verse</emph> his daughter <lb/>
8. his wives daughter <lb/>
9. his father daughter. <lb/>
his mothers daughter. <lb/>
10. his mothers mother. <lb/>
his fathers mother. <lb/>
11. his father wives daughter. <lb/>
12. his sisters daughter. <lb/>
13. his brothers daughter. <lb/>
14. his wives sisters daughter. <lb/>
15. his wives <emph style="st">daughter</emph> mother. <lb/>
16. his wives sister. <lb/>
17. his fathers wife. <lb/>
(his fathers fathers wife) <lb/>
(his mothers fathers wife).
</s>
</p>
<pb file="add_6787_f549v" o="549v" n="1097"/>
<pb file="add_6787_f550" o="550" n="1098"/>
<pb file="add_6787_f550v" o="550v" n="1099"/>
<pb file="add_6787_f551" o="551" n="1100"/>
<pb file="add_6787_f551v" o="551v" n="1101"/>
<pb file="add_6787_f552" o="552" n="1102"/>
<pb file="add_6787_f552v" o="552v" n="1103"/>
<pb file="add_6787_f553" o="553" n="1104"/>
<pb file="add_6787_f553v" o="553v" n="1105"/>
<pb file="add_6787_f554" o="554" n="1106"/>
<pb file="add_6787_f554v" o="554v" n="1107"/>
<pb file="add_6787_f555" o="555" n="1108"/>
<pb file="add_6787_f555v" o="555v" n="1109"/>
<pb file="add_6787_f556" o="556" n="1110"/>
<pb file="add_6787_f556v" o="556v" n="1111"/>
<pb file="add_6787_f557" o="557" n="1112"/>
<pb file="add_6787_f557v" o="557v" n="1113"/>
<pb file="add_6787_f558" o="558" n="1114"/>
<pb file="add_6787_f558v" o="558v" n="1115"/>
<pb file="add_6787_f559" o="559" n="1116"/>
<pb file="add_6787_f559v" o="559v" n="1117"/>
<pb file="add_6787_f560" o="560" n="1118"/>
<pb file="add_6787_f560v" o="560v" n="1119"/>
<pb file="add_6787_f561" o="561" n="1120"/>
<pb file="add_6787_f561v" o="561v" n="1121"/>
<pb file="add_6787_f562" o="562" n="1122"/>
<pb file="add_6787_f562v" o="562v" n="1123"/>
<pb file="add_6787_f563" o="563" n="1124"/>
<pb file="add_6787_f563v" o="563v" n="1125"/>
<pb file="add_6787_f564" o="564" n="1126"/>
<pb file="add_6787_f564v" o="564v" n="1127"/>
<pb file="add_6787_f565" o="565" n="1128"/>
<p>
<s xml:id="echoid-s614" xml:space="preserve">
a.b.c.d. ... <lb/>
A.B.C.D. ...
</s>
</p>
<pb file="add_6787_f565v" o="565v" n="1129"/>
<pb file="add_6787_f566" o="566" n="1130"/>
<div xml:id="echoid-div133" type="page_commentary" level="2" n="133">
<p>
<s xml:id="echoid-s615" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s615" xml:space="preserve">
The reference to Viète is to his
<emph style="it">Variorum responsorum liber VIII</emph>,
Chapter 17, entitled 'Progressio geometrica' (see also Add MS 6786, f. 453v), and
Chapter 18, entitled 'Polygonorum circulo ordinate inscriptorum ad circulum ratio'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head128" xml:space="preserve">
Some considerations rising upon the 23. p. of <lb/>
Archimedes De Quadratura parabolæ &amp; the 17 &amp; 18 chap. of <lb/>
Viætus his responsorum. pa. 29.
</head>
<p>
<s xml:id="echoid-s617" xml:space="preserve">
If there be nombers in subduple proportion infinitely to find out <lb/>
their summe.
</s>
<lb/>
<s xml:id="echoid-s618" xml:space="preserve">
as 8.4.2.1.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>6</mn></mrow></mfrac></mstyle></math>.&amp;c.
</s>
<lb/>
<s xml:id="echoid-s619" xml:space="preserve">
The first doubled is the summe. that is in this example .8. doubled <emph style="st">is</emph> is 16.
</s>
<lb/>
<s xml:id="echoid-s620" xml:space="preserve">
In this <emph style="super">kind of</emph> progression if the number of places be finite. the summe is found <lb/>
thus. <emph style="st">waye</emph>.
</s>
<s xml:id="echoid-s621" xml:space="preserve">
Double the first as here .8. which maketh 16. from it subtract the <lb/>
last &amp; the remayne wilbe the summe.
</s>
<s xml:id="echoid-s622" xml:space="preserve">
for alwayes this <emph style="it">last</emph> doth lacke of <lb/>
the double of the first [???] <emph style="super">the</emph> quantity of the last.
</s>
<lb/>
<s xml:id="echoid-s623" xml:space="preserve">
all the quantityes after the first, are æquall to the first wanting <lb/>
the quantity of the last.
</s>
<s xml:id="echoid-s624" xml:space="preserve">
as 4 &amp; 2 are æquall to 8 wanting 2. <lb/>
4.2.1. are æquall wanting one. 4.2.1.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. are æquall wanting <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, &amp; so forth.
</s>
<lb/>
<s xml:id="echoid-s625" xml:space="preserve">
The summe of that whole progression the first being 8. &amp; the last <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>6</mn></mrow></mfrac></mstyle></math> <lb/>
thereby.
</s>
<lb/>
<s xml:id="echoid-s626" xml:space="preserve">
All the fractions under .1. if they <lb/>
be of a finite number of places <lb/>
are lesse than .1. <lb/>
if infinite æquall to .1.
</s>
</p>
<p>
<s xml:id="echoid-s627" xml:space="preserve">
Demonstration
</s>
<lb/>
<s xml:id="echoid-s628" xml:space="preserve">
They are neither greater nor lesse.
</s>
<lb/>
<s xml:id="echoid-s629" xml:space="preserve">
If greater, it is more than the progression; for the summe of all <lb/>
after the first, must be lesse than the first by the quantity of the last.
</s>
<lb/>
<s xml:id="echoid-s630" xml:space="preserve">
If lesse, then <emph style="st">the last must</emph> must it be lesse <emph style="st">th</emph> by a quantity <lb/>
&amp; the progression is not yet ended but it is supposed infinite.
</s>
<lb/>
<s xml:id="echoid-s631" xml:space="preserve">
Therefore &amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s632" xml:space="preserve">
In lines thus.
</s>
</p>
<p>
<s xml:id="echoid-s633" xml:space="preserve">
The demonstration of all <lb/>
is, it cannot be greater or <lb/>
lesse according to the <lb/>
condition and property of <lb/>
the progression argued.
</s>
</p>
<p>
<s xml:id="echoid-s634" xml:space="preserve">
The like waye in playnes or bodyes.
</s>
</p>
<p>
<s xml:id="echoid-s635" xml:space="preserve">
All fractions under <emph style="super">&amp; from</emph> a unite.
</s>
<lb/>
<s xml:id="echoid-s636" xml:space="preserve">
Subtriple are æquall to: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s637" xml:space="preserve">
Subquadruple: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s638" xml:space="preserve">
Subquintuple: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s639" xml:space="preserve">
&amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s640" xml:space="preserve">
Consider also of triangles inscribed from a square or triangles in a circle <lb/>
how to get the sum of their progression ratione [???]
</s>
<lb/>
<s xml:id="echoid-s641" xml:space="preserve">
Consider also what proportion or whether any <lb/>
obtain for those above a unite &amp; others &amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s642" xml:space="preserve">
see another paper
</s>
</p>
<pb file="add_6787_f566v" o="566v" n="1131"/>
<pb file="add_6787_f567" o="567" n="1132"/>
<div xml:id="echoid-div134" type="page_commentary" level="2" n="134">
<p>
<s xml:id="echoid-s643" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s643" xml:space="preserve">
The terminology and examples on this page are taken from Viète,
<emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter V,
Harriot has re-written Viète's examples in symbols, using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> signs and
the convention <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>A</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>-squared, and so on.
The same material appears again on Add MS 6784, f. 325, but there in Harriot's more usual lower case notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s645" xml:space="preserve">
Antithesis.
<lb/>[<emph style="it">tr:
Antithesis
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s646" xml:space="preserve">
ergo.
<lb/>[<emph style="it">tr:
therefore
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s647" xml:space="preserve">
Ratio.
<lb/>[<emph style="it">tr:
Ratio
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s648" xml:space="preserve">
adde:
<lb/>[<emph style="it">tr:
add
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s649" xml:space="preserve">
Hypobibasmus.
<lb/>[<emph style="it">tr:
Hypobibasmus
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s650" xml:space="preserve">
ergo:
<lb/>[<emph style="it">tr:
therefore:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s651" xml:space="preserve">
Parabolismus.
<lb/>[<emph style="it">tr:
Parabolismus
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s652" xml:space="preserve">
ergo:
<lb/>[<emph style="it">tr:
therefore:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s653" xml:space="preserve">
Nota. ex nostra observatione
<lb/>[<emph style="it">tr:
Note, from my observation
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s654" xml:space="preserve">
ergo per 14. p.6<emph style="super">i</emph>. vel 34. II<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
therefore by page 14, proposition 6, or page 34, proposition 11
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s655" xml:space="preserve">
ergo
<lb/>[<emph style="it">tr:
therefore
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s656" xml:space="preserve">
ergo etiam si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> æqualia.
<lb/>[<emph style="it">tr:
therefore also if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> are equal.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f567v" o="567v" n="1133"/>
<pb file="add_6787_f568" o="568" n="1134"/>
<div xml:id="echoid-div135" type="page_commentary" level="2" n="135">
<p>
<s xml:id="echoid-s657" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s657" xml:space="preserve">
This page refers to Proposition 53 from Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
Proposition (Problem) I.53 is an extension of Proposition (Problem) I.52.
</s>
<lb/>
<quote>
I.52.
Given a straight line in a plane bounded at one point, to find in the plane the section of a cone
called parabola, whose diameter is the given straight line, and whose vertex is the end of the straight line,
and where whatever straight line is dropped from the section to the diameter at a given angle,
will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section
and by some other given straight line.
</quote>
<lb/>
<s xml:id="echoid-s658" xml:space="preserve">
There is also a reference to the equivalent construction for an ellipse,
as demonstrated by Commandino in <emph style="it">Pappi Alexandrini mathematicae collectiones</emph> (1588).
The construction, and Commandino's commentary on it, are to be found on pages 320v–321.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s660" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> latus rectum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>h</mi></mstyle></math> angulus ordin: applic: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> centrum hyperboles <lb/>
oportet invenire axem sectionis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> <lb/>
et punctum verticis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
according to Apollonius, Book 1, Proposition 53, page 38.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s661" xml:space="preserve">
1) fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math> <lb/>
et: periferia agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>i</mi></mstyle></math> <lb/>
secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> productam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/>
fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/>
et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi><mo>=</mo><mi>a</mi><mi>p</mi></mstyle></math> <lb/>
agantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi></mstyle></math> productæ <lb/>
et sunt asymptotæ hyperboles
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math>, and take the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>i</mi></mstyle></math>, which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> produced at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>;
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi><mo>=</mo><mi>a</mi><mi>p</mi></mstyle></math>.
Taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi></mstyle></math> produced, these will be the asymptotes of the hyperbolas.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s662" xml:space="preserve">
2<emph style="super">a</emph>) fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>r</mi><mo>=</mo><mi>d</mi><mi>q</mi></mstyle></math> <lb/>
Agatur periferia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi><mi>p</mi></mstyle></math> <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>s</mi></mstyle></math> ad angulos rectos <lb/>
fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>t</mi><mo>=</mo><mi>d</mi><mi>s</mi></mstyle></math> <lb/>
et; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>u</mi><mo>=</mo><mi>d</mi><mi>t</mi></mstyle></math> <lb/>
agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>t</mi></mstyle></math> <lb/>
secetur ut per medium in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> <lb/>
et fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>d</mi><mo>=</mo><mi>d</mi><mi>l</mi></mstyle></math> <lb/>
Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math>, est axis <lb/>
et puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>, est vertex sectionis.
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>r</mi><mo>=</mo><mi>d</mi><mi>q</mi></mstyle></math>; take the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi><mi>p</mi></mstyle></math>, and set <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>s</mi></mstyle></math> at right angles.
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>t</mi><mo>=</mo><mi>d</mi><mi>s</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>u</mi><mo>=</mo><mi>d</mi><mi>t</mi></mstyle></math>; taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>t</mi></mstyle></math>, it will be cut in the middle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>; and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>d</mi><mo>=</mo><mi>d</mi><mi>l</mi></mstyle></math>.
I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> is the axis, and the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> is the vertex of the section.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s663" xml:space="preserve">
cætera sunt de constructione <lb/>
secundum Appol: lib. 1. pr: 53. pag: 38. <lb/>
pro Elipsi <emph style="super">sequenti</emph> vide pappum, pag: 321 <lb/>
et ibi Comandinum. Ubi nota quod ibi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> est <lb/>
dimidium lateris rect.
<lb/>[<emph style="it">tr:
The rest concerns the construction according to Apollonius, Book 1, Proposition 53, page 38.
For the ellipse following it, see Pappus, page 321, and that place in Commandinus, where he notes there that
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> is half the latus rectum.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f568v" o="568v" n="1135"/>
<pb file="add_6787_f569" o="569" n="1136"/>
<div xml:id="echoid-div136" type="page_commentary" level="2" n="136">
<p>
<s xml:id="echoid-s664" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s664" xml:space="preserve">
This page refers to Proposition 54 from Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.54.
Given two bounded straight lines perpendicular to each other, one of them being produced
on the side of the right angle, to find on the straight line produced the section of a cone
called hyperbola in the same plane with the straight line, so that the straight line produced
is a diameter of the section and the point at the angle is the vertex, and where whatever straight line
is dropped from the section to the diameter, making an angle equal to a given angle,
will equal in square the rectangle applied to the other straight line having as breadth
the straight line cut off the dropped straight line beginning with the vertex and projecting beyond
by a figure similar and similarly situated to that contained by the original straight lines.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s666" xml:space="preserve">
Nota quod in ista <lb/>
diagrammate præter <lb/>
constructionem pappi, <lb/>
notum etiam Apollonij: <lb/>
prop. 54. lib. 1. conicorum.
<lb/>[<emph style="it">tr:
Note that in this diagram besides is the construction of Pappus,
noted also by Apollonius in Proposition 54 of Book I on conics.
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f569v" o="569v" n="1137"/>
<pb file="add_6787_f570" o="570" n="1138"/>
<pb file="add_6787_f570v" o="570v" n="1139"/>
<pb file="add_6787_f571" o="571" n="1140"/>
<pb file="add_6787_f571v" o="571v" n="1141"/>
<pb file="add_6787_f572" o="572" n="1142"/>
<pb file="add_6787_f572v" o="572v" n="1143"/>
<pb file="add_6787_f573" o="573" n="1144"/>
<pb file="add_6787_f573v" o="573v" n="1145"/>
<pb file="add_6787_f574" o="574" n="1146"/>
<pb file="add_6787_f574v" o="574v" n="1147"/>
<pb file="add_6787_f575" o="575" n="1148"/>
<pb file="add_6787_f575v" o="575v" n="1149"/>
<pb file="add_6787_f576" o="576" n="1150"/>
<pb file="add_6787_f576v" o="576v" n="1151"/>
<pb file="add_6787_f577" o="577" n="1152"/>
<div xml:id="echoid-div137" type="page_commentary" level="2" n="137">
<p>
<s xml:id="echoid-s667" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s667" xml:space="preserve">
The proposition on page 37 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566), is I.52.
</s>
<lb/>
<quote>
I.52.
Given a straight line in a plane bounded at one point, to find in the plane the section of a cone
called parabola, whose diameter is the given straight line, and whose vertex is the end of the straight line,
and where whatever straight line is dropped from the section to the diameter at a given angle,
will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section
and by some other given straight line.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head129" xml:space="preserve" xml:lang="lat">
In Dato Cono: invenire datam parabolam	ad pag: 37. Appol.
<lb/>[<emph style="it">tr:
In a given cone, to find a given parabola, as page 37 of Apollonius.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s669" xml:space="preserve">
Dato angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>. (Coni) <lb/>
et linea. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. (recta.) <lb/>
Invenire Isoscelem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi><mi>E</mi></mstyle></math> <lb/>
ita ut fit:
<lb/>[<emph style="it">tr:
Given angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> in the cone and the stright line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, find the isosceles triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi><mi>E</mi></mstyle></math> such that:
</emph>]<lb/>
</s>
</p>
<pb file="add_6787_f577v" o="577v" n="1153"/>
<pb file="add_6787_f578" o="578" n="1154"/>
<pb file="add_6787_f578v" o="578v" n="1155"/>
</div>
</text>
</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Thu, 02 May 2013 11:14:40 +0200
parents	22d6a63640c6
children